Column I		Column II
$\left(\mathrm{A}\right)$	In a triangle $∆XYZ$ let $a,b$ and $\mathrm{c}$ be the lengths of the angles $X,Y$ and $\mathrm{Z}$, respectively. If $2\left(a^2-b^2\right)=c^2$ and $\lambda =\frac{\sin \left(X-Y\right)}{\mathrm{sinZ}}$then possible values of $n$ for which $\cos \left(n\pi \lambda \right)=0$ is are	$\left(\mathrm{P}\right)$	$1$
$\left(\mathrm{B}\right)$	In a triangle $\mathrm{\Delta }XYZ,$ let $a,b$ and $c$ be the lengths of the sides opposite to the angles $X,Y$ and $Z$, respectively. If $1+\cos 2x-2\cos 2y$$=2\sin x\sin y$, then possible value(s) of $\frac{a}{b}$ is (are)	$\left(\mathrm{Q}\right)$	$2$
$\left(\mathrm{C}\right)$	In ${\mathbb{R}}^2,$ let $\sqrt{3}\mathrm{ }\hat{\mathrm{i}}+\hat{\mathrm{j}},\mathrm{ }\hat{\mathrm{i}}+\sqrt{3}\mathrm{ }\hat{\mathrm{j}}$ and $\beta \hat{\mathrm{i}}+\left(1-\beta \right)\mathrm{ }\hat{\mathrm{j}}$ be the position vectors of $X,Y$ and $Z$ with respect to the origin $O$, respectively. If the distance of $Z$ from the bisector of the acute angle of $\overrightarrow{OX}$ with $\overrightarrow{OY}$ is $\frac{3}{\sqrt{2}}$, then possible value (s) of $\left|\beta \right|$ is (are)	$\left(\mathrm{R}\right)$	$3$
$\left(\mathrm{D}\right)$	Suppose that $F\left(\alpha \right)$ denotes the area of the region bounded by $x=0,\mathrm{ }x=2, y^2=4x$ and$y=\left|ax-1\right|+$$\left|\alpha x-2\right|+\alpha x,$  where $\alpha \in \left\{0, 1\right\}.$ Then the value (s) of $F\left(\alpha \right)+\frac{8}{3} \sqrt{2},$ when $\alpha =0$ and $\alpha =1$, is (are)	$\left(\mathrm{S}\right)$	$5$
		$\left(\mathrm{T}\right)$	$6$

 

1. A a-p,r,s;b-p;c-p,q;d-s,t;
2. B a-q,r,s,t;b-q,r,s,t;c-r,s,t;d-q,r,s;
3. C a-q;b-r;c-r,s;d-r,s;
4. D a-q,r,s,t;b-t;c-r,s,t;d-q,r,s,t;