(a) Use the trapezium rule with five ordinates to... | WAEC SSCE - Further Mathematics - 2009 (Essay)

Question 1 Report

(a) Use the trapezium rule with five ordinates to evaluate \(\int_{0} ^{1} \frac{3}{1 + x^{2}} \mathrm {d} x\), 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.

Answer

(a) Five ordinates means four strips of width \(h = \dfrac{1 - 0}{4} = 0.25\). With \(f(x) = \dfrac{3}{1 + x^2}\):

\(x\)	0	0.25	0.5	0.75	1
\(f(x)\)	3.0000	2.8235	2.4000	1.9200	1.5000

\[\int_0^1 f\,dx \approx \frac{h}{2}\Big[f_0 + f_4 + 2(f_1 + f_2 + f_3)\Big] = \frac{0.25}{2}\Big[3 + 1.5 + 2(2.8235 + 2.4 + 1.92)\Big]\] \[= 0.125\big[4.5 + 2(7.1435)\big] = 0.125(18.787) = 2.348\ (\text{4 s.f.})\]

(b) \(A = \begin{pmatrix} 2 & 1 \\ 3 & 2 \end{pmatrix}\). First \(A^2\):

\[A^2 = \begin{pmatrix} 2 & 1 \\ 3 & 2 \end{pmatrix}\begin{pmatrix} 2 & 1 \\ 3 & 2 \end{pmatrix} = \begin{pmatrix} 7 & 4 \\ 12 & 7 \end{pmatrix}\] \[A^2 + A + 2I = \begin{pmatrix} 7 & 4 \\ 12 & 7 \end{pmatrix} + \begin{pmatrix} 2 & 1 \\ 3 & 2 \end{pmatrix} + \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix} = \begin{pmatrix} 11 & 5 \\ 15 & 11 \end{pmatrix}\]

Image of \((1, 2)\):

\[\begin{pmatrix} 11 & 5 \\ 15 & 11 \end{pmatrix}\begin{pmatrix} 1 \\ 2 \end{pmatrix} = \begin{pmatrix} 11 + 10 \\ 15 + 22 \end{pmatrix} = \begin{pmatrix} 21 \\ 37 \end{pmatrix}\]

The image is \((21,\ 37)\).

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Answer Details

(a) Five ordinates means four strips of width \(h = \dfrac{1 - 0}{4} = 0.25\). With \(f(x) = \dfrac{3}{1 + x^2}\):

\(x\)	0	0.25	0.5	0.75	1
\(f(x)\)	3.0000	2.8235	2.4000	1.9200	1.5000

(b) \(A = \begin{pmatrix} 2 & 1 \\ 3 & 2 \end{pmatrix}\). First \(A^2\):

Image of \((1, 2)\):

\[\begin{pmatrix} 11 & 5 \\ 15 & 11 \end{pmatrix}\begin{pmatrix} 1 \\ 2 \end{pmatrix} = \begin{pmatrix} 11 + 10 \\ 15 + 22 \end{pmatrix} = \begin{pmatrix} 21 \\ 37 \end{pmatrix}\]

The image is \((21,\ 37)\).

Read lesson note on Matrices And Linear Transformation (WAEC)

Read lesson note on Integration (WAEC)

Matrices And Linear Transformation

Integration

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(a) Use the trapezium rule with five ordinates to evaluate \(\int_{0} ^{1} \frac{3}{1 + x^{2}} \mathrm {d} x\), correct to four significant figures. (b) If ...

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(a) Use the trapezium rule with five ordinates to evaluate \(\int_{0} ^{1} \frac{3}{1 + x^{2}} \mathrm {d} x\), correct to four significant figures. (b) If ...

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