Indices — Assertion-Reason Type Questions

Assertion (A): Value of $\Big(\dfrac{8}{162}\Big)^{-1.5}$ is $\Big(\dfrac{729}{8}\Big)$.

Reason (R): x^m × yⁿ = (xy)^{m + 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,

$\Rightarrow \Big(\dfrac{8}{162}\Big)^{-1.5} \\[1em] \Rightarrow \Big(\dfrac{2^3}{2 \times 9^2}\Big)^{-1.5} \\[1em] \Rightarrow \Big(\dfrac{2^{3 - 1}}{9^2}\Big)^{-1.5} \\[1em] \Rightarrow \Big(\dfrac{2^2}{9^2}\Big)^{-1.5} \\[1em] \Rightarrow \Big[\Big(\dfrac{2}{9}\Big)^2\Big]^{-\dfrac{3}{2}} \\[1em] \Rightarrow \Big(\dfrac{2}{9}\Big)^{2 \times -\dfrac{3}{2}} \\[1em] \Rightarrow \Big(\dfrac{2}{9}\Big)^{-3} \\[1em] \Rightarrow \Big(\dfrac{9}{2}\Big)^{3} \\[1em] \Rightarrow \dfrac{729}{8}.$

Assertion (A) is true.

x^m × yⁿ = (xy)^{m + n} is not correct in general.

Correct identity:

⇒ x^m × y^m = (xy)^m

Or

⇒ x^m × xⁿ = x^m+n

Reason (R) is false.

A is true, R is false

Hence, option 1 is the correct option.

Assertion (A): (x^-1 - y^-1) × (x - y)^-1 = x^-1y^-1

Reason (R): For any non-zero number $x,\ x^{-1} = \dfrac{1}{x}$

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

Given,

(x^-1 - y^-1) × (x - y)^-1 = x^-1y^-1

Solving L.H.S,

$\Rightarrow (x^{-1} - y^{-1}) \times (x - y)^{-1} \\[1em] \Rightarrow \Big(\dfrac{1}{x} - \dfrac{1}{y}\Big) \times \Big(\dfrac{1}{x - y}\Big) \\[1em] \Rightarrow \Big(\dfrac{y - x}{xy}\Big) \times \dfrac{1}{x - y} \\[1em] \Rightarrow \dfrac{-(x - y)}{xy} \times \dfrac{1}{x - y} \\[1em] \Rightarrow \dfrac{-1}{xy} \\[1em] \Rightarrow -x^{-1}y^{-1}.$

L.H.S ≠ R.H.S

Thus, Assertion (A) is false.

For any non-zero number x, the notation x⁻¹ is defined as its multiplicative inverse, which is equal to $\dfrac{1}{x}$

Thus, Reason (R) is true.

A is false, R is true

Hence, option 2 is the correct option.

Assertion (A): If we divide 3^-2 by 3⁴, we get 9³.

Reason (R): $\Big(\dfrac{x}{y}\Big)^{-m} = \dfrac{y^{-m}}{x^{-m}}$

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,

$\Rightarrow \dfrac{3^{-2}}{3^4} \\[1em] \Rightarrow 3^{-2 - 4} \\[1em] \Rightarrow 3^{-6} \\[1em] \Rightarrow (3^2)^{-3} \\[1em] \Rightarrow 9^{-3}.$

∴ Assertion (A) is false.

According to reason (R) :

$\Big(\dfrac{x}{y}\Big)^{-m} = \dfrac{y^{-m}}{x^{-m}}$

Solving L.H.S,

$\Rightarrow \Big(\dfrac{x}{y}\Big)^{-m} \\[1em] \Rightarrow \Big(\dfrac{y}{x}\Big)^{m} \\[1em] \Rightarrow \Big(\dfrac{y^m}{x^m}\Big) \\[1em]$

Since, L.H.S ≠ R.H.S

∴ Reason (R) is false.

Both A and R are false.

Hence, option 4 is the correct option.