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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 1-50 of 303

  1. The absolute difference of the coefficients of $x^{10}$ and $x^{7}$ in the expansion of $\left(2x^{2}+\frac{1}{2x}\right)^{11}$ is equal to
  2. If the coefficients of $x^{7}$ in $\left(ax^{2}+\frac{1}{2bx}\right)^{11}$ and $x^{-7}$ in $\left(ax-\frac{1}{3bx^{2}}\right)^{11}$ are equal, then
  3. If $26\bigg(\frac{2^{3}}{3}\left({}^{12}C_{2}\right)+\frac{2^{5}}{5}\left({}^{12}C_{4}\right)+$ …
  4. If $\sum_{r=1}^{9}\left(\frac{r+3}{2^{r}}\right)\cdot{}^{9}C_r=\alpha\left(\frac{3}{2}\right)^{9}-\beta$, $\alpha,\beta\in\mathbb{N}$, then $(\alpha+\beta)^{2}$ is equal …
  5. Let $\left(a+bx+cx^{2}\right)^{10}=\sum_{i=0}^{20}p_ix^{i}$, $a,b,c\in\mathbb{N}$. If $p_1=20$ and $p_2=210$, then $2(a+b+c)$ is equal to
  6. Among the statements: (S1) : $2023^{2022}-1999^{2022}$ is divisible by $8$ (S2) : $13(13)^{n}-11n-13$ is divisible by $144$ for infinitely many $n\in\mathbb{N}$
  7. The coefficient of $x^{70}$ in $x^{2}(1+x)^{98}+x^{3}(1+x)^{97}+x^{4}(1+x)^{96}+\ldots+x^{54}(1+x)^{46}$ is ${}^{99}C_p-{}^{46}C_q$. Then a possible value of $p+q$ is :
  8. Let $m$ and $n$ be the coefficients of seventh and thirteenth terms respectively in the expansion of …
  9. The coefficient of $x^{2}$ in the expansion of $\left(2x^{2}+\frac{1}{x}\right)^{10}$, $x\neq0$, is :
  10. In the expansion of $\left(\sqrt[3]{2}+\frac{1}{\sqrt[3]{3}}\right)^{n}$, $n\in\mathbb{N}$, if the ratio of $15^{\text{th}}$ term from the beginning to the …
  11. The term independent of $x$ in the expansion of $\left(\frac{(x+1)}{\left(x^{2/3}+1-x^{1/3}\right)}-\frac{(x-1)}{\left(x-x^{1/2}\right)}\right)^{10}$, $x>1$, is :
  12. If the coefficients of three consecutive terms in the expansion of $(1+x)^{n}$ are in the ratio $1:5:20$, then the coefficient of the fourth term is
  13. Let the number $(22)^{2022}+(2022)^{22}$ leave the remainder $\alpha$ when divided by $3$ and $\beta$ when divided by $7$. Then $\left(\alpha^{2}+\beta^{2}\right)$ is …
  14. The remainder when $\left((64)^{(64)}\right)^{(64)}$ is divided by $7$ is equal to
  15. If $b$ is very small as compared to the value of $a$, so that the cube and other higher powers of $\frac{b}{a}$ can be neglected in the identity …
  16. If the term independent of $x$ in the expansion of $\left(\sqrt{a}x^{2}+\frac{1}{2x^{3}}\right)^{10}$ is $105$, then $a^{2}$ is equal to :
  17. The sum of the coefficients of three consecutive terms in the binomial expansion of $(1+x)^{n+2}$, which are in the ratio $1:3:5$, is equal to
  18. If the coefficients of $x$ and $x^{2}$ in $(1+x)^{p}(1-x)^{q}$ are $4$ and $-5$ respectively, then $2p+3q$ is equal to
  19. If the greatest value of the term independent of '$x$' in the expansion of $\left(x\sin\alpha+a\frac{\cos\alpha}{x}\right)^{10}$ is $\frac{10!}{(5!)^{2}}$, then the …
  20. The value of $-{}^{15}C_{1}+2\cdot{}^{15}C_{2}-3\cdot{}^{15}C_{3}+\ldots-15\cdot{}^{15}C_{15}+{}^{14}C_{1}+{}^{14}C_{3}+{}^{14}C_{5}+\ldots+{}^{14}C_{11}$ is :
  21. If …
  22. If the fourth term in the expansion of $\left(x+x^{\log_2x}\right)^{7}$ is $4480$, then the value of $x$ where $x\in\mathbf{N}$ is equal to :
  23. $\sum_{k=0}^{20}\left({}^{20}C_k\right)^{2}$ is equal to :
  24. The value of $\sum_{r=0}^{6}\left({}^{6}C_{r}\cdot{}^{6}C_{6-r}\right)$ is equal to :
