SATHEE: Trigonometrical Equations 3 Question 8

Trigonometrical Equations 3 Question 8

9. The number of all possible triplets $(a_{1}, a_{2}, a_{3})$ such that $a_{1} + a_{2} \cos (2 x) + a_{3} \sin^{2} (x) = 0, \forall x$ is

(1987, 2M)

(a) 0

(b) 1

(d) $\infty$

Show Answer

Answer:

Correct Answer: 9. (d)

Solution:

Given, $a_{1} + a_{2} \cos 2 x + a_{3} \sin^{2} x = 0, \forall x$

$\Rightarrow a_{1} + a_{2} \cos 2 x + a_{3} \frac{1 - \cos 2 x}{2} = 0, \forall x$

$\Rightarrow a_{1} + \frac{a_{3}}{2} + a_{2} - \frac{a_{3}}{2} \cos 2 x = 0, \forall x$

$\Rightarrow a_{1} + \frac{a_{3}}{2} = 0$ and $a_{2} - \frac{a_{3}}{2} = 0$

$\Rightarrow a_{1} = - \frac{k}{2}, a_{2} = \frac{k}{2}, a_{3} = k$ , where $k \in R$

Hence, the solutions, are $- \frac{k}{2}, \frac{k}{2}, k$ , where $k$ is any real number.

Thus, the number of triplets is infinite.

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Trigonometrical Equations 3 Question 8

9. The number of all possible triplets (a1,a2,a3) such that a1+a2cos⁡(2x)+a3sin2⁡(x)=0,∀x is

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9. The number of all possible triplets $(a_{1}, a_{2}, a_{3})$ such that $a_{1} + a_{2} \cos (2 x) + a_{3} \sin^{2} (x) = 0, \forall x$ is