First slide

Introduction to Determinants

Question

If $Δ (x) = |\begin{matrix} 1 & \cos x & 1 - \cos x \\ 1 + \sin x & \cos x & 1 + \sin x - \cos x \\ \sin x & \sin x & 1 \end{matrix}|$ then $\int_{0}^{π / 2} Δ (x) d x$ equal

Moderate

A

$\frac{1}{4}$
B

$\frac{1}{2}$
C

0
D

$\frac{- 1}{2}$

Solution

Using , $C_{1} \to C_{1} - C_{2} - C_{3},$ we get
$Δ (x) = |\begin{matrix} 0 & \cos x & 1 - \cos x \\ 0 & \cos x & 1 + \sin x - \cos x \\ - 1 & \sin x & 1 \end{matrix}|$
$= (- 1) |\begin{matrix} \cos x & 1 - \cos x \\ \cos x & 1 + \sin x - \cos x \end{matrix}|$
$= (- 1) \cos x [1 + \sin x - \cos x - 1 + \cos x]$
$= - \cos x \sin x = - \frac{1}{2} (\sin 2 x)$
${∴ \int_{0}^{π / 2} Δ (x) d x = - \frac{1}{2} \frac{(- 1)}{2} \cos 2 x]}_{0}^{π / 2}$
$= \frac{1}{4} (\cos π - \cos 0) = - \frac{1}{2}$

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