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Yukti Vault

Algebra

Binomial Theorem: JEE Main & Advanced PYQs

303 previous year questions on Binomial Theorem from JEE Main and JEE Advanced papers, each with its exam and year, an answer and a solution.

Topics covered: Binomial Theorem for Any Index, Coefficient of a Given Power / Term Independent of x, General Term and Middle Term, Greatest Term and Greatest Coefficient, Integral and Fractional Part of (√a+b)^n, Miscellaneous, Multinomial Expansion, Properties and Sums of Binomial Coefficients, Rational and Integral Terms, Remainder and Divisibility.

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Questions 251-300 of 303

  1. If the coefficient of $x$ in the expansion of $(ax^2+bx+c)(1-2x)^{26}$ is $-56$ and the coefficients of $x^2$ and $x^3$ are both zero, then $a+b+c$ is equal to :
  2. The sum of all possible values of $n\in\mathbf{N}$, so that the coefficients of $x$, $x^2$ and $x^3$ in the expansion of $(1+x^2)^2(1+x)^n$ are in arithmetic progression …
  3. The sum of the coefficients of $x^{499}$ and $x^{500}$ in $(1+x)^{1000}+x(1+x)^{999}+x^2(1+x)^{998}+\ldots+x^{1000}$ is :
  4. Coefficient of $x^{11}$ in the expansion of $(1+x^2)^4(1+x^3)^7(1+x^4)^{12}$ is
  5. For $r=0,1,\ldots,10$, let $A_r$, $B_r$ and $C_r$ denote, respectively, the coefficient of $x^r$ in the expansions of $(1+x)^{10}$, $(1+x)^{20}$ and $(1+x)^{30}$. Then …
  6. The coefficients of three consecutive terms of $(1+x)^{n+5}$ are in the ratio $5:10:14$. Then $n=$
  7. The coefficient of $x^9$ in the expansion of $(1+x)(1+x^2)(1+x^3)\ldots(1+x^{100})$ is
  8. Let $m$ be the smallest positive integer such that the coefficient of $x^2$ in the expansion of $(1+x)^2+(1+x)^3+\cdots+(1+x)^{49}+(1+mx)^{50}$ is $(3n+1)\,{}^{51}C_3$ …
  9. Let $$X=\left({}^{10}C_1\right)^2+2\left({}^{10}C_2\right)^2+3\left({}^{10}C_3\right)^2+\cdots+10\left({}^{10}C_{10}\right)^2,$$ where …
  10. If $\alpha=1+\sum_{r=1}^{6}(-3)^{r-1}\,{}^{12}C_{2r-1}$, then the distance of the point $(12,\sqrt{3})$ from the line $\alpha x-\sqrt{3}y+1=0$ is ____.
  11. If in the expansion of $(1+x)^p(1-x)^q$, the coefficients of $x$ and $x^2$ are $1$ and $-2$, respectively, then $p^2+q^2$ is equal to :
  12. The remainder left out when $8^{2n}-(62)^{2n+1}$ is divided by $9$ is :
  13. The total number of two digit numbers '$n$', such that $3^{n}+7^{n}$ is a multiple of $10$, is $\_\_\_\_\_\_$ .
  14. If $(m,n)=\frac{(1-x^{m})(1-x^{m-1})\cdots(1-x^{m-n+1})}{(1-x)(1-x^{2})\cdots(1-x^{n})}$, where $m$ and $n$ are positive integers $(n\le m)$, show that …
  15. If $(1+x)^{n}=C_0+C_1x+C_2x^{2}+\cdots+C_nx^{n}$, then show that the sum of the products of the $C_i$'s taken two at a time, represented by …
  16. Prove that $7^{2n}+\left(2^{3n-3}\right)\left(3^{n-1}\right)$ is divisible by $25$ for any natural number $n$.
  17. Given $s_n=1+q+q^{2}+\cdots+q^{n}$; $S_n=1+\dfrac{q+1}{2}+\left(\dfrac{q+1}{2}\right)^{2}+\cdots+\left(\dfrac{q+1}{2}\right)^{n}$, $q\neq1$. Prove that …
  18. Prove that $C_0-2^{2}C_1+3^{2}C_2-\cdots+(-1)^{n}(n+1)^{2}C_n=0$, $n>2$, where $C_r={}^{n}C_r$.
  19. Using mathematical induction, prove that ${}^{m}C_0\,{}^{n}C_k+{}^{m}C_1\,{}^{n}C_{k-1}+\cdots+{}^{m}C_k\,{}^{n}C_0={}^{(m+n)}C_k$, where $m,n,k$ are positive integers …
  20. If $\displaystyle\sum_{r=0}^{2n}a_r(x-2)^{r}=\sum_{r=0}^{2n}b_r(x-3)^{r}$ and $a_k=1$ for all $k\ge n$, then show that $b_n={}^{2n+1}C_{n+1}$.
  21. Let $n$ be a positive integer and $(1+x+x^{2})^{n}=a_0+a_1x+\cdots+a_{2n}x^{2n}$. Show that $a_0^{2}-a_1^{2}+a_2^{2}-\cdots+a_{2n}^{2}=a_n$.
  22. Prove that $\displaystyle\sum_{r=1}^{k}(-3)^{r-1}\,{}^{3n}C_{2r-1}=0$, where $k=(3n)/2$ and $n$ is an even positive integer.
  23. Let $n$ be any positive integer. Prove that …
