Previous Year Papers

Download Solved Question Papers Free for Offline Practice and view Solutions Online.

Test Series

Take Zigya Full and Sectional Test Series. Time it out for real assessment and get your results instantly.

Test Yourself

Practice and master your preparation for a specific topic or chapter. Check you scores at the end of the test.

Matrices

Multiple Choice Questions

271.

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

272.
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

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

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

273.

If A² - A + I = 0, then matrix A^-1 will be equal to

A + I
I - A
A
A - I

274.

For how many values of x in the interval [- 4, - 1] the matrix $[\begin{matrix} 3 & - 1 + x & 2 \\ 3 & - 1 & x + 2 \\ x + 3 & - 1 & 2 \end{matrix}]$ is singular ?

275.

If A = $[\begin{matrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 1 & 0 & 0 \end{matrix}]$ , then $A$ is a

singular
non-singular
symmetric
unit matrix

276.

Let $f (x) = |\begin{matrix} \sin (3 x) & 1 & 2 {(\cos (\frac{3 x}{2}) + \sin (\frac{3 x}{2}))}^{2} \\ \cos (3 x) & - 1 & 2 \cos^{2} (\frac{3 x}{2}) - \sin^{2} (\frac{3 x}{2}) \\ \tan (3 x) & 4 & 1 + 2 \tan (3 x) \end{matrix}|$ . Then, the value of f'(x) at x = (2n + 1) $π$ , n $\in I$ (the set of integers) is equal to

(- 1)ⁿ
(- 1)ⁿ^{+ 1}
3
9

277.

The value of $|\begin{matrix} 1 & a & b + c \\ 1 & b & c + a \\ 1 & c & a + b \end{matrix}|$ is

0
a + b + c
abc
1

278.

If 2x - $[\begin{matrix} 1 & 2 \\ 7 & 4 \end{matrix}] = [\begin{matrix} 3 & 2 \\ 0 & - 2 \end{matrix}]$ , then the value of x will be

$[\begin{matrix} 2 & 2 \\ 7 & 4 \end{matrix}]$
$[\begin{matrix} 1 & 2 \\ 7 / 2 & 2 \end{matrix}]$
$[\begin{matrix} 2 & 2 \\ 7 / 2 & 1 \end{matrix}]$
None of the above

279.

If $[\begin{matrix} 0 & 2 β & γ \\ α & β & - γ \\ α & - β & γ \end{matrix}]$ is orthagonal, then the values of $α, β and γ$ will be

$α = \pm \frac{1}{\sqrt{5}}, β = \pm \frac{1}{\sqrt{3}}, γ = \pm \frac{1}{\sqrt{2}}$
$α = \pm \frac{1}{\sqrt{6}}, β = \pm \frac{1}{\sqrt{7}}, γ = \pm \frac{1}{\sqrt{3}}$
$α = \pm \frac{1}{\sqrt{3}}, β = \pm \frac{1}{\sqrt{3}}, γ = \pm \frac{1}{\sqrt{3}}$
$α = \pm \frac{1}{\sqrt{2}}, β = \pm \frac{1}{\sqrt{6}}, γ = \pm \frac{1}{\sqrt{3}}$