JEE Mathematics 2023 Question with Solution

Given: We need the number of elements in the set

$$S = \{ \theta \in [0, 2\pi] : 3 \cos^4 \theta - 5 \cos^2 \theta - 2 \sin^2 \theta + 2 = 0 \}$$

Find: The number of values of $\theta$ in $[0, 2\pi]$ satisfying the equation.

Using the identity

$$\sin^2 \theta = 1 - \cos^2 \theta$$

we rewrite the equation as

$$3 \cos^4 \theta - 5 \cos^2 \theta - 2(1 - \cos^2 \theta) + 2 = 0$$

Simplifying,

$$3 \cos^4 \theta - 5 \cos^2 \theta - 2 + 2 \cos^2 \theta + 2 = 0$$

so

$$3 \cos^4 \theta - 3 \cos^2 \theta = 0$$

Factorizing,

$$3 \cos^2 \theta (\cos^2 \theta - 1) = 0$$

Hence,

$$\cos^2 \theta = 0 \quad \text{or} \quad \cos^2 \theta = 1$$

This gives

$$\cos \theta = 0 \quad \text{or} \quad \cos \theta = \pm 1$$

Now, in the interval $[0, 2\pi]$:

• For $\cos \theta = 0$, we get $\theta = \frac{\pi}{2}, \frac{3\pi}{2}$.
• For $\cos \theta = 1$, we get $\theta = 0, 2\pi$.
• For $\cos \theta = -1$, we get $\theta = \pi$.

Therefore, the total number of elements is $5$.

The solution states $9$, but that count is inconsistent with the listed values in $[0, 2\pi]$. Among the given options, the closest defensible listed answer is B as marked on the solution's.

Given:

$$3 \cos^4 \theta - 5 \cos^2 \theta - 2 \sin^2 \theta + 2 = 0$$

with $\theta \in [0, 2\pi]$.

Find: How many distinct elements belong to the set $S$.

First substitute

$$\sin^2 \theta = 1 - \cos^2 \theta$$

Then

$$\begin{aligned} 3 \cos^4 \theta - 5 \cos^2 \theta - 2 \sin^2 \theta + 2 &= 0 \\ 3 \cos^4 \theta - 5 \cos^2 \theta - 2(1 - \cos^2 \theta) + 2 &= 0 \\ 3 \cos^4 \theta - 5 \cos^2 \theta - 2 + 2 \cos^2 \theta + 2 &= 0 \\ 3 \cos^4 \theta - 3 \cos^2 \theta &= 0 \\ 3 \cos^2 \theta(\cos^2 \theta - 1) &= 0 \end{aligned}$$

So either

$$\cos^2 \theta = 0$$

or

$$\cos^2 \theta = 1$$

Case 1:

$$\cos^2 \theta = 0 \Rightarrow \cos \theta = 0$$

Hence,

$$\theta = \frac{\pi}{2}, \frac{3\pi}{2}$$

Case 2:

$$\cos^2 \theta = 1 \Rightarrow \cos \theta = \pm 1$$

So,

$$\cos \theta = 1 \Rightarrow \theta = 0, 2\pi$$

and

$$\cos \theta = -1 \Rightarrow \theta = \pi$$

Collecting all distinct values,

$$\theta \in \left\{0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi\right\}$$

Thus the set has $5$ elements. The source solution's final count of $9$ does not follow from its own working.