Questions - AQA FP3 - A-Level Maths

Find the values of the constants $a$, $b$ and $c$ for which $a + b \sin 2 x + c \cos 2 x$ is a particular integral of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + 4 y = 20 - 20 \cos 2 x$$ [4 marks]
Hence find the solution of this differential equation, given that $y = 4$ when $x = 0$.
[0pt] [4 marks]
\includegraphics[max width=\textwidth, alt={}]{0eb3e96e-528c-4a99-b164-31cc865f0d68-04_1974_1709_733_153}

AQA FP3 2014 June Q3

4 marks

3 A curve has polar equation $r ( 4 - 3 \cos \theta ) = 4$. Find its Cartesian equation in the form $y ^ { 2 } = \mathrm { f } ( x )$.
[0pt] [4 marks]

AQA FP3 2014 June Q4

10 marks

4 Solve the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } - 3 y = 2 \mathrm { e } ^ { - x }$$ given that $y \rightarrow 0$ as $x \rightarrow \infty$ and that $\frac { \mathrm { d } y } { \mathrm {~d} x } = - 3$ when $x = 0$.
[0pt] [10 marks]
\includegraphics[max width=\textwidth, alt={}, center]{0eb3e96e-528c-4a99-b164-31cc865f0d68-08_46_145_829_159}

AQA FP3 2014 June Q5

4 marks

Find $\int x \cos 8 x \mathrm {~d} x$.
Find $\lim _ { x \rightarrow 0 } \left[ \frac { 1 } { x } \sin 2 x \right]$.
Explain why $\int _ { 0 } ^ { \frac { \pi } { 4 } } \left( 2 \cot 2 x - \frac { 1 } { x } + x \cos 8 x \right) \mathrm { d } x$ is an improper integral.
Evaluate $\int _ { 0 } ^ { \frac { \pi } { 4 } } \left( 2 \cot 2 x - \frac { 1 } { x } + x \cos 8 x \right) \mathrm { d } x$, showing the limiting process used. Give your answer as a single term.
[0pt] [4 marks]

AQA FP3 2014 June Q6

8 marks

By using an integrating factor, find the general solution of the differential equation $$\frac { \mathrm { d } u } { \mathrm {~d} x } - \frac { 2 x } { x ^ { 2 } + 4 } u = 3 \left( x ^ { 2 } + 4 \right)$$ giving your answer in the form $u = \mathrm { f } ( x )$.
[0pt] [6 marks]
Show that the substitution $u = x ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x }$ transforms the differential equation $$x ^ { 2 } \left( x ^ { 2 } + 4 \right) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 8 x \frac { \mathrm {~d} y } { \mathrm {~d} x } = 3 \left( x ^ { 2 } + 4 \right) ^ { 2 }$$ into $$\frac { \mathrm { d } u } { \mathrm {~d} x } - \frac { 2 x } { x ^ { 2 } + 4 } u = 3 \left( x ^ { 2 } + 4 \right)$$
Hence, given that $x > 0$, find the general solution of the differential equation $$x ^ { 2 } \left( x ^ { 2 } + 4 \right) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 8 x \frac { \mathrm {~d} y } { \mathrm {~d} x } = 3 \left( x ^ { 2 } + 4 \right) ^ { 2 }$$ [2 marks]

AQA FP3 2014 June Q7

4 marks

It is given that $y = \ln ( \cos x + \sin x )$.
1. Show that $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = - \frac { 2 } { 1 + \sin 2 x }$.
2. Find $\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }$.
1. Hence use Maclaurin's theorem to show that the first three non-zero terms in the expansion, in ascending powers of $x$, of $\ln ( \cos x + \sin x )$ are $x - x ^ { 2 } + \frac { 2 } { 3 } x ^ { 3 }$.
2. Write down the first three non-zero terms in the expansion, in ascending powers of $x$, of $\ln ( \cos x - \sin x )$.
Hence find the first three non-zero terms in the expansion, in ascending powers of $x$, of $\ln \left( \frac { \cos 2 x } { \mathrm { e } ^ { 3 x - 1 } } \right)$.
[0pt] [4 marks]
\includegraphics[max width=\textwidth, alt={}]{0eb3e96e-528c-4a99-b164-31cc865f0d68-17_2484_1707_221_153}

