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Question (41)

QN0036512

If r is a non-zero rational number and x is irrational, then rx is:

(A) always rational
(B) always irrational
(C) always an integer
(D) always zero

Correct Answer: B

Explanation:

The product of a non-zero rational number and an irrational number is irrational.

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Question (42)

QN0036513

Which expression is necessarily irrational?

(A) √2 − √2
(B) 5 − √3
(C) √2 × 0
(D) √5/√5

Correct Answer: B

Explanation:

If 5 − √3 were rational, then √3 would be rational, which contradicts the fact that √3 is irrational.

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Question (43)

QN0036514

If 3√2 were rational, what contradiction would follow?

(A) 3 would be irrational
(B) √2 would be rational
(C) 3 would equal 2
(D) √2 would be an integer

Correct Answer: B

Explanation:

Dividing 3√2 by the non-zero rational number 3 would make √2 rational, which is a contradiction.

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Question (44)

QN0036515

If x = 4 + √7, which statement is correct?

(A) x is rational
(B) x is irrational
(C) x is an integer
(D) x is a natural number

Correct Answer: B

Explanation:

The sum of a rational number and an irrational number is irrational. Therefore, 4 + √7 is irrational.

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Question (45)

QN0036516

Let a and b be coprime positive integers. If a² = 7b², then which conclusion follows?

(A) 7 does not divide a
(B) 7 divides a, leading to 7 dividing b as well
(C) a = b
(D) a and b must both be prime

Correct Answer: B

Explanation:

Since 7 divides a², Theorem 1.2 gives 7 divides a. Substitution then forces 7 to divide b, contradicting the fact that a and b are coprime.

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Question (46)

QN0036517

A student claims that HCF × LCM equals the product of three positive integers. Which statement correctly evaluates this claim?

(A) It is always true
(B) It is true only when all three numbers are equal
(C) It is not generally true for three numbers
(D) It is true only for prime numbers

Correct Answer: C

Explanation:

For three numbers, HCF × LCM is not generally equal to their product.

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Question (47)

QN0036518

Three numbers have prime factorisations 2² × 3, 2 × 3² × 5, and 2³ × 3 × 5². Their LCM is:

(A) 2² × 3² × 5
(B) 2³ × 3² × 5²
(C) 2 × 3 × 5
(D) 2³ × 3 × 5

Correct Answer: B

Explanation:

For LCM, select the greatest exponent of each prime: 2³, 3² and 5².

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Question (48)

QN0036519

Three numbers have prime factorisations 2⁴ × 3², 2³ × 3³ × 5, and 2² × 3 × 5². Their HCF is:

(A) 2² × 3
(B) 2³ × 3
(C) 2² × 3² × 5
(D) 2⁴ × 3

Correct Answer: A

Explanation:

For HCF, take the smallest powers common to all three numbers: 2² and 3¹. Prime 5 is absent from the first number.

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Question (49)

QN0036520

Which reasoning correctly proves that √11 is irrational?

(A) 11 is odd, so √11 is irrational
(B) Assume √11 = a/b in lowest terms; then a² = 11b², forcing 11 to divide both a and b, contradicting their coprimality
(C) 11 is prime, so it cannot have a square root
(D) Every square root is irrational

Correct Answer: B

Explanation:

Assume √11 = a/b in lowest terms. Then a² = 11b², so 11 divides a. Writing a = 11c forces 11 to divide b as well, contradicting that a and b are coprime.

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Question (41)

If r is a non-zero rational number and x is irrational, then rx is:

Explanation:

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Question (42)

Which expression is necessarily irrational?

Explanation:

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Question (43)

If 3√2 were rational, what contradiction would follow?

Explanation:

Link:

Question (44)

If x = 4 + √7, which statement is correct?

Explanation:

Link:

Question (45)

Let a and b be coprime positive integers. If a² = 7b², then which conclusion follows?

Explanation:

Link:

Question (46)

A student claims that HCF × LCM equals the product of three positive integers. Which statement correctly evaluates this claim?

Explanation:

Link:

Question (47)

Three numbers have prime factorisations 2² × 3, 2 × 3² × 5, and 2³ × 3 × 5². Their LCM is:

Explanation:

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Question (48)

Three numbers have prime factorisations 2⁴ × 3², 2³ × 3³ × 5, and 2² × 3 × 5². Their HCF is:

Explanation:

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Question (49)

Which reasoning correctly proves that √11 is irrational?

Explanation:

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