First slide

Introduction to Determinants

Question

Let $f (x) = |\begin{array}{ccc} 2 \cos^{2} x & \sin 2 x & - \sin x \\ \sin 2 x & 2 \sin^{2} x & \cos x \\ \sin x & - \cos x & 0 \end{array}|$ . Then the value of $\int_{0}^{π / 2} [f (x) + f' (x)] dx$ is

Moderate

A

$π$
B

$π$ /2
C

2 $π$
D

3 $π$ /2

Solution

Applying $C_{1} \to C_{1} - 2 \sin {xC}_{3} and C_{2} \to C_{2} + 2 \cos {xC}_{3}$ , we get
$\begin{matrix} f (x) = |\begin{array}{ccc} 2 & 0 & - \sin x \\ 0 & 2 & \cos x \\ \sin x & - \cos x & 0 \end{array}| \\ = 2 \cos^{2} x + 2 \sin^{2} x = 2 \end{matrix}$
$∴ f' (x) = 0$
$∴ \int_{0}^{π / 2} [f (x) + f' (x)] = dx = \int_{0}^{π / 2} 2 dx = π$

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