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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 51-100 of 303

  1. The value of $\sum_{r=0}^{20}{}^{50-r}C_6$ is equal to :
  2. If the remainder when $x$ is divided by $4$ is $3$, then the remainder when $(2020+x)^{2022}$ is divided by $8$ is $\_\_\_\_\_\_$ .
  3. If the constant term, in binomial expansion of $\left(2x^{r}+\frac{1}{x^{2}}\right)^{10}$ is $180$, then $r$ is equal to $\_\_\_\_\_\_$ .
  4. The coefficient of $x^{7}$ in $\left(1-x+2x^{3}\right)^{10}$ is $\_\_\_\_\_\_$ .
  5. If the constant term in the expansion of $\left(3x^{3}-2x^{2}+\frac{5}{x^{5}}\right)^{10}$ is $2^{k}\cdot l$, where $l$ is an odd integer, then the value of $k$ is equal …
  6. For two positive real numbers $a$ and $b$ such that $\frac{1}{a^{2}}+\frac{1}{b^{3}}=4$, the minimum value of the constant term in the expansion of …
  7. Let $\binom{n}{k}$ denote ${}^{n}C_k$ and …
  8. The coefficient of $x^{1011}$ in the expression $(1+2x)^{2022}+2x(1+2x)^{2021}+(2x)^2(1+2x)^{2020}+\ldots+(2x)^{2022}$ is
  9. The mean of the coefficients of $x,x^{2},\ldots,x^{7}$ in the binomial expansion of $(2+x)^{9}$ is $\_\_\_\_\_\_$ .
  10. Let $n\in\mathbf{N}$ and $[x]$ denote the greatest integer less than or equal to $x$. If the sum of $(n+1)$ terms …
  11. If $\left(\frac{3^{6}}{4^{4}}\right)k$ is the term, independent of $x$, in the binomial expansion of $\left(\frac{x}{4}-\frac{12}{x^{2}}\right)^{12}$, then $k$ is equal …
  12. The coefficient of $x^{18}$ in the expansion of $\left(x^{4}-\frac{1}{x^{3}}\right)^{15}$ is $\_\_\_\_\_\_$ .
  13. If $\frac{1}{2\cdot3^{10}}+\frac{1}{2^{2}\cdot3^{9}}+\ldots+\frac{1}{2^{10}\cdot3}=\frac{K}{2^{10}\cdot3^{10}}$, then the remainder when $K$ is divided by $6$ is :
  14. If the term independent of $x$ in the expansion of $\left(\frac32x^2-\frac1{3x}\right)^9$ is $k$, then $18k$ is equal to :
  15. If $a,b$ and $c$ are the greatest values of ${}^{19}C_p$, ${}^{20}C_q$ and ${}^{21}C_r$ respectively, then :
  16. The term independent of $x$ in the expansion of $\left(1-x^{2}+3x^{3}\right)\left(\frac{5}{2}x^{3}-\frac{1}{5x^{2}}\right)^{11}$, $x\neq0$ is :
  17. For integers $n$ and $r$, let $\binom{n}{r}=\begin{cases}{}^{n}C_{r}, & \text{if } n\ge r\ge0\\ 0, & \text{otherwise}\end{cases}$ The maximum value of $k$ for which the …
  18. The coefficient of $x^{101}$ in the expression $(5+x)^{500}+x(5+x)^{499}+x^{2}(5+x)^{498}+\ldots+x^{500}$, $x>0$, is
  19. The remainder when $(2021)^{2023}$ is divided by $7$ is :
  20. If $(2021)^{3762}$ is divided by $17$, then the remainder is $\_\_\_\_\_\_$ .
  21. Let $n\ge5$ be an integer. If $9^{n}-8n-1=64\alpha$ and $6^{n}-5n-1=25\beta$, then $\alpha-\beta$ is equal to
  22. If the coefficient of $a^{7}b^{8}$ in the expansion of $(a+2b+4ab)^{10}$ is $K\cdot2^{16}$, then $K$ is equal to $\_\_\_\_\_\_$ .
  23. If the constant term in the binomial expansion of $\left(\sqrt x-\frac k{x^2}\right)^{10}$ is $405$, then $|k|$ equals :
  24. If $\alpha$ and $\beta$ be the coefficients of $x^4$ and $x^2$ respectively in the expansion of $\left(x+\sqrt{x^2-1}\right)^6+\left(x-\sqrt{x^2-1}\right)^6$, then :
  25. The remainder when $7^{2022}+3^{2022}$ is divided by $5$ is :
  26. Let the coefficients of third, fourth and fifth terms in the expansion of $\left(x+\frac{a}{x^{2}}\right)^{n}$, $x\neq0$, be in the ratio $12:8:3$. Then the term …
  27. $3\times7^{22}+2\times10^{22}-44$ when divided by $18$ leaves the remainder $\_\_\_\_\_\_$ .
