Question

The number of all possible triplets $\left( a_{1},a_{2},a_{3} \right)$ such than $a_{1} + a_{2}\cos 2x + a_{3}\sin^{2}x = 0$ for all $x$is

• A. 0
• B. 1
• C. 3
• D. Infinite

Answer 1

Answer: (D)

Infinite

Answer 2

The given equation can be written as

$a_{1} + a_{2}\cos 2x + \frac{a_{3}}{2}\left( 1 - \cos 2x \right) = 0$

⇒ $\left( a_{1} + \frac{a_{3}}{2} \right) + \left( a_{2} - \frac{a_{3}}{2} \right)\cos 2x = 0$which is zero for all values of x.

If $a_{1} = - \frac{a_{3}}{2} = - a_{2}$

Or $a_{1} = - \frac{k}{2},a_{2} = \frac{k}{2}$, $a_{3} = k$for any $k \in R$

Hence the required number of triplets is infinite.