Given that $(2x + y)^3 - (x + 2y)^3 = ( x - y) [A(x^2 +y^2)+Bxy]$, the value of (2A - B) is :
Answer & explanation
Correct answer: option 4
We know that,
(a + b)3 = a3 + b3 + 3ab(a + b)
(2x + y)3 - (x + 2y)3 = (x - y) [A(x2 + y2) + Bxy]
= (2x)3 + y3 + 3 × 2x × y(2x + y) - (x)3 - (2y)3 - 3 × x × 2y (x + 2y) = (x - y) [Ax2 + Ay2 + Bxy]
= 8x3 + y3 + 12x2y + 6xy2 - x3 - 8y3 - 6x2y - 12xy2 = [Ax3 + Axy2+ Bx2y - Ax2y - Ay3 - Bxy2]
= 7x3 - 7y3 + 6x2y - 6xy2 = Ax3 - Ay3 + (B - A)x2y + (A - B)xy2
By comparing LHS and RHS
= A = 7, (B - A) = 6, (A - B) = - 6
= (B - 7) = 6
= B = 13
Now put the values of A and B in the required equation,
(2A - B) = (2 × 7 - 13)
=1