Quick recall · 237 cards
Short MCQ-style retrieval prompts. Tap a card to reveal the answer.
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What type of number is 22?
A · Natural Number
PYQ
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What type of number is 2.43?
D · Rational Number
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Which one of the following is not a cube number? a) 1 b) 27 c) 64 d) 81
D · 81
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The number 4 is not included in which group of numbers? A. Integers B. Whole numbers C. Irrational numbers D. Natural numbers
C · Irrational numbers
PYQ
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Ravi has three wooden logs of lengths 91m, 112m, and 49m. If he wants to cut the wood into equal planks, what is the greatest possible length of each plank?
A · 7m
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If the HCF of two numbers is 12 and their LCM is 360, find the numbers.
B · 36 and 120
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A six-digit number X40Y84 is divisible by 72. How many distinct values can X assume?
B · 3
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A number 416 + 1 is divisible by x. Which among the following is also divisible by x?
D · 464 + 1
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If x + y = 21 and xy = 110, find the value of \( x^2 + y^2 \).
A · 221
Use the identity \( (x + y)^2 = x^2 + y^2 + 2xy \).\( x^2 + y^2 = (x + y)^2 - 2xy = 21^2 - 2 \times 110 = 441 - 220 = 221 \).Option A matches 221.
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A man has Rs. 480 in the denominations of one-rupee notes, five-rupee notes and ten-rupee notes. The number of notes of each denomination is equal. What is the total number of notes that he has?
A · 64
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The cube root of 0.000216 is:
B · 0.06
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What is the cube root of 2197?
B · 13
Prime factorization of 2197: \(2197 = 13 \times 13 \times 13 = 13^3\). Therefore, \(\sqrt[3]{2197} = 13\), which corresponds to option B. Verification by multiplication confirms \(13^3 = 2197\).[5]
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The square root of 73.96 is:
A · 8.6
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If (a + b√n) is the positive square root of (29 - 12√5), where a and b are integers, and n is a natural number, then the maximum possible value of (a + b + n) is:
(A) 4
(B) 18
(C) 6
(D) 22
B · 18
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What is the square root of \( (8 + 2\sqrt{15}) \)?
A. \( \sqrt{5} + \sqrt{3} \)
B. \( 2\sqrt{2} + 2\sqrt{6} \)
C. \( 2\sqrt{5} + 2\sqrt{3} \)
D. \( \sqrt{10} + \sqrt{6} \)
C · \( 2\sqrt{5} + 2\sqrt{3} \)
PYQ
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Find the missing number in the series: 2, 5, 12.5, ?, 78.125, 195.3125
A · 31.25
PYQ
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In the following number series, find the wrong number: 2800, 700, 4200, 1100, 6300, 1575
D · 1575
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What is the remainder when \( 91 \times 92 \times 93 \times 94 \times ... \times 99 \) is divided by 1261?
A · 0
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What is the remainder when \( 9^{100} \) is divided by 18?
C · 9
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How many factors of the number \(2^8 \times 3^6 \times 5^4 \times 10^5\) are multiples of 120?
C · 594
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If both \(11^2\) and \(3^3\) are factors of the number \(a \times 4^3 \times 6^2 \times 13^{11}\), then what is the smallest possible value of 'a'?
C · \(36^3\)
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Which of the following numbers is a whole number?
B · 0
Whole numbers include all natural numbers and zero, but not negative numbers or fractions.
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Identify the type of number \( -7 \).
C · Integer
Integers include negative and positive whole numbers including zero. \( -7 \) is an integer but not a natural or whole number.
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Which of the following numbers is irrational?
C · \( \sqrt{3} \)
\( \sqrt{3} \) is an irrational number because it cannot be expressed as a ratio of two integers.
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Which of the following is a prime number?
B · 23
23 is a prime number as it has only two factors: 1 and 23.
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Which of the following numbers is composite?
C · 39
39 is composite because it has factors other than 1 and itself (3 and 13).
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Which of the following is an odd prime number?
D · 11
11 is an odd prime number. 2 is prime but even, 4 and 9 are not prime.
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Which of the following is a perfect square?
A · 49
49 is a perfect square since \( 7^2 = 49 \). 64 is also a perfect square but since only one option is correct, 49 is the best choice here.
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Which of the following numbers is both a perfect square and a perfect cube?
A · 64
64 is \( 8^2 \) (perfect square) and \( 4^3 \) (perfect cube).
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Which of the following is a perfect cube number?
A · 27
27 is a perfect cube since \( 3^3 = 27 \).
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Which of the following decimals is a terminating decimal?
B · 0.125
0.125 is a terminating decimal. 0.333... and 0.272727... are non-terminating repeating decimals. \( \sqrt{2} \) is irrational.
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Which of the following numbers is rational?
C · \( \frac{7}{9} \)
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Which of the following decimals is non-terminating and non-repeating, hence irrational?
B · 0.1010010001...
0.1010010001... is a non-terminating, non-repeating decimal, which makes it irrational. 0.666... and 0.333... are repeating decimals (rational), and 0.25 is terminating decimal (rational).
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Which of the following sets is a subset of the real numbers?
A · Natural numbers
Natural numbers are a subset of real numbers. Complex and imaginary numbers are not subsets of real numbers.
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Which of the following correctly represents the relationship between number sets?
B · Natural numbers \( \subseteq \) Whole numbers \( \subseteq \) Integers
Natural numbers are a subset of whole numbers, which are a subset of integers. Rational and irrational numbers are disjoint subsets of real numbers.
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Which of the following numbers is a whole number but not a natural number?
A · 0
Whole numbers include all natural numbers along with zero. Zero is a whole number but not a natural number.
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Identify the type of number \( -\frac{3}{4} \):
D · Rational number
Negative fractions are rational numbers but not natural numbers, whole numbers, or integers.
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Which of the following sets contains only irrational numbers?
A · \( \sqrt{2}, \pi, 0.1010010001... \)
\( \sqrt{2} \), \( \pi \), and the non-repeating, non-terminating decimal 0.1010010001... are irrational numbers.
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Which property is illustrated by the statement: For any two integers \( a \) and \( b \), \( a + b \) is also an integer?
