WAEC 2008 · Paper 2 · Q12✱✱

(a)
Given that $M\begin{pmatrix} 2 & 9 \\ 8 & 6 \end{pmatrix} = \begin{pmatrix} 14 & 3 \end{pmatrix}$ , find the matrix $M$ .
Separate values with commas, e.g. 3, −2
(b)(i)
Copy and complete the table below, giving each value to four decimal places.

$x$ 0.00 0.25 0.50 0.75 1.00

$\dfrac{1}{\sqrt{2 + x^2}}$ 0.6963

Separate values with commas, e.g. 3, −2
(b)(ii)
Use the trapezium rule to determine, correct to three significant figures, the value of $\displaystyle\int_0^1 \frac{dx}{\sqrt{2 + x^2}}$ .

Worked solution (try it first)

$M$ times a $2 \times 2$ matrix gives a $1 \times 2$ matrix, so $M$ is $1 \times 2$ .
Let $M = \begin{pmatrix} a & b \end{pmatrix}$ .
Multiply: $\begin{pmatrix} 2a + 8b & 9a + 6b \end{pmatrix} = \begin{pmatrix} 14 & 3 \end{pmatrix}$ .
So $2a + 8b = 14$ , which is $a + 4b = 7$ , and $9a + 6b = 3$ , which is $3a + 2b = 1$ .
From the first, $a = 7 - 4b$ .
Substitute: $21 - 12b + 2b = 1$ , so $b = 2$ .
Then $a = 7 - 8 = -1$ , so $M = \begin{pmatrix} -1 & 2 \end{pmatrix}$ .

Trapezium rule with $h = 0.25$ : $\frac{h}{2}\left[y_0 + y_4 + 2(y_1 + y_2 + y_3)\right]$ .
Substitute: $0.125\left[0.7071 + 0.5774 + 2(0.6963 + 0.6667 + 0.6247)\right] = 0.125 \times 5.2599$ .
$0.6575$ , which is $0.657$ to three significant figures.