If ${\tan }^{-1}\left(\frac{\mathrm{x} + 1}{\mathrm{x} - 1}\right) + {\tan }^{-1}\left(\frac{\mathrm{x} - 1}{\mathrm{x}}\right) = {\tan }^{-1}\left(- 7\right)$, then the value of x is

• zero

• - 2

• 1

• 2

Let the function $\mathrm{f} : \mathrm{R} \to \mathrm{R}$ be defined by f(x) = 2x + cos(x), then f

• has maximum at x = 0

• has minimum at x = $\mathrm{\pi }$

• is an increasing function

• is a decreasing

If ${\cos }^{-1}\left(\sqrt{\mathrm{p}}\right) + {\cos }^{-1}\left(\sqrt{1 - \mathrm{p}}\right) +\hspace{0.17em}{\cos }^{-1}\left(\sqrt{1 - \mathrm{q}}\right) = \frac{3\mathrm{\pi }}{4}$, then the value of q is

• $\frac{2}{\sqrt{2}}$

• 1

• 1/2

• 1/3

If 5 is one root of the equations $\left|\begin{array}{ccc}\mathrm{x}& 3& 7\\ 2& \mathrm{x}& - 2\\ 7& 8& \mathrm{x}\end{array}\right| = 0$, then the other two roots of the equation are

• - 2, - 7

• - 2, 7

• 2, - 7

• 2, 7

The equation to the tangent to the curve y = be^{- x/a} at the point where it crosses the Y-axis is

• ax + by = 1

• $\frac{\mathrm{x}}{\mathrm{a}} - \frac{\mathrm{y}}{\mathrm{b}} = 1$

• $\frac{\mathrm{x}}{\mathrm{a}} + \frac{\mathrm{y}}{\mathrm{b}} = 1$

• ax - by = 1

6.

If w = $- \frac{1}{2} + \frac{\sqrt{3}\mathrm{i}}{2}$, the value of the determinant $\left|\begin{array}{ccc}1& \mathrm{w}& {\mathrm{w}}^2\\ \mathrm{w}& {\mathrm{w}}^2& 1\\ {\mathrm{w}}^2& 1& \mathrm{w}\end{array}\right|$ is

• zero

• 3

• - 1

• 1

The height of the cylinder of maximum volume inscribed in a sphere of radius 'a' is

• $\frac{3\mathrm{a}}{2}$

• $\frac{\sqrt{2}\mathrm{a}}{3}$

• $\frac{\mathrm{a}}{\sqrt{3}}$

• $\frac{2\mathrm{a}}{\sqrt{3}}$

If f(x) = $\left|\begin{array}{ccc}\sin \left(\mathrm{x}\right)& \cos \left(\mathrm{x}\right)& \tan \left(\mathrm{x}\right)\\ {\mathrm{x}}^3& {\mathrm{x}}^2& \mathrm{x}\\ 2\mathrm{x}& 1& 1\end{array}\right|$, then $\underset{\mathrm{x}\to 0}{\lim }\frac{\mathrm{f}\left(\mathrm{x}\right)}{{\mathrm{x}}^2}$ is

• - 1

• 3

• 1

• zero

If for a triangle ABC, $\left|\begin{array}{ccc}1& \mathrm{a}& \mathrm{b}\\ 1& \mathrm{b}& \mathrm{a}\\ 1& \mathrm{c}& \mathrm{c}\end{array}\right| = 0$, then the value of ${\sin }^2\left(\mathrm{A}\right) + {\sin }^2\left(\mathrm{B}\right) + {\sin }^2\left(\mathrm{C}\right)$ is

• $\frac{4}{9}$

• $\frac{9}{4}$

• 1

• $\frac{3\sqrt{3}}{2}$

Let f(x) = $$\begin{gathered} \left\{\frac{\sin \left(\mathrm{\pi x}\right)}{5\mathrm{x}}, \mathrm{x} \ne 0 \\ \mathrm{k}, \mathrm{x} = 0\right\} \end{gathered}$$. If f(x) continuous at x = 0, the value of k is

• $\frac{5}{\mathrm{\pi }}$

• $\frac{\mathrm{\pi }}{5}$

• zero

• 1