Let f(x)=(x+1)10+(x+2)10⋯+(x+50)10×10+1010 Statement-1:limx→∞ f(x)=10Statement-2:f(x)=∑r=110 ∑α=150 10Crxr−10α10−r1+10×10−1

# Let $f\left(x\right)=\frac{\left(x+1{\right)}^{10}+\left(x+2{\right)}^{10}\cdots +\left(x+50{\right)}^{10}}{{x}^{10}+{10}^{10}}$ Statement-1:$\underset{x\to \mathrm{\infty }}{lim} f\left(x\right)=10$Statement-2:$f\left(x\right)=\sum _{r=1}^{10} \sum _{\alpha =1}^{50}{ }^{10}{C}_{r}{x}^{r-10}{\alpha }^{10-r}{\left(1+{\left(\frac{10}{x}\right)}^{10}\right)}^{-1}$

1. A

STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1

2. B

STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1

3. C

STATEMENT-1 is True, STATEMENT-2 is False

4. D

STATEMENT-1 is False, STATEMENT-2 is True

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### Solution:

$f\left(x\right)=\frac{{\left(1+\frac{1}{x}\right)}^{10}+{\left(1+\frac{2}{x}\right)}^{10}+\cdots +{\left(1+\frac{50}{x}\right)}^{10}}{1+{\left(\frac{10}{x}\right)}^{10}}$

so , $\underset{x\to \mathrm{\infty }}{lim} f\left(x\right)=\frac{1+\cdots +1}{1}$ (50 times)=50

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