WAEC Further Mathematics 2018 Paper

Concept:Partial fractions express a rational expression as a sum of simpler fractions.Since $(x-2)^2$ is a repeated factor, use the form $\frac{A}{x-2} + \frac{Bx}{(x-2)^2}$.Explanation:Write $\frac{3x-1}{(x-2)^2} = \frac{A}{x-2} + \frac{Bx}{(x-2)^2}$.Combine the right side over the common denominator $(x-2)^2$:$\frac{3x-1}{(x-2)^2} = \frac{A(x-2) + Bx}{(x-2)^2}$.Equate the numerators:$3x - 1 = A(x-2) + Bx = Ax - 2A + Bx$.Collect like terms:$(A+B)x - 2A = 3x - 1$.Compare coefficients of $x$ and the constant term:$A + B = 3$ and $-2A = -1$.From $-2A = -1$, get $A = \frac{1}{2}$.Then $B = 3 - \frac{1}{2} = \frac{5}{2}$.Substitute the values of $A$ and $B$:$\frac{3x-1}{(x-2)^2} = \frac{\frac{1}{2}}{x-2} + \frac{\frac{5}{2}x}{(x-2)^2}$.Therefore, $\frac{3x-1}{(x-2)^2} = \frac{1}{2(x-2)} + \frac{5x}{2(x-2)^2}$.Answer:Option C.