In the expansion of ${\left(7^{\frac{1}{3}}+{11}^{\frac{1}{9}}\right)}^{6561}$,

Important Questions on Binomial Theorem

HARD

JEE Main/Advance

IMPORTANT

Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 8 - Binomial Theorem>Exercises>Q 19

If ${\left(1+x\right)}^n=C_0+C_{1}x+C_2{x}^2+.....+C_n{x}^n$, then $C_0-\left(C_0+C_1\right)+\left(C_0+C_1+C_2\right)-\left(C_0+C_1+C_2+C_3\right)+....{\left(-1\right)}^{n-1}\left(C_0+C_1+....+C_{n-1}\right)$ is (where $n$ is an even integer and $C_r={}^{n}C_r$)

HARD

JEE Main/Advance

IMPORTANT

Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 8 - Binomial Theorem>Exercises>Q 20

In the expansion of ${\left(a+b\right)}^n$, if two consecutive terms are equal, then which of the following is/are always integer?

HARD

JEE Main/Advance

IMPORTANT

Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 8 - Binomial Theorem>Exercises>Q 21

If for $z$ as real or complex, ${\left(1+z^2+z^4\right)}^8=C_0+C_1{z}^2+C_2{z}^4+....+C_{16}{z}^{32}$, then

HARD

JEE Main/Advance

IMPORTANT

Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 8 - Binomial Theorem>Exercises>Q 22

If  $f\left(m\right)= \sum _{i=0}^m \left(\begin{array}{c}30\\ 30-i\end{array}\right) \left(\begin{array}{c}20\\ m-i\end{array}\right),$ where $\left(\begin{array}{c}p\\ q\end{array}\right)={}^{p}C_q$, then

HARD

JEE Main/Advance

IMPORTANT

Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 8 - Binomial Theorem>Exercises>Q 23

If ${\left(1+x\right)}^n=C_0+C_{1}x+C_2{x}^2+....+C_n{x}^n, n\in N$, then $C_0-C_1+C_2-....+{\left(-1\right)}^{m-1}{C}_{m-1}$ is equal to $\left(m<n\right)$

HARD

JEE Main/Advance

IMPORTANT

Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 8 - Binomial Theorem>Exercises>Q 24

If $\sum _{r=0}^n\left(pr+2\right)·{}^{n}C_r=\left(25\right)\left(64\right)$, where $n, p\in N$, then

HARD

JEE Main/Advance

IMPORTANT

Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 8 - Binomial Theorem>Exercises>Q 25

If ${\left(x+\frac{1}{x}+1\right)}^6=a_0+\left(a_{1}x+\frac{b_1}{x}\right)+\left(a_2{x}^2+\frac{b_2}{x^2}\right)+....+\left(a_6{x}^6+\frac{b_6}{x^6}\right)$, then

HARD

JEE Main/Advance

IMPORTANT

Mathematics for Joint Entrance Examination JEE (Advanced) Algebra>Chapter 8 - Binomial Theorem>Exercises>Q 26

The value of ${}^{n}C_0\times {}^{2n}C_r-{}^{n}C_1\times {}^{2n-2}C_r+{}^{n}C_2\times {}^{2n-4}C_r+....$ is equal to