Match List-I with List-II

[.] denotes the greatest integer function.

Choose the correct answer from the options given below:

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

The correct answer is Option (2) → (A)-(IV), (B)-(I), (C)-(II), (D)-(III)

Evaluating each integral:

(A) $\int_0^3 [x]\,dx$

$= \int_0^1 0\,dx + \int_1^2 1\,dx + \int_2^3 2\,dx = 0 + 1 + 2 = 3$

(A) → (IV)

(B) $\int_0^1 [2x]\,dx$

$2x \in [0, 2) \Rightarrow [2x] = 0$ on $[0, \frac{1}{2})$, $1$ on $[\frac{1}{2}, 1)$

$= \int_0^{1/2} 0\,dx + \int_{1/2}^{1} 1\,dx = 0 + \frac{1}{2} = \frac{1}{2}$

(B) → (I)

(C) $\int_0^1 [3x]\,dx$

$= \int_0^{1/3} 0\,dx + \int_{1/3}^{2/3} 1\,dx + \int_{2/3}^1 2\,dx$

$= 0 + \frac{1}{3} + \frac{2}{3} = 1$

(C) → (II)

(D) $\int_0^1 [4x]\,dx$

$= \int_0^{1/4} 0\,dx + \int_{1/4}^{1/2} 1\,dx + \int_{1/2}^{3/4} 2\,dx + \int_{3/4}^1 3\,dx$

$= 0 + \frac{1}{4} + \frac{1}{4} \cdot 2 + \frac{1}{4} \cdot 3 = \frac{1}{4} + \frac{1}{2} + \frac{3}{4} = \frac{3}{2}$

(D) → (III)