MCQ
$\int {\frac{{{e^{{{\tan }^{ - 1}}\sqrt x }}}}{{\sqrt x + x\sqrt x }}dx = } $
- A${e^{{{\tan }^{ - 1}}\sqrt x }} + c$
- B$\frac{1}{2}{e^{{{\tan }^{ - 1}}\sqrt x }} + c$
- C$\log {\tan ^{ - 1}}\sqrt x + c$
- ✓$2{e^{{{\tan }^{ - 1}}\sqrt x }} + c$
$\frac{1}{1+x} \times \frac{1}{2 \sqrt{x}} d x=d t$
$\frac{d x}{\sqrt{x}+x \sqrt{x}}=2 d t$
$\int \mathrm{e}^{\mathrm{t}} 2 \mathrm{dt}=2 \mathrm{e}^{\mathrm{t}}+\mathrm{c}$
$=2 \mathrm{e}^{\tan ^{-1} \sqrt{\mathrm{x}}}+\mathrm{c}$
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
| List-$I$ | List-$II$ |
| ($I$) Probability of $\left(X_2 \geq Y_2\right)$ is | ($P$) $\frac{3}{8}$ |
| ($II$) Probability of $\left(X_2>Y_2\right)$ is | ($Q$) $\frac{11}{16}$ |
| ($III$) Probability of $\left(X_3=Y_3\right)$ is | ($R$) $\frac{5}{16}$ |
| ($IV$) Probability of $\left(X_3>Y_3\right)$ is | ($S$) $\frac{355}{864}$ |
| ($T$) $\frac{77}{432}$ |
The correct option is:
where $\omega=\frac{-1+ i \sqrt{3}}{2},$ and $I _{3}$ be the identity matrix of order $3$. If the determinant of the matrix $\left( P ^{-1} AP - I _{3}\right)^{2}$ is $\alpha \omega^{2},$ then the value of $\alpha$ is equal to