Indices — Assertion-Reason Type Questions

Assertion (A): If 2^x.3^y.5^z = 16200, then x = 3, y = 4, z = 2.

Reason (R): If p, q are different prime numbers, then p^m.qⁿ = p^l.q^k ⇒ m = l and n = k

1. Assertion (A) is true, Reason (R) is false.

2. Assertion (A) is false, Reason (R) is true.

3. Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct reason for Assertion (A).

4. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct reason (or explanation) for Assertion (A).

Answer

If p, q are different prime numbers, then p^m.qⁿ = p^l.q^k ⇒ m = l and n = k.

This statement is true. This is a direct consequence of the Fundamental Theorem of Arithmetic.

∴ Reason (R) is true.

Given, 2^x.3^y.5^z = 16200

⇒ 2^x.3^y.5^z = 2³.3⁴.5²

⇒ x = 3, y = 4 and z = 2.

∴ Assertion (A) is true.

∴ Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct reason (or explanation) for Assertion (A).

Hence, option 3 is the correct option.

Assertion (A): If x = 9 and y = 2, then x^y = y^x.

Reason (R): a^-n = $\Big(\dfrac{1}{a}\Big)^n$ when a is a real positive number and n is a rational number.

1. Assertion (A) is true, Reason (R) is false.

2. Assertion (A) is false, Reason (R) is true.

3. Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct reason for Assertion (A).

4. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct reason (or explanation) for Assertion (A).

Answer

According to Assertion: If x = 9 and y = 2, then x^y = y^x.

Taking L.H.S. = x^y

= 9²

= 81.

Taking R.H.S. = y^x

= 2⁹

= 512.

∴ L.H.S. ≠ R.H.S.

So, x^y ≠ y^x

∴ Assertion (A) is false.

a^-n = $\Big(\dfrac{1}{a}\Big)^n$ when a is a real positive number and n is a rational number.

This statement is a fundamental property of exponents. It correctly defines how to handle negative exponents. For example, 2^-3 = $\Big(\dfrac{1}{2}\Big)^3$

∴ Reason (R) is true.

∴ Assertion (A) is false, Reason (R) is true.

Hence, option 2 is the correct option.

Assertion (A): If 3^x = $9\sqrt{3}$, then x = $\dfrac{5}{2}$.

Reason (R): If a is a real positive number and n is positive integer then $\sqrt[n]{a}$ is also written as $a^\frac{1}{n}$.

1. Assertion (A) is true, Reason (R) is false.

2. Assertion (A) is false, Reason (R) is true.

3. Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct reason for Assertion (A).

4. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct reason (or explanation) for Assertion (A).

Answer

If a is a real positive number and n is positive integer then $\sqrt[n]{a}$ is also written as $a^\frac{1}{n}$.

This statement is a fundamental property of exponents and radicals. It defines the relationship between roots and fractional exponents. For example, $\sqrt{a} = a^\frac{1}{2}, \sqrt[3]{a} = a^\frac{1}{3}$.

∴ Reason (R) is true.

Given,

$\Rightarrow 3^x = 9 \sqrt{3}\\[1em] \Rightarrow 3^x = 3^2 . 3^\dfrac{1}{2}\\[1em] \Rightarrow 3^x = (3)^{2 + \dfrac{1}{2}}\\[1em] \Rightarrow 3^x = (3)^{\dfrac{4 + 1}{2}}\\[1em] \Rightarrow 3^x = (3)^{\dfrac{5}{2}}\\[1em] \Rightarrow x = \dfrac{5}{2}.$

∴ Assertion (A) is true.

∴ Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct reason (or explanation) for Assertion (A).

Hence, option 3 is the correct option.