CBSE Class 10 Mathematics Real Numbers MCQs Set 17

Answer: (a) 42 and 21

Explanation:
1. The tree shows y = 3 × 7 = 21.
2. Then x = 2 × y = 2 × 21 = 42.
3. So x = 42 and y = 21: (a).

Teacher's Note:
1. Work from the bottom of the tree upwards.
2. Each box is the product of the two boxes below it.

Answer: (b) 4

Explanation:
1. 196 = 2 × 2 × 7 × 7 = 2² × 7².
2. Sum of exponents = 2 + 2 = 4.

Teacher's Note:
1. Write 196 as a product of primes first.
2. Add only the exponents, not the primes.

Answer: (c) 2

Explanation:
1. A prime number has exactly two factors, 1 and the number itself.
2. So the answer is 2.

Teacher's Note:
1. This is the definition of a prime number.
2. The number 1 has only one factor, so it is not prime.

Answer: (c) 3, 420

Explanation:
1. 12 = 2² × 3, 21 = 3 × 7 and 15 = 3 × 5.
2. H.C.F. = lowest power of the common prime = 3.
3. L.C.M. = 2² × 3 × 5 × 7 = 420.

Teacher's Note:
1. H.C.F. uses common primes with the smallest powers.
2. L.C.M. uses every prime with the greatest power.

Answer: (b) 7119

Explanation:
1. Product of the two numbers = H.C.F. × L.C.M.
2. 904 × other number = 113 × 56952.
3. Other number = (113 × 56952) ÷ 904 = 7119.

Teacher's Note:
1. Use H.C.F. × L.C.M. = product of the numbers.
2. Cancel 113 with 904 = 8 × 113 to make the division easy.

Answer: (a) 1

Explanation:
1. For n = 1, 24¹ = 24 and 24 = 8 × 3, which is divisible by 8.
2. n must be a natural number, so the least value is 1.

Teacher's Note:
1. Natural numbers start from 1.
2. Try the smallest value first.

Answer: (d) 18

Explanation:
1. 306 = 2 × 3² × 17 and 1314 = 2 × 3² × 73.
2. H.C.F. = 2 × 3² = 18.

Teacher's Note:
1. Factorise both numbers into primes.
2. Multiply the common primes with the smallest powers.

Answer: (c) 56

Explanation:
1. For two positive integers, H.C.F. × L.C.M. = a × b.
2. a × b = 2 × 28 = 56.

Teacher's Note:
1. Use the product rule for H.C.F. and L.C.M.
2. The H.C.F. must always divide the L.C.M.

Answer: (d) 45

Explanation:
1. Common primes are 3 and 5.
2. Smallest powers: 3² and 5¹.
3. H.C.F. = 3² × 5 = 9 × 5 = 45.

Teacher's Note:
1. Take the smaller exponent of each common prime.
2. Do not multiply the larger powers; that gives the L.C.M.

Answer: (c) co-primes

Explanation:
1. Two numbers whose only common factor is 1 are called co-primes.
2. So the answer is co-primes.

Teacher's Note:
1. Co-prime numbers need not be prime themselves, for example 8 and 9.
2. Twin primes are primes differing by 2.

Answer: (b) an irrational number

Explanation:
1. π = 3.14159... is a non-terminating, non-repeating decimal.
2. So π is irrational.

Teacher's Note:
1. The values 22/7 and 3.14 are only approximations of π.
2. A rational number has a terminating or repeating decimal.

Answer: (b) 4

Explanation:
1. A composite number has more than two factors.
2. 2 and 3 are prime and 1 is neither prime nor composite.
3. The factors of 4 are 1, 2 and 4, so the smallest composite number is 4.

Teacher's Note:
1. Check the numbers in order, starting from 1.
2. 1 is neither prime nor composite.

Answer: (b) 2

Explanation:
1. Smallest prime = 2 and smallest composite = 4.
2. H.C.F. (2, 4) = 2.

Teacher's Note:
1. Identify the two numbers first.
2. 2 divides 4, so the H.C.F. is the smaller number.

Answer: (c) 2m

Explanation:
1. Even integers are …, −2, 0, 2, 4, 6, …
2. Each is 2 times an integer, so every even integer has the form 2m.

Teacher's Note:
1. An even number is divisible by 2.
2. 2m + 1 gives the odd integers.

Answer: (c) an odd integer

Explanation:
1. Let n = 2k + 1 (odd). Then n² − 1 = 4k² + 4k = 4k(k + 1).
2. k(k + 1) is a product of two consecutive integers, so it is even, and 4k(k + 1) is divisible by 8.
3. For an even n, n² − 1 is odd. Answer: an odd integer.

Teacher's Note:
1. Test n = 3: 3² − 1 = 8, and n = 2: 2² − 1 = 3.
2. Odd n gives n² − 1 = 4k(k + 1).

Answer: (a) Both A and R are true, and R is the correct explanation of A.

Explanation:
1. 6250 = 2 × 5⁵, which has the form 2ᵐ × 5ⁿ with m = 1 and n = 5.
2. So the Reason holds and the fraction 123/6250 terminates, so the Assertion is true.
3. The Reason is the rule that proves the Assertion: option (a).

Teacher's Note:
1. Only the primes 2 and 5 may appear in the denominator.
2. Factorise the denominator before deciding.

Answer: (a) Both A and R are true, and R is the correct explanation of A.

Explanation:
1. −1 = −1/1, 0 = 0/1 and 2 = 2/1 are in the form p/q, and −4/3 is already in that form, so the Assertion is true.
2. The Reason is true.
3. The Reason is the definition that makes each listed number rational: option (a).

Teacher's Note:
1. A rational number is any number that can be written as p/q with q ≠ 0.
2. Every integer n can be written as n/1.