$\Delta=\left|\begin{array}{ccc} 1 & \cos x & 1 \\ -\cos x & 1 & \cos x \\ -1 & -\cos x & 1 \end{array}\right|$
(A) $\Delta=2\left(1-\cos ^2 x\right)$
(B) $\Delta=2\left(2-\sin ^2 x\right)$
(C) Minimum value of $\Delta$ is 2
(D) Maximum value of $\Delta$ is 4
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (B), (C) and (D) only
$\Delta=\left|\begin{array}{ccc} 1 & \cos x & 1 \\ -\cos x & 1 & \cos x \\ -1 & -\cos x & 1 \end{array}\right|$
$=1+\cos^2x+\cos x(-\cos x+\cos x)+1+\cos^2x$
$=2(1+\cos^2x)$
$=2(2-\sin^2x)=Δ$
$0≤\cos^2x≤1$
$1≤1+\cos^2x≤2$
$2≤2(1+\cos^2x)≤4$
min value = 2
max value = 4
B, C, D only correct