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Binomial Theorem

JEE Main 2026 (Online) 28th January Evening ShiftMCQMedium📈 Topic in 11% papers+4−1
Ques. 1JEE Main 2026, 28 January (Evening Shift)

Given below are two statements : In the light of the above statements, choose the correct answer from the options given below :

  1. Statement I is false but Statement II is true
  2. Both Statement I and Statement II are false
  3. Both Statement I and Statement II are true
  4. Statement I is true but Statement II is false

Answer

(3) Both Statement I and Statement II are true

Solution

Statement I: for odd n, an + bn is divisible by a + b.

2513 + 313 is divisible by 25 + 3 = 28, and 2013 + 813 is divisible by 20 + 8 = 28

the sum is divisible by 28, hence by 7. Statement I is true.

Statement II: let (7 + 4 3)25 = I + f, 0 f < 1, and (7 - 4 3)25 = f'

0 < 7 - 4 3 < 1 0 < f' < 1

I + f + f' = 2 [25C0 725 + 25C2 723(4 3)2 + ] = even integer

f + f' is an integer with 0 < f + f' < 2 f + f' = 1

I = even - 1 = odd. Statement II is true.

Answer: (3)

Keep practising — Binomial Theorem

Ques. 2JEE Main 2026, 28 January (Evening Shift)

The sum of the coefficients of x499 and x500 in (1 + x)1000 + x(1 + x)999 + x2(1 + x)998 + + x1000 is :

  1. 1002C501
  2. 1001C501
  3. 1000C501
  4. 1002C500
Ques. 3JEE Main 2026, 24 January (Morning Shift)

Let S = 125! + 13! 23! + 15! 21! + up to 13 terms. If 13 S = 2kn!, k N, then n + k is equal to

  1. 50
  2. 52
  3. 49
  4. 51
Ques. 4JEE Main 2026, 23 January (Morning Shift)

The sum of all possible values of n N, so that the coefficients of x, x2 and x3 in the expansion of (1 + x2)2(1 + x)n, are in arithmetic progression is :

  1. 12
  2. 9
  3. 3
  4. 7
Ques. 5JEE Main 2026, 23 January (Morning Shift)

The value of 100C5051 + 100C5152 + + 100C100101 is:

  1. 2101101
  2. 2100101
  3. 2100100
  4. 2101100
Ques. 6JEE Main 2026, 22 January (Morning Shift)

The coefficient of x48 in (1 + x) + 2(1 + x)2 + 3(1 + x)3 + + 100(1 + x)100 is equal to

  1. 100 100C49 - 100C48
  2. 100 101C49 - 101C50
  3. 100C50 + 101C49
  4. 100 100C49 - 100C50
Ques. 7JEE Main 2026, 22 January (Evening Shift)

Let Cr denote the coefficient of xr in the binomial expansion of (1 + x)n, n N, 0 r n. If Pn = C0 - C1 + 223C2 - 234C3 + + (-2)nn+1Cn, then the value of n=125 1P2n equals

  1. 675
  2. 580
  3. 525
  4. 650
Ques. 8JEE Main 2026, 21 January (Morning Shift)

If the coefficient of x in the expansion of (ax2 + bx + c)(1 - 2x)26 is -56 and the coefficients of x2 and x3 are both zero, then a + b + c is equal to :

  1. 1483
  2. 1300
  3. 1500
  4. 1403
Ques. 9JEE Main 2026, 21 January (Evening Shift)

If ( 115C0 + 115C1 ) ( 115C1 + 115C2 ) ( 115C12 + 115C13 ) = 1314C0 14C1 14C12, then 30 is equal to .

Ques. 10JEE Main 2025, 8 April (Evening Shift)

The number of integral terms in the expansion of (5 12 + 7 18 )1016 is:

  1. 127
  2. 128
  3. 130
  4. 129
Ques. 11JEE Main 2025, 8 April (Evening Shift)

The product of the last two digits of (1919)1919 is

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