# The minimum value of $2 \sin ^{2} \theta+3 \cos ^{2} \theta$ is ?

A. 0 B. 3 C. 2 D. 1 Answer: Option C
Let the mother’s present age be x years Then, the person’s present age $\begin{array}{l} \text { Let } x=2 \sin ^{2} \theta+3 \cos ^{2} \theta \\ \Rightarrow x=2 \sin ^{2} \theta+2 \cos ^{2} \theta+\cos ^{2} \theta \\ \Rightarrow x=2\left(\sin ^{2} \theta+\cos ^{2} \theta\right)+\cos ^{2} \theta \\ \Rightarrow x=2+\cos ^{2} \theta \quad\left[\text { since } \sin ^{2} \theta+\cos ^{2} \theta=1\right] \end{array}$ Therefore x will be the minimum when cosθ = 0. i.e. minimum value of x will 2
Alternative Solution: $2 \sin ^{2} \theta+3 \cos ^{2} \theta$ Minimum value is 2, [If x sin2θ + y cos2θ, If x > y, then x will be always maximum value and y is minimum if y > x, vice versa will happen]