Exponents — Assertion-Reason Type Questions

Assertion (A) : (9⁰ + 8⁰) ÷ (72)⁰ = 1

Reason (R) : Any non-zero base raised to the power zero is equal to unity (i.e. 1).

1. A is true, R is false.

2. A is false, R is true.

3. Both A and R are true.

4. Both A and R are false.

Answer

Solving,

⇒ (9⁰ + 8⁰) ÷ (72)⁰

⇒ (1 + 1) ÷ 1

⇒ 2 ÷ 1

⇒ 2.

Since the value is 2 and not 1,

Thus, Assertion (A) is false.

Any non-zero base raised to the power zero is equal to 1, which is a correct statement.

Thus, Reason (R) is true.

∴ A is false, R is true.

Hence, Option 2 is the correct option.

Assertion (A) : -1ⁿ is always equal to -1, n is even or odd whole number.

Reason (R) : (-1)ⁿ = -1ⁿ for all n ∈ W.

1. A is true, R is false.

2. A is false, R is true.

3. Both A and R are true.

4. Both A and R are false.

Answer

Here -1ⁿ means -(1ⁿ). Since 1ⁿ = 1 for every whole number n, -1ⁿ = -1 for all whole numbers n.

Thus, Assertion (A) is true.

(-1)ⁿ is +1 when n is even and -1 when n is odd. So (-1)ⁿ is not equal to -1ⁿ for all n ∈ W.

Thus, Reason (R) is false.

∴ A is true, R is false.

Hence, Option 1 is the correct option.

Assertion (A) : $\left(-\dfrac{3}{4}\right) \times \left(-\dfrac{3}{4}\right) \times \left(-\dfrac{3}{4}\right) \times$ .... up to 10 terms = $\dfrac{3^a}{2^b}$, then a - b = -10.

Reason (R) : If an expression is in the form of fraction with the same power, then that power will be taken for numerator and denominator both, i.e. $\left(\dfrac{p}{q}\right)^n = \dfrac{p^n}{q^n}$.

1. A is true, R is false.

2. A is false, R is true.

3. Both A and R are true.

4. Both A and R are false.

Answer

Solving,

$\left(-\dfrac{3}{4}\right) \times \left(-\dfrac{3}{4}\right) \times \left(-\dfrac{3}{4}\right) \times \dots \text{ up to 10 terms}\\[1em] = \left(-\dfrac{3}{4}\right)^{10}\\[1em] = \dfrac{(-3)^{10}}{4^{10}}\\[1em] = \dfrac{3^{10}}{(2^2)^{10}}\\[1em] = \dfrac{3^{10}}{2^{20}}$

Comparing with $\dfrac{3^a}{2^b}$, we get a = 10 and b = 20.

⇒ a - b = 10 - 20 = -10.

Thus, Assertion (A) is true.

For an expression in the form of a fraction with the same power, $\left(\dfrac{p}{q}\right)^n = \dfrac{p^n}{q^n}$, which is correct.

Thus, Reason (R) is true.

∴ Both A and R are true.

Hence, Option 3 is the correct option.