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Question (31)

QN0036502

Which of the following expressions is certainly composite?

(A) 7 × 11 × 13 + 1
(B) 7 × 11 × 13 + 13
(C) 7 × 11 + 1
(D) 11 × 13 − 1

Correct Answer: B

Explanation:

7 × 11 × 13 + 13 = 13(7 × 11 + 1), so it has factors other than 1 and itself.

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Question (32)

QN0036503

The number 2ⁿ × 5ⁿ is equal to:

(A) 7ⁿ
(B) 10ⁿ
(C) 25ⁿ
(D) 10n

Correct Answer: B

Explanation:

2ⁿ × 5ⁿ = (2 × 5)ⁿ = 10ⁿ.

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Question (33)

QN0036504

If a and b are coprime positive integers, then their HCF is:

(A) 0
(B) 1
(C) a + b
(D) ab

Correct Answer: B

Explanation:

Coprime integers have no common factor other than 1. Therefore, their HCF is 1.

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Question (34)

QN0036505

If a and b are coprime, then their LCM is:

(A) a + b
(B) a − b
(C) ab
(D) a/b

Correct Answer: C

Explanation:

Since HCF(a,b) = 1, HCF × LCM = ab gives LCM = ab.

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Question (35)

QN0036506

Suppose a and b are positive integers such that HCF(a,b) = 6 and LCM(a,b) = 180. Which pair could be a,b?

(A) 18 and 60
(B) 30 and 36
(C) 12 and 90
(D) 24 and 45

Correct Answer: A

Explanation:

HCF(18,60) = 6 and LCM(18,60) = 180. Hence 18 and 60 satisfy the conditions.

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Question (36)

QN0036507

If 2 divides a², which statement must be true?

(A) a is odd
(B) a is even
(C) a is prime
(D) a is a perfect square

Correct Answer: B

Explanation:

Since 2 is prime and 2 divides a², Theorem 1.2 gives 2 divides a. Therefore, a is even.

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Question (37)

QN0036508

Which step is essential when proving that √2 is irrational by contradiction?

(A) Assume √2 is an integer
(B) Assume √2 = a/b, where a and b are coprime integers
(C) Assume a = b
(D) Assume a and b are both even

Correct Answer: B

Explanation:

The proof assumes √2 is rational and writes it as a/b in lowest terms, where a and b are coprime.

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Question (38)

QN0036509

If √5 = a/b, where a,b are coprime integers and b ≠ 0, then squaring gives:

(A) 5a² = b²
(B) a² = 5b²
(C) a = 5b
(D) a² = b² + 5

Correct Answer: B

Explanation:

Squaring √5 = a/b gives 5 = a²/b², so a² = 5b².

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Question (39)

QN0036510

If 3 divides a², what can be concluded using Theorem 1.2?

(A) 3 does not divide a
(B) a divides 3
(C) 3 divides a
(D) a = 3

Correct Answer: C

Explanation:

Since 3 is prime and divides a², Theorem 1.2 states that 3 must divide a.

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Question (40)

QN0036511

Which of the following numbers is definitely irrational?

(A) 2 + √3
(B) 2 + 3/7
(C) 5/2
(D) 0.3333...

Correct Answer: A

Explanation:

If 2 + √3 were rational, subtracting the rational number 2 would make √3 rational, which is a contradiction.

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Question (31)

Which of the following expressions is certainly composite?

Explanation:

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Question (32)

The number 2ⁿ × 5ⁿ is equal to:

Explanation:

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Question (33)

If a and b are coprime positive integers, then their HCF is:

Explanation:

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Question (34)

If a and b are coprime, then their LCM is:

Explanation:

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Question (35)

Suppose a and b are positive integers such that HCF(a,b) = 6 and LCM(a,b) = 180. Which pair could be a,b?

Explanation:

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Question (36)

If 2 divides a², which statement must be true?

Explanation:

Link:

Question (37)

Which step is essential when proving that √2 is irrational by contradiction?

Explanation:

Link:

Question (38)

If √5 = a/b, where a,b are coprime integers and b ≠ 0, then squaring gives:

Explanation:

Link:

Question (39)

If 3 divides a², what can be concluded using Theorem 1.2?

Explanation:

Link:

Question (40)

Which of the following numbers is definitely irrational?

Explanation:

Link:

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