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AQA FP1 2005 June Q1

6 marks Easy -1.2

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

Calculate the matrices:
1. $\mathbf { A } + \mathbf { B }$;
2. $\mathbf { A B }$.
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)

AQA FP1 2005 June Q2

6 marks Moderate -0.3

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)

AQA FP1 2005 June Q3

7 marks Standard +0.3

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)
Use the result from part (a) to find the value of $$\sum _ { r = 4 } ^ { 11 } r ^ { 2 } ( r - 1 )$$ (3 marks)

AQA FP1 2005 June Q4

7 marks Moderate -0.8

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

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.
Use your answer to part (a) to find the value of $\mathrm { f } ^ { \prime } ( 2 )$.

AQA FP1 2005 June Q5

7 marks Moderate -0.3

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

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

AQA FP1 2005 June Q6

11 marks Standard +0.3

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

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.
Find in the form $p + \mathrm { i } q$ the values of:
1. $( \alpha + \mathrm { i } ) + ( \beta + \mathrm { i } )$;
2. $( \alpha + \mathrm { i } ) ( \beta + \mathrm { i } )$.
Hence find a quadratic equation with roots $( \alpha + \mathrm { i } )$ and $( \beta + \mathrm { i } )$.

AQA FP1 2005 June Q7

11 marks Standard +0.3

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}

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.
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.

AQA FP1 2005 June Q8

8 marks Standard +0.3

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}

Write down the coordinates of $P$.
The gradient of the chord $P Q$ is 2 . Find the coordinates of $Q$.

AQA FP1 2005 June Q9

13 marks Standard +0.3

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

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.
Show that the equation $\mathrm { f } ( x ) = k$ has two equal roots if $9 k ^ { 2 } - 9 k - 4 = 0$.
Hence find the coordinates of the two stationary points on the graph of $y = \mathrm { f } ( x )$.
Surname Other Names
Centre Number Candidate Number
Candidate Signature
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.
Fill in the boxes at the top of this page.
Fasten this insert securely to your answer book.

AQA FP1 2006 June Q1

9 marks Standard +0.3

1 The quadratic equation $$3 x ^ { 2 } - 6 x + 2 = 0$$ has roots $\alpha$ and $\beta$.

Write down the numerical values of $\alpha + \beta$ and $\alpha \beta$.
1. Expand $( \alpha + \beta ) ^ { 3 }$.
2. Show that $\alpha ^ { 3 } + \beta ^ { 3 } = 4$.
Find a quadratic equation with roots $\alpha ^ { 3 }$ and $\beta ^ { 3 }$, giving your answer in the form $p x ^ { 2 } + q x + r = 0$, where $p , q$ and $r$ are integers.

Find the matrix:
1. $\mathbf { M } ^ { 2 }$;
2. $\mathbf { M } ^ { 4 }$.
Describe fully the geometrical transformation represented by $\mathbf { M }$.
Find the matrix $\mathbf { M } ^ { 2006 }$.

AQA FP1 2006 June Q6

7 marks Moderate -0.3

6 It is given that $z = x + \mathrm { i } y$, where $x$ and $y$ are real numbers.

Write down, in terms of $x$ and $y$, an expression for $$( z + \mathrm { i } ) ^ { * }$$ where $( z + \mathrm { i } ) ^ { * }$ denotes the complex conjugate of $( z + \mathrm { i } )$.
Solve the equation $$( z + \mathrm { i } ) ^ { * } = 2 \mathrm { i } z + 1$$ giving your answer in the form $a + b \mathrm { i }$.

AQA FP1 2006 June Q7

6 marks Standard +0.3

Describe a geometrical transformation by which the hyperbola $$x ^ { 2 } - 4 y ^ { 2 } = 1$$ can be obtained from the hyperbola $x ^ { 2 } - y ^ { 2 } = 1$.
The diagram shows the hyperbola $H$ with equation $$x ^ { 2 } - y ^ { 2 } - 4 x + 3 = 0$$
\includegraphics[max width=\textwidth, alt={}]{e44987a7-2cdf-442a-aecb-abd3e889ecd4-4_951_1216_824_402}
By completing the square, describe a geometrical transformation by which the hyperbola $H$ can be obtained from the hyperbola $x ^ { 2 } - y ^ { 2 } = 1$.

