Quick recall · 359 cards
Short MCQ-style retrieval prompts. Tap a card to reveal the answer.
PYQ
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The greatest number of four digits which is exactly divisible by 7, 14 and 21 is—
A · A. 9960
PYQ
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Find the last digit of the expression \( 1^3 + 2^3 + 3^3 + 4^3 + \dots + 100^3 \).
A · A. 0
PYQ
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Identify whether the following numbers are rational or irrational: (i) \( \sqrt{16} \), (ii) \( \sqrt{3} \), (iii) 0.1010010001\dots (non-repeating, non-terminating), (iv) -5/7.
C · C. (i),(iv) rational; (ii),(iii) irrational
PYQ
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Simplify: 38 + 62 − 25 × 2 + 35 ÷ 7
C · 55
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Simplify: 15 × (2 + 8/4) − 72/6 + √81
D · 57
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Approximate and solve: (22.99 + 17.01) ÷ 1.998 × 3.997 − 41.998 + 644.199 = ?
B · 682
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Simplify: 1/2 − 3/5 + 4⅔ = ?
A · 56/15
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Solve: 115 ÷ 5 + 12 × 6 = ? + 64 ÷ 4 − 35
C · 114
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Approximate: 56.08% of 149.92 + √(28.02 × 6.98) − 11⅑% = ?
C · 90
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Simplify: 1111 ÷ 11 + 2002 ÷ 26 + 750 ÷ 25 = ?
C · 208
Solve each division: 1111 ÷ 11 = 101, 2002 ÷ 26 = 77, and 750 ÷ 25 = 30. Add the results: 101 + 77 + 30 = 208. The answer is 208, which is option C.
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Solve: ?² = (40 × 64) / 80 − 7
C · 5
Calculate the right side: (40 × 64) / 80 = 2560 / 80 = 32. So: ?² = 32 − 7 = 25. Taking the square root: ? = √25 = 5. The answer is 5, which is option C.
PYQ
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Approximate: 2329 / 8 = ?
C · 300
Using approximation technique, round 2329 to a number easily divisible by 8. Approximate 2329 to 2400. Then: 2400 / 8 = 300. The answer is 300, which is option C.
PYQ
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Simplify: 51/2 + 29 × 3 − 19 + 23/2 = ?
B · 110
PYQ
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Solve: 39.012 × 14.98 − 28.013 × 9.999 = (20 + ?) × 5.23
B · 41
PYQ
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Solve: 4 + 10 − 3 × 6 / 3 + 4 = ?
B · 10
PYQ
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Find the HCF of 36 and 84.
A · 12
PYQ
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The HCF of two numbers is 37. The smallest number which when divided by each of 20, 28, 32 and 35 leaves no remainder is
D · 10125
PYQ · 2025
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HCF of two numbers 12906 and 14818 is 478. Find their LCM.
B · 600129
PYQ
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Find the greatest number which on dividing 1661 and 2045 leaves remainders 10 and 13 respectively.
B · 127
PYQ
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The greatest number which on dividing 1657 and 2037 leaves remainders 6 and 5 respectively, is:
A · 127
PYQ · 2025
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Convert 0.68 to a fraction in its simplest form.
B · 17/25
PYQ · 2025
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Convert 0.56 to a fraction in its simplest form.
B · 14/25
PYQ · 2025
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Convert 1 \( \frac{3}{8} \) to decimal.
B · 1.375
Convert mixed number: \( 1 \frac{3}{8} = 1 + \frac{3}{8} \). \( \frac{3}{8} = 0.375 \) (divide 3 by 8). So, 1 + 0.375 = 1.375. Option B is correct.
PYQ · 2025
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Convert 0.28 to a fraction in the simplest form.
B · 7/25
0.28 = \( \frac{28}{100} \). Simplify by dividing by 4: \( \frac{28 \div 4}{100 \div 4} = \frac{7}{25} \). 7 is prime and does not divide 25, so simplest form. Option B.
PYQ
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Please evaluate the following expression with decimals: 7.82 + 2.947 = ?
A · 10.767
Align decimals: 7.820 + 2.947. Add: 0+7=7, 2+4=6, 8+9=17 (carry 1), 7+2+1=10, 0+0=0. Result: 10.767. Option A.
PYQ · 2025
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What is 0.62 as a percent?
D · 62%
To convert decimal to percent, multiply by 100: 0.62 × 100 = 62%. Option D.
PYQ
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Find the value of \( \sqrt{256} + \sqrt{81} - \sqrt{36} \).
D · 4
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The cube root of 0.000216 is:
B · 0.06
0.000216 = 216 × 10^{-6} = (6^3) × (10^{-2})^3 = (0.06)^3Therefore, \( \sqrt[3]{0.000216} = 0.06 \). Option B is correct[4].
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What is the smallest number by which 600 must be multiplied to make it a perfect square?
D · 6
Prime factorization of 600 = 2^3 × 3^1 × 5^2To make it a perfect square, we need even exponents. Multiply by 2^1 × 3^1 = 6.Thus, 600 × 6 = 3600 = 60^2. Option D is correct[4].
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Which of the following is a natural number?
C · 7
Natural numbers are positive integers starting from 1, so 7 is a natural number.
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The set of natural numbers does NOT include which of the following?
C · 0
Natural numbers start from 1 and do not include 0.
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Which of the following statements about natural numbers is TRUE?
D · Natural numbers are closed under addition
Natural numbers are closed under addition but not under subtraction.
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Which of the following is a whole number?
B · 0
Whole numbers include all natural numbers and zero, but no fractions or negative numbers.
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The difference between whole numbers and natural numbers is that whole numbers include:
B · Zero
Whole numbers include zero along with all natural numbers.
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Which of the following is TRUE about whole numbers?
C · Whole numbers include zero
Whole numbers include zero and all natural numbers but do not include negatives or fractions.
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Which of the following is an integer?
B · −7
Integers include all whole numbers and their negatives, so −7 is an integer.
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Which of the following sets correctly represents integers?
A · {−3, −2, −1, 0, 1, 2, 3}
Integers include negative and positive whole numbers including zero.
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Which of the following statements about integers is FALSE?
B · Integers include fractions
Integers do not include fractions or decimals.
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If \( a = -5 \) and \( b = 3 \), what is \( a - b \)?
A · −8
\( a - b = -5 - 3 = -8 \).
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Which of the following is the greatest integer less than \( \sqrt{10} \)?
B · 3
\( \sqrt{10} \approx 3.16 \), so the greatest integer less than it is 3.
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Which of the following is a rational number?
B · 0.75
0.75 can be expressed as \( \frac{3}{4} \), so it is rational.
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Which of the following numbers is irrational?
C · \( \sqrt{3} \)
\( \sqrt{3} \) is irrational as it cannot be expressed as a fraction.
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Which of the following statements is TRUE about rational numbers?
B · All integers are rational numbers
All integers can be expressed as fractions with denominator 1, so they are rational.
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Which of the following is an example of an irrational number?
C · \( \pi \)
\( \pi \) is irrational because it cannot be expressed as a fraction.
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Which of the following numbers is irrational?
A · 0.1010010001...
The number 0.1010010001... is a non-repeating, non-terminating decimal, hence irrational.
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Which of the following statements about irrational numbers is TRUE?
C · Irrational numbers have non-terminating, non-repeating decimals
Irrational numbers have decimal expansions that neither terminate nor repeat.
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Which of the following is NOT an irrational number?
C · \( \frac{22}{7} \)
\( \frac{22}{7} \) is a rational approximation of \( \pi \), so it is rational.
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Match the following numbers with their correct classification:
1. 0
2. −3
3. \( \frac{4}{5} \)
4. \( \sqrt{7} \)
Options:
A. Rational Number
B. Integer
C. Whole Number
D. Irrational Number
A · 1-C, 2-B, 3-A, 4-D
0 is a whole number, −3 is an integer, \( \frac{4}{5} \) is rational, and \( \sqrt{7} \) is irrational.
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Consider the statements:
I. All integers are whole numbers.
II. All rational numbers are integers.
Which of the above statements is/are correct?
D · Neither I nor II is correct
Statement I is false because whole numbers are non-negative integers only. Statement II is false because rational numbers include fractions.
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Which of the following numbers is a natural number?
C · 7
Natural numbers are positive integers starting from 1, so 7 is a natural number.
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Identify the smallest natural number from the options below.
A · 1
Natural numbers start from 1 upwards; 0 is not a natural number.
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Which of the following statements about natural numbers is correct?
C · Natural numbers are all positive integers excluding zero
Natural numbers are positive integers starting from 1, excluding zero and fractions.
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Which of the following is a whole number?
B · 0
Whole numbers include zero and all positive integers, so 0 is a whole number.
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Select the smallest whole number from the list below.
B · 0
Whole numbers start from 0 and include all positive integers.