  24. Prove that $$2^k\binom{n}{0}\binom{n}{k}-2^{k-1}\binom{n}{1}\binom{n-1}{k-1}+2^{k-2}\binom{n}{2}\binom{n-2}{k-2}-\cdots+(-1)^k\binom{n}{k}\binom{n-k}{0}=\binom{n}{k}$$
  25. If $n$ is a positive integer, then $\left(\sqrt3+1\right)^{2n}-\left(\sqrt3-1\right)^{2n}$ is :
  26. For any positive integers $m,n$ (with $n\ge m$), let $\dbinom{n}{m}={}^{n}C_m$. Prove that …
  27. If $C_r$ $(r=0,1,2,\ldots,11)$ are the Binomial coefficients in the expansion of $(1+x)^{11}$, then the value of …
  28. If $n$ is the number of irrational terms in the expansion of $\left(3^{1/4}+5^{1/8}\right)^{60}$, then $(n-1)$ is divisible by :
  29. Let $[x]$ denote greatest integer less than or equal to $x$. If for $n\in\mathbf{N}$, $\left(1-x+x^{3}\right)^{n}=\displaystyle\sum_{j=0}^{3n}a_jx^{j}$, then …
  30. Let $a=1+\dfrac{{}^{2}C_{2}}{3!}+\dfrac{{}^{3}C_{2}}{4!}+\dfrac{{}^{4}C_{2}}{5!}+\ldots$ …
  31. The sum of all rational terms in the expansion of $\left(2^{\frac{1}{5}}+5^{\frac{1}{3}}\right)^{15}$ is equal to :
  32. If the coefficients of $x^{4}$, $x^{5}$ and $x^{6}$ in the expansion of $(1+x)^{n}$ are in the arithmetic progression, then the maximum value of $n$ is:
  33. If the constant term in the expansion of $\left(1+2x-3x^{3}\right)\left(\dfrac{3}{2}x^{2}-\dfrac{1}{3x}\right)^{9}$ is $p$, then $108p$ is equal to $\_\_\_\_\_\_\_\_$ .
  34. If the constant term in the expansion of $\left(\dfrac{\sqrt[5]{3}}{x}+\dfrac{2x}{\sqrt[3]{5}}\right)^{12}$, $x\neq0$, is $\alpha\times2^{8}\times\sqrt[5]{3}$, then …
  35. The coefficient of $x^{4}$ in the expansion of $\left(1+x+x^{2}+x^{3}\right)^{6}$ in powers of $x$, is $\_\_\_\_\_\_\_\_$ .
  36. If $\{p\}$ denotes the fractional part of the number $p$, then $\left\{\dfrac{3^{200}}{8}\right\}$, is equal to :
  37. If $\displaystyle\sum_{r=0}^{5}\dfrac{{}^{11}C_{2r+1}}{2r+2}=\dfrac{m}{n}$, $\gcd(m,n)=1$, then $m-n$ is equal to $\_\_\_\_\_\_\_\_$ .
  38. Let $\alpha,\beta,\gamma$ and $\delta$ be the coefficients of $x^{7},x^{5},x^{3}$ and $x$ respectively in the expansion of …
  39. If $\displaystyle\sum_{r=1}^{30}\dfrac{r^{2}\left({}^{30}C_r\right)^{2}}{{}^{30}C_{r-1}}=\alpha\times2^{29}$, then $\alpha$ is equal to $\_\_\_\_\_\_\_\_$ .
  40. The sum of all rational terms in the expansion of $\left(1+2^{1/3}+3^{1/2}\right)^{6}$ is equal to $\_\_\_\_\_\_\_\_$ .
  41. Suppose $A$ and $B$ are the coefficients of $30^{\text{th}}$ and $12^{\text{th}}$ terms respectively in the binomial expansion of $(1+x)^{2n-1}$. If $2A=5B$, then $n$ is …
  42. Let the coefficients of three consecutive terms $T_r$, $T_{r+1}$ and $T_{r+2}$ in the binomial expansion of $(a+b)^{12}$ be in a G.P. and let $p$ be the number of all …
  43. If the sum of the coefficients in the expansion of $(a+b)^{n}$ is $4096$, then the greatest coefficient in the expansion is
  44. $r$ and $n$ are positive integers, $r>1$, $n>2$ and the coefficients of the $(r+2)^{th}$ term and the $3r^{th}$ term in the expansion of $(1+x)^{2n}$ are equal, then $n$ …
  45. If $S_n=\sum\limits_{r=0}^{n}\frac{1}{{}^{n}C_r}$ and $t_n=\sum\limits_{r=0}^{n}\frac{r}{{}^{n}C_r}$, then $\frac{t_n}{S_n}$ is equal to
  46. For natural numbers $m,n$, if $(1-y)^{m}(1+y)^{n}=1+a_1y+a_2y^{2}+\cdots$ and $a_1=a_2=10$, then $(m,n)$ is
  47. The sum of the series ${}^{20}C_0-{}^{20}C_1+{}^{20}C_2-{}^{20}C_3+\cdots+{}^{20}C_{10}$ is
  48. Statement - 1 : $\sum\limits_{r=0}^{n}(r+1)\,{}^{n}C_r=(n+2)2^{n-1}$. Statement - 2 : $\sum\limits_{r=0}^{n}(r+1)\,{}^{n}C_rx^{r}=(1+x)^{n}+nx(1+x)^{n-1}$.
  49. The coefficient of $x^{7}$ in the expansion of $\left(1-x-x^{2}+x^{3}\right)^{6}$ is
  50. Let $S_1=\sum\limits_{j=1}^{10}j(j-1)\,{}^{10}C_j$, $S_2=\sum\limits_{j=1}^{10}j\,{}^{10}C_j$ and $S_3=\sum\limits_{j=1}^{10}j^{2}\,{}^{10}C_j$. Statement - 1 …

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