JEE Mathematics 2023 Question with Solution

Given:

with $\theta \in [-\pi, \pi]$.

Find: The value of $mn$, where $m$ is the number of positive solutions and $n$ is the number of negative solutions.

Using product-to-sum identities,

and

Therefore,

so

Now, from $\cos A = \cos B$,

Hence,

which gives

So,

In the interval $[-\pi, \pi]$, the solution set listed in the extracted solution is

Thus the number of positive values is $m = 5$ and the number of negative values is $n = 5$. Therefore,

So the required answer is $25$.

After reducing the equation to

use the standard result $\cos A = \cos B \Rightarrow A = 2k\pi \pm B$.

Case 1:

which gives

Case 2:

which gives

The extracted solution also states the equivalent family $\theta = \frac{2k\pi}{5}$, and the final listed points in the interval remain the same.

Now count the values in $[-\pi, \pi]$: negative values are $-\pi, -\frac{4\pi}{5}, -\frac{3\pi}{5}, -\frac{2\pi}{5}, -\frac{\pi}{5}$, so $n = 5$. Positive values are $\frac{\pi}{5}, \frac{2\pi}{5}, \frac{3\pi}{5}, \frac{4\pi}{5}, \pi$, so $m = 5$. Zero is neither positive nor negative.

Hence,

The answer is $25$.