Questions - AQA FP2 - A-Level Maths

Show that the length of arc $s$ of the curve between the points where $x = 0$ and $x = 1$ is given by $$s = \int _ { 0 } ^ { 1 } \sqrt { \frac { x + 4 } { x } } \mathrm {~d} x$$
1. Use the substitution $x = 4 \sinh ^ { 2 } \theta$ to show that $$\int \sqrt { \frac { x + 4 } { x } } \mathrm {~d} x = \int 8 \cosh ^ { 2 } \theta \mathrm {~d} \theta$$
2. Hence show that $$s = 4 \sinh ^ { - 1 } 0.5 + \sqrt { 5 }$$

AQA FP2 2007 June Q8

1. Given that $z ^ { 6 } - 4 z ^ { 3 } + 8 = 0$, show that $z ^ { 3 } = 2 \pm 2 \mathrm { i }$.
2. Hence solve the equation $$z ^ { 6 } - 4 z ^ { 3 } + 8 = 0$$ giving your answers in the form $r \mathrm { e } ^ { \mathrm { i } \theta }$, where $r > 0$ and $- \pi < \theta \leqslant \pi$.
Show that, for any real values of $k$ and $\theta$, $$\left( z - k \mathrm { e } ^ { \mathrm { i } \theta } \right) \left( z - k \mathrm { e } ^ { - \mathrm { i } \theta } \right) = z ^ { 2 } - 2 k z \cos \theta + k ^ { 2 }$$
Express $z ^ { 6 } - 4 z ^ { 3 } + 8$ as the product of three quadratic factors with real coefficients.

AQA FP2 2009 June Q1

1 Given that $z = 2 \mathrm { e } ^ { \frac { \pi \mathrm { i } } { 12 } }$ satisfies the equation $$z ^ { 4 } = a ( 1 + \sqrt { 3 } i )$$ where $a$ is real:

find the value of $a$;
find the other three roots of this equation, giving your answers in the form $r \mathrm { e } ^ { \mathrm { i } \theta }$, where $r > 0$ and $- \pi < \theta \leqslant \pi$.

AQA FP2 2009 June Q2

Given that $$\frac { 1 } { 4 r ^ { 2 } - 1 } = \frac { A } { 2 r - 1 } + \frac { B } { 2 r + 1 }$$ find the values of $A$ and $B$.
Use the method of differences to show that $$\sum _ { r = 1 } ^ { n } \frac { 1 } { 4 r ^ { 2 } - 1 } = \frac { n } { 2 n + 1 }$$
Find the least value of $n$ for which $\sum _ { r = 1 } ^ { n } \frac { 1 } { 4 r ^ { 2 } - 1 }$ differs from 0.5 by less than 0.001 .

AQA FP2 2009 June Q3

3 The cubic equation $$z ^ { 3 } + p z ^ { 2 } + 25 z + q = 0$$ where $p$ and $q$ are real, has a root $\alpha = 2 - 3 \mathrm { i }$.

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

AQA FP2 2009 June Q4

Sketch the graph of $y = \tanh x$.
Given that $u = \tanh x$, use the definitions of $\sinh x$ and $\cosh x$ in terms of $\mathrm { e } ^ { x }$ and $\mathrm { e } ^ { - x }$ to show that $$x = \frac { 1 } { 2 } \ln \left( \frac { 1 + u } { 1 - u } \right)$$
1. Show that the equation $$3 \operatorname { sech } ^ { 2 } x + 7 \tanh x = 5$$ can be written as $$3 \tanh ^ { 2 } x - 7 \tanh x + 2 = 0$$
2. Show that the equation $$3 \tanh ^ { 2 } x - 7 \tanh x + 2 = 0$$ has only one solution for $x$.
  Find this solution in the form $\frac { 1 } { 2 } \ln a$, where $a$ is an integer.