AQA FP1 2006 June Q8

10 marks Standard +0.3

The function f is defined for all real values of $x$ by $$\mathrm { f } ( x ) = x ^ { 3 } + x ^ { 2 } - 1$$
1. Express $\mathrm { f } ( 1 + h ) - \mathrm { f } ( 1 )$ 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)(i) to find the value of $f ^ { \prime } ( 1 )$.
The diagram shows the graphs of $$y = \frac { 1 } { x ^ { 2 } } \quad \text { and } \quad y = x + 1 \quad \text { for } \quad x > 0$$
\includegraphics[max width=\textwidth, alt={}]{e44987a7-2cdf-442a-aecb-abd3e889ecd4-5_643_791_1160_596}
The graphs intersect at the point $P$.
1. Show that the $x$-coordinate of $P$ satisfies the equation $\mathrm { f } ( x ) = 0$, where f is the function defined in part (a).
2. Taking $x _ { 1 } = 1$ as a first approximation to the root of the equation $\mathrm { f } ( x ) = 0$, use the Newton-Raphson method to find a second approximation $x _ { 2 }$ to the root.
  (3 marks)
The region enclosed by the curve $y = \frac { 1 } { x ^ { 2 } }$, the line $x = 1$ and the $x$-axis is shaded on the diagram. By evaluating an improper integral, find the area of this region.
(3 marks)

AQA FP1 2006 June Q9

16 marks Standard +0.8

9 A curve $C$ has equation $$y = \frac { ( x + 1 ) ( x - 3 ) } { x ( x - 2 ) }$$

1. Write down the coordinates of the points where $C$ intersects the $x$-axis. (2 marks)
2. Write down the equations of all the asymptotes of $C$.
1. Show that, if the line $y = k$ intersects $C$, then $$( k - 1 ) ( k - 4 ) \geqslant 0$$
2. Given that there is only one stationary point on $C$, find the coordinates of this stationary point.
  (No credit will be given for solutions based on differentiation.)
Sketch the curve $C$.

AQA FP2 2006 January Q1

6 marks Standard +0.3

Show that $$\frac { 1 } { r ^ { 2 } } - \frac { 1 } { ( r + 1 ) ^ { 2 } } = \frac { 2 r + 1 } { r ^ { 2 } ( r + 1 ) ^ { 2 } }$$
Hence find the sum of the first $n$ terms of the series $$\frac { 3 } { 1 ^ { 2 } \times 2 ^ { 2 } } + \frac { 5 } { 2 ^ { 2 } \times 3 ^ { 2 } } + \frac { 7 } { 3 ^ { 2 } \times 4 ^ { 2 } } + \ldots$$

AQA FP2 2006 January Q2

10 marks Standard +0.3

2 The cubic equation $$x ^ { 3 } + p x ^ { 2 } + q x + r = 0$$ where $p , q$ and $r$ are real, has roots $\alpha , \beta$ and $\gamma$.

Given that $$\alpha + \beta + \gamma = 4 \quad \text { and } \quad \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 20$$ find the values of $p$ and $q$.
Given further that one root is $3 + \mathrm { i }$, find the value of $r$.

