The length of the normal to the curve $\mathrm{x} = \mathrm{a}\left(\mathrm{\theta } + \sin \left(\mathrm{\theta }\right)\right)$, y =  $\mathrm{a}\left(1 - \cos \left(\mathrm{\theta }\right)\right) \mathrm{at} \mathrm{\theta } = \frac{\mathrm{\pi }}{2}$ is

• 2a

• $\frac{\mathrm{a}}{2}$

• $\frac{\mathrm{a}}{\sqrt{2}}$

• $\sqrt{2}\mathrm{a}$

The maximum value of $\frac{\log \left(\mathrm{x}\right)}{\mathrm{x}}$ is

• e

• 2e

• $\frac{1}{\mathrm{e}}$

• $\frac{2}{\mathrm{e}}$

In the interval $\left[- \frac{\mathrm{\pi }}{4}, \frac{\mathrm{\pi }}{4}\right]$, the number of real solutions of the equations $\left|\begin{array}{ccc}\sin \left(\mathrm{x}\right)& \cos \left(\mathrm{x}\right)& \cos \left(\mathrm{x}\right)\\ \cos \left(\mathrm{x}\right)& \sin \left(\mathrm{x}\right)& \cos \left(\mathrm{x}\right)\\ \cos \left(\mathrm{x}\right)& \cos \left(\mathrm{x}\right)& \sin \left(\mathrm{x}\right)\end{array}\right|$ = 0 is

• 0

• 2

• 1

• 3

If f(x) = $$\begin{gathered} \left\{\mathrm{xsin}\left(\frac{1}{\mathrm{x}}\right), \mathrm{x} \ne 0 \\ \mathrm{k} , \mathrm{x} = 0\right\} \end{gathered}$$ is continuous at x = 0, then the value of k will be

• 1

• - 1

• 0

• None of these

If $\cot \left({\cos }^{-1}\left(\mathrm{x}\right)\right) = \sec \left({\tan }^{-1}\left(\frac{\mathrm{a}}{\sqrt{{\mathrm{b}}^2 - {\mathrm{a}}^2}}\right)\right)$, then x is equal to

• $\frac{\mathrm{b}}{\sqrt{2{\mathrm{b}}^2 - {\mathrm{a}}^2}}$

• $\frac{\mathrm{a}}{\sqrt{2{\mathrm{b}}^2 - {\mathrm{a}}^2}}$

• $\frac{\sqrt{2{\mathrm{b}}^2 - {\mathrm{a}}^2}}{\mathrm{a}}$

• $\frac{\sqrt{2{\mathrm{b}}^2 - {\mathrm{a}}^2}}{\mathrm{b}}$

6.

Number of solutions ofthe equation

${\tan }^{-1}\left(\frac{1}{2\mathrm{x} + 1}\right) + {\tan }^{-1}\left(\frac{1}{4\mathrm{x} + 1}\right) = {\tan }^{-1}\left(\frac{2}{{\mathrm{x}}^2}\right)$ is

• 1

• 2

• 3

• 4

If f(x) = $\frac{4^{\mathrm{x}}}{4^{\mathrm{x}} + 2}$, then $\mathrm{f}\left(\frac{1}{97}\right) + \mathrm{f}\left(\frac{2}{97}\right) + ... + \mathrm{f}\left(\frac{96}{97}\right)$ is equal to

• 1

• 48

• - 48

• - 1

If f(x) = $$\begin{gathered} \left\{{\mathrm{x}}^{\mathrm{p}}\cos \left(\frac{1}{\mathrm{x}}\right), \mathrm{x} \ne 0 \\ 0 , \mathrm{x} = 0\right\} \end{gathered}$$ is differentiable at x = 0, then

• p < 0

• 0 < p < 1

• p = 1

• p > 1

If $\left|\begin{array}{ccc}\mathrm{x}& {\mathrm{x}}^2& 1 +\hspace{0.17em}{\mathrm{x}}^3\\ \mathrm{y}& {\mathrm{y}}^2& 1 + {\mathrm{y}}^3\\ \mathrm{z}& {\mathrm{z}}^2& 1 + {\mathrm{z}}^3\end{array}\right| = 0$ and x, y, z are all distinct, then xyz is equal to

• - 1

• 1

• 0

• 3

If A = $\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$, then A¹⁰⁰ is equal to

• 100 A

• 2⁹⁹ A

• 2¹⁰⁰ A

• 99 A