B · Closure
Closure property states that the sum of any two integers is also an integer.
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Which of the following statements is true regarding multiplication of rational numbers?
B · Multiplication of two rational numbers is always rational
The product of two rational numbers is always rational, demonstrating closure under multiplication.
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If \( a = 3 \), \( b = 5 \), and \( c = 7 \), which property justifies that \( a + (b + c) = (a + b) + c \)?
A · Associative property of addition
The associative property states that the way numbers are grouped in addition does not change the sum.
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Which of the following numbers is a composite number?
C · 21
21 has factors other than 1 and itself (3 and 7), so it is composite.
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Which of the following is an odd prime number?
C · 7
7 is a prime number and is odd. 2 is prime but even; 4 and 9 are not prime.
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Which decimal representation corresponds to a rational number?
A · 0.121212...
0.121212... is a repeating decimal, which represents a rational number.
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A number is defined as an integer but not a natural number. Which of the following could it be?
B · 0
0 is an integer and a whole number but not a natural number.
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If \( x \) is an irrational number and \( y \) is a rational number (\( y eq 0 \)), which of the following is always irrational?
D · All of the above
Adding, multiplying, or dividing an irrational number by a non-zero rational number always results in an irrational number.
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Consider the set of integers \(S = \{n \in \mathbb{Z}^+ : n^3 + 3n^2 + 3n + 1 \text{ is a perfect square}\}\). Which of the following is true about the elements of \(S\)?
B · All elements of \(S\) satisfy \(n+1\) is a perfect square
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Let \(p\) be a prime number greater than 3. Consider the integer \(N = p^2 + (p+2)^2 + (p+4)^2\). Which of the following statements is correct about \(N\)?
C · N is divisible by 24
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If \(x\) and \(y\) are positive integers such that \(x^2 - y^2 = 221\) and \(\gcd(x,y) = 1\), which of the following must be true?
B · Exactly one of \(x\) or \(y\) is even
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Let \(n\) be a positive integer such that \(n\) divides \(2^n - 2\). Which of the following must be true about \(n\)?
B · n is a Carmichael number or prime
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For positive integers \(a,b,c\) satisfying \(\gcd(a,b,c) = 1\) and \(a^2 + b^2 = c^2\), if \(a\) and \(b\) are consecutive integers, which of the following is true?
D · The product \(abc\) is divisible by 60
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If \(m,n\) are positive integers such that \(m^3 - n^3 = 91\) and \(\gcd(m,n) = 1\), which of the following pairs \((m,n)\) is possible?
C · (6,5)
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Which of the following integers \(n\) satisfies that \(n^2 + 1\) is divisible by 5 but \(n\) is not divisible by 5?
A · Any integer \(n \equiv 2 \pmod{5}\)
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Let \(x\) be an integer such that \(x^2 \equiv 1 \pmod{p}\), where \(p\) is an odd prime. Which of the following must be true?
A · Either \(x \equiv 1 \pmod{p}\) or \(x \equiv -1 \pmod{p}\)
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If \(a,b\) are positive integers such that \(a+b\) divides \(a^3 + b^3\), which of the following must be true?
C · \(a+b\) divides \(3ab\)
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Let \(n\) be a positive integer such that \(n^2 + n + 1\) is divisible by 7. Which of the following is true about \(n\)?
C · n \equiv 4 \pmod{7}
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Assertion (A): Every perfect square is congruent to 0, 1, or 4 modulo 8.
Reason (R): Squares of odd numbers are congruent to 1 modulo 8.
B · Both A and R are true but R is not the correct explanation of A
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If \(x\) is an integer such that \(x^4 \equiv 1 \pmod{5}\), which of the following is true?
D · x \equiv 0,1,2,3,4 \pmod{5}
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If \(a,b\) are positive integers such that \(a^2 + b^2\) is divisible by 13, which of the following must be true?
D · At least one of \(a,b\) is divisible by 13 or both are congruent to 0 modulo 13
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Let \(n\) be a positive integer such that \(n^2 + n + 41\) is prime. Which of the following is true?
A · This holds for all positive integers \(n < 41\)
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What is the Highest Common Factor (HCF) of 24 and 36?
B · 12
The factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24 and factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36. The highest common factor is 12.
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Which of the following is NOT a property of HCF of two numbers?
D · HCF is always greater than the LCM
HCF is always less than or equal to the smaller number and cannot be greater than the LCM. Hence, option D is not a property of HCF.
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If the HCF of two numbers is 5 and one of the numbers is 35, which of the following can be the other number?
A · 10
Since HCF is 5, the other number must be divisible by 5. Among the options, only 10 is divisible by 5.
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What is the Least Common Multiple (LCM) of 6 and 8?
B · 24
Multiples of 6 are 6, 12, 18, 24, 30... and multiples of 8 are 8, 16, 24, 32... The smallest common multiple is 24.
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Which of the following is NOT a property of LCM of two numbers?
C · LCM of two numbers is always their product
LCM of two numbers is not always their product; it is true only when the numbers are co-prime. Hence, option C is incorrect.
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If the LCM of two numbers is 60 and one number is 12, which of the following can be the other number?
C · 15
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Find the HCF of 48 and 18 using the Euclidean Algorithm.
A · 6
Using Euclidean Algorithm:48 ÷ 18 = 2 remainder 1218 ÷ 12 = 1 remainder 612 ÷ 6 = 2 remainder 0So, HCF is 6.
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Using prime factorization, find the LCM of 36 and 48.
A · 144
Prime factors:36 = 2^2 × 3^248 = 2^4 × 3LCM = 2^4 × 3^2 = 16 × 9 = 144Correction: 144 is correct, so option A is correct.
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If the HCF of two numbers is 7 and their LCM is 168, and one number is 28, what is the other number?
A · 42
Using the relation: \( \text{HCF} \times \text{LCM} = \text{Product of two numbers} \)So, \(7 \times 168 = 28 \times x \Rightarrow x = \frac{7 \times 168}{28} = 42\).
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Which of the following statements correctly represents the relationship between HCF and LCM of two numbers \(a\) and \(b\)?
