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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 101-150 of 303

  1. If $\sum_{k=1}^{10}K^2\left({}^{10}C_K\right)^2=22000L$, then $L$ is equal to ________.
  2. The remainder when $428^{2024}$ is divided by $21$ is $\_\_\_\_\_\_$ .
  3. The number of elements in the set $\{n\in\mathbb{N}:10\le n\le100$ and $3^{n}-3$ is a multiple of $7\}$ is $\_\_\_\_\_\_$ .
  4. The product of the last two digits of $(1919)^{1919}$ is $\_\_\_\_\_\_$ .
  5. Let $\alpha$ be the constant term in the binomial expansion of $\left(\sqrt{x}-\frac{6}{x^{3/2}}\right)^{n}$, $n\le15$. If the sum of the coefficients of the remaining …
  6. Number of integral terms in the expansion of $\left\{7^{\frac{1}{2}}+11^{\frac{1}{6}}\right\}^{824}$ is equal to $\_\_\_\_\_\_$ .
  7. Let $\alpha=\sum_{r=0}^{n}\left(4r^{2}+2r+1\right){}^{n}C_r$ and $\beta=\left(\sum_{r=0}^{n}\frac{{}^{n}C_r}{r+1}\right)+\frac{1}{n+1}$. If …
  8. Let $f(x)=\sum_{k=1}^{10}kx^{k}$, $x\in\mathbb{R}$. If $2f(2)+f'(2)=119(2)^{n}+1$ then $n$ is equal to $\_\_\_\_\_\_$ .
  9. If the sum of the coefficients of all the positive powers of $x$, in the Binomial expansion of $\left(x^{n}+\frac{2}{x^{5}}\right)^{7}$ is $939$, then the sum of all the …
  10. If the maximum value of the term independent of $t$ in the expansion of $\left(t^2x^{\frac15}+\frac{(1-x)^{\frac1{10}}}{t}\right)^{15}$, $x\ge0$, is $K$, then $8K$ is …
  11. If $C_r={}^{25}C_r$ and $C_0+5\cdot C_1+9\cdot C_2+\ldots+(101)\cdot C_{25}=2^{25}\cdot k$, then $k$ is equal to ________.
  12. Let the coefficients of $x^{-1}$ and $x^{-3}$ in the expansion of $\left(2x^{\frac{1}{5}}-\frac{1}{x^{\frac{1}{5}}}\right)^{15}$, $x>0$, be $m$ and $n$ respectively. If …
  13. If $1+(2+{}^{49}C_1+{}^{49}C_2+\ldots+{}^{49}C_{49})({}^{50}C_2+{}^{50}C_4+\ldots+{}^{50}C_{50})$ is equal to $2^n\cdot m$, where $m$ is odd, then $n+m$ is equal to …
  14. Let the ratio of the fifth term from the beginning to the fifth term from the end in the binomial expansion of $\left(\sqrt[4]{2}+\frac{1}{\sqrt[4]{3}}\right)^n$, in the …
  15. If the sum of the co-efficients of all the positive even powers of $x$ in the binomial expansion of $\left(2x^{3}+\frac{3}{x}\right)^{10}$ is $5^{10}-\beta\cdot3^{9}$ …
  16. If the Coefficient of $x^{30}$ in the expansion of $\left(1+\frac{1}{x}\right)^{6}\left(1+x^{2}\right)^{7}\left(1-x^{3}\right)^{8}$; $x\neq0$ is $\alpha$, then …
  17. Let $(2x^2+3x+4)^{10}=\sum_{r=0}^{20}a_rx^r$. Then $\frac{a_7}{a_{13}}$ is equal to ________.
  18. The remainder, when $7^{103}$ is divided by $17$, is $\_\_\_\_\_\_$ .
  19. Let the coefficient of $x^{r}$ in the expansion of $(x+3)^{n-1}+(x+3)^{n-2}(x+2)+(x+3)^{n-3}(x+2)^{2}+\ldots+(x+2)^{n-1}$ be $\alpha_r$. If …
  20. If the coefficients of $x$ and $x^2$ in the expansion of $(1+x)^p(1-x)^q$, $p,q\le15$, are $-3$ and $-5$ respectively, then the coefficient of $x^3$ is equal to ________.
  21. If $p_1$ and $p_2$, respectively, are the second largest and the fourth largest prime numbers in the factorization of the coefficient of $x^{30}$ in the expansion of …
  22. If the second, third and fourth terms in the expansion of $(x+y)^{n}$ are $135$, $30$ and $\frac{10}{3}$, respectively, then $6\left(n^{3}+x^{2}+y\right)$ is equal to …
  23. The number of positive integers $k$ such that the constant term in the binomial expansion of $\left(2x^{3}+\frac{3}{x^{k}}\right)^{12}$, $x\neq0$ is $2^{8}\cdot\ell$ …
  24. If the coefficient of $x^{10}$ in the binomial expansion of $\left(\frac{\sqrt{x}}{5^{1/4}}+\frac{\sqrt{5}}{x^{1/3}}\right)^{60}$ is $5^{k}\cdot l$, where …
  25. If $S(x)=(1+x)+2(1+x)^{2}+3(1+x)^{3}+\ldots+60(1+x)^{60}$, $x\neq0$, and $(60)^{2}S(60)=a(b)^{b}+b$, where $a,b\in\mathbf{N}$, then $(a+b)$ equal to $\_\_\_\_\_\_$ .
