JEE Mathematics 2024 Question with Solution

Given:

• $A$ is the sum of all coefficients in the expansion of $(1 - 3x + 10x^2)^n$.
• $B$ is the sum of all coefficients in the expansion of $(1 + x^2)^n$.

Find: The relation between $A$ and $B$.

The sum of coefficients of a polynomial is obtained by substituting $x = 1$.

So,

$$A = (1 - 3 \cdot 1 + 10 \cdot 1^2)^n = (1 - 3 + 10)^n = 8^n$$

Also,

$$B = (1 + 1^2)^n = 2^n$$

Now,

$$A = 8^n = (2^3)^n = (2^n)^3 = B^3$$

Therefore, the correct option is A, that is, $A = B^3$.

Given: The expressions are $(1 - 3x + 10x^2)^n$ and $(1 + x^2)^n$.

Find: Values of $A$ and $B$ in terms of $n$, and then compare them.

1. For $(1 - 3x + 10x^2)^n$, substitute $x = 1$ to get the sum of coefficients.

$$A = (1 - 3 \cdot 1 + 10 \cdot 1^2)^n$$ $$A = (1 - 3 + 10)^n = 8^n$$

2. For $(1 + x^2)^n$, again substitute $x = 1$.

$$B = (1 + 1^2)^n = 2^n$$

3. Compare the two values.

$$8^n = (2^3)^n = (2^n)^3$$

Hence,

$$A = B^3$$

Therefore, the required relation is $A = B^3$.