Questions - AQA - 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 $$4 \sqrt { x ^ { 5 } } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + ( 2 \sqrt { x } ) y = \sqrt { x } ( \ln x ) ^ { 2 } + 5 , \quad x > 0$$
\includegraphics[max width=\textwidth, alt={}]{7b4a1237-bb28-4cba-84b1-35fa91d87408-14_1634_1709_1071_153}

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}
  \includegraphics[max width=\textwidth, alt={}, center]{7b4a1237-bb28-4cba-84b1-35fa91d87408-23_2486_1728_221_141}
  \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]
\includegraphics[max width=\textwidth, alt={}]{0b9b947d-824b-4d3a-b66d-4bfd8d49be17-16_2039_1719_671_148}

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]
\includegraphics[max width=\textwidth, alt={}]{0b9b947d-824b-4d3a-b66d-4bfd8d49be17-23_2488_1709_219_153}
\section*{DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED}

AQA D1 2005 January Q1

1 A student is using the algorithm below.

LINE 10	INPUT $A , B$
LINE 20	LET $C = A - B$
LINE 30	LET $D = A + B$
LINE 40	LET $E = ( D * D ) - ( C * C )$
LINE 50	LET $F = E / 4$
LINE 60	PRINT $F$
LINE 70	END

Trace the algorithm in the case where $A = 5$ and $B = 3$.

AQA D1 2005 January Q2

Use a bubble sort algorithm to rearrange the following numbers into ascending order, showing the new arrangement after each pass. $$\begin{array} { l l l l l l l l } 19 & 3 & 7 & 20 & 2 & 6 & 5 & 15 \end{array}$$
Write down the number of comparisons and the number of swaps during the first pass.
(2 marks)

AQA D1 2005 January Q3

3 A local council is responsible for gritting roads. The diagram shows the length, in miles, of the roads that have to be gritted.
\includegraphics[max width=\textwidth, alt={}, center]{76bccb26-f2ec-4798-bb6b-89c922f9651a-03_671_686_488_669} Total length $= 87$ miles The gritter is based at $A$, and must travel along all the roads, at least once, before returning to $A$.

Explain why it is not possible to start from $A$ and, by travelling along each road only once, return to $A$.
Find an optimal 'Chinese postman' route around the network, starting and finishing at $A$. State the length of your route.

LINE 10	INPUT \(A , B\)
LINE 20	LET \(C = A - B\)
LINE 30	LET \(D = A + B\)
LINE 40	LET \(E = ( D * D ) - ( C * C )\)
LINE 50	LET \(F = E / 4\)
LINE 60	PRINT \(F\)
LINE 70	END

Questions — AQA (3508 questions)

Browse by module

Browse by board

AQA FP3 2014 June Q2

AQA FP3 2014 June Q3

AQA FP3 2014 June Q4

AQA FP3 2014 June Q5

AQA FP3 2014 June Q6

AQA FP3 2014 June Q7

AQA FP3 2014 June Q8

AQA FP3 2015 June Q1

AQA FP3 2015 June Q2

AQA FP3 2015 June Q3

AQA FP3 2015 June Q4

AQA FP3 2015 June Q5

AQA FP3 2015 June Q6

AQA FP3 2015 June Q7

AQA FP3 2016 June Q1

AQA FP3 2016 June Q2

AQA FP3 2016 June Q3

AQA FP3 2016 June Q4

AQA FP3 2016 June Q5

AQA FP3 2016 June Q6

AQA FP3 2016 June Q7

AQA FP3 2016 June Q8

AQA D1 2005 January Q1

AQA D1 2005 January Q2

AQA D1 2005 January Q3