Binomial Theorem Questions - JEE

Let P(n) = 10n + 3·4n+2 + k is divisible by 9, ∀n ∈ N. The least positive integral value of k is:

In the expansion of $(1 + x)^{2m}\left(\dfrac{x}{1-x}\right)^{-2m}$, the term independent of $x$ is

If $\displaystyle\sum_{r=0}^{n}\dfrac{r+2}{r+1}\,{}^nC_r = \dfrac{2^8 - 1}{6}$, then $n$ is

For all $n \in \mathbb{N}$, $3 \cdot 5^{2n+1} + 2^{3n+1}$ is divisible by __________.

If $1 + x^4 + x^5 = a_0 + a_1(1+x) + a_2(1+x)^2 + a_3(1+x)^3 + a_4(1+x)^4 + a_5(1+x)^5$, then $a_2$ equals:

We have ${}^nC_{r+1} + {}^nC_{r-1} + 2 \times {}^nC_r$. Which of the following is equal to this expression?

The value of $({}^{21}C_1 - {}^{10}C_1) + ({}^{21}C_2 - {}^{10}C_2) + ({}^{21}C_3 - {}^{10}C_3) + ({}^{21}C_4 - {}^{10}C_4) + \ldots + ({}^{21}C_{10} - {}^{10}C_{10})$ is

If ${}^{n+1}C_{r+1} : {}^nC_r : {}^{n-1}C_{r-1} = 11:6:3$, then $nr =$

If $\sum_{r=0}^{n} {}^n C_r a^r b^{n-r} = na(a+b)^{n-1}$, then find the value of the expression. We have,\[\sum_{r=0}^{n} r \cdot {}^n C_r a^r b^{n-r}\]

Find the number of irrational terms in the expansion of $(5^{1/6} + 2^{1/8})^{100}$.

Let $\left[(5+\sqrt{b})^n\right] = N$ where $b$, $n$ are natural numbers and $|5-\sqrt{b}|<1$. If $b$ and $n$ are picked randomly, then the probability that $N$ is odd belongs to the set (where $[\cdot]$ denotes the greatest integer function)

In the expansion of $(1 + 2x + x^2)^9$ there is exactly one term whose coefficient is not equal to the coefficient of any other term. (State whether true or false.)

The value of $\displaystyle\sum_{r=0}^{10} r\,{}^{20}C_r$ is equal to

The first three terms in the expansion of $(1 + ax)^n$ $(n \neq 0)$ are $1, 6x,$ and $16x^2$. Then find the value of $a$ and $n$.

If $\binom{30}{0}\binom{30}{20}-\binom{30}{1}\binom{30}{19}+\binom{30}{2}\binom{30}{18}-\cdots+\binom{30}{20}\binom{30}{0}={}^nC_r$, then maximum possible value of $n+r$ equals

The coefficient of $x^{1012}$ in the expansion of $(1+x^n+x^{253})^{10}$ where $n\leq22$ is any positive integer, is

Find last three digits of the number $27^{27}$.

If $p = (8 + 3\sqrt{7})^n$ and $f = p - [p]$, where $[\cdot]$ denotes the greatest integer function, then the value of $p(1-f)$ is equal to

Maximum sum of coefficient in the expansion of $(1 - x\sin\theta + x^2)^n$ is

In the expansion of $\left(\dfrac{x^2}{y} + \dfrac{y^2}{x}\right)^{15}$, there is a term independent of $x$ and a term independent of $y$ but no term independent of $x$ and $y$ both. (State whether true or false.)

If the coefficients of 5th, 6th and 7th terms in the expansion of $(1 + x)^n$ are in A.P., then $n =$

The smallest integer larger than $(\sqrt{3} + \sqrt{2})^6$ is

The coefficient of $x^2$ in $(1+ax+bx^2)(1-3x)^{15}$ is equal to the coefficient of $x^3$ in the same expansion. Given this condition (along with another condition), find the values of $a$ and $b$.

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

If the fractional part of the number $\dfrac{2^{403}}{15}$ is $\dfrac{k}{15}$, then $k$ is equal to ___________.

Sum the series: $1 + \dfrac{1}{3} + \dfrac{1 \cdot 3}{3 \cdot 6} + \dfrac{1 \cdot 3 \cdot 5}{3 \cdot 6 \cdot 9} + \ldots$ to $\infty$.

