Matrices & Determinants
Determinant Identity — Testing Two Statements
nta_pyq_2026_jan
Grade 12
Question:
Among the statements:
I: If $\begin{vmatrix}1&\cos\alpha&\cos\beta\\\cos\alpha&1&\cos\gamma\\\cos\beta&\cos\gamma&1\end{vmatrix}=\begin{vmatrix}0&\cos\alpha&\cos\beta\\\cos\alpha&0&\cos\gamma\\\cos\beta&\cos\gamma&0\end{vmatrix}$, then $\cos^2\alpha+\cos^2\beta+\cos^2\gamma=\dfrac{3}{2}$
II: If $\begin{vmatrix}x^2+x&x+1&x-2\\2x^2+3x-1&3x&3x-3\\x^2+2x+3&2x-1&2x-1\end{vmatrix}=px+q$, then $p^2=196q^2$
only II is true
both are false
both are true
only I is true
Step-by-Step Solution
Key Concept: Statement I: $D_1=1-\cos^2\alpha-\cos^2\beta-\cos^2\gamma+2\cos\alpha\cos\beta\cos\gamma$, $D_2=2\cos\alpha\cos\beta\cos\gamma$. $D_1=D_2\Rightarrow\cos^2\alpha+\cos^2\beta+\cos^2\gamma=1\neq\tfrac{3}{2}$. Statement I is false.
Both statements are false.
Correct Answer: 2