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AQA FP2 2013 January Q6

8 marks Standard +0.8

6 A curve is defined parametrically by $$x = t ^ { 3 } + 5 , \quad y = 6 t ^ { 2 } - 1$$ The arc length between the points where $t = 0$ and $t = 3$ on the curve is $s$.

Show that $s = \int _ { 0 } ^ { 3 } 3 t \sqrt { t ^ { 2 } + A } \mathrm {~d} t$, stating the value of the constant $A$.
Hence show that $s = 61$. $7 \quad$ The polynomial $\mathrm { p } ( n )$ is given by $\mathrm { p } ( n ) = ( n - 1 ) ^ { 3 } + n ^ { 3 } + ( n + 1 ) ^ { 3 }$.
1. Show that $\mathrm { p } ( k + 1 ) - \mathrm { p } ( k )$, where $k$ is a positive integer, is a multiple of 9 .
2. Prove by induction that $\mathrm { p } ( n )$ is a multiple of 9 for all integers $n \geqslant 1$.
Using the result from part (a)(ii), show that $n \left( n ^ { 2 } + 2 \right)$ is a multiple of 3 for any positive integer $n$.

AQA FP2 2013 January Q8

14 marks Challenging +1.2

Express $- 4 + 4 \sqrt { 3 } \mathrm { i }$ in the form $r \mathrm { e } ^ { \mathrm { i } \theta }$, where $r > 0$ and $- \pi < \theta \leqslant \pi$.
1. Solve the equation $z ^ { 3 } = - 4 + 4 \sqrt { 3 } \mathrm { i }$, giving your answers in the form $r \mathrm { e } ^ { \mathrm { i } \theta }$, where $r > 0$ and $- \pi < \theta \leqslant \pi$.
2. The roots of the equation $z ^ { 3 } = - 4 + 4 \sqrt { 3 } \mathrm { i }$ are represented by the points $P , Q$ and $R$ on an Argand diagram. Find the area of the triangle $P Q R$, giving your answer in the form $k \sqrt { 3 }$, where $k$ is an integer.
By considering the roots of the equation $z ^ { 3 } = - 4 + 4 \sqrt { 3 } \mathrm { i }$, show that $$\cos \frac { 2 \pi } { 9 } + \cos \frac { 4 \pi } { 9 } + \cos \frac { 8 \pi } { 9 } = 0$$

AQA FP2 2008 June Q1

6 marks Standard +0.3

Express $$5 \sinh x + \cosh x$$ in the form $A \mathrm { e } ^ { x } + B \mathrm { e } ^ { - x }$, where $A$ and $B$ are integers.
Solve the equation $$5 \sinh x + \cosh x + 5 = 0$$ giving your answer in the form $\ln a$, where $a$ is a rational number.

AQA FP2 2008 June Q2

7 marks Standard +0.8

Given that $$\frac { 1 } { r ( r + 1 ) ( r + 2 ) } = \frac { A } { r ( r + 1 ) } + \frac { B } { ( r + 1 ) ( r + 2 ) }$$ show that $A = \frac { 1 } { 2 }$ and find the value of $B$.
Use the method of differences to find $$\sum _ { r = 10 } ^ { 98 } \frac { 1 } { r ( r + 1 ) ( r + 2 ) }$$ giving your answer as a rational number.

AQA FP2 2008 June Q3

12 marks Standard +0.8

3 The cubic equation $$z ^ { 3 } + q z + ( 18 - 12 i ) = 0$$ where $q$ is a complex number, has roots $\alpha , \beta$ and $\gamma$.

Write down the value of:
1. $\alpha \beta \gamma$;
2. $\alpha + \beta + \gamma$.
Given that $\beta + \gamma = 2$, find the value of:
1. $\alpha$;
2. $\quad \beta \gamma$;
3. $q$.
Given that $\beta$ is of the form $k \mathrm { i }$, where $k$ is real, find $\beta$ and $\gamma$.

AQA FP2 2008 June Q4

12 marks Standard +0.3

A circle $C$ in the Argand diagram has equation $$| z + 5 - \mathrm { i } | = \sqrt { 2 }$$ Write down its radius and the complex number representing its centre.
A half-line $L$ in the Argand diagram has equation $$\arg ( z + 2 \mathrm { i } ) = \frac { 3 \pi } { 4 }$$ Show that $z _ { 1 } = - 4 + 2 \mathrm { i }$ lies on $L$.
1. Show that $z _ { 1 } = - 4 + 2 \mathrm { i }$ also lies on $C$.
2. Hence show that $L$ touches $C$.
3. Sketch $L$ and $C$ on one Argand diagram.
The complex number $z _ { 2 }$ lies on $C$ and is such that $\arg \left( z _ { 2 } + 2 \mathrm { i } \right)$ has as great a value as possible. Indicate the position of $z _ { 2 }$ on your sketch.

