WAEC 2018 · Paper 2 · Q9

(a)
The radius of a sphere increased by $2\frac12\%$ . Find the percentage increase in the volume.
(b)
If the coefficients of $x^2$ and $x^3$ in the binomial expansion of $(p + qx)^7$ are equal, express $q$ in terms of $p$ .

Worked solution (try it first)

(a)

$V = \frac43\pi r^3$ , so for a small change $\dfrac{\delta V}{V} \approx 3\dfrac{\delta r}{r}$ .
With $\dfrac{\delta r}{r} = 2\frac12\%$ : $\dfrac{\delta V}{V} \approx 3 \times 2\frac12\%$
$= 7\frac12\%$ .
(Exactly, $1.025^3 = 1.0769$ , an increase of about $7.69\%$ .)

(b)

The $x^2$ term of $(p + qx)^7$ is $21p^5q^2x^2$ and the $x^3$ term is $35p^4q^3x^3$ .
Set the coefficients equal: $21p^5q^2 = 35p^4q^3$ .
Divide by $7p^4q^2$ : $3p = 5q$ , so $q = \frac35p$ .

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