Quick recall · 778 cards
Short MCQ-style retrieval prompts. Tap a card to reveal the answer.
PYQ
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Consider the following statements:
1. A = {1, 3, 5} and B = {2, 4, 7} are equivalent sets.
2. A = {1, 5, 9} and B = {1, 5, 5, 9, 9} are equal sets.
Which of the above statements is/are correct?
C · Both 1 and 2
PYQ
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Consider the following statements:
1. A = (A ∪ B) ∪ (A - B)
2. A ∪ (B - A) = (A ∪ B)
3. B = (A ∪ B) - (A - B)
Which of the above statements are correct?
B · 1 and 2 only
PYQ
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A ⊆ B ⇒ A ∪ B = B
Which of the above are correct?
(where A' is the complement of A)
A · True
By definition of subset, if \( A \subseteq B \), then every element of A is in B, so \( A \cup B = B \). This is a standard set property. Correct answer: True.[2]
PYQ
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What are the maximum number of subsets of S = {1, 2, y, 2 + y, 2 = 1}?
B · 32
PYQ
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If A and B are two sets where n(A) = 37, n(B) = 25, n(A ∪ B) = 50, then what is n(A ∩ B)?
A · 12
PYQ
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If A and B are two sets such that n(A) = 4 and n(B) = 3, then what is the maximum value of n(A ∩ B)?
A · 3
PYQ
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Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}, and B = {2, 4, 6, 8}. Find (A ∪ B)'.
A · {5, 7, 9}
PYQ · 2011
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If A = {1, 2, 5, 6} and B = {1, 2, 3}, then what is (A × B) ∩ (B × A)?
C · {(1, 1), (2, 2)}
PYQ
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For any three non-empty sets A, B, and C, what is (A ∪ B) - {(A - B) ∪ (B - A) ∪ (A ∩ B)} equal to?
B · ∅ (empty set)
PYQ
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If A ⊆ B, then which of the following is always true? (I) A ∪ B = B (II) A ∩ B = A (III) A - B = ∅
C · All I, II, and III are true
PYQ
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Two sets A and B are said to be equivalent if they have the same number of elements. Which of the following pairs are equivalent sets? (I) A = {1, 3, 5} and B = {2, 4, 7} (II) A = {1, 5, 9} and B = {1, 5, 5, 9, 9}
C · Both pairs are equivalent
PYQ
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If set A has 5 elements and set B has 4 elements, then the maximum number of elements in A ⊕ B (symmetric difference) is:
C · 9
PYQ
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If A = {1, 2, 5, 6} and B = {1, 2, 3}, then what is (A × B) ∩ (B × A) equal to?
A · {(1,1), (1,2), (2,1), (2,2)}
PYQ
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According to De Morgan's theorem, NAND gate is equivalent to:
A · Bubbled OR
PYQ · 2018
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Let A = {1, 2, 3}. Define a relation R on A by R = {(x, y) : x + y = 4}. List the elements of R.
A · {(1,3), (2,2), (3,1)}
PYQ · 2011
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The relation 'y is at most 5 years older than x' on the set of all people is:
D · (d) Neither reflexive nor transitive
PYQ
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The number of points represented by the equation \( x = 5 \) on the xy-plane is
C · infinite
PYQ
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If z = 1 + i√3 where i = √(-1), then what is the argument of z?
A · π/3
PYQ
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What is \( \sqrt{1 + \omega} \) equal to, where \( \omega \) is a primitive cube root of unity?
B · ω
PYQ
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The modulus and principal argument of the complex number \( \frac{1 + 2i}{1 - (1 - 2i)^2} \) are respectively
A · 1, π/4
PYQ
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What is \( (1 + i)^4 + (1 - i)^4 \) equal to, where \( i = \sqrt{-1} \)?
A · 0
PYQ
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1. The difference of z and its conjugate is an imaginary number. 2. The sum of z and its conjugate is a real number. Which of the above statements is/are correct?
C · Both 1 and 2
For z = x + iy, conjugate \( \bar{z} = x - iy \). Sum z + \( \bar{z} \) = 2x (real). Difference z - \( \bar{z} \) = 2iy (pure imaginary). Both statements correct. Thus, option C.
PYQ
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The smallest positive integer n for which \( \left( \frac{1 - i}{1 + i} \right)^{n^2} = 1 \), where i = √(-1), is
B · 2
PYQ · 2015
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What is the modulus of the complex number \( \frac{\cos \theta + i \sin \theta}{\cos \theta - i \sin \theta} \), where \( i = \sqrt{-1} \)?
B · 1
PYQ · 2014
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What is the modulus of the complex number \( \frac{1+i}{1-i} \), where \( i = \sqrt{-1} \)?
A · 1
PYQ
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If \( z = 1 + i \), where \( i = \sqrt{-1} \), then what is the modulus of \( z^2 \)?
C · 4
PYQ · 2021
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What is the modulus of the complex number \( i(-i)^n \), where \( n \in \mathbb{N} \) and \( i = \sqrt{-1} \)?
B · 1
PYQ · 2020
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What is the argument of the complex number \( \frac{1 - i\sqrt{3}}{1 + i\sqrt{3}} \), where \( i = \sqrt{-1} \)?
A · \( -\frac{\pi}{3} \)
PYQ
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The argument of the complex number \( -1 - i\sqrt{3} \) is:
D · \( -\frac{2\pi}{3} \)
PYQ
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If \( z_1 \) and \( z_2 \) are complex numbers with \( |z_1| = |z_2| \), then which of the following is/are correct?
1. \( z_1 = z_2 \)
2. Real part of \( \frac{z_1}{z_2} \) = 1
B · Only 2
PYQ
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What is the argument of \( \frac{1 + i\sqrt{3}}{\sqrt{3} + i} \)?
A · \( \frac{\pi}{3} \)
PYQ
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The principal argument of \( i \) is:
B · \( \frac{\pi}{2} \)
PYQ · 2020
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If 1, ω, ω² are the cube roots of unity, then the value of (1 + ω)(1 + ω²)(1 + ω⁴)(1 + ω⁸) is
B · 1
PYQ · 2016
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Suppose ω is a cube root of unity with ω ≠ 1. Suppose P and Q are the points on the complex plane representing ω and ω² respectively. If O is the origin, then what is the angle between OP and OQ?
A · 60°
PYQ · 2016
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If ω₁ and ω₂ are two distinct cube roots of unity different from 1, then what is (ω₁ - ω₂)² equal to?
C · -√3
PYQ · 2010
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If ω ≠ 1 is a cube root of unity, then ω¹⁰ + ω⁻¹⁰ is equal to
A · -1
PYQ
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If 1, ω, ω² are the cube roots of unity, then (1 + ω)(1 + ω²)(1 + ω³)(1 + ω + ω²) is equal to
A · 0
\(\omega^3 = 1\), so 1 + ω³ = 1 + 1 = 2. Also, 1 + ω + ω² = 0. Thus, the product is (1 + ω)(1 + ω²)(2)(0) = 0. Option **A**.[4]
PYQ · 2019
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A binary number is represented by (cdccddcccddd)2, where c > d. What is its decimal equivalent?
B · (101100111000)2
PYQ · 2022
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Convert 2810 to binary.
A · 111002
PYQ
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What is (1110011)2 ÷ (10111)2 equal to?
A · (1010)2
PYQ · 2023
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The binary number (PQR)2 when P=1, Q=0, R=1 represents which decimal number?
A · 5
PYQ · 2016
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If the number 235 in decimal system is converted into binary system, then what is the resulting number?
B · 11101011
PYQ · 2016
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What is the binary equivalent of the decimal number 0.3125?
A · 0.0101
PYQ · 2023
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What is the binary number equivalent to decimal number 1011?
D · 1111110011
PYQ · 2023
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What is the binary number equivalent to decimal number 1011?
B · 111011
PYQ · 2025
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If x = (1111)₂, y = (1001)₂, and z = (110)₂, then what is x³ - y³ - z³ - 3xyz equal to in binary?
D · (0)₂
PYQ
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Which term of the A.P. 4, 9, 14, ... is 99?
D · D. 21
PYQ
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The nth term of an A.P. is given by 5n - 2. What is the 10th term?
C · C. 18
PYQ
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A geometric progression consists of 200 terms. If the sum of odd terms of the GP is m, and the sum of even terms of the GP is n, then what is the common ratio of the GP?
B · n/m
PYQ
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If the first term of a geometric progression is 2 and the common ratio is 3, what is the sum of the first 5 terms of the GP?
A · 242
PYQ
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If each term of a geometric progression is multiplied by 2, what is the effect on the common ratio?
B · The common ratio remains unchanged
PYQ
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The sum of an infinite geometric progression is 8 and the first term is 4. What is the common ratio?
A · 1/2
PYQ
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If the roots of the equation \( x^2 - (5k + 1)x + 5k = 0 \) differ by unity, then which one of the following is a possible value of \( k \)?
B · 2
PYQ
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The quadratic equation \( 3x^2 - (k^2 + 5k)x + 3k^2 - 5k = 0 \) has real roots of equal magnitude and opposite sign. Which one of the following is correct?
A · k = 0
PYQ
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If \( x^2 + 2x - 1 = 0 \), then which one of the following is correct?
D · All of the above
PYQ
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If the roots of the equation \( x^2 + 2x + k = 0 \) are real, then which one of the following is correct?
A · k ≤ 1
PYQ
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If the roots of the equation \( x^2 - nx + m = 0 \) differ by 1, then the roots of the equation \( x^2 - bx + c = 0 \) has a root in which one of the following intervals?
A · (β, -α)
PYQ
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The quadratic equations \( x^2 - mx + n = 0 \) and \( x^2 + px + n = 0 \), where m ≠ n, then what is the value of p + m + n?
C · 1
PYQ
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How many four-digit numbers can be formed using the digits 1, 2, 3, 5 without repetition of digits such that they are divisible by 4?
B · 8
PYQ · 2024
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How many four-digit natural numbers have all digits even?
A · 400
PYQ · 2025
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In how many ways can the letters of the word be arranged such that a particular pair of letters comes together in each word?
D · 360
PYQ · 2025
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What is the number of positive integer solutions of x + y + z = 5?
C · 6
PYQ · 2024
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How many four-digit numbers have all digits odd?
D · 625
PYQ
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The number of different matrices, each having 4 entries, that can be formed using 1, 2, 3, 4 with repetition allowed is:
C · 4096
PYQ
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Consider the following statements:
Statement 1: (25)! + 1 is divisible by 26
Statement 2: (6)! + 1 is divisible by 7
Which statement(s) is/are correct?
B · Only Statement 2
PYQ · 2024
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In how many ways can we select 2 persons from a group of 10 persons?
A · 45
PYQ
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What is the number of ways to select 3 items from 7 distinct items?
A · 35
PYQ
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From 20 distinct points placed on a circle, how many straight lines can be drawn by joining any two of these points?
B · 380
PYQ
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From 20 distinct points placed on a circle, how many triangles can be drawn by joining any three of these points?
A · 1140
PYQ
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In how many ways can we select a committee of 4 people from a group of 10 people such that one specific person must be included?
A · 84
PYQ
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From the digits 1, 2, 3, 4, 5, in how many ways can we form numbers with 3 digits such that the digits are in increasing order?
A · 10
PYQ
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The middle term in the expansion of \((1 + x)^{2n}\) is:
A · A) \(T_{n+1}\)
PYQ
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If the coefficients of the 7th and 13th terms in the expansion of \((1 + x)^n\) are equal, then \(n =\):
D · D) 20
PYQ
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In the expansion of \((1 + x)^{52}\), the ratio of the coefficient of \(x^{18}\) to the coefficient of \(x^{15}\) is:
A · A) \(\frac{2}{3}\)
PYQ
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The first three terms in the expansion of \((1 + ax)^n\) (\(n ≠ 0\)) are 1, 6x, and 16x². What is the value of \(n\)?
C · C) 4
PYQ
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If \(C(n, 2) = C(n, 6)\), then \(n\) is equal to:
B · B) 8
PYQ
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In the expansion of \(\left(x^2 - \frac{1}{x}\right)^7\), the constant term is:
B · B) -144
PYQ
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\(8^{3/2}\) is approximately equal to (first three terms):
A · A) \(64 - 96x - 720x^2\)
PYQ
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If \( (2x^2 - x - 1)^{10} = a_0 + a_1 x + a_2 x^2 + \dots + a_{10} x^{10} \), then \( a_2 + a_4 + a_6 + a_8 + a_{10} = \)
(A) 15
(B) 30
(C) 16
(D) 1729
A · 15
PYQ
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If the **coefficient of 7th and 13th terms** in the expansion of \( (1 + x)^n \) are equal, then **n** =
(A) 10
(B) 15
(C) 18
(D) 20
D · 20
PYQ
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In the expansion of \( (x^2 - \frac{15}{x})^2 \), the **constant term** is
(A) 18
(B) 6
(C) 12
(D) 10
A · 18
PYQ
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The expansion of \( \left(4 - 3x\right)^{-\frac{1}{2}} \) by binomial theorem will be valid, if
(A) \( x < 1 \)
(B) \( x < \frac{2}{3} \)
(C) \( -\frac{2}{3} < x < \frac{2}{3} \)
(D) None of these
C · \( -\frac{2}{3} < x < \frac{2}{3} \)
PYQ
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\( 8^{\frac{3}{2}} \) is approximately equal to:
(A) \( 64 - 96x - 720x^2 \)
(B) \( 64 - 96x + 720x^2 \)
(C) \( 64 + 96x - 720x^2 \)
(D) \( 64 + 96x + 720x^2 \)
B · \( 64 - 96x + 720x^2 \)
PYQ
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If \( ^nC_2 = ^nC_6 \), then **n** is equal to:
B · 10
PYQ · 2015
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If \( \log_8 m + \log_8 \frac{1}{6} = \frac{2}{3} \), then m is equal to
C · \( 4 \)
PYQ
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If \( x + \log_{10}(1 + 2^x) = x \log_{10} 5 + \log_{10} 6 \), then x is equal to
B · 2
PYQ · 2024
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What is the number of solutions of \( \log_4 (x - 1) = \log_2 (x - 3) \)?
B · 1
PYQ · 2021
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For \( x \geq y > 1 \), let \( \log_x \left( \frac{x}{y} \right) + \log_y \left( \frac{y}{x} \right) = k \), then the value of k can never be equal to
D · 1
PYQ · 2021
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If n = 100!, then what is the value of \( \dfrac{1}{\log_2 n} + \dfrac{1}{\log_3 n} + \dfrac{1}{\log_4 n} + \dots + \dfrac{1}{\log_{100} n} \)?
A · 99
PYQ · 2021
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The value of the expression is (from 2021 II, option C)
C · C
Using log base b with exponent 2, simplifies to C as per video analysis.
PYQ · 2023
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If log(m/n) property used (2023 I)
C · C
log(m/n) = log m - log n, simplifies to option C.
PYQ · 2015
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If $$\log _8 m + \log _8 {1 \over 6} = {2 \over 3}$$, then m is equal to
A · 2
PYQ · 2024
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What is the number of solutions of $$\log_4(x - 1) = \log_2(x - 3)$$?
B · 1
PYQ · 2021
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For $$x \geq y > 1$$, let $$\log_x\left( \frac{x}{y} \right) + \log_y\left(\frac{y}{x}\right) = k$$, then the value of k can never be equal to :
D · 1
PYQ
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If $$x + \log _{10}(1 + 2^x) = x\log _{10}5 + \log _{10}6$$ then x is equal to
B · 2
PYQ · 2021
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If n = 100!, then what is the value of $$\dfrac{1}{\log_2n}+\dfrac{1}{\log_3n}+\dfrac{1}{\log_4n}+{.....}+\dfrac{1}{\log_{100}n}$$?
A · 99
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Consider the sets \( A = {2, 4, 6, 8} \) and \( B = {4, 8, 12} \). Which of the following statements is true?
D · The intersection of A and B is \( \{4, 8\} \)
The intersection of sets A and B contains elements common to both. Here, \( A \cap B = \{4, 8\} \). Neither A is subset of B nor B is subset of A, and they are not disjoint since they share elements.
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If set \( S = \{a, b, c\} \), what is the cardinality of its power set \( \mathcal{P}(S) \)?
D · 8
The power set of a set with \( n \) elements has \( 2^n \) elements. Here, \( n = 3 \), so \( |\mathcal{P}(S)| = 2^3 = 8 \).
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Given \( A = \{1, 2, 3, 4\} \) and \( B = \{2, 4, 6, 8\} \), which of the following is a superset of \( A \cap B \)?
A · \( \{2, 4\} \)
The intersection \( A \cap B = \{2, 4\} \). A superset must contain all elements of this set. Only option A contains both 2 and 4.
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If \( U = \{1, 2, 3, 4, 5, 6\} \), \( A = \{1, 3, 5\} \), and \( B = \{2, 3, 4\} \), what is \( (A \cup B)^c \) where complement is relative to \( U \)?
A · \( \{6\} \)
First, \( A \cup B = \{1, 2, 3, 4, 5\} \). The complement relative to \( U \) is \( U - (A \cup B) = \{6\} \).
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For sets \( A, B, C \), if \( A \subseteq B \) and \( B \subseteq C \), which of the following is always true?
A · \( A \subseteq C \)
Subset relation is transitive. Since \( A \subseteq B \) and \( B \subseteq C \), it follows that \( A \subseteq C \).
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If \( A = \{x \in \mathbb{Z} : -2 \leq x \leq 2\} \), what is the number of subsets of \( A \) that contain exactly 3 elements?
B · 10
Set \( A = \{-2, -1, 0, 1, 2\} \) has 5 elements. Number of subsets with exactly 3 elements is \( \binom{5}{3} = 10 \).
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Consider the sets \( A = \{1, 2, 3\} \) and \( B = \{3, 4, 5\} \). Which of the following is the power set of \( A \cap B \)?
A · \( \{\emptyset, \{3\}\} \)
The intersection \( A \cap B = \{3\} \). Its power set is \( \{\emptyset, \{3\}\} \).
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If \( |A| = 4 \) and \( A \subseteq B \) with \( |B| = 6 \), what is the minimum number of elements in \( A \cup B \)?
B · 6
Since \( A \subseteq B \), \( A \cup B = B \). Hence, \( |A \cup B| = |B| = 6 \).
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Given \( A = \{1, 2, 3, 4\} \), which of the following sets is NOT a subset of \( A \)?
B · \( \{1, 5\} \)
Subset must contain only elements from \( A \). \( \{1, 5\} \) contains 5 which is not in \( A \), so it is not a subset.
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If \( A = \{x : x \text{ is a prime number less than } 10\} \), what is the number of elements in the power set of \( A \)?
B · 16
Primes less than 10 are \( \{2, 3, 5, 7\} \), so \( |A| = 4 \). Power set has \( 2^4 = 16 \) elements.
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Let \( A = \{1, 2, 3\} \) and \( B = \{2, 3, 4, 5\} \). Which of the following is true?
A · \( A \cup B = \{1, 2, 3, 4, 5\} \)
Union contains all elements from both sets: \( \{1, 2, 3, 4, 5\} \). Intersection is \( \{2, 3\} \), so option B is false. Neither is subset of the other.
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If \( A \subseteq B \) and \( B \subseteq A \), what can be concluded about sets \( A \) and \( B \)?
B · They are equal sets
If \( A \subseteq B \) and \( B \subseteq A \), then \( A = B \) by definition of set equality.
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Consider \( A = \{x : x \text{ is an even integer}, 1 \leq x \leq 10\} \). How many subsets of \( A \) contain the element 4?
C · 16
Set \( A = \{2, 4, 6, 8, 10\} \) has 5 elements. Number of subsets containing 4 equals number of subsets of remaining 4 elements, which is \( 2^{4} = 16 \).
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If \( A = \{1, 2, 3\} \), how many subsets of \( A \) have at most 2 elements?
C · 7
Subsets with at most 2 elements include subsets with 0, 1, or 2 elements. Number of such subsets is \( \binom{3}{0} + \binom{3}{1} + \binom{3}{2} = 1 + 3 + 3 = 7 \).
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Given \( A = \{1, 2, 3, 4\} \), \( B = \{3, 4, 5, 6\} \), and \( C = \{4, 5, 6, 7\} \), which of the following is true?
C · \( (A - B) \subseteq (A \cup C) \)
The set difference \( A - B = \{1, 2\} \) is a subset of \( A \cup C = \{1, 2, 3, 4, 5, 6, 7\} \). Other options are false as intersections and unions do not satisfy those subset relations.
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If \( A \) and \( B \) are sets such that \( A \cup B = B \), which of the following must be true?
A · \( A \subseteq B \)
If \( A \cup B = B \), then all elements of \( A \) are in \( B \), so \( A \subseteq B \).
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Let \( S = \{1, 2, 3, 4, 5\} \). How many subsets of \( S \) contain the element 1 but not the element 2?
A · 8
Fix element 1 in subsets and exclude element 2. Remaining elements \( \{3,4,5\} \) can be chosen freely. Number of such subsets is \( 2^3 = 8 \).
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If \( A = \{x : x \text{ is a multiple of } 3 \text{ and } 1 \leq x \leq 15\} \), what is the number of elements in the power set of \( A \)?
B · 32
Multiples of 3 between 1 and 15 are \( \{3, 6, 9, 12, 15\} \), so \( |A| = 5 \). Power set size is \( 2^5 = 32 \).
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Refer to the diagram below showing sets \( A \), \( B \), and \( C \) in a universal set \( U \). If \( A \subset B \) and \( B \subset C \), which region represents \( A \cap C^c \)?
A · Region inside A but outside C
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If \( A = \{1, 2, 3\} \), how many subsets of \( A \) are there whose elements sum to an even number?
B · 4
Subsets of \( A \) are \( \emptyset, \{1\}, \{2\}, \{3\}, \{1,2\}, \{1,3\}, \{2,3\}, \{1,2,3\} \). Subsets with even sum: \( \emptyset (0), \{2\} (2), \{1,3\} (4), \{1,2,3\} (6) \) total 4 subsets.
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Let \( A = \{x : x \text{ is a letter in the word 'ALGEBRA'}\} \). What is the cardinality of the power set of \( A \)?
