The value of m[sinθ + 2cos²∅ + 3sinθ + 4cos²∅ + . . . . . . . . + 18cos²∅] is a perfect square of an integer, θ = 30°, ∅ = 45° and 150 ≤ m ≤ 180. Find the value of m.

Solution(By Examveda Team)

$$\eqalign{ & m\left[ {\sin \theta + 2{{\cos }^2}\phi + 3\sin \theta + 4{{\cos }^2}\phi + \,.....\,18{{\cos }^2}\phi } \right] \cr & \theta = {30^ \circ },\,\,\phi = {45^ \circ } \cr & = m\left[ {\left( {\sin \theta + 3\sin \theta + 5\sin \theta + \,.....\,17\sin \theta } \right) + \left( {2{{\cos }^2}\phi + 4{{\cos }^2}\phi + \,.....\,18{{\cos }^2}\phi } \right)} \right] \cr & \Rightarrow {\text{Sum of odd number}} = {\left( 9 \right)^2}, \cr & {\text{Sum of even number}} = 9 \times 10 \cr & = m\left[ {81\sin \theta + 90{{\cos }^2}\phi } \right] \cr & = m\left[ {81 \times \sin 30 + 90 \times {{\cos }^2}45} \right] \cr & = m\left[ {81 \times \frac{1}{2} + 90 \times \frac{1}{2}} \right] \cr & = m \times \frac{{171}}{2} \cr & {\text{Now, we will check through the option,}} \cr & {\text{Putting }}m = 152 \cr & = 152 \times \frac{{171}}{2} \cr & = 4 \times 19 \times 19 \times 9 \cr & = {\left( {2 \times 19 \times 3} \right)^2} \cr & = {\text{ perfect square}}{\text{. Ans}}{\text{.}} \cr} $$