The system of equations 3x + 2y + z = 6, 3x + 4y + 3z = 14 and 6x + 10y + 8z = a, has infinite number of solutions, If a is equal to

• 8

• 12

• 24

• 36

The number of real values of t such that the system of homogeneous equation

$$\begin{gathered} \mathrm{tx} + \left(\mathrm{t} + 1\right)\mathrm{y} + \left(\mathrm{t} - 1\right)\mathrm{z} = 0 \\ \left(\mathrm{t} + 1\right)\mathrm{x} + \mathrm{ty} + \left(\mathrm{t} + 2\right)\mathrm{z} = 0 \\ \left(\mathrm{t} - 1\right)\mathrm{x} + \left(\mathrm{t} + 2\right)\mathrm{y} + \mathrm{tz} = 0 \end{gathered}$$

has non-trivial solutions is 

• 3

• 2

• 1

• None of these

If a, b, c and d are real numbers such that ${\mathrm{a}}^2 + {\mathrm{b}}^2 + {\mathrm{c}}^2 + {\mathrm{d}}^2 = 1 \mathrm{and} \mathrm{A} = \left[\begin{array}{cc}\mathrm{a} + \mathrm{id}& \mathrm{c} + \mathrm{id}\\ - \mathrm{c} + \mathrm{id}& \mathrm{a} - \mathrm{ib}\end{array}\right], \mathrm{then} {\mathrm{A}}^{ - 1} = ?$

• $\left[\begin{array}{cc}\mathrm{a} +ib& - \mathrm{c} - \mathrm{id}\\ - \mathrm{c} - \mathrm{id}& \mathrm{a} - \mathrm{ib}\end{array}\right]$

• $\left[\begin{array}{cc}\mathrm{a} - \mathrm{ib}& \mathrm{c} - \mathrm{id}\\ - \mathrm{c} + \mathrm{id}& \mathrm{a} + \mathrm{ib}\end{array}\right]$

• $\left[\begin{array}{cc}\mathrm{a} - \mathrm{ib}& - \mathrm{c} - \mathrm{id}\\ \mathrm{c} - \mathrm{id}& \mathrm{a} + \mathrm{ib}\end{array}\right]$

• $\left[\begin{array}{cc}\mathrm{a} +\mathrm{id}& \mathrm{c} + \mathrm{id}\\ \mathrm{c} - \mathrm{id}& \mathrm{a} - \mathrm{ib}\end{array}\right]$

$\mathrm{If} \mathrm{the} \mathrm{matrix} \mathrm{A} = \left[\begin{array}{cccc}1& 2& 3& 0\\ 2& 4& 3& 2\\ 3& 2& 1& 3\\ 6& 8& 7& \mathrm{\alpha }\end{array}\right] \mathrm{is} \mathrm{of} \mathrm{rank} 3, \mathrm{then} \mathrm{\alpha } = ?$

• - 5

• 5

• 4

• 1

$\mathrm{If} \mathrm{k} > 1 \mathrm{and} \mathrm{the} \mathrm{determinant} \mathrm{of} \mathrm{the} \mathrm{matrix} {\mathrm{A}}^2, \mathrm{where} \mathrm{A} = \left[\begin{array}{ccc}\mathrm{k}& \mathrm{k\alpha }& \mathrm{\alpha }\\ 0& \mathrm{\alpha }& \mathrm{k\alpha }\\ 0& 0& \mathrm{k}\end{array}\right], \mathrm{is} {\mathrm{k}}^2, \mathrm{then} \left|\mathrm{\alpha }\right| = ?$

• $\frac{1}{{\mathrm{k}}^2}$

• k

• k²

• $\frac{1}{\mathrm{k}}$

If A is a square matrix of order 3, then $\left|\mathrm{adj} (\mathrm{adj} {\mathrm{A}}^2)\right|$ is equal to

• ${\left|\mathrm{A}\right|}^2$

• ${\left|\mathrm{A}\right|}^4$

• ${\left|\mathrm{A}\right|}^8$

• ${\left|\mathrm{A}\right|}^{16}$

The system 2x + 3y + z = 5, 3x + y + 5z = 7 and x + 4y - 2z = 3 has

• unique solution

• finite number of solution

• Infinite solutions

• No solution

318.

If $∆ = \left|\begin{array}{ccc}1& \cos \left(\mathrm{\theta }\right)& 1\\ - \cos \left(\mathrm{\theta }\right)& 1& \cos \left(\mathrm{\theta }\right)\\ - 1& - \cos \left(\mathrm{\theta }\right)& 1\end{array}\right|$, then $∆$ lies in the interval

• [2, 4]

• (2, 4)

• [1, 4]

• [- 1, 1]

If a, b, c are non-zero real numbers and if the equations (a - 1) x = y + z, (b - 1)y = z + x, (c - 1)z = x + y have a non-trivial solution, then ab + be + ca = ?

• a²b²c²

• 0

• abc

• a + b + c

If a system of three linear equations in three unknowns, which is in the matrix equation form of AX = D, is in consistent, then rank of A/rank of AD is

• Less than one

• Greater than or equal to one

• One

• Greater than one