JAMIA 2022 — Mathematics PYQ

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Question

JAMIA 2022 — Mathematics PYQ

JAMIA | Mathematics | 2022

The coefficient of $y$ in the expansion of $(y^{2} + \frac{c}{y})^{5}$ is

Choose the correct answer:

Explanation

Step 1: General Term Formula

The general term $T_{r+1}$ in the expansion of $(a + b)^n$ is given by:

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

Here:

$n = 5$
$a = y^2$
$b = \dfrac{c}{y} = cy^{-1}$

Step 2: Substitute the Values

$T_{r+1} = \binom{5}{r} (y^2)^{5-r} (cy^{-1})^r$

$T_{r+1} = \binom{5}{r} y^{10-2r} \cdot c^r y^{-r}$

$T_{r+1} = \binom{5}{r} c^r y^{10-3r}$

Step 3: Find $r$ for the Coefficient of $y$

We need the power of $y$ to be $1$ . Therefore, set the exponent of $y$ equal to $1$ :

$10 - 3r = 1$

$3r = 9$

$r = 3$

Step 4: Calculate the Coefficient

Now, substitute $r = 3$ into the coefficient part of the general term:

$\text{Coefficient} = \binom{5}{3} c^3$

Calculate $\binom{5}{3}$ :

$\binom{5}{3} = \frac{5 \times 4}{2 \times 1} = 10$

So, the coefficient is:

$10c^3$

Final Answer:

The coefficient of $y$ is $10c^3$ .