AQA FP2 2009 June Q5

Prove by induction that, if $n$ is a positive integer, $$( \cos \theta + \mathrm { i } \sin \theta ) ^ { n } = \cos n \theta + \mathrm { i } \sin n \theta$$
Hence, given that $$z = \cos \theta + \mathrm { i } \sin \theta$$ show that $$z ^ { n } + \frac { 1 } { z ^ { n } } = 2 \cos n \theta$$
Given further that $z + \frac { 1 } { z } = \sqrt { 2 }$, find the value of $$z ^ { 10 } + \frac { 1 } { z ^ { 10 } }$$

AQA FP2 2009 June Q6

Two points, $A$ and $B$, on an Argand diagram are represented by the complex numbers $2 + 3 \mathrm { i }$ and $- 4 - 5 \mathrm { i }$ respectively. Given that the points $A$ and $B$ are at the ends of a diameter of a circle $C _ { 1 }$, express the equation of $C _ { 1 }$ in the form $\left| z - z _ { 0 } \right| = k$.
A second circle, $C _ { 2 }$, is represented on the Argand diagram by the equation $| z - 5 + 4 \mathrm { i } | = 4$. Sketch on one Argand diagram both $C _ { 1 }$ and $C _ { 2 }$.
The points representing the complex numbers $z _ { 1 }$ and $z _ { 2 }$ lie on $C _ { 1 }$ and $C _ { 2 }$ respectively and are such that $\left| z _ { 1 } - z _ { 2 } \right|$ has its maximum value. Find this maximum value, giving your answer in the form $a + b \sqrt { 5 }$.

AQA FP2 2009 June Q7

7 The diagram shows a curve which starts from the point $A$ with coordinates ( 0,2 ). The curve is such that, at every point $P$ on the curve, $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 2 } s$$ where $s$ is the length of the $\operatorname { arc } A P$.
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1. Show that $$\frac { \mathrm { d } s } { \mathrm {~d} x } = \frac { 1 } { 2 } \sqrt { 4 + s ^ { 2 } }$$ (3 marks)
2. Hence show that $$s = 2 \sinh \frac { x } { 2 }$$
3. Hence find the cartesian equation of the curve.
Show that $$y ^ { 2 } = 4 + s ^ { 2 }$$

AQA FP2 2015 June Q1

5 marks

Express $\frac { 1 } { ( r + 2 ) r ! }$ in the form $\frac { A } { ( r + 1 ) ! } + \frac { B } { ( r + 2 ) ! }$, where $A$ and $B$ are integers.
[0pt] [3 marks]
Hence find $\sum _ { r = 1 } ^ { n } \frac { 1 } { ( r + 2 ) r ! }$.
[0pt] [2 marks]

AQA FP2 2015 June Q2

5 marks

Sketch the graph of $y = \tanh x$ and state the equations of its asymptotes.
Use the definitions of $\sinh x$ and $\cosh x$ in terms of $\mathrm { e } ^ { x }$ and $\mathrm { e } ^ { - x }$ to show that $$\operatorname { sech } ^ { 2 } x + \tanh ^ { 2 } x = 1$$
Solve the equation $6 \operatorname { sech } ^ { 2 } x = 4 + \tanh x$, giving your answers in terms of natural logarithms.
[0pt] [5 marks] \section*{Answer space for question 2}
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AQA FP2 2015 June Q3

5 marks

3 A curve $C$ is defined parametrically by $$x = \frac { t ^ { 2 } + 1 } { t } , \quad y = 2 \ln t$$

Show that $\left( \frac { \mathrm { d } x } { \mathrm {~d} t } \right) ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} t } \right) ^ { 2 } = \left( 1 + \frac { 1 } { t ^ { 2 } } \right) ^ { 2 }$.
The arc of $C$ from $t = 1$ to $t = 2$ is rotated through $2 \pi$ radians about the $x$-axis. Find the area of the surface generated, giving your answer in the form $\pi ( m \ln 2 + n )$, where $m$ and $n$ are integers.
[0pt] [5 marks]

