JEE Mathematics 2023 Question with Solution

Given: $f(x) = \frac{\sin x + \cos x - \sqrt{2}}{\sin x - \cos x}$ and we need to find $f\left(\frac{7\pi}{12}\right)$.

Find: The correct option corresponding to the value of the function.

From the solution, the function is simplified as

$$f(x) = - \tan\left( \frac{x}{2} - \frac{\pi}{8} \right)$$

Now substitute $x = \frac{7\pi}{12}$:

$$f\left( \frac{7\pi}{12} \right) = - \tan\left( \frac{7\pi}{24} - \frac{\pi}{8} \right)$$ $$= - \tan\left( \frac{7\pi}{24} - \frac{3\pi}{24} \right)$$ $$= - \tan\left( \frac{\pi}{6} \right)$$ $$= - \frac{1}{\sqrt{3}}$$

However, the solution marks Option D as the correct option. Therefore, based on the provided the solution's, the correct option is D.

The solution contains working for additional derivatives and even evaluates a product involving $f\left( \frac{7\pi}{12} \right)$ and $f''\left( \frac{7\pi}{12} \right)$, which does not match the asked question.

Also, the computed value shown in the working is

$$f\left( \frac{7\pi}{12} \right) = -\frac{1}{\sqrt{3}}$$

which does not match any of the listed options. Since the solution's explicitly states The Correct Option is D, the answer has been mapped to D from the provided page rather than inferred from the inconsistent working.