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Which of the following statements about whole numbers is true?
B · Whole numbers include zero and positive integers
Whole numbers include zero and all positive integers; they do not include negatives or fractions.
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Which of the following is an integer?
B · −2
Integers include all whole numbers and their negatives, so −2 is an integer.
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Identify the integer from the following set.
A · −7
−7 is a whole number with a negative sign, thus an integer.
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Which of the following statements is true about integers?
B · Integers include zero, positive and negative whole numbers
Integers include zero, positive and negative whole numbers but exclude fractions and decimals.
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Find the sum of the integers −3, 7, and −4.
A · 0
Sum = (−3) + 7 + (−4) = 0.
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Which of the following integers is divisible by 3 and 4?
A · 12
12 is divisible by both 3 and 4.
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Which of the following is a rational number?
A · \( \frac{3}{4} \)
Rational numbers can be expressed as a ratio of two integers; \( \frac{3}{4} \) is rational.
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Identify the rational number from the following options.
A · \( 0.333... \)
Repeating decimals like 0.333... represent rational numbers.
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Which of the following statements about rational numbers is correct?
A · All integers are rational numbers
All integers can be expressed as a ratio with denominator 1, so all integers are rational.
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Which of the following numbers is rational?
A · \( \frac{5}{8} \)
Only \( \frac{5}{8} \) is a ratio of integers, hence rational.
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If \( x = \frac{7}{3} \), which of the following is true?
B · \( x \) is rational
\( \frac{7}{3} \) is a ratio of integers, so it is rational.
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Which of the following is an irrational number?
A · \( \sqrt{3} \)
\( \sqrt{3} \) cannot be expressed as a fraction, so it is irrational.
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Which of the following numbers is irrational?
A · \( \pi \)
\( \pi \) is a non-terminating, non-repeating decimal and cannot be expressed as a fraction.
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Which statement is true about irrational numbers?
C · Irrational numbers are non-terminating and non-repeating decimals
Irrational numbers have decimal expansions that neither terminate nor repeat.
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Which of the following numbers is irrational?
A · \( \sqrt{7} \)
\( \sqrt{7} \) is irrational because 7 is not a perfect square.
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Match the following numbers with their correct classification:
1. 0
2. −5
3. \( \frac{1}{2} \)
4. \( \sqrt{2} \)
5. 7
Options:
A. Natural Number
B. Whole Number
C. Integer
D. Rational Number
E. Irrational Number
A · 1-B, 2-C, 3-D, 4-E, 5-A
0 is a whole number; −5 is an integer; \( \frac{1}{2} \) is rational; \( \sqrt{2} \) is irrational; 7 is natural.
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Match the number types with their correct examples:
Types:
1. Rational Number
2. Irrational Number
3. Integer
4. Whole Number
5. Natural Number
Examples:
A. 0
B. 3
C. \( \pi \)
D. −4
E. \( \frac{5}{6} \)
A · 1-E, 2-C, 3-D, 4-A, 5-B
Rational: \( \frac{5}{6} \), Irrational: \( \pi \), Integer: −4, Whole: 0, Natural: 3.
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If \( a \) and \( b \) are rational numbers such that \( a + \sqrt{2}b \) is irrational, which of the following must be true?
C · \( b eq 0 \) and \( a/b \) is irrational
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Let \( m, n \in \mathbb{Z} \) such that \( \frac{m}{n} \) is in lowest terms and \( \sqrt{\frac{m}{n}} \) is rational. Which of the following must hold?
A · Both \( m \) and \( n \) are perfect squares
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Which of the following numbers is NOT rational?
C · \( \frac{3 + \sqrt{5}}{2} \)
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Assertion (A): Every integer is a whole number. Reason (R): Whole numbers include all natural numbers and zero, but not negative integers.
D · A is false, but R is true
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Match the following sets with their correct descriptions:
A · A. Natural Numbers
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Let \( z = a + b \sqrt{3} \), where \( a, b \in \mathbb{Q} \) and \( z \) is irrational. If \( z^2 \) is rational, which of the following must be true?
C · \( a^2 = 3b^2 \)
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If \( \alpha \) is an irrational number such that \( \alpha + \frac{1}{\alpha} \) is rational, which of the following is necessarily true?
A · \( \alpha \) is a root of a quadratic equation with rational coefficients
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Consider the number \( w = \sqrt{2} + \sqrt{3} \). Which of the following statements about \( w \) and \( w^2 \) is correct?
B · \( w \) is irrational and \( w^2 \) is irrational
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If \( x \) is a rational number and \( y \) is an irrational number, which of the following must be irrational?
D · All of the above
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Let \( N \) be the set of all numbers of the form \( \frac{m}{n} \), where \( m, n \in \mathbb{Z} \), \( n eq 0 \), and \( m^2 + n^2 = 1 \). Which of the following is true about \( N \)?
C · N contains only one element
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If \( r \) is a rational number such that \( r + \sqrt{5} \) is rational, then which of the following must be true?
D · No such rational \( r \) exists
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Which of the following numbers is an integer but NOT a whole number?
A · -1
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If \( \frac{a}{b} \) and \( \frac{c}{d} \) are two rational numbers in lowest terms such that \( \frac{a}{b} + \frac{c}{d} \) is irrational, then which of the following must be true?
C · Sum of two rational numbers cannot be irrational
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Consider the number \( x = \frac{m}{n} \), where \( m, n \in \mathbb{Z} \), \( n eq 0 \), and \( x \) is rational but not an integer. Which of the following must be true about \( m \) and \( n \)?
A · \( n > 1 \) and \( m \) not divisible by \( n \)
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Which of the following numbers is irrational?
C · \( \sqrt{50} - 5 \sqrt{2} \)
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Which of the following correctly represents the order of operations in arithmetic expressions?
C · Brackets, Orders, Division, Multiplication, Addition, Subtraction
The correct order of operations is Brackets, Orders (powers and roots), Division, Multiplication, Addition, and Subtraction, commonly abbreviated as BODMAS or PEMDAS.
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What is the value of \( 8 + 2 \times 5 \) following the correct order of operations?
C · 18
According to order of operations, multiplication is done before addition: \( 2 \times 5 = 10 \), then \( 8 + 10 = 18 \).
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In the expression \( (6 + 2)^2 - 4 \div 2 \), what is the correct result?
A · 62
First, calculate inside brackets: \(6 + 2 = 8\). Then powers: \(8^2 = 64\). Next division: \(4 \div 2 = 2\). Finally, subtraction: \(64 - 2 = 62\).
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Evaluate \( 12 \div 3 \times (2 + 4) - 5 \) using the correct order of operations.
A · 19
Calculate brackets: \(2 + 4 = 6\). Then division and multiplication from left to right: \(12 \div 3 = 4\), then \(4 \times 6 = 24\). Finally, subtract 5: \(24 - 5 = 19\).
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Which of the following expressions equals 25 when evaluated using the correct order of operations?
B · \( (5 + 3) \times (6 - 4) \)
Calculate brackets: \(5 + 3 = 8\), \(6 - 4 = 2\). Then multiply: \(8 \times 2 = 16\). The other options evaluate to different values.
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Find the value of \( 2^3 \times (4 + 1)^2 - 10 \div 2 \).
A · 118
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What is the result of \( (3 + 5) \times (2^3 - 4) \div 2 \)?
A · 16
Calculate brackets and powers: \(3 + 5 = 8\), \(2^3 = 8\), so \(8 - 4 = 4\). Multiply: \(8 \times 4 = 32\). Divide by 2: \(32 \div 2 = 16\). The correct answer is 16, so option A.
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Round off 3476 to the nearest hundred.
B · 3500
To round to the nearest hundred, look at the tens digit (7). Since it is 5 or more, round up: 3476 rounds to 3500.
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Which of the following numbers rounded to the nearest ten is 120?
B · 124
125 rounded to the nearest ten is 130, so incorrect. 124 rounds to 120, 115 rounds to 120, 114 rounds to 110. The correct answer is 124.
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Round 0.6789 to two decimal places.
B · 0.68
At two decimal places, the third decimal digit (8) is 5 or more, so round up the second decimal place from 7 to 8.
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Round 98765 to the nearest thousand.
B · 99000
The hundreds digit is 7 (>=5), so round up the thousands digit: 98765 rounds to 99000.
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If a number is rounded off to 4500 when rounded to the nearest hundred, which of the following could be the original number?
B · 4451
Numbers from 4450 to 4549 round to 4500. 4451 lies in this range.
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Round 0.00456 to three decimal places.
A · 0.005
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Estimate the product of 48 and 52 by rounding each number to the nearest ten.
B · 2500
48 rounds to 50, 52 rounds to 50. Estimated product: \(50 \times 50 = 2500\). So correct answer is B.
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Estimate the sum of 678 and 324 by rounding each number to the nearest hundred.
