CBSE Class 10 Mathematics Real Numbers MCQs Set 20

Answer: (b) non-terminating repeating

Explanation:
1. The denominator is \( 2^2 \times 5 \times 7 \), and 29 has no common factor with it, so the fraction is already in lowest terms.
2. The prime factor 7 is present in the denominator, so it is not of the form \( 2^m \times 5^n \).
3. Hence the decimal expansion is non-terminating repeating.

Teacher's Note:
1. A fraction in lowest terms terminates only when its denominator has no prime factor other than 2 and 5.
2. A factor 7 in the denominator always gives a repeating block.

Answer: (d) x²y²

Explanation:
1. For the HCF take the smallest power of each common prime: x² and y².
2. HCF (a, b) = x²y².

Teacher's Note:
1. HCF uses the smallest powers, LCM uses the greatest powers.
2. Compare each prime separately.

Answer: (c) m × n

Explanation:
1. For two positive integers, HCF × LCM = product of the two numbers.
2. Hence the product of the LCM and HCF of m and n is m × n.

Teacher's Note:
1. Check it with 4 and 6: HCF 2, LCM 12 and 2 × 12 = 4 × 6.
2. This relation is used to find the LCM when the HCF is known.

Answer: (c) 87

Explanation:
1. The required number divides 615 − 6 = 609 and 963 − 6 = 957 exactly, so it is HCF (609, 957).
2. 609 = 3 × 7 × 29 and 957 = 3 × 11 × 29, so the HCF = 3 × 29 = 87.
3. Check: 87 is greater than the remainder 6, as required.

Teacher's Note:
1. Subtract the remainder from each number before taking the HCF.
2. The divisor must be larger than the remainder.

Answer: (c) 5

Explanation:
1. 70 = 2 × 5 × 7, 80 = 2⁴ × 5, 105 = 3 × 5 × 7 and 160 = 2⁵ × 5.
2. The only prime common to all four numbers is 5, so the HCF is 5.

Teacher's Note:
1. Look for the primes common to all the numbers.
2. The largest exact divisor is the HCF.

Answer: (c) 120

Explanation:
1. The first five even natural numbers are 2, 4, 6, 8 and 10.
2. 2 = 2, 4 = 2², 6 = 2 × 3, 8 = 2³, 10 = 2 × 5.
3. LCM = 2³ × 3 × 5 = 120.

Teacher's Note:
1. Take the highest power of every prime that appears.
2. The LCM is the smallest number divisible by all of them.

Answer: (d) 105

Explanation:
1. The number divides 318 − 3 = 315 and 739 − 4 = 735 exactly, so it is HCF (315, 735).
2. 315 = 3² × 5 × 7 and 735 = 3 × 5 × 7², so the HCF = 3 × 5 × 7 = 105.

Teacher's Note:
1. Subtract the respective remainders first.
2. Check that 105 is larger than both remainders 3 and 4.

Answer: (b) a rational number

Explanation:
1. Let x = 6.666... Then 10x = 66.666...
2. Subtracting, 9x = 60, so x = \( \frac{20}{3} \).
3. So \( 6.\overline{6} \) is a rational number.

Teacher's Note:
1. Every non-terminating repeating decimal can be written as p/q.
2. Multiply by 10 when one digit repeats.

Answer: (d) 10

Explanation:
1. The number divides 85 − 5 = 80 and 77 − 7 = 70 exactly.
2. 80 = 2⁴ × 5 and 70 = 2 × 5 × 7, so HCF (80, 70) = 2 × 5 = 10.

Teacher's Note:
1. Subtract the remainders before finding the HCF.
2. 10 is greater than both remainders.

Answer: (c) 60

Explanation:
1. The numbers 1 to 5 contain the primes 2, 3 and 5, with the highest powers 2² (from 4), 3 and 5.
2. LCM = 4 × 3 × 5 = 60.

Teacher's Note:
1. Use the highest power of each prime up to 5.
2. 60 is divisible by 1, 2, 3, 4 and 5.

Answer: (c) either terminating or non-terminating recurring

Explanation:
1. A rational number p/q has either a terminating decimal expansion or a non-terminating repeating one.
2. It is never non-terminating non-repeating, and it is not always terminating (for example 1/3).

Teacher's Note:
1. Non-terminating non-repeating decimals are irrational.
2. 1/4 terminates but 1/3 does not, so options (a) and (b) are wrong.

Answer: (b) two decimal places

Explanation:
1. 47/(2 × 5²) = 47/50 and 47 is not a multiple of 2 or 5.
2. 47/50 = 0.94, so the expansion ends after two decimal places.

Teacher's Note:
1. The number of decimal places equals the larger of the powers of 2 and 5 in the denominator.
2. Here the larger power is 2.

Answer: (a) always irrational

Explanation:
1. Suppose r is rational and s is irrational, and r + s (or r − s) were rational.
2. Then s = (r + s) − r would be rational, which is a contradiction.
3. So the sum or difference of a rational and an irrational number is always irrational.

Teacher's Note:
1. This is proved by contradiction.
2. The same reasoning works for a non-zero rational multiple of an irrational number.

Answer: (d) 30

Explanation:
1. The number divides 71 − 11 = 60 and 97 − 7 = 90 exactly, so it is HCF (60, 90).
2. 60 = 2² × 3 × 5 and 90 = 2 × 3² × 5, so the HCF = 2 × 3 × 5 = 30.

Teacher's Note:
1. Subtract the remainders first.
2. 30 is greater than the remainders 11 and 7.

Answer: (c) If the prime factorisation of q is of the form 2ⁿ5ᵐ, then p/q has a terminating decimal expansion.

Explanation:
1. If q = 2ⁿ5ᵐ, then p/q can be written with denominator 10 raised to the larger of n and m, so it terminates. Statement (c) is true.
2. \( \sqrt{5} \) is irrational, so its expansion is non-terminating non-repeating. Statement (a) is false.
3. \( \frac{\sqrt{72}}{\sqrt{50}} = \frac{6\sqrt{2}}{5\sqrt{2}} = \frac{6}{5} \), which is rational. Statement (b) is false.
4. \( \frac{7}{2^4 \times 5} \) has only 2 and 5 in the denominator, so it terminates. Statement (d) is false.

Teacher's Note:
1. Test each statement separately and write down a reason.
2. Simplify a quotient of roots before deciding that it is irrational.

Answer: (c) The largest number which exactly divides 12 and 60 is 4.

Explanation:
1. HCF (12, 60) = 12, because 12 divides 60. So the largest number dividing both is 12, not 4. Statement (c) is false.
2. Statement (a) is the Fundamental Theorem of Arithmetic, which is true.
3. Dividing an odd number by 6 leaves remainder 1, 3 or 5, so (b) is true.
4. HCF × LCM = 18 × x gives 6 × 36 = 18x, so x = 12 and (d) is true.

Teacher's Note:
1. When one number divides the other, the smaller number is the HCF.
2. Use HCF × LCM = product of the two numbers to find a missing number.

Answer: (b) The prime factorisation of 300 is 2² × 3 × 5².

Explanation:
1. 2² × 3 × 5² = 4 × 3 × 25 = 300, so (b) is true.
2. 18 = 2 × 3², and the factor 3 means \( \frac{17}{18} \) does not terminate, so (a) is false.
3. 1 has only one factor, so it is neither prime nor composite, and (c) is false.
4. LCM (8, 12) = 2³ × 3 = 24, not 48, so (d) is false.

Teacher's Note:
1. Multiply the factors back to check a factorisation.
2. The LCM of 8 and 12 is 24 because 24 is the first common multiple.