Assertion (A):

$\dfrac{1}{2}$ log₂₅ $5+\dfrac{1}{3}$ log₈2 = $\dfrac{13}{36}$.

Reason (R):

$\dfrac{1}{2}$ log₂₅ $5+\dfrac{1}{3}$ log₈2 = $\dfrac{1}{4}+\dfrac{1}{9}=\dfrac{13}{36}$

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.

Both A and R are true.

Explanation

Given,

$\dfrac{1}{2}$ log₂₅ $5+\dfrac{1}{3}$ log₈2

Using the property, log_nm = $\dfrac{log${a}m}{log{a}n}

$\dfrac{1}{2}$ log₂₅ $5+\dfrac{1}{3}$ log₈2

$= \dfrac{1}{2}\dfrac{log${10}5}{log{10}25} + \dfrac{1}{3}\dfrac{log{10}2}{log{10}8}\\[1em] = \dfrac{1}{2}\dfrac{log{10}5}{log{10}5^2} + \dfrac{1}{3}\dfrac{log{10}2}{log{10}2^3}\\[1em] = \dfrac{1}{2 \times 2}\dfrac{log{10}5}{log{10}5} + \dfrac{1}{3 \times 3}\dfrac{log{10}2}{log{10}2}\\[1em] = \dfrac{1}{4}\dfrac{\cancel{log{10}5}}{\cancel{log{10}5}} + \dfrac{1}{9}\dfrac{\cancel{log{10}2}}{\cancel{log{10}2}}\\[1em] = \dfrac{1}{4} + \dfrac{1}{9}\\[1em] = \dfrac{9 + 4}{36}\\[1em] = \dfrac{13}{36}\\[1em]

∴ Assertion (A) is true.

From the above calculations,

$\dfrac{1}{2}$ log₂₅ $5+\dfrac{1}{3}$ log₈2 = $\dfrac{13}{36}$

∴ Reason (R) is true.

Hence, both Assertion (A) and Reason (R) are true.