81.

If e$\left[\begin{array}{cc}{\mathrm{e}}^{\mathrm{x}}& {\mathrm{e}}^{\mathrm{y}}\\ {\mathrm{e}}^{\mathrm{y}}& {\mathrm{e}}^{\mathrm{x}}\end{array}\right] = \left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$, [he the values of x and y are respectively

• - 1, - 1

• 1, 1

• 0, 1

• 1, 0

If f(x) = $\left|\begin{array}{cc}\mathrm{x}& \mathrm{\lambda }\\ 2\mathrm{\lambda }& \mathrm{x}\end{array}\right|$, then $\mathrm{f}\left(\mathrm{\lambda x}\right) - \mathrm{f}\left(\mathrm{x}\right)$ is equal to

• $\mathrm{x}\left({\mathrm{\lambda }}^2 - 1\right)$

• $2\mathrm{\lambda }\left({\mathrm{x}}^2 - 1\right)$

• ${\mathrm{\lambda }}^2\left({\mathrm{x}}^2 - 1\right)$

• ${\mathrm{x}}^2\left({\mathrm{\lambda }}^2 - 1\right)$

If $\mathrm{A} = {\left({\mathrm{a}}_{\mathrm{ij}}\right)}_{\mathrm{m} \times \mathrm{n}}, \mathrm{B} = {\left({\mathrm{b}}_{\mathrm{ij}}\right)}_{\mathrm{n} \times \mathrm{p}}$, and C = ${\left({\mathrm{C}}_{\mathrm{ij}}\right)}_{\mathrm{p} \times \mathrm{q}}$, then the product (BC) A is possible only when

• m = q

• n = q

• p = q

• m = p

If a, b, c are non-zero and different from 1, then the value of $\left|\begin{array}{ccc}{\log }_{\mathrm{a}}\left(1\right)& {\log }_{\mathrm{a}}\left(\mathrm{b}\right)& {\log }_{\mathrm{a}}\left(\mathrm{c}\right)\\ {\log }_{\mathrm{a}}\left(\frac{1}{\mathrm{b}}\right)& {\log }_{\mathrm{b}}\left(1\right)& {\log }_{\mathrm{a}}\left(\frac{1}{\mathrm{c}}\right)\\ {\log }_{\mathrm{a}}\left(\frac{1}{\mathrm{c}}\right)& {\log }_{\mathrm{a}}\left(\mathrm{c}\right)& {\log }_{\mathrm{c}}\left(1\right)\end{array}\right|$ is

• 0

• $1 + {\log }_{\mathrm{a}}\left(\mathrm{a} +\hspace{0.17em}\mathrm{b}\hspace{0.17em}+ \mathrm{c}\right)$

• ${\log }_{\mathrm{a}}\left(\mathrm{ab} + \mathrm{bc} + \mathrm{ca}\right)$

• 1

If $∆ = \left|\begin{array}{ccc}1& 2& 3\\ 2& 3& 5\\ 3& 6& 12\end{array}\right|$ and $∆\text{'} = \left|\begin{array}{ccc}4& 8& 15\\ 3& 6& 12\\ 2& 3& 5\end{array}\right|$ then

• $∆\text{'} = 2∆$

• $∆\text{'} = - 2∆$

• $∆\text{'} = ∆$

• $∆ = - ∆\text{'}$

The roots of the equation $\left|\begin{array}{ccc}1 +\hspace{0.17em}\mathrm{x}& 3& 5\\ 2& 2 +\hspace{0.17em}\mathrm{x}& 5\\ 2& 3& \mathrm{x} +\hspace{0.17em}4\end{array}\right|$ =  0 are

• 2, 1, - 9

• 1, 1, - 9

• - 1, 1, - 9

• - 2, - 1, - 8

If $\left|\begin{array}{ccc}0& 3& 2\mathrm{b}\\ 2& 0& 1\\ 4& - 1& 6\end{array}\right|$ is singular, then the value of  b is

• - 3

• 3

• - 6

• 6

If A and B are square matrices of the same order and if A = A^T, B = B^T, then (ABA)^T is equal to

• BAB

• ABA

• ABAB

• AB^T

If the square of the matrix $\left[\begin{array}{cc}\mathrm{a}& \mathrm{b}\\ \mathrm{a}& - \mathrm{a}\end{array}\right]$ is the unit matrix, then b is equal to

• $\frac{\mathrm{a}}{1 + {\mathrm{a}}^2}$

• $\frac{1 - {\mathrm{a}}^2}{\mathrm{a}}$

• $\frac{1 + {\mathrm{a}}^2}{\mathrm{a}}$

• $\frac{\mathrm{a}}{1 - {\mathrm{a}}^2}$

If [1 x 1]$\left[\begin{array}{ccc}1& 3& 2\\ 0& 5& 1\\ 0& 2& 0\end{array}\right]\left[\begin{array}{c}1\\ 1\\ \mathrm{x}\end{array}\right] = 0$, then the values of x are

• 1, 5

• - 1, - 5

• 1, 6

• - 1, - 6