Let 2x2 3x 4 10 sumr 020 arxr Then a7 a13 is equal to

Question

Let ( 2x^2 + 3x + 4 )^10 = sum limits_r = 0^20 a_rx^r Then a_7 / a_13 is equal to ______.

DivineJEE · Accepted Answer

bfNote: Multinomial Theorem [1em] The general term of (x_1 + x_2 + s + x_n)^n in the expansion is: \ displaystyle (n! / n_1! n_2! s n_n!) x_1^n_1 x_2^n_2 s x_n^n_n [1em] where n_1 + n_2 + s + n_n = n [1em] Here, in (2x^2 + 3x + 4)^10 the general term is: \ = (10! / n_1! n_2! n_3!) (2x^2)^n_1 (3x)^n_2 (4)^n_3 \ = (10! / n_1! n_2! n_3!) * 2^n_1 * 3^n_2 * 4^n_3 * x^2n_1 + n_2 [1em] Therefore Coefficient of x^2n_1 + n_2 is: \ displaystyle (10! / n_1! n_2! n_3!) * 2^n_1 * 3^n_2 * 4^n_3 [1em] where n_1 + n_2 + n_3 = 10 [1em] For the coefficient of x^7: \ 2n_1 + n_2 = 7 [1em] Possible values of n_1, n_2, and n_3 are: \ \|c|c|c| hline n_1 n_2 n_3 \ hline 3 1 6 \ 2 3 5 \ 1 5 4 \ 0 7 3 \ hline [1em] Therefore Coefficient of x^7 \ = (10! / 3!1!6!) (2)^3 (3)^1 (4)^6 + (10! / 2!3!5!) (2)^2 (3)^3 (4)^5 \ + (10! / 1!5!4!) (2)^1 (3)^5 (4)^4 + (10! / 0!7!3!) (2)^0 (3)^7 (4)^3 [1em] Coefficient of x^13 = a_13 \ Here 2n_1 + n_2 = 13 [1em] Possible values of n_1, n_2, and n_3 are: \ \|c|c|c| hline n_1 n_2 n_3 \ hline 6 1 3 \ 5 3 2 \ 4 5 1 \ 3 7 0 \ hline [1em] Therefore Coefficient of x^13 \ = (10! / 6!1!3!) (2)^6 (3)^1 (4)^3 + (10! / 5!3!2!) (2)^5 (3)^3 (4)^2 \ + (10! / 4!5!1!) (2)^4 (3)^5 (4)^1 + (10! / 3!7!0!) (2)^3 (3)^7 (4)^0 [1em] Therefore fraca_7a_13 = 8

Explanation

The minimum number of elements that must be added to the relation R = {(a,b),(b,c),(b,d)} on the set { a,b,c,d} so that it is an equivalence relation is

For integers n and r let n r nCr ifn ge r ge 0 0 otherwise

The integral int_0^1 \frac{1}{7^{\left[\frac{1}{x}\right]}} {dx}, where [.] denotes the greatest integer function is equal to

The area of the region enclosed by the parabolas y=4x-x^2 and 3y=(x-4)^2 is equal to

For (2 leq r leq n,binom{n}{r} + 2binom{n}{r - 1} + binom{n}{r - 2} =)

Ten different letters of an alphabet are given. Words with five letters are formed from these given letters. Then the number of words which have atleast one letter repeated, is

Area of the region (x,y):x^2+(y-2)^2≤4,x^2≥2y is

Let R be the set of real numbers. end{enumerate} Statement-1: A = {(x,y) in R X R:y - x is an integer } is an equivalence . relation on R. Statement-2: B = {(x,y) in R X R:x = alpha y for some rational number } is . an equivalence relation on R.

In a college of 300 students, every student reads 5 newspapers and every newspaper is read by 60 students. The number of newspaper is

Download Divine JEE App

Quick Links

Exam Categories

Contact Info

Divine JEE