  28. The remainder when $(2020)^{2022}$ is divided by $337$ is $\_\_\_\_\_\_$ .
  29. Let $n$ be a positive integer. Let $A=\sum_{k=0}^{n}(-1)^{k}\,{}^{n}C_k\bigg[\left(\frac{1}{2}\right)^{k}+\left(\frac{3}{4}\right)^{k}+$ …
  30. If $(20)^{19}+2(21)(20)^{18}+3(21)^{2}(20)^{17}+\ldots+20(21)^{19}=k(20)^{19}$, then $k$ is equal to $\_\_\_\_\_\_$ .
  31. If the co-efficients of $x^{7}$ and $x^{8}$ in the expansion of $\left(2+\frac{x}{3}\right)^{n}$ are equal, then the value of $n$ is equal to $\_\_\_\_\_\_$ .
  32. $\sum_{\substack{i,j=0\\ i\ne j}}^{n}{}^nC_i\,{}^nC_j$ is equal to
  33. If $\sum_{k=1}^{31}\left({}^{31}C_k\right)\left({}^{31}C_{k-1}\right)-\sum_{k=1}^{30}\left({}^{30}C_k\right)\left({}^{30}C_{k-1}\right)=\frac{\alpha(60!)}{(30!)(31!)}$ …
  34. The remainder when $3^{2022}$ is divided by $5$ is :
  35. In the expansion of $\left(\frac{x}{\cos\theta}+\frac1{x\sin\theta}\right)^{16}$, if $l_1$ is the least value of the term independent of $x$ when …
  36. Let $[t]$ denote the greatest integer $\le t$. If the constant term in the expansion of $\left(3x^{2}-\frac{1}{2x^{5}}\right)^{7}$ is $\alpha$, then $[\alpha]$ is equal …
  37. If $(1-x^{3})^{10}=\sum_{r=0}^{10}a_{r}x^{r}(1-x)^{30-2r}$, then $\frac{9a_{9}}{a_{10}}$ is equal to $\_\_\_\_\_\_$ .
  38. The term independent of $x$ in the expansion of $\left[\frac{x+1}{x^{2/3}-x^{1/3}+1}-\frac{x-1}{x-x^{1/2}}\right]^{10}$, $x\neq1$, is equal to $\_\_\_\_\_\_$ .
  39. Let $m,n\in\mathbf{N}$ and $\gcd(2,n)=1$. If $30\binom{30}{0}+29\binom{30}{1}+\ldots+2\binom{30}{28}+1\binom{30}{29}=n\cdot2^{m}$, then $n+m$ is equal to $\_\_\_\_\_\_$ …
  40. If the coefficients of $x^7$ in $\left(x^2+\frac1{bx}\right)^{11}$ and $x^{-7}$ in $\left(x-\frac1{bx^2}\right)^{11}$, $b\ne0$, are equal, then the value of $b$ is equal …
  41. Let $\alpha>0$, $\beta>0$ be such that $\alpha^3+\beta^2=4$. If the maximum value of the term independent of $x$ in the binomial expansion of …
  42. The number of elements in the set $\{n\in\{1,2,3,\ldots,100\}\mid(11)^{n}>(10)^{n}+(9)^{n}\}$ is $\_\_\_\_\_\_$ .
  43. If for some positive integer $n$, the coefficients of three consecutive terms in the binomial expansion of $(1+x)^{n+5}$ are in the ratio $5:10:14$, then the largest …
  44. A possible value of $x$, for which the ninth term in the expansion of $\left\{3^{\log_3\sqrt{25^{x-1}+7}}+3^{\left(-\frac18\right)\log_3(5^{x-1}+1)}\right\}^{10}$ in the …
  45. Let ${}^{n}C_r$ denote the binomial coefficient of $x^{r}$ in the expansion of $(1+x)^{n}$. If …
  46. The coefficient of $x^4$ in the expansion of $(1+x+x^2)^{10}$ is ________.
  47. Let $\left(1+x+x^{2}\right)^{10}=a_0+a_1x+a_2x^{2}+\ldots+a_{20}x^{20}$. If $\left(a_1+a_3+a_5+\ldots+a_{19}\right)-11a_2=121k$, then $k$ is equal to $\_\_\_\_\_\_$ .
  48. Let $C_r$ denote the binomial coefficient of $x^{r}$ in the expansion of $(1+x)^{10}$. If for $\alpha,\beta\in\mathbf{R}$, $C_1+3\cdot2C_2+5\cdot3C_3+\ldots$ upto $10$ …
  49. For a positive integer $n$, $\left(1+\frac1x\right)^n$ is expanded in increasing powers of $x$. If three consecutive coefficients in this expansion are in the ratio …
  50. Let the coefficients of the middle terms in the expansion of $\left(\frac1{\sqrt6}+\beta x\right)^4$, $(1-3\beta x)^2$ and $\left(1-\frac\beta2x\right)^6$, $\beta>0$ …

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