JEE Main 2014 — Trigonometry

If $2 \cos \theta+\sin \theta=1\left(\theta \neq \frac{\pi}{2}\right)$, then $7 \cos \theta+6 \sin \theta$ is equal to:

If $\text{cosec }\theta =\frac{\text{p}+\text{q}}{\text{ p}-\text{q }}(\text{p}\neq \text{q, p}\neq 0),$then $|\text{cot}(\frac{\pi }{4}+\frac{\theta }{2})|$ is equals to:

Let ${f}_{k}(x)=\frac{1}{k}({\mathrm{sin}}^{k}x+{\mathrm{cos}}^{k}x)$ where $x\in R$ and $k\geq 1$. Then ${f}_{4}(x)-{f}_{6}(x)$ equals

Statement I: The equation $\left(\sin ^{-1} \mathrm{x}\right)^3+$ $\left(\cos ^{-1} \mathrm{x}\right)^3-\mathrm{a} \pi^3=0$ has a solution for all $\mathrm{a} \geq \frac{1}{32}$. Statement II: For any $\mathrm{x} \in \mathrm{R}$, $\sin ^{-1} x+\cos ^{-1} x=\frac{\pi}{2}$ and $0 \leq\left(\sin ^{-1} x-\frac{\pi}{4}\right)^2 \leq \frac{9 \pi^2}{16}$

The principal value of ${\mathrm{tan}}^{-1}(cot\frac{43\pi }{4})$ is