JAMIA 2024 — Mathematics PYQ

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Question

JAMIA 2024 — Mathematics PYQ

JAMIA | Mathematics | 2024

The coefficient of the middle term in the expansion of $(2 + 3 x)^{4}$ is

Choose the correct answer:

Explanation

Step 1: Middle term ki sankhya pata karein

Yahan $n = 4$ hai, jo ek even (sam) sankhya hai.

Is expansion mein total terms $n + 1 = 4 + 1 = 5$ honge.

Middle term $(\frac{n}{2} + 1)^{th}$ term hota hai.

\text{Middle term} = (\frac{4}{2} + 1)^{th} = 3^{rd} \text{ term}

Step 2: General term ka formula use karein

Binomial expansion $(a + b)^n$ ka general term hota hai:

T_{r+1} = \binom{n}{r} a^{n-r} b^r

Yahan $a = 2$ , $b = 3x$ , aur $n = 4$ hai. $3^{rd}$ term ke liye hum $r = 2$ lenge:

T_3 = T_{2+1} = \binom{4}{2} (2)^{4-2} (3x)^2

Step 3: Calculation karein

Ab values calculate karte hain:

$\binom{4}{2} = \frac{4 \times 3}{2 \times 1} = 6$
$(2)^{4-2} = 2^2 = 4$
$(3x)^2 = 9x^2$

Inhe multiply karne par:

T_3 = 6 \times 4 \times 9x^2

T_3 = 216x^2

Step 4: Final Answer

Middle term ka coefficient (gunank) 216 hai.

Choose the correct answer:

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