Consider the vectors

$$ \vec{x}=\hat{\imath}+2 \hat{\jmath}+3 \hat{k}, \quad \vec{y}=2 \hat{\imath}+3 \hat{\jmath}+\hat{k}, \quad \text { and } \quad \vec{z}=3 \hat{\imath}+\hat{\jmath}+2 \hat{k} $$

For two distinct positive real numbers $\alpha$ and $\beta$, define

$$ \vec{X}=\alpha \vec{x}+\beta \vec{y}-\vec{z}, \quad \vec{Y}=\alpha \vec{y}+\beta \vec{z}-\vec{x}, \quad \text { and } \quad \vec{Z}=\alpha \vec{z}+\beta \vec{x}-\vec{y} . $$

If the vectors $\vec{X}, \vec{Y}$, and $\vec{Z}$ lie in a plane, then the value of $\alpha+\beta-3$ is ____________.

For any two points $M$ and $N$ in the $XY$-plane, let $\overrightarrow{MN}$ denote the vector from $M$ to $N$, and $\vec{0}$ denote the zero vector. Let $P, Q$ and $R$ be three distinct points in the $XY$-plane. Let $S$ be a point inside the triangle $\triangle PQR$ such that

$$\overrightarrow{SP} + 5\; \overrightarrow{SQ} + 6\; \overrightarrow{SR} = \vec{0}.$$

Let $E$ and $F$ be the mid-points of the sides $PR$ and $QR$, respectively. Then the value of

$\frac{\text { length of the line segment } E F}{\text { length of the line segment } E S}$

is ________________.

Let $\vec{p}=2 \hat{i}+\hat{j}+3 \hat{k}$ and $\vec{q}=\hat{i}-\hat{j}+\hat{k}$. If for some real numbers $\alpha, \beta$, and $\gamma$, we have

$$ 15 \hat{i}+10 \hat{j}+6 \hat{k}=\alpha(2 \vec{p}+\vec{q})+\beta(\vec{p}-2 \vec{q})+\gamma(\vec{p} \times \vec{q}), $$

then the value of $\gamma$ is ________.

Let $\overrightarrow{O P}=\frac{\alpha-1}{\alpha} \hat{i}+\hat{j}+\hat{k}, \overrightarrow{O Q}=\hat{i}+\frac{\beta-1}{\beta} \hat{j}+\hat{k}$ and $\overrightarrow{O R}=\hat{i}+\hat{j}+\frac{1}{2} \hat{k}$ be three vectors, where $\alpha, \beta \in \mathbb{R}-\{0\}$ and $O$ denotes the origin. If $(\overrightarrow{O P} \times \overrightarrow{O Q}) \cdot \overrightarrow{O R}=0$ and the point $(\alpha, \beta, 2)$ lies on the plane $3 x+3 y-z+l=0$, then the value of $l$ is ____________.

Let $\vec{w} = \hat{i} + \hat{j} - 2\hat{k}$, and $\vec{u}$ and $\vec{v}$ be two vectors, such that $\vec{u} \times \vec{v} = \vec{w}$ and $\vec{v} \times \vec{w} = \vec{u}$. Let $\alpha, \beta, \gamma$, and $t$ be real numbers such that

$\vec{u} = \alpha \hat{i} + \beta \hat{j} + \gamma \hat{k},\ \ \ - t \alpha + \beta + \gamma = 0,\ \ \ \alpha - t \beta + \gamma = 0,\ \ \ \alpha + \beta - t \gamma = 0.$

Match each entry in List-I to the correct entry in List-II and choose the correct option.

$$\,\,\,\,$$ $$\,\,\,\,$$ $$\,\,\,\,$$ 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)$$\,\,\,\,$$ $$5$$

