Solutions: Real Numbers PYQ - Vardaan Learning Institute

Q10. \(\sqrt{0.4}\) is a/an

(A) natural number
(B) integer
(C) rational number
(D) irrational number

Q11. Which of the following cannot be the unit digit of \(8^n\), where \(n\) is a natural number?

(A) 4
(B) 2
(C) 0
(D) 6

Q20. Assertion (A): For two odd prime numbers \(x\) and \(y\), \((x \neq y)\), \(LCM(2x, 4y) = 4xy\)
Reason (R): \(LCM(x, y)\) is a multiple of \(HCF(x, y)\).

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is correct explanation of Assertion (A).
(B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).
(C) Assertion (A) is true, but Reason (R) is false.
(D) Assertion (A) is false, but Reason (R) is true.

Q30 (a). Prove that \(\sqrt{5}\) is an irrational number.

Let us assume, to the contrary, that \(\sqrt{5}\) is rational. So, we can find coprime integers \(a\) and \(b\) (\(b \neq 0\)) such that \(\sqrt{5} = \frac{a}{b}\).
Squaring: \(5b^2 = a^2\). Therefore, 5 divides \(a^2\), so 5 divides \(a\). Let \(a = 5c\).
Then \(5b^2 = 25c^2 \Rightarrow b^2 = 5c^2\). So 5 divides \(b\).
This contradicts the fact that \(a\) and \(b\) are coprime. Hence, \(\sqrt{5}\) is irrational.

Q30 (b). Let \(p, q\) and \(r\) be three distinct prime numbers. Check whether \(p \cdot q \cdot r + q\) is a composite number or not. Further, give an example for 3 distinct primes \(p, q, r\) such that: (i) \(p \cdot q \cdot r + 1\) is composite. (ii) \(p \cdot q \cdot r + 1\) is prime.

\(p \cdot q \cdot r + q = q(p \cdot r + 1)\). Since \(q > 1\) and \(pr + 1 > 1\), it is a composite number.
(i) For composite: \(p=3, q=5, r=7 \Rightarrow 105 + 1 = 106\) (even, composite).
(ii) For prime: \(p=2, q=3, r=5 \Rightarrow 30 + 1 = 31\) (prime).