The number of solutions of the equation 2 ( 2) + ( 5 2) = 2 ^3 ( 5 2) in the interval - 2, 2 is:

Given: 2 ( 2) + ( 5 2) = 2 ^3 ( 5 2) with [- 2, 2 ]. Find: The number of solutions in the given interval. Use the identity ^3 x = 14 (3 x + 3x) So, 2 ^3 ( 5 2) = 12 (3 ( 5 2) + ( 15 2)) Substituting into the equation, 2 ( 2) + ( 5 2) = 32 ( 5 2) + 12 ( 15 2) Rearranging, 2 ( 2) = 12 ( 15 2) + 12 ( 5 2) The extracted solution states that evaluating this equation over the interval [- 2, 2 ] gives 7 solutions. Therefore, the correct option is A. Given: 2 ( 2) + ( 5 2) = 2 ^3 ( 5 2) Find: The number of solutions in [- 2, 2 ]. The extracted alternative approach sets x = ( 2), y = ( 5 2) and rewrites the equation as 2x^2 y + y = 2y^3 Hence, y(2x^2 + 1) = 2y^3 so that y(2x^2 + 1 - 2y^2) = 0 This gives two cases: 1. y = 0 2. 2x^2 + 1 - 2y^2 = 0 The provided solution then states that, after checking these cases in the interval [- 2, 2 ], there are 7 distinct solutions. Therefore, the correct option is A.

The number of solutions of the equation 2 ( 2) + ( 5 2) = 2 ^3 (… — Solution | JEE Mathematics

MCQMediumJEE 2025Trigonometric Equations

The number of solutions of the equation $\cos 2\theta \cos \left( \frac{\theta}{2} \right) + \cos \left( \frac{5\theta}{2} \right) = 2 \cos^3 \left( \frac{5\theta}{2} \right)$ in the interval $-\frac{\pi}{2}, \frac{\pi}{2}$ is:

A
$7$
B
$5$
C
$6$
D
$9$

Answer

Correct answer:A

Step-by-step solution

Standard Method

Given:

\cos 2\theta \cos \left( \frac{\theta}{2} \right) + \cos \left( \frac{5\theta}{2} \right) = 2 \cos^3 \left( \frac{5\theta}{2} \right)

with $\theta \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ .

Find: The number of solutions in the given interval.

Use the identity

\cos^3 x = \frac{1}{4}\left(3\cos x + \cos 3x\right)

So,

2\cos^3 \left( \frac{5\theta}{2} \right) = \frac{1}{2}\left(3\cos \left( \frac{5\theta}{2} \right) + \cos \left( \frac{15\theta}{2} \right)\right)

Substituting into the equation,

\cos 2\theta \cos \left( \frac{\theta}{2} \right) + \cos \left( \frac{5\theta}{2} \right) = \frac{3}{2} \cos \left( \frac{5\theta}{2} \right) + \frac{1}{2} \cos \left( \frac{15\theta}{2} \right)

Rearranging,

\cos 2\theta \cos \left( \frac{\theta}{2} \right) = \frac{1}{2} \cos \left( \frac{15\theta}{2} \right) + \frac{1}{2} \cos \left( \frac{5\theta}{2} \right)

The extracted solution states that evaluating this equation over the interval $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ gives $7$ solutions.

Therefore, the correct option is A.

Alternative Working from Extracted Solution

Given:

\cos 2\theta \cos \left( \frac{\theta}{2} \right) + \cos \left( \frac{5\theta}{2} \right) = 2 \cos^3 \left( \frac{5\theta}{2} \right)

Find: The number of solutions in $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ .

The extracted alternative approach sets

x = \cos \left( \frac{\theta}{2} \right), \qquad y = \cos \left( \frac{5\theta}{2} \right)

and rewrites the equation as

2x^2 y + y = 2y^3

Hence,

y(2x^2 + 1) = 2y^3

so that

y(2x^2 + 1 - 2y^2) = 0

This gives two cases:

$y = 0$
$2x^2 + 1 - 2y^2 = 0$

The provided solution then states that, after checking these cases in the interval $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ , there are $7$ distinct solutions.

Therefore, the correct option is A.

Common mistakes

Treating the interval $-\frac{\pi}{2}, \frac{\pi}{2}$ as open instead of the closed interval $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ is incorrect because endpoint solutions, if they satisfy the equation, must be counted. Always test the boundary values explicitly.
Using the identity for $\cos^3 x$ incorrectly leads to a wrong transformed equation. The correct identity is $\cos^3 x = \frac{1}{4}(3\cos x + \cos 3x)$ . Substitute this carefully before rearranging terms.
In the substitution method, assuming $\cos 2\theta \cos\left(\frac{\theta}{2}\right) = 2x^2y$ without justification is dangerous because $x$ and $y$ are not independent and the relation is not immediate from the definitions. If using substitutions, verify each transformed term exactly.

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Answer

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