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Binomial Theorem

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  1. 51.If the coefficient of x^7 in (ax-1bx^2)^13 and the coefficient of x^-5 in (ax+1bx^2)^13 ar…
  2. 52.25^190-19^190-8^190+2^190 is divisible by :
  3. 53.The absolute difference of the coefficients of x^10 and x^7 in the expansion of [(2 x^2+(1…
  4. 54.If the coefficients of three consecutive terms in the expansion of (1+x)^n are in the rati…
  5. 55.If the coefficient of x^7 in (ax^2+(1)/(2bx))^11 and x^-7 in (ax-13bx^2)^11 are equal, the…
  6. 56.Among the statements : (S1) : 2023^2022-1999^2022 is divisible by 8 (S2) : 13(13)^n-12 n-1…
  7. 57.If ^2nC_3:^nC_3=10:1, then the ratio (n^2+3n):(n^2-3n+4) is :
  8. 58.If the ratio of the fifth term from the beginning to the fifth term from the end in the ex…
  9. 59.Let x=(8√3+13)^13 and y=(7√2+9)^9.
  10. 60.The sum of all those terms which are rational numbers in the expansion of (21/3 + 31/4)12 …
  11. 61.The value of -15C1 + 2.15C2 -- 3.15C3 + ...
  12. 62.The sum of the series 2.20C0 + 5.20C1 + 8.20C2 + 11.20C3 + ...
  13. 63.The total number of irrational terms in the binomial expansion of (71/5 -- 31/10)60 is :
  14. 64.If (1-x^3)^10= _r=0^10a_rx^r(1-x)^30-2r, then 9a_9a_10 is equal to ____ .
  15. 65.If the sum of the coefficients of x^7 and x^14 in the expansion of (1x^3-x^4)^n,x 0, is ze…
  16. 66.If (1^15C_0+1^15C_1)(1^15C_1+1^15C_2) (1^15C_12+1^15C_13)= ^13^14C_0^14C_1 ^14C_12, then 3…
  17. 67.The product of the last two digits of (1919)^1919 is Answer: 63
  18. 68.The sum of the series 2 1 ^20C_4-3 2 ^20C_5+4 3 ^20C_6-5 4 ^20C_7+ +18 17 ^20C_20, is equa…
  19. 69.Let (1+x+x^2)^10=a_0+a_1x+a_2x^2+ +a_20x^20.
  20. 70.If =1+ _r=1^6(-3)^r-1, ^12C_2r-1, then the distance of the point (12,√3) from the line x-√…
  21. 71.The sum of all rational terms in the expansion of (1+2^1/3+3^1/2)^6 is equal to _________.
  22. 72.If _r=1^30r^2(^30C_r)^2^30C_r-1= 2^29, then is equal to _________.
  23. 73.If _r=0^5^11C_2r+12r+2=(m)/(n),gcd(m,n)=1, then m-n is equal to __________.
  24. 74.The remainder when 428^2024 is divided by 21 is __________.
  25. 75.If the second, third and fourth terms in the expansion of (x+y)^n are 135, 30 and [(10)/(3…
  26. 76.If the constant term in the expansion of (1+2x-3x^3)((3)/(2)x^2-(1)/(3x))^9 is p, then 108…
  27. 77.Let a=1+^2C_23!+^3C_24!+^4C_25!+....,b=1+^1C_0+^1C_11!+^2C_0+^2C_1+^2C_22!+^3C_0+^3C_1+^3C…
  28. 78.If the Coefficient of x^30 in the expansion of (1+(1)/(x))^6(1+x^2)^7(1-x^3)^8;x 0 is , th…
  29. 79.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+…
  30. 80.In the expansion of (1+x)(1-x^2)(1+(3)/(x)+3x^2+1x^3)^5,x 0, the sum of the coefficients o…
  31. 81.Let = _k=0^n((^nC_k)^2k+1) and = _k=0^n-1(^nC_k^nC_k+1k+2) If 5 =6 , then n equals _______…
  32. 82.Answer: 138
  33. 83.Remainder when 64^32^32 is divided by 9 is equal to ________.
  34. 84.^11C_12+^11C_23+ +^11C_910=(n)/(m)gcd(n,m)=1n+m _______.
  35. 85.The coefficient of x^2012 in the expansion of (1-x)^2008(1+x+x^2)^2007 is equal to _______…
  36. 86.The remainder, when 7^103 is divided by 17, is __________ Answer: 12
  37. 87.Let be the constant term in the binomial expansion of (x-6x^(3)/(2))^n,n 15.
  38. 88.The mean of the coefficients of [x, x^2, , x^7] in the binomial expansion of (2+x)^9 is __…
  39. 89.The number of integral terms in the expansion of (3^(1)/(2)+5^(1)/(4))^680 is equal to ___…
  40. 90.The coefficient of x^7 in (1 - x + 2x^3)^10 is ___________.
  41. 91.Let [t] denote the greatest integer t.
  42. 92.The largest natural number n such that 3^n divides 66 ! is ___________.
  43. 93.The coefficient of x^18 in the expansion of (x^4-1x^3)^15 is __________.
  44. 94.The coefficient of x^-6, in the expansion of ((4x)/(5)+52x^2)^9, is Answer: 5040
  45. 95.If the constant term in the binomial expansion of (x^(5)/(2)2-4x^l)^9 is -84 and the coeff…
  46. 96.50^ root of a number x is 12 and 50^ root of another number y is 18 .
  47. 97.If (1-x^3)^10 = _r=0^10 a_r,x^r(1-x)^30-2r, then 9a_9a_10 is equal to ____.
  48. 98.Let C_r denote the coefficient of x^r in the binomial expansion of (1+x)^n, n N, 0 r n.
  49. 99.The value of _r=50^100^100C_rr+1 is:
  50. 100.Let S=11!,25!+13!,23!+15!,21!+ up to 13 terms.