  25. If the coefficient of $x^{7}$ in $\left(ax-\frac{1}{bx^{2}}\right)^{13}$ and the coefficient of $x^{-5}$ in $\left(ax+\frac{1}{bx^{2}}\right)^{13}$ are equal, then …
  26. The sum of the coefficient of $x^{2/3}$ and $x^{-2/5}$ in the binomial expansion of $\left(x^{2/3}+\frac{1}{2}x^{-2/5}\right)^{9}$ is
  27. The coefficient of $x^{5}$ in the expansion of $\left(2x^{3}-\frac{1}{3x^{2}}\right)^{5}$ is
  28. Fractional part of the number $\frac{4^{2022}}{15}$ is equal to
  29. In the expansion of $\left(9x-\frac{1}{3\sqrt{x}}\right)^{18}$, $x>0$, if the term independent of $x$ is $(221)k$, then $k$ is equal to:
  30. If $(1+x+x^{2})^{4}=a_0+a_1x+a_2x^{2}+\ldots+a_8x^{8}$, then the value of $a_0+4a_2+16a_4+\ldots+2^{8}a_8$ is equal to
  31. If $\frac{1}{n+1}{}^{n}C_n+\frac{1}{n}{}^{n}C_{n-1}+\ldots+\frac{1}{2}{}^{n}C_1+{}^{n}C_0=\frac{1023}{10}$ then $n$ is equal to
  32. If the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of $\left(\sqrt[4]{2}+\frac{1}{\sqrt[4]{3}}\right)^{n}$ is …
  33. $25^{190}-19^{190}-8^{190}+2^{190}$ is divisible by
  34. If $\sum_{r=0}^{10}\left(\frac{10^{r+1}-1}{10^{r}}\right)\cdot{}^{11}C_{r+1}=\frac{\alpha^{11}-11^{11}}{10^{10}}$, then $\alpha$ is equal to :
  35. The maximum value of the term independent of '$t$' in the expansion of $\left(tx^{\frac{1}{5}}+\frac{(1-x)^{\frac{1}{10}}}{t}\right)^{10}$ where $x\in(0,1)$ is :
  36. For an integer $n\ge2$, if the arithmetic mean of all coefficients in the binomial expansion of $(x+y)^{2n-3}$ is $16$, then the distance of the point $P(2n-1,n^{2}-4n)$ …
  37. Let $(1+x+2x^{2})^{20}=a_0+a_1x+a_2x^{2}+\ldots+a_{40}x^{40}$. Then, $a_1+a_3+a_5+\ldots+a_{37}$ is equal to :
  38. Suppose $2-p,\ p,\ 2-\alpha,\ \alpha$ are the coefficients of four consecutive terms in the expansion of $(1+x)^{n}$. Then the value of $p^{2}-\alpha^{2}+6\alpha+2p$ …
  39. The sum, of the coefficients of the first $50$ terms in the binomial expansion of $(1-x)^{100}$, is equal to
  40. The number of integral terms in the expansion of $\left(5^{\frac{1}{2}}+7^{\frac{1}{8}}\right)^{1016}$ is
  41. The sum of all those terms which are rational numbers in the expansion of $\left(2^{1/3}+3^{1/4}\right)^{12}$ is :
  42. If $A$ denotes the sum of all the coefficients in the expansion of $\left(1-3x+10x^{2}\right)^{n}$ and $B$ denotes the sum of all the coefficients in the expansion of …
  43. The sum of all rational terms in the expansion of $\left(2+\sqrt{3}\right)^{8}$ is
  44. If ${}^{20}C_r$ is the co-efficient of $x^{r}$ in the expansion of $(1+x)^{20}$, then the value of $\sum_{r=0}^{20}r^{2}\,{}^{20}C_r$ is equal to :
  45. In the Binomial expansion of $\left(2x\sqrt{x}+3y^{\frac{1}{16}}\right)^{33}$, the number of terms, having positive integral powers of $x$ and $y$ is :
  46. The remainder when $(2021)^{2022}+(2022)^{2021}$ is divided by $7$ is
  47. The remainder when $(11)^{1011}+(1011)^{11}$ is divided by $9$ is
  48. The ratio of the coefficient of the middle term in the expansion of $(1+x)^{20}$ and the sum of the coefficients of two middle terms in expansion of $(1+x)^{19}$ is …
  49. The term independent of '$x$' in the expansion of $\left(\frac{x+1}{x^{2/3}-x^{1/3}+1}-\frac{x-1}{x-x^{1/2}}\right)^{10}$, where $x\neq0,1$ is equal to $\_\_\_\_\_\_$ .
  50. If the number of integral terms in the expansion of $\left(3^{\frac12}+5^{\frac18}\right)^n$ is exactly $33$, then the least value of $n$ is :

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