Matrices & Determinants
Determinant with sine entries; sum of values of parameter
MJMT_Full_Test_11
Grade 12
Question:
If $\begin{vmatrix}\sin x+1&\sin 2x&\sin 3x\\ \sin 2x&\sin 3x+a&\sin 4x\\ \sin 3x&\sin 4x&\sin 5x+a^2\end{vmatrix}=2025(f(x)+45)$ where $f(x)$ is a function of $x$ and $a$ is complex, then sum of all possible values of $a$ is
Step-by-Step Solution
Key Concept: The sine sub-matrix has determinant 0 (rank $\leq2$). Expanding the full determinant: $\det = a^2\cdot a + \ldots = a^3 + \text{(trig terms)}$. For the expression to equal $2025(f(x)+45)$: $a^3=45^3$.
Sum of cube roots of $45^3$ is $0$.
Correct Answer: 1