CBSE Class 8 Mathematics Number System MCQs Set 39

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Multiple Choice Questions

Question 1: On multiplying the numerator and denominator by 2, the rational number \( -\frac{3}{8} \) takes the form:
(a) \( \frac{-6}{16} \)
(b) \( \frac{-3}{16} \)
(c) \( \frac{-6}{8} \)
(d) \( \frac{-5}{10} \)
Show Answer & Explanation

Answer: (a) \( \frac{-6}{16} \)

Explanation:
1. Multiply both -3 and 8 by 2.
2. -3/8 = (-3 × 2)/(8 × 2) = -6/16.

Teacher's Note:
1. Multiplying numerator and denominator by the same number gives an equivalent fraction.
2. Never multiply only one of them.

Question 2: If \( \frac{-5}{-8}=\frac{\square}{16}=\frac{25}{\square} \), the missing numbers are (in order):
(a) 8 and 40
(b) 5 and 25
(c) 10 and 20
(d) 10 and 40
Show Answer & Explanation

Answer: (d) 10 and 40

Explanation:
1. -5/-8 = 5/8.
2. 5/8 = 10/16 (multiply by 2) and 5/8 = 25/40 (multiply by 5).
3. The missing numbers are 10 and 40.

Teacher's Note:
1. Find the multiplier from the known part of each equivalent fraction.
2. Check: 10/16 = 5/8 and 25/40 = 5/8.

Question 3: Evaluate \( \left(\sqrt{\frac{225}{729}}-\sqrt{\frac{25}{144}}\right)\div\sqrt{\frac{16}{81}} \).
(a) \( \frac{5}{16} \)
(b) \( \frac{5}{9} \)
(c) \( \frac{16}{5} \)
(d) \( \frac{5}{36} \)
Show Answer & Explanation

Answer: (a) \( \frac{5}{16} \)

Explanation:
1. Square roots: 15/27 = 5/9, 5/12 and 4/9.
2. 5/9 - 5/12 = 20/36 - 15/36 = 5/36.
3. (5/36) ÷ (4/9) = (5/36) × (9/4) = 5/16.

Teacher's Note:
1. Take the roots of the numerator and denominator separately.
2. Dividing by 4/9 is multiplying by 9/4.

Question 4: The value of \( \left(5\left(8^{1/3}+27^{1/3}\right)\right)^{1/4} \) is:
(a) \( 5^{1/4} \)
(b) 5
(c) \( \sqrt{5} \) (about 2.236)
(d) 25
Show Answer & Explanation

Answer: (c) \( \sqrt{5} \) (about 2.236)

Explanation:
1. 8^(1/3) = 2 and 27^(1/3) = 3, so the bracket is 5(2 + 3) = 25.
2. 25^(1/4) = (5²)^(1/4) = 5^(1/2) = √5 ≈ 2.236.

Teacher's Note:
1. Evaluate the cube roots first.
2. 25^(1/4) is the square root of 25^(1/2) = 5.

Question 5: Which of the following is the greatest?
(a) All are equal
(b) \( 12^{9} \)
(c) \( 10^{11} \)
(d) \( 11^{10} \)
Show Answer & Explanation

Answer: (c) \( 10^{11} \)

Explanation:
1. 12⁹ ≈ 5.16 × 10⁹.
2. 11¹⁰ ≈ 2.59 × 10¹⁰.
3. 10¹¹ = 1 × 10¹¹, which is the greatest.

Teacher's Note:
1. Compare approximate sizes with scientific notation.
2. Higher powers of 10 dominate here.

Question 6: The value of \( \frac{1}{2^2-1}+\frac{1}{4^2-1}+\frac{1}{6^2-1}+\cdots+\frac{1}{20^2-1} \) is:
(a) \( \frac{10}{19} \)
(b) \( \frac{9}{19} \)
(c) \( \frac{10}{21} \)
(d) \( \frac{11}{21} \)
Show Answer & Explanation

Answer: (c) \( \frac{10}{21} \)

Explanation:
1. 1/((2k)² - 1) = 1/((2k - 1)(2k + 1)) = (1/2)(1/(2k - 1) - 1/(2k + 1)).
2. The sum telescopes: (1/2)(1 - 1/21) for k = 1 to 10.
3. = (1/2)(20/21) = 10/21.

Teacher's Note:
1. Split each term using partial fractions.
2. Intermediate terms cancel.

Question 7: If \( \sqrt{1+\frac{x}{169}}=1\frac{1}{13} \), then x equals:
(a) 13
(b) 27
(c) 1
(d) 15
Show Answer & Explanation

Answer: (b) 27

Explanation:
1. 1 1/13 = 14/13, so 1 + x/169 = 196/169.
2. x/169 = 27/169, so x = 27.