$$\,\,\,\,$$ $$\,\,\,\,$$ $$\,\,\,\,$$ List $$I$$
(P.)$$\,\,\,\,$$ Volume of parallelopiped determined by vectors $$\overrightarrow a ,\overrightarrow b $$ and $$\overrightarrow c $$ is $$2.$$ Then the volume of the parallelepiped determined by vectors $$2\left( {\overrightarrow a \times \overrightarrow b } \right),3\left( {\overrightarrow b \times \overrightarrow c } \right)$$ and $$\left( {\overrightarrow c \times \overrightarrow a } \right)$$ is
(Q.)$$\,\,\,\,$$ Volume of parallelopiped determined by vectors $$\overrightarrow a ,\overrightarrow b $$ and $$\overrightarrow c $$ is $$5.$$ Then the volume of the parallelepiped determined by vectors $$3\left( {\overrightarrow a + \overrightarrow b } \right),\left( {\overrightarrow b + \overrightarrow c } \right)$$ and $$2\left( {\overrightarrow c + \overrightarrow a } \right)$$ is
(R.)$$\,\,\,\,$$ Area of a triangle with adjacent sides determined by vectors $${\overrightarrow a }$$ and $${\overrightarrow b }$$ is $$20.$$ Then the area of the triangle with adjacent sides determined by vectors $$\left( {2\overrightarrow a + 3\overrightarrow b } \right)$$ and $$\left( {\overrightarrow a - \overrightarrow b } \right)$$ is
(S.)$$\,\,\,\,$$ Area of a parallelogram with adjacent sides determined by vectors $${\overrightarrow a }$$ and $${\overrightarrow b }$$ is $$30.$$ Then the area of the parallelogram with adjacent sides determined by vectors $$\left( {\overrightarrow a + \overrightarrow b } \right)$$ and $${\overrightarrow a }$$ is

$$\,\,\,\,$$ $$\,\,\,\,$$ $$\,\,\,\,$$ List $$II$$
(1.)$$\,\,\,\,$$ $$100$$
(2.)$$\,\,\,\,$$ $$30$$
(3.)$$\,\,\,\,$$ $$24$$
(4.)$$\,\,\,\,$$ $$60$$

$$\,\,\,\,$$ $$\,\,\,\,$$ $$\,\,\,\,$$ Column-$$I$$
(A) $$\,\,\,\,$$If $$\overrightarrow a = \widehat j + \sqrt 3 \widehat k,\overrightarrow b = - \widehat j + \sqrt 3 \widehat k$$ and $$\overrightarrow c = 2\sqrt 3 \widehat k$$ form a triangle, then the internal angle of the triangle between $$\overrightarrow a $$ and $$\overrightarrow b $$ is
(B)$$\,\,\,\,$$ If $$\int\limits_a^b {\left( {f\left( x \right) - 3x} \right)dx = {a^2} - {b^2},} $$ then the value of $$f$$ $$\left( {{\pi \over 6}} \right)$$ is
(C)$$\,\,\,\,$$ The value of $${{{\pi ^2}} \over {\ell n3}}\int\limits_{7/6}^{5/6} {\sec \left( {\pi x} \right)dx} $$ is
(D)$$\,\,\,\,$$ The maximum value of $$\left| {Arg\left( {{1 \over {1 - z}}} \right)} \right|$$ for $$\left| z \right| = 1,\,z \ne 1$$ is given by

$$\,\,\,\,$$ $$\,\,\,\,$$ $$\,\,\,\,$$ Column-$$II$$
(p)$$\,\,\,\,$$ $${{\pi \over 6}}$$
(q)$$\,\,\,\,$$ $${{2\pi \over 3}}$$
(r)$$\,\,\,\,$$ $${{\pi \over 3}}$$
(s)$$\,\,\,\,$$ $$\pi $$
(t) $$\,\,\,\,$$ $${{\pi \over 2}}$$

If $$\overrightarrow a ,\overrightarrow b ,\overrightarrow c $$ and $$\overrightarrow d $$ are unit vectors such that $$(\overrightarrow a \times \overrightarrow b )\,.\,(\overrightarrow c \times \overrightarrow d ) = 1$$ and $$\overrightarrow a \,.\,\overrightarrow c = {1 \over 2}$$, then

The shortest distance between $${L_1}$$ and $${L_2}$$ is :

The unit vector perpendicular to both $${L_1}$$ and $${L_2}$$ is :

STATEMENT-1: $$\overrightarrow {PQ} \times \left( {\overrightarrow {RS} + \overrightarrow {ST} } \right) \ne \overrightarrow 0 .$$ because
STATEMENT-2: $$\overrightarrow {PQ} \times \overrightarrow {RS} = \overrightarrow 0 $$ and $$\overrightarrow {PQ} \times \overrightarrow {ST} \ne \overrightarrow 0 \,\,.$$

(p)$$\,\,\,$$ $$2$$
(q)$$\,\,\,$$ $${4 \over 3}$$
(r)$$\,\,\,$$ $$\left| {\int\limits_0^1 {\sqrt {1 - xdx} } } \right| + \left| {\int\limits_{ - 1}^0 {\sqrt {1 + xdx} } } \right|$$
(s)$$\,\,\,$$ $$1$$

The point $$D,$$ then, is the ................ of the triangle $$ABC.$$