Questions - AQA FP3 - A-Level Maths

Find the area of the region bounded by the curve $C$.
The circle with cartesian equation $x ^ { 2 } + y ^ { 2 } = 9$ intersects the curve $C$ at the points $A$ and $B$.
1. Find the polar coordinates of $A$ and $B$.
2. Find, in surd form, the length of $A B$.

AQA FP3 2006 June Q5

Show that $\lim _ { a \rightarrow \infty } \left( \frac { 3 a + 2 } { 2 a + 3 } \right) = \frac { 3 } { 2 }$.
Evaluate $\int _ { 1 } ^ { \infty } \left( \frac { 3 } { 3 x + 2 } - \frac { 2 } { 2 x + 3 } \right) \mathrm { d } x$, giving your answer in the form $\ln k$, where $k$ is a rational number.
(5 marks)

AQA FP3 2006 June Q6

Show that the substitution $$u = \frac { \mathrm { d } y } { \mathrm {~d} x } + 2 y$$ transforms the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 4 y = \mathrm { e } ^ { - 2 x }$$ into $$\frac { \mathrm { d } u } { \mathrm {~d} x } + 2 u = \mathrm { e } ^ { - 2 x }$$ (4 marks)
By using an integrating factor, or otherwise, find the general solution of $$\frac { \mathrm { d } u } { \mathrm {~d} x } + 2 u = \mathrm { e } ^ { - 2 x }$$ giving your answer in the form $u = \mathrm { f } ( x )$.
Hence find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 4 y = \mathrm { e } ^ { - 2 x }$$ giving your answer in the form $y = \mathrm { g } ( x )$.

AQA FP3 2006 June Q7

1. Write down the first three terms of the binomial expansion of $( 1 + y ) ^ { - 1 }$, in ascending powers of $y$.
2. By using the expansion $$\cos x = 1 - \frac { x ^ { 2 } } { 2 ! } + \frac { x ^ { 4 } } { 4 ! } - \ldots$$ and your answer to part (a)(i), or otherwise, show that the first three non-zero terms in the expansion, in ascending powers of $x$, of $\sec x$ are $$1 + \frac { x ^ { 2 } } { 2 } + \frac { 5 x ^ { 4 } } { 24 }$$
By using Maclaurin's theorem, or otherwise, show that the first two non-zero terms in the expansion, in ascending powers of $x$, of $\tan x$ are $$x + \frac { x ^ { 3 } } { 3 }$$
Hence find $\lim _ { x \rightarrow 0 } \left( \frac { x \tan 2 x } { \sec x - 1 } \right)$.

AQA FP3 2008 June Q1

1 The function $y ( x )$ satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { f } ( x , y )$$ where $$\mathrm { f } ( x , y ) = \ln ( x + y )$$ and $$y ( 2 ) = 3$$ 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 $y ( 2.1 )$, giving your answer to four decimal places.
(6 marks)

AQA FP3 2008 June Q2

Find the values of the constants $a , b , c$ and $d$ for which $a + b x + c \sin x + d \cos x$ is a particular integral of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } - 3 y = 10 \sin x - 3 x$$ (4 marks)
Hence find the general solution of this differential equation.

AQA FP3 2008 June Q3

Show that $x ^ { 2 } = 1 - 2 y$ can be written in the form $x ^ { 2 } + y ^ { 2 } = ( 1 - y ) ^ { 2 }$.
A curve has cartesian equation $x ^ { 2 } = 1 - 2 y$. Find its polar equation in the form $r = \mathrm { f } ( \theta )$, given that $r > 0$.

AQA FP3 2008 June Q4

A differential equation is given by $$x \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - \frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 }$$ Show that the substitution $$u = \frac { \mathrm { d } y } { \mathrm {~d} x }$$ transforms this differential equation into $$\frac { \mathrm { d } u } { \mathrm {~d} x } - \frac { 1 } { x } u = 3 x$$
By using an integrating factor, find the general solution of $$\frac { \mathrm { d } u } { \mathrm {~d} x } - \frac { 1 } { x } u = 3 x$$ giving your answer in the form $u = \mathrm { f } ( x )$.
Hence find the general solution of the differential equation $$x \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - \frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 }$$ giving your answer in the form $y = \mathrm { g } ( x )$.