AQA FP2 2015 June Q4

4 The expression $\mathrm { f } ( n )$ is given by $\mathrm { f } ( n ) = 2 ^ { 4 n + 3 } + 3 ^ { 3 n + 1 }$.

Show that $\mathrm { f } ( k + 1 ) - 16 \mathrm { f } ( k )$ can be expressed in the form $A \times 3 ^ { 3 k }$, where $A$ is an integer.
Prove by induction that $\mathrm { f } ( n )$ is a multiple of 11 for all integers $n \geqslant 1$.

AQA FP2 2015 June Q5

2 marks

5 The locus of points, $L$, satisfies the equation $$| z - 2 + 4 \mathrm { i } | = | z |$$

Sketch $L$ on the Argand diagram below.
The locus $L$ cuts the real axis at $A$ and the imaginary axis at $B$.
1. Show that the complex number represented by $C$, the midpoint of $A B$, is $$\frac { 5 } { 2 } - \frac { 5 } { 4 } \mathrm { i }$$
2. The point $O$ is the origin of the Argand diagram. Find the equation of the circle that passes through the points $O , A$ and $B$, giving your answer in the form $| z - \alpha | = k$.
  [0pt] [2 marks] \section*{(a)}
  \includegraphics[max width=\textwidth, alt={}]{bc3aaed2-4aef-4aec-b657-098b1e581e55-10_1173_1242_1217_463}

AQA FP2 2015 June Q6

3 marks

Given that $y = ( x - 2 ) \sqrt { 5 + 4 x - x ^ { 2 } } + 9 \sin ^ { - 1 } \left( \frac { x - 2 } { 3 } \right)$, show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = k \sqrt { 5 + 4 x - x ^ { 2 } }$$ where $k$ is an integer.
Hence show that $$\int _ { 2 } ^ { \frac { 7 } { 2 } } \sqrt { 5 + 4 x - x ^ { 2 } } \mathrm {~d} x = p \sqrt { 3 } + q \pi$$ where $p$ and $q$ are rational numbers.
[0pt] [3 marks]

AQA FP2 2015 June Q7

5 marks

7 The cubic equation $27 z ^ { 3 } + k z ^ { 2 } + 4 = 0$ has roots $\alpha , \beta$ and $\gamma$.

Write down the values of $\alpha \beta + \beta \gamma + \gamma \alpha$ and $\alpha \beta \gamma$.
1. In the case where $\beta = \gamma$, find the roots of the equation.
2. Find the value of $k$ in this case.
1. In the case where $\alpha = 1 - \mathrm { i }$, find $\alpha ^ { 2 }$ and $\alpha ^ { 3 }$.
2. Hence find the value of $k$ in this case.
In the case where $k = - 12$, find a cubic equation with integer coefficients which has roots $\frac { 1 } { \alpha } + 1 , \frac { 1 } { \beta } + 1$ and $\frac { 1 } { \gamma } + 1$.
[0pt] [5 marks]

AQA FP2 2015 June Q8

8 The complex number $\omega$ is given by $\omega = \cos \frac { 2 \pi } { 5 } + \mathrm { i } \sin \frac { 2 \pi } { 5 }$.

1. Verify that $\omega$ is a root of the equation $z ^ { 5 } = 1$.
2. Write down the three other non-real roots of $z ^ { 5 } = 1$, in terms of $\omega$.
1. Show that $1 + \omega + \omega ^ { 2 } + \omega ^ { 3 } + \omega ^ { 4 } = 0$.
2. Hence show that $\left( \omega + \frac { 1 } { \omega } \right) ^ { 2 } + \left( \omega + \frac { 1 } { \omega } \right) - 1 = 0$.
Hence show that $\cos \frac { 2 \pi } { 5 } = \frac { \sqrt { 5 } - 1 } { 4 }$.

Questions — AQA FP2 (142 questions)

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