\includegraphics[max width=\textwidth, alt={}]{0eb3e96e-528c-4a99-b164-31cc865f0d68-19_2484_1707_221_153}

AQA FP3 2014 June Q8

1 marks

8 The diagram shows a sketch of a curve $C$, the pole $O$ and the initial line. The curve $C$ intersects the initial line at the point $P$.
\includegraphics[max width=\textwidth, alt={}, center]{0eb3e96e-528c-4a99-b164-31cc865f0d68-20_432_949_402_525} The polar equation of $C$ is $r = \left( 1 - \tan ^ { 2 } \theta \right) \sec \theta , - \frac { \pi } { 4 } \leqslant \theta \leqslant \frac { \pi } { 4 }$.

Show that the area of the region bounded by the curve $C$ is $\frac { 8 } { 15 }$.
The curve whose polar equation is $$r = \frac { 1 } { 2 } \sec ^ { 3 } \theta , \quad - \frac { \pi } { 4 } \leqslant \theta \leqslant \frac { \pi } { 4 }$$ intersects $C$ at the points $A$ and $B$.
1. Find the polar coordinates of $A$ and $B$.
2. Given that angle $O A P =$ angle $O B P = \alpha$, show that $\tan \alpha = k \sqrt { 3 }$, where $k$ is an integer.
3. Using your value of $k$ from part (b)(ii), state whether the point $A$ lies inside or lies outside the circle whose diameter is $O P$. Give a reason for your answer.
  [0pt] [1 mark]
  \includegraphics[max width=\textwidth, alt={}]{0eb3e96e-528c-4a99-b164-31cc865f0d68-21_2484_1707_221_153}
  
  \includegraphics[max width=\textwidth, alt={}]{0eb3e96e-528c-4a99-b164-31cc865f0d68-23_2484_1707_221_153}

AQA FP3 2015 June Q1

3 marks

1 It is given that $y ( x )$ satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { f } ( x , y )$$ where $$\mathrm { f } ( x , y ) = \frac { x + y ^ { 2 } } { x }$$ and $$y ( 2 ) = 5$$

Use the Euler formula $$y _ { r + 1 } = y _ { r } + h \mathrm { f } \left( x _ { r } , y _ { r } \right)$$ with $h = 0.05$, to obtain an approximation to $y ( 2.05 )$.
Use the formula $$y _ { r + 1 } = y _ { r - 1 } + 2 h \mathrm { f } \left( x _ { r } , y _ { r } \right)$$ with your answer to part (a), to obtain an approximation to $y ( 2.1 )$, giving your answer to three significant figures.
[0pt] [3 marks]

AQA FP3 2015 June Q2

9 marks

2 By using an integrating factor, find the solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + ( \tan x ) y = \tan ^ { 3 } x \sec x$$ given that $y = 2$ when $x = \frac { \pi } { 3 }$.
[0pt] [9 marks]

AQA FP3 2015 June Q3

1. Write down the expansion of $\ln ( 1 + 2 x )$ in ascending powers of $x$ up to and including the term in $x ^ { 4 }$.
2. Hence, or otherwise, find the first two non-zero terms in the expansion of $$\ln [ ( 1 + 2 x ) ( 1 - 2 x ) ]$$ in ascending powers of $x$ and state the range of values of $x$ for which the expansion is valid.
Find $\lim _ { x \rightarrow 0 } \left[ \frac { 3 x - x \sqrt { 9 + x } } { \ln [ ( 1 + 2 x ) ( 1 - 2 x ) ] } \right]$.

AQA FP3 2015 June Q4

Explain why $\int _ { 2 } ^ { \infty } ( x - 2 ) \mathrm { e } ^ { - 2 x } \mathrm {~d} x$ is an improper integral.
Evaluate $\int _ { 2 } ^ { \infty } ( x - 2 ) \mathrm { e } ^ { - 2 x } \mathrm {~d} x$, showing the limiting process used.

