If θ is the angle of intersection of two curves, then,
(tan \ θ = \left | \frac{m_{1}-m_{2}}{1 + m_{1}m_{2}} \right |)
Comprehension
Let $\mathrm{z} \in \mathrm{C}$ and $\mathrm{z}_{\mathrm{r}}(\mathrm{r}=\mathbf{1}, 2,3,4)$ satisfy the equation $|z+\bar{z}-4|+3|z-\bar{z}-2 i|=12$ such that $\left|z_r-(2+i)\right|$ is least.
$\sum_{r=1}^4\left|\operatorname{Re}\left(z_r\right)\right|$ is equal to
- 8
- $12 / 5$
- $8 / 5$
- $4 / 5$
- The value of definite integral $\int_{\frac{-1}{\sqrt{3}}}^{\frac{1}{\sqrt{3}}} \frac{\cos ^{-1}\left(\frac{2 x}{1+x^{2}}\right)+\tan ^{-1}\left(\frac{2 x}{1-x^{2}}\right)}{e^{x}+1} d x$ is equal to:
(b) $\frac{\pi}{\sqrt{3}}$
(c) $\frac{\pi}{4 \sqrt{3}}$
(d) $\frac{\pi}{3 \sqrt{3}}$
- $12 / 5$
- $4 / 5$ ✔️
- $8 / 5$
- (a) $\frac{\pi}{2 \sqrt{3}}$
Solution
The value of definite integral $\int_{\frac{-1}{\sqrt{3}}}^{\frac{1}{\sqrt{3}}} \frac{\cos ^{-1}\left(\frac{2 x}{1+x^{2}}\right)+\tan ^{-1}\left(\frac{2 x}{1-x^{2}}\right)}{e^{x}+1} d x$ is equal to:
(b) $\frac{\pi}{\sqrt{3}}$
(c) $\frac{\pi}{4 \sqrt{3}}$
(d) $\frac{\pi}{3 \sqrt{3}}$