Binomial Theorem Questions - JEE

If n is odd, the value of ∑r=1(n+1)/2 a2r-1 is

The sum of the coefficients of all odd degree terms in the expansion of $\left(x + \sqrt{x^3 - 1}\right)^5 + \left(x - \sqrt{x^3 - 1}\right)^5$, $(x > 1)$ is

For $r = 0, 1, \ldots, 10$, let $A_r$, $B_r$, and $C_r$ denote, respectively, the coefficients of $x^r$ in the expansions of $(1+x)^{10}$, $(1+x)^{20}$ and $(1+x)^{30}$. Then $\displaystyle\sum_{r=1}^{10} A_r(B_{10}B_r - C_{10}A_r)$ is equal to

In the expansion of $\left(x^3 - \dfrac{1}{x^2}\right)^{15}$, the constant term is

The term independent of $x$ in $(1+x)^m \left(1 + \frac{1}{x}\right)^n$ is

The coefficient of $x^9$ in the expansion of $(1+x)(1+x^2)(1+x^3)\cdots(1+x^{100})$ is _______.

If $(1 - x + x^2)^n = a_0 + a_1 x + a_2 x^2 + \cdots + a_{2n} x^{2n}$, then $a_0 + a_2 + a_4 + \cdots + a_{2n}$ is equal to

Find the value of $(\sqrt{2} + 1)^6 - (\sqrt{2} - 1)^6$.

Find the value of $\dfrac{18^3 + 7^3 + 3 \times 18 \times 7 \times 25}{3^6 + 6/243 \times 2 + 15/81 \times 4 + 20/27 \times 8 + 15/9 \times 16 + 6/3 \times 32 + 64}$.

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 ${}^{99}C_0, {}^{99}C_1, {}^{99}C_2, \ldots, {}^{99}C_{99}$.9. The value of $({}^{99}C_0)^2 + ({}^{99}C_1)^2 + \cdots + ({}^{99}C_{99})^2$ is equal to

If $(1-x^2)^n = \displaystyle\sum_{r=0}^{n} a_r x^r (1-x)^{2n-r}$, then $a_r$ is equal to

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

If $C_r = {}^nC_r$, the sum of the series $\dfrac{2\left[\left(\dfrac{n}{2}\right)!\right]^2}{n!}\left[C_0^2 - 2C_1^2 + 3C_2^2 - \ldots + (-1)^n(n+1)C_n^2\right]$, where $n$ is an even integer, is

In the binomial expansion of $(a - b)^n$, $n \geq 5$, the sum of 5th and 6th terms is zero, then $\dfrac{a}{b}$ equals

The remainder when the number $3^{256} - 3^{12}$ is divided by 8 is

If $\left(2+\dfrac{x}{3}\right)^{55}$ is expanded in the ascending powers of $x$ and the coefficients of powers of $x$ in two consecutive terms of the expansion are equal, then these terms are

The value of (21C1 × 10C1) + (21C2 × 10C2) + (21C3 × 10C3) + (21C4 × 10C4) + ... + (21C10 × 10C10) is

The term independent of x in the expansion of \[\left(\frac{x+1}{x^{2/3}-x^{1/3}+1}-\frac{x-1}{x-x^{1/2}}\right)^{10}\] is:

In the expansion of $(1 + 3x + 2x^2)^6$, find the coefficient of x11.

If $(1-x)^{-n} = a_0 + a_1 x + a_2 x^2 + \cdots + a_r x^r + \cdots$, then $a_0 + a_1 + a_2 + \cdots + a_r$ is equal to

The coefficient of $x^{1012}$ in the expansion of $(1+x^n+x^{253})^{10}$, (where $n\leq 22$ is any positive integer), is

Find the remainder when $7^{103}$ is divided by 25.

The total number of irrational terms in the binomial expansion of $(7^{1/5} - 3^{1/10})^{60}$ is:

A ratio of the 5th term from the beginning to the 5th term from the end in the binomial expansion of $\left(2^{1/3} + \dfrac{1}{2(3)^{1/3}}\right)^{10}$ is:

If Tr+1 is the general term of $\left[ax^2 + \frac{1}{bx}\right]^{11}$, and the coefficient of $x^7$ in $\left[ax^2 + \frac{1}{bx}\right]^{11}$ equals the coefficient of $x^{-7}$ in $\left[ax - \frac{1}{bx^2}\right]^{11}$, then $ab$ equals:

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

$1 + \dfrac{1}{4} + \dfrac{1 \times 3}{4 \times 8} + \dfrac{1 \times 3 \times 5}{4 \times 8 \times 12} + \cdots =$

If $\displaystyle\sum_{r=0}^{n}\{a_r(x-\alpha+2)^r - b_r(\alpha-x-1)^r\} = 0$, then $b_n$ is

Consider (1 + x + x2)n = ∑arxr, where a0, a1, a2, ..., a2n are real numbers and n is a positive integer. The value of ∑r=0n rar is

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

If $10^m$ divides the number $101^{100} - 1$, then find the greatest value of $m$.

