Binomial Theorem Questions - JEE

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

If $a_n = \sum_{r=0}^{n} \frac{1}{C_r}$, then $\sum_{r=0}^{n} \frac{r^2}{C_r} = P(n)a_{n+2} + Q(n)a_{n+1} + a_n + R(n)$, where $P(n), Q(n), R(n)$ are the polynomial functions of $n$, then $P(5) =$

Let $K$ be the sum of the coefficients of the odd powers of $x$ in the expansion of $(1+x)^{99}$. Let $a$ be the middle term in the expansion of $\left(2+\dfrac{1}{\sqrt{2}}\right)^{200}$. If $\dfrac{{}^{200}C_{99}\,K}{a}=\dfrac{2^\ell\,m}{n}$, where $m$ and $n$ are odd numbers, then the ordered pair $(\ell,n)$ is equal to:

$(1+x)^n = \sum_{r=0}^{n} C_r x^r$

Let $n \in \mathbb{N}, n \geq 4$ and $P = \prod_{r=0}^{n} \, ^nC_r$, then:

Let $\sum_{r=0}^{200} a_r(1+x)^r = \sum_{r=0}^{200} b_r x^r$, where $a_r = 1$ ∀ $r ≥ 98$, then the greatest coefficient in the expansion of $(1+x)^{201}$ is :

In the expansion of $\left(z^2 + z^{-2}\right)^{15}$, the coefficients of the $8^{th}$ and $19^{th}$ terms are equal. The term independent of $z$ is given by

The number of different terms in the expansion of $(1 - z^{30})\left(1 + z + x^2\right)^{30}$ is

The coefficient of $x^4$ in the expansion of $(1 + x + x^2)^7$ is

The value of $\displaystyle\sum_{r=0}^{22}\,{}^{22}C_r\cdot{}^{23}C_r$ is:

If $a_n = \sum_{r=0}^{n} \frac{1}{C_r}$, then $\sum_{r=0}^{n} \frac{r^2}{C_r} = P(n)a_{n+2} + Q(n)a_{n+1} + a_n + R(n)$, where $P(n), Q(n), R(n)$ are the polynomial functions of $n$, then $Q(5) =$

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)

The value of x in the expression $\left[x+x^{\log _{10}(x)}\right]^{5}$, if the third term in the expansion is 10,00,000, is

Let a and b are two positive real numbers such that a2 + b = 2, then the maximum value of term independent of x in the expansion of $\left(a x^{\frac{1}{6}}+b x^{\frac{-1}{3}}\right)^{9}$ is:

The number of rational terms in the binomial expansion of $(\sqrt{2}+\sqrt[3]{3})^{18}$ is:

If x is positive, the first negative term in the expansion of $(1+x)^{\frac{27}{5}}$ is

The absolute difference of the coefficients of x10 and x7 in the expansion of $\left(2 x^2+\frac{1}{2 x}\right)^{11}$ is equal to

If for some $m, n ;{ }^{6} C_{m}+2\left({ }^{6} C_{m+1}\right)+{ }^{6} C_{m+2} \gt { }^{8} C_{3}$ and ${ }^{n-1} P_{3}:{ }^{n} P_{4}=1: 8$, then ${ }^{n} P_{m+1}+{ }^{n+1} C_{m}$ is equal to

The term independent of x in the expression of (1 - x2 + 3x3)$\left(\frac{5}{2} x^3-\frac{1}{5 x^2}\right)^{11}$, x $\neq$ 0 is

If $(1+x)^{2010}=C_0+C_1x+\cdots+C_{2010}x^{2010}$, then $C_2+C_5+C_8+\cdots+C_{2009}$ equals

The value of $-{ }^{15} \mathrm{C}_1+2 .{ }^{15} \mathrm{C}_2-3 .{ }^{15} \mathrm{C}_3+\ldots$ $ -15 .{ }^{15} \mathrm{C}_{15}+{ }^{14} \mathrm{C}_1+{ }^{14} \mathrm{C}_3+{ }^{14} \mathrm{C}_5+\ldots +{ }^{14} \mathrm{C}_{11}$ is:

If C0, C1, C2, ..., Cn are the binomial coefficients, then 2$\cdot$C1 + 23$\cdot$C3 + 25$\cdot$C5 + ... equals

The greatest coefficient in the expansion of (1 + x)2n+2 is

If (1 + x)n = C0 + C1x + C2x2 + ... + Cnxn, then $\frac{C_{1}}{C_{0}}+\frac{2 C_{2}}{C_{1}}+\frac{3 C_{3}}{C_{2}}+\ldots \frac{n C_{n}}{C_{n-1}}$ =

Let $\alpha=\displaystyle\sum_{r=0}^{n}(4r^2+2r+1)\binom{n}{r}$ and $\beta=\left(\displaystyle\sum_{r=0}^{n}\dfrac{\binom{n}{r}}{r+1}\right)+\dfrac{1}{n+1}$. If $140<\dfrac{2\alpha}{\beta}<281$, then the value of $n$ is _______.

