WAEC 2009 · Paper 2 · Q12

(a)
Use the trapezium rule with five ordinates to evaluate $\displaystyle\int_0^1 \frac{x}{1 + x^2}\,dx$ , correct to four significant figures.
(b)
If $A = \begin{pmatrix} 2 & 1 \\ 3 & 2 \end{pmatrix}$ , find the image of the point $(1, 2)$ under the linear transformation $A^2 + A + 2I$ , where $I$ is the $2 \times 2$ unit matrix.
Separate values with commas, e.g. 3, −2

Worked solution (try it first)

Five ordinates means four strips, so $h = \frac{1 - 0}{4} = 0.25$ .
Ordinates of $y = \dfrac{x}{1 + x^2}$ at $x = 0, 0.25, 0.5, 0.75, 1$ : $0$ , $0.23529$ , $0.4$ , $0.48$ , $0.5$ .
Trapezium rule: $\frac{h}{2}[y_0 + y_4 + 2(y_1 + y_2 + y_3)] = 0.125[0 + 0.5 + 2(1.11529)]$ .
$= 0.125 \times 2.73059 = 0.34132$ , which is $0.3413$ to 4 significant figures.

$A^2 = \begin{pmatrix} 2 & 1 \\ 3 & 2 \end{pmatrix}\begin{pmatrix} 2 & 1 \\ 3 & 2 \end{pmatrix}$
$= \begin{pmatrix} 7 & 4 \\ 12 & 7 \end{pmatrix}$ .
Add $A$ and $2I$ : $A^2 + A + 2I = \begin{pmatrix} 7 + 2 + 2 & 4 + 1 \\ 12 + 3 & 7 + 2 + 2 \end{pmatrix}$
$= \begin{pmatrix} 11 & 5 \\ 15 & 11 \end{pmatrix}$ .
Multiply by $\begin{pmatrix} 1 \\ 2 \end{pmatrix}$ : $\begin{pmatrix} 11 + 10 \\ 15 + 22 \end{pmatrix} = \begin{pmatrix} 21 \\ 37 \end{pmatrix}$ .
The image is $(21, 37)$ .