What must be the matrix X if $2\mathrm{X} + \left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right] = \left[\begin{array}{cc}3& 8\\ 7& 2\end{array}\right]$ ?

• $\left[\begin{array}{cc}1& 3\\ 2& - 1\end{array}\right]$

• $\left[\begin{array}{cc}1& - 3\\ 2& - 1\end{array}\right]$

• $\left[\begin{array}{cc}2& 6\\ 4& - 2\end{array}\right]$

• $\left[\begin{array}{cc}2& - 6\\ 4& - 2\end{array}\right]$

The value of $\left|\begin{array}{ccc}1& 1& 1\\ \mathrm{bc}& \mathrm{ca}& \mathrm{ab}\\ \mathrm{b} + \mathrm{c}& \mathrm{c} + \mathrm{a}& \mathrm{a} + \mathrm{b}\end{array}\right|$ is

• 1

• 0

• (a - b)(b - c)(c - a)

• (a + b)(b + c)(c + a)

The value of $\left|\begin{array}{ccc}441& 442& 443\\ 445& 446& 447\\ 449& 450& 451\end{array}\right|$ is :

• $441 \times 446 \times 4510$

• 0

• - 1

• 1

4.

Inverse of the matrix $\left[\begin{array}{cc}\cos \left(2\mathrm{\theta }\right)& - \sin \left(2\mathrm{\theta }\right)\\ \sin \left(2\mathrm{\theta }\right)& \cos \left(2\mathrm{\theta }\right)\end{array}\right]$ is

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

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

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

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

If the distance 's' metres traversed by a particle int seconds is given by s = t³ - 3t², then the velocity of the particle when the acceleration is zero, in m/s is

• 3

• - 2

• - 3

• 2

If x = $\mathrm{Acos}\left(4\mathrm{t}\right) + \mathrm{Bsin}\left(4\mathrm{t}\right)$, then $\frac{{\mathrm{d}}^2\mathrm{x}}{{\mathrm{dt}}^2}$ is equal to

• - 16x

• 16x

• x

• - x

If tangent to the curve x = at², y = 2at is perpendicular to X - axis, then its point of contact is

• (a, a)

• (0, a)

• (0, 0)

• (a, 0)

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

• 0

• $\frac{1}{2}$

• $\frac{1}{4}$

• $\frac{1}{2}$

If ${\cos }^{-1}\left(\mathrm{p}\right) +\hspace{0.17em}{\cos }^{-1}\left(\mathrm{q}\right) +\hspace{0.17em}{\cos }^{-1}\left(\mathrm{r}\right) = \mathrm{\pi }$, then p² + q² + r² + 2pqr is equal to

• 3

• 1

• 2

• - 1

If ${\sin }^{-1}\left(\frac{\mathrm{x}}{5}\right) +\hspace{0.17em}{\csc }^{-1}\left(\frac{5}{4}\right) = \frac{\mathrm{\pi }}{2}$, then x is equal to

• 1

• 4

• 3

• 5