B · \( \text{HCF}(a,b) \times \text{LCM}(a,b) = a \times b \)
The fundamental relationship between HCF and LCM of two numbers is \( \text{HCF}(a,b) \times \text{LCM}(a,b) = a \times b \).
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Three bells ring at intervals of 12, 15, and 20 minutes respectively. If they ring together at 8:00 AM, at what time will they ring together again?
C · 9:00 AM
LCM of 12, 15, and 20 is calculated as:12 = 2^2 × 315 = 3 × 520 = 2^2 × 5LCM = 2^2 × 3 × 5 = 60 minutes = 1 hour.So, they will ring together again at 9:00 AM.
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A factory produces two types of products every 8 and 12 days respectively. If both products were produced today, after how many days will both products be produced again on the same day?
A · 24
The problem requires finding the LCM of 8 and 12.Prime factors:8 = 2^312 = 2^2 × 3LCM = 2^3 × 3 = 8 × 3 = 24 days.
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Which of the following is always a divisor of both numbers when finding their HCF?
C · A common factor
HCF is the highest number that divides both numbers exactly, so it must be a common factor.
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If the HCF of two numbers is 6, which of the following cannot be their LCM?
C · 12
LCM must be a multiple of both numbers and divisible by their HCF. 12 is less than 6 times any number, so it cannot be the LCM if HCF is 6.
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Which of the following statements is true about the HCF of two prime numbers?
A · It is always 1
Two distinct prime numbers have no common factors other than 1, so their HCF is always 1.
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The LCM of 4 and 6 is:
A · 12
Multiples of 4: 4, 8, 12, 16... Multiples of 6: 6, 12, 18... The smallest common multiple is 12.
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Which property is true for the LCM of any two numbers?
A · It is always less than or equal to the product of the two numbers
LCM of two numbers is always a divisor of their product, so it cannot be greater than the product.
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If the LCM of two numbers is 60 and one of the numbers is 12, which of the following can be the other number?
C · 15
LCM(12, x) = 60. 15 is a multiple of 3 and 5, and LCM(12,15) = 60.
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If two numbers are 18 and 24, what is the product of their HCF and LCM?
A · 432
HCF(18,24) = 6, LCM(18,24) = 72. Product = 6 × 72 = 432.
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For two numbers \( a \) and \( b \), which of the following equations correctly represents the relationship between their HCF and LCM?
B · \( HCF(a,b) \times LCM(a,b) = a \times b \)
The product of the HCF and LCM of two numbers equals the product of the numbers themselves.
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If the HCF of two numbers is 8 and their LCM is 96, what is the sum of the two numbers?
A · 56
Let the numbers be 8x and 8y with HCF 8 and LCM 96. Then \( x \) and \( y \) are co-prime and \( x \times y = \frac{LCM}{HCF} = \frac{96}{8} = 12 \). Possible pairs (3,4) or (4,3). Sum = 8(3+4) = 56.
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Find the HCF of 48 and 180 using the division method.
A · 12
Using division method: 180 ÷ 48 = 3 remainder 36; 48 ÷ 36 = 1 remainder 12; 36 ÷ 12 = 3 remainder 0. So HCF is 12.
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Using prime factorization, find the LCM of 36 and 48.
A · 144
Prime factors: 36 = 2^2 × 3^2, 48 = 2^4 × 3. LCM = 2^4 × 3^2 = 16 × 9 = 144.
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Two machines operate on cycles of 15 and 20 minutes respectively. If they start together at 9:00 AM, after how many minutes will they next start together?
A · 60 minutes
The time after which they start together is the LCM of 15 and 20, which is 60 minutes.
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Three friends start jogging around a circular track at the same time. Their jogging times to complete one round are 12, 15, and 20 minutes respectively. After how many minutes will they meet again at the starting point?
A · 60 minutes
They will meet again after the LCM of 12, 15, and 20 minutes. LCM(12,15,20) = 60 minutes.
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Which of the following is a fraction?
C · \( \frac{3}{4} \)
A fraction is a number expressed as \( \frac{numerator}{denominator} \). Option C is in fraction form.
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Which of the following represents a fraction less than 1?
B · \( \frac{2}{5} \)
A fraction less than 1 has numerator smaller than denominator. \( \frac{2}{5} \) satisfies this.
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If \( \frac{a}{b} \) is a fraction where \( a > b \), which type of fraction is it?
B · Improper fraction
When numerator is greater than denominator, the fraction is called an improper fraction.
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Which of the following is a proper fraction?
B · \( \frac{3}{8} \)
Proper fractions have numerator less than denominator. \( \frac{3}{8} \) fits this definition.
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Identify the mixed fraction from the following options.
B · \( 2 \frac{1}{3} \)
Mixed fractions consist of a whole number and a proper fraction. Option B is a mixed fraction.
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Which of the following is an improper fraction?
B · \( \frac{9}{4} \)
Improper fractions have numerator greater than denominator. \( \frac{9}{4} \) is improper.
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Convert the fraction \( \frac{3}{5} \) into decimal form.
A · 0.6
Dividing 3 by 5 gives 0.6 as decimal equivalent.
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Express 0.125 as a fraction in simplest form.
A · \( \frac{1}{8} \)
0.125 = \( \frac{125}{1000} = \frac{1}{8} \) after simplification.
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Which decimal corresponds to the fraction \( \frac{7}{20} \)?
A · 0.35
Dividing 7 by 20 gives 0.35.
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Convert the repeating decimal 0.\( \overline{3} \) to a fraction.
A · \( \frac{1}{3} \)
The repeating decimal 0.333... equals \( \frac{1}{3} \).
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Calculate \( \frac{2}{5} + \frac{3}{10} \).
A · \( \frac{7}{10} \)
Common denominator is 10, so \( \frac{4}{10} + \frac{3}{10} = \frac{7}{10} \).
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Find the product of \( \frac{3}{4} \) and \( \frac{2}{5} \).
A · \( \frac{3}{10} \)
Multiply numerators and denominators: \( \frac{3 \times 2}{4 \times 5} = \frac{6}{20} = \frac{3}{10} \).
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Calculate \( \frac{5}{6} - \frac{1}{4} \).
