Questions - AQA FP2 - A-Level Maths

Write down the value of:
1. $\alpha + \beta + \gamma$;
2. $\alpha \beta + \beta \gamma + \gamma \alpha$;
3. $\alpha \beta \gamma$.
Find the value of:
1. $\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }$;
2. $\alpha ^ { 2 } \beta ^ { 2 } + \beta ^ { 2 } \gamma ^ { 2 } + \gamma ^ { 2 } \alpha ^ { 2 }$;
3. $\alpha ^ { 2 } \beta ^ { 2 } \gamma ^ { 2 }$.
Hence write down a cubic equation whose roots are $\alpha ^ { 2 } , \beta ^ { 2 }$ and $\gamma ^ { 2 }$.

AQA FP2 2008 January Q5

5 Prove by induction that for all integers $n \geqslant 1$ $$\sum _ { r = 1 } ^ { n } \left( r ^ { 2 } + 1 \right) ( r ! ) = n ( n + 1 ) !$$

AQA FP2 2008 January Q6

1. By applying De Moivre's theorem to $( \cos \theta + \mathrm { i } \sin \theta ) ^ { 3 }$, show that $$\cos 3 \theta = \cos ^ { 3 } \theta - 3 \cos \theta \sin ^ { 2 } \theta$$
2. Find a similar expression for $\sin 3 \theta$.
3. Deduce that $$\tan 3 \theta = \frac { \tan ^ { 3 } \theta - 3 \tan \theta } { 3 \tan ^ { 2 } \theta - 1 }$$
1. Hence show that $\tan \frac { \pi } { 12 }$ is a root of the cubic equation $$x ^ { 3 } - 3 x ^ { 2 } - 3 x + 1 = 0$$
2. Find two other values of $\theta$, where $0 < \theta < \pi$, for which $\tan \theta$ is a root of this cubic equation.
Hence show that $$\tan \frac { \pi } { 12 } + \tan \frac { 5 \pi } { 12 } = 4$$

AQA FP2 2008 January Q7

Given that $y = \ln \tanh \frac { x } { 2 }$, where $x > 0$, show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \operatorname { cosech } x$$
A curve has equation $y = \ln \tanh \frac { x } { 2 }$, where $x > 0$. The length of the arc of the curve between the points where $x = 1$ and $x = 2$ is denoted by $s$.
1. Show that $$s = \int _ { 1 } ^ { 2 } \operatorname { coth } x \mathrm {~d} x$$
2. Hence show that $s = \ln ( 2 \cosh 1 )$.

AQA FP2 2009 January Q1

Use the definitions $\sinh \theta = \frac { 1 } { 2 } \left( \mathrm { e } ^ { \theta } - \mathrm { e } ^ { - \theta } \right)$ and $\cosh \theta = \frac { 1 } { 2 } \left( \mathrm { e } ^ { \theta } + \mathrm { e } ^ { - \theta } \right)$ to show that $$1 + 2 \sinh ^ { 2 } \theta = \cosh 2 \theta$$
Solve the equation $$3 \cosh 2 \theta = 2 \sinh \theta + 11$$ giving each of your answers in the form $\ln p$.

AQA FP2 2009 January Q2

Indicate on an Argand diagram the region for which $| z - 4 \mathrm { i } | \leqslant 2$.
The complex number $z$ satisfies $| z - 4 \mathrm { i } | \leqslant 2$. Find the range of possible values of $\arg z$.

AQA FP2 2009 January Q3

Given that $\mathrm { f } ( r ) = \frac { 1 } { 4 } r ^ { 2 } ( r + 1 ) ^ { 2 }$, show that $$\mathrm { f } ( r ) - \mathrm { f } ( r - 1 ) = r ^ { 3 }$$
Use the method of differences to show that $$\sum _ { r = n } ^ { 2 n } r ^ { 3 } = \frac { 3 } { 4 } n ^ { 2 } ( n + 1 ) ( 5 n + 1 )$$

AQA FP2 2009 January Q4

4 It is given that $\alpha , \beta$ and $\gamma$ satisfy the equations $$\begin{aligned} & \alpha + \beta + \gamma = 1
& \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = - 5
& \alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } = - 23 \end{aligned}$$

Show that $\alpha \beta + \beta \gamma + \gamma \alpha = 3$.
Use the identity $$( \alpha + \beta + \gamma ) \left( \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } - \alpha \beta - \beta \gamma - \gamma \alpha \right) = \alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } - 3 \alpha \beta \gamma$$ to find the value of $\alpha \beta \gamma$.
Write down a cubic equation, with integer coefficients, whose roots are $\alpha , \beta$ and $\gamma$.
Explain why this cubic equation has two non-real roots.
Given that $\alpha$ is real, find the values of $\alpha , \beta$ and $\gamma$.

