WAEC 2010 · Paper 2 · Q1

A binary operation $*$ is defined on the set $\mathbb{R}$ of real numbers by $x * y = \dfrac{x^2}{x + y}$ , where $x, y \in \mathbb{R}$ .

(a)
Evaluate $(2 * 3) * 5$ .
(b)
If $(x + 1) * (x + 2) = \dfrac13$ , find the values of $x$ .
Separate values with commas, e.g. 3, −2

Worked solution (try it first)

(a)

Work out the bracket first: $2 * 3 = \dfrac{2^2}{2 + 3} = \dfrac45$ .
Now use $x = \frac45$ and $y = 5$ : $\dfrac45 * 5 = \dfrac{\left(\frac45\right)^2}{\frac45 + 5}$
$= \dfrac{\frac{16}{25}}{\frac{29}{5}}$ .
Divide the fractions: $\dfrac{16}{25} \times \dfrac{5}{29} = \dfrac{16}{145}$ .

(b)

Put $x + 1$ for $x$ and $x + 2$ for $y$ : $\dfrac{(x + 1)^2}{(x + 1) + (x + 2)} = \dfrac13$ .
Cross-multiply: $3(x + 1)^2 = 2x + 3$ .
Expand: $3x^2 + 6x + 3 = 2x + 3$ , so $3x^2 + 4x = 0$ .
Factorise: $x(3x + 4) = 0$ .
So $x = 0$ or $x = -\dfrac43$ .

Report a problem with this question