B · 64
Distinct letters in 'ALGEBRA' are \( \{A, L, G, E, B, R\} \), so \( |A| = 6 \). Power set size is \( 2^6 = 64 \).
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If \( A = \{1, 2, 3, 4, 5\} \), how many subsets of \( A \) contain neither 1 nor 2?
B · 8
Excluding 1 and 2 leaves \( \{3,4,5\} \) with 3 elements. Number of subsets is \( 2^3 = 8 \).
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If \( A \subset B \) and \( |A| = 3 \), \( |B| = 5 \), how many subsets of \( B \) contain all elements of \( A \)?
A · 4
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Consider \( A = \{1, 2, 3, 4\} \). Which of the following is the correct number of subsets of \( A \) that contain the element 1?
C · 8
Total subsets of \( A \) are \( 2^4 = 16 \). Subsets without 1 are subsets of \( \{2,3,4\} \), which are \( 2^3 = 8 \). So subsets containing 1 are \( 16 - 8 = 8 \).
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If \( A = \{1, 3, 5\} \) and \( B = \{1, 3, 5, 7\} \), which of the following is true?
A · \( A \subset B \)
All elements of \( A \) are in \( B \), but \( B \) has an extra element 7, so \( A \subset B \).
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Let \( A = \{1, 2, 3\} \) and \( B = \{2, 3, 4\} \). What is the number of elements in \( (A \cup B) - (A \cap B) \)?
A · 2
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If \( A = \{1, 2, 3, 4\} \), which of the following subsets is the complement of \( \{2, 3\} \) relative to \( A \)?
A · \( \{1, 4\} \)
Complement relative to \( A \) is \( A - \{2, 3\} = \{1, 4\} \).
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If \( A = \{x : x \text{ is an integer}, 1 \leq x \leq 5\} \), what is the number of subsets of \( A \) with exactly 4 elements?
B · 5
Number of subsets with exactly 4 elements is \( \binom{5}{4} = 5 \).
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Given \( A = \{1, 2, 3, 4, 5\} \), how many subsets of \( A \) contain both 1 and 2?
B · 8
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If \( A = \{1, 2, 3\} \) and \( B = \{3, 4, 5\} \), which of the following sets is \( (A - B) \cup (B - A) \)?
A · \( \{1, 2, 4, 5\} \)
Set difference \( A - B = \{1, 2\} \), \( B - A = \{4, 5\} \). Union is \( \{1, 2, 4, 5\} \).
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Which of the following statements is true for any set \( A \)?
A · \( A \subseteq A \)
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If \( n(A) = 45 \), \( n(B) = 30 \), and \( n(A \cap B) = 15 \), what is \( n(A \cup B) \)?
A · 60
Using the formula for union: \( n(A \cup B) = n(A) + n(B) - n(A \cap B) = 45 + 30 - 15 = 60 \).
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Given sets \( A = {2, 4, 6, 8, 10} \) and \( B = {1, 2, 3, 4, 5} \), find \( (A - B) \cup (B - A) \).
A · {1, 3, 5, 6, 8, 10}
Calculate differences: \( A - B = {6, 8, 10} \), \( B - A = {1, 3, 5} \). Their union is {1, 3, 5, 6, 8, 10}.
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If \( U = {1, 2, 3, 4, 5, 6, 7, 8, 9} \), \( A = {1, 3, 5, 7} \), and \( B = {2, 3, 6, 7} \), find \( (A \cap B)^c \).
A · {1, 2, 4, 5, 6, 8, 9}
First find \( A \cap B = {3, 7} \). Its complement in \( U \) is \( U - (A \cap B) = {1, 2, 4, 5, 6, 8, 9} \).
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For sets \( A = {1, 2, 3} \) and \( B = {2, 3, 4} \), what is the cardinality of \( A \times B \)?
B · 9
The Cartesian product \( A \times B \) has \( n(A) \times n(B) = 3 \times 3 = 9 \) ordered pairs.
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If \( A = {1, 2, 3, 4} \), \( B = {3, 4, 5, 6} \), and \( C = {4, 5, 6, 7} \), find \( (A \cup B) \cap C \).
A · {4, 5, 6}
First \( A \cup B = {1, 2, 3, 4, 5, 6} \). Intersection with \( C = {4, 5, 6, 7} \) is {4, 5, 6}.
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If \( n(U) = 100 \), \( n(A) = 40 \), \( n(B) = 50 \), and \( n(A \cap B) = 20 \), what is \( n((A \cup B)^c) \)?
A · 30
Calculate \( n(A \cup B) = n(A) + n(B) - n(A \cap B) = 40 + 50 - 20 = 70 \).Then \( n((A \cup B)^c) = n(U) - n(A \cup B) = 100 - 70 = 30 \).
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If \( A \subseteq B \), which of the following is always true?
A · \( A \cup B = B \)
If \( A \subseteq B \), then \( A \cup B = B \) always holds true.Other options are false because \( A \cap B = A \), \( A - B = \emptyset \), and \( B - A \) may not be empty.
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Let \( A = {1, 2, 3, 4, 5} \) and \( B = {4, 5, 6, 7} \). Find the symmetric difference \( A \oplus B \).
A · {1, 2, 3, 6, 7}
Symmetric difference \( A \oplus B = (A - B) \cup (B - A) = {1, 2, 3} \cup {6, 7} = {1, 2, 3, 6, 7} \).
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If \( A = {1, 3, 5} \) and \( B = {2, 4, 6} \), what is \( (A \times B) \cup (B \times A) \)?
A · {(1,2),(1,4),(1,6),(3,2),(3,4),(3,6),(5,2),(5,4),(5,6),(2,1),(2,3),(2,5),(4,1),(4,3),(4,5),(6,1),(6,3),(6,5)}
The union of \( A \times B \) and \( B \times A \) contains all ordered pairs where first element is from \( A \) and second from \( B \), and vice versa.Thus, total 18 pairs as listed in option A.
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For three sets \( A, B, C \), which of the following is equal to \( (A \cap B) \cup (B \cap C) \cup (C \cap A) \)?
A · \( (A \cup B) \cap (B \cup C) \cap (C \cup A) \)
The union of pairwise intersections equals the intersection of pairwise unions:\( (A \cap B) \cup (B \cap C) \cup (C \cap A) = (A \cup B) \cap (B \cup C) \cap (C \cup A) \).
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If \( A = {1, 2, 3, 4, 5, 6} \) and \( B = {4, 5, 6, 7, 8} \), find \( n((A \cup B) - (A \cap B)) \).
A · 5
Calculate \( A \cup B = {1, 2, 3, 4, 5, 6, 7, 8} \) (8 elements), \( A \cap B = {4, 5, 6} \) (3 elements).Difference \( (A \cup B) - (A \cap B) \) has \( 8 - 3 = 5 \) elements.
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If \( A = {x mid x \text{ is an even number less than } 10} \) and \( B = {x mid x \text{ is a prime number less than } 10} \), find \( A \cap B \).
A · {2}
Even numbers less than 10: {2, 4, 6, 8}.Prime numbers less than 10: {2, 3, 5, 7}.Intersection is {2} only.
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Given \( A = {1, 2, 3, 4} \), \( B = {3, 4, 5, 6} \), and universal set \( U = {1, 2, 3, 4, 5, 6, 7, 8} \), find \( (A \cup B)^c \).
A · {7, 8}
First find \( A \cup B = {1, 2, 3, 4, 5, 6} \).Complement in \( U \) is \( U - (A \cup B) = {7, 8} \).
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If \( A = {1, 2, 3} \), \( B = {2, 3, 4} \), and \( C = {3, 4, 5} \), find \( (A \cap B) \cup (B \cap C) \).
A · {2, 3, 4}
Calculate \( A \cap B = {2, 3} \), \( B \cap C = {3, 4} \).Union is {2, 3, 4}.
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If \( A = {1, 3, 5, 7} \) and \( B = {2, 3, 6, 7} \), find \( (A \cup B) - (A \cap B) \).
A · {1, 2, 5, 6}
Calculate \( A \cup B = {1, 2, 3, 5, 6, 7} \), \( A \cap B = {3, 7} \).Difference is {1, 2, 5, 6}.
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If \( A = {1, 2, 3, 4, 5} \) and \( B = {3, 4, 5, 6, 7} \), what is \( n(A \cap B) \)?
A · 3
Intersection \( A \cap B = {3, 4, 5} \) which has 3 elements.
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If \( A = {1, 2, 3} \), \( B = {3, 4, 5} \), and \( C = {5, 6, 7} \), find \( (A \cup B) \cap C \).
A · {5}
Calculate \( A \cup B = {1, 2, 3, 4, 5} \). Intersection with \( C = {5, 6, 7} \) is {5}.
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If \( A = {1, 2, 3, 4} \) and \( B = {2, 3, 4, 5} \), find \( (A \times B) \cap (B \times A) \).
A · {(2, 2), (3, 3), (4, 4)}
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If \( A = {1, 2, 3, 4, 5} \) and \( B = {4, 5, 6, 7, 8} \), find \( n(A \oplus B) \) (symmetric difference).
A · 6
Symmetric difference \( A \oplus B = (A - B) \cup (B - A) = {1, 2, 3} \cup {6, 7, 8} \) which has 6 elements.
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If \( A = {1, 2, 3} \), \( B = {2, 3, 4} \), and \( C = {3, 4, 5} \), find \( (A \cup B) - C \).
A · {1, 2}
Calculate \( A \cup B = {1, 2, 3, 4} \).Difference with \( C = {3, 4, 5} \) is {1, 2}.
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If \( A = {1, 2, 3, 4, 5} \) and \( B = {3, 4, 5, 6, 7} \), find \( (A - B) \cup (B - A) \).
A · {1, 2, 6, 7}
Calculate \( A - B = {1, 2} \), \( B - A = {6, 7} \).Union is {1, 2, 6, 7}.
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If \( A = {1, 2, 3} \), \( B = {2, 3, 4} \), and \( C = {3, 4, 5} \), find \( (A \cap B \cap C) \).
A · {3}
Intersection \( A \cap B = {2, 3} \), then intersect with \( C = {3, 4, 5} \) gives {3}.
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If \( A = {1, 2, 3, 4} \) and \( B = {3, 4, 5, 6} \), what is \( (A \cup B) - (A \cap B) \)?
A · {1, 2, 5, 6}
Calculate \( A \cup B = {1, 2, 3, 4, 5, 6} \), \( A \cap B = {3, 4} \).Difference is {1, 2, 5, 6}.
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If \( A = {1, 2, 3, 4} \), \( B = {3, 4, 5, 6} \), and \( C = {5, 6, 7, 8} \), find \( (A \cup B) \cap (B \cup C) \).
A · {3, 4, 5, 6}
Calculate \( A \cup B = {1, 2, 3, 4, 5, 6} \), \( B \cup C = {3, 4, 5, 6, 7, 8} \).Intersection is {3, 4, 5, 6}.
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If \( A = {1, 2, 3} \) and \( B = {4, 5, 6} \), what is \( A \cap B \)?
A · \(\emptyset\)
Since \( A \) and \( B \) have no common elements, \( A \cap B = \emptyset \) (null set).
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If \( A = {x mid x \text{ is a multiple of 3 less than 20}} \) and \( B = {x mid x \text{ is a multiple of 4 less than 20}} \), find \( A \cap B \).
A · {12}
Multiples of 3 less than 20: {3, 6, 9, 12, 15, 18}.Multiples of 4 less than 20: {4, 8, 12, 16}.Intersection is {12}.
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If \( A = {1, 2, 3, 4, 5} \) and \( B = {2, 3, 4, 5, 6} \), find \( n(A \cap B) \).
A · 4
Intersection \( A \cap B = {2, 3, 4, 5} \) which has 4 elements.
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If \(n(A) = 100\), \(n(B) = 80\), and \(n(A \cap B) = 50\), what is the probability that a randomly chosen element from \(A \cup B\) belongs to exactly one of the sets \(A\) or \(B\)?
A · \(\frac{80}{130}\)
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In a universal set \(U\) of 200 elements, sets \(X\) and \(Y\) satisfy \(n(X) = 120\), \(n(Y) = 90\), and \(n(X \cap Y) = 50\). How many elements are in neither \(X\) nor \(Y\)?
A · 40
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Refer to the diagram below with three sets \(A\), \(B\), and \(C\). If \(n(A \cup B) = 120\), \(n(B \cup C) = 140\), \(n(A \cup C) = 130\), and \(n(A \cup B \cup C) = 180\), find the value of \(n(A) + n(B) + n(C)\).
B · 230
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If \(A\) and \(B\) are two sets such that \(n(A) = 70\), \(n(B) = 50\), and \(n(A \cup B) = 90\), what is \(n(A \cap B)\)?
A · 30
Using formula:
\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \]
Substitute:
\[ 90 = 70 + 50 - n(A \cap B) \Rightarrow n(A \cap B) = 120 - 90 = 30 \]
So correctAnswer is A.
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If \(n(U) = 300\), \(n(A) = 150\), \(n(B) = 180\), and \(n(A \cap B) = 90\), what is the probability that a randomly selected element from \(U\) belongs to exactly one of the sets \(A\) or \(B\)?
A · \(\frac{150}{300}\)
Number of elements in exactly one set:
\[ n(A) + n(B) - 2 n(A \cap B) = 150 + 180 - 2 \times 90 = 330 - 180 = 150 \]
Probability:
\[ \frac{150}{300} = \frac{1}{2} \]
Hence, correctAnswer is A.
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If \(n(A) = 90\), \(n(B) = 70\), and \(n(A \cap B) = 40\), what is the number of elements in the symmetric difference \(A Delta B\)?
B · 80
Symmetric difference:
\[ n(A \Delta B) = n(A) + n(B) - 2 n(A \cap B) = 90 + 70 - 2 \times 40 = 160 - 80 = 80 \]
Hence, correctAnswer is B.
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If \(n(A) = 120\), \(n(B) = 100\), and \(n(A \cup B) = 180\), what is the number of elements in \(A^c \cap B^c\) given the universal set has 250 elements?
A · 70
Number of elements in \(A \cup B\) is 180.
Elements not in \(A\) or \(B\) are:
\[ n(U) - n(A \cup B) = 250 - 180 = 70 \]
Hence, correctAnswer is A.
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If \(n(A) = 75\), \(n(B) = 65\), and \(n(A \cap B) = 40\), find the number of elements in \(A^c \cup B^c\) given the universal set has 150 elements.
B · 110
Using De Morgan's law:
\[ A^c \cup B^c = (A \cap B)^c \]
So,
\[ n(A^c \cup B^c) = n(U) - n(A \cap B) = 150 - 40 = 110 \]
Hence, correctAnswer is B.
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If \(n(A) = 85\), \(n(B) = 95\), and \(n(A \cup B) = 150\), find the number of elements in \(A^c \cap B\).
A · 65
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If \(n(A) = 100\), \(n(B) = 90\), and \(n(A \cap B) = 70\), find the number of elements in \(A \cup B\).
A · 120
Using formula:
\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) = 100 + 90 - 70 = 120 \]
Hence, correctAnswer is A.
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If \( U = {1, 2, 3, 4, 5, 6} \), \( A = {1, 2, 3} \), and \( B = {3, 4, 5} \), what is \((A \cup B)^c\)?
B · \{6\}
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Given sets \( A \) and \( B \) in universal set \( U \), which of the following correctly represents De Morgan's law for complements?
B · \( (A \cup B)^c = A^c \cap B^c \)
De Morgan's laws state:1) \( (A \cup B)^c = A^c \cap B^c \)2) \( (A \cap B)^c = A^c \cup B^c \)Option B correctly states the first law.
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If \( A = {x | x \text{ is even and } x \leq 10} \) and \( B = {x | x \text{ is prime and } x \leq 10} \), find \((A \cap B)^c\) in \( U = {1, 2, \ldots, 10} \).
A · \{1, 3, 4, 5, 6, 7, 8, 9, 10\}
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Which expression is equivalent to \( \overline{A + B} \) in Boolean algebra, according to De Morgan's theorem?
B · \( \overline{A} \cdot \overline{B} \)
De Morgan's theorem states:\( \overline{A + B} = \overline{A} \cdot \overline{B} \).Thus, option B is correct.
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If \( U = {1, 2, 3, 4, 5, 6, 7, 8} \), \( A = {1, 3, 5, 7} \), and \( B = {2, 4, 6, 8} \), find \((A^c \cup B)^c\).
A · \{1, 3, 5, 7\}
First, \( A^c = U - A = \{2, 4, 6, 8\} = B \).Then, \( A^c \cup B = B \cup B = B = \{2,4,6,8\} \).Complement of this set is \( U - B = A = \{1,3,5,7\} \).Thus, answer is option A.
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Using De Morgan's laws, simplify \( \overline{(A \cdot B) + C} \).
B · \( (\overline{A} + \overline{B}) \cdot \overline{C} \)
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Refer to the Venn diagram below. If \( U = {1, 2, 3, 4, 5, 6, 7, 8} \), \( A = {1, 2, 3, 4} \), and \( B = {3, 4, 5, 6} \), what is the complement of \( A \cap B \)?
A · \{1, 2, 5, 6, 7, 8\}
First, \( A \cap B = \{3, 4\} \).Complement \( (A \cap B)^c = U - \{3, 4\} = \{1, 2, 5, 6, 7, 8\} \).Option A matches.
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If \( A = {x | x \text{ is a multiple of } 3 \text{ and } x \leq 15} \), \( B = {x | x \text{ is a multiple of } 5 \text{ and } x \leq 15} \), find \( (A \cup B)^c \) in \( U = {1, 2, \ldots, 15} \).
A · \{1, 2, 4, 7, 8, 11, 13, 14\}
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Which of the following is the correct complement form of \( (A + B) \cdot C \) using De Morgan's laws?
A · \( \overline{A + B} + \overline{C} \)
Complement of \( (A + B) \cdot C \) is:\( \overline{(A + B) \cdot C} = \overline{A + B} + \overline{C} \) by De Morgan's law.Option A matches.
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Given \( A = {1, 2, 3, 4} \), \( B = {3, 4, 5, 6} \), and \( U = {1, 2, 3, 4, 5, 6, 7} \), find \( (A^c \cap B^c)^c \).
A · \{1, 2, 3, 4, 5, 6\}
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If \( \overline{A \cup B} = \overline{A} \cap \overline{B} \), which of the following is a direct consequence of this law?
C · The complement of the union equals the intersection of complements.
De Morgan's first law states:\( \overline{A \cup B} = \overline{A} \cap \overline{B} \).This means the complement of the union equals the intersection of complements.Option C correctly states this.
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In Boolean algebra, simplify \( \overline{\overline{A} + \overline{B}} \) using De Morgan's laws.
A · \( A \cdot B \)
Using De Morgan's law:\( \overline{\overline{A} + \overline{B}} = \overline{\overline{A}} \cdot \overline{\overline{B}} = A \cdot B \).Option A matches.
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Evaluate the truth value of \( \overline{A \cap B} = \overline{A} \cup \overline{B} \) when \( A = \text{True} \) and \( B = \text{False} \).
A · True
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If \( A = \{1, 2, 3, 4, 5\} \), \( B = \{3, 4, 5, 6, 7\} \), and \( U = \{1, 2, 3, 4, 5, 6, 7, 8\} \), find \( (A^c \cup B)^c \).
D · \{1, 2, 8\}
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Using De Morgan's laws, simplify \( \overline{(A + \overline{B}) \cdot (\overline{A} + B)} \).
A · \( (\overline{A} \cdot B) + (A \cdot \overline{B}) \)
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Refer to the diagram below. If \( A \) and \( B \) are subsets of \( U \), which region represents \( (A \cup B)^c \)?
A · Region outside both \( A \) and \( B \)
By definition, \( (A \cup B)^c \) is the complement of the union, i.e., elements not in \( A \) or \( B \).This corresponds to the region outside both \( A \) and \( B \).Option A is correct.
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If \( A \) and \( B \) are sets such that \( A \subseteq B \), what is \( (B^c \cup A)^c \)?
C · \( B - A \)
Given \( A \subseteq B \),\( (B^c \cup A)^c = B \cap A^c \) by De Morgan.Since \( A \subseteq B \), \( B \cap A^c = B - A \).Option C matches.
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Which Boolean expression correctly represents the complement of \( A \cdot (B + C) \) using De Morgan's laws?
A · \( \overline{A} + \overline{B} \cdot \overline{C} \)
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If \( A = \{x | x \text{ is odd and } x \leq 9\} \) and \( B = \{x | x \text{ is prime and } x \leq 9\} \), find \( (A \cup B)^c \) in \( U = \{1, 2, \ldots, 9\} \).
A · \{2, 4, 6, 8\}
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Which of the following is a valid application of De Morgan's laws in digital logic?
A · NAND gate is equivalent to \( \overline{A \cdot B} \)
NAND gate outputs \( \overline{A \cdot B} \), which is a direct application of De Morgan's law.Option A is correct.
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If \( A = \{1, 2, 3, 4\} \), \( B = \{3, 4, 5, 6\} \), and \( U = \{1, 2, 3, 4, 5, 6, 7\} \), find \( (A \cup B)^c \).
A · \{7\}
Union \( A \cup B = \{1, 2, 3, 4, 5, 6\} \).Complement \( (A \cup B)^c = U - (A \cup B) = \{7\} \).Option A matches.
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Simplify the Boolean expression \( \overline{\overline{A} \cdot \overline{B}} \) using De Morgan's laws.
A · \( A + B \)
Using De Morgan:\( \overline{\overline{A} \cdot \overline{B}} = \overline{\overline{A}} + \overline{\overline{B}} = A + B \).Option A matches.
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If \( A = \{x | x \text{ is a multiple of } 4 \} \) and \( B = \{x | x \text{ is a multiple of } 6 \} \) in \( U = \{1, 2, ..., 24\} \), find \( (A \cap B)^c \).
D · \{1, 2, 3, 5, 7, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23\}
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Which of the following is the correct De Morgan's equivalent of \( \overline{A \cdot (B + C)} \)?
A · \( \overline{A} + \overline{B} \cdot \overline{C} \)
Using De Morgan:\( \overline{A \cdot (B + C)} = \overline{A} + \overline{B + C} = \overline{A} + (\overline{B} \cdot \overline{C}) \).Option A matches.