A · \( \frac{7}{12} \)
Common denominator 12: \( \frac{10}{12} - \frac{3}{12} = \frac{7}{12} \).
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Divide \( \frac{3}{5} \) by \( \frac{2}{7} \).
A · \( \frac{21}{10} \)
Division of fractions: \( \frac{3}{5} \div \frac{2}{7} = \frac{3}{5} \times \frac{7}{2} = \frac{21}{10} \).
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Calculate 1.2 + 3.45.
A · 4.65
Adding decimals: 1.2 + 3.45 = 4.65.
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Subtract 5.75 from 9.3.
A · 3.55
9.3 - 5.75 = 3.55.
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Find the product of 0.6 and 0.25.
A · 0.15
0.6 \( \times \) 0.25 = 0.15.
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Divide 4.8 by 0.6.
A · 8
4.8 \( \div \) 0.6 = 8.
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Which of the following fractions is the greatest?
D · \( \frac{1}{2} \)
\( \frac{1}{2} = 0.5 \) is greater than others when converted to decimals.
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Arrange the decimals in ascending order: 0.45, 0.405, 0.54, 0.5.
A · 0.405, 0.45, 0.5, 0.54
Ordering decimals from smallest to largest: 0.405 < 0.45 < 0.5 < 0.54.
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Which of the following fractions is equivalent to \( \frac{4}{6} \)?
A · \( \frac{2}{3} \)
\( \frac{4}{6} = \frac{2}{3} \) after simplification by dividing numerator and denominator by 2.
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Simplify the fraction \( \frac{18}{24} \).
A · \( \frac{3}{4} \)
Dividing numerator and denominator by 6, \( \frac{18}{24} = \frac{3}{4} \).
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Which fraction is equivalent to \( \frac{5}{8} \)?
D · All of the above
All options are multiples of \( \frac{5}{8} \) and hence equivalent fractions.
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Which decimal is a terminating decimal?
A · 0.75
Terminating decimals have finite digits after decimal point. 0.75 terminates.
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Which of the following decimals is repeating?
C · 0.\( \overline{6} \)
0.\( \overline{6} \) means 0.666... which is a repeating decimal.
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Identify the fraction that corresponds to the repeating decimal 0.\( \overline{09} \).
A · \( \frac{1}{11} \)
0.090909... equals \( \frac{1}{11} \) is incorrect; the correct fraction is \( \frac{1}{11} \) for 0.\( \overline{09} \). Actually, 0.\( \overline{09} \) = \( \frac{1}{11} \). So correct answer is A.
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A tank is \( \frac{3}{4} \) full of water. If \( \frac{1}{3} \) of the water is used, what fraction of the tank remains filled?
A · \( \frac{1}{2} \)
Water used = \( \frac{1}{3} \times \frac{3}{4} = \frac{1}{4} \). Remaining = \( \frac{3}{4} - \frac{1}{4} = \frac{1}{2} \).
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If a decimal number is 0.8 and it is increased by \( \frac{1}{5} \), what is the new value?
A · 1.0
Convert \( \frac{1}{5} = 0.2 \). New value = 0.8 + 0.2 = 1.0 (Option A). Correction: 0.8 + 0.2 = 1.0, so correct answer is A.
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A shopkeeper sold \( \frac{2}{5} \) of his stock of 150 items. How many items are left?
B · 90
Items sold = \( \frac{2}{5} \times 150 = 60 \). Left = 150 - 60 = 90.
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A car travels \( \frac{3}{4} \) of a journey in 1.5 hours. How long will it take to complete the entire journey?
A · 2 hours
If \( \frac{3}{4} \) journey takes 1.5 hours, full journey time = \( \frac{1.5}{3/4} = 1.5 \times \frac{4}{3} = 2 \) hours. Correction: 1.5 \( \times \frac{4}{3} = 2 \) hours, so correct answer is A.
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Which of the following represents a proper fraction?
B · \( \frac{3}{4} \)
A proper fraction has a numerator smaller than the denominator. \( \frac{3}{4} \) fits this definition.
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What is the decimal equivalent of \( \frac{1}{2} \)?
B · 0.5
Dividing 1 by 2 gives 0.5 as the decimal equivalent.
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Which of the following is a mixed fraction?
B · \( 1 \frac{2}{5} \)
A mixed fraction consists of a whole number and a proper fraction. \( 1 \frac{2}{5} \) is a mixed fraction.
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Calculate \( \frac{3}{4} + \frac{2}{5} \).
A · \( \frac{23}{20} \)
LCM of 4 and 5 is 20. \( \frac{3}{4} = \frac{15}{20} \), \( \frac{2}{5} = \frac{8}{20} \). Sum = \( \frac{15+8}{20} = \frac{23}{20} \).
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Which of the following decimals is equivalent to \( \frac{7}{8} \)?
A · 0.875
\( \frac{7}{8} = 0.875 \) when converted to decimal.
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Identify the fraction type of \( \frac{12}{4} \).
B · Improper fraction
Since numerator (12) is greater than denominator (4), it is an improper fraction.
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Convert 0.375 to a fraction in simplest form.
A · \( \frac{3}{8} \)
0.375 = \( \frac{375}{1000} = \frac{3}{8} \) after simplification.
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Calculate \( \frac{5}{6} \times \frac{3}{4} \).
A · \( \frac{15}{24} \)
Multiply numerators and denominators: \( \frac{5 \times 3}{6 \times 4} = \frac{15}{24} \).
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Which decimal is the result of dividing 7 by 8?
D · Both A and C
7 divided by 8 is 0.875, which can be written as 0.875 or 0.8750.
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Which of the following decimals is the greatest?
B · 0.652
0.652 is greater than 0.625, 0.562, and 0.526.
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Arrange the following fractions in ascending order: \( \frac{3}{7}, \frac{2}{5}, \frac{4}{9} \).
D · \( \frac{2}{5} < \frac{3}{7} < \frac{4}{9} \)
Converting to decimals: \( \frac{2}{5} = 0.4, \frac{3}{7} \approx 0.4286, \frac{4}{9} \approx 0.4444 \). So ascending order is \( \frac{2}{5} < \frac{3}{7} < \frac{4}{9} \).
