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Matrices

Multiple Choice Questions

471.

If $|\begin{matrix} - a^{2} & ab & ac \\ ab & - b^{2} & bc \\ ac & bc & - c^{2} \end{matrix}|$ = ka²b²c², then k is equal to

472.

If $|\begin{matrix} 6 i & - 3 i & 1 \\ 4 & 3 i & - 1 \\ 20 & 3 & i \end{matrix}|$ = x + iy, then (x, y) is

(3, 1)
(1, 3)
(0, 3)
(0, 0)

473.

If $[\begin{matrix} 1 & 3 & λ + 2 \\ 2 & 4 & 8 \\ 3 & 5 & 10 \end{matrix}]$ is a singular matrix, then the value of $λ$ is

474.

$|\begin{matrix} 1 + x & 1 & 1 \\ 1 & 1 + y & 1 \\ 1 & 1 & 1 + z \end{matrix}|$ is equal to

$xyz (1 + \frac{1}{x} + \frac{1}{y} + \frac{1}{z})$
xyz
$1 + \frac{1}{x} + \frac{1}{y} + \frac{1}{z}$
$\frac{1}{x} + \frac{1}{y} + \frac{1}{z}$

475.

If $|\begin{matrix} x & x^{2} & 1 + x^{3} \\ y & y^{2} & 1 + y^{3} \\ z & z^{2} & 1 + z^{3} \end{matrix}| = 0$ , then

z = xy
$z = \frac{1}{xy}$
$z = - \frac{1}{xy}$
None of these

476.

A square matrix A is called an orthogonal matrix, if

$A A = I$
AA' = I
${AA}^{θ} = I$
A² = I

477.

If the equations 2x - y + 2z = 2, x - 2y + z = - 4 and x + y + $λ$ z = 4 have no solution, then the value of $λ$ will be

478.

$If A = [\begin{matrix} α & 0 \\ 1 & 1 \end{matrix}], B = [\begin{matrix} 1 & 0 \\ 5 & 1 \end{matrix}]$ , then the value of $α$ , if A² = B, will be

4
3
5
None of these

479.

If A = $[\begin{matrix} 2 & 3 & 1 \\ 0 & - 1 & 5 \end{matrix}]$ and B = $[\begin{matrix} 1 & 2 & - 6 \\ 0 & - 1 & 3 \end{matrix}]$ , then 3A - 4B is equal to

$[\begin{matrix} 2 & 1 & 27 \\ 0 & 1 & 3 \end{matrix}]$
$[\begin{matrix} 2 & 1 & 27 \\ 0 & 1 & 3 \end{matrix}]$
$[\begin{matrix} 3 & 5 & - 5 \\ 0 & - 2 & 8 \end{matrix}]$
None of these

480.
If x is a complex root of the equation $|\begin{matrix} 1 & x & x \\ x & 1 & x \\ x & x & 1 \end{matrix}| + |\begin{matrix} 1 - x & 1 & 1 \\ 1 & 1 - x & 1 \\ 1 & 1 & 1 - x \end{matrix}| = 0$ , then $x^{2005} + \frac{1}{x^{2005}}$ is equal to

1

- 1

i

w

$Given, |\begin{matrix} 1 & x & x \\ x & 1 & x \\ x & x & 1 \end{matrix}| + |\begin{matrix} 1 - x & 1 & 1 \\ 1 & 1 - x & 1 \\ 1 & 1 & 1 - x \end{matrix}| = 0 Applying C_{1} \to C_{1} + C_{2} + C_{3} and taking common from C_{1} \Rightarrow (1 + 2 x) |\begin{matrix} 1 & x & x \\ x & 1 & x \\ x & x & 1 \end{matrix}| + (3 - x) |\begin{matrix} 1 - x & 1 & 1 \\ 1 & 1 - x & 1 \\ 1 & 1 & 1 - x \end{matrix}| = 0 Applying R_{2} \to R_{2} - R_{1} and R_{3} \to R_{3} - R_{1} \Rightarrow (1 + 2 x) |\begin{matrix} 1 & x & x \\ x & 1 & x \\ x & x & 1 \end{matrix}| + (3 - x) |\begin{matrix} 1 & 1 & 1 \\ 0 & - x & 0 \\ 0 & 0 & - x \end{matrix}| = 0$

$\Rightarrow (1 + 2 x) {(1 - x)}^{2} + (3 - x) x^{2} = 0 \Rightarrow 1 + 2 x + x^{2} + 2 x^{3} - 2 x - 4 x^{2} + 3 x^{2} - x^{3} = 0 \Rightarrow x^{3} + 1 = 0 \Rightarrow x^{3} = - 1 ∴ x = - 1, - w, - w^{2} x^{2005} + \frac{1}{x^{2005}} = x^{2004} . x + \frac{1}{x^{2004} . x} = {(x^{3})}^{668} . x + \frac{1}{{(x^{3})}^{668} . x} = x {(- 1)}^{668} + \frac{1}{{(- 1)}^{668} . x} = x + \frac{1}{x} \times \frac{x^{2}}{x^{2}} = x + \frac{x^{2}}{- 1} = x - x^{2} When x = - w x - x^{2} = - w - {(- w)}^{2} = - w - w^{2} = 1 When x - x^{2} = - w^{2} - w^{4} = - w^{2} - w = 1$

Previous Year Papers

Test Series

Test Yourself

Multiple Choice Questions

480. If x is a complex root of the equation 1xxx1xxx1 + 1 - x1111 - x1111 - x = 0, then x2005 + 1x2005 is equal to 1 - 1 i w

480.
If x is a complex root of the equation $|\begin{matrix} 1 & x & x \\ x & 1 & x \\ x & x & 1 \end{matrix}| + |\begin{matrix} 1 - x & 1 & 1 \\ 1 & 1 - x & 1 \\ 1 & 1 & 1 - x \end{matrix}| = 0$ , then $x^{2005} + \frac{1}{x^{2005}}$ is equal to

1

- 1

i

w