Question bank/JEE Main/Mathematics
JEE MainMathematics
If the number of positive integral solutions of v + u + w = n
be denoted by p_n, then the absolute value of determinant
Show answer and solution
Given, u + v + w = n, u, v, w ≥ 1
Coefficient of x^n in (x^1 + x^2 + x^3 +.....+x^n - 2^)3^
Coefficient of x^n-3 in (1+ x + x^2 +.....+x^n - 1^)3^
= (1 - x^n)^3 (1 - x)^-3
^= n^-3+3-1C~3 -1~ = ^n-1C_2 = P_n (given)
p_n+1 = ^nC_2, p_n+2 = ^n+1C_2, p_n+3 = ^n+2C_2, p_n+4 =
^4+3C_2,
Put these values into given determinant
[IMAGE] Simplify ∆ = -1
Coefficient of x^n in (x^1 + x^2 + x^3 +.....+x^n - 2^)3^
Coefficient of x^n-3 in (1+ x + x^2 +.....+x^n - 1^)3^
= (1 - x^n)^3 (1 - x)^-3
^= n^-3+3-1C~3 -1~ = ^n-1C_2 = P_n (given)
p_n+1 = ^nC_2, p_n+2 = ^n+1C_2, p_n+3 = ^n+2C_2, p_n+4 =
^4+3C_2,
Put these values into given determinant
[IMAGE] Simplify ∆ = -1
Source: Grass-G / M-02 Mathematics Paper-2.docx
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