JEE Mathematics 2025 Question with Solution

Given: $(1 + x)^p (1 - x)^q$ has coefficient of $x$ equal to $1$ and coefficient of $x^2$ equal to $-2$.

Find: $p^2 + q^2$.

Expand the two binomials up to the second power of $x$:

The coefficient of $x$ in the product is

So,

For the coefficient of $x^2$, combine the $x^2$ terms and the cross term:

Hence,

Substitute $p = q+1$:

Therefore,

Now calculate:

Therefore, the correct option is C.

Given: coefficient of $x$ is $1$ and coefficient of $x^2$ is $-2$ in $(1+x)^p(1-x)^q$.

Find: $p^2 + q^2$.

Use the direct coefficient relations:

• coefficient of $x$ is $p-q$
• coefficient of $x^2$ is $\frac{p(p-1)}{2} - pq + \frac{q(q-1)}{2}$

From the first condition,

So write

Then substitute into the second condition and solve to get

Hence,

Therefore, the correct option is C.