AQA FP3 2015 June Q5

3 marks

Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 6 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 9 y = 36 \sin 3 x$$
It is given that $y = \mathrm { f } ( x )$ is the solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 6 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 9 y = 36 \sin 3 x$$ such that $\mathrm { f } ( 0 ) = 0$ and $\mathrm { f } ^ { \prime } ( 0 ) = 0$.
1. Show that $f ^ { \prime \prime } ( 0 ) = 0$.
2. Find the first two non-zero terms in the expansion, in ascending powers of $x$, of $\mathrm { f } ( x )$.
  [0pt] [3 marks]

AQA FP3 2015 June Q6

6 A differential equation is given by $$4 \sqrt { x ^ { 5 } } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + ( 2 \sqrt { x } ) y = \sqrt { x } ( \ln x ) ^ { 2 } + 5 , \quad x > 0$$

Show that the substitution $x = \mathrm { e } ^ { 2 t }$ transforms this differential equation into $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } - 2 \frac { \mathrm {~d} y } { \mathrm {~d} t } + 2 y = 4 t ^ { 2 } + 5 \mathrm { e } ^ { - t }$$
Hence find the general solution of the differential equation $$4 \sqrt { x ^ { 5 } } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + ( 2 \sqrt { x } ) y = \sqrt { x } ( \ln x ) ^ { 2 } + 5 , \quad x > 0$$
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AQA FP3 2015 June Q7

7 The diagram shows the sketch of a curve $C _ { 1 }$.
\includegraphics[max width=\textwidth, alt={}, center]{7b4a1237-bb28-4cba-84b1-35fa91d87408-18_362_734_360_635} The polar equation of the curve $C _ { 1 }$ is $$r = 1 + \cos 2 \theta , \quad - \frac { \pi } { 2 } \leqslant \theta \leqslant \frac { \pi } { 2 }$$

Find the area of the region bounded by the curve $C _ { 1 }$.
The curve $C _ { 2 }$ whose polar equation is $$r = 1 + \sin \theta , \quad - \frac { \pi } { 2 } \leqslant \theta \leqslant \frac { \pi } { 2 }$$ intersects the curve $C _ { 1 }$ at the pole $O$ and at the point $A$. The straight line drawn through $A$ parallel to the initial line intersects $C _ { 1 }$ again at the point $B$.
1. Find the polar coordinates of $A$.
2. Show that the length of $O B$ is $\frac { 1 } { 4 } ( \sqrt { 13 } + 1 )$.
3. Find the length of $A B$, giving your answer to three significant figures. \includegraphics[max width=\textwidth, alt={}, center]{7b4a1237-bb28-4cba-84b1-35fa91d87408-22_2486_1728_221_141}
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  \includegraphics[max width=\textwidth, alt={}, center]{7b4a1237-bb28-4cba-84b1-35fa91d87408-24_2488_1728_219_141}

AQA FP3 2016 June Q1

3 marks

Find the values of the constants $a$ and $b$ for which $a x + b$ is a particular integral of the differential equation $$2 \frac { \mathrm {~d} y } { \mathrm {~d} x } - 5 y = 10 x$$
Hence find the general solution of $2 \frac { \mathrm {~d} y } { \mathrm {~d} x } - 5 y = 10 x$.
[0pt] [3 marks]

AQA FP3 2016 June Q2

Write down the expansion of $\sin 2 x$ in ascending powers of $x$ up to and including the term in $x ^ { 5 }$.
It is given that the first non-zero term in the expansion of $$\sin 2 x - 2 x \left( 1 - p x ^ { 2 } \right) \left( 1 - x ^ { 2 } \right) ^ { - 1 }$$ in ascending powers of $x$ is $q x ^ { 5 }$.
Find the values of the rational numbers $p$ and $q$.

AQA FP3 2016 June Q3

1 marks

It is given that $y ( x )$ satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { f } ( x , y )$$ where $$\mathrm { f } ( x , y ) = ( 2 x + 1 ) \ln ( x + y )$$ and $$y ( 0 ) = 2$$ Use the improved Euler formula $$y _ { r + 1 } = y _ { r } + \frac { 1 } { 2 } \left( k _ { 1 } + k _ { 2 } \right)$$ where $k _ { 1 } = h \mathrm { f } \left( x _ { r } , y _ { r } \right)$ and $k _ { 2 } = h \mathrm { f } \left( x _ { r } + h , y _ { r } + k _ { 1 } \right)$ and $h = 0.1$, to obtain an approximation to $y ( 0.1 )$, giving your answer to three decimal places.
It is given that $y ( x )$ satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = ( 2 x + 1 ) \ln ( x + y )$$ and $y = 2$ when $x = 0$.
1. Use implicit differentiation to find $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$, giving your answer in terms of $x$ and $y$.
2. Hence find the first three non-zero terms in the expansion, in ascending powers of $x$, of $y ( x )$. Give your answer in an exact form.
3. Use your answer to part (b)(ii) to obtain an approximation to $y ( 0.1 )$, giving your answer to three decimal places.
  [0pt] [1 mark]