AQA FP2 2008 June Q5

10 marks Challenging +1.3

Use the definition $\cosh x = \frac { 1 } { 2 } \left( \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } \right)$ to show that $\cosh 2 x = 2 \cosh ^ { 2 } x - 1$.
(2 marks)
1. The arc of the curve $y = \cosh x$ between $x = 0$ and $x = \ln a$ is rotated through $2 \pi$ radians about the $x$-axis. Show that $S$, the surface area generated, is given by $$S = 2 \pi \int _ { 0 } ^ { \ln a } \cosh ^ { 2 } x \mathrm {~d} x$$
2. Hence show that $$S = \pi \left( \ln a + \frac { a ^ { 4 } - 1 } { 4 a ^ { 2 } } \right)$$

AQA FP2 2008 June Q6

5 marks Standard +0.8

6 By using the substitution $u = x - 2$, or otherwise, find the exact value of $$\int _ { - 1 } ^ { 5 } \frac { \mathrm {~d} x } { \sqrt { 32 + 4 x - x ^ { 2 } } }$$

AQA FP2 2008 June Q7

9 marks Standard +0.8

Explain why $n ( n + 1 )$ is a multiple of 2 when $n$ is an integer.
1. Given that $$\mathrm { f } ( n ) = n \left( n ^ { 2 } + 5 \right)$$ show that $\mathrm { f } ( k + 1 ) - \mathrm { f } ( k )$, where $k$ is a positive integer, is a multiple of 6 .
2. Prove by induction that $\mathrm { f } ( n )$ is a multiple of 6 for all integers $n \geqslant 1$.

AQA FP2 2008 June Q8

14 marks Challenging +1.2

1. Expand $$\left( z + \frac { 1 } { z } \right) \left( z - \frac { 1 } { z } \right)$$
2. Hence, or otherwise, expand $$\left( z + \frac { 1 } { z } \right) ^ { 4 } \left( z - \frac { 1 } { z } \right) ^ { 2 }$$
1. Use De Moivre's theorem to show that if $z = \cos \theta + \mathrm { i } \sin \theta$ then $$z ^ { n } + \frac { 1 } { z ^ { n } } = 2 \cos n \theta$$
2. Write down a corresponding result for $z ^ { n } - \frac { 1 } { z ^ { n } }$.
Hence express $\cos ^ { 4 } \theta \sin ^ { 2 } \theta$ in the form $$A \cos 6 \theta + B \cos 4 \theta + C \cos 2 \theta + D$$ where $A , B , C$ and $D$ are rational numbers.
Find $\int \cos ^ { 4 } \theta \sin ^ { 2 } \theta d \theta$.

AQA FP2 2010 June Q1

9 marks Standard +0.3

Show that $$9 \sinh x - \cosh x = 4 \mathrm { e } ^ { x } - 5 \mathrm { e } ^ { - x }$$
Given that $$9 \sinh x - \cosh x = 8$$ find the exact value of $\tanh x$.

AQA FP2 2010 June Q2

8 marks Standard +0.3

Express $\frac { 1 } { r ( r + 2 ) }$ in partial fractions.
Use the method of differences to find $$\sum _ { r = 1 } ^ { 48 } \frac { 1 } { r ( r + 2 ) }$$ giving your answer as a rational number.

AQA FP2 2010 June Q3

9 marks Standard +0.3

3 Two loci, $L _ { 1 }$ and $L _ { 2 }$, in an Argand diagram are given by $$\begin{aligned} & L _ { 1 } : | z + 1 + 3 \mathrm { i } | = | z - 5 - 7 \mathrm { i } | \\ & L _ { 2 } : \arg z = \frac { \pi } { 4 } \end{aligned}$$

Verify that the point represented by the complex number $2 + 2 \mathrm { i }$ is a point of intersection of $L _ { 1 }$ and $L _ { 2 }$.
Sketch $L _ { 1 }$ and $L _ { 2 }$ on one Argand diagram.
Shade on your Argand diagram the region satisfying
both $$| z + 1 + 3 i | \leqslant | z - 5 - 7 i |$$ and $$\frac { \pi } { 4 } \leqslant \arg z \leqslant \frac { \pi } { 2 }$$

AQA FP2 2010 June Q4

13 marks Standard +0.8

4 The roots of the cubic equation $$z ^ { 3 } - 2 z ^ { 2 } + p z + 10 = 0$$ are $\alpha , \beta$ and $\gamma$.
It is given that $\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } = - 4$.

