Binomial Theorem · Question 5

Let the smallest value of k N, for which the coefficient of x3 in (1+x)3+(1+x)4+(1+x)5+ +(1+x)99+(1+kx)100, x 0, is (43n+ 1014 ) (100C3 ) for some n N, be p. Then the value of p+n is :

1. 10
2. 11
3. 12
4. 13

In the expansion of (9x- 13 x )18, x>0, if the term independent of x is (221)k, then k is equal to :

1. 84
2. 78
3. 168
4. 198

The number of elements in the set S= \(r,k) : k Z and 36Cr+1= 6 (35Cr ) (k2-3 ) \, is :

1. 2
2. 4
3. 8
4. 16

If for 3 r 30, (30C30-r )+3 (30C31-r )+3 (30C32-r )+ (30C33-r )=mCr, then m equals :

1. 31
2. 32
3. 33
4. 34

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

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

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

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

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

1. 2101101
2. 2100101
3. 2100100
4. 2101100

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