Sum of Binomial Coefficients of Odd Numbered Terms

Q. If the coefficients of three consecutive terms in the expansion of $(1+x{)}^n$ are $165,330$ and $462$ respectively, then the value of $n$ is

Q. ${}^{10}{C}_1{+}^{10}{C}_3{+}^{10}{C}_5{+}^{10}{C}_7{+}^{10}{C}_9=$

1. $2^9$
2. $2^{10}$
3. $2^{10}-1$
4. $Noneofthese$

Q.

The sum of the coefficients of all odd degree terms in the expansion of ${\left(x+\sqrt{\left(x^3-1\right)}\right)}^5+{\left(x-\sqrt{\left(x^3-1\right)}\right)}^5$ is

1. $1$

2. $2$

3. $-1$

4. $0$

Q. In the expansion of $(1+x{)}^{50}$, the sum of the coefficient of odd powers of x is

1. 2⁴⁹
2. 
3. 
4. 0

Q. Let $P=\sum _{r=1}^{50}\frac{{}^{50+r}{C}_r(2r-1)}{{}^{50}{C}_r(50+r)},Q=\sum _{r=0}^{50}{(}^{50}{C}_r{)}^2$ and $R=\sum _{r=0}^{100}(-1{)}^r{(}^{100}{C}_r{)}^2$. Then

1. $Q=R$
2. $P-Q=-1$
3. $Q+R=2P+1$
4. $P-R=1$

Q.

In the expansion of $(1+x{)}^n$ the sum of coefficients

of odd powers of x is

1. 

2. 

3. 

4. 

Q.

If the sum of odd numbered terms and the sum of even numbered terms in the expansion of $(x+a{)}^n$ are A and B respectively, then the value of $(x^2-a^2{)}^n$ is

1. $A^2-B^2$

2. $A^2+B^2$

3. 4AB

4. None of these

Q.

If A and B are the sums of odd and even terms respectively in the expansion of $(x+a{)}^n$, then $(x+a{)}^{2n}-(x-a{)}^{2n}$ is equal to

1. 4(A+B)

2. 4(A-B)

3. 4AB

4. AB

Q. Let the coefficients of powers of $x$ in the second, third and fourth terms in the binomial expansion of $(1+x{)}^n,$ where $n$ is a positive integer, be in arithmetic progression. The sum of the coefficients of odd powers of $x$ in the expansion is

1. $32$
2. $256$
3. $128$
4. $64$

Q. If in the expansion of ${(2^x+\frac{1}{4^x})}^n,$ the ratio of the third term and the second term is $7$ and the sum of the coefficients of the second term and the third term is $36,$ then the value of $x$ is

1. $-\frac{1}{3}$
2. $-\frac{1}{2}$
3. $\frac{1}{3}$
4. $\frac{1}{2}$

Q.

Write ${\sum }_{r=0}^m{}^{n+r}{C}_r$ in the simplified form.

Q. In the expansion of $(1+x{)}^{70}$, the sum of coefficients of odd powers of x is

1. 
2. 
3. 
4. 

Q.

. The minimum value of the sum of real numbers $a^{-5},a^{-4},3a^{-3},1,a^8,a^{10}$ , where $a>0$ is

1. $5$

2. $6$

3. $8$

4. $7$

Q. ${}^{10}{C}_1{+}^{10}{C}_3{+}^{10}{C}_5{+}^{10}{C}_7{+}^{10}{C}_9=$

1. 2⁹
2. 2¹⁰
3. None of these
4. 2¹⁰-1

Q. The anti derivative of $(\sqrt{x}+\frac{1}{\sqrt{x}})$ equals

1. $\frac{1}{3}{x}^{\frac{1}{3}}+2x^{\frac{1}{2}}+C$
2. $\frac{2}{3}{x}^{\frac{2}{3}}+\frac{1}{2}{x}^2+C$
3. $\frac{3}{2}{x}^{\frac{3}{2}}+\frac{1}{2}{x}^{\frac{1}{2}}+C$
4. $\frac{2}{3}{x}^{\frac{3}{2}}+2x^{\frac{1}{2}}+C$

Q. Find the value of ${}^n{C}_1{+}^{n}C5{+}^{n}C9{+}^n{C}_{13}.....$

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

Q. If $n$ is a positive integer, then $(\sqrt{3}+1{)}^{2n}-(\sqrt{3}-1{)}^{2n}$ is :

1. an irrational number
2. an odd positive interger
3. an even positive interger
4. a rational number other than positive integers

Q. The term independent of $x$ in the binomial expansion of $(1-\frac{1}{x}+3x^5){(2x^2-\frac{1}{x})}^8$ is

1. $-496$
2. $-400$
3. $496$
4. $400$

Q. Consider the binomial expansion of ${(\sqrt{x}+\frac{1}{2\cdot \sqrt[4]{4x}})}^n,n\in N$ where the terms of the expansion are written in decreasing powers of $x.$ If the coefficients of the first three terms form an arithmetic progression, then which of the following statements hold(s) good?

