Question

If a is a positive integer, then the number of values of a satisfying$\int _ { 0 } ^ { \pi / 2 } \left\{ a \left( \frac { \cos 3 x } { 4 } + \frac { 3 } { 4 } \cos x \right) + a \sin x - 20 \cos x \right\}$dx£ $\frac { \mathrm { a } ^ { 2 } } { 3 }$ are -

• A. One
• B. Two
• C. Three
• D. Four

Answer 1

Answer: (D)

Four

Answer 2

The L.H.S. of the above inequality is equal to

a² – a cos x – 20 $\left. \sin x \right| _ { 0 } ^ { \pi / 2 }$

= a² $\left( - \frac { 1 } { 12 } + \frac { 3 } { 4 } \right)$ – a (0 – 1) – 20 = + a – 20.

Thus the given inequality is (2a²/3) + a – 20 £ – a²/3

i.e. a² + a – 20 £ 0 Ū – 5 £ a £ 4

Since a is a positive integer so a = 1, 2, 3, 4.