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Subtract \( 0.75 \) from \( 1.25 \). What is the result?
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Which of the following is the product of 0.6 and 0.7?
A · 0.42
0.6 \( \times \) 0.7 = 0.42
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Which of the following is an equivalent fraction of \( \frac{4}{6} \)?
A · \( \frac{2}{3} \)
Simplifying \( \frac{4}{6} \) gives \( \frac{2}{3} \), which is equivalent.
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Which of the following fractions is a unit fraction?
A · \( \frac{1}{5} \)
A unit fraction has numerator 1. \( \frac{1}{5} \) is a unit fraction.
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Convert the decimal 0.2 recurring (0.222...) into a fraction.
A · \( \frac{2}{9} \)
0.222... = \( \frac{2}{9} \) as a recurring decimal fraction.
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Calculate \( \frac{7}{8} - \frac{3}{5} \).
A · \( \frac{11}{40} \)
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What is the sum of 0.45 and 0.55?
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Which of the following decimals is the smallest?
B · 0.303
0.303 < 0.313 < 0.330 < 0.333
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If \( \frac{5}{x} = 0.25 \), what is the value of \( x \)?
A · 20
0.25 = \( \frac{1}{4} \), so \( \frac{5}{x} = \frac{1}{4} \) implies \( x = 20 \).
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A recipe requires \( \frac{3}{4} \) cup of sugar. If you want to make half the recipe, how much sugar do you need?
A · \( \frac{3}{8} \) cup
Half of \( \frac{3}{4} \) is \( \frac{3}{4} \times \frac{1}{2} = \frac{3}{8} \).
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If \( 0.6x = 1.8 \), what is the value of \( x \)?
A · 3
Divide both sides by 0.6: \( x = \frac{1.8}{0.6} = 3 \).
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Which of the following is the result of dividing \( \frac{3}{5} \) by \( \frac{2}{7} \)?
A · \( \frac{21}{10} \)
Dividing fractions: \( \frac{3}{5} \div \frac{2}{7} = \frac{3}{5} \times \frac{7}{2} = \frac{21}{10} \).
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A car travels \( \frac{5}{8} \) of a journey in 1 hour. How long will it take to complete the entire journey at the same speed?
D · Both A and B
Time for full journey = \( \frac{1}{(5/8)} = \frac{8}{5} = 1.6 \) hours.
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Which of the following decimals is equal to \( \frac{11}{16} \)?
A · 0.6875
\( \frac{11}{16} = 0.6875 \) when converted to decimal.
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Add \( 0.125 + 0.375 + 0.5 \). What is the sum?
A · 1.0
Sum is 0.125 + 0.375 + 0.5 = 1.0
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Which of the following fractions is the odd one out?
D · \( \frac{5}{7} \)
The first three fractions are equivalent to \( \frac{3}{4} \), but \( \frac{5}{7} \) is not.
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If the decimal expansion of \(\frac{1}{n}\) has a non-repeating part of length 4 and a repeating part of length 0, which of the following must be true about \(n\)?
A · \(n = 2^4 \times 5^4\)
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The decimal expansion of \(\frac{1}{n}\) has a non-repeating part of length 2 and a repeating part of length 5. If \(n\) divides \(10^7 - 1\), which of the following could be the value of \(n\)?
A · 2^2 \times 5^2 \times 41
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If \(\frac{1}{n}\) has a decimal expansion with a repeating cycle of length 15 and a non-repeating part of length 0, which of the following must be true about \(n\)?
A · \(n\) divides \(10^{15} - 1\) but not \(10^k - 1\) for any \(k < 15\)
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The decimal expansion of \(\frac{1}{n}\) has a non-repeating part of length 1 and a repeating part of length 1. Which of the following could be the denominator \(n\)?
A · 2 \times 5 \times 3
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Which of the following numbers is divisible by 3?
B · 135
A number is divisible by 3 if the sum of its digits is divisible by 3. For 135, 1 + 3 + 5 = 9, which is divisible by 3.
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Which of the following is divisible by 5?
B · 140
A number is divisible by 5 if it ends with 0 or 5. 140 ends with 0, so it is divisible by 5.
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Which of the following numbers is divisible by 9?
A · 729
A number is divisible by 9 if the sum of its digits is divisible by 9. For 729, 7 + 2 + 9 = 18, which is divisible by 9.
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Which of the following numbers is divisible by 7?
A · 203
203 divided by 7 equals 29 exactly, so 203 is divisible by 7.
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Which number is divisible by 11?
A · 121
A number is divisible by 11 if the difference between the sum of digits in odd places and even places is a multiple of 11. For 121: (1 + 1) - 2 = 0, which is divisible by 11.
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Which of the following numbers is divisible by 14?
B · 210
14 = 2 × 7. A number divisible by 14 must be divisible by both 2 and 7. 210 is even and divisible by 7 (210 ÷ 7 = 30).
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Find the smallest positive number divisible by both 6 and 15.
A · 30
The smallest number divisible by both 6 and 15 is their LCM. LCM of 6 and 15 is 30.
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Which of the following numbers is a composite number divisible by 12?
A · 36
36 is divisible by 12 (36 ÷ 12 = 3) and is composite.
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Which of the following numbers is divisible by 18?
B · 162
18 = 2 × 9. Number must be divisible by 2 and 9. 162 is even and sum of digits (1+6+2=9) is divisible by 9.
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Which of the following numbers is divisible by 15?
A · 135
A number divisible by 15 must be divisible by both 3 and 5. 135 ends with 5 and sum of digits is 9, divisible by 3.
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Which of the following numbers is divisible by 20?
A · 200
20 = 4 × 5. Number must be divisible by 4 and 5. 200 ends with 00, divisible by both 4 and 5.
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Which of the following numbers is divisible by 24?
D · 192
24 = 8 × 3. Number must be divisible by 8 and 3. 192 is divisible by 8 (192 ÷ 8 = 24) and sum of digits (1+9+2=12) divisible by 3.
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A number leaves a remainder 3 when divided by 7 and remainder 4 when divided by 5. Which of the following numbers satisfies this?
