Binomial Theorem Questions - JEE

C0 - C1 + C2 - C3 + ... + (-1)n Cn is equal to

If $(1+x+x^2)^n=\sum_{r=0}^{2n}a_r x^r$, then $a_r-{}^nC_1 a_{r-1}+{}^nC_2 a_{r-2}-\cdots+(-1)^r{}^nC_r$ equals (r not multiple of 3)

If the coefficient of $x^7$ in the expansion of $\left(ax^2 + \frac{1}{bx}\right)^{11}$ and coefficient of $x^{-7}$ in the expansion of $\left(ax - \frac{1}{bx^2}\right)^{11}$ are equal, then the minimum value of $a^2 + b^2$ is

If the coefficient of x7 in $\left(a x-\frac{1}{b x^2}\right)^{13}$ and the coefficient of x-5 in $\left(a x+\frac{1}{b x^2}\right)^{13}$ are equal, then a4b4 is equal to:

Let p be the coefficient of x2 in the expansion (1 + x)(1 – 3x)(1 + 5x)(1 – 7x)...(1 – 23x)(1 + 25x), then the sum of the digits of |p| is equal to

The term independent of $x$ in $(1+x+x^{-2}+x^{-3})^{10}$ is $n$. Then the last digit of $(n+2)^n$ is

The remainder when $4^{28^{2024}}$ is divided by 21 is __________.

The sum of the co-efficients of all even degree terms in $x$ in the expansion of $(x + \sqrt{x^3 - 1})^6 + (x - \sqrt{x^3 - 1})^6$, $(x > 1)$ is equal to ___________.

In the binomial expansion of $\left(\dfrac{x}{3}\right)^r$ in $\left(2^{55-r}\right)$, two consecutive terms are equal. Find the value of $r$.Equivalently: In the expansion of $\left(\dfrac{x}{3}\right)$ using $^{32}C_r \cdot 2^{55-r}$, if two consecutive terms $T_{r+1}$ and $T_{r+2}$ are equal, then $r$ equals:

The coefficient of $x^4$ in the expansion of $\{1+(x^n+x^{253})\}^{10}$ is $^{10}C_4$. Find the value of $n$.

Given $\left(\dfrac{2}{x} + x^{\log_8 x}\right)^6$, if the 4th term of the binomial expansion is $20 \times 8^7$, find $x$.

The general term of $\left(x^2 + \dfrac{1}{x^3}\right)^n$ is $T_{r+1} = {}^nC_r (x^2)^{n-r} \left(\dfrac{1}{x^3}\right)^r = {}^nC_r \, x^{2n-5r}$. Given ${}^nC_r = {}^nC_{23}$ and the coefficient of $x$ exists, find the value of $n$.

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

If 7103 is divided by 25, find the remainder.

The sum $\displaystyle\sum_{r=1}^{10} (r^2 + 1) \times (r!)$ is equal to

In $\left(\sqrt[3]{2} + \dfrac{1}{\sqrt[3]{3}}\right)^n$, if the ratio of 7th term from the beginning to the 7th term from the end is $1/6$, then find the value of $n$.

The sum of the coefficients in the expansion of (x + y)n is 4096. The greatest coefficient in the expansion is

Let P(n) = 3·52n+1 + 23n+1. Then P(n) is divisible by:

The coefficient of $x^{10}$ in the expansion of $(1+x)^2(1+x^2)^3(1+x^3)^4$ is equal to

If f(x) = 1 - x + x2 - x3 + ... - x15 + x16 - x17, then the coefficient of x2 in f(x - 1) is

$\displaystyle\sum_{r=0}^{n} {}^nC_r \sin rx \cos(n-r)x$ is equal to

The number of terms in the expansion of $(1 + x)^{101}(1 + x^2 - x)^{100}$ in powers of $x$ is

In the expansion $\left(\sqrt{2} \sqrt[6]{3}+\frac{1}{\sqrt[3]{3}}\right)^{n}$, if the ratio of 7th term from the beginning to the 7th term from the end is $\frac{1}{6}$, then n =

If the expansion in powers of $x$ of the function $\dfrac{1}{(1-ax)(1-bx)}$ is $a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \cdots$, then $a_n$ is

If the coefficient of x15 in the expansion of $\left(a x^3+\frac{1}{b x^{\frac{1}{3}}}\right)^{15}$ is equal to the coefficient of x-15 in the expansion of $\left(\mathrm{ax}^{\frac{1}{3}}-\frac{1}{\mathrm{bx}^3}\right)^{15}$, where a and b are positive real numbers, then for each such ordered pair (a, b):

The value of x, for which the 6th term in the expansion of $\left(2^{\log_2(9^x-1)} + \frac{1}{2(3^x-1+1))}\right)^7$ is 84, is equal to

If $\displaystyle\sum_{r=0}^{n} \frac{r}{{}^nC_r} = \displaystyle\sum_{r=0}^{n} \frac{n^2 - 3n + 3}{2 \cdot {}^nC_r}$, then

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 ${}^{99}C_0, {}^{99}C_1, {}^{99}C_2, \ldots, {}^{99}C_{99}$.7. The value of $a_{99}$ is equal to

For Problems 12–14: Consider the expansion of $(a + b + c + d)^6$. Then the sum of all the coefficients of the terms13. Which contains $a$ but not $b$ is

If x4 occurs in the (r + 1)th term in the expansion of $\left(x^4 + \dfrac{1}{x^3}\right)^{15}$, then find the value of r.

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

The value of the sum ${}^{1000}C_{50} + {}^{999}C_{49} + {}^{998}C_{48} + \cdots + {}^{950}C_0$ is

The value of $\displaystyle\sum_{r=0}^{10} r\,{}^{10}C_r\,3^r(-2)^{10-r}$ is

$(n+2)\,{}^nC_0 \cdot 2^{n-1} - (n+1)\,{}^nC_1 \cdot 2^n + n\,{}^nC_2 \cdot 2^{n-1} - \cdots$ is equal to

If $f(x) = {}^{40}C_1 \cdot x(1-x)^{39} + 2 \cdot {}^{40}C_2 \cdot x^2(1-x)^{38} + 3 \cdot {}^{40}C_3 \cdot x^3(1-x)^{37} + \cdots + 40 \cdot {}^{40}C_{40} \cdot x^{40}$, then the value of $f(3)$ is

If the 6th term in the expansion of $\left(\dfrac{1}{x^{8/3}} + x^2 \log_{10} x\right)^8$ is 5600, then $x$ equals

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

Coefficient of $x^2$ in the expansion of $(x^3 + 2x^2 + x + 4)^{15}$ is

The term independent of $x$ in $(1+x+x^{-2}+x^{-3})^{10}$ is $n$. Then the last digit of $(n+2)^n$ is

If the coefficient of 4th term in the expansion of $(a + b)^n$ is 56, then find the value of $n$.

Represent $\cos 6\theta$ in terms of $\cos \theta$.

If $1+x^4+x^5=\displaystyle\sum_{i=0}^{5}a_i(1+x)^i$ for all $x\in R$, then $a_2$ is

Find the remainder when $34562^{222}$ is divided by 7.

Coefficient of $x^{11}$ in the expansion of $(1+x^2)^4(1+x)^7(1+x^4)^{12}$ is

If $\dfrac{x^2 + x + 1}{1 - x} = a_0 + a_1 x + a_2 x^2 + \cdots$, then $\displaystyle\sum_{r=1}^{50} a_r$ is equal to

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:

Binomial Theorem Questions (605)