Questions - AQA - A-Level Maths

Find the cartesian equation of $C$, giving your answer in the form $y ^ { 2 } = \mathrm { f } ( x )$.
The straight line with polar equation $4 r = 3 \sec \theta$ intersects the curve $C$ at the points $P$ and $Q$. Find the length of $P Q$.

AQA FP3 2011 January Q4

4 By using an integrating factor, find the solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } - \frac { 2 } { x } y = 2 x ^ { 3 } \mathrm { e } ^ { 2 x }$$ given that $y = \mathrm { e } ^ { 4 }$ when $x = 2$. Give your answer in the form $y = \mathrm { f } ( x )$.

AQA FP3 2011 January Q5

Write $\frac { 4 } { 4 x + 1 } - \frac { 3 } { 3 x + 2 }$ in the form $\frac { C } { ( 4 x + 1 ) ( 3 x + 2 ) }$, where $C$ is a constant.
(l mark)
Evaluate the improper integral $$\int _ { 1 } ^ { \infty } \frac { 10 } { ( 4 x + 1 ) ( 3 x + 2 ) } d x$$ showing the limiting process used and giving your answer in the form $\ln k$, where $k$ is a constant.
(6 marks)

AQA FP3 2011 January Q6

6 The diagram shows a sketch of a curve $C$.
\includegraphics[max width=\textwidth, alt={}, center]{8cb4b110-274e-47ec-a31b-ee8f84434a65-3_305_556_1078_721} The polar equation of the curve is $$r = 2 \sin 2 \theta \sqrt { \cos \theta } , \quad 0 \leqslant \theta \leqslant \frac { \pi } { 2 }$$ Show that the area of the region bounded by $C$ is $\frac { 16 } { 15 }$.

AQA FP3 2011 January Q7

Write down the expansions in ascending powers of $x$ up to and including the term in $x ^ { 3 }$ of:
1. $\cos x + \sin x$;
2. $\quad \ln ( 1 + 3 x )$.
It is given that $y = \mathrm { e } ^ { \tan x }$.
1. Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ and show that $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = ( 1 + \tan x ) ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x }$.
2. Find the value of $\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }$ when $x = 0$.
3. Hence, by using Maclaurin's theorem, show that the first four terms in the expansion, in ascending powers of $x$, of $\mathrm { e } ^ { \tan x }$ are $$1 + x + \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 2 } x ^ { 3 }$$
Find $$\lim _ { x \rightarrow 0 } \left[ \frac { \mathrm { e } ^ { \tan x } - ( \cos x + \sin x ) } { x \ln ( 1 + 3 x ) } \right]$$

AQA FP3 2011 January Q8

Given that $x = \mathrm { e } ^ { t }$ and that $y$ is a function of $x$, show that $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } = \frac { \mathrm { d } y } { \mathrm {~d} t }$$
Hence show that the substitution $x = \mathrm { e } ^ { t }$ transforms the differential equation $$x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 3 x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 4 y = 2 \ln x$$ into $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } - 4 \frac { \mathrm {~d} y } { \mathrm {~d} t } + 4 y = 2 t$$
Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } - 4 \frac { \mathrm {~d} y } { \mathrm {~d} t } + 4 y = 2 t$$
Hence solve the differential equation $x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 3 x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 4 y = 2 \ln x$, given that $y = \frac { 3 } { 2 }$ and $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 2 }$ when $x = 1$.
(5 marks)

AQA FP3 2012 January Q1

1 The function $y ( x )$ satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { f } ( x , y )$$ where $$\mathrm { f } ( x , y ) = \frac { y - x } { y ^ { 2 } + x }$$ and $$y ( 1 ) = 2$$

Use the Euler formula $$y _ { r + 1 } = y _ { r } + h \mathrm { f } \left( x _ { r } , y _ { r } \right)$$ with $h = 0.1$, to obtain an approximation to $y ( 1.1 )$.
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 $y ( 1.2 )$, giving your answer to three decimal places.

AQA FP3 2012 January Q2

2 Find $$\lim _ { x \rightarrow 0 } \left[ \frac { \sqrt { 4 + x } - 2 } { x + x ^ { 2 } } \right]$$ (3 marks)

AQA FP3 2012 January Q3

3 Solve the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 10 y = 26 \mathrm { e } ^ { x }$$ given that $y = 5$ and $\frac { \mathrm { d } y } { \mathrm {~d} x } = 11$ when $x = 0$. Give your answer in the form $y = \mathrm { f } ( x )$.
(10 marks)

AQA FP3 2012 January Q4

By using an integrating factor, find the general solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { 2 } { x } y = \ln x$$
Hence, given that $y \rightarrow 0$ as $x \rightarrow 0$, find the value of $y$ when $x = 1$.