B · 24
Check each option: 39 ÷ 7 leaves remainder 4, so not correct. 31 ÷ 7 = 4 remainder 3 and 31 ÷ 5 = 6 remainder 1, no. 24 ÷ 7 = 3 remainder 3 and 24 ÷ 5 = 4 remainder 4, correct.
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Find the smallest positive number which when divided by 3, 4, and 5 leaves remainders 2, 3, and 4 respectively.
A · 59
The number is 1 less than the LCM of 3,4,5 because remainders are one less than divisors. LCM of 3,4,5 is 60, so number = 60 -1 = 59.
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Which of the following numbers is divisible by 2 according to the basic divisibility rule?
A · 1358
A number is divisible by 2 if its last digit is even. 1358 ends with 8, which is even, so it is divisible by 2.
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What is the divisibility rule for 5?
A · Number ends with 0 or 5
A number is divisible by 5 if its last digit is either 0 or 5.
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Which of the following numbers is divisible by 3?
B · 456
A number is divisible by 3 if the sum of its digits is divisible by 3. For 456, sum = 4+5+6=15, which is divisible by 3.
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Which of the following numbers is divisible by 11?
A · 2728
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Find the smallest number divisible by both 7 and 13.
A · 91
The smallest number divisible by both 7 and 13 is their LCM. Since 7 and 13 are prime, LCM = 7 × 13 = 91.
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If a number is divisible by 9, which of the following must be true?
A · Sum of digits is divisible by 9
A number is divisible by 9 if the sum of its digits is divisible by 9.
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What is the HCF of 84 and 126?
A · 42
Prime factorization: 84 = 2^2 × 3 × 7, 126 = 2 × 3^2 × 7. Common factors: 2 × 3 × 7 = 42.
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If a number \( n \) is divisible by 4 and 6, which of the following must be true?
A · \( n \) is divisible by 12
LCM of 4 and 6 is 12, so any number divisible by both 4 and 6 must be divisible by 12.
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If \( a \) divides \( b \) and \( b \) divides \( c \), which property of divisibility is demonstrated?
A · Transitive property
The transitive property states that if \( a \mid b \) and \( b \mid c \), then \( a \mid c \).
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Which of the following is a valid divisibility test for the composite number 15?
A · Number divisible by both 3 and 5
A number is divisible by 15 if it is divisible by both 3 and 5.
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Evaluate \( 8 + 2 \times 5 \) using the correct order of operations.
B · 18
According to BODMAS, multiplication comes before addition: \(2 \times 5 = 10\), then \(8 + 10 = 18\).
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Find the value of \( (15 - 3) \div 4 + 2^2 \).
A · 7
First, evaluate inside parentheses: \(15 - 3 = 12\). Then division: \(12 \div 4 = 3\). Next, exponent: \(2^2 = 4\). Finally, add: \(3 + 4 = 7\). The correct answer is 7.
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Calculate \( 6 + 3 \times (4^2 - 10) \).
D · 36
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Evaluate \( \frac{5 + 3 \times 2}{4} - 1 \).
A · 2
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Simplify \( 12 - 3 \times 2 + 4 \div 2 \).
B · 8
Multiply and divide first: \(3 \times 2 = 6\), \(4 \div 2 = 2\). Then perform addition and subtraction: \(12 - 6 + 2 = 8\). The correct answer is 8.
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Simplify \( \frac{3}{4} + \frac{5}{8} \).
B · \( \frac{11}{8} \)
Find common denominator 8: \(\frac{3}{4} = \frac{6}{8}\). Add: \(\frac{6}{8} + \frac{5}{8} = \frac{11}{8}\).
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Calculate \( 7 - \frac{2}{3} \).
A · \( \frac{19}{3} \)
Convert 7 to fraction: \( \frac{21}{3} \). Subtract: \( \frac{21}{3} - \frac{2}{3} = \frac{19}{3} \).
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Simplify \( \frac{5}{6} \times \frac{3}{10} + \frac{1}{2} \).
D · \( \frac{11}{20} \)
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Simplify \( \frac{3}{5} + \frac{2}{3} \times \frac{15}{8} \).
D · \( \frac{11}{8} \)
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Evaluate \( 2^3 \times 3^2 \).
A · 72
\(2^3 = 8\), \(3^2 = 9\), multiply: \(8 \times 9 = 72\). Correct answer is 72.
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Simplify \( \sqrt{81} + 2^4 \).
A · 40
\( \sqrt{81} = 9 \), \(2^4 = 16\), sum is \(9 + 16 = 25\). None of the options is 25, so options need correction.
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Calculate \( (3^2 + 4^2)^{\frac{1}{2}} \).
A · 5
Calculate inside parentheses: \(3^2 = 9\), \(4^2 = 16\), sum is 25. Then square root: \(\sqrt{25} = 5\).
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Simplify \( 16^{\frac{3}{4}} \).
A · 8
\(16^{\frac{3}{4}} = (16^{\frac{1}{4}})^3 = 2^3 = 8\).
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What is 25% of 240?
A · 60
25% of 240 = \( \frac{25}{100} \times 240 = 60 \).
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Simplify \( 0.75 + 0.2 \times 0.4 \).
A · 0.83
Multiply first: \(0.2 \times 0.4 = 0.08\). Add: \(0.75 + 0.08 = 0.83\). Correct answer is 0.83, option A.
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Calculate \( 120 - 15\% \text{ of } 200 \).
A · 90
15% of 200 = 30. Subtract: 120 - 30 = 90. Correct answer is 90, option A.
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Simplify \( \sqrt{49} + \sqrt[3]{27} \).
C · 10
\( \sqrt{49} = 7 \), \( \sqrt[3]{27} = 3 \), sum is 10.
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Evaluate \( \sqrt{64} \times \sqrt[3]{8} \).
D · 16
\( \sqrt{64} = 8 \), \( \sqrt[3]{8} = 2 \), product is 16.
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Simplify \( \sqrt{50} + \sqrt{18} \).
B · \( 8\sqrt{2} \)
\( \sqrt{50} = 5\sqrt{2} \), \( \sqrt{18} = 3\sqrt{2} \), sum is \( 8\sqrt{2} \).
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Find the value of \( |-7| + (-3) \).