AQA FP2 2006 January Q3

12 marks Moderate -0.3

3 The complex numbers $z _ { 1 }$ and $z _ { 2 }$ are given by $$z _ { 1 } = \frac { 1 + \mathrm { i } } { 1 - \mathrm { i } } \quad \text { and } \quad z _ { 2 } = \frac { 1 } { 2 } + \frac { \sqrt { 3 } } { 2 } \mathrm { i }$$

Show that $z _ { 1 } = \mathrm { i }$.
Show that $\left| z _ { 1 } \right| = \left| z _ { 2 } \right|$.
Express both $z _ { 1 }$ and $z _ { 2 }$ in the form $r \mathrm { e } ^ { \mathrm { i } \theta }$, where $r > 0$ and $- \pi < \theta \leqslant \pi$.
Draw an Argand diagram to show the points representing $z _ { 1 } , z _ { 2 }$ and $z _ { 1 } + z _ { 2 }$.
Use your Argand diagram to show that $$\tan \frac { 5 } { 12 } \pi = 2 + \sqrt { 3 }$$

AQA FP2 2006 January Q4

9 marks Challenging +1.2

Prove by induction that $$2 + ( 3 \times 2 ) + \left( 4 \times 2 ^ { 2 } \right) + \ldots + ( n + 1 ) 2 ^ { n - 1 } = n 2 ^ { n }$$ for all integers $n \geqslant 1$.
Show that $$\sum _ { r = n + 1 } ^ { 2 n } ( r + 1 ) 2 ^ { r - 1 } = n 2 ^ { n } \left( 2 ^ { n + 1 } - 1 \right)$$

AQA FP2 2006 January Q5

9 marks Standard +0.8

5 The complex number $z$ satisfies the relation $$| z + 4 - 4 i | = 4$$

Sketch, on an Argand diagram, the locus of $z$.
Show that the greatest value of $| z |$ is $4 ( \sqrt { 2 } + 1 )$.
Find the value of $z$ for which $$\arg ( z + 4 - 4 \mathrm { i } ) = \frac { 1 } { 6 } \pi$$ Give your answer in the form $a + \mathrm { i } b$.

AQA FP2 2006 January Q6

12 marks Challenging +1.2

6 It is given that $z = \mathrm { e } ^ { \mathrm { i } \theta }$.

1. Show that $$z + \frac { 1 } { z } = 2 \cos \theta$$ (2 marks)
2. Find a similar expression for $$z ^ { 2 } + \frac { 1 } { z ^ { 2 } }$$ (2 marks)
3. Hence show that $$z ^ { 2 } - z + 2 - \frac { 1 } { z } + \frac { 1 } { z ^ { 2 } } = 4 \cos ^ { 2 } \theta - 2 \cos \theta$$ (3 marks)
Hence solve the quartic equation $$z ^ { 4 } - z ^ { 3 } + 2 z ^ { 2 } - z + 1 = 0$$ giving the roots in the form $a + \mathrm { i } b$.

AQA FP2 2006 January Q7

17 marks Challenging +1.2

Use the definitions $$\sinh \theta = \frac { 1 } { 2 } \left( \mathrm { e } ^ { \theta } - \mathrm { e } ^ { - \theta } \right) \quad \text { and } \quad \cosh \theta = \frac { 1 } { 2 } \left( \mathrm { e } ^ { \theta } + \mathrm { e } ^ { - \theta } \right)$$ to show that:
1. $2 \sinh \theta \cosh \theta = \sinh 2 \theta$;
2. $\cosh ^ { 2 } \theta + \sinh ^ { 2 } \theta = \cosh 2 \theta$.
A curve is given parametrically by $$x = \cosh ^ { 3 } \theta , \quad y = \sinh ^ { 3 } \theta$$
1. Show that $$\left( \frac { \mathrm { d } x } { \mathrm {~d} \theta } \right) ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} \theta } \right) ^ { 2 } = \frac { 9 } { 4 } \sinh ^ { 2 } 2 \theta \cosh 2 \theta$$
2. Show that the length of the arc of the curve from the point where $\theta = 0$ to the point where $\theta = 1$ is $$\frac { 1 } { 2 } \left[ ( \cosh 2 ) ^ { \frac { 3 } { 2 } } - 1 \right]$$

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Questions — AQA (3548 questions)

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AQA FP1 2005 June Q1

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