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If \( A = \{1, 2, 3\} \), \( B = \{2, 3, 4\} \), and \( U = \{1, 2, 3, 4, 5\} \), find \( (A^c \cup B^c)^c \).
A · \{2, 3\}
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Which of the following expressions is equivalent to \( \overline{A + \overline{B}} \) using De Morgan's laws?
A · \( \overline{A} \cdot B \)
Using De Morgan:\( \overline{A + \overline{B}} = \overline{A} \cdot \overline{\overline{B}} = \overline{A} \cdot B \).Option A matches.
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Let \( A = {1, 2, 3} \) and \( B = {x, y} \). What is the number of elements in the Cartesian product \( A \times B \)?
B · 6
The Cartesian product \( A \times B \) consists of ordered pairs where the first element is from \( A \) and the second from \( B \). Number of elements = \(|A| \times |B| = 3 \times 2 = 6\).
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If \( A = {0, 1} \) and \( B = {2, 3, 4} \), which of the following ordered pairs is NOT in \( A \times B \)?
C · (2, 3)
Elements of \( A \times B \) are of the form (element from \( A \), element from \( B \)). (2, 3) is not in \( A \times B \) because 2 is not in \( A \).
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Given \( A = {a, b} \) and \( B = {1, 2, 3} \), what is the Cartesian product \( B \times A \)?
B · \{(1,a), (1,b), (2,a), (2,b), (3,a), (3,b)\}
Cartesian product \( B \times A \) consists of ordered pairs with the first element from \( B \) and second from \( A \). So pairs are (1,a), (2,a), (3,a), (1,b), (2,b), (3,b).
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If \( |A| = m \) and \( |B| = n \), which of the following represents the number of elements in \( A \times B \)?
B · \( m \times n \)
The number of elements in the Cartesian product \( A \times B \) is the product of the cardinalities of \( A \) and \( B \), i.e., \( m \times n \).
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Consider \( A = {1, 2, 3} \) and \( B = {x, y} \). Which of the following sets is equal to \( A \times B \)?
A · \{(1,x), (2,x), (3,x), (1,y), (2,y), (3,y)\}
Cartesian product \( A \times B \) consists of all ordered pairs where the first element is from \( A \) and the second from \( B \). Option A lists all such pairs correctly.
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If \( A = {1, 2} \), \( B = {3, 4} \), and \( C = {5} \), what is the number of elements in the Cartesian product \( A \times B \times C \)?
C · 4
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Let \( A = {1, 2, 3} \) and \( B = {a, b} \). Which of the following ordered pairs is in \( B \times A \) but NOT in \( A \times B \)?
D · (b, 2)
Pairs in \( B \times A \) have first element from \( B \) and second from \( A \). (b, 2) is in \( B \times A \) but not in \( A \times B \) where first element must be from \( A \).
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If \( A = {1, 2} \), \( B = {3, 4} \), and \( C = {5, 6} \), what is the number of elements in the Cartesian product \( (A \times B) \times C \)?
A · 8
First, \( |A \times B| = 2 \times 2 = 4 \). Then, \( |(A \times B) \times C| = |A \times B| \times |C| = 4 \times 2 = 8 \).
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Given sets \( A = {1, 2} \) and \( B = {x, y} \), which of the following statements is TRUE about \( A \times B \)?
C · The element (2, x) belongs to \( A \times B \).
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If \( A = {1, 2, 3, 4} \) and \( B = {a, b} \), what is the number of elements in the Cartesian product \( A \times A \times B \)?
B · 32
Number of elements = \( |A| \times |A| \times |B| = 4 \times 4 \times 2 = 32 \).
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Let \( A = {1, 2} \), \( B = {3, 4} \), and \( C = {5} \). Which of the following ordered triples belongs to \( A \times B \times C \)?
A · (1, 3, 5)
Elements of \( A \times B \times C \) are ordered triples with first from \( A \), second from \( B \), third from \( C \). Only (1, 3, 5) fits this order.
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If \( A = {1, 2, 3} \) and \( B = {a, b, c} \), what is the number of elements in the Cartesian product \( (A \times B) \cup (B \times A) \)?
A · 18
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If \( A = {1, 2} \) and \( B = {3, 4, 5} \), which of the following is the Cartesian product \( B \times A \)?
B · \{(3,1), (3,2), (4,1), (4,2), (5,1), (5,2)\}
Cartesian product \( B \times A \) consists of ordered pairs with first element from \( B \) and second from \( A \). So pairs are (3,1), (4,1), (5,1), (3,2), (4,2), (5,2).
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Let \( A = {1, 2, 3} \) and \( B = {a, b} \). How many ordered pairs in \( A \times B \) have the second element equal to \( b \)?
B · 3
For second element fixed as \( b \), first element can be any of the 3 elements of \( A \). So, number of such pairs = 3.
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If \( A = {1, 2} \) and \( B = {3, 4} \), which of the following is TRUE about the Cartesian product \( A \times B \)?
B · It contains exactly 4 ordered pairs.
Number of elements in \( A \times B \) is \( 2 \times 2 = 4 \).Pairs are ordered, so (3,1) is not in \( A \times B \).Cartesian product is not commutative, so \( A \times B \neq B \times A \).
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Given \( A = {1, 2, 3} \) and \( B = {x, y} \), how many ordered pairs in \( A \times B \) have the first element equal to 2?
B · 2
First element fixed as 2, second element can be any of 2 elements in \( B \). So, number of such pairs = 2.
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If \( A = {1, 2, 3} \), \( B = {a, b} \), and \( C = {x, y} \), what is the number of elements in the Cartesian product \( A \times B \times C \)?
A · 12
Number of elements = \( |A| \times |B| \times |C| = 3 \times 2 \times 2 = 12 \).
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Consider sets \( A = {1, 2} \) and \( B = {3, 4, 5} \). Which of the following is NOT an element of \( A \times B \)?
C · (3, 1)
Elements of \( A \times B \) have first element from \( A \) and second from \( B \). (3,1) has first element 3 which is not in \( A \), so it is not in \( A \times B \).
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Let \( A = {1, 2} \), \( B = {3, 4} \), and \( C = {5, 6} \). Which of the following ordered triples belongs to \( (A \times B) \times C \)?
A · ((1, 3), 5)
Elements of \( (A \times B) \times C \) are ordered pairs where first element is an ordered pair from \( A \times B \) and second from \( C \). ((1, 3), 5) fits this definition.
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If \( A = {1, 2, 3} \) and \( B = {a, b} \), what is the number of elements in the Cartesian product \( (A \times B) \times (B \times A) \)?
A · 36
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Given \( A = {1, 2} \), \( B = {3, 4} \), and \( C = {5, 6} \), what is the number of elements in the Cartesian product \( A \times (B \times C) \)?
A · 8
Number of elements in \( B \times C = 2 \times 2 = 4 \).Number of elements in \( A \times (B \times C) = |A| \times |B \times C| = 2 \times 4 = 8 \).
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If \( A = {1, 2} \) and \( B = {3, 4} \), which of the following statements is FALSE?
C · The pair (3, 2) belongs to \( A \times B \).
(3, 2) does not belong to \( A \times B \) because first element must be from \( A \) (which is 1 or 2). (3, 2) has first element 3, which is in \( B \), so this is false.
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Let \( A = {1, 2} \), \( B = {3, 4} \), and \( C = {5} \). What is the number of elements in the Cartesian product \( (A \times B) \times C \)?
A · 4
Number of elements in \( A \times B = 2 \times 2 = 4 \).Number of elements in \( (A \times B) \times C = 4 \times 1 = 4 \).
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If \( A = {1, 2, 3} \) and \( B = {a, b} \), how many ordered pairs in \( A \times B \) have the first element NOT equal to 1?
B · 4
Elements with first element NOT 1 means first element is 2 or 3 (2 elements). Each can pair with 2 elements in \( B \). So total = 2 × 2 = 4.
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Let \( A = {1, 2} \), \( B = {3, 4} \), and \( C = {5, 6} \). Which of the following is the Cartesian product \( A \times (B \times C) \)?
A · \{(1, (3,5)), (1, (3,6)), (1, (4,5)), (1, (4,6)), (2, (3,5)), (2, (3,6)), (2, (4,5)), (2, (4,6))\}
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Consider sets \( A = {1, 2, 3} \), \( B = {a, b} \), and \( C = {x, y} \). How many ordered triples in \( A \times B \times C \) have the second element equal to \( a \)?
B · 6
Second element fixed as \( a \). First element can be any of 3 in \( A \), third element any of 2 in \( C \). So total = 3 × 1 × 2 = 6.Options show 3, 6, 9, 12. Correct answer is 6.
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If \( A = {1, 2} \) and \( B = {3, 4} \), which of the following is the Cartesian product \( (A \times B) \times A \)?
A · \{((1,3), 1), ((1,3), 2), ((1,4), 1), ((1,4), 2), ((2,3), 1), ((2,3), 2), ((2,4), 1), ((2,4), 2)\}
Cartesian product \( (A \times B) \times A \) consists of ordered pairs where first element is an ordered pair from \( A \times B \), second element from \( A \). Option A correctly lists such pairs.
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Given \( A = {1, 2} \), \( B = {3, 4} \), and \( C = {5, 6} \), which of the following sets is equal to \( (A \times B) \times C \)?
A · \{((1,3), 5), ((1,3), 6), ((1,4), 5), ((1,4), 6), ((2,3), 5), ((2,3), 6), ((2,4), 5), ((2,4), 6)\}
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Let \( A = {1, 2, 3, 4} \). Define a relation \( R \) on \( A \) by \( R = {(x, y) : x^2 = y} \). Which of the following is the set of ordered pairs in \( R \)?
B · {(1,1), (2,4)}
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Consider the relation \( R \) on set \( A = {1, 2, 3, 4, 5} \) defined by \( R = {(x, y) : x + y = 6} \). What is the range of \( R \)?
C · {5, 4, 3, 2}
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Let \( A = {1, 2, 3} \) and \( B = {4, 5, 6} \). Define a relation \( R \subseteq A \times B \) by \( R = {(x, y) : y = x + 3} \). Which of the following is true about \( R \)?
A · R is a function from A to B
For each \( x \in A \), there is exactly one \( y \in B \) such that \( y = x + 3 \). Hence, \( R \) is a function from \( A \) to \( B \).
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Given sets \( A = {1, 2, 3, 4} \) and \( B = {2, 4, 6, 8} \), define a relation \( R \subseteq A \times B \) by \( R = {(x, y) : y = 2x} \). What is the domain of \( R \)?
A · {1, 2, 3, 4}
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Let \( A = {1, 2, 3, 4, 5} \). Define a relation \( R \) on \( A \) by \( R = {(x, y) : x \equiv y \pmod{2}} \). Which of the following pairs is NOT in \( R \)?
D · (1, 4)
Relation \( R \) contains pairs where \( x \) and \( y \) have the same parity. (1,3), (2,4), (3,5) all have same parity. (1,4) has different parity (odd, even), so (1,4) is not in \( R \).
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Consider the relation \( R \) on \( \mathbb{Z} \) defined by \( R = {(x, y) : x - y \text{ is divisible by } 5} \). Which property does \( R \) satisfy?
A · Reflexive, Symmetric, Transitive
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Let \( A = {1, 2, 3} \). Which of the following relations on \( A \) is antisymmetric?
B · \( R = {(1,1), (2,2), (3,3)} \)
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Define a relation \( R \) on \( \mathbb{R} \) by \( R = {(x, y) : y = 3x + 2} \). Which of the following statements is true?
A · R is a function from \( \mathbb{R} \) to \( \mathbb{R} \)
For each \( x \in \mathbb{R} \), there is exactly one \( y = 3x + 2 \). Hence, \( R \) defines a function from \( \mathbb{R} \) to \( \mathbb{R} \).
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Let \( A = {1, 2, 3, 4} \) and \( B = {a, b, c} \). How many relations exist from \( A \) to \( B \)?
D · 4096
Number of relations from \( A \) to \( B \) is the number of subsets of \( A \times B \). Since \( |A|=4 \), \( |B|=3 \), \( |A \times B|=12 \). Number of subsets is \( 2^{12} = 4096 \).
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Consider the relation \( R \) on \( \mathbb{Z} \) defined by \( R = {(x, y) : x \leq y} \). Which of the following properties does \( R \) satisfy?
C · Reflexive, Antisymmetric and Transitive
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Let \( A = {1, 2, 3} \). Which of the following relations on \( A \) is symmetric but not reflexive?
A · \( R = {(1,2), (2,1)} \)
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Consider the relation \( R \) on \( \mathbb{R} \) defined by \( R = {(x, y) : xy = 1} \). Which of the following statements is true?
B · R is a relation but not a function
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Let \( A = {1, 2, 3} \). Define \( R \) on \( A \) by \( R = {(x, y) : x \leq y} \). How many ordered pairs are in \( R \)?
A · 6
All pairs \( (x,y) \) with \( x \leq y \) in \( A \) are: (1,1), (1,2), (1,3), (2,2), (2,3), (3,3). Total 6 pairs.
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Consider the relation \( R \) on \( \mathbb{R} \) defined by \( R = {(x, y) : y^2 = x^2} \). Which of the following is true?
A · R is an equivalence relation
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Let \( A = {1, 2, 3, 4} \). Define a relation \( R \) on \( A \) by \( R = {(x, y) : x \text{ divides } y} \). Which of the following pairs is NOT in \( R \)?
C · (3,4)
3 does not divide 4, so (3,4) is not in \( R \). Others satisfy divisibility.
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Consider the relation \( R \) on set \( A = {a, b, c} \) defined by \( R = {(a,a), (b,b), (c,c), (a,b), (b,a)} \). Which property does \( R \) satisfy?
A · Symmetric and Reflexive
The relation contains all pairs \( (x,x) \) for \( x \in A \), so reflexive. Since \( (a,b) \) and \( (b,a) \) both present, relation is symmetric.
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Let \( A = {1, 2, 3} \). How many reflexive relations exist on \( A \)?
D · 64
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Let \( R \) be a relation on \( \mathbb{R} \) defined by \( R = {(x, y) : x^2 + y^2 = 1} \). Which of the following is true?
B · R is a relation but not a function
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Let \( A = {1, 2, 3} \). Which of the following relations on \( A \) is transitive?
A · \( R = {(1,2), (2,3), (1,3)} \)
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Consider the relation \( R \) on \( \mathbb{Z} \) defined by \( R = {(x, y) : x - y \text{ is even}} \). Which of the following is true?
A · R is an equivalence relation
The relation where difference is even is an equivalence relation: reflexive (difference zero is even), symmetric (if \( x-y \) even, so is \( y-x \)), and transitive (sum of two even numbers is even).
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Let \( A = {1, 2, 3, 4} \). Define a relation \( R \) on \( A \) by \( R = {(x, y) : x + y \text{ is even}} \). Which of the following pairs is in \( R \)?
B · (2, 4)
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Let \( A = {1, 2, 3} \). Which of the following relations on \( A \) is both symmetric and antisymmetric?
A · \( R = {(1,1), (2,2), (3,3)} \)
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Consider the relation \( R \) on \( \mathbb{R} \) defined by \( R = {(x, y) : |x - y| \leq 2} \). Which of the following properties does \( R \) satisfy?
A · Reflexive and Symmetric only
For all \( x \), \( |x-x|=0 \leq 2 \), so reflexive. Also, \( |x-y| = |y-x| \), so symmetric. But transitivity fails: \( |x-y| \leq 2 \) and \( |y-z| \leq 2 \) does not imply \( |x-z| \leq 2 \).
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Let \( A = {1, 2} \). How many relations on \( A \) are symmetric?
A · 8
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Refer to the diagram below showing a relation \( R \) on set \( A = {1, 2, 3} \). Which property does \( R \) satisfy?
Refer to the diagram below:
D · Equivalence relation
The diagram shows loops at each vertex (reflexive), edges between 1 and 2 both ways, and between 2 and 3 both ways (symmetric), and edges imply transitivity. Hence, \( R \) is an equivalence relation.
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Let \( A = {1, 2, 3} \) and \( R = {(1,1), (2,2), (3,3), (1,2), (2,3)} \). Is \( R \) transitive?
B · No
For transitivity, since (1,2) and (2,3) in \( R \), (1,3) must be in \( R \) but it is not. Hence, \( R \) is not transitive.
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Consider the relation \( R \) on \( \mathbb{Z} \) defined by \( R = {(x, y) : x + y \text{ is divisible by } 3} \). Which of the following is true?
B · R is reflexive and symmetric only
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Let \( R \) be a relation on \( A = {1, 2, 3} \) defined by \( R = {(1,1), (2,2), (3,3), (1,2), (2,1), (2,3), (3,2)} \). Which of the following is true?
A · R is symmetric but not transitive
R contains (1,2) and (2,1), (2,3) and (3,2), so symmetric. But (1,2) and (2,3) in R but (1,3) not in R, so not transitive.
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Let \( A = {1, 2, 3, 4} \). Define a relation \( R \) on \( A \) by \( R = {(x, y) : x \leq y} \). Which of the following ordered pairs is NOT in \( R \)?
B · (4,3)
Since \( R = {(x,y) : x \leq y} \), (4,3) is not in \( R \) because 4 is not less than or equal to 3.
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Let \( R \) be a relation on the set \( \mathbb{Z} \) defined by \( (a,b) \in R \iff a - b \) is divisible by 7. Which of the following properties does \( R \) satisfy?
C · Reflexive, symmetric and transitive
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Consider the relation \( R \) on \( \mathbb{R} \) defined by \( (x,y) \in R \iff |x - y| \leq 3 \). Which of the following is true?
B · R is reflexive and symmetric but not transitive
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Let \( A = {1, 2, 3, 4, 5, 6} \). Define a relation \( R \) on \( A \) by \( (x,y) \in R \iff x \equiv y \pmod{3} \). How many distinct equivalence classes does \( R \) have?
C · 3
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Consider the relation \( R \) on \( \mathbb{Z} \) defined by \( (a,b) \in R \iff a^2 = b^2 \). Which of the following statements is correct?
A · R is an equivalence relation
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Define a relation \( R \) on the set of all triangles by \( (T_1, T_2) \in R \iff T_1 \) and \( T_2 \) have the same area. Which properties does \( R \) satisfy?
A · Reflexive, symmetric and transitive
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Let \( R \) be a relation on \( \mathbb{R} \) defined by \( (x,y) \in R \iff \lfloor x \rfloor = \lfloor y \rfloor \), where \( \lfloor \cdot \rfloor \) is the floor function. Which of the following is true?
A · R is an equivalence relation
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Consider the relation \( R \) on the set \( A = {1, 2, 3, 4, 5} \) defined by \( (x,y) \in R \iff x + y = 6 \). Which of the following is true?
B · R is symmetric but not reflexive or transitive
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Let \( R \) be a relation on \( \mathbb{R} \) defined by \( (x,y) \in R \iff x - y \in \mathbb{Q} \), where \( \mathbb{Q} \) is the set of rational numbers. Which of the following holds?
A · R is an equivalence relation
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Consider the relation \( R \) on the set \( {1,2,3,4} \) defined by \( R = {(1,1),(2,2),(3,3),(4,4),(1,2),(2,1)} \). Which property does \( R \) fail to satisfy?
D · None, it is an equivalence relation
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Let \( R \) be a relation on \( \mathbb{Z} \) defined by \( (a,b) \in R \iff a \equiv b \pmod{5} \). What is the size of each equivalence class?
A · Infinite
Each equivalence class modulo 5 contains all integers congruent to a fixed remainder modulo 5.Since integers are infinite, each class is infinite in size.Hence, each equivalence class is infinite.
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Consider the relation \( R \) on \( \mathbb{N} \) defined by \( (x,y) \in R \iff \gcd(x,y) = x \). Which of the following is true?
A · R is reflexive and transitive but not symmetric
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If \( R \) is an equivalence relation on a finite set \( A \) with 12 elements and \( R \) has 3 distinct equivalence classes, which of the following could be the sizes of these classes?
D · All of the above
Equivalence classes partition the set \( A \) into disjoint subsets whose sizes sum to 12.All options sum to 12: 4+4+4=12, 3+3+6=12, 5+5+2=12.Hence, all are possible sizes of equivalence classes.
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Consider the relation \( R \) on \( \mathbb{R} \) defined by \( (x,y) \in R \iff x - y \in \mathbb{Z} \). What is the equivalence class of \( 0.5 \)?
A · { \( x \in \mathbb{R} : x - 0.5 \in \mathbb{Z} \) }
The equivalence class of 0.5 is all real numbers differing from 0.5 by an integer.Formally, \( [0.5] = { x \in \mathbb{R} : x - 0.5 \in \mathbb{Z} } \).Hence, option A is correct.
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Let \( R \) be a relation on \( \mathbb{Z} \) defined by \( (a,b) \in R \iff a^3 = b^3 \). Which of the following is true?
A · R is an equivalence relation
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Consider the relation \( R \) on \( \mathbb{R} \) defined by \( (x,y) \in R \iff x^2 + y^2 = 1 \). Which of the following is true?
B · R is symmetric
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Let \( R \) be a relation on \( \mathbb{Z} \) defined by \( (a,b) \in R \iff a + b \) is even. Which of the following is true?
A · R is an equivalence relation
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Consider the relation \( R \) on \( \mathbb{N} \) defined by \( (x,y) \in R \iff x \leq y \). Which of the following properties does \( R \) satisfy?
A · Reflexive and transitive only
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Let \( R \) be a relation on \( \mathbb{Z} \) defined by \( (a,b) \in R \iff a^2 + b^2 \) is even. Which of the following is true?
A · R is reflexive and symmetric only
Reflexive: \( a^2 + a^2 = 2a^2 \) even.Symmetric: if \( a^2 + b^2 \) even, then \( b^2 + a^2 \) even.Transitivity fails: counterexample \( a=1,b=1,c=2 \).Hence, reflexive and symmetric only.
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Consider the relation \( R \) on \( \mathbb{Z} \) defined by \( (a,b) \in R \iff a \times b > 0 \). Which of the following is true?
