Trigonometric Functions and Equations

If
$$\frac{\sin^4x}2+\frac{\cos^4x}3=\frac15,$$
then

If
$$I_n=\int_{-\pi}^{\pi}\frac{\sin nx}{(1+\pi^x)\sin x}\,dx,\qquad n=0,1,2,\ldots,$$
then

For $0<\theta<\dfrac\pi2$, the solution(s) of
$$\sum_{m=1}^6\operatorname{cosec}\left(\theta+\frac{(m-1)\pi}4\right)\operatorname{cosec}\left(\theta+\frac{m\pi}4\right)=4\sqrt2$$
is(are)

Match the statements/expressions given in Column I with the values given in Column II.

Let $ABC$ and $ABC'$ be two non-congruent triangles with sides $AB=4$, $AC=AC'=2\sqrt2$ and angle $B=30°$. The absolute value of the difference between the areas of these triangles is

If $0<x<1$, then
$$\sqrt{1+x^2}\left[\{x\cos(\cot^{-1}x)+\sin(\cot^{-1}x)\}^2-1\right]^{1/2}=$$

Let the function $g:(-\infty,\infty)\to\left(-\frac\pi2,\frac\pi2\right)$ be given by $g(u)=2\tan^{-1}(e^u)-\frac\pi2$. Then, $g$ is

Let a solution $y=y(x)$ of the differential equation
$$x\sqrt{x^2-1}\,dy-y\sqrt{y^2-1}\,dx=0$$
satisfy $y(2)=\frac2{\sqrt3}$.

STATEMENT-1: $y(x)=\sec\left(\sec^{-1}x-\frac\pi6\right)$

and

STATEMENT-2: $y(x)$ is given by $\dfrac1y=\dfrac{2\sqrt3}{x}-\sqrt{1-\dfrac1{x^2}}$

Match the Statements / Expressions in Column I with the Statements / Expressions in Column II and indicate your answer by darkening the appropriate bubbles in the 4 × 4 matrix given in the ORS.

The number of solutions of the pair of equations
$$2\sin^2\theta-\cos2\theta=0,\qquad2\cos^2\theta-3\sin\theta=0$$
in the interval $[0,2\pi]$ is

Let $f(x)=2+\cos x$ for all real $x$.

STATEMENT-1: For each real $t$, there exists a point $c$ in $[t,t+\pi]$ such that $f'(c)=0$.

because

STATEMENT-2: $f(t)=f(t+2\pi)$ for each real $t$.

Let $(x,y)$ be such that
$$\sin^{-1}(ax)+\cos^{-1}(y)+\cos^{-1}(bxy)=\frac\pi2.$$
Match the statements in Column I with statements in Column II and indicate your answer by darkening the appropriate bubbles in the 4 × 4 matrix given in the ORS.