Let O be the vertex of the parabola $y^2=4 x$ and its chords OP and OQ are perpendicular to each other. If the locus of the mid-point of the line segment PQ is a conic C , then the length of its latus rectum is :

Let $\alpha=3 \sin ^{-1}\left(\frac{6}{11}\right)$ and $\beta=3 \cos ^{-1}\left(\frac{4}{9}\right)$, where inverse trigonometric functions take only the principal values.

Given below are two statements :

Statement I : $\quad \cos (\alpha+\beta)>0$.

Statement II : $\quad \cos (\alpha)<0$.

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

For the function $f(x)=\mathrm{e}^{\sin |x|}-|x|, x \in \mathbf{R}$, consider the following statements :

Statement I : $ f$ is differentiable for all $x \in \mathbf{R}$.

Statement II : $ f$ is increasing in $\left(-\pi,-\frac{\pi}{2}\right)$.

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

Let $\overrightarrow{\mathrm{a}}=4 \hat{i}-\hat{j}+3 \hat{k}, \overrightarrow{\mathrm{~b}}=10 \hat{i}+2 \hat{j}-\hat{k}$ and a vector $\overrightarrow{\mathrm{c}}$ be such that $2(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}})+3(\overrightarrow{\mathrm{~b}} \times \overrightarrow{\mathrm{c}})=\overrightarrow{0}$.

If $\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{c}}=15$, then $\overrightarrow{\mathrm{c}} \cdot(\hat{i}+\hat{j}-3 \hat{k})$ is equal to :