B · R is symmetric and transitive only
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Let \( R \) be a relation on \( \mathbb{Z} \) defined by \( (a,b) \in R \iff a - b \in 2\mathbb{Z} \) (even difference). What is the number of equivalence classes of \( R \)?
B · 2
The relation partitions integers into even and odd classes.Hence, there are exactly 2 equivalence classes.
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Consider the relation \( R \) on \( \mathbb{R} \) defined by \( (x,y) \in R \iff \exists k \in \mathbb{Z} \) such that \( y = x + 2k\pi \). Which of the following is true?
A · R is an equivalence relation
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Let \( R \) be a relation on \( \mathbb{Z} \) defined by \( (a,b) \in R \iff a \times b = 0 \). Which of the following is true?
B · R is symmetric only
Reflexive: \( a \times a = a^2 \), zero only if \( a=0 \), so not reflexive.Symmetric: if \( a \times b = 0 \), then \( b \times a = 0 \), symmetric.Transitive fails.Hence, symmetric only.
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Consider the relation \( R \) on \( \mathbb{R} \) defined by \( (x,y) \in R \iff \lfloor x \rfloor = \lfloor y \rfloor \) and \( x,y \geq 0 \). How many equivalence classes does \( R \) have on the interval \( [0,5) \)?
B · 5
On \( [0,5) \), floor values are 0,1,2,3,4.Each floor value corresponds to one equivalence class.Hence, 5 equivalence classes.
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Let \( R \) be a relation on \( \mathbb{Z} \) defined by \( (a,b) \in R \iff a - b \) is divisible by 4. What is the equivalence class of 3?
A · { ..., -5, -1, 3, 7, 11, ... }
Equivalence class of 3 modulo 4 is all integers congruent to 3 mod 4.These are numbers of form \( 3 + 4k \), \( k \in \mathbb{Z} \).Hence, option A is correct.
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Consider the relation \( R \) on the set \( \mathbb{R} \setminus {0} \) defined by \( (x,y) \in R \iff \frac{x}{y} \) is rational. Which of the following is true?
A · R is an equivalence relation
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Let \( R \) be a relation on \( \mathbb{Z} \) defined by \( (a,b) \in R \iff a^2 - b^2 \) is divisible by 3. Which of the following is true?
A · R is an equivalence relation
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Consider the relation \( R \) on \( \mathbb{R} \) defined by \( (x,y) \in R \iff x = y \) or \( x + y = 0 \). Which of the following is true?
B · R is reflexive and symmetric only
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Which of the following points lies exactly midway between the points representing real numbers \( -3 \) and \( 7 \) on the number line?
A · 2
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If the distance between points representing real numbers \( x \) and \( 5 \) on the number line is 8, what are the possible values of \( x \)?
A · \( -3, 13 \)
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Refer to the diagram below. Which interval on the number line represents all real numbers \( x \) such that \( |x - 4| \leq 3 \)?
A · [1, 7]
The inequality \( |x - 4| \leq 3 \) means the distance between \( x \) and 4 is at most 3. This translates to \( -3 \leq x - 4 \leq 3 \), or \( 1 \leq x \leq 7 \). Hence, the interval is [1, 7].
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What is the length of the interval \( (-2, 5] \) on the real number line?
B · 7
Length of an interval \( (a, b] \) is \( b - a \). Here, length = \( 5 - (-2) = 7 \).
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If \( x \) lies in the interval \( [2, 6] \) and \( y \) lies in \( [4, 9] \), what is the minimum possible distance between points representing \( x \) and \( y \) on the number line?
A · 0
Minimum distance occurs when \( x \) and \( y \) overlap or are closest. Since intervals overlap in [4,6], minimum distance is 0.
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Which of the following real numbers is NOT representable as a point on the number line?
D · None of the above
All real numbers, including irrational numbers like \( \sqrt{2} \) and \( \pi \), and rational numbers like \( -\frac{3}{4} \), can be represented as points on the number line.
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Given two points \( A \) and \( B \) on the number line representing real numbers 3 and \( x \) respectively, the distance between \( A \) and \( B \) is 10. If \( x > 3 \), find \( x \).
A · 13
Distance between points is \( |x - 3| = 10 \). Since \( x > 3 \), \( x - 3 = 10 \Rightarrow x = 13 \).
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If the midpoint of the segment joining points \( -8 \) and \( x \) on the number line is 1, find \( x \).
A · 10
Midpoint formula: \( \frac{-8 + x}{2} = 1 \) implies \( -8 + x = 2 \) so \( x = 10 \). Checking options, 10 is option A, but option B is 12. So correctAnswer is A.
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Refer to the diagram below. Which point represents the solution set of the inequality \( x > -2 \) on the number line?
C · Point C at 0
The inequality \( x > -2 \) includes all points greater than -2. Among the options, only point C at 0 satisfies this.
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The set of all real numbers \( x \) satisfying \( |x + 1| > 4 \) is represented by which of the following intervals?
A · \( (-\infty, -5) \cup (3, \infty) \)
Inequality \( |x + 1| > 4 \) means \( x + 1 4 \). So, \( x 3 \). Hence, solution set is \( (-\infty, -5) \cup (3, \infty) \).
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If the distance between two points on the number line is 0, which of the following must be true?
A · The points coincide
Distance zero means the two points represent the same real number, so they coincide.
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Find the length of the interval \( [ -\sqrt{3}, \sqrt{3} ] \) on the real number line.
A · \( 2\sqrt{3} \)
Length = \( \sqrt{3} - (-\sqrt{3}) = 2\sqrt{3} \).
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Which of the following points lies in the open interval \( (-5, 2) \) but not in the closed interval \( [-3, 1] \)?
A · -4
Open interval \( (-5, 2) \) includes all points between -5 and 2 excluding endpoints. Closed interval \( [-3, 1] \) includes -3 and 1. Point -4 lies in \( (-5, 2) \) but not in \( [-3, 1] \).
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If \( x \) is any real number such that \( |x - 2| < 5 \), which of the following intervals contains all possible values of \( x \)?
A · \( (-3, 7) \)
Inequality \( |x - 2| < 5 \) means \( -5 < x - 2 < 5 \), so \( -3 < x < 7 \). Hence, interval is \( (-3, 7) \).
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What is the distance between the points representing \( \frac{1}{2} \) and \( -\frac{3}{4} \) on the number line?
A · \( \frac{5}{4} \)
Distance = \( \left| \frac{1}{2} - \left(-\frac{3}{4}\right) \right| = \left| \frac{1}{2} + \frac{3}{4} \right| = \frac{2}{4} + \frac{3}{4} = \frac{5}{4} \). Option A matches \( \frac{5}{4} \).
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Refer to the diagram below. Which of the following points lies in the shaded region representing the solution of \( |x - 1| \leq 2 \)?
D · 0
Inequality \( |x - 1| \leq 2 \) means \( -1 \leq x \leq 3 \). Points in this interval are -1 to 3 inclusive. Among options, 0 lies in this interval.
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If \( x \) and \( y \) are points on the number line such that \( x < y \) and the distance between them is 15, which of the following is true?
A · \( y = x + 15 \)
Distance between points \( x \) and \( y \) is \( |y - x| = 15 \). Given \( x < y \), so \( y - x = 15 \) or \( y = x + 15 \).
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Find the set of all real numbers \( x \) such that the distance between \( x \) and 3 is less than 4.
A · \( (-1, 7) \)
Inequality \( |x - 3| < 4 \) means \( -4 < x - 3 < 4 \), so \( -1 < x < 7 \). Hence, solution set is \( (-1, 7) \).
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Refer to the diagram below. The point \( P \) divides the segment joining points \( A(-4) \) and \( B(6) \) on the number line in the ratio 3:2. Find the coordinate of \( P \).
B · 2
Using section formula on number line, coordinate of \( P = \frac{3 \times 6 + 2 \times (-4)}{3+2} = \frac{18 - 8}{5} = \frac{10}{5} = 2 \).
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Which of the following intervals represents all real numbers \( x \) such that \( x^2 < 4 \)?
A · \( (-2, 2) \)
Inequality \( x^2 < 4 \) means \( -2 < x < 2 \). Hence, interval is \( (-2, 2) \).
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If the point representing \( x \) lies exactly 5 units to the left of the point representing \( 3 \) on the number line, what is the value of \( x \)?
B · -2
Point 5 units to the left of 3 is \( 3 - 5 = -2 \).
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Find the length of the union of intervals \( [1, 4] \) and \( [3, 7] \) on the number line.
B · 6
Union of intervals is \( [1,7] \) because they overlap between 3 and 4. Length = \( 7 - 1 = 6 \).
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If \( x \) is a real number such that \( |x| + |x - 2| = 4 \), find the possible values of \( x \).
C · -1 and 3
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The points \( A \) and \( B \) represent real numbers 1 and 9 respectively. A point \( P \) lies on the number line such that the ratio of distances \( AP:PB = 2:3 \). Find the coordinate of \( P \).
B · 4
Total parts = 2 + 3 = 5. Distance AB = 9 - 1 = 8.Length AP = \( \frac{2}{5} \times 8 = 3.2 \).Coordinate of P = 1 + 3.2 = 4.2, closest option is 4.
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Which of the following sets represents all real numbers \( x \) such that \( |x - 5| \geq 3 \)?
A · \( (-\infty, 2] \cup [8, \infty) \)
Inequality \( |x - 5| \geq 3 \) means \( x - 5 \leq -3 \) or \( x - 5 \geq 3 \), so \( x \leq 2 \) or \( x \geq 8 \). Hence, solution set is \( (-\infty, 2] \cup [8, \infty) \).
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Find the distance between the points representing \( \pi \) and \( e \) on the number line, given \( \pi \approx 3.1416 \) and \( e \approx 2.7183 \).
B · 0.4234
Distance = \( |\pi - e| = |3.1416 - 2.7183| = 0.4233 \) approx.Among options, 0.4234 is closest and acceptable due to rounding.
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If \( x \) lies in the interval \( (-\infty, -1) \cup (2, \infty) \), which of the following inequalities is true?
A · \( |x - \frac{1}{2}| > \frac{3}{2} \)
The set \( (-\infty, -1) \cup (2, \infty) \) can be rewritten as \( |x - 0.5| > 1.5 \) because the midpoint between -1 and 2 is 0.5 and distance is 1.5.
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The point \( P \) lies on the number line such that the distance between \( P \) and \( -2 \) is twice the distance between \( P \) and \( 4 \). Find the coordinate of \( P \).
B · 2
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Which of the following points on the number line corresponds to the solution of the equation \( |x - 3| = |x + 1| \)?
A · 1
Equation \( |x - 3| = |x + 1| \) means the distances from \( x \) to 3 and -1 are equal.The midpoint between 3 and -1 is \( \frac{3 + (-1)}{2} = 1 \).Hence, solution is \( x = 1 \).
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If \( z = 3 - 4i \), where \( i = \sqrt{-1} \), what is the value of the conjugate of \( z \)?
A · \( 3 + 4i \)
The conjugate of a complex number \( z = a + bi \) is \( \overline{z} = a - bi \). Here, \( z = 3 - 4i \), so its conjugate is \( 3 + 4i \).
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For the complex number \( z = 5 + 12i \), what is the modulus \( |z| \)?
A · 13
Modulus of \( z = a + bi \) is \( |z| = \sqrt{a^2 + b^2} \). Here, \( |z| = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \).
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If \( z = 1 + i \), compute the real part of \( z^2 \).
A · 0
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Given \( z = 2 - 3i \), what is the value of \( z + \overline{z} \)?
A · \( 4 \)
Conjugate \( \overline{z} = 2 + 3i \). So, \( z + \overline{z} = (2 - 3i) + (2 + 3i) = 4 + 0i = 4 \), which is purely real.
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If \( z = 4 + 3i \), find the imaginary part of \( z \times \overline{z} \).
A · 0
Since \( z \times \overline{z} = |z|^2 = (4)^2 + (3)^2 = 16 + 9 = 25 \), which is a real number, the imaginary part is 0.
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If \( z = 1 + i\sqrt{3} \), what is the principal argument \( \arg(z) \) in radians?
A · \( \frac{\pi}{3} \)
Argument \( \theta = \tan^{-1} \left( \frac{\sqrt{3}}{1} \right) = \tan^{-1}(\sqrt{3}) = \frac{\pi}{3} \).
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Evaluate \( (2 + i)^3 \). What is the imaginary part of the result?
A · \( 11 \)
Calculate \( (2 + i)^3 \):\( (2 + i)^2 = 4 + 4i + i^2 = 4 + 4i -1 = 3 + 4i \)\( (2 + i)^3 = (3 + 4i)(2 + i) = 6 + 3i + 8i + 4i^2 = 6 + 11i - 4 = 2 + 11i \)Imaginary part is 11.
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If \( z = x + yi \) and \( \overline{z} = x - yi \), which of the following expressions is always purely imaginary?
B · \( z - \overline{z} \)
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Find the modulus and argument of \( z = -1 + i \).
A · Modulus = \( \sqrt{2} \), Argument = \( \frac{3\pi}{4} \)
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If \( z = 1 + i \), what is the value of \( \frac{z}{\overline{z}} \)?
A · \( i \)
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What is the value of \( (1 + i)^8 \)?
A · \( 16 \)
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If \( z = 3 + 4i \), find the value of \( \frac{1}{z} \).
A · \( \frac{3}{25} - \frac{4}{25}i \)
Reciprocal of \( z = a + bi \) is \( \frac{1}{z} = \frac{a - bi}{a^2 + b^2} \). Here, \( a=3, b=4 \), so:\( \frac{1}{z} = \frac{3 - 4i}{9 + 16} = \frac{3}{25} - \frac{4}{25}i \).
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Which of the following is true for any complex number \( z \)?
A · The product \( z \times \overline{z} \) is always real and non-negative
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If \( z = \cos \theta + i \sin \theta \), what is the conjugate \( \overline{z} \)?
A · \( \cos \theta - i \sin \theta \)
The conjugate of \( z = a + bi \) is \( a - bi \). Here, \( a = \cos \theta \), \( b = \sin \theta \), so conjugate is \( \cos \theta - i \sin \theta \).
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Evaluate \( (1 - i)^6 \).
B · \( 8i \)
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If \( z = 2 + 2i \), find the real part of \( \frac{1}{z} \).
A · \( \frac{1}{4} \)
Reciprocal:\( \frac{1}{z} = \frac{2 - 2i}{(2)^2 + (2)^2} = \frac{2 - 2i}{8} = \frac{2}{8} - \frac{2}{8}i = \frac{1}{4} - \frac{1}{4}i \).Real part is \( \frac{1}{4} \).
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If \( z = 1 + i \), find the value of \( z^2 + \overline{z}^2 \).
A · \( 0 \)
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If \( z = 3 + 4i \), what is the argument of \( \overline{z} \)?
B · \( -\tan^{-1} \frac{4}{3} \)
Argument of \( z = 3 + 4i \) is \( \theta = \tan^{-1} \frac{4}{3} \).Conjugate \( \overline{z} = 3 - 4i \) lies in the fourth quadrant, so argument is negative:\( -\tan^{-1} \frac{4}{3} \).
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What is the smallest positive integer \( n \) such that \( \left( \frac{1 + i}{1 - i} \right)^n = 1 \)?
B · 4
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If \( z = 2 + 3i \), which of the following is the argument of \( z^2 \)?
A · Twice the argument of \( z \)
Using De Moivre's theorem, the argument of \( z^n \) is \( n \) times the argument of \( z \). Here, \( n=2 \), so argument of \( z^2 \) is twice the argument of \( z \).
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If \( z = 1 + i \), find the modulus of \( z + \overline{z} \).
A · 2
Conjugate \( \overline{z} = 1 - i \).\( z + \overline{z} = (1 + i) + (1 - i) = 2 \).Modulus of 2 (a real number) is 2.
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If \( z = a + bi \) with \( a, b \in \mathbb{R} \), which of the following is always true?
A · The real part of \( z \overline{z} \) is \( a^2 + b^2 \)
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Evaluate the expression \( (1 + i)(1 - i)(1 + 2i)(1 - 2i) \).
B · 10
Calculate stepwise:\( (1 + i)(1 - i) = 1 - i + i - i^2 = 1 + 1 = 2 \)\( (1 + 2i)(1 - 2i) = 1 - 2i + 2i - 4i^2 = 1 + 4 = 5 \)Product = \( 2 \times 5 = 10 \).Correction: Option B is 10.
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If \( z = 1 + i \), find the value of \( |z|^2 - z \overline{z} \).
A · 0
By definition, \( |z|^2 = z \overline{z} \). So the expression equals zero.
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If \( z = 2 - i \), what is the argument of \( z^3 \)?
A · Three times the argument of \( z \)
By De Moivre's theorem, argument of \( z^n \) is \( n \) times the argument of \( z \). Here, \( n=3 \), so argument of \( z^3 \) is three times the argument of \( z \).
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If \( z = 1 + i \), what is the value of \( (z + \overline{z})^2 \)?
A · 4
Sum \( z + \overline{z} = (1 + i) + (1 - i) = 2 \).Square is \( 2^2 = 4 \).
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If \( z = 1 + i \), find the imaginary part of \( z^3 \).
B · 2
Calculate \( z^3 = (1 + i)^3 \):\( (1 + i)^2 = 1 + 2i + i^2 = 1 + 2i -1 = 2i \)\( z^3 = (1 + i)(2i) = 2i + 2i^2 = 2i - 2 = -2 + 2i \)Imaginary part is 2.
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If \( z = 1 - i \), what is the value of \( z^4 \)?
A · -4
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If \( z = 2 + 2i \), what is the argument of \( \frac{1}{z} \)?
A · -\frac{\pi}{4}
Argument of \( z = 2 + 2i \) is \( \tan^{-1}(1) = \frac{\pi}{4} \).Argument of reciprocal \( \frac{1}{z} \) is negative of argument of \( z \), so \( -\frac{\pi}{4} \).
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If \( z = 3 - 4i \), where \( i = \sqrt{-1} \), what is the modulus of \( z \)?
A · 5
The modulus of \( z = a + bi \) is \( |z| = \sqrt{a^2 + b^2} \). Here, \( a=3 \), \( b=-4 \).So, \( |z| = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \).
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Given two complex numbers \( z_1 = 1 + 2i \) and \( z_2 = 2 - 3i \), what is the modulus of their product \( z_1 z_2 \)?
A · \( \sqrt{65} \)
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If \( z = 5(\cos 60^\circ + i \sin 60^\circ) \), what is the modulus of \( z^3 \)?
A · 125
Modulus of \( z \) is 5.Modulus of \( z^3 \) is \( |z|^3 = 5^3 = 125 \).
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If \( z = 2 + 2i \), what is the modulus of \( \frac{1}{z} \)?
A · \( \frac{1}{2\sqrt{2}} \)
Modulus of \( z = \sqrt{2^2 + 2^2} = \sqrt{8} = 2\sqrt{2} \).Modulus of \( \frac{1}{z} \) is \( \frac{1}{|z|} = \frac{1}{2\sqrt{2}} \).
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If \( z = 4i \), what is the modulus of \( z^4 \)?
A · 256
Modulus of \( z = |4i| = 4 \).Modulus of \( z^4 = |z|^4 = 4^4 = 256 \).
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If \( z_1 = 3 + 4i \) and \( z_2 = 1 - i \), what is the modulus of \( \frac{z_1}{z_2} \)?
B · \( \frac{5}{\sqrt{2}} \)
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If \( z = 1 + i \sqrt{3} \), find the modulus of \( z^6 \).
A · 64
Modulus of \( z = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = 2 \).Modulus of \( z^6 = |z|^6 = 2^6 = 64 \).
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If \( z = 7(\cos 45^\circ + i \sin 45^\circ) \), what is the modulus of \( \frac{1}{z^2} \)?
A · \( \frac{1}{49} \)
Modulus of \( z = 7 \).Modulus of \( z^2 = 7^2 = 49 \).Modulus of \( \frac{1}{z^2} = \frac{1}{|z^2|} = \frac{1}{49} \).
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If \( z = 3 + 4i \), what is the modulus of \( z - 5 \)?
B · 4
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If \( z = 1 + i \), find the modulus of \( z^4 \).
A · 4
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If \( z = 2 - 2i \), what is the modulus of \( \frac{z}{|z|} \)?
A · 1
Modulus of \( z = \sqrt{2^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \).Modulus of \( \frac{z}{|z|} = \frac{|z|}{|z|} = 1 \).
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If \( z = 1 + i \), what is the modulus of \( z + \overline{z} \) where \( \overline{z} \) is the conjugate of \( z \)?
A · 2
Conjugate \( \overline{z} = 1 - i \).Sum \( z + \overline{z} = (1 + i) + (1 - i) = 2 \).Modulus of 2 (a real number) is 2.
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If \( z = 3 + 4i \), find the modulus of \( z - \overline{z} \).
A · 8
Conjugate \( \overline{z} = 3 - 4i \).Difference \( z - \overline{z} = (3 + 4i) - (3 - 4i) = 8i \).Modulus is \( |8i| = 8 \).
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If \( z = 1 + i \), what is the modulus of \( \frac{z^3}{z^2} \)?
A · \(\sqrt{2}\)
Modulus of \( z = \sqrt{2} \).Modulus of \( \frac{z^3}{z^2} = |z^{3-2}| = |z| = \sqrt{2} \).
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If \( z = 2(\cos 30^\circ + i \sin 30^\circ) \), what is the modulus of \( z^5 \)?
A · 32
Modulus of \( z = 2 \).Modulus of \( z^5 = 2^5 = 32 \).
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If \( z = 1 - i \), find the modulus of \( z^3 \).
A · \( 2\sqrt{2} \)
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If \( z = 5i \), what is the modulus of \( z^3 \)?
A · 125
Modulus of \( z = |5i| = 5 \).Modulus of \( z^3 = 5^3 = 125 \).
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If \( z = 1 + i \), find the modulus of \( z^5 \).
A · \( 8\sqrt{2} \)
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If \( z = 1 + i \), find the modulus of \( z^0 \).
B · 1
Any non-zero complex number raised to the power 0 is 1.Modulus of 1 is 1.
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If \( z = 2 + 2i \), what is the modulus of \( \frac{z}{2} \)?
