Let X be a random variable. Let E (X) and Var (X) denote the mean and the variance of X respectively. Then match List-I with List-II

List-I	List-II
(A) If Var (X) = $a$, then Var (2X + 3) is	(I) 11$a$
(B) If E (X) = $a$, then E (2X) is	(II) 6$a$
(C) If Var (X) = $a$, then Var(3X - $a$) + Var ($\sqrt{2}x + \beta$) is	(III) 4$a$
(D) If E (X) = $\frac{5a}{12}$, then E (12X + $a$) is	(IV) 2$a$

Choose the correct answer from the options given below:

For Expectation: E(aX + b) = a·E(X) + b

For Variance: Var(aX + b) = a²·Var(X), where constants vanish in variance calculations.

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(A) If Var(X) = $a$, then Var(2X + 3) is

The coefficient of X is 2.

Var(2X + 3) = $(2)^2 \cdot \text{Var}(X)$

$= 4 \times a$

$= 4a$

Match: (A) → (III)

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(B) If E(X) = $a$, then E(2X) is

E(2X) = $2 \cdot E(X)$

$= 2 \times a$

$= 2a$

Match: (B) → (IV)

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(C) If Var(X) = $a$, then Var(3X - $a$) + Var($\sqrt{2}X + \beta$) is

The constant $-a$ vanishes in variance:

Var(3X - $a$) = $(3)^2 \cdot \text{Var}(X)$

$= 9a$

The constant $+\beta$ vanishes in variance:

Var($\sqrt{2}X + \beta$) = $(\sqrt{2})^2 \cdot \text{Var}(X)$

$= 2a$

Total = $9a + 2a$

$= 11a$

Match: (C) → (I)

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(D) If E(X) = $\frac{5a}{12}$, then E(12X + $a$) is

E(12X + $a$) = $12 \cdot E(X) + a$

$= 12 \times \frac{5a}{12} + a$

$= 5a + a$

$= 6a$

Match: (D) → (II)

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Therefore, the correct matching is:

(A) - (III), (B) - (IV), (C) - (I), (D) - (II)