Mathematics : Determinants

If the lines x + 2ay + a = 0, x + 3by + b = 0, 32x + 4cy + c = 0 are concurrent, then a, b and c are in

• arithmetic progression

• geometric progression

• harmonc progression

• arithmetico-geometric progression

If a, b, c are distinct positive real numbers, then the value of the determinant $\left|\begin{array}{ccc}\mathrm{a}& \mathrm{b}& \mathrm{c}\\ \mathrm{b}& \mathrm{c}& \mathrm{a}\\ \mathrm{c}& \mathrm{a}& \mathrm{b}\end{array}\right| \mathrm{is}$

• $< 0$

• $> 0$

• 0

• $\ge 0$

$$\begin{gathered} \mathrm{Let} \mathrm{A} \mathrm{be} \mathrm{a} 3 \times 3 \mathrm{matrix} \mathrm{such} \mathrm{that} \\ \mathrm{adj} \mathrm{A} = \left[\begin{array}{ccc}2& - 1& 1\\ - 1& 0& 2\\ 1& - 2& - 1\end{array}\right] \mathrm{and} \mathrm{B} = \mathrm{adj}\left(\mathrm{adj} \mathrm{A}\right) \end{gathered}$$

$\mathrm{If} \left|\mathrm{A}\right| = \mathrm{\lambda } \mathrm{and} \left|{\left({\mathrm{B}}^{ - 1}\right)}^{\mathrm{T}}\right| = \mathrm{\mu }, \mathrm{Then} \mathrm{the} \mathrm{ordered} \mathrm{pair}, \left(\left|\mathrm{\lambda }\right|, \mathrm{\mu }\right) \mathrm{is} \mathrm{equal} \mathrm{to}$

• $\left(3, \frac{1}{81}\right)$

• $\left(9, \frac{1}{9}\right)$

• (3, 81)

• $\left(9, \frac{1}{81}\right)$

If the system of equations

$$\begin{gathered} \mathrm{x} + \mathrm{y} + \mathrm{z} = 2 \\ 2\mathrm{x} + 4\mathrm{y} – \mathrm{z} = 6 \\ 3\mathrm{x} + 2\mathrm{y} + \mathrm{\lambda z} = \mathrm{\mu } \end{gathered}$$

• $\mathrm{\lambda } + 2\mathrm{\mu } = 14$

• $2\mathrm{\lambda } - \mathrm{\mu } = 5$

• $2\mathrm{\lambda } + \mathrm{\mu } = 14$

• $\mathrm{\lambda } - 2\mathrm{\mu } = - 5$

$$\begin{gathered} \mathrm{Let} \mathrm{\lambda } \in \mathrm{R}. \mathrm{The} \mathrm{system} \mathrm{of} \mathrm{linear} \mathrm{equations} \\ 2{\mathrm{x}}_1 - 4{\mathrm{x}}_2 + {\mathrm{\lambda x}}_3 = 1 \\ {\mathrm{x}}_1 - 6{\mathrm{x}}_2 + {\mathrm{x}}_3 = 2 \\ {\mathrm{\lambda x}}_1 - 10{\mathrm{x}}_2 + 4{\mathrm{x}}_3 = 3 \\ \mathrm{is} \mathrm{inconsistant} \mathrm{for} : \end{gathered}$$

• every value of $\mathrm{\lambda }$

• exactly two values of $\mathrm{\lambda }$

• $\mathrm{exactly} \mathrm{one} \mathrm{positive} \mathrm{value} \mathrm{of \lambda }$

• exactly one negative value of $\mathrm{\lambda }$

If the system of linear equations

x + y + 3z = 0

x + 3y + k²z = 0

3x + y + 3z = 0

has a non–zero solution (x, y, z) for some k $\in$ R, then x $\mathrm{x} + \left(\frac{\mathrm{y}}{\mathrm{z}}\right)$ is equal to :

•  - 9

•  - 3

• 9

• 3

The values of $\mathrm{\lambda } \mathrm{and} \mathrm{\mu }$ for which the system of linear equations 

x + y + z = 2

x + 2y + 3z = 5

x + 3y + $\mathrm{\lambda z} = \mathrm{\mu }$

has infinitely many solutions are respectively :

• 5 and 8

• 4 and 9

• 6 and 8

• 5 and 7

$\mathrm{Let} \mathrm{\theta } = \frac{\mathrm{\theta }}{5} \mathrm{and} \mathrm{A} = \left[\begin{array}{cc}\cos \left(\mathrm{\theta }\right)& \sin \left(\mathrm{\theta }\right)\\ - \sin \left(\mathrm{\theta }\right)& \cos \left(\mathrm{\theta }\right)\end{array}\right]. \mathrm{If} \mathrm{B} = \mathrm{A} + {\mathrm{A}}^4, \mathrm{Then} \det \left(\mathrm{B}\right) :$

• 0

• 1

• lies in (2, 3)

• lies in (1, 2)

$$\begin{gathered} \mathrm{The} \mathrm{sum} \mathrm{of} \mathrm{distinct} \mathrm{values} \mathrm{of} \mathrm{l} \mathrm{for} \mathrm{which} \mathrm{the} \mathrm{system} \mathrm{of} \mathrm{equations}: \\ (\mathrm{\lambda } – 1)\mathrm{x} + (3\mathrm{\lambda } + 1)\mathrm{y} + 2\mathrm{\lambda z} = 0 \\ (\mathrm{\lambda }– 1)\mathrm{x} + (4\mathrm{\lambda } – 2)\mathrm{y} + (\mathrm{\lambda } + 3)\mathrm{z} = 0 \\ 2\mathrm{x} + (3\mathrm{\lambda } + 1)\mathrm{y} + 3(\mathrm{\lambda } – 1)\mathrm{z} = 0, \\ \mathrm{Has} \mathrm{non}-\mathrm{zero} \mathrm{solutions}, \mathrm{is}... \end{gathered}$$