AQA FP1 (Further Pure Mathematics 1) 2005 June

1 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are defined by $$\mathbf { A } = \left[ \begin{array} { l l } 3 & 4
4 & 3 \end{array} \right] \quad \mathbf { B } = \left[ \begin{array} { l l } 0 & 2
2 & 0 \end{array} \right]$$

1. Calculate the matrices:
   1. \(\mathbf { A } + \mathbf { B }\);
   2. \(\mathbf { A B }\).
2. Show that \(\mathbf { A } + \mathbf { B } - \mathbf { A B } = k \mathbf { I }\), where \(k\) is an integer and \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.
   (2 marks)

2 A curve satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \sin 2 x$$ where the angle \(2 x\) is measured in radians.
Starting at the point \(( 0.5,1 )\) on the curve, use a step-by-step method with a step length of 0.1 to estimate the value of \(y\) at \(x = 0.7\). Give your answer to three significant figures.
(6 marks)

3

1. Use the formulae $$\begin{gathered} \sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )
   \sum _ { r = 1 } ^ { n } r ^ { 3 } = \frac { 1 } { 4 } n ^ { 2 } ( n + 1 ) ^ { 2 } \end{gathered}$$ and $$\sum _ { r = 1 } ^ { n } r ^ { 2 } ( r - 1 ) = \frac { 1 } { 12 } n \left( n ^ { 2 } - 1 \right) ( 3 n + 2 )$$ (4 marks)
2. Use the result from part (a) to find the value of $$\sum _ { r = 4 } ^ { 11 } r ^ { 2 } ( r - 1 )$$ (3 marks)

4 The function f is defined for all real values of \(x\) by $$\mathrm { f } ( x ) = x ^ { 3 } + x$$

1. Express \(\mathrm { f } ( 2 + h ) - \mathrm { f } ( 2 )\) in the form $$p h + q h ^ { 2 } + r h ^ { 3 }$$ where \(p , q\) and \(r\) are integers.
2. Use your answer to part (a) to find the value of \(\mathrm { f } ^ { \prime } ( 2 )\).

5 Find the general solutions of the following equations, giving your answers in terms of \(\pi\) :

1. \(\quad \tan 3 x = \sqrt { 3 }\);
2. \(\quad \tan \left( 3 x - \frac { \pi } { 3 } \right) = - \sqrt { 3 }\).

6 The equation $$x ^ { 2 } - 4 x + 13 = 0$$ has roots \(\alpha\) and \(\beta\).

1. 1. Write down the values of \(\alpha + \beta\) and \(\alpha \beta\).
   2. Deduce that \(\alpha ^ { 2 } + \beta ^ { 2 } = - 10\).
   3. Explain why the statement \(\alpha ^ { 2 } + \beta ^ { 2 } = - 10\) implies that \(\alpha\) and \(\beta\) cannot both be real.
2. Find in the form \(p + \mathrm { i } q\) the values of:
   1. \(( \alpha + \mathrm { i } ) + ( \beta + \mathrm { i } )\);
   2. \(( \alpha + \mathrm { i } ) ( \beta + \mathrm { i } )\).
3. Hence find a quadratic equation with roots \(( \alpha + \mathrm { i } )\) and \(( \beta + \mathrm { i } )\).

7 [Figure 1, printed on the insert, is provided for use in this question.]
The diagram shows a triangle with vertices \(A ( 1,1 ) , B ( 3,1 )\) and \(C ( 3,2 )\).
\includegraphics[max width=\textwidth, alt={}, center]{5bfb4d19-8772-43d7-b667-bd124d2504a8-04_1114_1141_552_360}

1. The triangle \(D E F\) is obtained by applying to triangle \(A B C\) the transformation T represented by the matrix $$\left[ \begin{array} { r r } 2 & 2
   - 2 & 2 \end{array} \right]$$
   1. Calculate the coordinates of \(D , E\) and \(F\).
   2. Draw the triangle \(D E F\) on Figure 1.
2. Given that T is a combination of an enlargement and a rotation, find the exact value of:
   1. the scale factor of the enlargement;
   2. the magnitude of the angle of the rotation.

8 The diagram shows a part of the curve $$\frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 6 } = 1$$ and a chord \(P Q\) of the curve, where \(P\) lies on the \(x\)-axis.
\includegraphics[max width=\textwidth, alt={}, center]{5bfb4d19-8772-43d7-b667-bd124d2504a8-05_751_1072_680_459}

1. Write down the coordinates of \(P\).
2. The gradient of the chord \(P Q\) is 2 . Find the coordinates of \(Q\).

9 The function f is defined by $$f ( x ) = \frac { x ^ { 2 } + 4 x } { x ^ { 2 } + 9 }$$

1. 1. The graph of \(y = \mathrm { f } ( x )\) has an asymptote which is parallel to the \(x\)-axis. Find the equation of this asymptote.
   2. Explain why the graph of \(y = \mathrm { f } ( x )\) has no asymptotes parallel to the \(y\)-axis.
2. Show that the equation \(\mathrm { f } ( x ) = k\) has two equal roots if \(9 k ^ { 2 } - 9 k - 4 = 0\).
3. Hence find the coordinates of the two stationary points on the graph of \(y = \mathrm { f } ( x )\).

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   General Certificate of Education
   June 2005
   Advanced Subsidiary Examination MATHEMATICS
   MFP1
   Unit Further Pure 1 ASSESSMENT and
   QUALIFICATIONS
   ALLIANCE Wednesday 22 June 2005 Afternoon Session Insert for use in Question 7.
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