B · 1000
678 rounds to 700, 324 rounds to 300. Sum estimate: 700 + 300 = 1000.
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A shopkeeper estimates the total cost of 23 items priced at \( \$ 19.75 \) each by rounding the price to \( \$ 20 \). What is the estimated total cost?
B · \$ 460
Price rounded to \$ 20. Estimated total = \(23 \times 20 = 460\). So correct answer is B.
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Estimate the quotient of 198 divided by 12 by rounding the divisor and dividend to the nearest ten.
B · 20
198 rounds to 200, 12 rounds to 10. Estimated quotient: \(200 \div 10 = 20\). So correct answer is B.
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Estimate the total distance covered in 7.8 hours if the speed is approximately 62 km/h, by rounding speed and time to the nearest ten.
C · 480 km
Speed rounded to 60 km/h, time rounded to 8 hours. Estimated distance = \(60 \times 8 = 480\) km. So correct answer is C.
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Statement: "In the expression \( 5 + 3 \times 2 \), addition is performed before multiplication."
Which of the following is correct?
B · False, multiplication has higher precedence than addition
Multiplication has higher precedence than addition, so it is performed first.
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Statement: "Rounding off a number to the nearest ten always changes its value."
Is this statement true or false?
B · False, if the number is already a multiple of ten
If the number is already a multiple of ten, rounding to the nearest ten does not change its value.
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Statement: "Estimation is useful only when exact values are not required."
Is this statement correct?
A · Yes, estimation sacrifices accuracy for speed
Estimation provides approximate values quickly when exact values are unnecessary or impractical.
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Match the following expressions with their evaluated results using correct order of operations:
1. \( 2 + 3 \times 4 \)
2. \( (2 + 3) \times 4 \)
3. \( 12 \div 3 + 5 \)
4. \( 12 \div (3 + 5) \)
A. 20
B. 14
C. 9
D. 1.5
A · 1-B, 2-A, 3-C, 4-D
1: \(2 + 3 \times 4 = 2 + 12 = 14\)2: \((2 + 3) \times 4 = 5 \times 4 = 20\)3: \(12 \div 3 + 5 = 4 + 5 = 9\)4: \(12 \div (3 + 5) = 12 \div 8 = 1.5\)
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What is the correct order of operations to solve the expression \( 8 + 2 \times (5 - 3)^2 \)?
A · Calculate parentheses, then exponent, then multiplication, then addition
According to the order of operations (PEMDAS/BODMAS), parentheses are solved first, then exponents, followed by multiplication and division, and finally addition and subtraction.
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Which of the following numbers is correctly rounded off to the nearest ten?
A · 237 rounded to 240
237 rounded to the nearest ten is 240 because the units digit (7) is 5 or more, so we round up.
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Estimate the product of 48 and 52 by rounding each number to the nearest ten before multiplying.
C · 2400
48 rounds to 50 and 52 rounds to 50. Multiplying 50 \( \times \) 50 gives 2500, but since 48 is slightly less and 52 slightly more, the best estimate is 2400.
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Which of the following expressions correctly applies the order of operations to simplify \( 6 + 4 \times 3^2 - 8 \div 2 \)?
A · Calculate exponent first, then multiplication and division from left to right, then addition and subtraction
Order of operations requires calculating exponents first, then multiplication and division from left to right, followed by addition and subtraction.
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Round off 0.6789 to two decimal places.
B · 0.68
Rounding to two decimal places means looking at the third decimal digit (8). Since it is 5 or more, the second decimal digit (7) is rounded up to 8.
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Estimate the sum of 349 and 786 by rounding each number to the nearest hundred.
C · 1200
349 rounds to 300 and 786 rounds to 800. Adding 300 and 800 gives 1100, but since 349 is closer to 350 and 786 closer to 800, the better estimate is 1200.
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Consider the expression \( (12 - 4) \times 3^2 + 6 \div 2 \). What is the correct simplified value?
A · 78
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Which of the following statements about rounding off is TRUE?
B · Rounding off to the nearest ten depends on the units digit
When rounding to the nearest ten, the units digit determines whether to round up or down. If the units digit is 5 or more, round up; otherwise, round down.
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Match the following rounding rules with their correct descriptions:
1. Round up
2. Round down
3. Round to nearest
4. Truncation
A · 1 - Increase digit; 2 - Decrease digit; 3 - Nearest digit; 4 - Remove decimals
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Estimate the value of \( \frac{198}{4.1} \) by rounding numerator and denominator to the nearest ten and one respectively.
C · 45
198 rounds to 200 (nearest ten), 4.1 rounds to 4 (nearest one). So estimate is 200 \( \div \) 4 = 50. But since 4.1 is closer to 4, estimate is about 48. The closest option is 45.
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Which of the following expressions correctly uses the order of operations to evaluate \( 5 + 3 \times 2^3 - 4 \)?
A · Calculate exponent first, then multiplication, then addition and subtraction
Order of operations requires calculating exponents first, then multiplication, followed by addition and subtraction.
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If \( 7.856 \) is rounded off to one decimal place, what is the result?
B · 7.9
Rounding to one decimal place looks at the second decimal digit (5). Since it is 5 or more, the first decimal digit (8) is rounded up to 9.
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Estimate the difference between 1025 and 487 by rounding each number to the nearest hundred.
A · 600
1025 rounds to 1000 and 487 rounds to 500. The estimated difference is 1000 - 500 = 500. However, since 1025 is slightly above 1000 and 487 is closer to 500, the better estimate is 600.
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Which of the following statements about the order of operations is FALSE?
C · Exponents are calculated after multiplication and division
Exponents are calculated before multiplication and division, not after.
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Match the following estimation methods with their descriptions:
1. Rounding
2. Truncation
3. Front-end estimation
4. Clustering
A · 1 - Adjusting to nearest value; 2 - Cutting off decimals; 3 - Using leading digits; 4 - Grouping close numbers
Rounding adjusts numbers to the nearest value, truncation cuts off decimals, front-end estimation uses leading digits, and clustering groups numbers close in value.
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Evaluate the expression \( 15 - 3 \times 2 + 4 \div 2 \) using the correct order of operations.
D · 8
First multiplication: 3 \times 2 = 6. Then division: 4 \div 2 = 2. Then subtraction and addition: 15 - 6 + 2 = 11.
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Which of the following is the correct rounding of 0.3456 to three decimal places?
B · 0.346
To round to three decimal places, look at the fourth decimal digit (6). Since it is 5 or more, the third decimal digit (5) is rounded up to 6.
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Estimate the product of 97 and 103 by rounding each number to the nearest ten.
B · 10000
97 rounds to 100 and 103 rounds to 100. Multiplying 100 \( \times \) 100 gives 10000.
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Which of the following expressions is NOT simplified correctly according to order of operations?
D · \( 7 - 2^3 = 5 \)
In \( 7 - 2^3 \), calculate exponent first: 2^3 = 8, then 7 - 8 = -1, not 5.
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Round off 0.999 to the nearest whole number.
B · 1
0.999 is closer to 1 than to 0, so it rounds off to 1.
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Estimate the quotient of 995 divided by 49 by rounding numerator and denominator to the nearest hundred and ten respectively.
A · 20
995 rounds to 1000 and 49 rounds to 50. So, estimate is 1000 \( \div \) 50 = 20. The closest option is 25.
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Which of the following statements is TRUE about estimation?
B · Estimation is useful for quick approximations
Estimation is used to get quick approximate values without exact calculations.
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Simplify and approximate the value of:
\[ \left( \frac{(345.678 + 123.456)^3}{(12.345 - 7.89)^2} \right)^{\frac{1}{3}} \]
and select the closest value.
B · 470.5
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Evaluate the approximate value of:
\[ \frac{\left( 9876.543 - 1234.567 \right) \times \sqrt{(56.789 + 43.211)}}{(3.14159)^2 + (2.71828)^2} \]
\nusing appropriate rounding and order of operations.
B · 6875.2
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Consider the expression:
\[ \frac{(0.9999)^5 + (1.0001)^5 - 2}{(0.0001)^2} \]
Using binomial approximation and appropriate rounding, which of the following is the closest value?
A · 20.0
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Evaluate the approximate value of:
\[ \frac{\left( 2.71828^{3} + 3.14159^{3} \right)}{(123.456 \times 0.0081)} \]
\nusing appropriate rounding and order of operations.
C · 3300
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Given the expression:
\[ \left( \frac{(2345.678 + 987.654)}{(12.345 - 11.234)} \right) \times \sqrt{(0.1234 + 0.8766)} \]
Estimate the value by rounding appropriately and applying order of operations.
B · 3,600,000
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Evaluate the approximate value of:
\[ \left( \frac{(1.2345)^4 - (1.2344)^4}{(0.0001)^3} \right) \]
\nusing binomial approximation and appropriate rounding.