A · 4
Absolute value of -7 is 7. Then \(7 + (-3) = 4\).
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Simplify \( -5 + | -2 + 3 | \).
A · -4
Inside absolute value: \(-2 + 3 = 1\), absolute value is 1. Then \(-5 + 1 = -4\).
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Evaluate \( | -4 \times 2 | - 3 \).
A · 5
Multiply: \(-4 \times 2 = -8\), absolute value is 8. Then \(8 - 3 = 5\).
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Evaluate \( 8 + 2 \times (15 - 5) \div 5 \). What is the result?
A · 12
First, calculate inside the parentheses: 15 - 5 = 10.Then multiply: 2 \times 10 = 20.Divide by 5: 20 \div 5 = 4.Finally, add 8 + 4 = 12.
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Find the value of \( (24 \div 6) + 3^2 - 4 \times 2 \).
C · 5
Calculate stepwise:24 \div 6 = 43^2 = 94 \times 2 = 8Then, 4 + 9 - 8 = 5.
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Simplify \( 18 - 3 \times (4 + 2)^2 \div 9 \).
B · 6
Inside parentheses: 4 + 2 = 6.Square: 6^2 = 36.Multiply: 3 \times 36 = 108.Divide: 108 \div 9 = 12.Subtract: 18 - 12 = 6.
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What is the value of \( \frac{5}{8} + 0.375 \)?
A · 1
Convert \( \frac{5}{8} = 0.625 \).Add: 0.625 + 0.375 = 1.0.
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Simplify \( 2.5 \times \frac{4}{5} + 1.2 \).
A · 3.2
Multiply: 2.5 \times \frac{4}{5} = 2.5 \times 0.8 = 2.0.Add: 2.0 + 1.2 = 3.2.
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Calculate \( \frac{7}{12} \div \frac{14}{18} \).
A · \( \frac{3}{4} \)
Division of fractions: \( \frac{7}{12} \times \frac{18}{14} = \frac{7 \times 18}{12 \times 14} = \frac{126}{168} = \frac{3}{4} \).
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Simplify \( 3.6 - 1.2 \times 2 + 0.4 \div 0.2 \).
B · 4.4
Multiply: 1.2 \times 2 = 2.4.Divide: 0.4 \div 0.2 = 2.Calculate: 3.6 - 2.4 + 2 = 3.2 + 2 = 4.4.
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Evaluate \( (2^3)^2 \div 4^3 \).
A · 1
Calculate numerator: \( (2^3)^2 = 2^{3 \times 2} = 2^6 = 64 \).Denominator: \( 4^3 = (2^2)^3 = 2^{6} = 64 \).Division: 64 \div 64 = 1.
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Simplify \( \sqrt{81} + 2^4 - \sqrt{16} \).
A · 21
\( \sqrt{81} = 9 \), \( 2^4 = 16 \), \( \sqrt{16} = 4 \).Sum: 9 + 16 - 4 = 21.
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Find the value of \( 5^{3} \div 5^{1} + \sqrt{49} \).
C · 134
Calculate powers: \( 5^{3} \div 5^{1} = 5^{3-1} = 5^{2} = 25 \).\( \sqrt{49} = 7 \).Sum: 25 + 7 = 32.
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Simplify \( \frac{LCM(6, 8)}{HCF(12, 18)} \).
A · 4
LCM of 6 and 8 is 24.HCF of 12 and 18 is 6.Divide: 24 \div 6 = 4.
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If \( HCF(24, 36) = 12 \) and \( LCM(24, 36) = 72 \), what is \( \frac{LCM}{HCF} + 5 \)?
A · 11
\( \frac{72}{12} = 6 \).Add 5: 6 + 5 = 11.
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Calculate \( \frac{LCM(9, 15)}{HCF(18, 24)} \times 3 \).
B · 18
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Simplify \( |-7| + (-3) \times 4 \).
A · -19
Absolute value: \( |-7| = 7 \).Multiply: (-3) \times 4 = -12.Add: 7 + (-12) = -5.
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What is the value of \( -5 + | -3 + 2 | \)?
B · -4
Inside absolute value: -3 + 2 = -1.Absolute value: |-1| = 1.Add: -5 + 1 = -4.
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Evaluate \( |-8| - (-4) + 3 \).
A · 15
Absolute value: |-8| = 8.Subtract negative: 8 - (-4) = 8 + 4 = 12.Add 3: 12 + 3 = 15.
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Simplify the algebraic expression: \( 3x + 5 - 2x + 7 \).
A · \( x + 12 \)
Combine like terms:\( 3x - 2x = x \), constants: 5 + 7 = 12.Result: \( x + 12 \).
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Simplify \( 4(a - 2) + 3a \).
A · \( 7a - 8 \)
Distribute: 4a - 8 + 3a = 7a - 8.
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Simplify \( 2x + 3 - (x - 5) \).
A · \( x + 8 \)
Remove parentheses: 2x + 3 - x + 5 = (2x - x) + (3 + 5) = x + 8.
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Which property justifies the equality: \( 7 + (3 + 5) = (7 + 3) + 5 \)?
B · Associative Property
The grouping of numbers changes without changing the sum, which is the Associative Property of Addition.
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Simplify \( 5 \times (2 + 4) \) using the distributive property.
A · \( 5 \times 2 + 5 \times 4 \)
Distributive property states \( a(b + c) = ab + ac \), so \( 5(2 + 4) = 5 \times 2 + 5 \times 4 \).
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Which of the following is a simplified form of \( \sqrt{50} \)?
A · \( 5\sqrt{2} \)
Since \( \sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2} \), option A is correct.
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Which of the following is an irrational surd?
C · \( \sqrt{7} \)
\( \sqrt{7} \) is irrational because 7 is not a perfect square, while the others simplify to rational numbers.
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Calculate \( \sqrt{18} + \sqrt{8} \).
A · \( 3\sqrt{2} + 2\sqrt{2} = 5\sqrt{2} \)
Simplify each surd: \( \sqrt{18} = 3\sqrt{2} \), \( \sqrt{8} = 2\sqrt{2} \). Adding gives \( 5\sqrt{2} \).