AQA FP3 2012 January Q5

Explain why $\int _ { \frac { 1 } { 2 } } ^ { \infty } \frac { x ( 1 - 2 x ) } { x ^ { 2 } + 3 \mathrm { e } ^ { 4 x } } \mathrm {~d} x$ is an improper integral.
(1 mark)
By using the substitution $u = x ^ { 2 } \mathrm { e } ^ { - 4 x } + 3$, find $$\int \frac { x ( 1 - 2 x ) } { x ^ { 2 } + 3 \mathrm { e } ^ { 4 x } } \mathrm {~d} x$$
Hence evaluate $\int _ { \frac { 1 } { 2 } } ^ { \infty } \frac { x ( 1 - 2 x ) } { x ^ { 2 } + 3 \mathrm { e } ^ { 4 x } } \mathrm {~d} x$, showing the limiting process used.

AQA FP3 2012 January Q6

Given that $y = \ln \cos 2 x$, find $\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } }$.
Use Maclaurin's theorem to show that the first two non-zero terms in the expansion, in ascending powers of $x$, of $\ln \cos 2 x$ are $- 2 x ^ { 2 } - \frac { 4 } { 3 } x ^ { 4 }$.
Hence find the first two non-zero terms in the expansion, in ascending powers of $x$, of $\ln \sec ^ { 2 } 2 x$.

AQA FP3 2012 January Q7

7 It is given that, for $x \neq 0 , y$ satisfies the differential equation $$x \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 ( 3 x + 1 ) \frac { \mathrm { d } y } { \mathrm {~d} x } + 3 y ( 3 x + 2 ) = 18 x$$

Show that the substitution $u = x y$ transforms this differential equation into $$\frac { \mathrm { d } ^ { 2 } u } { \mathrm {~d} x ^ { 2 } } + 6 \frac { \mathrm {~d} u } { \mathrm {~d} x } + 9 u = 18 x$$
Hence find the general solution of the differential equation $$x \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 ( 3 x + 1 ) \frac { \mathrm { d } y } { \mathrm {~d} x } + 3 y ( 3 x + 2 ) = 18 x$$ giving your answer in the form $y = \mathrm { f } ( x )$.
(8 marks)

AQA FP3 2012 January Q8

8 The diagram shows a sketch of the curve $C$ with polar equation $$r = 3 + 2 \cos \theta , \quad 0 \leqslant \theta \leqslant 2 \pi$$ \includegraphics[max width=\textwidth, alt={}, center]{80c4336c-b0ca-46de-a871-812e6923f7f2-4_463_668_1468_699}

Find the area of the region bounded by the curve $C$.
A circle, whose cartesian equation is $( x - 4 ) ^ { 2 } + y ^ { 2 } = 16$, intersects the curve $C$ at the points $A$ and $B$.
1. Find, in surd form, the length of $A B$.
2. Find the perimeter of the segment $A O B$ of the circle, where $O$ is the pole.

AQA FP3 2013 January Q1

1 It is given that $y ( x )$ satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { f } ( x , y )$$ where $$\mathrm { f } ( x , y ) = \sqrt { 2 x + y }$$ and $$y ( 3 ) = 5$$

Use the Euler formula $$y _ { r + 1 } = y _ { r } + h \mathrm { f } \left( x _ { r } , y _ { r } \right)$$ with $h = 0.2$, to obtain an approximation to $y ( 3.2 )$, giving your answer to four decimal places.
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 $y ( 3.4 )$, giving your answer to three decimal places.

AQA FP3 2013 January Q2

Write down the expansion of $\mathrm { e } ^ { 3 x }$ in ascending powers of $x$ up to and including the term in $x ^ { 2 }$.
Hence, or otherwise, find the term in $x ^ { 2 }$ in the expansion, in ascending powers of $x$, of $\mathrm { e } ^ { 3 x } ( 1 + 2 x ) ^ { - \frac { 3 } { 2 } }$.
(4 marks)

AQA FP3 2013 January Q3

3 It is given that the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = 0$$ is $y = \mathrm { e } ^ { x } ( A x + B )$. Hence find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = 6 \mathrm { e } ^ { x }$$

AQA FP3 2013 January Q4

Explain why $\int _ { 0 } ^ { 1 } x ^ { 4 } \ln x \mathrm {~d} x$ is an improper integral.
(l mark)
Evaluate $\int _ { 0 } ^ { 1 } x ^ { 4 } \ln x \mathrm {~d} x$, showing the limiting process used.
(6 marks)

AQA FP3 2013 January Q5

Show that $\tan x$ is an integrating factor for the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { \sec ^ { 2 } x } { \tan x } y = \tan x$$ (2 marks)
Hence solve this differential equation, given that $y = 3$ when $x = \frac { \pi } { 4 }$.
(6 marks)

