Let $\vec{a}, \vec{b}, \vec{c}$ be three vectors such that $\vec{a} \times \vec{b}=2(\vec{a} \times \vec{c})$. If $|\vec{a}|=1,|\vec{b}|=4,|\vec{c}|=2$, and the angle between $\vec{b}$ and $\vec{c}$ is $60^{\circ}$, then $|\vec{a} \cdot \vec{c}|$ is equal to

Consider two sets $\mathrm{A}=\{x \in \mathrm{Z}:|(|x-3|-3)| \leq 1\}$ and

$\mathrm{B}=\left\{x \in \mathbb{R}-\{1,2\}: \frac{(x-2)(x-4)}{x-1} \log _e(|x-2|)=0\right\}$. Then the number of

onto functions $f: \mathrm{A} \rightarrow \mathrm{B}$ is equal to :

Let $\mathrm{A}=\{0,1,2, \ldots, 9\}$. Let R be a relation on A defined by $(x, y) \in \mathrm{R}$ if and only if $|x-y|$ is a multiple of 3.

Given below are two statements :

Statement I : $n(\mathrm{R})=36$.

Statement II : R is an equivalence relation.

In the light of the above statements, choose the correct answer from the options given below :

Bag A contains 9 white and 8 black balls, while bag B contains 6 white and 4 black balls. One ball is randomly picked up from the bag B and mixed up with the balls in the bag A . Then a ball is randomly drawn from the bag A . If the probability, that the ball drawn is white, is $\frac{\mathrm{p}}{\mathrm{q}}, \operatorname{gcd}(\mathrm{p}, \mathrm{q})=1$, then $\mathrm{p}+\mathrm{q}$ is equal to