The equation of the tangent to the curve $\mathrm{y} = {\left(1 + \mathrm{x}\right)}^{\mathrm{y}} + {\sin }^{-1}\left({\sin }^2\left(\mathrm{x}\right)\right)$

• x - y + 1 = 0

• x + y + 1 = 0

• 2x - y + 1 = 0

• x + 2y + 2 = 0

If $\left[\begin{array}{cc}2& 1\\ 3& 2\end{array}\right]\mathrm{A}\left[\begin{array}{cc}- 3& 2\\ 5& - 3\end{array}\right] = \left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$, then A is equal to :

• $- \left[\begin{array}{cc}1& 1\\ 1& 0\end{array}\right]$

• $\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]$

• $\left[\begin{array}{cc}1& 0\\ 1& 1\end{array}\right]$

• $\left[\begin{array}{cc}0& 1\\ 1& 1\end{array}\right]$

The point on the curve y = 2x² - 6x - 4 at which the tangent is parallel to the x-axis, is :

• $\left(\frac{3}{2}, \frac{13}{2}\right)$

• $\left(- \frac{5}{2}, - \frac{17}{2}\right)$

• $\left(\frac{3}{2}, \frac{17}{2}\right)$

• $\left(\frac{3}{2}, - \frac{17}{2}\right)$

The value of $\left|\begin{array}{ccc}\cos \left(\mathrm{x} - \mathrm{a}\right)& \cos \left(\mathrm{x} + \mathrm{a}\right)& \cos \left(\mathrm{x}\right)\\ \sin \left(\mathrm{x} + \mathrm{a}\right)& \sin \left(\mathrm{x} - \mathrm{a}\right)& \sin \left(\mathrm{x}\right)\\ \cos \left(\mathrm{a}\right)\tan \left(\mathrm{x}\right)& \cos \left(\mathrm{a}\right)\cot \left(\mathrm{x}\right)& \csc \left(2\mathrm{x}\right)\end{array}\right|$ is equal to :

• 1

• $\sin \left(\mathrm{a}\right)\cos \left(\mathrm{a}\right)$

• 0

• $\sin \left(\mathrm{x}\right)\cos \left(\mathrm{x}\right)$

Let f be twice differentiable function such that f"(x) = - f(x) and f'(x) = g(x), h(x) = {f(x)}² + {g(x)}². If h(5) = 11, then h(10) is equal to :

• 22

• 11

• 0

• 20

Suppose A is a matrix of order 3 and $\mathrm{B} = \left|\mathrm{A}\right|{\mathrm{A}}^{-1}$. If $\left|\mathrm{A}\right| = - 5$, then $\left|\mathrm{B}\right|$ is equal to :

• 1

• - 5

• - 1

• 25

7.

A differentiable function f(x) is defined for all x > 0 and satisfies f(x³) = 4x⁴ for all x > 0. The value of f'(8) is :

• $\frac{16}{3}$

• $\frac{32}{3}$

• $\frac{16\sqrt{2}}{3}$

• $\frac{32\sqrt{2}}{3}$

The value of $\cos \left[2{\tan }^{-1}\left(- 7\right)\right]$ is :

• $\frac{49}{50}$

• $- \frac{49}{50}$

• $\frac{24}{25}$

• $- \frac{24}{25}$

If $\left|\begin{array}{ccc}2& - 3\mathrm{i}& 1\\ 3& 3\mathrm{i}& - 1\\ 4& 3& \mathrm{i}\end{array}\right|$ = x + iy, then :

• x = 3, y = 1

• x = 2, y = 5

• x = 0, y = 0

• x = 1, y = 1

The value of the $\left|\begin{array}{ccc}{}^{10}\mathrm{C}_4& {}^{10}\mathrm{C}_5& {}^{11}\mathrm{C}_{\mathrm{m}}\\ {}^{11}\mathrm{C}_6& {}^{11}\mathrm{C}_7& {}^{12}\mathrm{C}_{\mathrm{m} + 2}\\ {}^{12}\mathrm{C}_8& {}^{12}\mathrm{C}_9& {}^{13}\mathrm{C}_{\mathrm{m} + 4}\end{array}\right| = 0$ when m is equal to :

• 6

• 5

• 4

• 1