Match the following. List – I List – II (A) The number of...

\left(P\right)f\left(x\right) = a{x}^{2} + bx, \mathop{∫}_{0}^{1}f\left(x\right)dx = 1$<br />$⇒ 2a + 3b = 6$<br />$⇒ \left(a, b\right) ≡ \left(0, 2\right)and\left(3, 0\right)$<br />\left(Q\right)f\left(x\right) = \sqrt{2}\cos \left({x}^{2} - \dfrac{\pi}{4}\right)$
For maximum value, ${x}^{2} - \dfrac{\pi}{4} = 2n\pi$
$⇒ {x}^{2} = 2n\pi + \dfrac{\pi}{4}$
$⇒ x = \pm \sqrt{\dfrac{\pi}{4}}, \pm \sqrt{\dfrac{9\pi}{4}}$ as $xϵ\left[- \sqrt{3}, \sqrt{13}\right]$
\left(R\right)\mathop{∫}_{0}^{2}\left(\dfrac{3{x}^{2}}{1 + {e}^{x}} + \dfrac{3{x}^{2}}{1 + {e}^{- x}}\right)dx =\mathop{∫}_{0}^{2}3{x}^{2}dx = 8$<br />\left(S\right)\mathop{∫}_{- \dfrac{1}{2}}^{\dfrac{1}{2}}\cos 2xIn\left(\dfrac{1 + x}{1 - x}\right)dx = 0$ as it is an odd function.