Binomial Theorem

JEE Main 2025 (Online) 28th January Evening ShiftMCQMedium📈 Topic in 21% papers+4−1

Ques. 1JEE 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:

Answer

(2) 283

Solution

Coefficients of Tr, Tr+1, Tr+2: 12Cr-1, 12Cr, 12Cr+1

G.P. 12Cr12Cr-1 = 12Cr+112Cr

13 - rr = 12 - rr + 1

(13 - r)(r + 1) = r(12 - r)

13 + 12r - r2 = 12r - r2 13 = 0, impossible

p = 0

General term of (31/4 + 41/3)12: Tk+1 = 12Ck 3 12 - k4 4 k3, 0 k 12

Rational 3 k and 4 (12 - k) 12 k k = 0, 12

q = 33 + 44 = 27 + 256 = 283

p + q = 0 + 283 = 283

Answer: (2)

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Ques. 2JEE Main 2025, 24 January (Morning Shift)

For some n 10, let the coefficients of the 5 th, 6 th and 7 th terms in the binomial expansion of (1+x)n+4 be in A.P. Then the largest coefficient in the expansion of (1+x)n+4 is:

Ques. 3JEE Main 2025, 24 January (Evening Shift)

Suppose A and B are the coefficients of 30th and 12th terms respectively in the binomial expansion of (1+x)2n-1. If 2A=5B, then n is equal to:

Ques. 4JEE Main 2025, 23 January (Morning Shift)

The sum of all rational terms in the expansion of (1+21/3+31/2 )6 is equal to .

Ques. 5JEE Main 2025, 23 January (Evening Shift)

If in the expansion of (1+x)p(1-x)q, the coefficients of x and x2 are 1 and -2, respectively, then p2+q2 is equal to :

Ques. 6JEE Main 2025, 22 January (Morning Shift)

If r=05 11C2r+12r+2= mn, (m,n)=1, then m-n is equal to .

Ques. 7JEE Main 2025, 22 January (Evening Shift)

Let , , and be the coefficients of x7,x5,x3 and x respectively in the expansion of (x+ x3-1 )5+ (x- x3-1 )5, x>1. If u and v satisfy the equations u+ v=18, u+ v=20, then u+v equals :

Ques. 8JEE Main 2025, 22 January (Evening Shift)

If r=130 r2 (30Cr )230Cr-1= 229, then is equal to .

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