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Binomial Theorem

JEE Main 2025 (Online) 4th April Morning ShiftMCQMedium📈 Topic in 21% papers+4−1
Ques. 1JEE Main 2025, 4 April (Morning Shift)

In the expansion of ( [3]2 + 1 [3]3 )n, n N, if the ratio of 15th term from the beginning to the 15th term from the end is 16, then the value of nC3 is

  1. 4960
  2. 2300
  3. 1040
  4. 4060

Answer

(2) 2300

Solution

Let a = 21/3, b = 3-1/3.

15th term from the beginning: T15 = nC14 an - 14 b14

15th term from the end = 15th term of (b + a)n = nC14 bn - 14 a14

Ratio = an - 28bn - 28 = ( ab )n - 28 = (21/3 31/3 )n - 28 = 6 n - 283

6 n - 283 = 16 = 6-1

n - 28 = -3 n = 25

25C3 = 25 24 236 = 2300

Answer: (2)

Keep practising — Binomial Theorem

Ques. 2JEE Main 2025, 4 April (Evening Shift)

If 12 (15C1 ) + 22 (15C2 ) + 32 (15C3 ) + + 152 (15C15 ) = 2m 3n 5k, where m, n, k N, then m + n + k is equal to :

  1. 20
  2. 19
  3. 18
  4. 21
Ques. 3JEE Main 2025, 3 April (Morning Shift)

The sum of all rational terms in the expansion of (2+ 3)8 is :

  1. 16923
  2. 18817
  3. 3763
  4. 33845
Ques. 4JEE Main 2025, 3 April (Morning Shift)

If r=19 ( r+32r ) 9Cr = ( 32 )9 - , , N, then ( + )2 is equal to

  1. 27
  2. 81
  3. 18
  4. 9
Ques. 5JEE Main 2025, 3 April (Evening Shift)

Let (1 + x + x2 )10 = a0 + a1x + a2x2 + + a20x20. If (a1 + a3 + a5 + + a19) - 11a2 = 121k, then k is equal to .

Ques. 6JEE Main 2025, 2 April (Morning Shift)

The term independent of x in the expansion of ( (x + 1)(x2/3 + 1 - x1/3) - (x - 1)(x - x1/2) )10, x > 1, is :

  1. 240
  2. 120
  3. 150
  4. 210
Ques. 7JEE Main 2025, 2 April (Evening Shift)

If r=010 ( 10r+1 - 110r ) 11Cr+1 = 11 - 11111010, then is equal to :

  1. 11
  2. 20
  3. 24
  4. 15
Ques. 8JEE Main 2025, 29 January (Morning Shift)

The least value of n for which the number of integral terms in the Binomial expansion of ( [3]7 + [12]11)n is 183, is :

  1. 2184
  2. 2172
  3. 2196
  4. 2148
Ques. 9JEE Main 2025, 29 January (Evening Shift)

The remainder, when 7103 is divided by 23, is equal to:

  1. 9
  2. 6
  3. 14
  4. 17
Ques. 10JEE Main 2025, 28 January (Morning Shift)

If = 1 + r=16(-3)r-1 12C2r-1, then the distance of the point (12, 3) from the line x - 3y + 1 = 0 is .

Ques. 11JEE Main 2025, 28 January (Evening Shift)

Let the coefficients of three consecutive terms Tr, Tr+1 and Tr+2 in the binomial expansion of (a + b)12 be in a G.P. and let p be the number of all possible values of r. Let q be the sum of all rational terms in the binomial expansion of ( [4]3 + [3]4)12. Then p + q is equal to:

  1. 295
  2. 283
  3. 299
  4. 287

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