WAEC 2014 · Paper 2 · Q10

(a)
If $P = \begin{pmatrix} 4 & 2 \\ c & 3 \end{pmatrix}$ , $Q = \begin{pmatrix} 6 & 2 \\ 4 & d \end{pmatrix}$ and $PQ = QP$ , find the values of $c$ and $d$ .
Separate values with commas, e.g. 3, −2
(b)
Using the trapezium rule with five ordinates, calculate, correct to two decimal places, the approximate value of $\displaystyle\int_2^6 \frac{dx}{x}$ .

Worked solution (try it first)

(a)

$PQ = \begin{pmatrix} 32 & 8 + 2d \\ 6c + 12 & 2c + 3d \end{pmatrix}$ and $QP = \begin{pmatrix} 24 + 2c & 18 \\ 16 + cd & 8 + 3d \end{pmatrix}$ .
Match the top-left entries: $32 = 24 + 2c$ , so $c = 4$ .
Match the top-right entries: $8 + 2d = 18$ , so $d = 5$ .
Check the others: $6c + 12 = 36 = 16 + cd$ ✓ and $2c + 3d = 23 = 8 + 3d$ ✓.

(b)

Five ordinates means four strips, so $h = \dfrac{6 - 2}{4} = 1$ .
The ordinates of $\frac1x$ at $x = 2, 3, 4, 5, 6$ are $0.5,\ 0.3333,\ 0.25,\ 0.2,\ 0.1667$ .
$\frac12[(0.5 + 0.1667) + 2(0.3333 + 0.25 + 0.2)] = \frac12(0.6667 + 1.5667)$
$= 1.1167$ , about $1.12$ .

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