First slide

Introduction to Determinants

Question

If the determinant $|\begin{array}{lll} \cos 2 x \sin^{2} x \cos 4 x \\ \sin^{2} x \cos 2 x \cos^{2} x \\ \cos 4 x \cos^{2} x \cos 2 x \end{array}|$ is expanded in powers of sin x, then the constant term in that expression is

Moderate

A

1
B

0
C

-1
D

2

Solution

$f (x) = |\begin{array}{ccc} 1 - 2 \sin^{2} x & \sin^{2} x & 1 - 8 \sin^{2} x (1 - \sin^{2} x) \\ \sin^{2} x & 1 - 2 \sin^{2} x & 1 - \sin^{2} x \\ 1 - 8 \sin^{2} x (1 - \sin^{2} x) & 1 - \sin^{2} x & 1 - 2 \sin^{2} x \end{array}|$
The required constant term is
$f (0) = |\begin{array}{lll} 1 0 1 \\ 0 1 1 \\ 1 1 1 \end{array}| = |\begin{array}{lll} 1 0 0 \\ 0 1 1 \\ 1 1 0 \end{array}| = 1 (0 - 1) = - 1$

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