Questions - AQA FP2 - A-Level Maths

Show that $s = \int _ { 2 } ^ { 9 } \sqrt { \frac { x } { x - 1 } } \mathrm {~d} x$.
1. Show that $\cosh ^ { - 1 } 3 = 2 \ln ( 1 + \sqrt { 2 } )$.
2. Use the substitution $x = \cosh ^ { 2 } \theta$ to show that $$s = m \sqrt { 2 } + \ln ( 1 + \sqrt { 2 } )$$ where $m$ is an integer.
  [0pt] [6 marks]
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AQA FP2 2016 June Q1

4 marks

Given that $\mathrm { f } ( r ) = \frac { 1 } { 4 r - 1 }$, show that $$\mathrm { f } ( r ) - \mathrm { f } ( r + 1 ) = \frac { A } { ( 4 r - 1 ) ( 4 r + 3 ) }$$ where $A$ is an integer.
Use the method of differences to find the value of $\sum _ { r = 1 } ^ { 50 } \frac { 1 } { ( 4 r - 1 ) ( 4 r + 3 ) }$, giving your answer as a fraction in its simplest form.
[0pt] [4 marks]

AQA FP2 2016 June Q2

2 The cubic equation $3 z ^ { 3 } + p z ^ { 2 } + 17 z + q = 0$, where $p$ and $q$ are real, has a root $\alpha = 1 + 2 \mathrm { i }$.

1. Write down the value of another non-real root, $\beta$, of this equation.
2. Hence find the value of $\alpha \beta$.
Find the value of the third root, $\gamma$, of this equation.
Find the values of $p$ and $q$.

AQA FP2 2016 June Q3

6 marks

3 The arc of the curve with equation $y = 4 - \ln \left( 1 - x ^ { 2 } \right)$ from $x = 0$ to $x = \frac { 3 } { 4 }$ has length $s$.

Show that $s = \int _ { 0 } ^ { \frac { 3 } { 4 } } \left( \frac { 1 + x ^ { 2 } } { 1 - x ^ { 2 } } \right) \mathrm { d } x$.
Find the value of $s$, giving your answer in the form $p + \ln N$, where $p$ is a rational number and $N$ is an integer.
[0pt] [6 marks]
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AQA FP2 2016 June Q4

4 marks

Given that $y = \tan ^ { - 1 } \sqrt { ( 3 x ) }$, find $\frac { \mathrm { d } y } { \mathrm {~d} x }$, giving your answer in terms of $x$.
Hence, or otherwise, show that $\int _ { \frac { 1 } { 3 } } ^ { 1 } \frac { 1 } { ( 1 + 3 x ) \sqrt { x } } \mathrm {~d} x = \frac { \sqrt { 3 } \pi } { n }$, where $n$ is an integer.
[0pt] [4 marks]

AQA FP2 2016 June Q5

5 marks

Find the modulus of the complex number $- 4 \sqrt { 3 } + 4 \mathrm { i }$, giving your answer as an integer.
The locus of points, $L$, satisfies the equation $| z + 4 \sqrt { 3 } - 4 \mathrm { i } | = 4$.
1. Sketch the locus $L$ on the Argand diagram below.
2. The complex number $w$ lies on $L$ so that $- \pi < \arg w \leqslant \pi$. Find the least possible value of arg $w$, giving your answer in terms of $\pi$.
Solve the equation $z ^ { 3 } = - 4 \sqrt { 3 } + 4 \mathrm { i }$, giving your answers in the form $r \mathrm { e } ^ { \mathrm { i } \theta }$, where $r > 0$ and $- \pi < \theta \leqslant \pi$.
[0pt] [5 marks]

AQA FP2 2016 June Q6

11 marks

Given that $y = \sinh x$, use the definition of $\sinh x$ in terms of $\mathrm { e } ^ { x }$ and $\mathrm { e } ^ { - x }$ to show that $x = \ln \left( y + \sqrt { y ^ { 2 } + 1 } \right)$.
A curve has equation $y = 6 \cosh ^ { 2 } x + 5 \sinh x$.
1. Show that the curve has a single stationary point and find its $x$-coordinate, giving your answer in the form $\ln p$, where $p$ is a rational number.
2. The curve lies entirely above the $x$-axis. The region bounded by the curve, the coordinate axes and the line $x = \cosh ^ { - 1 } 2$ has area $A$. Show that $$A = a \cosh ^ { - 1 } 2 + b \sqrt { 3 } + c$$ where $a$, $b$ and $c$ are integers.
  [0pt] [5 marks] $7 \quad$ Given that $p \geqslant - 1$, prove by induction that, for all integers $n \geqslant 1$, $$( 1 + p ) ^ { n } \geqslant 1 + n p$$ [6 marks]

