Mathematics questions
18011 published questions with worked solutions.
- Let (a+bx+cx^2)^ 10 = _ i=0 ^ 20 p_ix^i, a,b,c N. If p_1=20 and p_2=210, then 2(a+b+c) is equal to
- Let K be the sum of the coefficients of the odd powers of x in the expansion of (1+x)^ 99 . Let a be the middle term in the...
- If the 1011^ th term from the end in the binomial expansion of ( 4x 5- 5 2x )^ 2022 is 1024 times 1011^ th term from the...
- If the coefficient of x^7 in (ax- 1 bx^2 )^ 13 and the coefficient of x^ -5 in (ax+ 1 bx^2 )^ 13 are equal, then a^4b^4 is equal...
- _ k=0 ^6 ^ 51-k C_3 is equal to
- If ( ^ 30 C_1 )^2+2 ( ^ 30 C_2 )^2+3 ( ^ 30 C_3 )^2+ +30 ( ^ 30 C_ 30 )^2= ,60! (30!)^2 , then is equal to
- If the sum of the coefficients of all even powers of x in the product (1+x+x^2+ +x^ 2n ) (1-x+x^2-x^3+ +x^ 2n ) is 61, then n is...
- If C_r= ^ 25 C_r and C_0+5 C_1+9 C_2+ +(101) C_ 25 =2^ 25 k, then k is equal to
- For a positive integer n, (1+ 1x )^n is expanded in increasing powers of x. If three consecutive coefficients in this expansion...
- Let (2x^2+3x+4 )^ 10 = _ r=0 ^ 20 a_rx^r. Then a_7 a_ 13 is equal to
- The natural number m, for which the coefficient of x in the binomial expansion of (x^m+ 1 x^2 )^ 22 is 1540, is
- The coefficient of x^4 in the expansion of (1+x+x^2+x^3 )^6 in powers of x, is
- For integers n and r, let nr= cases ^nC_r,& if n r 0 0,& otherwise cases . The maximum value of k for which the sum _ i=0 ^k 10 i...
- If the remainder when x is divided by 4 is 3, then the remainder when (2020+x)^ 2022 is divided by 8 is
- Let m,n N and (2,n)=1. If 30 30 0+29 30 1+ +2 30 28 +1 30 29 =n 2^m, then n+m is equal to (Here nk= ^nC_k)
- Let n be a positive integer. Let A= _ k=0 ^n(-1)^k , ^nC_k [ ( 12 )^k+ ( 34 )^k+ ( 78 )^k+ ( 15 16 )^k+ ( 31 32 )^k ]. If 63A=1-...
- Let the coefficients of third, fourth and fifth terms in the expansion of (x+ a x^2 )^n, x 0, be in the ratio 12:8:3. Then the...
- Let ^nC_r denote the binomial coefficient of x^r in the expansion of (1+x)^n. If _ k=0 ^ 10 (2^2+3k ) ^ 10 C_k= 3^ 10 + 2^ 10 , ,...
- The ratio of the coefficient of the middle term in the expansion of (1+x)^ 20 and the sum of the coefficients of two middle terms...
- Let nk denote ^nC_k and [ n k ]= cases nk,& if 0 k n 0,& otherwise. cases If A_k= _ i=0 ^9 9i [ 12 12-k+i ]+ _ i=0 ^8 8i [ 13...