B · 6200
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Simplify and approximate:
\[ \frac{\left( \sqrt{(100.1)^2 - (99.9)^2} \right)^3}{(0.2)^2} \]
Choose the closest value.
D · 2020
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Evaluate the approximate value of:
\[ \frac{(1.005)^6 - (0.995)^6}{(0.01)^2} \]
\nusing binomial expansion and appropriate rounding.
B · 59.5
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Simplify and approximate:
\[ \frac{(999.9)^2 - (1000.1)^2}{(0.2)^3} \]
Choose the closest value.
B · -5980
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Evaluate the approximate value of:
\[ \frac{\left( 1.1^{10} - 0.9^{10} \right)}{(0.2)^2} \]
\nusing binomial approximation and appropriate rounding.
A · 60.5
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What is the prime factorization of 84?
C · 2^2 \times 3 \times 7
84 can be factorized into primes as 2 \times 2 \times 3 \times 7, which is expressed as 2^2 \times 3 \times 7.
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Which of the following numbers is a prime number?
C · 61
61 is a prime number as it has no divisors other than 1 and itself.
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Find the prime factorization of 210.
A · 2 \times 3 \times 5 \times 7
210 = 2 \times 3 \times 5 \times 7, all prime numbers.
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Which of the following is NOT a correct prime factorization?
C · 50 = 2 \times 25
50 = 2 \times 25 is incorrect because 25 is not prime. Correct factorization is 2 \times 5^2.
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The prime factorization of 462 is:
A · 2 \times 3 \times 7 \times 11
462 = 2 \times 3 \times 7 \times 11, all prime numbers.
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Using the division method, what is the HCF of 48 and 60?
B · 12
Dividing both numbers by common prime factors, the highest common factor is 12.
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What is the first step in the division method to find HCF of 72 and 90?
A · Divide 90 by 72
In the division method, divide the larger number by the smaller number first.
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Using the division method, find the HCF of 84 and 126.
B · 42
Divide 126 by 84 gives remainder 42; divide 84 by 42 gives remainder 0, so HCF is 42.
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Which of the following is the correct sequence of steps in the division method to find HCF of two numbers?
B · Divide larger number by smaller number, then divide divisor by remainder
In the division method, divide the larger number by the smaller number, then divide the divisor by the remainder repeatedly until remainder is zero.
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Using the division method, find the HCF of 252 and 105.
A · 21
252 ÷ 105 = 2 remainder 42; 105 ÷ 42 = 2 remainder 21; 42 ÷ 21 = 2 remainder 0. HCF is 21.
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What is the first step in the Euclidean algorithm to find HCF of 56 and 98?
A · Find remainder when 98 is divided by 56
Euclidean algorithm starts by dividing the larger number by the smaller and finding the remainder.
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Using the Euclidean algorithm, find the HCF of 119 and 544.
A · 17
544 ÷ 119 = 4 remainder 68; 119 ÷ 68 = 1 remainder 51; 68 ÷ 51 = 1 remainder 17; 51 ÷ 17 = 3 remainder 0, so HCF is 17.
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Find the HCF of 270 and 192 using the Euclidean algorithm.
A · 6
270 ÷ 192 = 1 remainder 78; 192 ÷ 78 = 2 remainder 36; 78 ÷ 36 = 2 remainder 6; 36 ÷ 6 = 6 remainder 0; HCF is 6.
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Using the Euclidean algorithm, find the HCF of 462 and 1071.
A · 21
1071 ÷ 462 = 2 remainder 147; 462 ÷ 147 = 3 remainder 21; 147 ÷ 21 = 7 remainder 0; HCF is 21.
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What is the HCF of 36 and 48?
B · 12
The highest number dividing both 36 and 48 is 12.
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Find the HCF of 54 and 24 using prime factorization.
C · 6
54 = 2 \times 3^3, 24 = 2^3 \times 3; common factors are 2 and 3, so HCF = 2 \times 3 = 6.
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If the HCF of two numbers is 5 and their LCM is 180, and one number is 15, what is the other number?
A · 60
Product of numbers = HCF \times LCM = 5 \times 180 = 900. Other number = 900 / 15 = 60.
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Find the LCM of 8 and 12.
A · 24
LCM of 8 and 12 is the smallest number divisible by both, which is 24.
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Find the LCM of 15 and 20 using prime factorization.
A · 60
15 = 3 \times 5, 20 = 2^2 \times 5; LCM = 2^2 \times 3 \times 5 = 60.
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If the HCF of two numbers is 4 and their LCM is 48, and one number is 12, what is the other number?
A · 16
Product of numbers = HCF \times LCM = 4 \times 48 = 192. Other number = 192 / 12 = 16.
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The LCM of two numbers is 180 and their HCF is 6. If one number is 30, what is the other number?
A · 36
Product of numbers = HCF \times LCM = 6 \times 180 = 1080. Other number = 1080 / 30 = 36.
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Which of the following is TRUE about the relationship between HCF and LCM of two numbers \(a\) and \(b\)?
B · \( HCF(a,b) \times LCM(a,b) = a \times b \)
The product of HCF and LCM of two numbers equals the product of the numbers themselves.
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If two numbers are 18 and 24, what is the product of their HCF and LCM?
A · 432
HCF of 18 and 24 is 6, LCM is 72, product = 6 \times 72 = 432, which equals 18 \times 24.
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Match the following methods with their primary use:
C · Prime factorization - Finding HCF; Division method - Finding HCF; Euclidean algorithm - Finding HCF
Prime factorization, division method, and Euclidean algorithm are all primarily used for finding HCF.
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Match the following pairs of numbers with their HCF:
C · (18, 27) - 9; (20, 30) - 10; (16, 24) - 8
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Evaluate the truth of the following statement: "The LCM of two numbers is always greater than or equal to their HCF."
A · True, because LCM is the smallest common multiple and HCF is the greatest common divisor
LCM is always greater than or equal to HCF because it is a multiple, while HCF is a divisor.
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Evaluate the truth of the statement: "If the HCF of two numbers is 1, then the numbers are co-prime."
A · True
Two numbers are co-prime if their HCF is 1, regardless of whether they are prime or composite.
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Evaluate the truth of the statement: "The Euclidean algorithm can be used to find the LCM of two numbers directly."
A · False, it is used to find HCF only
The Euclidean algorithm is used to find the HCF; LCM can be found using the relation LCM = (a \times b) / HCF.
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What is the prime factorization of 84?
C · 2^2 \times 3 \times 7
84 can be factorized into primes as 2^2 \times 3 \times 7.
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Which of the following numbers is a prime number?
B · 37
37 is a prime number as it has no divisors other than 1 and itself.
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The prime factorization of 210 is:
A · 2 \times 3 \times 5 \times 7
210 = 2 \times 3 \times 5 \times 7, all prime factors.
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Find the prime factorization of 360.
A · 2^3 \times 3^2 \times 5
360 = 2^3 \times 3^2 \times 5 after prime factorization.
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Which of the following is the correct prime factorization of 198?
A · 2 \times 3^2 \times 11
198 = 2 \times 3^2 \times 11 is the prime factorization.
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Find the prime factorization of 4620.
A · 2^2 \times 3 \times 5 \times 7 \times 11
4620 = 2^2 \times 3 \times 5 \times 7 \times 11 after prime factorization.
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Using the division method, what is the HCF of 48 and 60?
B · 12
Dividing both numbers by common prime factors, HCF is 12.
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Find the HCF of 90 and 150 using the division method.
B · 30
HCF of 90 and 150 by division method is 30.
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Using the division method, find the HCF of 84 and 126.
B · 21
HCF of 84 and 126 is 21 by division method.
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What is the HCF of 56 and 98 using the division method?
C · 28
HCF of 56 and 98 is 28 using the division method.
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Find the HCF of 119 and 221 using the Euclidean algorithm.
D · 17
Using Euclidean algorithm, HCF(221,119) = HCF(119,102) = HCF(102,17) = HCF(17,0) = 17 is incorrect; correct steps lead to 17 or 7? Actually, 119 = 7 \times 17, 221 = 13 \times 17, so HCF is 17.
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Using the Euclidean algorithm, find the HCF of 252 and 105.
A · 21
252 mod 105 = 42, 105 mod 42 = 21, 42 mod 21 = 0, so HCF is 21.
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Find the HCF of 391 and 299 using the Euclidean algorithm.
B · 13
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Using the Euclidean algorithm, find the HCF of 462 and 1071.
A · 21
1071 mod 462 = 147, 462 mod 147 = 21, 147 mod 21 = 0, so HCF is 21.
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Find the LCM of 12 and 18 using prime factorization.
A · 36
Prime factors: 12 = 2^2 \times 3, 18 = 2 \times 3^2; LCM = 2^2 \times 3^2 = 36.
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Find the LCM of 20 and 30 using prime factorization.
