Binomial Expression

Q.

Find the coefficient of $x^6{y}^3$ in the expansion of $(x+2y{)}^9$

Q.

Use the summation formulas to rewrite the expression without the summation notation. Use the result to find the sums for $n=10,100,1000\text{and}10,000$.

$\sum _{k=1}^n\frac{6k\left(k-1\right)}{n^3}$

Q.

Using binomial theorem, find the value of $(103{)}^4$

Q. Expand using binomial theorem
${[1+\frac{x}{2}-\frac{2}{x}]}^4,x\ne 0$.

Q. Using binomial theorem, evaluate the following:
${\left(102\right)}^5$

Q.

Expand ${(x^3-\frac{2}{x^2})}^6$ using binomial expansion.

Q.

Find the coefficient of $x^4$ in the product $(1+2x{)}^4\times (2-x{)}^5$

Q. The coefficient of $x^7$in the expansion of $(1-x-x^2+x^3{)}^6$ is

1. -132
2. -144
3. 132
4. 144

Q.

Square roots of - 7 - 24i are :

1. 

2. 

3. 

4. 

Q. Find the expansion of $(3x^2-2ax+3a^2{)}^3$ using binomial theorem.

Q.

When x is so small that its square and higher powers may be neglected, find the value of
$\frac{{(1+\frac{2}{3}x)}^{-5}+\sqrt{4+2x}}{\sqrt{(4+x{)}^3}}$.

1. $=\frac{1}{4}(3-\frac{95}{24}x)$

2. $=\frac{1}{8}(3-\frac{95}{24}x)$

3. $=\frac{1}{8}(3-\frac{24}{95}x)$

4. $=\frac{1}{4}(3-\frac{9}{24}x)$

Q. If $p$ and $q$ are positive integers, then the absolute difference of the coefficients of $x^p$ and $x^q$ in the expansion of $(1+x{)}^{p+q}$ is

1. $0$
2. $p,$ when $p>q$
3. $q,$ when $p<q$
4. $|p-q|$

Q.

Find the sum of the series ${}^n{C}_0{}^n{C}_2+{}^n{C}_1{}^n{C}_3+{}^n{C}_2{}^n{C}_4.....{}^n{C}_{n-2}{}^n{C}_n$

1. 

2. 

3. 

4. 

Q. Expand using Binomial Theorem .

Q.

Find the value of ${}^n{C}_1+2^n{C}_2+3^n{C}_3.........n^n{C}_n$

1. $(n-1{)}^{2n}$

2. $(n-1)2^{n-1}$

3. $n\cdot 2^{n-1}$

4. $n\cdot 2^n$

Q. If $n$ is a positive integer and $C_k={}^n{C}_k,$ then the value of $\sum _{k=1}^n{k}^3{\left(\frac{C_k}{C_{k-1}}\right)}^2$ is

1. $\frac{n(n+2)(n+1{)}^2}{12}$
2. $\frac{n(n+1)(n+2{)}^2}{12}$
3. $\frac{n(n+1)(n+2)}{12}$
4. None of these

Q. In the binomial expansion of $(a-b{)}^n,n\ge 5,$ the sum of the $5^{th}and{6}^{th}$ terms is zero. Then a/b equals

1. 
2. 
3. 
4. 

Q. Find an approximation of $(0.99{)}^5$ using the first three terms of its expansion.

Q.

If n is a positive integer, find the coefficient of $x^{-1}$ in the expansion of $(1+x{)}^n{(1+\frac{1}{x})}^n$

Q.

Find the sum of the series
${\sum }_n^{r=0}(-1{)}^r{}^n{C}_r[\frac{1}{2^r}+\frac{3^r}{2^{2r}}+\frac{7^r}{2^{3r}}+\frac{{15}^r}{2^{4r}}\cdots uptomterms]$

1. $\frac{2^{mn}-1}{2^{mn}(2^n-1)}$

2. $\frac{2^{mn}-1}{2^{mn}(2^m-1)}$

3. $\frac{2^n-1}{2^{mn}(2^n-1)}$

4. $\frac{2^m-1}{2^n(2^n-1)}$

Q. Find a , b and n in the expansion of ( a + b ) n if the first three terms of the expansion are 729, 7290 and 30375, respectively.

Q.

The sum of the following series, $\frac{1}{3}{}^n{c}_1+\frac{(1^2+2^2)}{5}{}^n{c}_2+\frac{(1^2+2^2+3^2)}{7}{}^n{c}_3$ + ...... upto n terms is

1. 

2. 

3. 

4. 

Q. 12. Find (r + 1)° + (x - 1)°. Hence or otherwise evaluate (V2 + )2

Q.

Find the coefficient of $x^{17}$ in the expansion $(x+x^2+x^3+..............x^6{)}^6(1+x+x^2+x^3+.........)$

1. 7756

2. 9604

3. 3493

4. 11914

Q.

For a positive integer n, find the value of $(1-i{)}^n{(1-\frac{1}{i})}^n$

Q. If $(1+x{)}^n=\sum _{r=0}^n{a}_r{x}^r$, then $(1+\frac{a_1}{a_0})(1+\frac{a_2}{a_1})\dots (1+\frac{a_n}{a_{n-1}})$ is equal to

1. $\frac{(n+1{)}^{n+1}}{n!}$
2. $\frac{(n+1{)}^n}{n!}$
3. $\frac{n^{n-1}}{(n-1)!}$
4. $\frac{(n+1{)}^{n-1}}{(n-1)!}$

Q. If $(1+x)(1+x+x^2).....(1+x+.....+x^n)=a_0+a_{1}x+a_2{x}^2+a_3{x}^3+.....$, then the value of $a_0+a_2+a_4+.....$ is

1. 
2. 
3. 
4. 

Q. Sum of the series ${}^n{C}_1+2\cdot 5{}^n{C}_2+3\cdot 5^2{}^n{C}_3+\cdots$ upto $n$ terms is

1. $n\cdot 6^{n-1}$
2. $6^{n-1}$
3. $6^n$
4. $n\cdot 6^n$

Q. R=$(5\sqrt{5}+11{)}^{2n+1}$ and $f=R-\left[R\right]$, where [] denotes the greatest integer function. Then the value of $Rf$ is

1. $4^{2n+1}$
2. $2^{2n+1}$
3. $2^n$
4. $8^{n+1}$

Q.

Coefficient of $y^3.x^6$ in the binomial expansion $(x+2y{)}^9$

1. 276

2. 672

3. 762

4. 267