Question

Match the Statements / Expressions in List 1 with the Statements / Expressions in List 2 and indicate your answer by darkening the appropriate bubbles in the $4\times 4$ matrix given in the $ORS.$

Answer 1

$\left(A\right)y=\frac{x^2+2x+4}{x+2}$
$\Rightarrow x^2+(2-y)x+4-2y=0$
$\Rightarrow y^2+4y-12\ge 0$
$y\le -6$ or $y\ge 2$
minimum value is 2.
$\left(B\right)$ $(A+B)(A-B)=(A-B)(A+B)$
$\Rightarrow AB=BA$
as $A$ is symmetric and $B$ is skew symmetric
$\Rightarrow (AB{)}^t=-AB$
$\Rightarrow k=1$ and $k=3$
$\left(C\right)$ $a={\log }_3{\log }_{3}2\Rightarrow 3^{-a}={\log }_{2}3$
Now $1<2^{-k+{\log }_2^3}<2$
$\Rightarrow 1<{3.2}^{-k}<2$
$\Rightarrow {\log }_2\left(\frac{3}{2}\right)<k<{\log }_2\left(3\right)$
$\Rightarrow k=1$ or $k<2$ and $k<3$
$\left(D\right)$ $\sin \theta =\cos \varphi \Rightarrow \cos (\frac{\pi }{2}-\theta )=\cos \varphi$
$\frac{\pi }{2}-\theta =2n\pi \pm \varphi$
$\frac{1}{\pi }(\theta \pm \varphi -\frac{\pi }{2})=-2n$
$\Rightarrow 0$ and 2 are possible.