  26. Let for the $9^{\text{th}}$ term in the binomial expansion of $(3+6x)^n$, in the increasing powers of $6x$, to be the greatest for $x=\frac32$, the least value of $n$ is …
  27. The sum of the series $2\times1\times{}^{20}C_4-3\times2\times{}^{20}C_5+4\times3\times{}^{20}C_6-5\times4\times{}^{20}C_7+\ldots+18\times17\times{}^{20}C_{20}$ is equal …
  28. Let $\alpha=\sum_{k=0}^{n}\frac{\left({}^{n}C_k\right)^{2}}{k+1}$ and $\beta=\sum_{k=0}^{n-1}\frac{{}^{n}C_k\,{}^{n}C_{k+1}}{k+2}$. If $5\alpha=6\beta$, then $n$ equals …
  29. In the expansion of $(1+x)(1-x^{2})\left(1+\frac{3}{x}+\frac{3}{x^{2}}+\frac{1}{x^{3}}\right)^{5}$, $x\neq0$, the sum of the coefficients of $x^{3}$ and $x^{-13}$ is …
  30. If $\frac{{}^{11}C_1}{2}+\frac{{}^{11}C_2}{3}+\ldots+\frac{{}^{11}C_9}{10}=\frac{n}{m}$ with $\gcd(n,m)=1$, then $n+m$ is equal to $\_\_\_\_\_\_$ .
  31. The natural number $m$, for which the coefficient of $x$ in the binomial expansion of $\left(x^m+\frac1{x^2}\right)^{22}$ is $1540$, is ________.
  32. The number of integral terms in the expansion of $\left(3^{\frac{1}{2}}+5^{\frac{1}{4}}\right)^{680}$ is equal to $\_\_\_\_\_\_$ .
  33. If $\left({}^{40}C_0\right)+\left({}^{41}C_1\right)+\left({}^{42}C_2\right)+\ldots+\left({}^{60}C_{20}\right)=\frac{m}{n}\,{}^{60}C_{20}$, $m$ and $n$ are coprime, then …
  34. The value of $r$ for which ${}^{20}C_r\,{}^{20}C_0+{}^{20}C_{r-1}\,{}^{20}C_1+{}^{20}C_{r-2}\,{}^{20}C_2+\ldots+{}^{20}C_0\,{}^{20}C_r$ is maximum, is :
  35. Let $(x+10)^{50}+(x-10)^{50}=a_0+a_1x+a_2x^{2}+\ldots+a_{50}x^{50}$, for all $x\in\mathbf R$; then $\frac{a_2}{a_0}$ is equal to :
  36. The positive value of $\lambda$ for which the co-efficient of $x^2$ in the expansion $x^2\left(\sqrt x+\frac{\lambda}{x^2}\right)^{10}$ is $720$, is :
  37. The coefficient of $x^7$ in the expression $(1+x)^{10}+x(1+x)^9+x^2(1+x)^8+\ldots+x^{10}$ is :
  38. If the number of terms in the expansion of $\left(1-\frac2x+\frac{4}{y^{2}}\right)^{n}$, $x,y\neq0$, is $28$, then the sum of the coefficients of all the terms in this …
  39. Let $S_n=1+q+q^{2}+\ldots+q^{n}$ and $T_n=1+\left(\frac{q+1}{2}\right)+\left(\frac{q+1}{2}\right)^{2}+\ldots+\left(\frac{q+1}{2}\right)^{n}$ where $q$ is a real number …
  40. If $a_r$ is the coefficient of $x^{10-r}$ in the Binomial expansion of $(1+x)^{10}$, then $\sum_{r=1}^{10}r^{3}\left(\frac{a_r}{a_{r-1}}\right)^{2}$ is equal to
  41. If the fourth term in the Binomial expansion of $\left(\frac2x+x^{\log_8x}\right)^{6}$, $(x>0)$ is $20\times8^{7}$, then a value of $x$ is
  42. The coefficient of $t^{4}$ in the expansion of $\left(\frac{1-t^{6}}{1-t}\right)^{3}$ is
  43. The term independent of $x$ in the binomial expansion of $\left(1-\frac1x+3x^{5}\right)\left(2x^{2}-\frac1x\right)^{8}$ is
  44. The coefficient of $x^{18}$ in the product $(1+x)(1-x)^{10}\left(1+x+x^2\right)^9$ is
  45. If $(27)^{999}$ is divided by $7$, then the remainder is :
  46. If $\sum_{r=0}^{25}\left\{{}^{50}C_r\cdot{}^{50-r}C_{25-r}\right\}=K\left({}^{50}C_{25}\right)$, then $K$ is equal to :
  47. For $x\in R$, $x\neq-1$, if $(1+x)^{2016}+x(1+x)^{2015}+x^{2}(1+x)^{2014}+\ldots+x^{2016}=\sum_{i=0}^{2016}a_{i}x^{i}$, then $a_{17}$ is equal to
  48. If some three consecutive coefficients in the binomial expansion of $(x+1)^{n}$ in powers of $x$ are in the ratio $2:15:70$, then the average of these three coefficients …
  49. If the fractional part of the number $\frac{2^{403}}{15}$ is $\frac{k}{15}$, then $k$ is equal to
  50. The coefficient of $x^{10}$ in the expansion of $(1+x)^{2}\left(1+x^{2}\right)^{3}\left(1+x^{3}\right)^{4}$ is equal to

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