AQA FP3 2008 June Q5

Find $\int x ^ { 3 } \ln x \mathrm {~d} x$.
Explain why $\int _ { 0 } ^ { \mathrm { e } } x ^ { 3 } \ln x \mathrm {~d} x$ is an improper integral.
Evaluate $\int _ { 0 } ^ { \mathrm { e } } x ^ { 3 } \ln x \mathrm {~d} x$, showing the limiting process used.

AQA FP3 2008 June Q6

Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } - 3 y = 10 \mathrm { e } ^ { - 2 x } - 9$$ (10 marks)
Hence express $y$ in terms of $x$, given that $y = 7$ when $x = 0$ and that $\frac { \mathrm { d } y } { \mathrm {~d} x } \rightarrow 0$ as $x \rightarrow \infty$.

AQA FP3 2008 June Q7

Write down the expansion of $\sin 2 x$ in ascending powers of $x$ up to and including the term in $x ^ { 3 }$.
1. Given that $y = \sqrt { 3 + \mathrm { e } ^ { x } }$, find the values of $\frac { \mathrm { d } y } { \mathrm {~d} x }$ and $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ when $x = 0$.
2. Using Maclaurin's theorem, show that, for small values of $x$, $$\sqrt { 3 + \mathrm { e } ^ { x } } \approx 2 + \frac { 1 } { 4 } x + \frac { 7 } { 64 } x ^ { 2 }$$
Find $$\lim _ { x \rightarrow 0 } \left[ \frac { \sqrt { 3 + \mathrm { e } ^ { x } } - 2 } { \sin 2 x } \right]$$

AQA FP3 2008 June Q8

8 The polar equation of a curve $C$ is $$r = 5 + 2 \cos \theta , \quad - \pi \leqslant \theta \leqslant \pi$$

Verify that the points $A$ and $B$, with polar coordinates ( 7,0 ) and ( $3 , \pi$ ) respectively, lie on the curve $C$.
Sketch the curve $C$.
Find the area of the region bounded by the curve $C$.
The point $P$ is the point on the curve $C$ for which $\theta = \alpha$, where $0 < \alpha \leqslant \frac { \pi } { 2 }$. The point $Q$ lies on the curve such that $P O Q$ is a straight line, where the point $O$ is the pole. Find, in terms of $\alpha$, the area of triangle $O Q B$.

AQA FP3 2009 June Q1

1 The function $y ( x )$ satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { f } ( x , y )$$ where $$\mathrm { f } ( x , y ) = \sqrt { x ^ { 2 } + y + 1 }$$ and $$y ( 3 ) = 2$$

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 $y ( 3.1 )$, 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 $y ( 3.2 )$, giving your answer to three decimal places.

AQA FP3 2009 June Q2

2 By using an integrating factor, find the solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } - y \tan x = 2 \sin x$$ given that $y = 2$ when $x = 0$.
(9 marks)

AQA FP3 2009 June Q3

3 The diagram shows a sketch of a circle which passes through the origin $O$.
\includegraphics[max width=\textwidth, alt={}, center]{13cfb9ca-9495-4b69-80c5-9fb7e8e0f957-3_423_451_356_794} The equation of the circle is $( x - 3 ) ^ { 2 } + ( y - 4 ) ^ { 2 } = 25$ and $O A$ is a diameter.

Find the cartesian coordinates of the point $A$.
Using $O$ as the pole and the positive $x$-axis as the initial line, the polar coordinates of $A$ are $( k , \alpha )$.
1. Find the value of $k$ and the value of $\tan \alpha$.
2. Find the polar equation of the circle $( x - 3 ) ^ { 2 } + ( y - 4 ) ^ { 2 } = 25$, giving your answer in the form $r = p \cos \theta + q \sin \theta$.

AQA FP3 2009 June Q4

4 Evaluate the improper integral $$\int _ { 1 } ^ { \infty } \left( \frac { 1 } { x } - \frac { 4 } { 4 x + 1 } \right) \mathrm { d } x$$ showing the limiting process used and giving your answer in the form $\ln k$, where $k$ is a constant to be found.

