JEE Main 2022 · 27 Jun · Shift 1 · Mathematics Q25

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MathematicsInteger+4 / -1Question 25 of 90

If the coefficient of $x^{10}$ in the binomial expansion of $\left(\dfrac{\sqrt x}{5^{1/4}}+\dfrac{\sqrt5}{x^{1/3}}\right)^{60}$ is $5^k\cdot l$ , where $l,k\in\mathbb N$ and $l$ is co-prime to 5, then $k$ is equal to ______.

Your answer

Answer and worked solution

Official answer: 5

Term. $\binom{60}r\left(x^{1/2}5^{-1/4}\right)^{60-r}\left(5^{1/2}x^{-1/3}\right)^r$ ; the power of $x$ is $\frac{60-r}2-\frac r3=10$ , so $r=24$ .

Coefficient. $\binom{60}{24}5^{-9}5^{12}=\binom{60}{24}5^3$ .

Power of 5 in $\binom{60}{24}$ . $14-4-8=2$ . So $k=3+2=5$ .

Answer: 5

Topics:General and middle terms Number systems

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JEE Main 2022 · 27 Jun · Shift 1, Mathematics question 25