The angle between the tangents at those points on the curve x = t² + 1 and y = t² - t - 6 where it meets x-axis is :

• $\pm {\tan }^{-1}\left(\frac{4}{29}\right)$

• $\pm {\tan }^{-1}\left(\frac{5}{49}\right)$

• $\pm {\tan }^{-1}\left(\frac{10}{49}\right)$

• $\pm {\tan }^{-1}\left(\frac{8}{29}\right)$

If $\mathrm{A}\left(\mathrm{\theta }\right) = \left[\begin{array}{cc}1& \tan \left(\mathrm{\theta }\right)\\ - \tan \left(\mathrm{\theta }\right)& 1\end{array}\right] \mathrm{and} \mathrm{AB} = \mathrm{I}, \mathrm{then} \left({\sec }^2\left(\mathrm{\theta }\right)\right)\mathrm{B}$ is equal to :

• $\mathrm{A}\left(\mathrm{\theta }\right)$

• $\mathrm{A}\left(\frac{\mathrm{\theta }}{2}\right)$

• $\mathrm{A}\left(- \mathrm{\theta }\right)$

• $\mathrm{A}\left(\frac{- \mathrm{\theta }}{2}\right)$

23.

If x = - 5 is a root of $\left|\begin{array}{ccc}2\mathrm{x} + 1& 4& 8\\ 2& 2\mathrm{x}& 2\\ 7& 6& 2\mathrm{x}\end{array}\right|$ = 0, then the other roots are :

• 3, 3.5

• 1, 3.5

• 1, 7

• 2, 7

The simultaneous equations Kx + 2y - z = 1, (K - I)y - 2z = 2 and (K + 2)z = 3 have only one solution when :

• K = - 2

• K = - 1

• K = 0

• K = 1

If the rank of the matrix $\left[\begin{array}{ccc}- 1& 2& 5\\ 2& - 4& \mathrm{a} - 4\\ 1& - 2& \mathrm{a} +\hspace{0.17em}1\end{array}\right]$ is 1, then the value of a is :

• - 1

• 2

• - 6

• 4

If ${\sin }^{-1}\left(\mathrm{x}\right) + {\sin }^{-1}\left(\mathrm{y}\right) = \frac{\mathrm{\pi }}{2}$, then ${\cos }^{-1}\left(\mathrm{x}\right) + {\cos }^{-1}\left(\mathrm{y}\right)$ is equal to :

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{4}$

• $\mathrm{\pi }$

• $\frac{3\mathrm{\pi }}{4}$

The value of ${\int }_1^4\left|\mathrm{x} - 3\right|\mathrm{dx}$ is equal to :

• 1

• $\frac{5}{2}$

• $\frac{1}{2}$

• $\frac{3}{2}$

The area of the region bounded by the straight lines x = 0 and x = 2^x and the curves y = 2 and y = 2x - x² is equal to

• $\frac{2}{\log \left(2\right)} - \frac{4}{3}$

• $\frac{3}{\log \left(2\right)} - \frac{4}{3}$

• $\frac{1}{\log \left(2\right)} - \frac{4}{3}$

• $\frac{4}{\log \left(2\right)} - \frac{3}{2}$

The value of ${\int }_0^{\infty }\frac{\mathrm{dx}}{\left({\mathrm{a}}^2 + {\mathrm{x}}^2\right)}$ is equal to

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{2\mathrm{a}}$

• $\frac{\mathrm{\pi }}{\mathrm{a}}$

• $\frac{1}{2\mathrm{a}}$

The value of the integral $\int {\mathrm{e}}^{\mathrm{x}}{\left(\frac{1 - \mathrm{x}}{1 + {\mathrm{x}}^2}\right)}^2\mathrm{dx}$ is

• ${\mathrm{e}}^{\mathrm{x}}\left(\frac{1 - \mathrm{x}}{1 + {\mathrm{x}}^2}\right) + \mathrm{c}$

• ${\mathrm{e}}^{\mathrm{x}}\left(\frac{1 + \mathrm{x}}{1 + {\mathrm{x}}^2}\right)$

• $\frac{{\mathrm{e}}^{\mathrm{x}}}{1 + {\mathrm{x}}^2} + \mathrm{c}$

• ${\mathrm{e}}^{\mathrm{x}}\left(1 - \mathrm{x}\right) + \mathrm{c}$