AQA FP2 2009 January Q5

Given that $u = \cosh ^ { 2 } x$, show that $\frac { \mathrm { d } u } { \mathrm {~d} x } = \sinh 2 x$.
Hence show that $$\int _ { 0 } ^ { 1 } \frac { \sinh 2 x } { 1 + \cosh ^ { 4 } x } \mathrm {~d} x = \tan ^ { - 1 } \left( \cosh ^ { 2 } 1 \right) - \frac { \pi } { 4 }$$

AQA FP2 2009 January Q6

6 Prove by induction that $$\frac { 2 \times 1 } { 2 \times 3 } + \frac { 2 ^ { 2 } \times 2 } { 3 \times 4 } + \frac { 2 ^ { 3 } \times 3 } { 4 \times 5 } + \ldots + \frac { 2 ^ { n } \times n } { ( n + 1 ) ( n + 2 ) } = \frac { 2 ^ { n + 1 } } { n + 2 } - 1$$ for all integers $n \geqslant 1$.

AQA FP2 2009 January Q7

Show that $$\frac { \mathrm { d } } { \mathrm {~d} x } \left( \cosh ^ { - 1 } \frac { 1 } { x } \right) = \frac { - 1 } { x \sqrt { 1 - x ^ { 2 } } }$$ (3 marks)
A curve has equation $$y = \sqrt { 1 - x ^ { 2 } } - \cosh ^ { - 1 } \frac { 1 } { x } \quad ( 0 < x < 1 )$$ Show that:
1. $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { \sqrt { 1 - x ^ { 2 } } } { x }$;
  (4 marks)
2. the length of the arc of the curve from the point where $x = \frac { 1 } { 4 }$ to the point where $$x = \frac { 3 } { 4 } \text { is } \ln 3 .$$ (5 marks)

AQA FP2 2009 January Q8

Show that $$\left( z ^ { 4 } - \mathrm { e } ^ { \mathrm { i } \theta } \right) \left( z ^ { 4 } - \mathrm { e } ^ { - \mathrm { i } \theta } \right) = z ^ { 8 } - 2 z ^ { 4 } \cos \theta + 1$$ (2 marks)
Hence solve the equation $$z ^ { 8 } - z ^ { 4 } + 1 = 0$$ giving your answers in the form $\mathrm { e } ^ { \mathrm { i } \phi }$, where $- \pi < \phi \leqslant \pi$.
Indicate the roots on an Argand diagram.

AQA FP2 2006 June Q1

Given that $$\frac { r ^ { 2 } + r - 1 } { r ( r + 1 ) } = A + B \left( \frac { 1 } { r } - \frac { 1 } { r + 1 } \right)$$ find the values of $A$ and $B$.
Hence find the value of $$\sum _ { r = 1 } ^ { 99 } \frac { r ^ { 2 } + r - 1 } { r ( r + 1 ) }$$

AQA FP2 2006 June Q2

2 A curve has parametric equations $$x = t - \frac { 1 } { 3 } t ^ { 3 } , \quad y = t ^ { 2 }$$

Show that $$\left( \frac { \mathrm { d } x } { \mathrm {~d} t } \right) ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} t } \right) ^ { 2 } = \left( 1 + t ^ { 2 } \right) ^ { 2 }$$
The arc of the curve between $t = 1$ and $t = 2$ is rotated through $2 \pi$ radians about the $x$-axis. Show that $S$, the surface area generated, is given by $S = k \pi$, where $k$ is a rational number to be found.

AQA FP2 2006 June Q3

3 The curve $C$ has equation $$y = \cosh x - 3 \sinh x$$

1. The line $y = - 1$ meets $C$ at the point $( k , - 1 )$. Show that $$\mathrm { e } ^ { 2 k } - \mathrm { e } ^ { k } - 2 = 0$$
2. Hence find $k$, giving your answer in the form $\ln a$.
1. Find the $x$-coordinate of the point where the curve $C$ intersects the $x$-axis, giving your answer in the form $p \ln a$.
2. Show that $C$ has no stationary points.
3. Show that there is exactly one point on $C$ for which $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 0$.

AQA FP2 2006 June Q4

On one Argand diagram, sketch the locus of points satisfying:
1. $| z - 3 + 2 \mathrm { i } | = 4$;
2. $\quad \arg ( z - 1 ) = - \frac { 1 } { 4 } \pi$.
Indicate on your sketch the set of points satisfying both $$| z - 3 + 2 i | \leqslant 4$$ and $$\arg ( z - 1 ) = - \frac { 1 } { 4 } \pi$$ (1 mark)

AQA FP2 2006 June Q5

5 The cubic equation $$z ^ { 3 } - 4 \mathrm { i } z ^ { 2 } + q z - ( 4 - 2 \mathrm { i } ) = 0$$ 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 $\alpha = \beta + \gamma$, show that:
1. $\alpha = 2 \mathrm { i }$;
2. $\quad \beta \gamma = - ( 1 + 2 \mathrm { i } )$;
3. $\quad q = - ( 5 + 2 \mathrm { i } )$.
Show that $\beta$ and $\gamma$ are the roots of the equation $$z ^ { 2 } - 2 \mathrm { i } z - ( 1 + 2 \mathrm { i } ) = 0$$
Given that $\beta$ is real, find $\beta$ and $\gamma$.