AQA FP3 2013 January Q6

It is given that $y = \ln \left( \mathrm { e } ^ { 3 x } \cos x \right)$.
1. Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 - \tan x$.
2. Find $\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } }$.
Hence use Maclaurin's theorem to show that the first three non-zero terms in the expansion, in ascending powers of $x$, of $\ln \left( \mathrm { e } ^ { 3 x } \cos x \right)$ are $3 x - \frac { 1 } { 2 } x ^ { 2 } - \frac { 1 } { 12 } x ^ { 4 }$.
(3 marks)
Write down the expansion of $\ln ( 1 + p x )$, where $p$ is a constant, in ascending powers of $x$ up to and including the term in $x ^ { 2 }$.
1. Find the value of $p$ for which $\lim _ { x \rightarrow 0 } \left[ \frac { 1 } { x ^ { 2 } } \ln \left( \frac { \mathrm { e } ^ { 3 x } \cos x } { 1 + p x } \right) \right]$ exists.
2. Hence find the value of $\lim _ { x \rightarrow 0 } \left[ \frac { 1 } { x ^ { 2 } } \ln \left( \frac { \mathrm { e } ^ { 3 x } \cos x } { 1 + p x } \right) \right]$ when $p$ takes the value found in part (d)(i).

AQA FP3 2013 January Q7

Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } - 6 \frac { \mathrm {~d} y } { \mathrm {~d} t } + 10 y = \mathrm { e } ^ { 2 t }$$ giving your answer in the form $y = \mathrm { f } ( t )$.
Given that $x = t ^ { \frac { 1 } { 2 } } , x > 0 , t > 0$ and $y$ is a function of $x$, show that $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 4 t \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} t }$$ (5 marks)
Hence show that the substitution $x = t ^ { \frac { 1 } { 2 } }$ transforms the differential equation $$x \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - \left( 12 x ^ { 2 } + 1 \right) \frac { \mathrm { d } y } { \mathrm {~d} x } + 40 x ^ { 3 } y = 4 x ^ { 3 } \mathrm { e } ^ { 2 x ^ { 2 } }$$ into $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } - 6 \frac { \mathrm {~d} y } { \mathrm {~d} t } + 10 y = \mathrm { e } ^ { 2 t }$$ (2 marks)
Hence write down the general solution of the differential equation $$x \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - \left( 12 x ^ { 2 } + 1 \right) \frac { \mathrm { d } y } { \mathrm {~d} x } + 40 x ^ { 3 } y = 4 x ^ { 3 } \mathrm { e } ^ { 2 x ^ { 2 } }$$ (l mark)

AQA FP3 2013 January Q8

8 The diagram shows a sketch of a curve.
\includegraphics[max width=\textwidth, alt={}, center]{f05737eb-adb1-4228-aebf-6b5c7f26a434-5_464_574_402_726} The polar equation of the curve is $$r = \sin 2 \theta \sqrt { \left( 2 + \frac { 1 } { 2 } \cos \theta \right) } , \quad 0 \leqslant \theta \leqslant \frac { \pi } { 2 }$$ The point $P$ is the point of the curve at which $\theta = \frac { \pi } { 3 }$. The perpendicular from $P$ to the initial line meets the initial line at the point $N$.

1. Find the exact value of $r$ when $\theta = \frac { \pi } { 3 }$.
2. Show that the polar equation of the line $P N$ is $r = \frac { 3 \sqrt { 3 } } { 8 } \sec \theta$.
3. Find the area of triangle $O N P$ in the form $\frac { k \sqrt { 3 } } { 128 }$, where $k$ is an integer.
1. Using the substitution $u = \sin \theta$, or otherwise, find $\int \sin ^ { n } \theta \cos \theta \mathrm {~d} \theta$, where $n \geqslant 2$.
2. Find the area of the shaded region bounded by the line $O P$ and the arc $O P$ of the curve. Give your answer in the form $a \pi + b \sqrt { 3 } + c$, where $a , b$ and $c$ are constants.
  (8 marks)

AQA FP3 2006 June Q1

1 It is given that $y$ satisfies the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 5 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 4 y = 8 x - 10 - 10 \cos 2 x$$

Show that $y = 2 x + \sin 2 x$ is a particular integral of the given differential equation.
Find the general solution of the differential equation.
Hence express $y$ in terms of $x$, given that $y = 2$ and $\frac { \mathrm { d } y } { \mathrm {~d} x } = 0$ when $x = 0$.

AQA FP3 2006 June Q2

2 The function $y ( x )$ satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { f } ( x , y )$$ where $$\mathrm { f } ( x , y ) = \frac { x ^ { 2 } + y ^ { 2 } } { x y }$$ and $$y ( 1 ) = 2$$

Use the Euler formula $$y _ { r + 1 } = y _ { r } + h \mathrm { f } \left( x _ { r } , y _ { r } \right)$$ with $h = 0.1$, to obtain an approximation to $y ( 1.1 )$.
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 $y ( 1.1 )$, giving your answer to four decimal places.

AQA FP3 2006 June Q3

Show that $\sin x$ is an integrating factor for the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + ( \cot x ) y = 2 \cos x$$
Solve this differential equation, given that $y = 2$ when $x = \frac { \pi } { 2 }$.

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