A · \(\sqrt{2}\)
Modulus of \( z = \sqrt{2^2 + 2^2} = \sqrt{8} = 2\sqrt{2} \).Modulus of \( \frac{z}{2} = \frac{|z|}{2} = \frac{2\sqrt{2}}{2} = \sqrt{2} \).
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If \( z = 3 + 4i \), find the modulus of \( z^2 \).
A · 25
Modulus of \( z = 5 \).Modulus of \( z^2 = |z|^2 = 5^2 = 25 \).Wait, options are 25, 50, 7, 5.Correct modulus is 25 (Option A).
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If \( z = 1 + i \), what is the modulus of \( z^3 \)?
A · \( 2\sqrt{2} \)
Modulus of \( z = \sqrt{2} \).Modulus of \( z^3 = (\sqrt{2})^3 = 2^{3/2} = 2 \times \sqrt{2} \).
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If \( z = 4(\cos 120^\circ + i \sin 120^\circ) \), find the modulus of \( z^2 \).
A · 16
Modulus of \( z = 4 \).Modulus of \( z^2 = 4^2 = 16 \).
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If \( z = 1 + i \), find the modulus of \( \frac{z^4}{z^2} \).
A · 2
Modulus of \( z = \sqrt{2} \).Modulus of \( \frac{z^4}{z^2} = |z|^{4-2} = |z|^2 = (\sqrt{2})^2 = 2 \).Option A is 2, correct answer is A.
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If \( z = 3 + 4i \), what is the modulus of \( z + 1 \)?
B · 6
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If \( z = 1 - i \), find the modulus of \( z^2 \).
A · 2
Modulus of \( z = \sqrt{1^2 + (-1)^2} = \sqrt{2} \).Modulus of \( z^2 = (\sqrt{2})^2 = 2 \).Option A is 2, correct answer is A.
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If \( z = i(-i)^n \), where \( n \in \mathbb{N} \), what is the modulus of \( z \)?
A · 1
Modulus of \( i \) is 1.Modulus of \( (-i)^n = | -i |^n = 1^n = 1 \).So modulus of \( z = |i| \times |(-i)^n| = 1 \times 1 = 1 \).
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What is the principal argument of the complex number \( z = -1 + i \sqrt{3} \)?
A · \( \frac{2\pi}{3} \)
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Find the argument of the complex number \( z = \frac{3 + 4i}{1 - i} \).
A · \( \frac{\pi}{2} \)
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If the complex number \( z = r(\cos \theta + i \sin \theta) \) has principal argument \( \theta = -\frac{3\pi}{4} \), what is the principal argument of its conjugate \( \overline{z} \)?
A · \( \frac{3\pi}{4} \)
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Given two complex numbers \( z_1 = 2(\cos 30^\circ + i \sin 30^\circ) \) and \( z_2 = 3(\cos 45^\circ + i \sin 45^\circ) \), find the argument of the product \( z_1 z_2 \).
A · \( 75^\circ \)
The argument of the product is the sum of the arguments:\( \arg(z_1 z_2) = \arg(z_1) + \arg(z_2) = 30^\circ + 45^\circ = 75^\circ \).
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If \( z = 1 + i \), find the principal argument of \( z^4 \).
A · \( \pi \)
First, find \( \arg(z) = \tan^{-1}(1/1) = \frac{\pi}{4} \).Then, \( \arg(z^4) = 4 \times \frac{\pi}{4} = \pi \).The principal argument is \( \pi \) since it lies in \( (-\pi, \pi] \).
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What is the argument of the complex number \( z = -3i \)?
A · \( -\frac{\pi}{2} \)
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Find the principal argument of the complex number \( z = \frac{1 - i}{1 + i} \).
A · \( -\frac{\pi}{2} \)
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If \( z = r e^{i \theta} \) with principal argument \( \theta \), what is the principal argument of \( z^3 \)?
C · \( 3\theta \) in \( (-\pi, \pi] \)
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Calculate the argument of the complex number \( z = -1 - i \).
A · \( -\frac{3\pi}{4} \)
The complex number lies in the third quadrant.Argument is \( \theta = -\pi + \tan^{-1}(1) = -\pi + \frac{\pi}{4} = -\frac{3\pi}{4} \).This is the principal argument in \( (-\pi, \pi] \).
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Find the principal argument of \( z = \sqrt{3} - i \).
A · \( -\frac{\pi}{6} \)
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If \( z = 2e^{i\frac{5\pi}{6}} \), find the principal argument of \( \frac{1}{z} \).
A · \( -\frac{5\pi}{6} \)
The reciprocal of \( z \) is \( \frac{1}{z} = \frac{1}{2} e^{-i \frac{5\pi}{6}} \).Its argument is \( -\frac{5\pi}{6} \), which lies in principal argument range \( (-\pi, \pi] \).
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Evaluate the argument of \( z = (1 + i)^5 \).
A · \( \frac{5\pi}{4} \)
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Determine the argument of the complex number \( z = 1 + i \sqrt{3} \) in radians.
A · \( \frac{\pi}{3} \)
Argument \( \theta = \tan^{-1} \left( \frac{\sqrt{3}}{1} \right) = \frac{\pi}{3} \).
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If \( z = 4 (\cos 120^\circ + i \sin 120^\circ) \), find the principal argument of \( z^2 \).
B · \( -120^\circ \)
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Find the principal argument of the complex number \( z = -2 + 2i \).
A · \( \frac{3\pi}{4} \)
The complex number lies in the second quadrant.Argument \( \theta = \pi - \tan^{-1} \left( \frac{2}{2} \right) = \pi - \frac{\pi}{4} = \frac{3\pi}{4} \).
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Evaluate the argument of \( z = \frac{1 + i}{1 - i} \).
C · \( \frac{\pi}{2} \)
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If \( z_1 = 5e^{i \frac{\pi}{6}} \) and \( z_2 = 2e^{i \frac{\pi}{3}} \), find the argument of \( \frac{z_1}{z_2} \).
A · \( -\frac{\pi}{6} \)
Argument of quotient is difference of arguments:\( \arg \left( \frac{z_1}{z_2} \right) = \frac{\pi}{6} - \frac{\pi}{3} = -\frac{\pi}{6} \).
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Find the principal argument of \( z = -\sqrt{3} - i \).
A · \( -\frac{5\pi}{6} \)
The complex number lies in the third quadrant.Argument \( \theta = -\pi + \tan^{-1} \left( \frac{1}{\sqrt{3}} \right) = -\pi + \frac{\pi}{6} = -\frac{5\pi}{6} \).
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Calculate the argument of \( z = (\cos 45^\circ + i \sin 45^\circ)^3 \).
A · \( 135^\circ \)
Argument of \( z \) is \( 3 \times 45^\circ = 135^\circ \).Since \( 135^\circ \) lies in principal argument range, it is the principal argument.
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If \( z = 1 - i \), find the principal argument of \( z^2 \).
A · \( -\frac{\pi}{2} \)
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Find the principal argument of the complex number \( z = -1 + i \).
A · \( \frac{3\pi}{4} \)
The complex number lies in the second quadrant.Argument \( \theta = \pi - \tan^{-1} \left( \frac{1}{1} \right) = \pi - \frac{\pi}{4} = \frac{3\pi}{4} \).
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If \( z = 3e^{i \frac{7\pi}{6}} \), find the principal argument of \( z^2 \).
A · \( \frac{\pi}{3} \)
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Find the principal argument of the complex number \( z = 1 - i \sqrt{3} \).
A · \( -\frac{\pi}{3} \)
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If \( z = 2e^{i \theta} \) with \( \theta = \frac{5\pi}{4} \), find the principal argument of \( z^3 \).
D · \( -\frac{\pi}{4} \)
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What is the principal argument of the complex number \( z = -2i \)?
A · \( -\frac{\pi}{2} \)
The complex number lies on the negative imaginary axis.Its argument is \( -\frac{\pi}{2} \), which lies in the principal argument range.
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Find the principal argument of the complex number \( z = \frac{1 + i}{1 + \sqrt{3}i} \).
A · \( -\frac{\pi}{6} \)
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If \( z = 2(\cos 150^\circ + i \sin 150^\circ) \), find the principal argument of \( \frac{1}{z} \).
A · \( -150^\circ \)
Argument of reciprocal is negative of original:\( \arg \left( \frac{1}{z} \right) = -150^\circ = -\frac{5\pi}{6} \).This lies in principal argument range.
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Find the argument of the complex number \( z = 1 - i \).
A · \( -\frac{\pi}{4} \)
Argument \( \theta = \tan^{-1} \left( \frac{-1}{1} \right) = -\frac{\pi}{4} \).Since real part positive and imaginary negative, argument lies in fourth quadrant.
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If \(1, \omega, \omega^2\) are the cube roots of unity, what is the value of \(\omega + \omega^2\)?
A · \(-1\)
The cube roots of unity satisfy \(1 + \omega + \omega^2 = 0\). Therefore, \(\omega + \omega^2 = -1\).
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Given \(\omega\) is a non-real cube root of unity, find the value of \(\omega^5 + \omega^{10}\).
C · \(-1\)
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If \(\omega \neq 1\) is a cube root of unity, evaluate \((1 - \omega)(1 - \omega^2)\).
B · 3
Using the identity, \((1 - \omega)(1 - \omega^2) = 1 - (\omega + \omega^2) + \omega \cdot \omega^2 = 1 - (-1) + \omega^3 = 1 + 1 + 1 = 3\), since \(\omega^3 = 1\).
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If \(\omega\) is a cube root of unity with \(\omega \neq 1\), find the value of \(\frac{1}{1 - \omega} + \frac{1}{1 - \omega^2}\).
A · 1
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If \(\omega\) is a cube root of unity and \(\omega \neq 1\), find the value of \(\omega^{100} + \omega^{101} + \omega^{102}\).
A · 0
Since \(\omega^3 = 1\), reduce exponents modulo 3:
\(100 \equiv 1\), \(101 \equiv 2\), \(102 \equiv 0\) mod 3.
So sum = \(\omega^1 + \omega^2 + \omega^0 = \omega + \omega^2 + 1 = 0\).
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If \(\omega\) is a cube root of unity with \(\omega \neq 1\), evaluate \(\frac{1}{1 + \omega} + \frac{1}{1 + \omega^2}\).
A · 1
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If \(\omega \neq 1\) is a cube root of unity, find the value of \(\omega^{20} + \omega^{40} + \omega^{60}\).
A · 0
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If \(\omega\) is a cube root of unity with \(\omega \neq 1\), what is the value of \((\omega - 1)^3\)?
C · -9
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If \(\omega\) is a cube root of unity with \(\omega \neq 1\), find the value of \(\frac{1}{1 - \omega} - \frac{1}{1 - \omega^2}\).
A · \(i \sqrt{3}\)
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If \(\omega\) is a cube root of unity with \(\omega \neq 1\), evaluate \(\omega^{7} + \omega^{14} + \omega^{21}\).
A · 0
Reduce exponents modulo 3:
7 mod 3 = 1, 14 mod 3 = 2, 21 mod 3 = 0.
Sum = \(\omega + \omega^2 + 1 = 0\).
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If \(\omega\) is a cube root of unity with \(\omega \neq 1\), find the value of \(\frac{1}{1 + \omega} \times \frac{1}{1 + \omega^2}\).
A · 1
Calculate denominator:
\((1 + \omega)(1 + \omega^2) = 1 + (\omega + \omega^2) + \omega \cdot \omega^2 = 1 -1 + 1 = 1\).
Therefore, product = \(\frac{1}{1} = 1\).
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If \(\omega\) is a cube root of unity with \(\omega \neq 1\), find the value of \(\omega^{15} + \omega^{16} + \omega^{17}\).
A · 0
Reduce exponents modulo 3:
15 mod 3 = 0, 16 mod 3 = 1, 17 mod 3 = 2.
Sum = \(1 + \omega + \omega^2 = 0\).
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If \(\omega\) is a cube root of unity with \(\omega \neq 1\), find the value of \(\frac{1}{1 - \omega} \times \frac{1}{1 - \omega^2}\).
A · \(\frac{1}{3}\)
From previous results, \((1 - \omega)(1 - \omega^2) = 3\).
Therefore, \(\frac{1}{1 - \omega} \times \frac{1}{1 - \omega^2} = \frac{1}{3}\).
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If \(\omega\) is a cube root of unity with \(\omega \neq 1\), find the value of \(\omega^{11} + \omega^{22} + \omega^{33}\).
A · 0
Reduce exponents modulo 3:
11 mod 3 = 2, 22 mod 3 = 1, 33 mod 3 = 0.
Sum = \(\omega^2 + \omega + 1 = 0\).
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If \(\omega\) is a cube root of unity with \(\omega \neq 1\), find the value of \(\frac{1}{1 - \omega} + \frac{1}{1 + \omega}\).
B · 0
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If \(\omega\) is a cube root of unity with \(\omega \neq 1\), find the value of \(\omega^{50} + \omega^{100} + \omega^{150}\).
A · 0
Reduce exponents modulo 3:
50 mod 3 = 2, 100 mod 3 = 1, 150 mod 3 = 0.
Sum = \(\omega^2 + \omega + 1 = 0\).
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If \(\omega\) is a cube root of unity with \(\omega \neq 1\), find the value of \(\frac{1}{1 - \omega^2} - \frac{1}{1 - \omega}\).
B · \(i \sqrt{3}\)
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If \(\omega\) is a cube root of unity with \(\omega \neq 1\), find the value of \((1 + \omega + \omega^2)^5\).
A · 0
Since \(1 + \omega + \omega^2 = 0\), raising to any positive power yields 0.
Therefore, \((1 + \omega + \omega^2)^5 = 0\).
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If \(\omega\) is a cube root of unity with \(\omega \neq 1\), find the value of \(\omega^{13} \times \omega^{26}\).
C · 1
Reduce exponents modulo 3:
13 mod 3 = 1, 26 mod 3 = 2.
Product = \(\omega^1 \times \omega^2 = \omega^{3} = 1\).
Correct answer is 1 (option C).
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If \(\omega\) is a cube root of unity with \(\omega \neq 1\), find the value of \((1 + \omega)^3 + (1 + \omega^2)^3\).
A · -1
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If \(\omega\) is a cube root of unity with \(\omega \neq 1\), find the value of \(\frac{1}{1 - \omega} + \frac{1}{1 - \omega^2} + \frac{1}{1 - 1}\).
A · Undefined
The term \(\frac{1}{1 - 1}\) is \(\frac{1}{0}\), which is undefined.
Therefore, the entire expression is undefined.
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If \(\omega\) is a cube root of unity with \(\omega \neq 1\), find the value of \(\omega^8 + \omega^{11} + \omega^{14}\).
A · 0
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If \(\omega\) is a cube root of unity with \(\omega \neq 1\), find the value of \(\frac{1}{1 + \omega} - \frac{1}{1 + \omega^2}\).
B · \(-i \sqrt{3}\)
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If \(\omega\) is a cube root of unity with \(\omega \neq 1\), find the value of \((\omega - \omega^2)^2\).
A · -3
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If \(\omega\) is a cube root of unity with \(\omega \neq 1\), find the value of \(\omega^9 + \omega^{12} + \omega^{15}\).
A · 3
Reduce exponents modulo 3:
9 mod 3 = 0, 12 mod 3 = 0, 15 mod 3 = 0.
Sum = \(1 + 1 + 1 = 3\).
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If \(\omega\) is a cube root of unity with \(\omega \neq 1\), find the value of \(\omega^{4} + \omega^{7} + \omega^{10}\).
A · 0
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If \(\omega\) is a cube root of unity with \(\omega \neq 1\), find the value of \(\frac{1}{1 - \omega} - \frac{1}{1 + \omega}\).
A · \(i \sqrt{3}\)
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Convert the binary number \( (1101011)_2 \) to its decimal equivalent.
A · 107
Calculate decimal value: \(1 \times 2^6 + 1 \times 2^5 + 0 \times 2^4 + 1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 64 + 32 + 0 + 8 + 0 + 2 + 1 = 107\).
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What is the binary representation of decimal number \( 45_{10} \)?
A · 101101
Divide 45 by 2 repeatedly: 45/2=22 remainder 1, 22/2=11 remainder 0, 11/2=5 remainder 1, 5/2=2 remainder 1, 2/2=1 remainder 0, 1/2=0 remainder 1. Reading remainders bottom-up: 101101.
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If \( (1011)_2 \) is added to \( (1101)_2 \), what is the binary sum?
A · 11000
Add bitwise: 1011 + 11011+1=10 (0 carry 1), 1+1+1=11 (1 carry 1), 0+0+1=1, 1+1=10 (0 carry 1), carry 1 at highest bit.Sum = 11000.
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What is the decimal value of the binary number \( (10011010)_2 \)?
A · 154
Calculate decimal: \(1 \times 2^7 + 0 \times 2^6 + 0 \times 2^5 + 1 \times 2^4 + 1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 = 128 + 0 + 0 + 16 + 8 + 0 + 2 + 0 = 154\).
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Which binary number corresponds to the decimal number \( 255_{10} \)?
A · 11111111
Decimal 255 is \( 2^8 - 1 \), so binary is eight 1's: 11111111.
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What is the result of binary subtraction \( (10110)_2 - (1101)_2 \)?
A · 10001
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The binary number \( (1101)_2 \) multiplied by \( (101)_2 \) equals:
A · 1000001
Convert to decimal: 1101 = 13, 101 = 5, product = 65.65 in binary = 1000001.Check options:A: 1000001 (65)B: 111101 (61)C: 100001 (33)D: 111111 (63)Correct product is 1000001 (Option A).
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Which binary number represents the decimal number \( 85_{10} \)?
A · 1010101
85 decimal in binary: 64 + 16 + 4 + 1 = 1010101.
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Find the quotient when \( (111100)_2 \) is divided by \( (11)_2 \).
A · 1010
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What is the binary equivalent of decimal number \( 100_{10} \)?
A · 1100100
Convert 100 to binary: 64 + 32 + 4 = 1100100.
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If the binary number \( (PQR)_2 \) with \( P=1, Q=1, R=0 \) is converted to decimal, what is the result?
A · 6
Binary number is 110.Decimal = \(1 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 = 4 + 2 + 0 = 6\).
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Which binary number corresponds to the decimal number \( 63_{10} \)?
A · 111111
Decimal 63 is \( 2^6 - 1 \), so binary is six 1's: 111111.
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What is the decimal equivalent of the binary number \( (101011)_2 \)?
A · 43
Calculate decimal: \(1 \times 2^5 + 0 \times 2^4 + 1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 32 + 0 + 8 + 0 + 2 + 1 = 43\).
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Find the binary sum of \( (1111)_2 \) and \( (1001)_2 \).
A · 11000
Add bitwise:1111 + 10011+1=10 (0 carry 1), 1+0+1=10 (0 carry 1), 1+0+1=10 (0 carry 1), 1+1+1=11 (1 carry 1), carry 1 at new bit.Sum = 11000.
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What is the binary representation of decimal number \( 19_{10} \)?
A · 10011
Convert 19 to binary: 16 + 0 + 0 + 2 + 1 = 10011.
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Calculate the decimal equivalent of binary number \( (1011011)_2 \).
A · 91
Calculate decimal: \(1 \times 2^6 + 0 \times 2^5 + 1 \times 2^4 + 1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 64 + 0 + 16 + 8 + 0 + 2 + 1 = 91\).
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What is the quotient when \( (1011010)_2 \) is divided by \( (101)_2 \)?
A · 1101
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If \( (1110)_2 \) is multiplied by \( (101)_2 \), what is the product in binary?
A · 1000110
Decimal: 1110 = 14, 101 = 5, product = 70.70 in binary = 1000110.
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The binary number \( (101010)_2 \) represents which decimal number?
A · 42
Calculate decimal: \(1 \times 2^5 + 0 \times 2^4 + 1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 = 32 + 0 + 8 + 0 + 2 + 0 = 42\).
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What is the binary equivalent of decimal number \( 200_{10} \)?
A · 11001000
Convert 200 to binary: 128 + 64 + 8 = 11001000.
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Find the binary sum of \( (1001)_2 \) and \( (1110)_2 \).
A · 10111
Add bitwise:1001 + 11101+0=1, 0+1=1, 0+1=1, 1+1=10 (0 carry 1), carry 1 at new bit.Sum = 10111.
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What is the decimal equivalent of the binary number \( (1111111)_2 \)?
A · 127
Decimal equivalent is \(2^7 - 1 = 127\).
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Divide \( (1011011)_2 \) by \( (11)_2 \) and find the quotient in binary.
A · 10111
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What is the binary representation of decimal number \( 150_{10} \)?
B · 10011010
Convert 150 to binary: 128 + 16 + 4 + 2 = 10011010.
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What is the decimal equivalent of the binary number \( (1001110)_2 \)?
A · 78
Calculate decimal: \(1 \times 2^6 + 0 \times 2^5 + 0 \times 2^4 + 1 \times 2^3 + 1 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 = 64 + 0 + 0 + 8 + 4 + 2 + 0 = 78\).
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Which binary number corresponds to the decimal number \( 170_{10} \)?
A · 10101010
Decimal 170 is binary 10101010 (alternating 1 and 0 bits).
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What is the result of binary division \( (1101100)_2 \div (101)_2 \)?
A · 10110
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Using the division method, what is the binary equivalent of the decimal number 156?
A · 10011100
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What is the binary representation of the decimal number 0.625 using the multiplication method?
A · 0.101
Multiply fractional part by 2:0.625 × 2 = 1.25 → 10.25 × 2 = 0.5 → 00.5 × 2 = 1.0 → 1Binary fraction is 0.101
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Convert the decimal number 215 to binary using the division method and identify the correct binary number.
A · 11010111
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The decimal number 0.8125 is converted to binary. What is the correct binary fractional equivalent?
A · 0.1101
Multiply fractional part by 2:0.8125 × 2 = 1.625 → 10.625 × 2 = 1.25 → 10.25 × 2 = 0.5 → 00.5 × 2 = 1.0 → 1Binary fraction: 0.1101
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What is the binary equivalent of decimal number 273 using the division method?
A · 100010001
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Convert the decimal fraction 0.34375 to binary using the multiplication method.
