$\mathrm{The} \mathrm{inverse} \mathrm{of} \mathrm{the} \mathrm{matrix} \left[\begin{array}{ccc}7& - 3& - 3\\ - 1& 1& 0\\ - 1& 0& 1\end{array}\right] \mathrm{is}$

• $\left[\begin{array}{ccc}1& 3& 3\\ 1& 4& 3\\ 1& 3& 4\end{array}\right]$

• $\left[\begin{array}{ccc}1& 3& 1\\ 4& 3& 8\\ 3& 4& 1\end{array}\right]$

• $\left[\begin{array}{ccc}1& 1& 1\\ 3& 3& 4\\ 3& 4& 3\end{array}\right]$

• $\left[\begin{array}{ccc}1& 1& 1\\ 3& 4& 3\\ 3& 3& 4\end{array}\right]$

$\mathrm{If} {\sin }^{-1}\left(\frac{3}{\mathrm{x}}\right) + {\sin }^{-1}\left(\frac{4}{\mathrm{x}}\right) = \frac{\mathrm{\pi }}{2}, \mathrm{then} \mathrm{x} = ?$

• 3

• 5

• 7

• 11

$$\begin{gathered} \mathrm{If} \mathrm{f} : \mathrm{R} \to \mathrm{R} \mathrm{is} \mathrm{defined} \mathrm{by} \\ \mathrm{f}\left(\mathrm{x}\right) = \left\{\frac{\cos \left(3\mathrm{x}\right) - \cos \left(\mathrm{x}\right)}{{\mathrm{x}}^2}, \mathrm{for} \mathrm{x} \ne 0 \\ \mathrm{\lambda }, \mathrm{for} \mathrm{x} = 0\right\} \\ \mathrm{and} \mathrm{if} \mathrm{f} \mathrm{is} \mathrm{continuous} \mathrm{at} \mathrm{x} = 0, \mathrm{then} \mathrm{\lambda } = ? \end{gathered}$$

• - 2

• - 4

• - 6

• - 8

$$\begin{gathered} \mathrm{If} \mathrm{f}\left(2\right) = 4 \mathrm{and} \mathrm{f}\text{'}\left(2\right) = 1, \mathrm{then} \\ \underset{\mathrm{x}\to 2}{\lim }\frac{\mathrm{xf}\left(2\right) - 2\mathrm{f}\left(\mathrm{x}\right)}{\mathrm{x} - 2} = ? \end{gathered}$$

• - 2

• 1

• 2

• 3

55.

If $\mathrm{y} = \sin \left({\log }_{\mathrm{e}}\left(\mathrm{x}\right)\right)$, then ${\mathrm{x}}^2\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} + \mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to

• $\mathrm{y} = \sin \left({\log }_{\mathrm{e}}\left(\mathrm{x}\right)\right)$

• $\cos \left({\log }_{\mathrm{e}}\left(\mathrm{x}\right)\right)$

• y²

• - y

The equation of the normal to the curve y⁴ = ax³ at (a, a) is

• x + 2y = 3a

• 3x - 4y + a = 0

• 4x + 3y = 7a

• 4x - 3y = 0

The angle between the curves y = 4x + 4 and y² = 36(9 - x) is

• 30°

• 45°

• 60°

• 90°

If m and M respectively denote the minimum and maximum of f(x) = (x - 1)² + 3 for x $\in$ [- 3, 1], then the ordered pair (m, M) is equal to

• (- 3, 19)

• (3, 19)

• (- 19, 3)

• (- 19, - 3)

The length of the subtangent at (2, 2) to thecurve x⁵ = 2y⁴ is

• $\frac{5}{2}$

• $\frac{8}{5}$

• $\frac{2}{5}$

• $\frac{5}{8}$

If z = ${\sec }^{-1}\left(\frac{{\mathrm{x}}^4 + {\mathrm{y}}^4 - 8{\mathrm{x}}^2{\mathrm{y}}^2}{{\mathrm{x}}^2 + {\mathrm{y}}^2}\right)$, then $\mathrm{x}\frac{\partial \mathrm{z}}{\partial \mathrm{y}} + \mathrm{y}\frac{\partial \mathrm{z}}{\partial \mathrm{y}}$ is equal to

• $\cot \left(\mathrm{z}\right)$

• $2\cot \left(\mathrm{z}\right)$

• $2\tan \left(\mathrm{z}\right)$

• $2\sec \left(\mathrm{z}\right)$