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Column-I
Column-II
(A) In a triangle \( \triangle X Y Z \), let a, b and \( c \) be the lengths of the sides
[JEE. Advanced-2015] opposite to the angles \( X, Y \) and \( Z \), respectively. If \( 2\left(a^{2}-b^{2}\right)=c^{2} \)
\( (\mathrm{P}) \) and \( \lambda=\frac{\sin (X-Y)}{\sin Z} \), then possible values of \( n \) for which \( \cos (\mathrm{n} \pi \lambda)=0 \) is (are)
(B) In a triangle \( \triangle \mathrm{XYZ} \), let a, b and \( \mathrm{c} \) be the lengths of the sides
(Q) 2 opposite to the angles \( X, Y \) and \( Z \), respectively. If
\( 1+\cos 2 \mathrm{X}-2 \cos 2 \mathrm{Y}=2 \sin \mathrm{X} \sin \mathrm{Y} \), then possible value(s) of \( \frac{\mathrm{a}}{\mathrm{b}} \) is (are)
(C) In \( R^{2} \), let \( \sqrt{3} \hat{i}+\hat{j}, \hat{i}+\sqrt{3} \hat{j} \) and \( \beta \hat{i}+(1-\beta) \hat{j} \) be the position vectors
(R) 3 of \( \mathrm{X}, \mathrm{Y} \) and \( \mathrm{Z} \) with respect to the origin \( \mathrm{O} \), respectively. If the distance of \( Z \) from the bisector of the acute angle of \( \overrightarrow{O X} \) and \( \overrightarrow{O Y} \) is \( \frac{3}{\sqrt{2}} \), then possible value(s) of \( \beta \mid \) is (are)
(D) Suppose that \( F(\alpha) \) denotes the area of the region bounded by
(S) 5 \( x=0, x=2, y^{2}=4 x \) and \( y=|\alpha x-1|+|\alpha x-2|+\alpha x \), where \( \alpha \in\{0,1\} \). Then the value(s) of \( F(\alpha)+\frac{8}{3} \sqrt{2} \), when \( \alpha=0 \) and \( \alpha=1 \), is (are)
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