Mathematics : Matrices

$\mathrm{If} \left[\begin{array}{ccc}1& 2& \mathrm{x}\\ 4& - 1& 7\\ 2& 4& - 6\end{array}\right] \mathrm{is} \mathrm{a} \mathrm{singular} \mathrm{matrix}, \mathrm{then} \mathrm{x} \mathrm{is}$

• 0

• 1

• - 3

• 3

The number of non-trivial solutions of the system x - y + z = 0, x + 2y - z = 0, 2x + y + 3z = 0 is :

• 0

• 1

• 2

• 3

$$\begin{gathered} \mathrm{If} \mathrm{A} =\left[\begin{array}{cc}1& - 2\\ 4& 5\end{array}\right] \mathrm{and} \mathrm{f}\left(\mathrm{t}\right) = {\mathrm{t}}^2 - 3\mathrm{t} +\hspace{0.17em}7, \mathrm{then} \\ \mathrm{f}\left(\mathrm{A}\right) + \left[\begin{array}{cc}3& 6\\ - 12& - 9\end{array}\right] = ? \end{gathered}$$

• $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

• $\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$

• $\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$

• $\left[\begin{array}{cc}1& 1\\ 0& 0\end{array}\right]$

Let A and B be two symmetric matrices of same order. Then, the matrix AB - BA is

• a symmetnc matrix

• a skew-symmetric matrix

• a null matrix

• the identity matrix

If one of the roots of $\left|\begin{array}{ccc}3& 5& \mathrm{x}\\ 7& \mathrm{x}& 7\\ \mathrm{x}& 5& 3\end{array}\right|$ = 0 is - 10, then the other roots are

• 3, 7

• 4, 7

• 3, 9

• 3, 4

$\mathrm{If} \frac{3{\mathrm{x}}^2 + \mathrm{x} + 1}{{\left(\mathrm{x} - 1\right)}^4} = \frac{\mathrm{a}}{\left(\mathrm{x} - 1\right)} + \frac{\mathrm{b}}{{\left(\mathrm{x} - 1\right)}^2} +\hspace{0.17em}\frac{\mathrm{c}}{{\left(\mathrm{x} - 1\right)}^3} + \frac{\mathrm{d}}{{\left(\mathrm{x} - 1\right)}^4}, \mathrm{then} \left[\begin{array}{cc}\mathrm{a}& \mathrm{b}\\ \mathrm{c}& \mathrm{d}\end{array}\right] = ?$

• $\left[\begin{array}{cc}3& 7\\ 5& 0\end{array}\right]$

• $\left[\begin{array}{cc}0& 3\\ 7& 5\end{array}\right]$

• $\left[\begin{array}{cc}0& 7\\ 3& 5\end{array}\right]$

• $\left[\begin{array}{cc}3& 5\\ 7& 0\end{array}\right]$

$\mathrm{If} \left|\begin{array}{ccc}\mathrm{x}& {\mathrm{x}}^2& 1 + {\mathrm{x}}^3\\ \mathrm{y}& {\mathrm{y}}^2& 1 + {\mathrm{y}}^3\\ \mathrm{z}& {\mathrm{z}}^2& 1 + {\mathrm{z}}^3\end{array}\right| = 0, \mathrm{x}\ne \mathrm{y} \ne \mathrm{z}, \mathrm{then} 1 + \mathrm{xyz} = ?$

• 0

• - 1

• 1

• 2

$$\begin{gathered} \mathrm{If} \mathrm{the} \mathrm{system} \mathrm{of} \mathrm{equations} \\ {\left(\mathrm{k} + 1\right)}^3\mathrm{x} + {\left(\mathrm{k} + 2\right)}^3\mathrm{y} = {\left(\mathrm{k} + 3\right)}^3 \\ \left(\mathrm{k} + 1\right)\mathrm{x} + \left(\mathrm{k} + 2\right)\mathrm{y} = \mathrm{k} +\hspace{0.17em}3 \\ \mathrm{x} + \mathrm{y} = 1 \\ \mathrm{is} \mathrm{consistent}, \mathrm{then} \mathrm{the} \mathrm{value} \mathrm{of} \mathrm{k} \mathrm{is} \end{gathered}$$

• 2

• - 2

• - 1

• 1

If A is a non-zero square matrix of order n with det (I + A) $\ne$ 0 and A³ = 0, where I, 0 are unit and null matrices of order n x n respectively, then (I + A)^{- 1} is equal to

• I - A + A²

• I + A + A²

• I + A ^{- 1}

• I + A

$\mathrm{If} \mathrm{x} \mathrm{is} \mathrm{real}, \mathrm{then} \mathrm{the} \mathrm{value} \mathrm{of} \frac{{\mathrm{x}}^2 - 3\mathrm{x} + 4}{{\mathrm{x}}^2 + 3\mathrm{x} + 4} \mathrm{lies} \mathrm{in} \mathrm{the} \mathrm{interval}$

• $\left[\frac{1}{3}, 3\right]$

• $\left[\frac{1}{5}, 5\right]$

• $\left[\frac{1}{6}, 6\right]$

• $\left[\frac{1}{7}, 7\right]$