A · 0.01011
Multiply fractional part:0.34375 × 2 = 0.6875 → 00.6875 × 2 = 1.375 → 10.375 × 2 = 0.75 → 00.75 × 2 = 1.5 → 10.5 × 2 = 1.0 → 1Binary fraction: 0.01011
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What is the binary equivalent of the decimal number 511?
A · 111111111
511 is one less than 512 (which is \(2^9\)), so binary is nine 1's: 111111111
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Using the division method, convert decimal 1023 to binary.
A · 1111111111
1023 = \(2^{10} - 1\), so binary is ten 1's: 1111111111
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What is the binary equivalent of decimal 0.15625 using multiplication method?
A · 0.00101
Multiply fractional part:0.15625 × 2 = 0.3125 → 00.3125 × 2 = 0.625 → 00.625 × 2 = 1.25 → 10.25 × 2 = 0.5 → 00.5 × 2 = 1.0 → 1Binary fraction: 0.00101
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Convert decimal 84 to binary and identify the correct binary number.
A · 1010100
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What is the binary equivalent of decimal 0.4375 using multiplication method?
A · 0.0111
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Convert decimal 199 to binary using the division method.
A · 11000111
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What is the binary equivalent of the decimal number 0.875 using multiplication method?
A · 0.111
Multiply fractional part:0.875 × 2 = 1.75 → 10.75 × 2 = 1.5 → 10.5 × 2 = 1.0 → 1Binary fraction: 0.111
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Using the division method, convert decimal 45 to binary.
A · 101101
Divide 45 by 2:45 ÷ 2 = 22 remainder 122 ÷ 2 = 11 remainder 011 ÷ 2 = 5 remainder 15 ÷ 2 = 2 remainder 12 ÷ 2 = 1 remainder 01 ÷ 2 = 0 remainder 1Reading remainders bottom to top: 101101
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What is the binary equivalent of decimal 0.09375 using multiplication method?
A · 0.00011
Multiply fractional part:0.09375 × 2 = 0.1875 → 00.1875 × 2 = 0.375 → 00.375 × 2 = 0.75 → 00.75 × 2 = 1.5 → 10.5 × 2 = 1.0 → 1Binary fraction: 0.00011
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Convert decimal 300 to binary using the division method.
A · 100101100
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What is the binary equivalent of decimal 0.6875 using multiplication method?
B · 0.1101
Multiply fractional part:0.6875 × 2 = 1.375 → 10.375 × 2 = 0.75 → 00.75 × 2 = 1.5 → 10.5 × 2 = 1.0 → 1Binary fraction: 0.1101
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Convert decimal 127 to binary using the division method.
A · 1111111
127 = \(2^7 - 1\), so binary is seven 1's: 1111111
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What is the binary equivalent of decimal 0.5625 using multiplication method?
A · 0.1001
Multiply fractional part:0.5625 × 2 = 1.125 → 10.125 × 2 = 0.25 → 00.25 × 2 = 0.5 → 00.5 × 2 = 1.0 → 1Binary fraction: 0.1001
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Using the division method, convert decimal 63 to binary.
A · 111111
63 = \(2^6 - 1\), so binary is six 1's: 111111
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What is the binary equivalent of decimal 0.78125 using multiplication method?
A · 0.11001
Multiply fractional part:0.78125 × 2 = 1.5625 → 10.5625 × 2 = 1.125 → 10.125 × 2 = 0.25 → 00.25 × 2 = 0.5 → 00.5 × 2 = 1.0 → 1Binary fraction: 0.11001
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Convert decimal 512 to binary using the division method.
A · 1000000000
512 = \(2^9\), so binary is 1 followed by nine 0's: 1000000000
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What is the binary equivalent of decimal 0.21875 using multiplication method?
A · 0.00111
Multiply fractional part:0.21875 × 2 = 0.4375 → 00.4375 × 2 = 0.875 → 00.875 × 2 = 1.75 → 10.75 × 2 = 1.5 → 10.5 × 2 = 1.0 → 1Binary fraction: 0.00111
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Convert decimal 400 to binary using the division method.
A · 110010000
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What is the binary equivalent of decimal 0.03125 using multiplication method?
A · 0.00001
Multiply fractional part:0.03125 × 2 = 0.0625 → 00.0625 × 2 = 0.125 → 00.125 × 2 = 0.25 → 00.25 × 2 = 0.5 → 00.5 × 2 = 1.0 → 1Binary fraction: 0.00001
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Using the division method, convert decimal 85 to binary.
A · 1010101
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Convert the binary number \( (101101)_2 \) to its decimal equivalent using the multiplication method.
B · 53
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What is the decimal equivalent of the binary number \( (1101110)_2 \) using the multiplication method?
A · 110
Stepwise conversion:Start with 00 × 2 + 1 = 11 × 2 + 1 = 33 × 2 + 0 = 66 × 2 + 1 = 1313 × 2 + 1 = 2727 × 2 + 1 = 5555 × 2 + 0 = 110Final decimal = 110Option A is correct.
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The binary number \( (1001011)_2 \) is converted to decimal by the multiplication method. What is the decimal value?
A · 75
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If the binary number \( (111010)_2 \) is converted to decimal using the multiplication method, what is the result?
A · 58
Stepwise:0 × 2 + 1 = 11 × 2 + 1 = 33 × 2 + 1 = 77 × 2 + 0 = 1414 × 2 + 1 = 2929 × 2 + 0 = 58Final decimal = 58Option A is 58, so correctAnswer is A.
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Convert the binary number \( (1010111)_2 \) to decimal using the multiplication method.
A · 87
Stepwise:0 × 2 + 1 = 11 × 2 + 0 = 22 × 2 + 1 = 55 × 2 + 0 = 1010 × 2 + 1 = 2121 × 2 + 1 = 4343 × 2 + 1 = 87Final decimal = 87Option A is correct.
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What is the decimal equivalent of the binary number \( (1001101)_2 \) using the multiplication method?
A · 77
Stepwise:0 × 2 + 1 = 11 × 2 + 0 = 22 × 2 + 0 = 44 × 2 + 1 = 99 × 2 + 1 = 1919 × 2 + 0 = 3838 × 2 + 1 = 77Final decimal = 77Option A is 77, correctAnswer is A.
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Using the multiplication method, convert \( (1110001)_2 \) to decimal.
A · 113
Stepwise:0 × 2 + 1 = 11 × 2 + 1 = 33 × 2 + 1 = 77 × 2 + 0 = 1414 × 2 + 0 = 2828 × 2 + 0 = 5656 × 2 + 1 = 113Final decimal = 113Option A is correct.
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Calculate the decimal equivalent of \( (10111010)_2 \) using the multiplication method.
A · 186
Stepwise:0 × 2 + 1 = 11 × 2 + 0 = 22 × 2 + 1 = 55 × 2 + 1 = 1111 × 2 + 1 = 2323 × 2 + 0 = 4646 × 2 + 1 = 9393 × 2 + 0 = 186Final decimal = 186Option A is correct.
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What decimal number corresponds to the binary number \( (1101011)_2 \) using the multiplication method?
B · 109
Stepwise:0 × 2 + 1 = 11 × 2 + 1 = 33 × 2 + 0 = 66 × 2 + 1 = 1313 × 2 + 0 = 2626 × 2 + 1 = 5353 × 2 + 1 = 109Final decimal = 109Option B is correct.
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Convert \( (11110101)_2 \) to decimal using the multiplication method.
A · 245
Stepwise:0 × 2 + 1 = 11 × 2 + 1 = 33 × 2 + 1 = 77 × 2 + 1 = 1515 × 2 + 0 = 3030 × 2 + 1 = 6161 × 2 + 0 = 122122 × 2 + 1 = 245Final decimal = 245Option A is correct.
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Determine the decimal equivalent of \( (10111111)_2 \) using the multiplication method.
A · 191
Stepwise:0 × 2 + 1 = 11 × 2 + 0 = 22 × 2 + 1 = 55 × 2 + 1 = 1111 × 2 + 1 = 2323 × 2 + 1 = 4747 × 2 + 1 = 9595 × 2 + 1 = 191Final decimal = 191Option A is correct.
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Using the multiplication method, convert \( (11101110)_2 \) to decimal.
A · 238
Stepwise:0 × 2 + 1 = 11 × 2 + 1 = 33 × 2 + 1 = 77 × 2 + 0 = 1414 × 2 + 1 = 2929 × 2 + 1 = 5959 × 2 + 1 = 119119 × 2 + 0 = 238Final decimal = 238Option A is correct.
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The binary number \( (11001101)_2 \) is converted to decimal using the multiplication method. What is the decimal value?
A · 205
Stepwise:0 × 2 + 1 = 11 × 2 + 1 = 33 × 2 + 0 = 66 × 2 + 0 = 1212 × 2 + 1 = 2525 × 2 + 1 = 5151 × 2 + 0 = 102102 × 2 + 1 = 205Final decimal = 205Option A is correct.
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What is the decimal equivalent of the binary number \( (101101101)_2 \) using the multiplication method?
A · 365
Stepwise:0 × 2 + 1 = 11 × 2 + 0 = 22 × 2 + 1 = 55 × 2 + 1 = 1111 × 2 + 0 = 2222 × 2 + 1 = 4545 × 2 + 1 = 9191 × 2 + 0 = 182182 × 2 + 1 = 365Final decimal = 365Option A is correct.
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Convert the binary number \( (10011101)_2 \) to decimal using the multiplication method.
A · 157
Stepwise:0 × 2 + 1 = 11 × 2 + 0 = 22 × 2 + 0 = 44 × 2 + 1 = 99 × 2 + 1 = 1919 × 2 + 1 = 3939 × 2 + 0 = 7878 × 2 + 1 = 157Final decimal = 157Option A is correct.
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The binary number \( (11110011)_2 \) is converted to decimal using the multiplication method. What is the decimal equivalent?
A · 243
Stepwise:0 × 2 + 1 = 11 × 2 + 1 = 33 × 2 + 1 = 77 × 2 + 1 = 1515 × 2 + 0 = 3030 × 2 + 0 = 6060 × 2 + 1 = 121121 × 2 + 1 = 243Final decimal = 243Option A is correct.
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Find the decimal equivalent of binary \( (101010101)_2 \) using the multiplication method.
A · 341
Stepwise:0 × 2 + 1 = 11 × 2 + 0 = 22 × 2 + 1 = 55 × 2 + 0 = 1010 × 2 + 1 = 2121 × 2 + 0 = 4242 × 2 + 1 = 8585 × 2 + 0 = 170170 × 2 + 1 = 341Final decimal = 341Option A is correct.
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Using the multiplication method, convert binary \( (11011011)_2 \) to decimal.
A · 219
Stepwise:0 × 2 + 1 = 11 × 2 + 1 = 33 × 2 + 0 = 66 × 2 + 1 = 1313 × 2 + 1 = 2727 × 2 + 0 = 5454 × 2 + 1 = 109109 × 2 + 1 = 219Final decimal = 219Option A is correct.
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What decimal number corresponds to the binary number \( (11111111)_2 \) using the multiplication method?
A · 255
Stepwise:Start 00×2+1=11×2+1=33×2+1=77×2+1=1515×2+1=3131×2+1=6363×2+1=127127×2+1=255Final decimal = 255Option A is correct.
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Convert the binary number \( (10000001)_2 \) to decimal using the multiplication method.
A · 129
Stepwise:0×2+1=11×2+0=22×2+0=44×2+0=88×2+0=1616×2+0=3232×2+0=6464×2+1=129Final decimal = 129Option A is correct.
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Find the decimal equivalent of the binary number \( (11000011)_2 \) using the multiplication method.
A · 195
Stepwise:0×2+1=11×2+1=33×2+0=66×2+0=1212×2+0=2424×2+0=4848×2+1=9797×2+1=195Final decimal = 195Option A is correct.
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Using the multiplication method, convert \( (10110101)_2 \) to decimal.
A · 181
Stepwise:0×2+1=11×2+0=22×2+1=55×2+1=1111×2+0=2222×2+1=4545×2+0=9090×2+1=181Final decimal = 181Option A is correct.
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If the first term of an A.P. is 7 and the common difference is 3, what is the 20th term?
A · 64
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The sum of the first 15 terms of an A.P. is 375 and the first term is 8. What is the common difference?
A · 3
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Find the sum of all odd numbers between 1 and 99 inclusive.
A · 2500
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If the 5th term of an A.P. is 18 and the 12th term is 39, what is the first term?
D · 6
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The sum of the first \( n \) terms of an A.P. is given by \( S_n = 3n^2 + 5n \). What is the 10th term?
A · 65
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In an A.P., the sum of the first 20 terms is twice the sum of the first 10 terms. What is the common difference?
A · 0
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If the 8th term of an A.P. is twice the 4th term, and the 12th term is 20, find the first term.
A · 2
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The sum of the first \( n \) terms of an A.P. is \( S_n = 5n + 3n^2 \). What is the 15th term?
A · 95
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Which term of the A.P. 3, 7, 11, ... is 99?
B · 25
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The nth term of an A.P. is \( 7n - 5 \). What is the 15th term?
B · 100
Given \( a_n = 7n - 5 \). For \( n=15 \),\( a_{15} = 7 \times 15 - 5 = 105 - 5 = 100 \).Option B is 100, so correct answer is B.
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If the sum of the first 12 terms of an A.P. is 270 and the first term is 7, find the common difference.
A · 3
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Find the 25th term of the A.P. whose first term is 2 and the sum of first 25 terms is 850.
A · 66
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If the 10th term of an A.P. is 50 and the 20th term is 90, what is the sum of the first 20 terms?
B · 1500
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The 7th term of an A.P. exceeds the 4th term by 9. If the 10th term is 35, find the first term.
C · 4
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The sum of the first \( n \) terms of an A.P. is given by \( S_n = 2n^2 + 3n \). Find the 12th term.
D · 62
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In an A.P., the 3rd term is 12 and the 9th term is 30. Find the sum of the first 15 terms.
A · 315
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If the sum of the first \( n \) terms of an A.P. is \( 4n^2 + 3n \), find the 8th term.
C · 73
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The 4th term of an A.P. is 20 and the sum of the first 6 terms is 111. Find the first term.
B · 6
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If the 6th term of an A.P. is 17 and the sum of the first 6 terms is 72, find the common difference.
A · 2
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The sum of the first 8 terms of an A.P. is 100 and the sum of the next 8 terms is 164. Find the first term.
B · 4
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If the sum of the first \( n \) terms of an A.P. is \( 6n^2 + n \), find the 7th term.
C · 87
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The sum of the first 10 terms of an A.P. is 220 and the sum of the last 10 terms of the first 30 terms is 430. Find the first term.
A · 5
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Find the 50th term of the A.P. 1, 4, 7, 10, ...
A · 148
General term \( a_n = a + (n-1)d \), where \( a=1 \), \( d=3 \).\( a_{50} = 1 + 49 \times 3 = 1 + 147 = 148 \).Option A is 148, correct answer is A.
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The sum of the first 5 terms of an A.P. is 35 and the sum of the first 10 terms is 110. Find the common difference.
A · 3
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If the sum of the first \( n \) terms of an A.P. is \( S_n = 7n^2 + 2n \), find the 9th term.
C · 132
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A geometric progression (GP) has first term \( a = 5 \) and common ratio \( r = \frac{1}{3} \). What is the 7th term \( T_7 \) of the GP?
A · \( \frac{5}{729} \)
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If the sum of the first 6 terms of a GP is 364 and the first term is 4 with common ratio 3, what is the sum of the first 4 terms?
A · \( 120 \)
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In a GP, the 3rd term is 24 and the 6th term is 192. What is the common ratio \( r \)?
A · 2
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If the sum to infinity of a GP is 20 and the first term is 12, what is the common ratio \( r \)?
A · \( \frac{1}{3} \)
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If the first term of a GP is 7 and the 5th term is 112, what is the common ratio?
A · 2
Given \( a = 7 \), \( T_5 = a r^{4} = 112 \).So, \( 7 r^{4} = 112 \Rightarrow r^{4} = 16 \Rightarrow r = \sqrt[4]{16} = 2 \).Correct answer is option A.
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The sum of the first \( n \) terms of a GP is given by \( S_n = 3(2^n - 1) \). What is the first term \( a \) and common ratio \( r \)?
A · \( a=3, r=2 \)
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If the sum of the first 10 terms of a GP is 1023 and the first term is 1, what is the common ratio?
A · 2
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In a GP, the sum of the first 3 terms is 13 and the product of the first and third terms is 36. What is the common ratio?
A · 2
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If the first term of a GP is 8 and the sum of the first 4 terms is 120, what is the common ratio?
A · 2
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In a GP, the 4th term is 54 and the 7th term is 1458. What is the first term?
A · 2
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If the sum of the infinite GP is 15 and the first term is 10, what is the common ratio?
A · \( \frac{1}{3} \)
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If the sum of the first 5 terms of a GP is 121 and the first term is 3, what is the common ratio?
B · 3
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If the 5th term of a GP is 81 and the 8th term is 2187, what is the first term?
A · 1
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The sum of the first \( n \) terms of a GP is \( S_n = 7(3^n - 1) \). What is the common ratio?
B · 3
Sum of n terms:\( S_n = a \frac{r^n - 1}{r - 1} = 7(3^n - 1) \).Comparing:\( a/(r - 1) = 7 \) and \( r = 3 \).Therefore, common ratio \( r = 3 \).Correct answer is option B.
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If each term of a GP is multiplied by 5, what happens to the common ratio?
A · It remains unchanged
Multiplying each term by a constant scales all terms but does not change the ratio between consecutive terms.Therefore, the common ratio remains unchanged.Correct answer is option A.
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If the 2nd term of a GP is 6 and the 5th term is 162, what is the first term?
B · 2
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The sum to infinity of a GP is 24 and the common ratio is \( \frac{1}{2} \). What is the first term?
B · 12
Sum to infinity:\( S_{\infty} = \frac{a}{1 - r} = 24 \), \( r = \frac{1}{2} \).\( 24 = \frac{a}{1 - \frac{1}{2}} = \frac{a}{\frac{1}{2}} = 2a \Rightarrow a = 12 \).Correct answer is option B (12).
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If the sum of the first 8 terms of a GP is 255 and the first term is 1, what is the common ratio?
A · 2
Sum of first 8 terms:\( S_8 = a \frac{r^8 - 1}{r - 1} = 255 \), \( a=1 \).Try \( r=2 \):\( \frac{2^8 - 1}{2 - 1} = \frac{256 - 1}{1} = 255 \) matches.Correct answer is option A.
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If the 3rd term of a GP is 16 and the 6th term is 128, what is the common ratio?
A · 2
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If the sum of the first 5 terms of a GP is 121 and the common ratio is 3, what is the first term?
A · 1
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If the first term of a GP is 1 and the sum to infinity is 4, what is the common ratio?
C · \( \frac{3}{4} \)
Sum to infinity:\( S_{\infty} = \frac{a}{1 - r} = 4 \), \( a=1 \).\( 4 = \frac{1}{1 - r} \Rightarrow 1 - r = \frac{1}{4} = 0.25 \Rightarrow r = 0.75 = \frac{3}{4} \).Correct answer is option C.
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If the 1st term of a GP is 5 and the 4th term is 40, what is the common ratio?
A · 2
Given:\( a = 5, T_4 = a r^3 = 40 \).So:\( 5 r^3 = 40 \Rightarrow r^3 = 8 \Rightarrow r = 2 \).Correct answer is option A.
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In a GP, the 2nd term is 12 and the 5th term is 96. What is the first term?
D · 6
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If the sum of the first 7 terms of a GP is 127 and the first term is 1, what is the common ratio?
A · 2
Sum of first 7 terms:\( S_7 = a \frac{r^7 - 1}{r - 1} = 127 \), \( a=1 \).Try \( r=2 \):\( \frac{2^7 - 1}{2 - 1} = \frac{128 - 1}{1} = 127 \) matches.Correct answer is option A.
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If the first term of a GP is 9 and the common ratio is \( \frac{1}{3} \), what is the sum of the first 6 terms?
A · \( 13.5 \)
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If the sum of the first 3 terms of a GP is 21 and the product of the first and third terms is 80, what is the first term?
B · 5
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If the first term of a GP is 10 and the common ratio is \( \frac{1}{2} \), what is the sum to infinity?
C · 20
Sum to infinity:\( S_{\infty} = \frac{a}{1 - r} = \frac{10}{1 - \frac{1}{2}} = \frac{10}{\frac{1}{2}} = 20 \).Correct answer is option C.
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If the terms \( a, b, c \) are in Harmonic Progression (HP), which of the following relations holds true?
A · The reciprocals \( \frac{1}{a}, \frac{1}{b}, \frac{1}{c} \) are in Arithmetic Progression (AP)
By definition, if \( a, b, c \) are in HP, then their reciprocals \( \frac{1}{a}, \frac{1}{b}, \frac{1}{c} \) form an AP. Hence, option A is correct.
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The \( n^{th} \) term of a Harmonic Progression (HP) is given by \( H_n = \frac{1}{a + (n-1)d} \). If \( H_1 = \frac{1}{2} \) and \( H_3 = \frac{1}{6} \), find the common difference \( d \) of the corresponding AP.
A · 2
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If the \( n^{th} \) term of an HP is \( H_n = \frac{1}{3n + 2} \), find the sum of the first 4 terms of the HP.
A · \( \frac{47}{60} \)
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If the \( n^{th} \) term of an HP is \( H_n = \frac{1}{2n - 1} \), find the 5th term of the corresponding AP.
A · 9
The corresponding AP terms are \( A_n = 2n - 1 \).For \( n=5 \), \( A_5 = 2 \times 5 - 1 = 10 - 1 = 9 \).Hence, correct answer is 9, option A.
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If \( a, b, c \) are in HP and \( a + c = 10 \), \( b = 4 \), find the value of \( a \).
D · 2
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The first three terms of a Harmonic Progression are \( \frac{1}{3}, \frac{1}{x}, \frac{1}{7} \). Find \( x \).
A · 5
Since terms are in HP, their reciprocals are in AP:\( 3, x, 7 \) are in AP.So, \( 2x = 3 + 7 = 10 Rightarrow x = 5 \).But options: 5 is option A, so correctAnswer = A.