1. Total number of terms in the binomial expansion is $8$
2. Number of terms in the binomial expansion with integral power of $x$ is $3$
3. There is no term in the binomial expansion which is independent of $x$
4. Fourth and fifth terms are the middle terms of the expansion

Q. The term independent of $x$ in the expansion of
${(x^2+\frac{1}{x})}^9$ is

1. 1
2. -1
3. 48
4. 84

Q. Integrate the function $\frac{e^{2x}-e^{-2x}}{e^{2x}+e^{-2x}}$

Q. $\int \frac{(\sqrt{x}{)}^3}{(\sqrt{x}{)}^5+x^4}dx=A\log \left(\frac{x^k}{x^k+1}\right)+c$ then the values of $A$ and $k$ respectively are.

1. $\frac{3}{2}\&\frac{2}{3}$
2. $\frac{3}{2}$ & $2$
3. does not exist
4. $\frac{2}{3}\&\frac{3}{2}$

Q. Statement-$1$: If vertices of a triangle are $A\left(\overrightarrow{a}\right)$ , $B\left(\overrightarrow{b}\right)$ , $C\left(\overrightarrow{c}\right)$ then length of altitude through $A$ is $\frac{|\overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{b}\times \overrightarrow{c}+\overrightarrow{c}\times \overrightarrow{a}|}{|\overrightarrow{b}-\overrightarrow{c}|}$

1. Statement$1$ is true, statement-$2$ is true, statement-$2$ is a correct explanation for statement-$1$.
2. Statement-$1$ is true, statement-$2$ is false
3. Statement-$1$ is true, statement-$2$ is true, statement-$2$ is not a correct explanation for statement-$1$
4. Statement-$1$ is false, statement-$2$ is true

Q. The sum of coefficients of intergral power of $x$ in the binomial expansion ${(1-2\sqrt{x})}^{50}$is:

1. $\frac{1}{2}(3^{50}-1)$
2. $\frac{1}{2}(3^{50}+1)$
3. $\frac{1}{2}\left(3^{50}\right)$
4. $\frac{1}{2}(2^{50}+1)$

Q. The sum $\sum _{i=0}^m(\frac{10}{i})(\frac{20}{m-i})$ be maximum when m is

1. 15
2. 5
3. 20
4. 10

Q. Determine n if
(i) 2nC3:nC3 =12:1

Q. Let the coefficients of powers of $x$ in the second, third and fourth terms in the binomial expansion of $(1+x{)}^n,$ where $n$ is a positive integer, be in arithmetic progression. The sum of the coefficients of odd powers of $x$ in the expansion is

1. $32$
2. $64$
3. $128$
4. $256$

Q. Integrate the function $\frac{1}{\sqrt{x^2+2x+2}}$

Q. The function $f\left(x\right)=\{\begin{array}{cc}\frac{x^2}{a}& 0\le x<1\\ a& 1\le x<\sqrt{2}\\ (2b^2-4b)/x^2& \sqrt{2}\le x<\infty \end{array}$ is continuous for $0\le x<\infty$, then the most suitable values of $a$ and $b$ are

1. $a=-1,b=1$
2. $a=1,b=1$
3. $a=-1,b=1+\sqrt{2}$
4. none of these

Q. If $A(\frac{2c}{a},\frac{c}{b}),B(\frac{c}{a},0)$ and $C(\frac{1+c}{a},\frac{1}{b})$ are three points, then

1. $(AB{)}^2+(BC{)}^2-(CA{)}^2=2c$
2. $\frac{(AB{)}^2+(BC{)}^2}{(CA{)}^2}=\frac{c^2+1}{(c-1{)}^2}$
3. $(AB{)}^2+(BC{)}^2-(CA{)}^2=\frac{2c(a^2+b^2)}{a^2{b}^2}$
4. $(AB{)}^2-(BC{)}^2-(CA{)}^2=2b$