| List-I | List-II | | --- | --- | | (P) The number of polynomials $f(x)$ with non-negative integer coefficients of degree $\leq 2$, satisfying $f(0) = 0$ and $\int_{0}^{1} f(x)\,dx = 1$, is | (1) $8$ | | (Q) The number of points in the interval $[-\sqrt{13}, \sqrt{13}]$ at which $f(x) = \sin(x^{2}) + \cos(x^{2})$ attains its maximum value, is | (2) $2$ | | (R) $\int_{-2}^{2} \dfrac{3x^{2}}{1 + e^{x}}\,dx$ equals | (3) $4$ | | (S) $\dfrac{\int_{-1/2}^{1/2} \cos 2x \log\left(\dfrac{1+x}{1-x}\right) dx}{\int_{0}^{1/2} \cos 2x \log\left(\dfrac{1+x}{1-x}\right) dx}$ | (4) $0$ |

$\text{(P) } f(x) = ax^{2}+bx+c, \ a,b,c \ge 0 \text{ integers}, \ f(0)=0, \ \int_0^1 f = 1$

$\text{(Q) } f(x) = \sin(x^2)+\cos(x^2) \text{ on } [-\sqrt{13}, \sqrt{13}]$

$\text{(R) } \int_{-2}^{2} \dfrac{3x^2}{1+e^x}\,dx \qquad \text{(S) ratio of the two integrals}$

$\text{Match } P, Q, R, S \text{ with List-II, then choose the code}$

$1.\ \text{solve } (P), (Q), (R), (S)$

$2.\ \text{match each to List-II}$

$3.\ \text{select the correct code}$

$f(x) = ax^{2} + bx + c, \quad f(0) = 0 \Rightarrow c = 0$

$\int_0^1 (ax^2+bx)\,dx = \dfrac{a}{3} + \dfrac{b}{2} = 1$

$2a + 3b = 6, \quad a, b \in \mathbb{Z}_{\ge 0}$

$(a,b) = (3,0),\ (0,2) \Rightarrow P \to (2)$

$f(x) = \sqrt2\,\sin\!\left(x^2 + \tfrac{\pi}{4}\right), \quad \max = \sqrt2$

$\text{max when } x^2 + \tfrac{\pi}{4} = \tfrac{\pi}{2} + 2k\pi \Rightarrow x^2 = \tfrac{\pi}{4} + 2k\pi$

$x^2 = \dfrac{\pi}{4} \text{ or } \dfrac{\pi}{4} + 2\pi \quad (\text{next exceeds } 13)$

$\Rightarrow 4 \text{ points} \Rightarrow Q \to (3)$

$\int_{-a}^{a} g(x)\,dx = \int_0^a \big[g(x)+g(-x)\big]dx$

$I = \int_0^2 \left[\dfrac{3x^2}{1+e^x} + \dfrac{3x^2}{1+e^{-x}}\right]dx$

$\dfrac{3x^2}{1+e^x} + \dfrac{3x^2 e^x}{1+e^x} = 3x^2$

$I = \int_0^2 3x^2\,dx = \Big[x^3\Big]_0^2 = 8 \Rightarrow R \to (1)$

$h(x) = \cos 2x \log\!\left(\dfrac{1+x}{1-x}\right)$

$\log\!\left(\dfrac{1+x}{1-x}\right) \text{ odd}, \ \cos 2x \text{ even} \Rightarrow h \text{ odd}$

$\int_{-1/2}^{1/2} h(x)\,dx = 0 \ (\text{odd})$

$\text{ratio} = \dfrac{0}{\int_0^{1/2} h} = 0 \Rightarrow S \to (4)$

$P \to (2), \quad Q \to (3), \quad R \to (1), \quad S \to (4)$

$\text{code } 2\ 3\ 1\ 4 \Rightarrow \text{option (4)}$