Teacher's Note:
1. Square both sides.
2. Check: 1 + 27/169 = 196/169 and its root is 14/13.

Question 8: A positive number a satisfies \( a^{1/x}=b^{1/y}=c^{1/z} \) and \( b^2=ac \). The value of \( \frac{x+z}{2y} \) is:
(a) 1/2
(b) 2
(c) 0
(d) 1
Show Answer & Explanation

Answer: (d) 1

Explanation:
1. Let each expression equal k: a = k^x, b = k^y, c = k^z.
2. b² = ac gives k^(2y) = k^(x+z), so 2y = x + z.
3. Hence (x + z)/(2y) = 1.

Teacher's Note:
1. Express a, b, c as powers of the same number.
2. Equal powers of the same base have equal exponents.

Question 9: Find x if \( \frac{2^{x-1}\cdot 4^{2x+1}}{8^{x-1}}=64 \).
(a) 3
(b) 0
(c) 2
(d) 1
Show Answer & Explanation

Answer: (d) 1

Explanation:
1. Write in base 2: 2^(x-1) × 2^(4x+2) ÷ 2^(3x-3) = 2^(x-1+4x+2-3x+3).
2. The exponent is 2x + 4, and 64 = 2⁶, so 2x + 4 = 6.
3. x = 1.

Teacher's Note:
1. Convert 4 and 8 to powers of 2.
2. Add exponents when multiplying and subtract when dividing.

Question 10: The value of \( \sqrt{10+\sqrt{25+\sqrt{108+\sqrt{154+\sqrt{225}}}}} \) is:
(a) 3
(b) 5
(c) 6
(d) 4
Show Answer & Explanation

Answer: (d) 4

Explanation:
1. √225 = 15; √(154 + 15) = √169 = 13.
2. √(108 + 13) = √121 = 11; √(25 + 11) = √36 = 6.
3. √(10 + 6) = √16 = 4.

Teacher's Note:
1. Work from the innermost root outwards.
2. Each stage gives a perfect square.

Question 11: If \( x=\sqrt{3+2\sqrt{2}} \), the value of \( x^4+\frac{1}{x^4} \) is:
(a) 6
(b) 18
(c) 32
(d) 34
Show Answer & Explanation

Answer: (d) 34

Explanation:
1. 3 + 2√2 = (1 + √2)², so x = 1 + √2 and 1/x = √2 - 1.
2. x + 1/x = 2√2, so x² + 1/x² = 8 - 2 = 6.
3. x⁴ + 1/x⁴ = 36 - 2 = 34.

Teacher's Note:
1. Rewrite the nested root as a perfect square.
2. Use (a + b)² = a² + b² + 2ab repeatedly.

Question 12: If a + b + c = 0, then \( (a^3+b^3+c^3)^2 \) equals:
(a) \( 9a^2b^2c^2 \)
(b) \( a^2b^2c^2 \)
(c) \( 3abc \)
(d) \( 27a^2b^2c^2 \)
Show Answer & Explanation

Answer: (a) \( 9a^2b^2c^2 \)

Explanation:
1. When a + b + c = 0, a³ + b³ + c³ = 3abc.
2. Squaring gives 9a²b²c².

Teacher's Note:
1. Use a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ca).
2. With a + b + c = 0 the right side is zero.

Question 13: If \( x=2+2^{1/3}+2^{2/3} \), then \( x^3-6x^2+6x \) equals:
(a) 4
(b) 3
(c) 2
(d) 1
Show Answer & Explanation

Answer: (c) 2

Explanation:
1. Let y = x - 2 = t + t², where t = 2^(1/3) and t³ = 2.
2. Then y² = t² + 2t³ + t⁴ = t² + 2t + 4, and y³ = y·y² = 6 + 6t + 6t² = 6 + 6y.
3. x³ - 6x² + 6x = (y + 2)³ - 6(y + 2)² + 6(y + 2) = y³ - 6y - 4 = (6 + 6y) - 6y - 4 = 2.

Teacher's Note:
1. Shift the variable: put y = x - 2.
2. Use t³ = 2 to reduce the powers of t.

Question 14: The value of \( \sqrt{6+\sqrt{6+\sqrt{6+\cdots\infty}}} \) is:
(a) 1
(b) \( \pm 3 \)
(c) 3
(d) 2
Show Answer & Explanation

Answer: (c) 3

Explanation:
1. Let the value be x. Then x = √(6 + x), so x² = x + 6 and (x - 3)(x + 2) = 0.
2. x cannot be negative, so x = 3.