The value of $x$ for which the sixth term in the expansion of \[\left[2^{\log_2\sqrt{9^{x-1}+7}} + \dfrac{1}{2^{\frac{1}{5}\log_2(3^{x-1}+1)}}\right]^7\] is 84 is

The coefficients of three consecutive terms of $(1+x)^{n+5}$ are in the ratio $5:10:14$. Then $n =$ _______.

The value of $\displaystyle\sum_{r=0}^{40} r \cdot {}^{40}C_r \cdot {}^{30}C_r$ is

If $(1+x)^{20} = {}^{20}C_0 + {}^{20}C_1 x + \cdots + {}^{20}C_{20}x^{20}$, then ${}^{20}C_0 - {}^{20}C_1 + \cdots - {}^{20}C_9 + {}^{20}C_{10}$ equals:

If $(1+x)^{2016}\left[\dfrac{\left(\dfrac{x}{1+x}\right)^{2017}-1}{\dfrac{1}{1+x}-1}\right] = \sum_{i=0}^{2016} a_i x^i$, then $a_{17}$ equals:

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

Let T2−1, Tr, Tr+1 be the three successive terms of (1 + x)n. If nCr−1 : nCr−1 : nCr = 1 : 7 : 42, then the first of the three given terms will be the kth term. Find k.

The value of $\displaystyle\sum_{r=1}^{n}(-1)^{r+1}\dfrac{{}^nC_r}{r+1}$ is equal to

If the middle term in the expansion of $(x^2 + 1/x)^n$ is $924x^6$, then find the value of $n$.

The sum of series ${}^{20}C_0 - {}^{20}C_1 + {}^{20}C_2 - {}^{20}C_3 + \cdots + {}^{20}C_{10}$ is

The value of $\displaystyle\sum_{r=1}^{15} \dfrac{r \cdot 2^r}{(r+2)!}$ is equal to

For Problems 7–9: An equation $a_0 + a_1 x + a_2 x^2 + \cdots + a_{99} x^{99} + x^{100} = 0$ has roots ${}^{99}C_0, {}^{99}C_1, {}^{99}C_2, \ldots, {}^{99}C_{99}$.8. The value of $a_{98}$ is

For Problems 12–14: Consider the expansion of $(a + b + c + d)^6$. Then the sum of all the coefficients of the terms12. Which contains all of $a, b, c$ and $d$ is

Given $\sum_{r=0}^{2n} a_r \left(\frac{1}{x}\right)^r = \left(1 + \frac{1}{x} + \frac{1}{x^2}\right)^n$, find $a_r$ in terms of $a_{2n-r}$ for $0 \le r \le 2n$.

The value of $\displaystyle\sum_{r=0}^{10}(-1)^r \cdot 4^{10-r} \cdot {}^{30}C_r \cdot {}^{30}C_{10-r}$ is equal to

Given $x^2\left(\sqrt{x}+\dfrac{\lambda}{x^2}\right)^{10}$, the coefficient of $x^2$ in the expansion is 720. Find $\lambda$.

The value of $\displaystyle\sum_{r=0}^{50} (-1)^r \dfrac{{}^{50}C_r}{r+2}$ is equal to

If the middle term of $(1 + \alpha x)^4$ equals the middle term of $(1 - \alpha x)^6$ and $\alpha \neq 0$, then $\alpha$ equals:

Find the value of $\dfrac{1}{81^n} - \dfrac{10}{81^n}\,{}^{2n}C_1 + \dfrac{10^2}{81^n}\,{}^{2n}C_2 - \dfrac{10^3}{81^n}\,{}^{2n}C_3 + \cdots + \dfrac{10^{2n}}{81^n}$.

If $\binom{2n-1}{0} + \binom{2n-1}{3} + \binom{2n-1}{6} + \ldots = 170$, then $n$ equals

If the number of terms in the expansion of (x1/a + x-1/b)n where 0 is 28, then the sum of the coefficients of all the terms in this expansion is

If (1 + 2x + 3x^2)^{10} = a_0 + a_1 x + a_2 x^2 + \cdots + a_{20} x^{20}, then a_1 equals

Number of values of r satisfying the equation \binom{69}{3r-1} + \binom{69}{r} = 2\binom{69}{3r} + \binom{69}{r^2+1} is

If the fourth term in the expansion of $\left(\sqrt[3]{p}x + \frac{1}{\sqrt[3]{x}}\right)^{n}$ is 5, then $n + p$ is equal to

Binomial Theorem Questions (605)