MCQ
The integral $\int_{0}^{\frac{\pi}{2}} \frac{1}{3+2 \sin x+\cos x} d x$ is equal to.
- A$\tan ^{-1}(2)$
- ✓$\tan ^{-1}(2)-\frac{\pi}{4}$
- C$\frac{1}{2} \tan ^{-1}(2)-\frac{\pi}{8}$
- D$\frac{1}{2}$
Put $\tan \frac{x}{2}=t$, so
$I=\int_{0}^{1} \frac{d t}{(t+1)^{2}+1}=\left.\tan ^{-1}(x+1)\right|_{0} ^{1}=\tan ^{-1} 2-\frac{\pi}{4}$
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
| List $I$ | List $II$ |
| $P$ $\quad\left(\frac{1}{y^2}\left(\frac{\cos \left(\tan ^{-1} y\right)+y \sin \left(\tan ^{-1} y\right)}{\cot \left(\sin ^{-1} y\right)+\tan \left(\sin ^{-1} y\right)}\right)^2+y^4\right)^{1 / 2}$ takes value | $1.\quad$ $\frac{1}{2} \sqrt{\frac{5}{3}}$ |
| $Q.\quad$ If $\cos x+\cos y+\cos z=0=\sin x+\sin y+\sin z$ then possible value of $\cos \frac{x-y}{2}$ is | $2.\quad$ $\sqrt{2}$ |
| $R.\quad$ If $\cos \left(\frac{\pi}{4}-x\right) \cos 2 x+\sin x \sin 2 x \sec x=\cos x \sin 2 x \sec x+$ $\cos \left(\frac{\pi}{4}+x\right) \cos 2 x$ then possible value of $\sec x$ is | $3.\quad$ $\frac{1}{2}$ |
| $S.\quad$ If $\cot \left(\sin ^{-1} \sqrt{1-x^2}\right)=\sin \left(\tan ^{-1}(x \sqrt{6})\right), x \neq 0$, then possible value of $x$ is | $4.\quad$ $1$ |
Codes: $ \quad P \quad Q \quad R \quad S $
(where $C$ is constant of integration)
$x+y+z=1$ ; $2 x+N y+2 z=2$ ; $3 x+3 y+N z=3$
has unique solution is $\frac{k}{6}$, then the sum of value of $k$ and all possible values of $N$ is