If $\Delta \left( x \right) = \left| {\begin{array}{*{20}{c}} 1&{\cos x}&{1 - \cos x}\\ {1 + \sin x}&{\cos x}&{1 + \sin x - \cos x}\\ {\sin x}&{\sin x}&1 \end{array}} \right|$, then $\mathop \smallint \nolimits_0^{\pi /2} \Delta \left( x \right)dx$ is equal to

JEE MATHEMATICS MCQ: MATRICES

1 Medium

1 of 1

Analyzing +4 | -1

\Delta \left( x \right) = \left| {\begin{array}{cccccccccccccccccccc} 1&{\cos x}&{1 - \cos x}\\ {1 + \sin x}&{\cos x}&{1 + \sin x - \cos x}\\ {\sin x}&{\sin x}&1 \end{array}} \right|

, then

\int \nolimits_0^{\pi /2} \Delta \left( x \right)dx

is equal to

View answer and explanation

Correct option: D

- \frac{1}{2}

The correct answer is D (

- \frac{1}{2}

). The integral of the determinant, after simplification, evaluates to -1/2.

Practice All Chapter MCQs →

\left|\begin{array}{ccc}a & b-c & b+c \\ c+a & b & c-a \\ a-b & a+b & c\end{array}\right|=0

a, b, c

a x+b y+c=0

{x_2} - {x_3} = 1,\; - {x_1} + 2{x_3} = - 2,\;{x_1} - 2,\;{x_1} - 2{x_2} = 3

2x + 3y + 1 = 0,3x + y - 2 = 0

ax + 2y - b = 0

\left|\begin{array}{ccc}1 & \left(\alpha^{2 x}-\alpha^{-2 x}\right)^2 & \left(\alpha^{2 x}+\alpha^{-2 x}\right)^2 \\ 1 & \left(\beta^{2 x}-\beta^{-2 x}\right)^2 & \left(\beta^{2 x}+\beta^{-2 x}\right)^2 \\ 1 & \left(\gamma^{2 x}-\gamma^{-2 x}\right)^2 & \left(\gamma^{2 x}+\gamma^{-2 x}\right)^2\end{array}\right|

\omega

\left| {\begin{array}{cccccccccccccccccccc} 1&\omega &{{\omega ^2}}\\ \omega &{{\omega ^2}}&1\\ {{\omega ^2}}&1&\omega \end{array}} \right|

A

2 \times 2

A(\operatorname{adj} A)=\left[\begin{array}{cc}10 & 0 \\ 0 & 10\end{array}\right]

|A|