MHTCET 22 Apr 2025 Shift 1 Paper

Concept:Use substitution to convert the given exponential equation into a quadratic and then count solutions of $\sin x$ in the given interval.Explanation:Given equation: $16^{\sin ^{2} x}+16^{\cos ^{2} x}=10$.Let $y=16^{\sin ^{2} x}$.Since $\cos ^{2} x=1-\sin ^{2} x$, we get $16^{\cos ^{2} x}=16^{1-\sin ^{2} x}=\frac{16}{y}$.So the equation becomes $y+\frac{16}{y}=10$.Multiplying by $y$: $y^{2}-10y+16=0$.Factoring: $(y-2)(y-8)=0$.Thus $y=2$ or $y=8$.Case 1: $16^{\sin ^{2} x}=2$ → $2^{4\sin ^{2} x}=2^{1}$ → $\sin ^{2} x=\frac{1}{4}$.So $\sin x=\pm \frac{1}{2}$, giving 4 solutions in $[0,2\pi]$: $\frac{\pi}{6}, \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{11\pi}{6}$.Case 2: $16^{\sin ^{2} x}=8$ → $2^{4\sin ^{2} x}=2^{3}$ → $\sin ^{2} x=\frac{3}{4}$.So $\sin x=\pm \frac{\sqrt{3}}{2}$, giving 4 solutions in $[0,2\pi]$: $\frac{\pi}{3}, \frac{2\pi}{3}, \frac{4\pi}{3}, \frac{5\pi}{3}$.Total number of solutions $=4+4=8$.Answer:The number of solutions is $8$, i.e. Option A.