If a straight line drawn through the point of intersection of the lines $4 x+3 y-1=0$ and $3 x+4 y-1=0$, meets the co-ordinate axes at the points P and Q , then the locus of the mid point of PQ is :

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 :