Binomial Theorem — JEE Main PYQs

231 previous year questions on Binomial Theorem (Mathematics), newest first. Open any question for its complete step-by-step solution.

The value of $r$ for which $^{20} C_{r}^{20} C_{0} +^{20} C_{r - 1}^{20} C_{1} +^{20} C_{r - 2}^{20} C_{2} + \dots +^{20} C_{0}^{20} C_{r}$ is maximum i …
JEE Main 2019View solution →
The sum of the real values of x for which the middle term in the binomial expansion of $(\frac{x ^{3}}{3} + \frac{3}{x})^{8}$ equals 5670 is:
JEE Main 2019View solution →
A ratio of the $5^{th}$ term from the beginning to the $5^{th}$ term from the end in the binomial expansion of $(2^{1/3} + \frac{1}{2 ( 3 ) ^{1/3}})^{10}$ is:
JEE Main 2019View solution →
The term independent of $x$ in the expansion of $(\frac{1}{60} - \frac{x ^{8}}{81}) \cdot (2 x^{2} - \frac{3}{x ^{2}})^{6}$ is equal to:
JEE Main 2019View solution →
If $^{20} C_{1} + (2^{2})^{20} C_{3} + (3^{2})^{20} C_{3} + (2^{2}) + \dots\dots + (2 0^{2})^{20} C_{20} = A (2^{β})$ , then the ordered pair ( $A, β$ ) is equ …
JEE Main 2019View solution →
The total number or irrational terms in the binomial expansion of $(7^{\frac{1}{5}} - 3^{\frac{1}{10}})^{60}$ is :
JEE Main 2019View solution →
If some three consecutive coefficients in the binomial expansion of $(x + 1)^{n}$ in powers of $x$ are in the ratio $2 : 15 : 70$ , then the average of these three coefficients is:
JEE Main 2019View solution →
The coefficient of $t^{4}$ in the expansion of $(\frac{1 - t ^{6}}{1 - t})^{3}$ is
JEE Main 2019View solution →
If the fractional part of the number $\frac{2 ^{403}}{15}$ is $\frac{k}{15}$ , then k is equal to:
JEE Main 2019View solution →
If the coefficients of $x^{2}$ and $x^{3}$ are both zero, in the expansion of the expression $(1 + a x + b x^{2}) (1 - 3 x)^{15}$ in powers of $x$ , then the ordered pair $(a, b)$ is equal to
JEE Main 2019View solution →
The sum of the co-efficients of all odd degree terms in the expansion of $(x + x^{3} - 1)^{5} + (x - x^{3} - 1)^{5}, (x > 1)$ is :
JEE Main 2018View solution →
The value of $(^{21} C_{1} -^{10} C_{1}) + (^{21} C_{2} -^{10} C_{2}) + (^{21} C_{3} -^{10} C_{3}) + (^{21} C_{4} -^{10} C_{4}) + \dots .. + (^{21} C_{10} -^{10} C_{10})$ is :
JEE Main 2017View solution →
If the number of terms in the expansion of $(1 - \frac{2}{x} + \frac{4}{x ^{2}})^{n}, x \neq = 0$ , is 28, then the sum of the coefficients of all the terms in this expansion, is:
JEE Main 2016View solution →
The sum of coefficients of integral powers of $x$ in the binomial expansion of $(1 - 2 x)^{50}$ is :
JEE Main 2015View solution →
If the coefficients of $x^{3}$ and $x^{4}$ in the expansion of $(1 + a x + b x^{2}) (1 - 2 x)^{18}$ in powers of $x$ are both zero, then $(a, b)$ is equal to
JEE Main 2014View solution →
The term independent of x in expansion of $(\frac{x + 1}{x ^{2/3} - x ^{1/3} + 1} - \frac{x - 1}{x - x ^{1/2}})^{10}$ is
JEE Main 2013View solution →
If n is a positive integer, then $(3 + 1)^{2 n} - (3 - 1)^{2 n}$ is
JEE Main 2012View solution →
The coefficient of $x^{7}$ in the expansion of $(1 - x - x^{2} + x^{3})^{6}$ is
JEE Main 2011View solution →
Let $S_{1} = \sum_{j = 1}^{10} j (j - 1)^{10} C_{j}$ , $S_{2} = \sum_{j = 1}^{10} j^{10} C_{j}$ and $S_{3} = \sum_{j = 1}^{10} j^{2}^{10} C_{j}$ .

Statement-1: $S_{3} = 55 \times 2^{9}$
Statement-2: $S_{1} = 90 \times 2^{8}$ and $S_{2} = 10 \times 2^{8}$ .
JEE Main 2010View solution →
The remainder left out when $8^{2 n} - (62)^{2 n + 1}$ is divided by 9 is
JEE Main 2009View solution →
Statement -1: $\sum_{r = 0}^{n} (r + 1)^{n} C_{r} = (n + 2) 2^{n - 1}$ .

Statement-2: $\sum_{r = 0}^{n} (r + 1)^{n} C_{r} x^{r} = (1 + x)^{n} + n x (1 + x)^{n - 1}$ .
JEE Main 2008View solution →
In the binomial expansion of $(a - b)^{n}, n \geq 5$ , the sum of $5^{th}$ and $6^{th}$ terms is zero, then $\frac{a}{b}$ equals
JEE Main 2007View solution →
The sum of the series
$^{20} C_{0} -^{20} C_{1} +^{20} C_{2} -^{20} C_{3} + \dots - \dots +^{20} C_{10}$ is
JEE Main 2007View solution →
If the expansion in powers of $x$ of the function $\frac{1}{( 1 - a x ) ( 1 - b x )}$ is $a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{3} + \dots$ , then $a_{n}$ is
JEE Main 2006View solution →
For natural numbers $m, n$ if $(1 - y)^{m} (1 + y)^{n} = 1 + a_{1} y + a_{2} y^{2} + \dots$ , and $a_{1} = a_{2} = 10$ , then $(m, n)$ is
JEE Main 2006View solution →
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
JEE Main 2005View solution →
If the coefficient of $x^{7}$ in $[a x^{2} + (\frac{1}{b x})]^{11}$ equals the coefficient of $x^{- 7}$ in $[a x - (\frac{1}{b x ^{2}})]^{11}$ , then a and b satisfy the relation
JEE Main 2005View solution →
If $x$ is so small that $x^{3}$ and higher powers of $x$ may be neglected, then $\frac{( 1 + x ) ^{3/2} - ( 1 + \frac{1}{2} x ) ^{3}}{( 1 - x ) ^{1/2}}$ may be approximated as
JEE Main 2005View solution →
The coefficient of the middle term in the binomial expansion in powers of $x$ of $(1 + α x)^{4}$ and of $(1 - α x)^{6}$ is the same if $α$ equals
JEE Main 2004View solution →
The coefficient of $x^{n}$ in expansion of $(1 + x) (1 - x)^{n}$ is
JEE Main 2004View solution →
If $S_{n} = \sum_{r = 0}^{n} \frac{1}{^{n} C _{r}}$ and $t_{n} = \sum_{r = 0}^{n} \frac{r}{^{n} C _{r}}$ , then …
JEE Main 2004View solution →