Question

Match the statements given in List 1 with the values given in List 2

Answer 1

A) $\left|\overrightarrow{a}\right|=2,\left|\overrightarrow{b}\right|=2,\left|\overrightarrow{c}\right|=2\sqrt{3}$
Angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is
$\cos \theta =\frac{{\left|\overrightarrow{a}\right|}^2+{\left|\overrightarrow{b}\right|}^2-{\left|\overrightarrow{c}\right|}^2}{2\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|}=\frac{4+4-12}{2\times 2\times 2}=-\frac{1}{2}$
$\Rightarrow \theta =\frac{2\pi }{3}$
B) ${\int }_a^b\left(f\right(x)-3x)dx=a^2-b^2$
$\Rightarrow {\int }_a^{b}f\left(x\right)dx-{\int }_a^{b}3xdx=a^2-b^2$
$\Rightarrow {\int }_a^{b}f\left(x\right)dx=\frac{b^2-a^2}{2}$
$\Rightarrow f\left(x\right)=x$
$\Rightarrow f\left(\frac{\pi }{6}\right)=\frac{\pi }{6}$
C) $I=\frac{{\pi }^2}{\ln 3}{\int }_{7/6}^{5/6}\sec \left(\pi x\right)dx$
$=\frac{{\pi }^2}{\ln 3}\left[ln|\right(\sec \left(\pi x\right)+\tan \left(\pi x\right))|{]}_{7/6}^{5/6}$ $=\frac{{\pi }^2}{\ln 3}\times \frac{1}{\pi }\ln 3$
$=\pi$
D) Let $u=\frac{1}{1-z}$

$\Rightarrow z=1-\frac{1}{u}$

Given that $|z|=1\Rightarrow |1-\frac{1}{u}|=1$

$\Rightarrow |u-1|=|u|$

$\therefore$ locus of $u$ is perpendicular bisector of line segment joining $(1,0)$ and $(0,0)$ in complex plane.

$\Rightarrow$ maximum argument approaches $\frac{\pi }{2}$