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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 151-200 of 303

  1. If the third term in the binomial expansion of $\left(1+x^{\log_2x}\right)^{5}$ equals $2560$, then a possible value of $x$ is
  2. The sum of the co-efficient 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
  3. If the coefficients of three successive terms in the binomial expansion of $(1+x)^n$ are in the ratio $1:7:42$, then the first of these terms in the expansion is
  4. The coefficient of $x^{256}$ in the expansion of $(1-x)^{101}\left(x^{2}+x+1\right)^{100}$ is :
  5. The value of $\sum_{r=1}^{15}r^{2}\left(\frac{{}^{15}C_{r}}{{}^{15}C_{r-1}}\right)$ is equal to :
  6. Let $K$ be the sum of the coefficients of the odd powers of $x$ in the expansion of $(1+x)^{99}$. Let $a$ be the middle term in the expansion of …
  7. If the fourth term in the binomial expansion of $\left(\sqrt{x^{\frac{1}{1+\log_{10}x}}}+x^{\frac1{12}}\right)^6$ is equal to $200$, and $x>1$, then the value of $x$ is :
  8. $\sum_{k=0}^{6}{}^{51-k}C_3$ is equal to :
  9. The greatest positive integer $k$, for which $49^{k}+1$ is a factor of the sum $49^{125}+49^{124}+\ldots+49^{2}+49+1$, is
  10. If the coefficients of $x^{2}$ and $x^{3}$ are both zero, in the expansion of the expression $\left(1+ax+bx^{2}\right)(1-3x)^{15}$ in powers of $x$, then the ordered …
  11. If $\sum_{i=1}^{20}\left(\frac{{}^{20}C_{i-1}}{{}^{20}C_{i}+{}^{20}C_{i-1}}\right)^{3}=\frac{k}{21}$, then $k$ equals
  12. The sum of the real values of $x$ for which the middle term in the binomial expansion of $\left(\frac{x^{3}}{3}+\frac3x\right)^{8}$ equals $5670$ is :
  13. The coefficient of $x^{301}$ in $(1+x)^{500}+x(1+x)^{499}+x^2(1+x)^{498}+\ldots+x^{500}$ is:
  14. The total number of irrational terms in the binomial expansion of $\left(7^{\frac15}-3^{\frac1{10}}\right)^{60}$ is
  15. If ${}^{20}C_1+\left(2^{2}\right){}^{20}C_2+\left(3^{2}\right){}^{20}C_3+\ldots+\left(20^{2}\right){}^{20}C_{20}=A\left(2^{\beta}\right)$, then the ordered pair …
  16. The ratio of the coefficient of $x^{15}$ to the term independent of $x$ in the expansion of $\left(x^{2}+\frac{2}{x}\right)^{15}$ is :
  17. 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 A.P., then $m$ and $r$ satisfy …
  18. The sum of the rational terms in the binomial expansion of $\left(2^{\frac12}+3^{\frac15}\right)^{10}$ is :
  19. The coefficients of $x^{p}$ and $x^{q}$ in the expansion of $(1+x)^{p+q}$ are
  20. The value of $\frac1{1!50!}+\frac1{3!48!}+\frac1{5!46!}+\ldots+\frac1{49!2!}+\frac1{51!1!}$ is
  21. The number of integral terms in the expansion of $\left(\sqrt3+\sqrt[8]{5}\right)^{256}$ is
  22. The value of $\sum_{r=0}^{22}{}^{22}C_r\cdot{}^{23}C_r$ is
  23. The positive integer just greater than $(1+0.0001)^{10000}$ is
  24. The coefficient of $x^{n}$ in expansion of $(1+x)(1-x)^{n}$ is
  25. Let $a$ and $b$ be two nonzero real numbers. If the coefficient of $x^5$ in the expansion of $\left(ax^2+\frac{70}{27bx}\right)^4$ is equal to the coefficient of …
  26. The sum of the series $2\cdot{}^{20}C_0+5\cdot{}^{20}C_1+8\cdot{}^{20}C_2+11\cdot{}^{20}C_3+\ldots+62\cdot{}^{20}C_{20}$ is equal to
  27. If $x$ is positive, the first negative term in the expansion of $(1+x)^{27/5}$ is
