Question

$\begin{array}{cccc}& ListI& & ListII\\ P.& \text{The number of polynomials}f\left(x\right)\text{with non-negative integer coefficients of degree}\le 2,\text{satisfying}f\left(0\right)=0\text{and}\underset{0}{\overset{1}{\int }}f\left(x\right)dx=1,\text{is}& 1.& 8\\ Q.& \text{The number of points in the interval}[-\sqrt{13},\sqrt{13}]\text{at which}f\left(x\right)=\sin \left(x^2\right)+\cos \left(x^2\right)\text{attains its maximum value, is}& 2.& 2\\ R.& \underset{-2}{\overset{2}{\int }}\frac{3x^2}{1+e^x}dx\text{equals}& 3.& 4\\ S.& \frac{\left(\underset{-1/2}{\overset{1/2}{\int }}\cos 2x\log \left(\frac{1+x}{1-x}\right)dx\right)}{\left(\underset{0}{\overset{1/2}{\int }}\cos 2x\log \left(\frac{1+x}{1-x}\right)dx\right)}\text{equals}& 4.& 0\end{array}$

Which of the following option is correct?

• A. $\left(P\right)\to \left(3\right),\left(Q\right)\to \left(2\right)\left(R\right)\to \left(4\right),\left(S\right)\to \left(1\right)$
• B. $\left(P\right)\to \left(2\right),\left(Q\right)\to \left(3\right)\left(R\right)\to \left(4\right),\left(S\right)\to \left(1\right)$
• C. $\left(P\right)\to \left(3\right),\left(Q\right)\to \left(2\right)\left(R\right)\to \left(1\right),\left(S\right)\to \left(4\right)$
• D. $\left(P\right)\to \left(2\right),\left(Q\right)\to \left(3\right)\left(R\right)\to \left(1\right),\left(S\right)\to \left(4\right)$

Answer 1

The correct option is D $\left(P\right)\to \left(2\right),\left(Q\right)\to \left(3\right)\left(R\right)\to \left(1\right),\left(S\right)\to \left(4\right)$

$\left(P\right)$
$f\left(0\right)=0$
Let $f\left(x\right)=a{x}^2+bx$
$\underset{0}{\overset{1}{\int }}f\left(x\right)dx=\frac{a}{3}+\frac{b}{2}=1\Rightarrow a=0,b=2;a=3,b=0$
Two such polynomials is possible.

$\left(Q\right)$
$f\left(x\right)=\sin \left(x^2\right)+\cos \left(x^2\right)$
$x^2\in [0,13]$
Maximum value of $f\left(x\right)=\sqrt{2}$ and its attans at
$\sin \left(x^2\right)=\cos \left(x^2\right)$
$\Rightarrow \tan \left(x^2\right)=1$
$\Rightarrow x^2=\frac{\pi }{4},\frac{9\pi }{4}<13\Rightarrow x=\pm \frac{\sqrt{\pi }}{2},\pm \frac{3\sqrt{\pi }}{2}$

Hence, there are four points in $x\in [-\sqrt{13},\sqrt{13}]$

$\left(R\right)$
$I=\underset{-2}{\overset{2}{\int }}\frac{3x^2}{1+e^x}dx\Rightarrow I=\underset{-2}{\overset{2}{\int }}\frac{3x^2}{1+e^{-x}}dx\Rightarrow 2I=\underset{-2}{\overset{2}{\int }}\frac{3x^2(1+e^x)}{1+e^x}dx\Rightarrow 2I=\underset{-2}{\overset{2}{\int }}3x^{2}dx\Rightarrow I=8$

$\left(S\right)$
$\frac{\left(\underset{-1/2}{\overset{1/2}{\int }}\cos 2x\log \left(\frac{1+x}{1-x}\right)dx\right)}{\left(\underset{0}{\overset{1/2}{\int }}\cos 2x\log \left(\frac{1+x}{1-x}\right)dx\right)}$
$I_1=\underset{-1/2}{\overset{1/2}{\int }}\cos 2x\log \left(\frac{1+x}{1-x}\right)dx\Rightarrow I_1=\underset{-1/2}{\overset{1/2}{\int }}\cos 2x\log \left(\frac{1-x}{1+x}\right)dx\Rightarrow 2I_1=0$

Hence,
$\left(P\right)\to \left(2\right),\left(Q\right)\to \left(3\right)\left(R\right)\to \left(1\right),\left(S\right)\to \left(4\right)$