Write down the value of $\alpha + \beta + \gamma$.
1. Explain why $\alpha ^ { 3 } - 2 \alpha ^ { 2 } + p \alpha + 10 = 0$.
2. Hence show that $$\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = p + 13$$
3. Deduce that $p = - 3$.
1. Find the real root $\alpha$ of the cubic equation $z ^ { 3 } - 2 z ^ { 2 } - 3 z + 10 = 0$.
2. Find the values of $\beta$ and $\gamma$.

AQA FP2 2010 June Q5

18 marks Standard +0.8

Using the identities $$\cosh ^ { 2 } t - \sinh ^ { 2 } t = 1 , \quad \tanh t = \frac { \sinh t } { \cosh t } \quad \text { and } \quad \operatorname { sech } t = \frac { 1 } { \cosh t }$$ show that:
1. $\tanh ^ { 2 } t + \operatorname { sech } ^ { 2 } t = 1$;
2. $\frac { \mathrm { d } } { \mathrm { d } t } ( \tanh t ) = \operatorname { sech } ^ { 2 } t$;
3. $\frac { \mathrm { d } } { \mathrm { d } t } ( \operatorname { sech } t ) = - \operatorname { sech } t \tanh t$.
A curve $C$ is given parametrically by $$x = \operatorname { sech } t , y = 4 - \tanh t$$
1. Show that the arc length, $s$, of $C$ between the points where $t = 0$ and $t = \frac { 1 } { 2 } \ln 3$ is given by $$s = \int _ { 0 } ^ { \frac { 1 } { 2 } \ln 3 } \operatorname { sech } t \mathrm {~d} t$$
2. Using the substitution $u = \mathrm { e } ^ { t }$, find the exact value of $s$.

AQA FP2 2010 June Q6

8 marks Challenging +1.2

Show that $\frac { 1 } { ( k + 2 ) ! } - \frac { k + 1 } { ( k + 3 ) ! } = \frac { 2 } { ( k + 3 ) ! }$.
Prove by induction that, for all positive integers $n$, $$\sum _ { r = 1 } ^ { n } \frac { r \times 2 ^ { r } } { ( r + 2 ) ! } = 1 - \frac { 2 ^ { n + 1 } } { ( n + 2 ) ! }$$ (6 marks)
\includegraphics[max width=\textwidth, alt={}]{d417bc62-f92a-4c90-a4c2-435f38e46edc-7_2010_1711_693_152}

AQA FP2 2010 June Q7

10 marks Standard +0.3

1. Express each of the numbers $1 + \sqrt { 3 } \mathrm { i }$ and $1 - \mathrm { i }$ in the form $r \mathrm { e } ^ { \mathrm { i } \theta }$, where $r > 0$.
2. Hence express $$( 1 + \sqrt { 3 } i ) ^ { 8 } ( 1 - i ) ^ { 5 }$$ in the form $r \mathrm { e } ^ { \mathrm { i } \theta }$, where $r > 0$.
Solve the equation $$z ^ { 3 } = ( 1 + \sqrt { 3 } \mathrm { i } ) ^ { 8 } ( 1 - \mathrm { i } ) ^ { 5 }$$ giving your answers in the form $a \sqrt { 2 } \mathrm { e } ^ { \mathrm { i } \theta }$, where $a$ is a positive integer and $- \pi < \theta \leqslant \pi$.

AQA FP2 2011 June Q1

8 marks Moderate -0.3

Draw on the same Argand diagram:
1. the locus of points for which $$| z - 2 - 5 \mathrm { i } | = 5$$
2. the locus of points for which $$\arg ( z + 2 i ) = \frac { \pi } { 4 }$$
Indicate on your diagram the set of points satisfying both $$| z - 2 - 5 i | \leqslant 5$$ and $$\arg ( z + 2 \mathrm { i } ) = \frac { \pi } { 4 }$$ (2 marks)

AQA FP2 2011 June Q2

10 marks Moderate -0.3

Use the definitions of $\cosh \theta$ and $\sinh \theta$ in terms of $\mathrm { e } ^ { \theta }$ to show that $$\cosh x \cosh y - \sinh x \sinh y = \cosh ( x - y )$$
It is given that $x$ satisfies the equation $$\cosh ( x - \ln 2 ) = \sinh x$$
1. Show that $\tanh x = \frac { 5 } { 7 }$.
2. Express $x$ in the form $\frac { 1 } { 2 } \ln a$.