  28. Let $x=\left(8\sqrt3+13\right)^{13}$ and $y=\left(7\sqrt2+9\right)^{9}$. If $[t]$ denotes the greatest integer $\le t$, then
  29. The remainder, when $7^{103}$ is divided by $23$, is equal to :
  30. The coefficient of the middle term in the binomial expansion in powers of $x$ of $(1+\alpha x)^{4}$ and $(1-\alpha x)^{6}$ is the same if $\alpha$ equals
  31. If $\left({}^{30}C_1\right)^{2}+2\left({}^{30}C_2\right)^{2}+3\left({}^{30}C_3\right)^{2}+\ldots+30\left({}^{30}C_{30}\right)^{2}=\frac{\alpha\,60!}{(30!)^{2}}$, then …
  32. Suppose $$\det\begin{bmatrix}\sum_{k=0}^{n}k&\sum_{k=0}^{n}{}^nC_k k^2\\\sum_{k=0}^{n}{}^nC_k k&\sum_{k=0}^{n}{}^nC_k 3^k\end{bmatrix}=0$$ holds for some positive …
  33. If for positive integers $r>1$, $n>2$, the coefficients of the $(3r)^{\text{th}}$ and $(r+2)^{\text{th}}$ powers of $x$ in the expansion of $(1+x)^{2n}$ are equal, then …
  34. If the 7th term in the binomial expansion of $\left(\frac{3}{\sqrt[3]{84}}+\sqrt3\ln x\right)^{9}$, $x>0$, is equal to $729$, then $x$ can be :
  35. The number of terms in the expansion of $(1+x)^{101}\left(1-x+x^{2}\right)^{100}$ in powers of $x$ is
  36. The coefficient of $x^{1012}$ in the expansion of $\left(1+x^{n}+x^{253}\right)^{10}$, (where $n\le22$ is any positive integer), is
  37. The term independent of $x$ in the expansion of $\left(\frac1{60}-\frac{x^{8}}{81}\right)\cdot\left(2x^{2}-\frac3{x^{2}}\right)^{6}$ is equal to
  38. The term independent of $x$ in expansion of $\left(\frac{x+1}{x^{2/3}-x^{1/3}+1}-\frac{x-1}{x-x^{1/2}}\right)^{10}$ is
  39. For some $n\neq10$, let the coefficients of the $5$th, $6$th and $7$th terms in the binomial expansion of $(1+x)^{n+4}$ be in A.P. Then the largest coefficient in the …
  40. If the coefficient of $x^{15}$ in the expansion of $\left(ax^3+\frac{1}{bx^{1/3}}\right)^{15}$ is equal to the coefficient of $x^{-15}$ in the expansion of …
  41. The smallest natural number $n$, such that the coefficient of $x$ in the expansion of $\left(x^{2}+\frac{1}{x^{3}}\right)^{n}$ is ${}^{n}C_{23}$, is
  42. The least value of $n$ for which the number of integral terms in the Binomial expansion of $\left(\sqrt[3]7+\sqrt[12]{11}\right)^n$ is $183$, is :
  43. The coefficient of $x^{-5}$ in the binomial 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
  44. If the coefficients of $x^{-2}$ and $x^{-4}$, in the expansion of $\left(x^{\frac13}+\frac{1}{2x^{\frac13}}\right)^{18},\ (x>0)$, are $m$ and $n$ respectively, then …
  45. If the coefficient of $x^{7}$ in $\left[ax^{2}+\left(\frac{1}{bx}\right)\right]^{11}$ equals the coefficient of $x^{-7}$ in …
  46. The coefficient of $x^{2}$ in the expansion of the product $(2-x^{2})$ $\left\{(1+2x+3x^{2})^{6}+(1-4x^{2})^{6}\right\}$ is
  47. Let the smallest value of $k\in\mathbb{N}$, for which the coefficient of $x^{3}$ in $(1+x)^{3}+(1+x)^{4}+(1+x)^{5}+\ldots+(1+x)^{99}+(1+kx)^{100}$, $x\neq0$, is …
  48. If $x$ is so small that $x^{3}$ and higher powers of $x$ may be neglected, then $\frac{(1+x)^{3/2}-\left(1+\frac12x\right)^{3}}{(1-x)^{1/2}}$ may be approximated as
  49. A ratio of the $5^{th}$ term from the beginning to the $5^{th}$ term from the end in the binomial expansion of $\left(2^{\frac13}+\frac{1}{2(3)^{\frac13}}\right)^{10}$ is
  50. If $1+x^{4}+x^{5}=\sum_{i=0}^{5}a_i(1+x)^{i}$, for all $x$ in $R$, then $a_2$ is :

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