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Find the value of \( (\sqrt{3} + 2)(\sqrt{3} - 2) \).
A · \( -1 \)
Using the identity \( (a+b)(a-b) = a^2 - b^2 \), \( (\sqrt{3})^2 - 2^2 = 3 - 4 = -1 \).
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Simplify \( \frac{\sqrt{5}}{\sqrt{2}} \).
A · \( \sqrt{\frac{5}{2}} \)
Dividing surds: \( \frac{\sqrt{5}}{\sqrt{2}} = \sqrt{\frac{5}{2}} \).
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Rationalize the denominator of \( \frac{5}{\sqrt{3}} \).
A · \( \frac{5\sqrt{3}}{3} \)
Multiply numerator and denominator by \( \sqrt{3} \): \( \frac{5}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{5\sqrt{3}}{3} \).
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Rationalize the denominator of \( \frac{7}{2 + \sqrt{5}} \).
A · \( \frac{7(2 - \sqrt{5})}{-1} \)
Multiply numerator and denominator by conjugate \( 2 - \sqrt{5} \): denominator becomes \( (2)^2 - (\sqrt{5})^2 = 4 - 5 = -1 \).
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Simplify \( 2^{3} \times 2^{4} \).
A · \( 2^{7} \)
Using the law of indices: \( a^{m} \times a^{n} = a^{m+n} \), so \( 2^{3} \times 2^{4} = 2^{7} \).
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Simplify \( \frac{5^{7}}{5^{3}} \).
A · \( 5^{4} \)
Using the law of indices: \( \frac{a^{m}}{a^{n}} = a^{m-n} \), so \( \frac{5^{7}}{5^{3}} = 5^{4} \).
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Express \( \sqrt[3]{16} \) in terms of indices.
A · \( 16^{\frac{1}{3}} \)
The cube root of a number is the same as raising it to the power \( \frac{1}{3} \).
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Convert \( 81^{\frac{3}{4}} \) into surd form.
A · \( \left( \sqrt[4]{81} \right)^{3} \)
Using the property \( a^{\frac{m}{n}} = \left( \sqrt[n]{a} \right)^{m} \), so \( 81^{\frac{3}{4}} = \left( \sqrt[4]{81} \right)^{3} \).
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If \( x = \sqrt{2} \) and \( y = 2^{3} \), what is the value of \( x^{2} \times y^{\frac{1}{3}} \)?
B · \( 4 \)
Calculate \( x^{2} = (\sqrt{2})^{2} = 2 \), and \( y^{\frac{1}{3}} = (2^{3})^{\frac{1}{3}} = 2^{1} = 2 \). Then \( 2 \times 2 = 4 \). Correction: The correct answer is 4, not 8.
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What does the Remainder Theorem state for a polynomial \( f(x) \) divided by \( x - a \)?
A · The remainder is \( f(a) \)
The Remainder Theorem states that when a polynomial \( f(x) \) is divided by \( x - a \), the remainder is equal to \( f(a) \).
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If a polynomial \( f(x) = 2x^3 - 3x^2 + 4x - 5 \) is divided by \( x - 2 \), what is the remainder?
B · 7
By the Remainder Theorem, remainder = \( f(2) = 2(2)^3 - 3(2)^2 + 4(2) - 5 = 16 - 12 + 8 - 5 = 7 \).
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Find the remainder when \( f(x) = x^4 - 2x^3 + x - 1 \) is divided by \( x + 1 \).
C · 1
Remainder = \( f(-1) = (-1)^4 - 2(-1)^3 + (-1) - 1 = 1 + 2 - 1 - 1 = 1 \). Correction: \( 1 + 2 - 1 - 1 = 1 \). So correct answer is 1 (Option C).
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What is the remainder when \( f(x) = 3x^3 + 4x^2 - x + 6 \) is divided by \( x - 3 \)?
A · 72
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Is \( x - 2 \) a factor of the polynomial \( f(x) = x^3 - 4x^2 + 5x - 2 \)?
A · Yes, because \( f(2) = 0 \)
By the Factor Theorem (a consequence of the Remainder Theorem), \( x - a \) is a factor of \( f(x) \) if and only if \( f(a) = 0 \). Here, \( f(2) = 8 - 16 + 10 - 2 = 0 \), so \( x - 2 \) is a factor.
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Determine whether \( x + 1 \) is a factor of \( f(x) = 2x^3 + 3x^2 - x - 6 \).
B · No, because \( f(-1) eq 0 \)
Calculate \( f(-1) = 2(-1)^3 + 3(-1)^2 - (-1) - 6 = -2 + 3 + 1 - 6 = -4 eq 0 \), so \( x + 1 \) is not a factor.
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If \( f(x) = x^3 - 6x^2 + 11x - 6 \), find all factors of \( f(x) \) using the Remainder Theorem.
A · \( x - 1, x - 2, x - 3 \)
Check \( f(1) = 0 \), \( f(2) = 0 \), \( f(3) = 0 \), so factors are \( x - 1, x - 2, x - 3 \).
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Given \( f(x) = 2x^3 + 3x^2 - 2x - 3 \), if \( x - 1 \) is a factor, find the remainder when \( f(x) \) is divided by \( x + 1 \).
A · 0
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Divide \( f(x) = x^3 - 4x^2 + 5x - 2 \) by \( x - 1 \). What is the remainder?
A · 0
Remainder = \( f(1) = 1 - 4 + 5 - 2 = 0 \).
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If \( f(x) = x^4 - 5x^3 + 8x^2 - 4x + 1 \) is divided by \( x - 2 \), what is the remainder?
A · 1
Remainder = \( f(2) = 16 - 40 + 32 - 8 + 1 = 1 \). Correction: 16 - 40 = -24, -24 + 32 = 8, 8 - 8 = 0, 0 + 1 = 1. So remainder is 1 (Option A).
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Find the value of \( k \) such that \( x - 2 \) is a factor of \( f(x) = x^3 + kx^2 - 4x + 8 \).
C · 4
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If the remainder when \( f(x) = x^3 + ax^2 + bx + 6 \) is divided by \( x - 1 \) is 4 and the remainder when divided by \( x + 2 \) is 0, find \( a + b \).
C · 2
