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Binomial Theorem · Question 23

JEE Main 2025 (Online) 4th April Morning ShiftMCQMedium📈 Topic in 52% papers+4−1
1

For an integer n 2, if the arithmetic mean of all coefficients in the binomial expansion of (x + y)2n - 3 is 16, then the distance of the point P (2n - 1, n2 - 4n ) from the line x + y = 8 is

  1. 2
  2. 2 2
  3. 5 2
  4. 3 2

Answer

(4) 3 2

Solution

In (x + y)2n - 3 there are 2n - 2 coefficients and their sum is 22n - 3 (put x = y = 1).

Arithmetic mean = 22n - 32n - 2 = 16

22n - 3 = 32(n - 1)

Trying integers n 2: n = 5 gives 27 = 128 = 32 4 (the left side grows much faster, so n = 5 is the only solution).

P(2n - 1, n2 - 4n) = P(9, 5)

Distance of P from x + y - 8 = 0:

d = |9 + 5 - 8| 12 + 12 = 6 2 = 3 2

Answer: (4)

JEE Main 2025 (Online) 4th April Morning Shift

Keep practising — Binomial Theorem

2

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
JEE Main 2025 (Online) 4th April Morning Shift
3

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
JEE Main 2025 (Online) 4th April Evening Shift
4

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

  1. 16923
  2. 18817
  3. 3763
  4. 33845
JEE Main 2025 (Online) 3rd April Morning Shift
5

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

  1. 27
  2. 81
  3. 18
  4. 9
JEE Main 2025 (Online) 3rd April Morning Shift
6

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

JEE Main 2025 (Online) 3rd April Evening Shift
7

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
JEE Main 2025 (Online) 2nd April Morning Shift
8

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

  1. 11
  2. 20
  3. 24
  4. 15
JEE Main 2025 (Online) 2nd April Evening Shift
9

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
JEE Main 2025 (Online) 29th January Morning Shift
10

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 .

JEE Main 2025 (Online) 28th January Morning Shift
11

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
JEE Main 2025 (Online) 28th January Evening Shift

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