AQA FP2 2016 June Q8

By applying de Moivre's theorem to $( \cos \theta + i \sin \theta ) ^ { 4 }$, where $\cos \theta \neq 0$, show that $$( 1 + i \tan \theta ) ^ { 4 } + ( 1 - i \tan \theta ) ^ { 4 } = \frac { 2 \cos 4 \theta } { \cos ^ { 4 } \theta }$$
Hence show that $z = \mathrm { i } \tan \frac { \pi } { 8 }$ satisfies the equation $( 1 + z ) ^ { 4 } + ( 1 - z ) ^ { 4 } = 0$, and express the three other roots of this equation in the form $\mathrm { i } \tan \phi$, where $0 < \phi < \pi$.
Use the results from part (b) to find the values of:
1. $\tan ^ { 2 } \frac { \pi } { 8 } \tan ^ { 2 } \frac { 3 \pi } { 8 }$;
2. $\tan ^ { 2 } \frac { \pi } { 8 } + \tan ^ { 2 } \frac { 3 \pi } { 8 }$.
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AQA FP2 2006 January Q1

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

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

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

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

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

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

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 $$\frac { 1 } { 2 } \left[ ( \cosh 2 ) ^ { \frac { 3 } { 2 } } - 1 \right]$$

AQA FP2 2007 January Q1

Given that $$4 \cosh ^ { 2 } x = 7 \sinh x + 1$$ find the two possible values of $\sinh x$.
Hence obtain the two possible values of $x$, giving your answers in the form $\ln p$.

AQA FP2 2007 January Q2

Sketch on one diagram:
1. the locus of points satisfying $| z - 4 + 2 \mathrm { i } | = 2$;
2. the locus of points satisfying $| z | = | z - 3 - 2 \mathrm { i } |$.
Shade on your sketch the region in which
both $$| z - 4 + 2 i | \leqslant 2$$ and $$| z | \leqslant | z - 3 - 2 \mathrm { i } |$$

AQA FP2 2007 January Q3

3 The cubic equation $$z ^ { 3 } + 2 ( 1 - \mathrm { i } ) z ^ { 2 } + 32 ( 1 + \mathrm { i } ) = 0$$ has roots $\alpha , \beta$ and $\gamma$.

It is given that $\alpha$ is of the form $k \mathrm { i }$, where $k$ is real. By substituting $z = k \mathrm { i }$ into the equation, show that $k = 4$.
Given that $\beta = - 4$, find the value of $\gamma$.

AQA FP2 2007 January Q4

Given that $y = \operatorname { sech } t$, show that:
1. $\frac { \mathrm { d } y } { \mathrm {~d} t } = - \operatorname { sech } t \tanh t$;
2. $\left( \frac { \mathrm { d } y } { \mathrm {~d} t } \right) ^ { 2 } = \operatorname { sech } ^ { 2 } t - \operatorname { sech } ^ { 4 } t$.
The diagram shows a sketch of part of the curve given parametrically by $$x = t - \tanh t \quad y = \operatorname { sech } t$$
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The curve meets the $y$-axis at the point $K$, and $P ( x , y )$ is a general point on the curve. The arc length $K P$ is denoted by $s$. Show that:
1. $\left( \frac { \mathrm { d } x } { \mathrm {~d} t } \right) ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} t } \right) ^ { 2 } = \tanh ^ { 2 } t$;
2. $s = \ln \cosh t$;
3. $y = \mathrm { e } ^ { - s }$.
The arc $K P$ is rotated through $2 \pi$ radians about the $x$-axis. Show that the surface area generated is $$2 \pi \left( 1 - \mathrm { e } ^ { - S } \right)$$ (4 marks)

