IIT-JEE 2010 · Paper 1, Mathematics question 48

Denominator in double angles.
$$\sin^2\theta+3\sin\theta\cos\theta+5\cos^2\theta=3+2\cos2\theta+\tfrac32\sin2\theta.$$

Its minimum. $2\cos2\theta+\frac32\sin2\theta$ ranges over $\left[-\frac52,\frac52\right]$, so the denominator's minimum is $\frac12$ (positive).

Maximum of the expression. $\dfrac1{1/2}=2$.

Answer: 2