If $\dfrac{1}{\sqrt{4x+1}}\left\{\left(\dfrac{1+\sqrt{4x+1}}{2}\right)^n - \left(\dfrac{1-\sqrt{4x+1}}{2}\right)^n\right\} = a_0 + a_1 x + \cdots + a_5 x^5$, then find the possible values of $n$.

If $(4x^2 + 1)^n = \displaystyle\sum_{r=0}^{n} a_r(1+x^2)^{n-r} x^{2r}$, then the value of $\displaystyle\sum_{r=0}^{n} a_r$ is

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

Find the middle term in the expansion of $\left(x^2 + \dfrac{1}{x^2} + 2\right)^n$.

The coefficient of $x^{50}$ in the expansion of $(1+x)^{1000} + 2x(1+x)^{999} + 3x^2(1+x)^{998} + \cdots + 1001 x^{1000}$ is:

For Problems 15–17: Let $P = \displaystyle\sum_{r=1}^{50} \frac{{}^{50+r}C_r(2r-1)}{{}^{50}C_r(50+r)}$, $Q = \displaystyle\sum_{r=0}^{50} \left({}^{50}C_r\right)^2$, $R = \displaystyle\sum_{r=0}^{100} (-1)^r \left({}^{100}C_r\right)^2$16. The value of $P - R$ is equal to

Find the coefficient of $x^7$ in the expansion of $(1 + 3x - 2x^3)^{10}$.

If the sum of binomial coefficients in the expansion of $(x + y)^n$ is 1024 then the greatest binomial coefficient occurs in the ______ th term.

$\binom{30}{0} - \binom{30}{10} + \binom{30}{1} - \binom{30}{11} + \ldots + \binom{30}{20} - \binom{30}{30}$ is equal to

$\sum_{r=0}^{n} \frac{(-1)^r(\binom{n}{r})^2}{1+nx}$ is equal to

The coefficient of x-5 in the binomial expansion of $\left(\frac{x \ + \ 1}{x^{\frac{2}{3}} \ - \ x^{\frac{1}{3}} \ + \ 1}-\frac{x \ - \ 1}{x \ - \ x^{\frac{1}{2}}}\right)^{10}$ where x $\ne$ 0, 1, is:

If f(x) = \sum_{r=1}^{n} \left\{ r^2 \left( \binom{n}{r} + \binom{n}{r+1} \right) - (2r-1)\binom{n}{r} \right\} and f(30) = 30(2)^5, then the value of n is

If the rth term in the expansion of $(1 + x)^{20}$ has its coefficient equal to that of the $(r + 4)$th term, then $r$ is

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

Find the value of $({}^{10}C_0) + ({}^{10}C_0 + {}^{10}C_1) + ({}^{10}C_0 + {}^{10}C_1 + {}^{10}C_2) + \ldots + ({}^{10}C_0 + {}^{10}C_1 + {}^{10}C_2 + \ldots + {}^{10}C_9)$.

For Problems 15–17: Let $P = \displaystyle\sum_{r=1}^{50} \frac{{}^{50+r}C_r(2r-1)}{{}^{50}C_r(50+r)}$, $Q = \displaystyle\sum_{r=0}^{50} \left({}^{50}C_r\right)^2$, $R = \displaystyle\sum_{r=0}^{100} (-1)^r \left({}^{100}C_r\right)^2$15. The value of $P - Q$ is equal to

Let $m$ and $n$ be the coefficients of seventh and thirteenth terms respectively in the expansion of $\left(\frac{1}{3}x^{1/3} + \frac{1}{2x^{2/3}}\right)^{18}$. Then $\left(\frac{n}{m}\right)^{1/3}$ is:

For which of the following values of x, 5th term is the numerically greatest term in the expansion of $(1 + x/3)^{10}$:

We have $\dfrac{2^{403}}{15}$. Find the fractional part and the value of $k$ where the fractional part equals $\dfrac{k}{15}$.

Binomial Theorem Questions (605)