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JEE Main Binomial Theorem questions

41 Mathematics questions on Binomial Theorem as asked in JEE Main, newest first. 18 come with a worked solution.

  1. If a_r is the coefficient of x^10-r in the Binomial expansion of (1+x)^10, then displaystylesum_r=1^10 r^3 (dfraca_ra_r-1 )^2 is equal to

    2023

  2. If (^30C_1)^2 + 2(^30C_2)^2 + 3(^30C_3)^2 + ... + 30 (^30C_30)^2 = (α 60!)/((30!)^2) then α is equal to :

    2023

  3. sum_k=0^6 51-k C_3 is equal to

    2023 · with solution

  4. Let the coefficients of three consecutive terms in the binomial expansion of (1+2x)^n be in the ratio 2:5:8. Then the coefficient of the term, which is in the middle of these three terms, is _______.

    2023 · with solution

  5. The remainder when (2023)^2023 is divided by 35 is __________

    2023

  6. If the co-efficient of x^9 in (ax^3 + 1/(β x) )^11 and the co-efficient of x^-9 in (ax - 1/(β x^3) )^11 are equal, then (αβ)^2 is equal to ________.

    2023 · with solution

  7. Suppose sum_r=0^2023 r^2 · ^2023C_r = 2023 × α × 2^2022. Then the value of α is _______

    2023

  8. Let (x+10)^50 + (x-10)^50 = a_0 + a_1x + a_2x^2 + … + a_50x^50 for all x ∈ R; then a_2/a_0 is equal to

    2019

  9. The coefficient of x^8 in the product (1+x)^11(1-x)^10(1+x+x^2)^9 is

    2019

  10. If ^20C_1 + (2^2) ^20C_2 + (3^2) ^20C_3 + … + (20^2) ^20C_20 = A(2^β), then the ordered pair (A, β) is equal to

    2019

  11. If the fractional part of the number frac2^40315 is k/15, then k is equal to

    2019 · with solution

  12. If the coefficients of x^2 and x^3 are both zero, in the expansion of the expression (1 + ax + bx^2)(1 - 3x)^15 in powers of x, then the ordered pair (a, b) is equal to

    2019

  13. The term independent of x in the expansion of (1/60 - (x^8)/81 ) · (2x^2 - 3/x )^6 is equal to

    2019

  14. The smallest natural number n, such that the coefficient of x in the expansion of (x^2 + 1/(x^3) )^n is ^nC_23, is

    2019 · with solution

  15. 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 is

    2019

  16. If the fourth term in the binomial expansion of (2/x + x^log_8 x )^6 (x > 0) is 20 × 8^7, then the value of x is

    2019 · with solution

  17. If the fourth term in the binomial expansion of (sqrtfrac1x^1+log_10x+x^1/12 )^6 is equal to 200, and x>1, then the value of x is

    2019

  18. The sum of the coefficients of all even degree terms is x in the expansion of (x + √(x^3 - 1))^6 + (x - √(x^3 - 1))^6, (x > 1) is equal to

    2019 · with solution

  19. The total number of irrational terms in the binomial expansion of (7^1/5 - 3^1/10)^60 is

    2019

  20. The ratio of the 5th term from the beginning to the 5th term from the end in the binomial expansion of (2^1/3 + dfrac12(3)^1/3 )^10 is

    2019

  21. The sum of the real values of x for which the middle term in the binomial expansion of ((x^3)/3 + 3/x )^8 equals 5670 is

    2019 · with solution

  22. The positive value of λ for which the coefficient of x^2 in the expression x^2 (√(x) + (λ)/(x^2) )^10 is 720, is

    2019 · with solution

  23. If the third term in the binomial expansion of (1 + x^log_2 x)^5 equals 2560, then a possible value of x is

    2019

  24. The coefficient of t^4 in the expansion of ((1-t^6)/(1-t) )^3 is

    2019

  25. The sum of the coefficients of all odd degree terms in the expansion of (x + √(x^3 - 1) )^5 + (x - √(x^3 - 1) )^5, (x > 1) is

    2018 · with solution

  26. The value of binom21C_1 - binom10C_1 + binom21C_2 - binom10C_2 + binom21C_3 - binom10C_3 + binom21C_4 - binom10C_4 + … + binom21C_10 - binom10C_10 is

    2017

  27. If the number of terms in the expansion of (1 - 2/x + 4/(x^2) )^n, x ≠ 0, is 28, then the sum of the coefficients of all the terms in this expansion, is

    2016 · with solution

  28. The sum of coefficients of integral powers of x in the binomial expansion (1 - 2√(x))^50 is

    2015

  29. The coefficient of x^11 in the expansion of (1+x^2)^4(1+x^3)^7(1+x^4)^12 is

    2014

  30. The term independent of x in expansion of (dfracx+1x^2/3-x^1/3+1-dfracx-1x-x^1/2 )^10 is

    2013 · with solution

  31. For r = 0, 1, …, 10, if 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, sum_r=1^10 A_r (B_10B_r - C_10A_r) is equal to

    2010

  32. binom300binom3010 - binom301binom3011 + binom302binom3012 + ·s + binom3020binom3030 is equal to

    2005 · with solution

  33. If ^n-1C_r = (k^2 - 3) ^nC_r+1, then k belongs to

    2004

  34. Coefficient of t^24 in (1 + t^2)^12 (1 + t^12) (1 + t^24)

    2003

  35. The sum displaystylesum_i=0^m binom10ibinom20m-i, where binompq = 0 if p > q, is maximum when m is equal to

    2002 · with solution

  36. In the binomial expansion of (a - b)^n, n ≥ 5 the sum of the 5th and 6th terms is zero. Then, a/b equals

    2001 · with solution

  37. For 2 ≤ r ≤ n, binomnr + 2binomnr-1 + binomnr-2 is equal to

    2000 · with solution

  38. If in the expansion of (1 + x)^m (1 - x)^n, the coefficients of x and x^2 are 3 and -6 respectively, then m is equal to

    1999 · with solution

  39. The coefficient of x^4 in (x/2 - 3/x )^10 is

    1983

  40. Given positive integers r > 1, n > 2 and the coefficient of (3r)th and (r + 2)th terms in the binomial expansion of (1 + x)^2n are equal. Then,

    1980 · with solution

  41. If the coefficients of x^3 and x^4 in the expansion of (1 + ax + bx^2)(1 - 2x)^18 in powers of x are both zero, then (a, b) is equal to

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