The system of linear equations kx + y + z = 1, x + ky + z = 1 and x + y + kz = 1 has a unique solution under which one of the following conditions ?

• $\mathrm{k} \ne 1 \mathrm{and} \mathrm{k} \ne - 2$

• $\mathrm{k} \ne 1 \mathrm{and} \mathrm{k} \ne 2$

• $\mathrm{k} \ne - 1 \mathrm{and} \mathrm{k} \ne - 2$

• $\mathrm{k} \ne - 1 \mathrm{and} \mathrm{k} \ne 2$

For the system of linear equations 2x + 3y + 5z =9, 7x + 3y - 2z = 8 and 2x + 3y + $\mathrm{\lambda z} = \mathrm{\mu }$

Under what condition does the above system of equations have infinitely many solutions ?

• $\mathrm{\lambda } = 5 \mathrm{and} \mathrm{\mu } \ne 9$

• $\mathrm{\lambda } = 5 \mathrm{and} \mathrm{\mu } = 9$

• $\mathrm{\lambda } = 0 \mathrm{and} \mathrm{\mu } = 5$

• $\mathrm{\lambda } = 9 \mathrm{and} \mathrm{\mu } \ne 5$

For the system of linear equations 2x + 3y + 5z =9, 7x + 3y - 2z = 8 and 2x + 3y + $\mathrm{\lambda z} = \mathrm{\mu }$

Under what condition does the above system of equations have unique solution ?

• $\mathrm{\lambda } = 5, \mathrm{\mu } = 9$

• $\mathrm{\lambda } = 5 \mathrm{and} \mathrm{\mu } = 7 \mathrm{only}$

• $\mathrm{\lambda } \ne 5 \mathrm{and} \mathrm{\mu } \mathrm{has} \mathrm{any} \mathrm{real} \mathrm{value}$

• $\mathrm{\lambda }$ has any real value and $\mathrm{\mu } \ne 9$

If A is a square matrix of order 3 and detA = 5, then what is det [(2A)^-1] equal to ?

• $\frac{1}{10}$

• $\frac{2}{5}$

• $\frac{8}{5}$

• $\frac{1}{40}$

What is $\left[\begin{array}{ccc}\mathrm{x}& \mathrm{y}& \mathrm{z}\end{array}\right]\left[\begin{array}{ccc}\mathrm{a}& \mathrm{h}& \mathrm{g}\\ \mathrm{h}& \mathrm{b}& \mathrm{f}\\ \mathrm{g}& \mathrm{f}& \mathrm{c}\end{array}\right] = ?$

• $\left[\begin{array}{ccc}\mathrm{ax} +\hspace{0.17em}\mathrm{hy} + \mathrm{gz}& \mathrm{h} + \mathrm{b} + \mathrm{f}& \mathrm{g} + \mathrm{f} +\hspace{0.17em}\mathrm{c}\end{array}\right]$

• $\left[\begin{array}{ccc}\mathrm{a}& \mathrm{h}& \mathrm{g}\\ \mathrm{hx}& \mathrm{by}& \mathrm{fz}\\ \mathrm{g}& \mathrm{f}& \mathrm{c}\end{array}\right]$

• $\left[\begin{array}{c}\mathrm{ax} + \mathrm{hy} + \mathrm{gz}\\ \mathrm{hx} + \mathrm{by} +\hspace{0.17em}\mathrm{fz}\\ \mathrm{gx} + \mathrm{fy} +\hspace{0.17em}\mathrm{cz}\end{array}\right]$

• $\left[\begin{array}{ccc}\mathrm{ax} +\hspace{0.17em}\mathrm{hy} + \mathrm{gz}& \mathrm{hx} + \mathrm{by} + \mathrm{fz}& \mathrm{gx} + \mathrm{fy} +\hspace{0.17em}\mathrm{cz}\end{array}\right]$

Out of 15 points in a plane, n points are in the same straight line. 445 triangles can be formed by joining these points. What is the value of n ?

• 3

• 4

• 5

• 6

37.

If $\mathrm{f}\left(\mathrm{x}\right) = \left[\begin{array}{ccc}\cos \left(\mathrm{x}\right)& - \sin \left(\mathrm{x}\right)& 0\\ \sin \left(\mathrm{x}\right)& \cos \left(\mathrm{x}\right)& 0\\ 0& 0& 1\end{array}\right]$, then which of the following are correct ?

$$\begin{gathered} 1. \mathrm{f}\left(\mathrm{\theta }\right) \times \mathrm{f}\left(\mathrm{\varphi }\right) = \mathrm{f}\left(\mathrm{\theta } + \mathrm{\varphi }\right) \\ 2. \left|\mathrm{f}\left(\mathrm{\theta } + \mathrm{\varphi }\right)\right| = 1 \\ 3. \mathrm{The} \left|\mathrm{f}\left(\mathrm{x}\right)\right| \mathrm{is} \mathrm{an} \mathrm{even} \mathrm{function} \end{gathered}$$

Select the determinant of f(x) is an even function

• 1 and 2 only

• 2 and 3 only

• 1 and 3 only

• 1, 2 and 3

If $\mathrm{A} = \left[\begin{array}{cc}1& - 1\\ 2& 3\end{array}\right] \mathrm{and} \left[\begin{array}{cc}2& 3\\ - 1& - 2\end{array}\right],$ then which of the following is/are correct ?

$$\begin{gathered} 1. \mathrm{AB}\left({\mathrm{A}}^{-1}{\mathrm{B}}^{-1}\right) \mathrm{is} \mathrm{a} \mathrm{matrix}. \\ 2. {\left(\mathrm{AB}\right)}^{-1} = {\mathrm{A}}^{-1}{\mathrm{B}}^{-1}. \end{gathered}$$

Select the correct answer using the code given below :

• 1 only

• 2 only

• both 1 and 2

• Neither 1 nor 2

$\mathrm{If} \mathrm{A} = \left[\begin{array}{cc}1& 2\\ 2& 3\\ 3& 4\end{array}\right] \mathrm{and} \left(\begin{array}{cc}1& 2\\ 2& 1\end{array}\right), \mathrm{then} \mathrm{which} \mathrm{one} \mathrm{of} \mathrm{the} \mathrm{following} \mathrm{is} \mathrm{correct} ?$

• Both AB and BA exist

• Neither AB nor BA exists

• AB exists but BA does not exis

• AB does not exist but BA exists

If $\left[\begin{array}{cc}1& - 1\\ - 1& 1\end{array}\right]$, then the expression A³ - 2A² is 

• a null matrix

• an identity matrix

• equal to A

• equal to - A