JEE 2022 — Mathematics PYQ

Step 1: Simplify the equation

The given equation is:

We know the trigonometric identity:

Multiplying by $5$, we get:

Now, substitute this back into the original equation:

Step 2: Solve for $\cos 2\theta$

Factor out $\cos 2\theta$:

This gives us two possible cases:

1. $\cos 2\theta =0$

2. $\cos 2\theta =-1/3$

Step 3: Find the number of solutions in the interval

The given interval for $\theta$ is $[-4\pi ,4\pi ]$.

Since the variable in our functions is $2\theta$, the interval for $2\theta$ is:

Case 1: $\cos 2\theta =0$

In one full cycle (length $2\pi$), the cosine function equals $0$ exactly 2 times.

The total length of the interval $[-8\pi ,8\pi ]$ is $16\pi$.

Number of cycles = $16\pi /2\pi =8$ cycles.

Total solutions for Case 1 = $8\times 2=16$ solutions.

Case 2: $\cos 2\theta =-1/3$

Since $-1/3$ is between $-1$ and $1$, the cosine function equals $-1/3$ exactly 2 times in every cycle.

Using the same logic for 8 cycles:

Total solutions for Case 2 = $8\times 2=16$ solutions.

Final Calculation

Total number of elements in set $S$ = (Solutions from Case 1) + (Solutions from Case 2)

Answer: The number of elements in the set is 32.