In the expansion of ${\left(x^3+3·2^{-{{\log }_{\sqrt{2}}}^{\sqrt{x^3}}}\right)}^{11}$

Important Questions on Binomial Theorem

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Beta Question Bank for Engineering: Mathematics>Chapter 8 - Binomial Theorem>EXERCISE-2>Q 14

The sum of the series $\left(1^2+1\right)·1!+\left(2^2+1\right)·2!+\left(3^2+1\right)·3!+...+\left(n^2+1\right)·n!$ is

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Beta Question Bank for Engineering: Mathematics>Chapter 8 - Binomial Theorem>EXERCISE-2>Q 15

The binomial expansion of ${\left(x^k+\frac{1}{x^{2k}}\right)}^{3n}, n\in N$ contains a term independent of $x$

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Beta Question Bank for Engineering: Mathematics>Chapter 8 - Binomial Theorem>EXERCISE-2>Q 16

Let $n\in N$. If ${\left(1+x\right)}^n=a_o+a_{1}x+a_2{x}^2+....+a_n{x}^n$ and $a_{n-3}, a_{n-2}, a_{n-1}$ are in $A.P$, then

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Beta Question Bank for Engineering: Mathematics>Chapter 8 - Binomial Theorem>EXERCISE-2>Q 17

If the coefficients of three consecutive terms in the expansion of ${\left(1+x\right)}^n$ are in ratio of $1:7:42,$ then $n$ is divisible by

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Beta Question Bank for Engineering: Mathematics>Chapter 8 - Binomial Theorem>EXERCISE-2>Q 18

In the expansion of ${\left(\sqrt[3]{4}+\frac{1}{\sqrt[4]{6}}\right)}^{20}$

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Beta Question Bank for Engineering: Mathematics>Chapter 8 - Binomial Theorem>EXERCISE-2>Q 19

If ${\left(1+x+x^2+x^3\right)}^{100}=a_0+a_{1}x+a_2{x}^2+...+a_{300}{x}^{300}$, then

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Beta Question Bank for Engineering: Mathematics>Chapter 8 - Binomial Theorem>EXERCISE-2>Q 20

The number ${101}^{100}-1$ is divisible by

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Beta Question Bank for Engineering: Mathematics>Chapter 8 - Binomial Theorem>EXERCISE-2>Q 21

If ${\left(9+\sqrt{80}\right)}^n=\mathrm{l}+f$ where $\mathrm{l},n$ are integers and $0<f<1$, then