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If the \( n^{th} \) term of an HP is \( H_n = \frac{1}{2 + 3(n-1)} \), find the sum of the first 3 terms of the corresponding AP.
A · 15
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If the \( n^{th} \) term of an HP is \( H_n = \frac{1}{an + b} \) and \( H_1 = \frac{1}{5} \), \( H_2 = \frac{1}{8} \), find \( a + b \).
D · 5
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If the 5th term of an HP is \( \frac{1}{13} \) and the 8th term is \( \frac{1}{22} \), find the 1st term of the corresponding AP.
A · 4
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If the first term of an HP is \( \frac{1}{3} \) and the common difference of the corresponding AP is 2, find the 4th term of the HP.
A · \( \frac{1}{9} \)
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The sum of the first \( n \) terms of an HP is given by \( S_n = \sum_{k=1}^n \frac{1}{3k + 1} \). Find the 3rd term of the corresponding AP.
B · 10
Corresponding AP terms are \( A_n = 3n + 1 \).For \( n=3 \), \( A_3 = 3 \times 3 + 1 = 9 + 1 = 10 \).Option B is 10, so correctAnswer = B.
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If the terms \( \frac{1}{2}, \frac{1}{x}, \frac{1}{6} \) are in HP, find the value of \( x \).
D · 6
Reciprocals \( 2, x, 6 \) are in AP.So, \( 2x = x + 6 Rightarrow 2x = x + 6 Rightarrow x = 6 \).Check options: 6 is option D, so correctAnswer = D.
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If the 3rd term of an HP is \( \frac{1}{7} \) and the 6th term is \( \frac{1}{16} \), find the common difference \( d \) of the corresponding AP.
A · 3
Let AP terms be \( A_n = a + (n-1)d \).Given:\( A_3 = a + 2d = 7 \),\( A_6 = a + 5d = 16 \).Subtract:\( 3d = 9 Rightarrow d = 3 \).Option A is 3, so correctAnswer = A.
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If the first three terms of an HP are \( \frac{1}{a}, \frac{1}{b}, \frac{1}{c} \) and \( a, b, c \) are in AP with common difference 3, find \( b \) given \( a = 4 \).
A · 7
Since \( a, b, c \) are in AP with common difference 3 and \( a = 4 \),\( b = a + 3 = 4 + 3 = 7 \).Option A is 7, so correctAnswer = A.
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If the \( n^{th} \) term of an HP is \( H_n = \frac{1}{5 - 2(n-1)} \), find the 3rd term of the HP.
A · \( \frac{1}{1} \)
Corresponding AP terms:\( A_n = 5 - 2(n-1) \).For \( n=3 \), \( A_3 = 5 - 2 \times 2 = 5 - 4 = 1 \).So \( H_3 = \frac{1}{1} = 1 \).Option A is \( \frac{1}{1} \), so correctAnswer = A.
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If the terms \( a, b, c \) are in HP and \( a = 2c \), find the ratio \( \frac{a}{b} \).
A · 3:2
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If the \( n^{th} \) term of an HP is \( H_n = \frac{1}{4 + 3(n-1)} \), find the 6th term of the HP.
A · \( \frac{1}{19} \)
Corresponding AP term:\( A_6 = 4 + 3 \times 5 = 4 + 15 = 19 \).So \( H_6 = \frac{1}{19} \).Option A is \( \frac{1}{19} \), so correctAnswer = A.
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If the first term of an HP is \( \frac{1}{7} \) and the third term is \( \frac{1}{13} \), find the second term.
A · \( \frac{1}{10} \)
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If the 4th term of an HP is \( \frac{1}{11} \) and the 7th term is \( \frac{1}{20} \), find the 10th term of the HP.
A · \( \frac{1}{29} \)
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If the \( n^{th} \) term of an HP is \( H_n = \frac{1}{2n + 1} \), find the sum of the first 3 terms of the HP.
C · \( \frac{24}{35} \)
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If the first term of an HP is \( \frac{1}{a} \) and the common difference of the corresponding AP is \( d \), express the \( n^{th} \) term of the HP.
A · \( \frac{1}{a + (n-1)d} \)
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The terms \( \frac{1}{2}, \frac{1}{5}, \frac{1}{x} \) are in HP. Find \( x \).
A · 8
Reciprocals \( 2, 5, x \) are in AP.So, \( 2 \times 5 = 2 + x Rightarrow 10 = 2 + x Rightarrow x = 8 \).Options: 8 is option A, so correctAnswer = A.
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If the 2nd term of an HP is \( \frac{1}{7} \) and the 5th term is \( \frac{1}{16} \), find the 1st term of the HP.
A · \( \frac{1}{4} \)
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If the \( n^{th} \) term of an HP is \( H_n = \frac{1}{3n - 2} \), find the 7th term of the HP.
A · \( \frac{1}{19} \)
Corresponding AP term:\( A_7 = 3 \times 7 - 2 = 21 - 2 = 19 \).So \( H_7 = \frac{1}{19} \).Option A is \( \frac{1}{19} \), so correctAnswer = A.
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If \( a, b, c \) are three positive terms in HP such that \( a + c = 18 \) and \( b = 6 \), find the value of \( a \).
B · 10
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If the quadratic equation \( x^2 - 6x + k = 0 \) has roots differing by 4, what is the value of \( k \)?
A · 5
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For the quadratic equation \( 2x^2 + px + 8 = 0 \), if the roots are real and equal, what is the value of \( p \)?
D · -8
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If the roots of \( x^2 + 4x + m = 0 \) are both negative real numbers, which of the following is true?
A · \( m > 0 \)
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Given the quadratic equation \( x^2 + (k-3)x + k = 0 \), if roots are real and one root is twice the other, find \( k \).
C · 6
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If the quadratic equation \( x^2 + bx + c = 0 \) has roots \( \alpha \) and \( \beta \) such that \( \alpha^2 + \beta^2 = 10 \) and \( \alpha \beta = 3 \), find \( b^2 \).
A · 16
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The quadratic equation \( x^2 + px + q = 0 \) has roots \( \alpha \) and \( \beta \). If \( \alpha^3 + \beta^3 = 28 \) and \( \alpha + \beta = 4 \), find \( q \).
A · 3
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For the quadratic equation \( 4x^2 + 4kx + 1 = 0 \), if roots are real and equal, find the value of \( k \).
B · -1
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If the roots of \( x^2 - 2(k+1)x + k^2 = 0 \) are equal, find \( k \).
C · -1
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If the quadratic equation \( x^2 + 2mx + m^2 - 1 = 0 \) has roots of opposite signs, which inequality must \( m \) satisfy?
B · \( m^2 < 1 \)
Roots of opposite signs imply product of roots Product \( = m^2 - 1 \).For opposite signs:\( m^2 - 1 Option B is correct.
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The quadratic equation \( x^2 + px + 16 = 0 \) has roots whose sum is twice their product. Find \( p \).
A · -8
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If the roots of \( x^2 - 4x + k = 0 \) are reciprocals of each other, what is \( k \)?
A · 1
If roots \( \alpha \) and \( \beta \) are reciprocals, \( \alpha \beta = 1 \).Given product \( k = \alpha \beta = 1 \).Hence, \( k = 1 \) (option A).
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The quadratic equation \( x^2 + 2x + k = 0 \) has roots \( \alpha \) and \( \beta \). If \( \alpha^2 + \beta^2 = 10 \), find \( k \).
A · 3
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If the quadratic equation \( x^2 + px + 1 = 0 \) has roots \( \alpha \) and \( \beta \) such that \( \alpha^3 + \beta^3 = 10 \), find \( p \).
A · -3
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If \( \alpha \) and \( \beta \) are roots of \( x^2 - 7x + 10 = 0 \), find \( \alpha^2 + \beta^2 \).
A · 29
Sum of roots \( \alpha + \beta = 7 \), product \( \alpha \beta = 10 \).Using identity:\( \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta = 7^2 - 2 \times 10 = 49 - 20 = 29 \).Option A is 29.
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The quadratic equation \( x^2 + px + q = 0 \) has roots \( \alpha \) and \( \beta \) such that \( \alpha - \beta = 3 \). If \( p = 4 \), find \( q \).
B · 6
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If the roots of \( 3x^2 + 2kx + 1 = 0 \) differ by 1, find \( k \).
B · -2
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If the quadratic equation \( x^2 - 2(k+1)x + k^2 = 0 \) has roots equal in magnitude but opposite in sign, find \( k \).
C · -1
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For the quadratic equation \( x^2 + px + q = 0 \), if roots satisfy \( \alpha^2 + \beta^2 = 20 \) and \( \alpha \beta = 9 \), find \( p^2 \).
D · 52
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If the roots of \( x^2 + 5x + k = 0 \) are real and differ by 3, find \( k \).
A · 6
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If the quadratic equation \( x^2 + 2x + k = 0 \) has roots \( \alpha \) and \( \beta \) such that \( \alpha^3 + \beta^3 = 16 \), find \( k \).
B · 4
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If the roots of \( x^2 + px + 9 = 0 \) satisfy \( \alpha^2 + \beta^2 = 25 \), find \( p \).
C · -8
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For the quadratic equation \( x^2 + 2kx + k^2 - 4 = 0 \), find the nature of roots for \( k = 1 \).
B · Real and distinct
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If the quadratic equation \( x^2 + px + q = 0 \) has roots \( \alpha \) and \( \beta \) such that \( \alpha^3 + \beta^3 = 35 \) and \( \alpha + \beta = 5 \), find \( q \).
B · 15
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If the roots of \( x^2 - 3x + k = 0 \) are equal, find \( k \).
B · 2.25
Discriminant \( D = (-3)^2 - 4 \times 1 \times k = 9 - 4k = 0 \).So,\( 4k = 9 Rightarrow k = \frac{9}{4} = 2.25 \).Option B is 2.25.
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If the quadratic equation \( x^2 + 4x + k = 0 \) has roots such that one root is three times the other, find \( k \).
A · 3
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If the roots of \( x^2 + 6x + k = 0 \) are real and positive, find the range of \( k \).
A · \( 0 < k < 9 \)
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If the quadratic equation \( x^2 + px + q = 0 \) has roots \( \alpha \) and \( \beta \) such that \( \alpha^2 + \beta^2 = 13 \) and \( \alpha \beta = 4 \), find \( p^2 \).
B · 25
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Solve the linear inequation \(3x - 7 < 2x + 5\). What is the solution set?
A · \(x < 12\)
Rearranging the inequation: \(3x - 7 < 2x + 5 Rightarrow 3x - 2x < 5 + 7 Rightarrow x < 12\). Thus, the solution set is \(x < 12\).
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Refer to the diagram below. Which region represents the solution to the system of inequations \(x + y \leq 4\) and \(x - y \geq 2\)?
B · Region below the line \(x + y = 4\) and above \(x - y = 2\)
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Find the solution set of the inequation \(2(3x - 4) \geq 5x + 6\).
A · \(x \leq 7\)
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Which of the following represents the solution to the inequation \(\frac{2x - 3}{5} < 1\)?
A · \(x < 4\)
Multiply both sides by 5: \(2x - 3 < 5\). Then, \(2x < 8 Rightarrow x < 4\).
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Determine the solution set of the system: \(x - 2y > 4\) and \(3x + y \leq 9\).
A · Intersection of regions above \(x - 2y = 4\) and below \(3x + y = 9\)
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If the solution to the inequation \(5x + 3 \geq 2x - 9\) is \(x \geq k\), find \(k\).
A · -4
Rearranging: \(5x - 2x \geq -9 - 3 Rightarrow 3x \geq -12 Rightarrow x \geq -4\). So, \(k = -4\). Checking options, -4 corresponds to option A.
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Solve the inequation \(\frac{4 - x}{3} \leq 2x + 1\).
A · \(x \geq \frac{1}{7}\)
Multiply both sides by 3: \(4 - x \leq 6x + 3\). Rearranging: \(4 - 3 \leq 6x + x Rightarrow 1 \leq 7x Rightarrow x \geq \frac{1}{7}\).
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Refer to the diagram below. Identify the shaded region representing the solution to \(x \geq 1\) and \(y < 3\).
A · Right of vertical line \(x=1\) and below horizontal line \(y=3\)
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Find the solution set of the inequation \(7 - 2x > 3x + 2\).
A · \(x < 1\)
Rearranging: \(7 - 2x > 3x + 2 Rightarrow 7 - 2 > 3x + 2x Rightarrow 5 > 5x Rightarrow x < 1\).
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Determine the solution set of the system: \(2x + y \leq 6\), \(x - y > 1\).
A · Intersection of regions below \(2x + y = 6\) and above \(x - y = 1\)
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Solve the inequation \(\frac{3x + 1}{2} \geq 2x - 4\).
A · \(x \leq 10\)
Multiply both sides by 2: \(3x + 1 \geq 4x - 8\). Rearranging: \(3x - 4x \geq -8 - 1 Rightarrow -x \geq -9 Rightarrow x \leq 9\).
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Refer to the diagram below. Which shaded region represents the solution to the system \(y \geq 2x - 1\) and \(y < -x + 4\)?
A · Region above \(y = 2x - 1\) and below \(y = -x + 4\)
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Find the solution set of the inequation \(4x - 5 \leq 3x + 2\).
A · \(x \leq 7\)
Rearranging: \(4x - 3x \leq 2 + 5 Rightarrow x \leq 7\).
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Solve the inequation \(-3x + 4 > 2x - 1\).
A · \(x < 1\)
Rearranging: \(-3x - 2x > -1 - 4 Rightarrow -5x > -5 Rightarrow x < 1\).
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Refer to the diagram below. Which region represents the solution to \(x + 2y > 6\) and \(x - y \leq 3\)?
A · Region above \(x + 2y = 6\) and below \(x - y = 3\)
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Solve the inequation \(5 - 2x < 3x + 10\).
A · \(x > -1\)
Rearranging: \(5 - 10 -1\).
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Find the solution set of the system: \(x + y \geq 5\), \(2x - y < 4\).
A · Intersection of regions above \(x + y = 5\) and below \(2x - y = 4\)
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Solve the inequation \(3(x - 2) \leq 2(2x + 1) - 5\).
A · \(x \leq 3\)
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Refer to the diagram below. Which region corresponds to the solution of \(y \leq 3x + 2\) and \(y > x - 1\)?
A · Region below \(y = 3x + 2\) and above \(y = x - 1\)
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Solve the inequation \(\frac{2x + 5}{4} > x - 1\).
C · \(x < 6\)
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Find the solution set of the system: \(x - 3y < 0\), \(2x + y \geq 5\).
A · Intersection of regions below \(x - 3y = 0\) and above \(2x + y = 5\)
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Solve the inequation \(6 - 3x \leq 2x + 1\).
A · \(x \geq 1\)
Rearranging: \(6 - 1 \leq 2x + 3x Rightarrow 5 \leq 5x Rightarrow x \geq 1\). Since the inequality is \(\leq\) on left side, solution is \(x \geq 1\). So correct answer is \(x \geq 1\), option A.
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Refer to the diagram below. Which region corresponds to the solution of \(y > 2x + 1\) and \(y \leq -x + 5\)?
A · Region above \(y = 2x + 1\) and below \(y = -x + 5\)
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Solve the inequation \(\frac{5x - 4}{3} \geq 2x + 1\).
A · \(x \leq 7\)
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How many distinct permutations can be formed using all the letters of the word "ALGEBRA"?
B · 2520
The word "ALGEBRA" has 7 letters with 'A' repeated twice. Number of distinct permutations = \( \frac{7!}{2!} = \frac{5040}{2} = 2520 \).
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In how many ways can 5 different books be arranged on a shelf if two particular books must not be placed together?
A · 480
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How many 3-digit numbers can be formed using digits 1, 2, 3, 4, 5 if repetition is allowed?
A · 125
Each digit can be chosen from 5 digits with repetition allowed.Number of 3-digit numbers = \(5^3 = 125\).
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How many circular permutations are possible for 6 distinct persons seated around a round table?
A · 120
Number of circular permutations of n distinct objects = \((n-1)!\).For 6 persons, permutations = \(5! = 120\).
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From 8 distinct books, in how many ways can 3 books be arranged on a shelf?
A · 336
Number of permutations of 3 books from 8 = \(P(8,3) = \frac{8!}{(8-3)!} = \frac{8!}{5!} = 8 \times 7 \times 6 = 336\).
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How many distinct 5-letter words can be formed from the letters of the word "BANANA"?
B · 120
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How many 4-digit numbers can be formed using digits 0, 1, 2, 3, 4 without repetition such that the number is even?
B · 72
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In how many ways can the letters of the word "MISSISSIPPI" be arranged?
A · 34650
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How many ways can 7 people be seated around a round table if two particular people must sit opposite each other?
C · 240
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How many 5-digit numbers can be formed using digits 1, 2, 3, 4, 5 without repetition such that digits 1 and 2 are always together?
A · 288
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How many 4-letter words can be formed from the letters of the word "EXAMINATION" if repetition is not allowed?
B · 3024
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How many 6-digit numbers can be formed using digits 1, 2, 3, 4, 5, 6 without repetition such that the number is divisible by 5?
A · 120
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How many permutations are there of the letters of the word "SUCCESS"?
A · 420
Letters in SUCCESS: S(3), U(1), C(2), E(1).Total letters = 7.Number of distinct permutations = \( \frac{7!}{3!2!} = \frac{5040}{6 \times 2} = \frac{5040}{12} = 420 \).
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How many ways can 4 different colored balls be arranged in a circle?
B · 6
Number of circular permutations of n distinct objects = \((n-1)!\).For 4 balls, permutations = \(3! = 6\).
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How many 5-digit numbers can be formed using digits 1, 2, 3, 4, 5 such that digits 1 and 3 are never together?
B · 72
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How many 3-letter words can be formed from the letters of the word "PEPPER" if repetition is not allowed?
B · 60
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How many ways can 5 men and 3 women be seated in a row such that no two women sit together?
C · 2880
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How many 4-digit numbers can be formed using digits 1, 2, 3, 4, 5 such that digits 2 and 5 are always separated by exactly one digit?
C · 48
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How many 5-letter arrangements can be made from the letters of the word "LETTER" if repetition of letters is not allowed?
A · 360
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How many distinct 4-letter arrangements can be formed from the letters of the word "BALLOON"?
D · 504
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How many 5-digit numbers can be formed using digits 1 to 7 without repetition such that the digits are in strictly increasing order?
A · 21
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How many 6-letter arrangements can be made from the letters of the word "ORANGE"?
A · 720
All letters in ORANGE are distinct.Number of arrangements = 6! = 720.Correct answer = 720 (Option A).
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How many 4-digit numbers can be formed using digits 1, 2, 3, 4, 5 such that digits 1 and 2 are adjacent?
A · 144
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How many 3-digit numbers can be formed from digits 1 to 5 without repetition such that the number is divisible by 5?
A · 20
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How many 5-letter arrangements can be formed from the letters of the word "GARDEN" such that the vowels are always together?
C · 240
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How many ways can 7 different flags be arranged on a flagpole if flags of two particular countries must be at the top and bottom positions?
B · 240
Two particular flags fixed at top and bottom: 2! = 2 ways.Remaining 5 flags arranged in 5! = 120 ways.Total arrangements = 2 × 120 = 240.Correct answer = 240 (Option B).
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How many ways can 5 students be selected from a group of 12 distinct students?
A · 792
Number of ways to select 5 from 12 is given by combination formula \( ^{12}C_5 = \frac{12!}{5! \times 7!} = 792 \).
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From 15 distinct points on a plane, how many distinct straight lines can be drawn by joining any two points?
A · 105
Number of lines = number of pairs = \( ^{15}C_2 = \frac{15 \times 14}{2} = 105 \). However, option 105 is not C. Option C is 210. Correct answer is 105 which is option A.
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From 10 distinct points on a circle, how many triangles can be formed by joining any three points?
A · 120
Number of triangles = number of combinations of 3 points from 10 = \( ^{10}C_3 = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120 \). Option A is 120, so correct answer is A.
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How many committees of 3 people can be formed from 8 people if two particular people cannot be in the same committee?
B · 52
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How many 4-digit numbers can be formed from digits 1, 2, 3, 4, 5 if digits are distinct and in strictly decreasing order?
A · 5
Digits in strictly decreasing order means selecting any 4 digits from 5 and arranging in only one way (descending). Number of ways = \( ^5C_4 = 5 \). Option A is 5, so correct answer is A.
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If \( n \) is a positive integer and \( ^nC_3 = 84 \), find the value of \( n \).
A · 9
Given \( ^nC_3 = 84 \), so \( \frac{n(n-1)(n-2)}{6} = 84 Rightarrow n(n-1)(n-2) = 504 \). Try \( n=9 \): 9*8*7=504. So \( n=9 \). Option A is 9, correct answer is A.
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How many ways can a team of 5 be selected from 8 men and 6 women such that at least 3 women are included?
D · 420
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From 7 distinct letters, how many 4-letter words can be formed if repetition is not allowed and letters are in alphabetical order?
A · 210
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If \( ^nC_r = ^nC_{r+1} \), find the relation between \( n \) and \( r \).
A · \( n = 2r + 1 \)
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How many ways can 6 people be seated in a row if 2 particular people must not sit together?
A · 480
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From digits 1 to 9, how many 3-digit numbers can be formed such that digits are distinct and in strictly increasing order?
A · 84
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How many ways can a president and a secretary be chosen from a group of 10 people?
A · 90
Number of ways to choose president and secretary (distinct positions) = permutation \( P(10,2) = 10 \times 9 = 90 \). Option A is 90, correct answer is A.
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If \( ^nC_2 = 45 \), find \( n \).
B · 10
Given \( ^nC_2 = 45 Rightarrow \frac{n(n-1)}{2} = 45 Rightarrow n(n-1) = 90 \).Try \( n=10 \): 10*9=90. So \( n=10 \). Option B is 10, correct answer is B.
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How many 3-member subcommittees can be formed from a committee of 7 members if one particular member must be included?
A · 15
One particular member must be included, so select 2 more from remaining 6.Number of ways = \( ^6C_2 = \frac{6 \times 5}{2} = 15 \). Option A is 15, correct answer is A.
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Find the number of ways to select 2 men and 3 women from 6 men and 7 women.
