The solution of equation 

${\sin }^{-1}\left(1 - \frac{\mathrm{x}}{2}\right) - 2{\sin }^{-1}\left(\frac{\mathrm{x}}{2}\right) = \frac{\mathrm{\pi }}{2}$ is

• x = 0

• $\mathrm{x} = \frac{1}{2}$

• $\mathrm{x} = 0 \mathrm{and} \frac{1}{2}$

• None of these

Let f(x) = $$\begin{gathered} \left\{- 2\sin \left(\mathrm{x}\right), - \mathrm{\pi } \le \mathrm{x} \le - \frac{\mathrm{\pi }}{2} \\ \mathrm{asin}\left(\mathrm{x}\right) + \mathrm{b}, - \frac{\mathrm{\pi }}{2} \le \mathrm{x} \le \frac{\mathrm{\pi }}{2} \\ \cos \left(\mathrm{x}\right), \frac{\mathrm{\pi }}{2} \le \mathrm{x} \le \mathrm{\pi }\right\} \end{gathered}$$. If f(x) is continuous on $\left[- \mathrm{\pi }, \mathrm{\pi }\right]$, then

• a = 1, b = 1

• a = - 1, b = - 1

• a = - 1, b = 1

• a = 1, b = - 1

If f(x) = $$\begin{gathered} \left\{\frac{\mathrm{x}}{1 + \exp \left({\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\mathrm{x}$}\right.}\right) }, \mathrm{x} \ne 0 \\ 0 , \mathrm{x} = 0\right\} \end{gathered}$$, then f(x) at x = 0 is

• continuous

• not continuous

• differentiable

• not differentiable

The maximum value of f(x) = $2\sin \left(\mathrm{x}\right) + \cos \left(2\mathrm{x}\right), 0 \le \mathrm{x} \le \frac{\mathrm{\pi }}{2}$ occurs at x is equal to

• 0

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$6$}\right.$

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$

• None of these

If f(x) = $\left|\mathrm{x} - 2\right|$ and g(x) = f(f(x)), then $\frac{\mathrm{dg}\left(\mathrm{x}\right)}{\mathrm{dx}}$ for x > 10 is

• 1 or - 1

• 1

• - 1

• None of these

If y = ${\sin }^{-1}\left(\frac{\sqrt{1 + \mathrm{x}} + \sqrt{1 - \mathrm{x}}}{2}\right)$, then the value of $\frac{\mathrm{dy}}{\mathrm{dx}}$ at x = 0 is

• 1/2

• - 1/2

• 0

• None of these

The distance (in metre) travelled by a vehicle in time t (in seconds) is given by the equation s = 3t³ + 2t² + t + 1. The difference in the acceleration between t = 2 and t = 4 is

• 36 m/s²

• 38 m/s²

• 45 m/s²

• 46 m/s²

The value of $\left[\begin{array}{cc}1& - \tan \left(\frac{\mathrm{\theta }}{4}\right)\\ \tan \left(\frac{\mathrm{\theta }}{4}\right)& 1\end{array}\right]{\left[\begin{array}{cc}1& \tan \left(\frac{\mathrm{\theta }}{4}\right)\\ - \tan \left(\frac{\mathrm{\theta }}{4}\right)& 1\end{array}\right]}^{-1}$ is

• $\left[\begin{array}{cc}\cos \left(\frac{\mathrm{\theta }}{2}\right)& \sin \left(\frac{\mathrm{\theta }}{2}\right)\\ - \sin \left(\frac{\mathrm{\theta }}{2}\right)& \cos \left(\frac{\mathrm{\theta }}{2}\right)\end{array}\right]$

• $\left[\begin{array}{cc}\cos \left(\frac{\mathrm{\theta }}{2}\right)& - \sin \left(\frac{\mathrm{\theta }}{2}\right)\\ \sin \left(\frac{\mathrm{\theta }}{2}\right)& \cos \left(\frac{\mathrm{\theta }}{2}\right)\end{array}\right]$

• $\left[\begin{array}{cc}\sin \left(\frac{\mathrm{\theta }}{2}\right)& \cos \left(\frac{\mathrm{\theta }}{2}\right)\\ \cos \left(\frac{\mathrm{\theta }}{2}\right)& \sin \left(\frac{\mathrm{\theta }}{2}\right)\end{array}\right]$

• $\left[\begin{array}{cc}\sin \left(\frac{\mathrm{\theta }}{2}\right)& - \cos \left(\frac{\mathrm{\theta }}{2}\right)\\ \cos \left(\frac{\mathrm{\theta }}{2}\right)& \sin \left(\frac{\mathrm{\theta }}{2}\right)\end{array}\right]$

The value of $\mathrm{\lambda }$ and µ for whichi the simultaneous equation x + y + z = 6, x + 2y + 3z = 10 and x + 2y + $\mathrm{\lambda }$z = µ have a unique solution are

• $\mathrm{\lambda } \ne 3$

• $\mathrm{\mu } = 3 \mathrm{only}$

• $\mathrm{\lambda } = 3 \mathrm{and} \mathrm{\mu } = 3$

• $\mathrm{\lambda } \ne 3 \mathrm{and} \mathrm{\mu } \mathrm{can} \mathrm{take} \mathrm{any} \mathrm{value}.$

10.

If X is any matrix of order n x p (n and p are integers) and I is an identity matrix of order nxn, then the matrix M = I -  X(X'X)^-1X' is

(i) idempotent matrix (ii) MX = 0

Choose the correct answer

• (i) is correct

• (ii) is correct

• (i) is incorrect

• (ii) is incorrect