If a² + b² + c² = - 2 and f(x) = $\left|\begin{array}{ccc}1 +\hspace{0.17em}{\mathrm{a}}^2\mathrm{x}& \left(1 +\hspace{0.17em}{\mathrm{b}}^2\right)\mathrm{x}& \left(1 +\hspace{0.17em}{\mathrm{c}}^2\right)\mathrm{x}\\ \left(1 +\hspace{0.17em}{\mathrm{a}}^2\right)\mathrm{x}& 1 +\hspace{0.17em}{\mathrm{b}}^2\mathrm{x}& \left(1 +\hspace{0.17em}{\mathrm{c}}^2\right)\mathrm{x}\\ \left(1 +\hspace{0.17em}{\mathrm{a}}^2\right)\mathrm{x}& \left(1 +\hspace{0.17em}{\mathrm{b}}^2\right)\mathrm{x}& 1 +\hspace{0.17em}{\mathrm{c}}^2\mathrm{x}\end{array}\right|$ then f(x) is a polynomial of degree

• 3

• 2

• 1

• 0

If A = $\left|\begin{array}{ccc}1& - 1& 1\\ 0& 2& - 3\\ 2& 1& 0\end{array}\right|$ and B = (adj A), and C = 5A, then $\frac{\left|\mathrm{adj} \mathrm{B}\right|}{\left|\mathrm{C}\right|}$ is

• 5

• 25

• - 1

• 1

If matrix A = $\left|\begin{array}{ccc}1& 0& - 1\\ 3& 4& 5\\ 0& 6& 7\end{array}\right|$ and its inverse is denoted by A^-1 = $\left|\begin{array}{ccc}{\mathrm{a}}_{11}& {\mathrm{a}}_{12}& {\mathrm{a}}_{13}\\ {\mathrm{a}}_{21}& {\mathrm{a}}_{22}& {\mathrm{a}}_{23}\\ {\mathrm{a}}_{31}& {\mathrm{a}}_{32}& {\mathrm{a}}_{33}\end{array}\right|$, then the value of a₂₃ is

• $\frac{21}{20}$

• $\frac{1}{5}$

• $- \frac{2}{5}$

• $\frac{2}{5}$

The number of solutions of the system of equations 2x + y - z = 7, x - 3y + 2z = 1 and x + 4y - 3z = 5 is

• 3

• 2

• 1

• 0

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

• $\mathrm{\lambda } \ne 3$

• $\mathrm{\mu } = 3 \mathrm{only}$

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

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

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

If A is invertible matrix and B is any matrix, then

• Rank (AB) = Rank(A)

• Rank (AB) = Rank (B)

• Rank (AB) > Rank (A)

• Rank (AB) > Rank (B)

Rank of the matrix $\mathrm{A} = \left[\begin{array}{ccc}1& 2& 2\\ 2& 1& 2\\ 2& 2& 1\end{array}\right]$ is

• 0

• 1

• 2

• 3

380.

If w is a cube root of unity, then $\left|\begin{array}{ccc}1& \mathrm{w}& {\mathrm{w}}^2\\ \mathrm{w}& {\mathrm{w}}^2& 1\\ {\mathrm{w}}^2& 1& \mathrm{w}\end{array}\right|$ is equal to

• 1

• w

• w²

• 0