WAEC Further Mathematics 2014 Paper

Concept:This is a partial fractions problem where equating numerators gives two simple equations for $P$ and $Q$.Explanation:Combine the fractions on the right-hand side over a common denominator $(x-1)(x-3)$.$$\frac{x+P}{(x-1)(x-3)} = \frac{Q(x-3) + 2(x-1)}{(x-1)(x-3)}$$Since denominators are equal, equate the numerators.$$x + P = Q(x-3) + 2(x-1)$$Expanding the right side gives:$$x + P = Qx - 3Q + 2x - 2 = (Q + 2)x + (-3Q - 2)$$Now compare the coefficients of $x$ and the constant terms.Coefficient of $x$: $1 = Q + 2$, so $Q = -1$.Constant term: $P = -3Q - 2$.Substitute $Q = -1$: $P = -3(-1) - 2 = 3 - 2 = 1$.Therefore, $P + Q = 1 + (-1) = 0$.Answer:$P + Q = 0$, so option C is correct.