PYQSide ExamsMy practice

Binomial Theorem · Question 10

JEE Main 2026 (Online) 28th January Evening ShiftMCQMedium📈 Topic in 10% papers+4−1
1

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)

JEE Main 2026 (Online) 28th January Evening Shift

Keep practising — Binomial Theorem

2

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
JEE Main 2026 (Online) 28th January Evening Shift
3

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
JEE Main 2026 (Online) 24th January Morning Shift
4

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
JEE Main 2026 (Online) 23rd January Morning Shift
5

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

  1. 2101101
  2. 2100101
  3. 2100100
  4. 2101100
JEE Main 2026 (Online) 23rd January Morning Shift
6

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
JEE Main 2026 (Online) 22nd January Morning Shift
7

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
JEE Main 2026 (Online) 22nd January Evening Shift
8

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
JEE Main 2026 (Online) 21st January Morning Shift
9

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

JEE Main 2026 (Online) 21st January Evening Shift
10

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

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

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

JEE Main 2025 (Online) 8th April Evening Shift

All Binomial Theorem PYQs →

PDFDownload Binomial Theorem PYQs as PDF Free, with answer key and a QR code to every solution — print it or study offline⬇
All JEE Main Mathematics chapters
Class 11SetsRelations and FunctionsTrigonometric FunctionsComplex Numbers and Quadratic EquationsPermutations and CombinationsBinomial TheoremSequences and SeriesStraight LinesConic SectionsIntroduction to Three Dimensional GeometryLimits and DerivativesStatisticsProbability
Class 12Relations and Functions (Class 12)Inverse Trigonometric FunctionsMatricesDeterminantsContinuity and DifferentiabilityApplication of DerivativesIntegralsApplication of IntegralsDifferential EquationsVector AlgebraThree Dimensional GeometryProbability (Class 12)