AQA FP3 2016 June Q4

1 marks

The curve with Cartesian equation $\frac { x ^ { 2 } } { c } + \frac { y ^ { 2 } } { d } = 1$ is mapped onto the curve with polar equation $r = \frac { 10 } { 3 - 2 \cos \theta }$ by a single geometrical transformation. By writing the polar equation as a Cartesian equation in a suitable form, find the values of the constants $c$ and $d$.
Hence describe the geometrical transformation referred to in part (a).
[0pt] [1 mark]

AQA FP3 2016 June Q5

11 marks

Express $\frac { 1 } { ( 1 + x ) ( 2 + x ) }$ in the form $\frac { A } { 1 + x } + \frac { B } { 2 + x }$, where $A$ and $B$ are integers.
Use the substitution $u = \frac { \mathrm { d } y } { \mathrm {~d} x }$ to solve the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + \frac { 1 } { ( 1 + x ) ( 2 + x ) } \frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 + x } { 1 + x }$$ given that $y = 1$ and $\frac { \mathrm { d } y } { \mathrm {~d} x } = 4$ when $x = 0$. Give your answer in the form $y = \mathrm { f } ( x )$.
[0pt] [11 marks]

AQA FP3 2016 June Q6

4 marks

Use the substitution $a = \frac { 1 } { p }$ to find $\lim _ { p \rightarrow \infty } \left[ \frac { \ln p } { p ^ { k } } \right]$, where $k > 0$.
Evaluate the improper integral $\int _ { 1 } ^ { \infty } \frac { \ln x } { x ^ { 7 } } \mathrm {~d} x$, showing the limiting process used.
[0pt] [4 marks]
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AQA FP3 2016 June Q7

10 marks

7 Find the solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 y = 10 \mathrm { e } ^ { 4 x } + 8 \sin 2 x + 4 \cos 2 x$$ given that $y = 2.5$ when $x = 0$ and $y = \frac { \pi } { 4 }$ when $x = \frac { \pi } { 4 }$.
[0pt] [10 marks]

AQA FP3 2016 June Q8

7 marks

8 The diagram shows the sketch of part of a curve, the pole $O$ and the initial line.
\includegraphics[max width=\textwidth, alt={}, center]{0b9b947d-824b-4d3a-b66d-4bfd8d49be17-20_609_670_358_703} The polar equation of the curve is $r = 1 + \tan \theta$.
The point $A$ is the point on the curve at which $\theta = \frac { \pi } { 3 }$.
The perpendicular, $A N$, from $A$ to the initial line intersects the curve at the point $B$.

Find the exact length of $O A$.
1. Given that, at the point $B , \theta = \alpha$, show that $( \cos \alpha + \sin \alpha ) ^ { 2 } = 1 + \frac { \sqrt { 3 } } { 2 }$.
2. Hence, or otherwise, find $\alpha$ in terms of $\pi$.
Show that the area of triangle $O A B$ is $\frac { 3 + 2 \sqrt { 3 } } { 6 }$.
Find, in an exact simplified form, the area of the shaded region bounded by the curve and the line segment $A B$.
[0pt] [7 marks]
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\section*{DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED}

AQA FP3 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 $$\arg ( z + 4 - 4 i ) = \frac { 1 } { 6 } \pi$$ Give your answer in the form $a + \mathrm { i } b$.

AQA FP3 Q6

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

1. Show that $$z + \frac { 1 } { z } = 2 \cos \theta$$
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 FP3 2006 January Q1

Find the roots of the equation $m ^ { 2 } + 2 m + 2 = 0$ in the form $a + i b$.
(2 marks)
1. Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y = 4 x$$
2. Hence express $y$ in terms of $x$, given that $y = 1$ and $\frac { \mathrm { d } y } { \mathrm {~d} x } = 2$ when $x = 0$.

Questions — AQA FP3 (144 questions)

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