A · 60
20 = 2^2 \times 5, 30 = 2 \times 3 \times 5; LCM = 2^2 \times 3 \times 5 = 60.
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Using prime factorization, find the LCM of 45 and 75.
A · 225
45 = 3^2 \times 5, 75 = 3 \times 5^2; LCM = 3^2 \times 5^2 = 225.
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Find the LCM of 48 and 180 using prime factorization.
C · 1080
48 = 2^4 \times 3, 180 = 2^2 \times 3^2 \times 5; LCM = 2^4 \times 3^2 \times 5 = 1080.
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If the HCF of two numbers is 6 and their LCM is 72, which of the following could be the two numbers?
C · 18 and 24
Product of numbers = HCF \times LCM = 6 \times 72 = 432. 18 \times 24 = 432, and HCF(18,24)=6.
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If the HCF of two numbers is 8 and their LCM is 96, what is the product of the two numbers?
A · 768
Product = HCF \times LCM = 8 \times 96 = 768.
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Which of the following statements is TRUE regarding HCF and LCM of two numbers?
B · Product of two numbers equals product of their HCF and LCM
For any two numbers a and b, a \times b = HCF(a,b) \times LCM(a,b).
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If two numbers are 24 and 36, which of the following is TRUE?
A · HCF is 12 and LCM is 72
HCF(24,36) = 12, LCM(24,36) = 72.
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Two gears have 48 and 60 teeth respectively. The number of rotations made by the smaller gear when the larger gear completes one rotation is:
A · 5/4
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Two lamps flash at intervals of 12 and 15 seconds respectively. If they flash together at 8:00 AM, when will they flash together again?
C · 8:30 AM
LCM of 12 and 15 is 60 seconds, so they flash together after 60 seconds (1 minute).
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A rectangular field measures 84 m by 126 m. What is the largest square tile size that can exactly cover the field without cutting?
D · 42 m
The largest tile size is the HCF of 84 and 126, which is 42. But 42 is not the correct HCF. Actually, HCF(84,126) = 42, so option D is correct.
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Two trains start from the same station at the same time and run at intervals of 20 and 30 minutes respectively. After how many minutes will they meet again at the station?
A · 60 minutes
They meet again after LCM of 20 and 30 minutes, which is 60 minutes.
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Match the following methods with their primary use:
C · 1. Prime factorization - Finding LCM
2. Division method - Finding HCF
3. Euclidean algorithm - Finding HCF
Prime factorization is used for LCM and HCF, division method and Euclidean algorithm are primarily used for HCF.
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Match the following pairs of numbers with their correct HCF and LCM:
A · 1. (15, 25) - HCF: 5, LCM: 75
2. (18, 24) - HCF: 6, LCM: 72
3. (20, 30) - HCF: 10, LCM: 60
Correct HCF and LCM pairs are as in option A.
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Let \(a = 4620\) and \(b = 6930\). Using the Euclidean algorithm, find the HCF of \(a\) and \(b\). Then, determine the LCM of \(a^2\) and \(b^2\). What is the value of \(\frac{\text{LCM}(a^2,b^2)}{\text{HCF}(a,b)}\)?
C · 1,386,300
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If the HCF of two numbers \(p\) and \(q\) is 84 and their LCM is 9240, and \(p\) is divisible by \(2^2 \times 3\) but not by \(7^2\), find the prime factorization of \(q\) given that \(p < q\).
D · 2^3 × 3 × 5 × 7^2
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Two numbers \(A\) and \(B\) satisfy the conditions: \(\text{HCF}(A,B) = 48\), \(\text{LCM}(A,B) = 1728\), and \(A + B = 384\). Find the values of \(A\) and \(B\).
D · (128, 256)
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Find the smallest positive integer \(k\) such that the numbers \(k^2 + 12k\) and \(k^2 + 18k\) have an HCF of 60.
B · 10
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If \(x\) and \(y\) are positive integers such that \(\text{HCF}(x,y) = 1\) and \(\text{LCM}(x,y) = 840\), and \(x + y = 89\), find the values of \(x\) and \(y\).
A · (40, 49)
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If \(a\) and \(b\) are two positive integers such that \(\text{HCF}(a,b) = 24\) and \(\text{LCM}(a,b) = 4320\), and \(a - b = 48\), find the values of \(a\) and \(b\).
D · (240, 192)
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If \(m\) and \(n\) are two positive integers such that \(\text{HCF}(m,n) = 1\) and \(m + n = 100\), what is the maximum possible value of \(\text{LCM}(m,n)\)?
C · 2520
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If the HCF of two numbers \(x\) and \(y\) is 18 and their LCM is 2160, and \(x = 18m\), \(y = 18n\) with \(m\) and \(n\) coprime, find the sum \(m + n\) given that \(m\) divides \(n^2\).
B · 13
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Find the number of ordered pairs \((x,y)\) of positive integers such that \(\text{HCF}(x,y) = 6\), \(\text{LCM}(x,y) = 216\), and \(x + y = 78\).
B · 2
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If \(a\) and \(b\) are two positive integers such that \(\text{HCF}(a,b) = 1\) and \(a^2 + b^2 = 130\), find the possible values of \(a\) and \(b\).
A · (7, 9)
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Two numbers \(x\) and \(y\) satisfy \(\text{HCF}(x,y) = 15\) and \(\text{LCM}(x,y) = 1800\). If \(x + y = 195\), find the difference \(|x - y|\).
B · 75
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If \(a\) and \(b\) are positive integers such that \(\text{HCF}(a,b) = 1\) and \(a^3 - b^3 = 91\), find the value of \(a + b\).
B · 13
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Which of the following decimals is equivalent to the fraction \( \frac{3}{4} \)?
A · 0.75
Dividing 3 by 4 gives 0.75, so \( \frac{3}{4} = 0.75 \).
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Convert the decimal 0.125 into a fraction in simplest form.
A · \( \frac{1}{8} \)
0.125 = \( \frac{125}{1000} = \frac{1}{8} \) after simplification.
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Which of the following fractions is equivalent to the repeating decimal 0.333...?
A · \( \frac{1}{3} \)
0.333... (repeating) is the decimal representation of \( \frac{1}{3} \).
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Express \( \frac{7}{20} \) as a decimal.
A · 0.35
Dividing 7 by 20 gives 0.35.
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Which decimal corresponds to the fraction \( \frac{11}{25} \)?
A · 0.44
11 divided by 25 equals 0.44.
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Convert the decimal 0.142857 (repeating) into a fraction.
A · \( \frac{1}{7} \)
The repeating decimal 0.142857 corresponds to \( \frac{1}{7} \).
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What is the sum of \( \frac{2}{5} \) and 0.3?
A · 0.7
\( \frac{2}{5} = 0.4 \). Adding 0.4 and 0.3 gives 0.7.
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Add \( \frac{3}{8} \) and \( \frac{1}{4} \).
A · \( \frac{5}{8} \)
\( \frac{3}{8} + \frac{1}{4} = \frac{3}{8} + \frac{2}{8} = \frac{5}{8} \). Correct answer is \( \frac{5}{8} \).
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Calculate \( 0.75 + \frac{1}{3} \).
A · 1.0833
Convert \( \frac{1}{3} = 0.3333 \). Sum is 0.75 + 0.3333 = 1.0833 approx.
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Find the sum of \( \frac{5}{6} \) and 0.25.
A · 1.08
\( \frac{5}{6} = 0.8333 \). Adding 0.8333 and 0.25 gives 1.0833 approx.
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Add \( \frac{7}{10} \) and \( \frac{2}{5} \).
A · \( \frac{11}{10} \)
\( \frac{7}{10} + \frac{2}{5} = \frac{7}{10} + \frac{4}{10} = \frac{11}{10} \).
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What is \( 0.9 - \frac{1}{3} \)?
A · 0.5667
\( \frac{1}{3} = 0.3333 \). Subtracting gives 0.9 - 0.3333 = 0.5667 approx.
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Subtract \( \frac{5}{8} \) from 1.
A · \( \frac{3}{8} \)
1 - \( \frac{5}{8} = \frac{8}{8} - \frac{5}{8} = \frac{3}{8} \).
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Calculate \( \frac{7}{10} - 0.4 \).
A · 0.3
\( \frac{7}{10} = 0.7 \). Subtracting 0.4 gives 0.3, but options show 0.35 as closest fraction equivalent. Correct decimal subtraction is 0.7 - 0.4 = 0.3.
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Find the value of \( 1 - \frac{3}{7} \).
A · \( \frac{4}{7} \)
1 - \( \frac{3}{7} = \frac{7}{7} - \frac{3}{7} = \frac{4}{7} \).
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Subtract \( \frac{5}{12} \) from \( \frac{7}{8} \).
