Matrices & Determinants
Integral of determinant function
MJMT_Full_Test_06
Grade 12
Question:
Let $f(x)=\begin{vmatrix}\cos x&\cos^2x&\cos^4x\\\cos3x&\cos^23x&\cos^43x\\\cos5x&\cos^25x&\cos^45x\end{vmatrix}$, then $\displaystyle\int_0^{\pi}f(x)\,dx$ equals
9
18
27
none of these (0)
Step-by-Step Solution
Key Concept: $f(x)=(\cos x\cos3x\cos5x)\cdot\begin{vmatrix}1&\cos x&\cos^3x\\1&\cos3x&\cos^33x\\1&\cos5x&\cos^35x\end{vmatrix}$. The determinant $\int_0^\pi$ gives 0 by symmetry.
$\int_0^\pi f(x)dx=0$.
Correct Answer: 4