Binomial Theorem · JEE Main previous year questions

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

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

If the coefficints 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. A.$\left(14,\frac{272}{3}\right)$
2. B.$\left(16,\frac{272}{3}\right)$
3. C.$\left(16,\frac{251}{3}\right)$
4. D.$\left(14,\frac{251}{3}\right)$

The middle term in the expansion of ${\left(1-\frac{1}{x}\right)}^n(1-x{)}^n$ in powers of $x$ is

1. A.$-{}^{2n}{C}_{n-1}$
2. B.$-{}^{2n}{C}_n$
3. C.${}^{2n}{C}_{n-1}$
4. D.${}^{2n}{C}_n$

The number of terms in the expansion of $(y^{1/5}+x^{1/10}{)}^{55}$, in which powers of $x$ and $y$ are free from radical signs are

1. A.six
2. B.twelve
3. C.seven
4. D.five

If $f(y)=1-(y-1)+(y-1{)}^2-(y-1{)}^3+\cdots -(y-1{)}^{17}$ then the coefficient of $y^2$ in it is

1. A.${}^{17}{\mathrm{C}}_2$
2. B.${}^{17}{\mathrm{C}}_3$
3. C.${}^{18}{\mathrm{C}}_2$
4. D.${}^{18}{\mathrm{C}}_3$

If $x$ is so small that $x^3$ and higher powers of $x$ may be neglected, then $\frac{(1+x{)}^{\frac{3}{2}}-{\left(1+\frac{1}{2}x\right)}^3}{(1-x{)}^{\frac{1}{2}}}$ may be approximated as

1. A.$1-\frac{3}{8}{x}^2$
2. B.$3x+\frac{3}{8}{x}^2$
3. C.$-\frac{3}{8}{x}^2$
4. D.$\frac{x}{2}-\frac{3}{8}{x}^2$

If the coefficients of $r^{th}$, $(r+1{)}^{th}$, and $(r+2{)}^{th}$ terms in the binomial expansion of $(1+y{)}^m$ are in A.P., then m and r satisfy the equation

1. A.$m^2-m(4r-1)+4r^2-2=0$
2. B.$m^2-m(4r+1)+4r^2+2=0$
3. C.$m^2-m(4r+1)+4r^2-2=0$
4. D.$m^2-m(4r-1)+4r^2+2=0$

The coefficient of the middle term in the binomial expansion in powers of $x$ of $(1+\alpha x{)}^4$ and $(1-\alpha x{)}^6$ is the same if $\alpha$ equals

1. A.$\frac{3}{5}$
2. B.$\frac{10}{3}$
3. C.$\frac{-3}{10}$
4. D.$\frac{-5}{3}$

The coefficient of $x^n$ in expansion of $(1+x)(1-x{)}^n$ is

1. A.$(-1{)}^{n-1}n$
2. B.$(-1{)}^n(1-n)$
3. C.$(-1{)}^{n-1}(n-1{)}^2$
4. D.$(n-1)$

The coefficients of $x^p$ and $x^q$ in the expansion of $(1+x{)}^{p+q}$ are

1. A.equal
2. B.equal with opposite signs
3. C.reciprocals of each other
4. D.none of these

If the sum of the coefficients in the expansion of $(a+b{)}^n$ is 4096, then the greatest coefficient in the expansion is

1. A.1594
2. B.792
3. C.924
4. D.2924

$r$ and $n$ are positive integers $r>1,n>2$ and coefficient of $(r+2{)}^{th}$ term and $3r^{th}$ term in the expansion of $(1+x{)}^{2n}$ are equal, then $n$ equals

1. A.$3r$
2. B.$3r+1$
3. C.$2r$
4. D.$2r+1$