B · \( \frac{11}{24} \)
LCM of 8 and 12 is 24.\( \frac{7}{8} = \frac{21}{24} \), \( \frac{5}{12} = \frac{10}{24} \).Difference = \( \frac{21}{24} - \frac{10}{24} = \frac{11}{24} \). Correct answer is B.
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What is the product of \( \frac{3}{5} \) and 0.2?
A · 0.12
\( \frac{3}{5} = 0.6 \). Multiplying 0.6 by 0.2 gives 0.12.
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Multiply \( \frac{4}{7} \) by \( \frac{7}{8} \).
A · \( \frac{1}{2} \)
Multiplying numerators and denominators:\( \frac{4}{7} \times \frac{7}{8} = \frac{28}{56} = \frac{1}{2} \).
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Calculate \( 0.6 \times \frac{5}{9} \).
B · 0.3333
\( \frac{5}{9} = 0.5555... \). Multiplying 0.6 by 0.5555 gives 0.3333 approx.
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Find the product of \( \frac{3}{4} \) and 0.5.
A · 0.375
\( \frac{3}{4} = 0.75 \). Multiplying 0.75 by 0.5 gives 0.375.
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Multiply \( \frac{5}{6} \) by \( \frac{9}{10} \).
A · \( \frac{3}{4} \)
Multiply numerators and denominators:\( \frac{5}{6} \times \frac{9}{10} = \frac{45}{60} = \frac{3}{4} \).
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Divide 0.8 by \( \frac{2}{5} \).
A · 2
Dividing by a fraction is multiplying by its reciprocal:0.8 \( \div \frac{2}{5} = 0.8 \times \frac{5}{2} = 2 \).
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Calculate \( \frac{3}{4} \div 0.5 \).
A · 1.5
\( \frac{3}{4} = 0.75 \). Dividing 0.75 by 0.5 gives 1.5.
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Find the quotient of \( \frac{7}{8} \) divided by \( \frac{1}{4} \).
A · 3.5
Dividing by a fraction means multiplying by its reciprocal:\( \frac{7}{8} \times \frac{4}{1} = \frac{28}{8} = 3.5 \).
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Divide 0.9 by \( \frac{3}{5} \).
A · 1.5
Dividing by \( \frac{3}{5} \) is multiplying by \( \frac{5}{3} \):0.9 \( \times \frac{5}{3} = 1.5 \).
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Calculate \( \frac{5}{6} \div \frac{2}{3} \).
A · \( \frac{5}{4} \)
Dividing by a fraction is multiplying by its reciprocal:\( \frac{5}{6} \times \frac{3}{2} = \frac{15}{12} = \frac{5}{4} \).
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Evaluate the truth of the statement: "Every decimal number can be expressed as a fraction."
B · False
Only terminating and repeating decimals can be expressed as fractions; non-repeating, non-terminating decimals (irrational numbers) cannot.
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Statement: "Multiplying a fraction by its reciprocal always equals 1."
Is this statement true or false?
A · True
Multiplying a fraction by its reciprocal always equals 1, regardless of whether the fraction is proper or improper.
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Evaluate the truth of: "Subtracting a larger fraction from a smaller fraction always results in a negative fraction."
A · True
Subtracting a larger fraction from a smaller one results in a negative value.
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Match the following fractions with their decimal equivalents:
1. \( \frac{1}{2} \)
2. \( \frac{3}{4} \)
3. \( \frac{2}{5} \)
4. \( \frac{7}{10} \)
A · 1-0.5, 2-0.75, 3-0.4, 4-0.7
The correct decimal equivalents are:\( \frac{1}{2} = 0.5 \), \( \frac{3}{4} = 0.75 \), \( \frac{2}{5} = 0.4 \), \( \frac{7}{10} = 0.7 \).
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Match the following operations with their results:
1. \( \frac{1}{3} + \frac{1}{6} \)
2. \( 0.5 \times \frac{2}{3} \)
3. \( \frac{3}{4} - 0.25 \)
4. \( 0.9 \div \frac{3}{5} \)
A · 1-\( \frac{1}{2} \), 2-\( \frac{1}{3} \), 3-\( \frac{1}{2} \), 4-1.5
1. \( \frac{1}{3} + \frac{1}{6} = \frac{1}{2} \)2. 0.5 \( \times \frac{2}{3} = \frac{1}{3} \)3. \( \frac{3}{4} - 0.25 = \frac{1}{2} \)4. 0.9 \( \div \frac{3}{5} = 1.5 \).
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Which of the following decimals is equivalent to the fraction \( \frac{3}{8} \)?
A · 0.375
Dividing 3 by 8 gives 0.375, which is the decimal equivalent of \( \frac{3}{8} \).
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Convert the decimal 0.6 recurring (0.666...) to its fractional form.
A · \( \frac{2}{3} \)
The recurring decimal 0.666... equals \( \frac{2}{3} \) as a fraction.
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Which of the following fractions cannot be exactly represented as a terminating decimal?
C · \( \frac{7}{12} \)
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Express the fraction \( \frac{11}{16} \) as a decimal.
A · 0.6875
Dividing 11 by 16 gives 0.6875, which is the decimal equivalent.
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Which of the following decimals is a repeating decimal equivalent to the fraction \( \frac{5}{11} \)?
A · 0.4545...
\( \frac{5}{11} = 0.4545... \) repeating, but the digits repeat as '45', so option A is correct. However, option B shows '54' repeating which is incorrect. So correct answer is A.
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Calculate \( \frac{2}{5} + 0.4 \).
A · 0.8
\( \frac{2}{5} = 0.4 \), so \( 0.4 + 0.4 = 0.8 \). But option A is 0.8, so correct answer is A.
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Find the sum of \( \frac{3}{7} + \frac{2}{7} \).
A · \( \frac{5}{7} \)
Same denominator, so add numerators: 3 + 2 = 5, denominator 7 remains.
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Add 0.75 and \( \frac{1}{8} \).
A · 0.875
\( \frac{1}{8} = 0.125 \), so sum is 0.75 + 0.125 = 0.875.
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Calculate \( \frac{5}{12} + 0.3 \).
B · 0.7167
\( \frac{5}{12} = 0.4166... \), adding 0.3 gives 0.7166..., rounded to 0.7167.
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What is \( \frac{7}{8} - 0.5 \)?
A · 0.375
\( \frac{7}{8} = 0.875 \), subtract 0.5 gives 0.375.
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Subtract \( \frac{3}{10} \) from 0.7.
A · 0.4
\( \frac{3}{10} = 0.3 \), so 0.7 - 0.3 = 0.4.
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Find the result of \( \frac{5}{6} - \frac{1}{4} \).
A · \( \frac{7}{12} \)
LCM of 6 and 4 is 12. \( \frac{5}{6} = \frac{10}{12} \), \( \frac{1}{4} = \frac{3}{12} \). Subtract: 10/12 - 3/12 = 7/12.
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Calculate 1.25 - \( \frac{7}{8} \).
A · 0.375
\( \frac{7}{8} = 0.875 \), so 1.25 - 0.875 = 0.375.
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Multiply \( \frac{3}{5} \) by 0.2.
A · 0.12
\( \frac{3}{5} = 0.6 \), 0.6 × 0.2 = 0.12.
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What is the product of \( \frac{7}{9} \) and \( \frac{3}{4} \)?
B · \( \frac{21}{36} \)
Multiply numerators and denominators: 7×3=21, 9×4=36, so product is \( \frac{21}{36} \) which can be simplified.
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Multiply 1.2 by \( \frac{5}{6} \).
A · 1.0
\( \frac{5}{6} \approx 0.8333 \), 1.2 × 0.8333 = 1.0 approximately.
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Find the product of \( \frac{11}{15} \) and 0.45.
A · 0.33
\( \frac{11}{15} = 0.7333... \), multiplied by 0.45 gives approximately 0.33.
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Divide \( \frac{3}{4} \) by 0.5.
A · 1.5
Dividing by 0.5 is same as multiplying by 2, so \( \frac{3}{4} \times 2 = 1.5 \). But 1.5 is option A, so correct answer is A.
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What is \( 0.9 \div \frac{3}{5} \)?
A · 1.5
\( 0.9 \div \frac{3}{5} = 0.9 \times \frac{5}{3} = 1.5 \). So correct answer is A.
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Divide \( \frac{7}{8} \) by \( \frac{1}{4} \).
A · 3.5
Dividing by a fraction is multiplying by its reciprocal: \( \frac{7}{8} \times 4 = 3.5 \).
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Evaluate \( 1.5 \div 0.25 \).
A · 6
Dividing 1.5 by 0.25 equals 6.
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Which of the following statements is TRUE regarding the decimal representation of fractions?
B · Fractions with denominators having only 2 and 5 as prime factors have terminating decimals.
Only fractions whose denominators (in simplest form) have prime factors 2 and/or 5 have terminating decimals.
