Binomial Coefficients

Q. Let four different integers form an increasing A.P. If one of these numbers is equal to the sum of squares of the other three numbers, then which of the following is (are) CORRECT?

1. The product of all four numbers is $0$.
2. The sum of all four numbers is $0$.
3. The common difference of the A.P. is $1$.
4. The smallest number among the four numbers is $-1$.

Q. The sum of first four terms of a geometric progression (G.P.) is $\frac{65}{12}$ and the sum of their respective reciprocals is $\frac{65}{18}.$ If the product of first three terms of the G.P. is $1,$ and the third term is $\alpha ,$ then $2\alpha$ is

Q.

If $a_1,a_2,a_3,a_4$ are the coefficients of any four consecutive terms in the expansion of $(1+x{)}^n$, then $\frac{a_1}{a_1+a_2}+\frac{a_3}{a_3+a_4}=$

1. $\frac{a_2}{a_2+a_3}$

2. $\frac{1}{2}$ $\frac{a_2}{a_2+a_3}$

3. $\frac{2a_2}{a_2+a_3}$

4. $\frac{2a_3}{a_2+a_3}$

Q. The coefficient of $x^5$ in the expansion of $(1+x{)}^{21}+(1+x{)}^{22}+...+(1+x{)}^{30}$ is

1. ${}^{51}{C}_5$
2. ${}^9{C}_5$
3. ${}^{31}{C}_6{-}^{21}{C}_6$
4. ${}^{30}{C}_5{-}^{20}{C}_5$

Q.

The expression ${}^n{C}_r{+}^{n-1}{C}_r+.........{+}^r{C}_r$ is equal to :

1. ${}^{n+1}{C}_r$

2. ${}^{n+1}{C}_{r+1}$

3. ${}^{n+2}{C}_r$

4. $2^n$

Q. If the product of three consecutive numbers in A.P. is $224$ and the largest number is $7$ times the smallest, then which of the following is (are) CORRECT?

1. The smallest number is $4$.
2. The sum of all the three numbers is $10$.
3. The smallest number is $2$.
4. The sum of all the three numbers is $24$.

Q. The coefficient of $\frac{1}{x}$ in the expansion of $(1+x{)}^n{(1+\frac{1}{x})}^n$ is

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

Q. If $x=(\sqrt{3}+1{)}^n,$ where $n$ is odd positive integer, then $\left[x\right]$ is
(where $\left[x\right]$ denotes the greatest integer less than or equal to $x$)

1. $2k$ where $k\in I$
2. $2k+1$ where $k\in I$
3. $4^n$
4. $8^n$

Q. The coefficient of $x^5$ in the expansion of $(1+x{)}^{21}+(1+x{)}^{22}+...+(1+x{)}^{30}$ is

1. ${}^{51}{C}_5$
2. ${}^9{C}_5$
3. ${}^{31}{C}_6{-}^{21}{C}_6$
4. ${}^{30}{C}_5{-}^{20}{C}_5$

Q. If $p$ and $q$ be positive, then the coefficients of $x^p$ and $x^q$ in the expansion of $(1+x{)}^{p+q}$ will be:

1. Equal
2. Unequal
3. Reciprocal to each other
4. None of the above

Q. The coefficient of $x^{50}$ in the expansion of $(1+x{)}^{1000}+2x(1+x{)}^{999}+3x^2(1+x{)}^{998}+......+1001x^{1000}$

1. ${}^{1000}{C}_{50}$
2. ${}^{1001}{C}_{50}$
3. ${}^{1002}{C}_{50}$
4. $2^{1001}$

Q. The coefficient of $x^{203}$ in $(1-x)(2-x^2)(3-x^3)\cdots (20-x^{20})$ is
(correct answer + 1, wrong answer - 0.25)

1. $11$
2. $21$
3. $13$
4. $15$

Q. If ${}^{n-1}{C}_r=(k^2-3{)}^n{C}_{r+1}$, then $kϵ$

1. $(-\infty ,-2]$
2. $[2,\infty )$
3. $[-\sqrt{3},\sqrt{3}]$
4. $(\sqrt{3},2]$

Q. If $(2x-1{)}^{20}-(ax+b{)}^{20}={(x^2+px+q)}^{10}$ holds true $\forall x\in R$ where $a,b,p$ and $q$ are real numbers, then which of the following is (are) CORRECT?

