Question

Let $f(x)= \begin{vmatrix} 1+\sin ^2 x & \cos ^2 x & \sin 2 x \\\ \sin ^2 x & 1+\cos ^2 x & \sin 2 x \\\ \sin ^2 x & \cos ^2 x & 1+\sin 2 x\end{vmatrix}, x \in\left[\frac{\pi}{6}, \frac{\pi}{3}\right]$ If $\alpha$ and $\beta$ respectively are the maximum and the minimum values of $f$, then

• A. $\beta^2+2 \sqrt{\alpha}=\frac{19}{4}$
• B. $\alpha^2+\beta^2=\frac{9}{2}$
• C. $\alpha^2-\beta^2=4 \sqrt{3}$
• D. $\beta^2-2 \sqrt{\alpha}=\frac{19}{4}$

Answer 1

Answer: (D)

$\beta^2-2 \sqrt{\alpha}=\frac{19}{4}$

Answer 2

C1​→C1​+C2​+C3​
f(x)=∣∣​2+sin2x2+sin2x2+sin2x​cos2x1+cos2xcos2x​sin2xsin2x1+sin2x​∣∣​
f(x)=(2+sin2x)∣∣​111​cos2x1+cos2xcos2x​sin2xsin2x1+sin2x​∣∣​
R2​→R2​−R1​
R3​→R3​−R1​
f(x)=2+sin2x)∣∣​100​cos2x10​sin2x01​∣∣​
=(2+sin2x)(1)=2+sin2x
=sin2x∈[23​​,1]
Hence 2+sin2x∈[2+23​​,3]