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

If $\left(x - 2 + \dfrac{1}{x}\right)^{30} = a_0 x^{30} + a_1 x^{29} + \ldots + a_{29}x + a_{30} x^{-1} + \ldots + a_{60}x^{-30}$ and $k = a_0 + a_1 + \ldots + a_{60}$. If $k - a_{30} = -{}^nC_r$, then $n + r$ is equal to

Line $L_{1}$ of slope 2 and line $L_{2}$ of slope $\frac{1}{2}$ intersect at the origin $O$. In the first quadrant, $P_{1}, P_{2}, \ldots ., P_{12}$ are 12 points on line $L_{1}$ and $Q_{1}$, $Q_{2}, \ldots ., Q_{9}$ are 9 points on line $L_{2}$. Then the total number of triangles, that can be formed having vertices at three of the 22 points $O, P_{1}$, $P_{2}, \ldots, P_{12}, Q_{1}, Q_{2}, \ldots ., Q_{9}$, is:

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)

Find the coefficient of the term independent of x in the expansion of $\left(\dfrac{x+1}{x^{2/3} - x^{1/3} + 1} - \dfrac{x-1}{x - x^{1/2}}\right)^{10}$.

Given $ {}^{20}C_r \cdot {}^{20}C_0 + {}^{20}C_{r-1} \cdot {}^{20}C_1 + {}^{20}C_{r-2} \cdot {}^{20}C_2 + \cdots + {}^{20}C_0 \cdot {}^{20}C_r $, which equals $ {}^{40}C_r $. Find the value of $ r $ for which this expression is maximum.

If the sum of coefficients in the expansion of (1 + x3)n is p, that of (1 - 4x + 7x2)n is q and (1 - 5x + 10x2 + 2x3)n is r, then

The term independent of $x$ in the binomial expansion of $\left(1-\dfrac{1}{x}+3x^5\right)\left(2x^2-\dfrac{1}{x}\right)^8$ is

If the coefficient of 4th term in the expansion of (a + b)n is 56, then n is

The number of terms in the expansion of $\left(1 - \dfrac{2}{x} + \dfrac{4}{x^2}\right)^n$ is ${}^{n+2}C_2 = 28$. Find the sum of coefficients of the expansion.

Given $(1-x)^2(1+x^2)^3(1+x^3)^4$ Find the coefficient of $x^{10}$.

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)

The remainder when (11)1011 + (1011)11 is divided by 9 is

If $10^n + 3 \cdot 4^{n+2} + k$ is divisible by 9 for all $n \in \mathbb{N}$, then the least positive integral value of $k$ is __________.

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

Given $S = 2\,{}^{20}C_0 + 5\,{}^{20}C_1 + 8\,{}^{20}C_2 + \cdots + 62\,{}^{20}C_{20}$, find the value of $S$.

Consider: $(\sqrt{3}+1)^{2n} - (\sqrt{3}-1)^{2n}$. Which of the following is true?

If $(27)^{999}$ is divided by 7, then the remainder is

If $(1+x)^{2010}=C_0+C_1x+\cdots+C_{2010}x^{2010}$, then $C_2+C_5+C_8+\cdots+C_{2009}$ equals

If 93199 is divided by 162, the remainder is

If a is the remainder when 540 is divided by 11 and b is the remainder when 22011 is divided by 17, the value of a ⊕ b is

The sum of the real values of $x$ for which the middle term in the binomial expansion of $\left(\dfrac{x^3}{3} + \dfrac{3}{x}\right)^8$ equals 5670 is ___________.

Find $a$ if the 17th and 18th terms of the expansion $(2 + a)^{50}$ are equal.

Find the 6th term in the expansion of $(2x^2 - 1/3x^2)^{10}$.

If the coefficients of $(r-5)$th and $(2r-1)$th terms in the expansion of $(1+x)^{34}$ are equal, find $r$.

Binomial Theorem Questions (605)