AQA FP2 2007 January Q5

Prove by induction that, if $n$ is a positive integer, $$( \cos \theta + \mathrm { i } \sin \theta ) ^ { n } = \cos n \theta + \mathrm { i } \sin n \theta$$
Find the value of $\left( \cos \frac { \pi } { 6 } + \mathrm { i } \sin \frac { \pi } { 6 } \right) ^ { 6 }$.
Show that $$( \cos \theta + \mathrm { i } \sin \theta ) ( 1 + \cos \theta - \mathrm { i } \sin \theta ) = 1 + \cos \theta + \mathrm { i } \sin \theta$$
Hence show that $$\left( 1 + \cos \frac { \pi } { 6 } + i \sin \frac { \pi } { 6 } \right) ^ { 6 } + \left( 1 + \cos \frac { \pi } { 6 } - i \sin \frac { \pi } { 6 } \right) ^ { 6 } = 0$$

AQA FP2 2007 January Q6

Find the three roots of $z ^ { 3 } = 1$, giving the non-real roots in the form $\mathrm { e } ^ { \mathrm { i } \theta }$, where $- \pi < \theta \leqslant \pi$.
Given that $\omega$ is one of the non-real roots of $z ^ { 3 } = 1$, show that $$1 + \omega + \omega ^ { 2 } = 0$$
By using the result in part (b), or otherwise, show that:
1. $\frac { \omega } { \omega + 1 } = - \frac { 1 } { \omega }$;
2. $\frac { \omega ^ { 2 } } { \omega ^ { 2 } + 1 } = - \omega$;
3. $\left( \frac { \omega } { \omega + 1 } \right) ^ { k } + \left( \frac { \omega ^ { 2 } } { \omega ^ { 2 } + 1 } \right) ^ { k } = ( - 1 ) ^ { k } 2 \cos \frac { 2 } { 3 } k \pi$, where $k$ is an integer.

AQA FP2 2007 January Q7

Use the identity $\tan ( A - B ) = \frac { \tan A - \tan B } { 1 + \tan A \tan B }$ with $A = ( r + 1 ) x$ and $B = r x$ to show that $$\tan r x \tan ( r + 1 ) x = \frac { \tan ( r + 1 ) x } { \tan x } - \frac { \tan r x } { \tan x } - 1$$ (4 marks)
Use the method of differences to show that $$\tan \frac { \pi } { 50 } \tan \frac { 2 \pi } { 50 } + \tan \frac { 2 \pi } { 50 } \tan \frac { 3 \pi } { 50 } + \ldots + \tan \frac { 19 \pi } { 50 } \tan \frac { 20 \pi } { 50 } = \frac { \tan \frac { 2 \pi } { 5 } } { \tan \frac { \pi } { 50 } } - 20$$

AQA FP2 2008 January Q1

Express $4 + 4 \mathrm { i }$ in the form $r \mathrm { e } ^ { \mathrm { i } \theta }$, where $r > 0$ and $- \pi < \theta \leqslant \pi$.
Solve the equation $$z ^ { 5 } = 4 + 4 i$$ giving your answers in the form $r \mathrm { e } ^ { \mathrm { i } \theta }$, where $r > 0$ and $- \pi < \theta \leqslant \pi$.

AQA FP2 2008 January Q2

Show that $$( 2 r + 1 ) ^ { 3 } - ( 2 r - 1 ) ^ { 3 } = 24 r ^ { 2 } + 2$$
Hence, using the method of differences, show that $$\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )$$

AQA FP2 2008 January Q3

3 A circle $C$ and a half-line $L$ have equations $$| z - 2 \sqrt { 3 } - \mathrm { i } | = 4$$ and $$\arg ( z + i ) = \frac { \pi } { 6 }$$ respectively.

Show that:
1. the circle $C$ passes through the point where $z = - \mathrm { i }$;
2. the half-line $L$ passes through the centre of $C$.
On one Argand diagram, sketch $C$ and $L$.
Shade on your sketch the set of points satisfying both $$| z - 2 \sqrt { 3 } - \mathrm { i } | \leqslant 4$$ and $$0 \leqslant \arg ( z + i ) \leqslant \frac { \pi } { 6 }$$

Questions — AQA FP2 (142 questions)

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