AQA FP2 2011 June Q3

6 marks Challenging +1.2

Show that $$( r + 1 ) ! - ( r - 1 ) ! = \left( r ^ { 2 } + r - 1 \right) ( r - 1 ) !$$
Hence show that $$\sum _ { r = 1 } ^ { n } \left( r ^ { 2 } + r - 1 \right) ( r - 1 ) ! = ( n + 2 ) n ! - 2$$ (4 marks) The cubic equation $$z ^ { 3 } - 2 z ^ { 2 } + k = 0 \quad ( k \neq 0 )$$ has roots $\alpha , \beta$ and $\gamma$.
1. Write down the values of $\alpha + \beta + \gamma$ and $\alpha \beta + \beta \gamma + \gamma \alpha$.
2. Show that $\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 4$.
3. Explain why $\alpha ^ { 3 } - 2 \alpha ^ { 2 } + k = 0$.
4. Show that $\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } = 8 - 3 k$.
Given that $\alpha ^ { 4 } + \beta ^ { 4 } + \gamma ^ { 4 } = 0$ :
1. show that $k = 2$;
2. find the value of $\alpha ^ { 5 } + \beta ^ { 5 } + \gamma ^ { 5 }$.

AQA FP2 2011 June Q5

13 marks Challenging +1.2

The arc of the curve $y ^ { 2 } = x ^ { 2 } + 8$ between the points where $x = 0$ and $x = 6$ is rotated through $2 \pi$ radians about the $x$-axis. Show that the area $S$ of the curved surface formed is given by $$S = 2 \sqrt { 2 } \pi \int _ { 0 } ^ { 6 } \sqrt { x ^ { 2 } + 4 } \mathrm {~d} x$$
By means of the substitution $x = 2 \sinh \theta$, show that $$S = \pi \left( 24 \sqrt { 5 } + 4 \sqrt { 2 } \sinh ^ { - 1 } 3 \right)$$

AQA FP2 2011 June Q6

8 marks Standard +0.3

Show that $$( k + 1 ) \left( 4 ( k + 1 ) ^ { 2 } - 1 \right) = 4 k ^ { 3 } + 12 k ^ { 2 } + 11 k + 3$$
Prove by induction that, for all integers $n \geqslant 1$, $$1 ^ { 2 } + 3 ^ { 2 } + 5 ^ { 2 } + \ldots + ( 2 n - 1 ) ^ { 2 } = \frac { 1 } { 3 } n \left( 4 n ^ { 2 } - 1 \right)$$

AQA FP2 2011 June Q7

16 marks Challenging +1.2

1. Use de Moivre's Theorem to show that $$\cos 5 \theta = \cos ^ { 5 } \theta - 10 \cos ^ { 3 } \theta \sin ^ { 2 } \theta + 5 \cos \theta \sin ^ { 4 } \theta$$ and find a similar expression for $\sin 5 \theta$.
2. Deduce that $$\tan 5 \theta = \frac { \tan \theta \left( 5 - 10 \tan ^ { 2 } \theta + \tan ^ { 4 } \theta \right) } { 1 - 10 \tan ^ { 2 } \theta + 5 \tan ^ { 4 } \theta }$$
Explain why $t = \tan \frac { \pi } { 5 }$ is a root of the equation $$t ^ { 4 } - 10 t ^ { 2 } + 5 = 0$$ and write down the three other roots of this equation in trigonometrical form.
(3 marks)
Deduce that $$\tan \frac { \pi } { 5 } \tan \frac { 2 \pi } { 5 } = \sqrt { 5 }$$

AQA FP2 2012 June Q1

7 marks Standard +0.3

Sketch the curve $y = \cosh x$.
Solve the equation $$6 \cosh ^ { 2 } x - 7 \cosh x - 5 = 0$$ giving your answers in logarithmic form.

AQA FP2 2012 June Q2

7 marks Standard +0.3

Draw on the Argand diagram below:
1. the locus of points for which $$| z - 2 - 3 \mathrm { i } | = 2$$
2. the locus of points for which $$| z + 2 - \mathrm { i } | = | z - 2 |$$
Indicate on your diagram the points satisfying both $$| z - 2 - 3 \mathrm { i } | = 2$$ and $$| z + 2 - \mathrm { i } | \leqslant | z - 2 |$$ (l mark) \includegraphics[max width=\textwidth, alt={}, center]{ff63460d-0fa1-437d-bc08-3e7ce809e32b-3_1404_1431_1043_319}

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