M L Aggarwal Solutions for Chapter: Binomial Theorem, Exercise 1: EXERCISE 9.1

Using binomial theorem, evaluate the following:

$(\sqrt{3}+\sqrt{2}{)}^3+(\sqrt{3}-\sqrt{2}{)}^3$.

Using binomial theorem, evaluate the following:

$(3+\sqrt{2}{)}^5-(3-\sqrt{2}{)}^5$

Using Binomial theorem, expand $(x+y{)}^5+(x-y{)}^5.$ Hence, find the value of $(\sqrt{2}+1{)}^5+(\sqrt{2}-1{)}^5$.

Using Binomial theorem, find $(x+1{)}^6+(x-1{)}^6.$ Hence or otherwise evaluate $(\sqrt{2}+1{)}^6+(\sqrt{2}-1{)}^6$.

Using Binomial theorem, expand $(a+b{)}^4$. Hence or otherwise prove that $(1-x{)}^8=(1-2x{)}^4+4x^2(1-2x{)}^3+6x^4(1-2x{)}^2+4x^6(1-2x)+x^8$.

Prove that $\sum _{r=0}^n{3}^{rn}{C}_r=4^n$.

By using binomial theorem, prove that: $2^{3n}-7n-1$ is divisible by $49$, for all $n\in N$.

By using binomial theorem, prove that: $3^{3n}-26n-1$ is divisible by $676$, for all $n\in N$.