1. $2p+3q=1$
2. $a+2b=0$
3. $a=\sqrt[20]{202^{20}-1}$
4. $4q+p=0$

Q. The coefficient of $x^5$ in the expansion of $(1+x^2{)}^5(1+x{)}^4$ is

Q. For a triangle ABC it is given that $cosA+cosB+cosC=\frac{3}{2}$ Prove that the triangle is equilateral

1. True
2. False

Q. For $2\le r\le n,\left(\begin{array}{c}n\\ r\end{array}\right)+2\left(\begin{array}{c}n\\ r-1\end{array}\right)+\left(\begin{array}{c}n\\ r-2\end{array}\right)$ is equal to

1. $\left(\begin{array}{c}n+1\\ r-1\end{array}\right)$
2. $\left(\begin{array}{c}n+1\\ r+1\end{array}\right)$
3. $2\left(\begin{array}{c}n+2\\ r\end{array}\right)$
4. $\left(\begin{array}{c}n+2\\ r\end{array}\right)$

Q. Let $(2x^2+3x+4{)}^{10}=\sum _{r=0}^{20}{a}_r{x}^r.$ Then the value of $\frac{a_9}{a_{11}}$ is equal to

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

Q. The coefficient of $x^4$ in the expansion of $(2-x+3x^2{)}^6$ is

Q. If $(2x-1{)}^{20}-(ax+b{)}^{20}={(x^2+px+q)}^{10}$ holds true $\forall x\in R$ where $a,b,p$ and $q$ are real numbers, then which of the following is (are) CORRECT?

1. $2p+3q=1$
2. $a+2b=0$
3. $a=\sqrt[20]{202^{20}-1}$
4. $4q+p=0$

Q. If $a_n={\sum }_{r=0}^n\frac{1}{{}^n{C}_r}$ then ${\sum }_{r=0}^n\frac{r}{{}^n{C}_r}$ equals

1. $(n-1)a_n$
2. $n{a}_n$
3. $\frac{1}{2}n{a}_n$
4. None of these

Q. If the coefficients of $x^3$ and $x^4$ in the expansion of $(1+ax+b{x}^2$) $(1–2x{)}^{18}$ in powers of $x$ are both zero, then $(a,b)$ is equal to

1. $(16,\frac{251}{3})$
2. $(14,\frac{251}{3})$
3. $(14,\frac{272}{3})$
4. $(16,\frac{272}{3})$

Q. If the coefficients of the three successive terms in the binomial expansion of $(1+x{)}^n$ are in the ratio $1:7:42$, then the first of these terms in the expansion is

1. $6$th
2. $7$th
3. $8$th
4. $9$th

Q. The coefficient of $x^7$ in the expression $(1+x{)}^{10}+x(1+x{)}^9+x^2(1+x{)}^8+\dots +x^{10}$ is :

1. $420$
2. $330$
3. $210$
4. $120$

Q. In the expansion of $(1+x+x^3+x^4{)}^{10},$ the coefficient of $x^4$ is

1. ${}^{40}{C}_4$
2. ${}^{10}{C}_5$
3. $210$
4. $310$

Q. In the expansion of $(1+x{)}^2(1+y{)}^3(1+z{)}^4(1+w{)}^5,$ the sum of coefficients of the term of degree $12$ is $k.$ Then the value of $\frac{k}{13}$ is

1. $5$
2. $6$
3. $7$
4. $8$