$$\,\,\,\,$$ $$\,\,\,\,$$ $$\,\,\,\,$$ Column $$I$$
(A)$$\,\,\,\,$$ In $${R^2},$$ If the magnitude of the projection vector of the vector $$\alpha \widehat i + \beta \widehat j$$ on $$\sqrt 3 \widehat i + \widehat j$$ and If $$\alpha = 2 + \sqrt 3 \beta ,$$ then possible value of $$\left| \alpha \right|$$ is/are
(B)$$\,\,\,\,$$ Let $$a$$ and $$b$$ be real numbers such that the function $$f\left( x \right) = \left\{ {\matrix{
{ - 3a{x^2} - 2,} & {x < 1} \cr
{bx + {a^2},} & {x \ge 1} \cr
} } \right.$$ if differentiable for all $$x \in R$$. Then possible value of $$a$$ is (are)
(C)$$\,\,\,\,$$ Let $$\omega \ne 1$$ be a complex cube root of unity. If $${\left( {3 - 3\omega + 2{\omega ^2}} \right)^{4n + 3}} + {\left( {2 + 3\omega - 3{\omega ^2}} \right)^{4n + 3}} + {\left( { - 3 + 2\omega + 3{\omega ^2}} \right)^{4n + 3}} = 0,$$ then possible value (s) of $$n$$ is (are)
(D)$$\,\,\,\,$$ Let the harmonic mean of two positive real numbers $$a$$ and $$b$$ be $$4.$$ If $$q$$ is a positive real nimber such that $$a, 5, q, b$$ is an arithmetic progression, then the value(s) of $$\left| {q - a} \right|$$ is (are)
$$\,\,\,\,$$ $$\,\,\,\,$$ $$\,\,\,\,$$ Column $$II$$
(p)$$\,\,\,\,$$ $$1$$
(q)$$\,\,\,\,$$ $$2$$
(r)$$\,\,\,\,$$ $$3$$
(s)$$\,\,\,\,$$ $$4$$
(t)$$\,\,\,\,$$ $$5$$
| Column I | Column I | ||
|---|---|---|---|
| (A) | $\begin{array}{l}\text { In a triangle } \Delta X Y Z \text {, let } a, b \text { and } c \text { be the lengths of the sides } \\\text { opposite to the angles } X, Y \text { and } Z \text {, respectively. If } 2\left(a^2-b^2\right)=c^2 \\\text { and } \lambda=\frac{\sin (X-Y)}{\sin Z} \text {, then possible values of } n \text { for which } \cos (n \lambda) \\=0 \text { is (are) }\end{array}$ | (P) | 1 |
| (B) | $\begin{array}{l}\text { In a triangle } \triangle X Y Z \text {, let } a, b \text { and } c \text { be the lengths of the sides } \\\text { opposite to the angles } X, Y \text { and } Z \text {, respectively. If } 1+\cos 2 X-2 \\\cos 2 Y=2 \sin X \sin Y \text {, then possible value(s) of } \frac{a}{b} \text { is (are) }\end{array}$ | (Q) | 2 |
| (C) | $\begin{array}{l}\text { In } \mathbb{R}^2 \text {, let } \sqrt{3} \hat{i}+\hat{j}, \hat{i}+\sqrt{3} \hat{j} \text { and } \beta \hat{i}+(1-\beta) \hat{j} \text { be the position } \\\text { vectors of } X, Y \text { and } Z \text { with respect of the origin } \mathrm{O} \text {, respectively. If } \\\text { the distance of } \mathrm{Z} \text { from the bisector of the acute angle of } \overrightarrow{\mathrm{OX}} \text { with } \\\overrightarrow{\mathrm{OY}} \text { is } \frac{3}{\sqrt{2}} \text {, then possible value(s) of }|\beta| \text { is (are) }\end{array}$ | (R) | 3 |
| (D) | $\begin{array}{l}\text { Suppose that } F(\alpha) \text { denotes the area of the region bounded by } \\x=0, x=2, y^2=4 x \text { and } y=|\alpha x-1|+|\alpha x-2|+\alpha x \text {, } \\\text { where, } \alpha \in\{0,1\} \text {. Then the value(s) of } F(\alpha)+\frac{8}{2} \sqrt{2} \text {, when } \alpha=0 \\\text { and } \alpha=1 \text {, is (are) }\end{array}$ | (S) | 5 |
| (T) | 6 |