Matrices & Determinants
Non-Trivial Solution — Characteristic Equation
nta_pyq_2023_apr
Grade 12
Question:
Let $S$ be the set of all values of $\theta\in[-\pi,\pi]$ for which the system $x+y+\sqrt{3}z=0$, $-x+(\tan\theta)y+\sqrt{7}z=0$, $x+y+(\tan\theta)z=0$ has non-trivial solution. Then $\dfrac{120}{\pi}\displaystyle\sum_{\theta\in S}\theta$ is equal to
Step-by-Step Solution
Key Concept: Non-trivial solution requires determinant $=0$. Expand to get $\tan^2\theta+(1-\sqrt{3})\tan\theta-\sqrt{3}=0\Rightarrow(\tan\theta+1)(\tan\theta-\sqrt{3})=0$.
$\sum\theta=\frac{1}{3}-\frac{2}{3}-\frac{1}{4}+\frac{3}{4}=\frac{\pi}{6}$... $\frac{120}{\pi}\times\frac{\pi}{6}=20$.
Correct Answer: 1