AQA FP2 2006 June Q6

The function f is given by $$\mathrm { f } ( n ) = 15 ^ { n } - 8 ^ { n - 2 }$$ Express $$\mathrm { f } ( n + 1 ) - 8 \mathrm { f } ( n )$$ in the form $k \times 15 ^ { n }$.
Prove by induction that $15 ^ { n } - 8 ^ { n - 2 }$ is a multiple of 7 for all integers $n \geqslant 2$.

AQA FP2 2006 June Q7

Find the six roots of the equation $z ^ { 6 } = 1$, giving your answers in the form $\mathrm { e } ^ { \mathrm { i } \phi }$, where $- \pi < \phi \leqslant \pi$.
It is given that $w = \mathrm { e } ^ { \mathrm { i } \theta }$, where $\theta \neq n \pi$.
1. Show that $\frac { w ^ { 2 } - 1 } { w } = 2 \mathrm { i } \sin \theta$.
2. Show that $\frac { w } { w ^ { 2 } - 1 } = - \frac { \mathrm { i } } { 2 \sin \theta }$.
3. Show that $\frac { 2 \mathrm { i } } { w ^ { 2 } - 1 } = \cot \theta - \mathrm { i }$.
4. Given that $z = \cot \theta - \mathrm { i }$, show that $z + 2 \mathrm { i } = z w ^ { 2 }$.
1. Explain why the equation $$( z + 2 \mathrm { i } ) ^ { 6 } = z ^ { 6 }$$ has five roots.
2. Find the five roots of the equation $$( z + 2 \mathrm { i } ) ^ { 6 } = z ^ { 6 }$$ giving your answers in the form $a + \mathrm { i } b$.

AQA FP2 2007 June Q1

Given that $\mathrm { f } ( r ) = ( r - 1 ) r ^ { 2 }$, show that $$\mathrm { f } ( r + 1 ) - \mathrm { f } ( r ) = r ( 3 r + 1 )$$
Use the method of differences to find the value of $$\sum _ { r = 50 } ^ { 99 } r ( 3 r + 1 )$$ (4 marks)

AQA FP2 2007 June Q2

2 The cubic equation $$z ^ { 3 } + p z ^ { 2 } + 6 z + q = 0$$ has roots $\alpha , \beta$ and $\gamma$.

Write down the value of $\alpha \beta + \beta \gamma + \gamma \alpha$.
Given that $p$ and $q$ are real and that $\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = - 12$ :
1. explain why the cubic equation has two non-real roots and one real root;
2. find the value of $p$.
One root of the cubic equation is $- 1 + 3 \mathrm { i }$. Find:
1. the other two roots;
2. the value of $q$.

AQA FP2 2007 June Q3

3 Use De Moivre's Theorem to find the smallest positive angle $\theta$ for which $$( \cos \theta + \mathrm { i } \sin \theta ) ^ { 15 } = - \mathrm { i }$$ (5 marks)

AQA FP2 2007 June Q4

Differentiate $x \tan ^ { - 1 } x$ with respect to $x$.
Show that $$\int _ { 0 } ^ { 1 } \tan ^ { - 1 } x \mathrm {~d} x = \frac { \pi } { 4 } - \ln \sqrt { 2 }$$ (5 marks)

AQA FP2 2007 June Q5

5 The sketch shows an Argand diagram. The points $A$ and $B$ represent the complex numbers $z _ { 1 }$ and $z _ { 2 }$ respectively. The angle $A O B = 90 ^ { \circ }$ and $O A = O B$.
\includegraphics[max width=\textwidth, alt={}, center]{847295e3-d806-43b1-8d25-688c5558bfe1-3_533_869_852_632}

Explain why $z _ { 2 } = \mathrm { i } z _ { 1 }$.
On a single copy of the diagram, draw:
1. the locus $L _ { 1 }$ of points satisfying $\left| z - z _ { 2 } \right| = \left| z - z _ { 1 } \right|$;
2. the locus $L _ { 2 }$ of points satisfying $\arg \left( z - z _ { 2 } \right) = \arg z _ { 1 }$.
Find, in terms of $z _ { 1 }$, the complex number representing the point of intersection of $L _ { 1 }$ and $L _ { 2 }$.

AQA FP2 2007 June Q6

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

Questions — AQA FP2 (142 questions)

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