JEE Mathematics 2023 Question with Solution

Given: the solution states

Find: $5a^2 + b^2$

Using the expansion shown in the solution,

For the limit to exist, the coefficient of $x$ must vanish, so

Hence,

Now the solution gives

Substituting $c = 2a$,

So,

Therefore, the computed value is $68$. The solution also explicitly marks the correct option as A, which conflicts with the listed options. the correct option is A.

The source contains an internal inconsistency:

• the question shows $\lim_{x \to 0} \frac{e^{x} - \cos(bx) - cx}{1 - \cos(2x)} = 2$ and asks for $5a^2 + b^2$.
• the solution instead works with $e^{ax}$, $\frac{cxe^{-cx}}{2}$, and the limit equal to $17$.
• The solution concludes $5a^2 + b^2 = 68$, but the solution says the correct option is A.

Since answer resolution must prioritize the solution, the output answer is set to A even though the worked value matches option D = 68.