If the coefficients of x and x^2 in the expansion of (1+x)^p(1-x)^q, p, q 15, are -3 and -5 respectively, then the coefficient of x^3 is equal to _____________.

Coefficient of x in (1 + x)^p(1 - x)^q\\ - ^pC_0\,^qC_1 + ^pC_1\,^qC_0 = - 3 p - q = - 3\\ Coefficient of x 2 in (1 + x)^p(1 - x)^q\\ ^pC_0\,^qC_2 - ^pC_1\,^

If the coefficients of x and x2 in the expansion of 1xp1xq p: Binomial Theorem (Mathematics)

Coefficient of x in ${(1 + x)^p}{(1 - x)^q}\\$

$- {}^p{C_0}\,{}^q{C_1} + {}^p{C_1}\,{}^q{C_0} = - 3 \Rightarrow p - q = - 3\\$

Coefficient of x² in ${(1 + x)^p}{(1 - x)^q}\\$

${}^p{C_0}\,{}^q{C_2} - {}^p{C_1}\,{}^q{C_1} + {}^p{C_2}\,{}^q{C_0} = - 5\\$

${{q(q - 1)} \over 2} - pq + {{p(q - 1)} \over 2} = - 5\\$

${{{q^2} - q} \over 2} - (q - 3)q + {{(q - 3)(q - 4)} \over 2} = - 5\\$

$\Rightarrow q = 11,\,p = 8\\$

Coefficient of x³ in ${(1 + x)^8}{(1 - x)^{11}}\\$

$= - {}^{11}{C_3} + {}^8{C_1}\,{}^{11}{C_2} - {}^8{C_2}\,{}^{11}{C_1} + {}^8{C_3} = 23$

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.

Download Divine JEE App

Quick Links

Exam Categories

Contact Info

Divine JEE