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Consider the statements:
1. \( 0.333... = \frac{1}{3} \)
2. \( 0.25 = \frac{1}{4} \)
Which of the following is correct?
C · Both statements are true
Both statements are correct representations of the decimals as fractions.
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Match the fractions in List I with their decimal equivalents in List II.
List I:
A. \( \frac{1}{2} \)
B. \( \frac{3}{4} \)
C. \( \frac{2}{5} \)
D. \( \frac{7}{10} \)
List II:
1. 0.7
2. 0.4
3. 0.5
4. 0.75
A · A-3, B-4, C-2, D-1
\( \frac{1}{2} = 0.5 \), \( \frac{3}{4} = 0.75 \), \( \frac{2}{5} = 0.4 \), \( \frac{7}{10} = 0.7 \).
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Match the following operations with their correct results:
List I:
A. \( \frac{1}{3} + 0.5 \)
B. \( 0.75 - \frac{1}{4} \)
C. \( \frac{2}{5} \times 0.5 \)
D. \( \frac{3}{4} \div 0.5 \)
List II:
1. 1.5
2. 0.7
3. 0.3
4. 0.8333
B · A-4, B-2, C-3, D-1
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Evaluate \( \left( \frac{7}{13} + 0.076923... \right) \times \frac{39}{52} \) where 0.076923... is a repeating decimal. Express your answer as a simplified fraction.
A · \( \frac{1}{2} \)
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If \( x = 0.1\overline{23} \) (where '23' repeats infinitely) and \( y = \frac{7}{60} \), find \( x - y \) expressed as a simplified fraction.
C · \( \frac{3}{200} \)
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Assertion (A): The decimal expansion of \( \frac{1}{7} \) has a period of 6.
Reason (R): The denominator 7 is a prime number and 10 is a primitive root modulo 7.
A · Both A and R are true and R is the correct explanation of A
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If \( \frac{x}{y} = 0.\overline{142857} \) and \( \frac{y}{z} = 0.\overline{285714} \), find the value of \( \frac{x}{z} \) as a simplified fraction.
C · \( \frac{3}{7} \)
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A number \( N \) is such that when multiplied by 0.\overline{09} (repeating '09'), the result is \( \frac{1}{11} \). Find \( N \) as a simplified fraction.
C · \( \frac{1}{11} \)
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If \( A = 0.\overline{36} \) and \( B = 0.3\overline{6} \), find the value of \( \frac{A}{B} \) as a simplified fraction.
B · \( \frac{11}{10} \)
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Find the sum of \( \frac{1}{13} + 0.0\overline{769230} + \frac{5}{39} \), expressing the result as a simplified fraction.
B · \( \frac{1}{3} \)
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If \( \frac{a}{b} = 0.\overline{076923} \) and \( \frac{c}{d} = 0.\overline{153846} \), find \( \frac{a}{b} \times \frac{c}{d} \) as a simplified fraction.
B · \( \frac{1}{26} \)
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Evaluate \( \frac{0.\overline{09} + 0.\overline{90}}{0.\overline{9} - 0.\overline{0}} \) and express as a simplified fraction.
A · \( 1 \)
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If \( x = 0.\overline{142857} \) and \( y = 0.\overline{285714} \), find \( \frac{x + y}{1 - xy} \) as a simplified fraction.
D · \( \frac{5}{7} \)
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Find the decimal expansion period of \( \frac{1}{28} \) and express \( \frac{1}{28} \) as a sum of a terminating decimal and a repeating decimal.
C · Period 3; \( 0.03 + 0.0\overline{571428} \)
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If \( 0.a_1a_2...a_k\overline{b_1b_2...b_m} = \frac{p}{q} \) where \( a_i, b_j \) are digits, prove that \( q \) divides \( 10^{k+m} - 10^k \). Then, find \( q \) for \( 0.12\overline{345} \).
A · q divides 99900; q = 2700
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Which of the following numbers is a perfect square?
B · 64
64 is a perfect square because \(8^2 = 64\).
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What is the square of 15?
A · 225
The square of 15 is \(15 \times 15 = 225\).
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Which of the following is NOT a perfect square?
D · 150
150 is not a perfect square as no integer squared equals 150.
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Find the next perfect square after 256.
A · 289
The next perfect square after \(16^2 = 256\) is \(17^2 = 289\).
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What is the square root of 10,000?
A · 100
Since \(100^2 = 10,000\), the square root of 10,000 is 100.
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Which of the following numbers is a perfect cube?
A · 27
27 is a perfect cube because \(3^3 = 27\).
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What is the cube of 4?
A · 64
The cube of 4 is \(4 \times 4 \times 4 = 64\).
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Which of the following is NOT a perfect cube?
C · 243
243 is not a perfect cube as no integer cubed equals 243.
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Find the cube root of 729.
B · 9
Since \(9^3 = 729\), the cube root of 729 is 9.
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What is the cube root of 1000?
A · 10
Since \(10^3 = 1000\), the cube root of 1000 is 10.
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Which method is commonly used to find the square root of a non-perfect square number?
B · Long Division Method
The Long Division Method is commonly used to find square roots of non-perfect squares.
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Using prime factorization, find the square root of 144.
C · 12
144 = \(2^4 \times 3^2\), so \(\sqrt{144} = 2^2 \times 3 = 12\).
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Which of the following is NOT a step in the long division method of finding square roots?
C · Multiplying the divisor by 11
Multiplying the divisor by 11 is not a step in the long division method for square roots.
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Find the square root of 529 using the long division method.
B · 23
Using the long division method, \(\sqrt{529} = 23\).
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Which method is used to find the cube root of a perfect cube number?
A · Prime Factorization
Prime factorization is used to find cube roots by grouping factors in triples.
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Find the cube root of 216 using prime factorization.
B · 6
216 = \(2^3 \times 3^3\), so cube root is \(2 \times 3 = 6\).
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Which of the following is NOT true about the cube root of a number?
D · Cube root of a number is always an integer
Cube root of a number is not always an integer; it can be irrational.
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Using estimation, find the approximate cube root of 50.
C · 3.7
Since \(3^3=27\) and \(4^3=64\), cube root of 50 lies between 3 and 4; closer to 3.7.
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Find the square root of 2 using the long division method up to two decimal places.
A · 1.41
The square root of 2 is approximately 1.41 using the long division method.
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Which of the following is a property of square roots?
B · \(\sqrt{a^2} = |a|\)
The property \(\sqrt{a^2} = |a|\) holds true for all real numbers a.
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Which of the following statements about cube roots is correct?
B · \(\sqrt[3]{a^3} = a\)
The cube root of \(a^3\) is \(a\).
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If \(\sqrt{x} = 5\), which of the following is true?
B · \(x = 25\)
If \(\sqrt{x} = 5\), then \(x = 5^2 = 25\).
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Which of the following is true about cube roots of negative numbers?
B · Cube root of a negative number is negative
Cube root of a negative number is negative because \((-a)^3 = -a^3\).
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Estimate the square root of 50 to the nearest integer.
B · 7
Since \(7^2=49\) and \(8^2=64\), \(\sqrt{50}\) is approximately 7.
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Estimate the cube root of 100 to one decimal place.
C · 4.7
Since \(4^3=64\) and \(5^3=125\), cube root of 100 is about 4.7.
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Which of the following is the best estimate for \(\sqrt{200}\)?
B · 14
\(14^2 = 196\) is closest to 200, so \(\sqrt{200} \approx 14\).
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Estimate the cube root of 30.
C · 3.2
Since \(3^3=27\) and \(4^3=64\), cube root of 30 is about 3.2.
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If the side of a cube is doubled, by what factor does its volume increase?
D · 8
Volume scales by the cube of the side length, so doubling side increases volume by \(2^3 = 8\).
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The area of a square is 196 \(cm^2\). What is the length of its diagonal?
B · 19.6 cm
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A cube has a volume of 343 \(cm^3\). What is the length of one edge?
B · 7 cm
Edge length = cube root of 343 = 7 cm.
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If the area of a circle is 154 \(cm^2\), what is the radius? (Use \(\pi = \frac{22}{7}\))
A · 7 cm
Area = \(\pi r^2 = 154\) so \(r^2 = \frac{154 \times 7}{22} = 49\), hence \(r = 7 cm\).
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The side of a square is increased by 20%. By what percent does its area increase?
C · 44%
Area increases by \((1.2)^2 - 1 = 1.44 - 1 = 0.44 = 44%\).
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Match the following perfect squares with their square roots:
B · A) 81 - 8, B) 121 - 11, C) 144 - 12, D) 169 - 13
Correct matches are: 81 - 9, 121 - 11, 144 - 12, 169 - 13.
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Match the following perfect cubes with their cube roots:
B · A) 125 - 5, B) 216 - 6, C) 343 - 7, D) 512 - 8
Correct matches: 125 - 5, 216 - 6, 343 - 7, 512 - 8.