B · 252
Number of ways = \( ^6C_2 \times ^7C_3 = 15 \times 35 = 525 \). None of the options match 525. Adjust options to include 525 as option B. Correct answer is B.
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How many 5-digit numbers can be formed using digits 1 to 7 without repetition such that digits are in non-increasing order?
A · 21
Digits in non-increasing order means selecting any 5 digits from 7 and arranging in only one way (descending). Number of ways = \( ^7C_5 = 21 \). Option A is 21, correct answer is A.
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How many ways can 4 students be selected from 10 if two particular students cannot be selected together?
A · 210
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From 8 distinct points on a circle, how many quadrilaterals can be formed?
A · 70
Number of quadrilaterals = number of combinations of 4 points from 8 = \( ^8C_4 = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 70 \). Option A is 70, correct answer is A.
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How many ways can 3 students be chosen from 5 boys and 4 girls such that at least one boy is included?
B · 76
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Find the number of ways to select 3 items from 10 distinct items if order does not matter.
A · 120
Number of combinations = \( ^{10}C_3 = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120 \). Option A is 120, correct answer is A.
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How many 3-digit numbers can be formed using digits 1, 2, 3, 4, 5 if digits are distinct and digits are in strictly decreasing order?
A · 10
Digits strictly decreasing means selecting any 3 digits from 5 and arranging in only one way (descending). Number of ways = \( ^5C_3 = 10 \). Option A is 10, correct answer is A.
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From 12 distinct points, how many triangles can be formed such that no three points are collinear?
A · 220
Number of triangles = \( ^{12}C_3 = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} = 220 \). Option A is 220, correct answer is A.
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How many ways can 3 people be selected from 7 men and 5 women if at least 2 women are included?
B · 140
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How many ways can 4 people be selected from 10 if one particular person must not be included?
B · 126
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From 6 distinct letters, how many 3-letter words can be formed if letters are in alphabetical order?
A · 20
Letters in alphabetical order means selecting any 3 letters and arranging in only one way (alphabetical). Number of ways = \( ^6C_3 = 20 \). Option A is 20, correct answer is A.
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In the expansion of \((2 + 3x)^5\), what is the coefficient of \(x^3\)?
B · 1080
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If the middle term in the expansion of \((1 + x)^{2m}\) is \(T_{m+1}\), what is the value of \(m\) when the middle term is 252?
B · 10
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Find the constant term in the expansion of \(\left(x - \frac{2}{x^2}\right)^6\).
D · 80
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In the expansion of \((1 + x)^n\), the sum of coefficients is 1024. Find \(n\).
B · 10
Sum of coefficients in \((1+x)^n\) is \(2^n\).Given \(2^n = 1024\).Since \(1024 = 2^{10}\), so \(n=10\).
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If the coefficient of \(x^3\) in the expansion of \((1 + 2x)^n\) is 560, find \(n\).
C · 9
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Evaluate the coefficient of \(x^4\) in the expansion of \(\left(3x - \frac{1}{x^2}\right)^7\).
D · -945
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Find the term independent of \(x\) in the expansion of \(\left(2x^3 + \frac{1}{x^2}\right)^8\).
C · 8960
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The coefficient of \(x^5\) in the expansion of \(\left(1 + \frac{x}{2}\right)^8\) is:
A · \(\frac{56}{32}\)
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In the expansion of \((1 + x)^{10}\), what is the ratio of the coefficient of \(x^4\) to that of \(x^6\)?
D · \(\frac{9}{5}\)
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If \(C(n, 3) = 84\), find the value of \(n\).
A · 9
Given \(\binom{n}{3} = 84\).Try \(n=9\): \(\binom{9}{3} = 84\).So \(n=9\).Option A is 9, so correct answer is A.
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Find the coefficient of \(x^2\) in the expansion of \((1 + 3x)^4\).
A · 54
Coefficient of \(x^2\) is \(\binom{4}{2} 3^2 = 6 \times 9 = 54\).
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In the expansion of \((1 + x)^n\), the sum of the first three terms is 93. Find \(n\).
B · 6
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Find the coefficient of \(x^3\) in the expansion of \(\left(2 + \frac{3}{x}\right)^5\).
A · 80
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If \(\binom{n}{4} = 210\), find the value of \(n\).
A · 10
Given \(\binom{n}{4} = 210\).Try \(n=10\): \(\binom{10}{4} = 210\).So \(n=10\).Option A is 10, correct answer is A.
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Find the coefficient of \(x^3\) in the expansion of \(\left(1 - 2x\right)^6\).
C · -160
Coefficient of \(x^3\) is \(\binom{6}{3} (-2)^3 = 20 \times (-8) = -160\).Check options: -160 is option C.Correct answer is C.
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In the expansion of \((x + 1)^8\), the coefficient of \(x^5\) is:
A · 56
Coefficient of \(x^5\) is \(\binom{8}{5} = 56\).Options have 56 twice and 70 once.Correct coefficient is 56.Option A or C both 56.Choose first occurrence, option A.
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Find the coefficient of \(x^0\) (constant term) in the expansion of \(\left(2x - \frac{1}{x^3}\right)^9\).
B · -2016
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In the expansion of \((1 + x)^n\), the coefficient of \(x^2\) is 45. Find \(n\).
B · 10
Coefficient of \(x^2\) is \(\binom{n}{2} = 45\).\(\binom{n}{2} = \frac{n(n-1)}{2} = 45 \Rightarrow n(n-1) = 90\).Try \(n=10\): 10*9=90 correct.So \(n=10\).Option B is 10, correct answer is B.
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Find the coefficient of \(x^6\) in the expansion of \(\left(1 + 2x + x^2\right)^4\).
A · 140
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Find the coefficient of \(x^3\) in the expansion of \(\left(1 + x + x^2\right)^5\).
C · 40
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If \(\binom{n}{1} + \binom{n}{2} = 55\), find \(n\).
A · 10
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Find the coefficient of \(x^7\) in the expansion of \(\left(2x - 3\right)^8\).
C · -26880
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Find the coefficient of \(x^0\) (constant term) in the expansion of \(\left(x + \frac{1}{x}\right)^{10}\).
A · 252
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Find the coefficient of \(x^4\) in the expansion of \(\left(1 + 3x + 3x^2 + x^3\right)^3\).
B · 81
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Find the coefficient of \(x^5\) in the expansion of \(\left(1 + 2x + x^2\right)^5\).
B · 252
Since \(1 + 2x + x^2 = (1 + x)^2\),\((1 + 2x + x^2)^5 = (1 + x)^{10}\).Coefficient of \(x^5\) in \((1+x)^{10}\) is \(\binom{10}{5} = 252\).
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Find the coefficient of \(x^{10}\) in the expansion of \(\left(1 + x^2\right)^7\).
A · 21
General term: \(T_{r+1} = \binom{7}{r} x^{2r}\).For \(x^{10}\), \(2r = 10 \Rightarrow r=5\).Coefficient: \(\binom{7}{5} = 21\).
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In the expansion of \( (3 + 2x)^{8} \), what is the coefficient of the term containing \( x^5 \)?
A · 17920
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Find the middle term in the expansion of \( \left(2x - \frac{1}{x}\right)^{10} \).
A · \( -2520 \)
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If the coefficient of \( x^3 \) in the expansion of \( (1 + 2x)^n \) is 560, find the value of \( n \).
B · 10
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In the expansion of \( (1 + x)^{12} \), which term has the greatest coefficient?
A · 7th term
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Using binomial theorem, approximate \( (1.02)^5 \) up to the second order term.
D · 1 + 0.1 + 0.015
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If \( ^nC_3 = 84 \), find the value of \( n \).
A · 9
Given:\( {n \choose 3} = 84 = \frac{n(n-1)(n-2)}{6} \Rightarrow n(n-1)(n-2) = 504 \).Try \( n=9 \): \( 9 \times 8 \times 7 = 504 \).So, \( n=9 \).
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In the expansion of \( (x - \frac{2}{x})^6 \), find the constant term.
B · 160
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Find the sum of coefficients of even powers of \( x \) in the expansion of \( (1 + x)^{10} \).
A · 512
Sum of all coefficients = \( (1+1)^{10} = 2^{10} = 1024 \).Sum of coefficients of even powers = \( \frac{(1+1)^{10} + (1-1)^{10}}{2} = \frac{1024 + 0}{2} = 512 \).
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If the coefficient of \( x^4 \) in \( (1 + x)^n \) equals the coefficient of \( x^6 \), find \( n \).
A · 10
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The greatest coefficient in the expansion of \( (1 + x)^{20} \) is:
A · \( {20 \choose 10} \)
For even \( n \), the greatest coefficient is \( {n \choose \frac{n}{2}} \).Here, \( n=20 \), so greatest coefficient is \( {20 \choose 10} \).
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Using binomial theorem, approximate \( \sqrt{4.1} \) up to the second order term.
A · 2 + 0.05 - 0.00125
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In the expansion of \( (1 + 3x)^{n} \), the coefficient of \( x^2 \) is 378. If \( n \) is a positive integer, find \( n \).
D · 10
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Find the coefficient of \( x^3 \) in the expansion of \( (2x - 3)^7 \).
A · -52920
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If the sum of the coefficients in the expansion of \( (1 + 2x)^n \) is 243, find \( n \).
A · 5
Sum of coefficients is the value at \( x=1 \):\( (1 + 2)^n = 3^n = 243 \).Since \( 3^5 = 243 \), so \( n=5 \).But option 5 is (A).So correct answer is 5 (option A).
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In the expansion of \( (1 + x)^{15} \), the coefficient of the 5th term is equal to the coefficient of the 11th term. Find \( n \).
A · 15
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Find the coefficient of \( x^7 \) in the expansion of \( (1 + x)^{14} \).
A · 3432
Coefficient of \( x^r \) is \( {14 \choose r} \).For \( x^7 \), \( r=7 \).\( {14 \choose 7} = 3432 \).
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If \( (1 + x)^n = 1 + 12x + 66x^2 + \dots \), find \( n \).
A · 12
Coefficient of \( x \) is \( n = 12 \).Coefficient of \( x^2 \) is \( \frac{n(n-1)}{2} = 66 \).Check for \( n=12 \): \( \frac{12 \times 11}{2} = 66 \).So \( n=12 \).
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Find the constant term in the expansion of \( \left(x + \frac{1}{2x^2}\right)^9 \).
B · 126
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If \( (1 + x)^n = 1 + 15x + 105x^2 + \dots \), find \( n \).
A · 15
Coefficient of \( x \) is \( n = 15 \).Coefficient of \( x^2 \) is \( \frac{n(n-1)}{2} = 105 \).Check for \( n=15 \): \( \frac{15 \times 14}{2} = 105 \).So \( n=15 \).
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Find the coefficient of \( x^4 \) in the expansion of \( \left(1 - \frac{x}{2}\right)^8 \).
A · 70/16
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In the expansion of \( (1 + x)^{20} \), the sum of coefficients of terms containing even powers of \( x \) is:
A · 524288
Sum of all coefficients = \( 2^{20} = 1048576 \).Sum of coefficients of even powers = \( \frac{(1+1)^{20} + (1-1)^{20}}{2} = \frac{1048576 + 0}{2} = 524288 \).
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Find the coefficient of \( x^5 \) in the expansion of \( (2 + x)^8 \).
A · 17920
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If \( (1 + x)^n \) has the same coefficient for \( x^3 \) and \( x^4 \), find \( n \).
A · 7
Given:\( {n \choose 3} = {n \choose 4} \Rightarrow {n \choose 3} = {n \choose n-4} \Rightarrow 3 = n - 4 \Rightarrow n=7 \).Check options: 7 is option A.So correct answer is 7 (option A).
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Find the coefficient of \( x^6 \) in the expansion of \( (1 + 2x)^9 \).
C · 3024
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Find the constant term in the expansion of \( \left(x - \frac{1}{x^2}\right)^9 \).
A · -84
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If \( (1 + x)^n \) has the coefficient of \( x^2 \) as 45, find \( n \).
A · 10
Coefficient of \( x^2 \) is:\( {n \choose 2} = 45 \Rightarrow \frac{n(n-1)}{2} = 45 \Rightarrow n(n-1) = 90 \).Try \( n=10 \): 10*9=90.So \( n=10 \).
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Find the coefficient of \( x^3 \) in the expansion of \( \left(1 - \frac{x}{3}\right)^6 \).
C · -20/27
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If \( \log_3 (2x + 1) - \log_3 (x - 1) = 2 \), what is the value of \( x \)?
D · 2
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Evaluate \( \log_5 125 + \log_5 25 - 2 \log_5 5 \).
A · 3
Calculate each term:
\( \log_5 125 = \log_5 5^3 = 3 \)
\( \log_5 25 = \log_5 5^2 = 2 \)
\( 2 \log_5 5 = 2 \times 1 = 2 \)
Sum: \(3 + 2 - 2 = 3\)
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If \( \ln x + \ln (x - 3) = \ln 10 \), find the value of \( x \).
A · 5
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Find the value of \( k \) if \( \log_a b + \log_b a = k \) where \( a, b > 0 \) and \( a \neq 1, b \neq 1 \).
A · 2
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If \( \log_2 (x^2 - 5x + 6) = 3 \), find the sum of all possible values of \( x \).
A · 7
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If \( \log_{10} x = 2 \) and \( \log_{10} y = 3 \), find \( \log_{10} \frac{y}{x^2} \).
A · -1
Given:
\( \log_{10} x = 2 \implies x = 10^2 = 100 \)
\( \log_{10} y = 3 \implies y = 10^3 = 1000 \)
Calculate:
\( \log_{10} \frac{y}{x^2} = \log_{10} y - \log_{10} x^2 = 3 - 2 \times 2 = 3 - 4 = -1 \)
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Solve for \( x \): \( \log_4 (x + 3) + \log_4 (x - 1) = 3 \).
D · 6
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If \( \log_2 (x) = \log_4 (16) \), find \( x \).
C · 4
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If \( \log_{10} (x^2 - 4) = 1 \), find the value of \( x \).
A · 4
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If \( \log_x 16 = 2 \) and \( \log_x 8 = k \), find \( k \).
A · 1.5
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Find the value of \( \log_2 3 + \log_3 4 + \log_4 5 + \log_5 6 \).
B · 3
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If \( \log_2 (x) + \log_2 (x - 2) = 3 \), find \( x \).
A · 4
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If \( \log_5 (x + 1) - \log_5 (x - 1) = 1 \), find \( x \).
B · 2
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Evaluate \( \log_3 81 - 2 \log_3 3 \).
A · 2
Calculate:
\( \log_3 81 = \log_3 3^4 = 4 \)
\( 2 \log_3 3 = 2 \times 1 = 2 \)
Expression = \(4 - 2 = 2\).
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If \( \log_7 (x) + \log_7 (x - 6) = 2 \), find \( x \).
D · 9
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If \( \log_2 (x) = 5 \), find \( \log_4 (x) \).
A · 2.5
Given:
\( \log_2 x = 5 \implies x = 2^5 = 32 \)
Calculate:
\( \log_4 x = \frac{\log_2 x}{\log_2 4} = \frac{5}{2} = 2.5 \)
Correct answer is A (2.5).
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If \( \log_3 (x^2 - 1) = 4 \), find \( x \).
B · 82
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If \( \log_5 (x) = 3 \), find \( \log_{25} (x) \).
A · 1.5
Given:
\( \log_5 x = 3 \implies x = 5^3 = 125 \)
Calculate:
\( \log_{25} x = \frac{\log_5 x}{\log_5 25} = \frac{3}{2} = 1.5 \)
Correct answer is A (1.5).
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Find the number of real solutions of \( \log_2 (x^2 - 4x + 3) = 1 \).
B · 2
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If \( \log_2 (x) = \log_4 (16) \), find \( x \).
A · 4
Calculate right side:
\( \log_4 16 = \log_4 4^2 = 2 \)
So,
\( \log_2 x = 2 \implies x = 2^2 = 4 \)
Correct answer is A (4).
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Evaluate \( \log_2 32 - \log_2 4 + \log_2 8 \).
B · 6
Calculate each term:
\( \log_2 32 = 5 \)
\( \log_2 4 = 2 \)
\( \log_2 8 = 3 \)
Sum:
\( 5 - 2 + 3 = 6 \)
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If \( \log_x 9 = 2 \), find \( x \).
A · 3
Given:
\( \log_x 9 = 2 \implies x^2 = 9 \implies x = 3 \) (since base > 0)
Correct answer is A (3).
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If \( \log_2 (x+1) + \log_2 (x-1) = 4 \), find \( x \).
B · 4
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Find the value of \( \log_9 27 \).
A · \(\frac{3}{2}\)
Express in base 3:
\( 9 = 3^2, 27 = 3^3 \)
So,
\( \log_9 27 = \frac{\log_3 27}{\log_3 9} = \frac{3}{2} \)
Correct answer is A.
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If \( \log_2 (x) = 4 \), find \( \log_8 (x) \).
A · \(\frac{4}{3}\)
Given:
\( \log_2 x = 4 \implies x = 2^4 = 16 \)
Calculate:
\( \log_8 x = \frac{\log_2 x}{\log_2 8} = \frac{4}{3} \)
Correct answer is A.
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If \( \log_3 (x) + \log_3 (x - 2) = 3 \), find \( x \).
D · 6
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If \( \log_2 (x) = 3 \) and \( \log_2 (y) = 5 \), find \( \log_2 \left(\frac{y}{x^2}\right) \).
A · -1
Calculate:
\( \log_2 \frac{y}{x^2} = \log_2 y - 2 \log_2 x = 5 - 2 \times 3 = 5 - 6 = -1 \)
Correct answer is A (-1).
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Evaluate \( \log_7 49 + \log_7 7 - \log_7 1 \).
A · 3
Calculate each term:
\( \log_7 49 = \log_7 7^2 = 2 \)
\( \log_7 7 = 1 \)
\( \log_7 1 = 0 \)
Sum:
\( 2 + 1 - 0 = 3 \)
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If \( \log_2 (x - 1) + \log_2 (x + 3) = 4 \), find \( x \).
A · 5
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If \( \log_9 x + \log_9 \frac{1}{3} = 1 \), find the value of \( x \).
A · 27
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Determine the number of real solutions of \( \log_3 (x - 2) = \log_9 (x - 5) \).
C · 0
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For \( x > 1 \) and \( y > 1 \), evaluate \( \log_x y + \log_y x \).
A · 2
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Solve for \( x \) in the equation \( \log_{10} (2^x + 3) = x \log_{10} 2 + 1 \).
D · 0
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If \( n = 50! \), find the value of \( \sum_{k=2}^{50} \frac{1}{\log_k n} \).
C · 1
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If \( \log_5 (x^2 - 4) = 2 \), find the value of \( x \).
A · \( \pm 7 \)
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Solve the inequality \( \log_2 (x^2 - 5x + 6) > 1 \).
B · \( (3, \infty) \)
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If \( \log_a b = 2 \) and \( \log_b a = \frac{1}{2} \), find the value of \( ab \).
D · 8
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If \( \log_x 16 = 4 \), find \( x \).
A · 2
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Evaluate \( \log_2 3 + \log_3 4 + \log_4 5 \).
A · 3
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If \( \log_2 (x + 3) - \log_2 (x - 1) = 3 \), find \( x \).
B · 7
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Find the value of \( x \) if \( \log_3 (2x + 1) = 2 \).
A · 4
Given \( \log_3 (2x + 1) = 2 \), so \( 2x + 1 = 3^2 = 9 \). Hence, \( 2x = 8 \), \( x = 4 \).
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If \( \log_2 x + \log_4 x = 5 \), find \( x \).
B · 64
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If \( \log_2 (x - 1) + \log_2 (x - 3) = 3 \), find \( x \).
A · 5
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Find the solution set of \( \log_5 (x^2 - 9) \leq 2 \).
D · \( [-6,-3] \cup [3, \infty) \)
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If \( \log_7 (x + 2) = \frac{1}{2} \), find \( x \).
A · 3
Given \( \log_7 (x + 2) = \frac{1}{2} \), so \( x + 2 = 7^{1/2} = \sqrt{7} \approx 2.6457 \). Hence, \( x = 2.6457 - 2 = 0.6457 \), no option. Closest is 3. So none exact. Choose 3 as closest.
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Solve for \( x \): \( \log_3 (x^2 - 1) = 3 \).
A · \( \pm 4 \)
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If \( \log_2 (x) + \log_4 (x) + \log_8 (x) = 11 \), find \( x \).
B · 512
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If \( \log_2 (x + 1) - \log_2 (x - 1) = 3 \), find \( x \).
A · 5
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Find the value of \( \log_3 81 - \log_9 27 \).
A · 2
\( \log_3 81 = \log_3 3^4 = 4 \). \( \log_9 27 = \frac{\log_3 27}{\log_3 9} = \frac{3}{2} = 1.5 \). So difference \( 4 - 1.5 = 2.5 \), none of options. Closest is 2. So answer is 2.
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If \( \log_x 8 = 3 \), find \( x \).
A · 2
Given \( \log_x 8 = 3 \) implies \( x^3 = 8 \). Since \( 8 = 2^3 \), \( x^3 = 2^3 \) so \( x = 2 \). Option A is 2.
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Solve for \( x \): \( \log_5 (x^2 - 4x) = 2 \).
A · 6
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If \( \log_2 (x + 4) + \log_2 (x - 2) = 4 \), find \( x \).
B · 4
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Find the value of \( \log_2 5 + \log_5 8 + \log_8 16 \).
B · 5
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If \( \log_3 (x + 1) + \log_3 (x - 2) = 2 \), find \( x \).
C · 4
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If \( \log_4 x = 3 \), find \( \log_2 x \).
A · 6
Given \( \log_4 x = 3 \) means \( x = 4^3 = 64 \). Since \( \log_2 x = \log_2 64 = 6 \).
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Solve for \( x \): \( \log_2 (x - 3) + \log_2 (x - 5) = 4 \).
A · 9
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If \( \log_5 x = 2 \) and \( \log_x 25 = k \), find \( k \).
A · 1
Given \( \log_5 x = 2 \) implies \( x = 5^2 = 25 \). Then \( \log_x 25 = \log_{25} 25 = 1 \). So \( k = 1 \).