Teacher's Note:
1. The expression repeats itself inside the root.
2. A square root is non-negative, so -2 is rejected.

Question 15: If \( x=\sqrt{7}-\sqrt{5} \) and \( y=\sqrt{131}-\sqrt{11} \), then:
(a) \( x=y \)
(b) \( x>y \)
(c) \( x<y \)
(d) cannot be compared
Show Answer & Explanation

Answer: (c) \( x<y \)

Explanation:
1. x = 2/(√7 + √5) ≈ 2/4.88 ≈ 0.41.
2. y = 120/(√131 + √11) ≈ 120/14.76 ≈ 8.13.
3. So x < y.

Teacher's Note:
1. Rationalise each difference of square roots.
2. A small numerator over a large sum gives a small value.

Question 16: If \( \sqrt{\frac{6+2\sqrt{3}}{33-19\sqrt{3}}}=a+b\sqrt{3} \), then a + b equals:
(a) 10
(b) 8
(c) 6
(d) 12
Show Answer & Explanation

Answer: (b) 8

Explanation:
1. Rationalise: (6 + 2√3)(33 + 19√3)/(33² - 19²·3) = (312 + 180√3)/6 = 52 + 30√3.
2. 52 + 30√3 = (5 + 3√3)², since 25 + 27 = 52 and 2 × 5 × 3 = 30.
3. So a = 5, b = 3 and a + b = 8.

Teacher's Note:
1. Multiply numerator and denominator by the conjugate.
2. Express the result as a perfect square (a + b√3)².

Question 17: The digit in the unit place of \( 7^{343} \) is:
(a) 9
(b) 3
(c) 7
(d) 1
Show Answer & Explanation

Answer: (b) 3

Explanation:
1. Unit digits of powers of 7 repeat in a cycle of 4: 7, 9, 3, 1.
2. 343 = 4 × 85 + 3, so the digit is the third in the cycle, 3.

Teacher's Note:
1. Find the remainder when the power is divided by 4.
2. 7³ = 343 ends in 3.

Question 18: If A and B are two odd numbers with A > B, the largest number that divides A² - B² for every such pair is:
(a) 2
(b) 16
(c) 4
(d) 8
Show Answer & Explanation

Answer: (d) 8

Explanation:
1. A² - B² = (A - B)(A + B). Both factors are even.
2. With A = 2m + 1, B = 2n + 1 the product is 4(m - n)(m + n + 1), and (m - n) and (m + n + 1) have opposite parity, so one of them is even.
3. So A² - B² is always divisible by 8; 9² - 1² = 80 and 5² - 3² = 16 show 16 does not always work.

Teacher's Note:
1. Test pairs such as (3, 1) giving 8 and (5, 3) giving 16.
2. The common divisor of 8, 16, 24 ... is 8.

Question 19: If \( \sqrt{23+x\sqrt{10}}=\sqrt{18}+\sqrt{5} \), then x equals:
(a) 4
(b) 5
(c) 6
(d) 3
Show Answer & Explanation

Answer: (c) 6

Explanation:
1. Square both sides: 23 + x√10 = 18 + 5 + 2√90.
2. 2√90 = 2 × 3√10 = 6√10.
3. So x = 6.

Teacher's Note:
1. Expand (√18 + √5)².
2. Compare the coefficients of √10.

Question 20: The sum of the rational numbers \( -\frac{5}{16} \) and \( \frac{7}{12} \) is:
(a) \( \frac{1}{14} \)
(b) \( -\frac{13}{48} \)
(c) \( \frac{35}{192} \)
(d) \( \frac{13}{48} \)
Show Answer & Explanation

Answer: (d) \( \frac{13}{48} \)

Explanation:
1. LCM of 16 and 12 is 48: -15/48 + 28/48.
2. = 13/48.

Teacher's Note:
1. Convert both to the denominator 48.
2. Add numerators with their signs.

Question 21: The additive inverse of \( \frac{9}{-11} \) is:
(a) \( \frac{9}{11} \)
(b) \( -\frac{9}{11} \)
(c) \( -\frac{11}{9} \)
(d) \( \frac{11}{9} \)
Show Answer & Explanation

Answer: (a) \( \frac{9}{11} \)

Explanation:
1. 9/(-11) = -9/11.
2. Its additive inverse is 9/11.

Teacher's Note:
1. Additive inverse changes the sign of the number.
2. Reduce to standard form first.

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Practice MCQs for Class 8 Mathematics Number System

Class 8 Mathematics Number System Objective Test Questions

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FAQs

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