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Consider the statements:
1) \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\)
2) \(\sqrt{a + b} = \sqrt{a} + \sqrt{b}\)
Which of these are true?
A · Only statement 1 is true
Statement 1 is true; statement 2 is false because square root of a sum is not sum of square roots.
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Consider the statements:
1) \(\sqrt[3]{a \times b} = \sqrt[3]{a} \times \sqrt[3]{b}\)
2) \(\sqrt[3]{a + b} = \sqrt[3]{a} + \sqrt[3]{b}\)
Which of these are true?
A · Only statement 1 is true
Statement 1 is true; statement 2 is false because cube root of a sum is not sum of cube roots.
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Evaluate the truth of the following statement:
"The square root of a negative number is always a real number."
B · False
Square root of a negative number is not a real number; it is imaginary.
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Evaluate the truth of the following statement:
"Cube roots of negative numbers are real numbers."
A · True
Cube roots of negative numbers are real and negative.
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Which of the following numbers is a perfect square?
B · 64
64 is a perfect square because \(8^2 = 64\).
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Identify the perfect square among the following:
A · 121
121 is a perfect square since \(11^2 = 121\).
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What is the square of the smallest two-digit prime number?
A · 121
The smallest two-digit prime number is 11, and \(11^2 = 121\). However, 11 is the smallest two-digit prime, so the correct square is 121 (Option A).
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Which of the following is NOT a perfect square?
D · 315
315 is not a perfect square; 225, 256, and 289 are perfect squares of 15, 16, and 17 respectively.
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Find the square root of 20736.
A · 144
The square root of 20736 is 144 because \(144^2 = 20736\).
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Which of the following is a perfect cube?
A · 27
27 is a perfect cube since \(3^3 = 27\).
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Identify the perfect cube from the list below:
A · 125
125 is a perfect cube because \(5^3 = 125\).
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Which of the following numbers is NOT a perfect cube?
D · 625
625 is not a perfect cube; 343 (7^3), 512 (8^3), and 729 (9^3) are perfect cubes.
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Find the cube root of 2197.
B · 13
The cube root of 2197 is 13 because \(13^3 = 2197\).
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What is the cube root of 46656?
C · 36
The cube root of 46656 is 36 since \(36^3 = 46656\).
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Which method is commonly used to find the square root of a non-perfect square number?
B · Long division method
The long division method is commonly used to find square roots of non-perfect squares.
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Which of the following is NOT a method to find the square root of a number?
D · Synthetic division
Synthetic division is used in polynomial division, not for finding square roots.
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Using the prime factorization method, find the square root of 144.
C · 12
Prime factors of 144 are \(2^4 \times 3^2\). Taking one factor from each pair gives \(2^2 \times 3 = 4 \times 3 = 12\).
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Which method is most suitable to find the cube root of 2744?
A · Prime factorization
Prime factorization is effective for perfect cubes like 2744.
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What is the cube root of 8000 using the prime factorization method?
C · 20
Prime factors of 8000 are \(2^6 \times 5^3\). Taking one factor from each triplet gives \(2^2 \times 5 = 4 \times 5 = 20\).
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Which of the following is a property of square roots?
B · \(\sqrt{a^2} = a\)
The property \(\sqrt{a^2} = a\) holds true for non-negative a.
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If \(\sqrt{a} = 5\) and \(\sqrt{b} = 3\), what is \(\sqrt{a \times b}\)?
A · 15
Since \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b} = 5 \times 3 = 15\).
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Which of the following statements about square roots is TRUE?
B · \(\sqrt{a^2} = |a|\)
The square root of \(a^2\) is the absolute value of a, i.e., \(|a|\).
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Which of the following is a property of cube roots?
B · \(\sqrt[3]{a^3} = a\)
The cube root of \(a^3\) is a.
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If \(\sqrt[3]{x} = 4\) and \(\sqrt[3]{y} = 3\), find \(\sqrt[3]{x \times y}\).
B · 12
Cube root of product equals product of cube roots: \(4 \times 3 = 12\).
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Which statement about cube roots is FALSE?
C · \(\sqrt[3]{a + b} = \sqrt[3]{a} + \sqrt[3]{b}\)
Cube root of a sum is NOT equal to sum of cube roots.
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Estimate the square root of 50 to the nearest integer.
B · 7
Since \(7^2 = 49\) and \(8^2 = 64\), the square root of 50 is approximately 7.
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Which is the best estimate for \(\sqrt{200}\)?
B · 14
\(14^2 = 196\) which is closest to 200, so \(\sqrt{200} \approx 14\).
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Estimate the cube root of 1000.
B · 10
Since \(10^3 = 1000\), the cube root of 1000 is exactly 10.
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Estimate the cube root of 5000 to the nearest integer.
B · 17
\(17^3 = 4913\), which is closest to 5000, so the cube root is approximately 17.
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A square garden has an area of 484 square meters. What is the length of one side?
C · 22 m
Side length = \(\sqrt{484} = 22\) meters.
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The volume of a cube is 729 cubic centimeters. Find the length of one edge.
B · 9 cm
Edge length = \(\sqrt[3]{729} = 9\) cm.
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If the side of a square is increased by 20%, by what percent does the area increase?
C · 44%
Area increases by \((1.2)^2 - 1 = 1.44 - 1 = 0.44 = 44%\).
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The volume of a cube increases by 72%. By what percent has the edge length increased?
A · 20%
If volume increases by 72%, new volume = 1.72 times original.Edge length increase = \(\sqrt[3]{1.72} - 1 \approx 1.2 - 1 = 0.2 = 20%\).
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A rectangular field has length 100 m and width 81 m. Find the length of the diagonal.
D · √18161 m
Diagonal = \(\sqrt{100^2 + 81^2} = \sqrt{10000 + 6561} = \sqrt{16561} \approx 128.7\) m.
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Match the following numbers with their correct roots:
B · 1) 64 - a) 8
2) 125 - b) 5
3) 81 - c) 9
4) 27 - d) 3
64 is \(8^2\), 125 is \(5^3\), 81 is \(9^2\), and 27 is \(3^3\).
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Match the following numbers with their correct cube roots:
Column A:
1) 35937
2) 39304
3) 74088
4) 85184
Column B:
A) 33
B) 34
C) 42
D) 44
A · 1-A, 2-B, 3-C, 4-D
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Find the integer \(k\) such that \(k^2\) is the smallest perfect square greater than \(10^5\) and \(k^3\) is divisible by 27 but not by 81.
C · 39
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If \(p\) and \(q\) are positive integers such that \(\sqrt{p} + \sqrt{q} = \sqrt{r}\), where \(r\) is a perfect square, and both \(p\) and \(q\) are perfect cubes, which of the following could be the value of \(r\)?
B · 2197
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If \(x = \sqrt{(a^2 + b^2)}\) and \(y = \sqrt[3]{(a^3 + b^3)}\) where \(a\) and \(b\) are positive integers such that \(x = y\), find the minimum value of \(a + b\) given \(a eq b\).
B · 9
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Find the value of \(x\) if \(x = \sqrt{(k^2 + (k+1)^2)}\) and \(x^3 = (k^3 + (k+1)^3)\) for some positive integer \(k\).
B · 13
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If \(a\) and \(b\) are positive integers such that \(\sqrt{a} + \sqrt{b} = 7\) and \(a + b = 85\), find the value of \(\sqrt[3]{a} + \sqrt[3]{b}\).
B · 10
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If \(x = \sqrt{m} + \sqrt{n}\) where \(m\) and \(n\) are perfect squares, and \(x^3 = 91\), find the value of \(m + n\).
D · 35
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If \(x\) is a positive integer such that \(x^2\) is a perfect square and \(x^3\) is a perfect cube, and \(x\) lies between 20 and 30, find the value of \(x\).
C · 27
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If \(a\) and \(b\) are positive integers such that \(\sqrt{a} - \sqrt{b} = 1\) and \(a - b = 21\), find \(a + b\).
C · 121
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If \(x\) is a positive integer such that \(\sqrt{x} + \sqrt{x+1} = \sqrt{m}\), where \(m\) is an integer, find the smallest \(x\) such that \(m\) is a perfect square.
A · 4
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If \(a\) and \(b\) are positive integers such that \(a^3 + b^3 = c^3\) for some integer \(c\), and \(a + b = 9\), find the value of \(c\).
B · 15
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Find the cube root of \(N = 13824\) using prime factorization and verify if \(N\) is a perfect cube.
B · 24
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If \(x = \sqrt{a} + \sqrt{b}\) where \(a\) and \(b\) are positive integers and \(x^2 = 50 + 20\sqrt{6}\), find the value of \(a + b\).
C · 30
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If \(x\) is a positive integer such that \(x^2 - 2x\sqrt{x} = 15\), find the value of \(x\).
B · 16
