JEE 2022 — Mathematics PYQ

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Question

JEE 2022 — Mathematics PYQ

JEE | Mathematics | 2022

If the constant term in the expansion of $\left( 3x^3 - 2x^2 + \frac{5}{x^5} \right)^{10}$ is $2^k \cdot l$ , where $l$ is an odd integer, then the value of $k$ is equal to:

Choose the correct answer:

Explanation

Solution

1. General Term of Multinomial Expansion:

The general term is given by:

T = \frac{10!}{n_1! n_2! n_3!} (3x^3)^{n_1} (-2x^2)^{n_2} \left(\frac{5}{x^5}\right)^{n_3}

where $n_1 + n_2 + n_3 = 10$ .

2. Condition for Constant Term:

The power of $x$ must be zero:

3n_1 + 2n_2 - 5n_3 = 0

Substitute $n_1 = 10 - n_2 - n_3$ :

3(10 - n_2 - n_3) + 2n_2 - 5n_3 = 0 \implies 30 - n_2 - 8n_3 = 0 \implies n_2 + 8n_3 = 30

Possible integer values for $n_1, n_2, n_3$ :

If $n_3 = 3$ , then $n_2 = 6$ and $n_1 = 1$ .

3. Calculate the Constant Term:

C = \frac{10!}{1! 6! 3!} (3)^1 (-2)^6 (5)^3 = \frac{10 \cdot 9 \cdot 8 \cdot 7}{3 \cdot 2 \cdot 1} \cdot 3 \cdot 2^6 \cdot 125

C = (10 \cdot 3 \cdot 4 \cdot 7) \cdot 3 \cdot 2^6 \cdot 125 = (2 \cdot 5 \cdot 3 \cdot 2^2 \cdot 7) \cdot 3 \cdot 2^6 \cdot 5^3

C = 2^9 \cdot (5^4 \cdot 3^2 \cdot 7)

Here, $l = (5^4 \cdot 3^2 \cdot 7)$ , which is an odd integer. Comparing $2^9 \cdot l$ with $2^k \cdot l$ , we get $k = 9$ .

Correct Option: (D)