AQA FP3 2009 June Q5

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

Find the value of the constant $k$ for which $y = k \sin x$ is a particular integral of the given differential equation.
Solve the differential equation, expressing $y$ in terms of $x$, given that $y = 1$ and $\frac { \mathrm { d } y } { \mathrm {~d} x } = 4$ when $x = 0$.
(8 marks)

AQA FP3 2009 June Q6

6 The function f is defined by $$\mathrm { f } ( x ) = ( 9 + \tan x ) ^ { \frac { 1 } { 2 } }$$

1. Find $f ^ { \prime \prime } ( x )$.
2. By using Maclaurin's theorem, show that, for small values of $x$, $$( 9 + \tan x ) ^ { \frac { 1 } { 2 } } \approx 3 + \frac { x } { 6 } - \frac { x ^ { 2 } } { 216 }$$
Find $$\lim _ { x \rightarrow 0 } \left[ \frac { f ( x ) - 3 } { \sin 3 x } \right]$$

AQA FP3 2009 June Q7

7 The diagram shows the curve $C _ { 1 }$ with polar equation $$r = 1 + 6 \mathrm { e } ^ { - \frac { \theta } { \pi } } , \quad 0 \leqslant \theta \leqslant 2 \pi$$ \includegraphics[max width=\textwidth, alt={}, center]{13cfb9ca-9495-4b69-80c5-9fb7e8e0f957-4_300_513_1414_760}

Find, in terms of $\pi$ and e , the area of the shaded region bounded by $C _ { 1 }$ and the initial line.
The polar equation of a curve $C _ { 2 }$ is $$r = \mathrm { e } ^ { \frac { \theta } { \pi } } , \quad 0 \leqslant \theta \leqslant 2 \pi$$ Sketch the curve $C _ { 2 }$ and state the polar coordinates of the end-points of this curve.
The curves $C _ { 1 }$ and $C _ { 2 }$ intersect at the point $P$. Find the polar coordinates of $P$.

AQA FP3 2009 June Q8

Given that $x = t ^ { 2 }$, where $t \geqslant 0$, and that $y$ is a function of $x$, show that:
1. $2 \sqrt { x } \frac { \mathrm {~d} y } { \mathrm {~d} x } = \frac { \mathrm { d } y } { \mathrm {~d} t }$;
2. $\quad 4 x \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } = \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } }$.
Hence show that the substitution $x = t ^ { 2 }$, where $t \geqslant 0$, transforms the differential equation $$4 x \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 ( 1 + 2 \sqrt { x } ) \frac { \mathrm { d } y } { \mathrm {~d} x } - 3 y = 0$$ into $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} t } - 3 y = 0$$ (2 marks)
Hence find the general solution of the differential equation $$4 x \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 ( 1 + 2 \sqrt { x } ) \frac { \mathrm { d } y } { \mathrm {~d} x } - 3 y = 0$$ giving your answer in the form $y = \mathrm { g } ( x )$.

AQA FP3 2010 June Q1

1 The function $y ( x )$ satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { f } ( x , y )$$ where $$\mathrm { f } ( x , y ) = x + 3 + \sin y$$ and $$y ( 1 ) = 1$$

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 $y ( 1.1 )$, 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 $y ( 1.2 )$, giving your answer to three decimal places.

AQA FP3 2010 June Q2

Find the value of the constant $k$ for which $k \sin 2 x$ is a particular integral of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + y = \sin 2 x$$
Hence find the general solution of this differential equation.

AQA FP3 2010 June Q3

Explain why $\int _ { 1 } ^ { \infty } 4 x \mathrm { e } ^ { - 4 x } \mathrm {~d} x$ is an improper integral.
Find $\quad \int 4 x \mathrm { e } ^ { - 4 x } \mathrm {~d} x$.
Hence evaluate $\int _ { 1 } ^ { \infty } 4 x \mathrm { e } ^ { - 4 x } \mathrm {~d} x$, showing the limiting process used.

AQA FP3 2010 June Q4

4 By using an integrating factor, find the solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { 3 } { x } y = \left( x ^ { 4 } + 3 \right) ^ { \frac { 3 } { 2 } }$$ given that $y = \frac { 1 } { 5 }$ when $x = 1$.
(9 marks)

AQA FP3 2010 June Q5

Write down the expansion of $\cos 4 x$ in ascending powers of $x$ up to and including the term in $x ^ { 4 }$. Give your answer in its simplest form.
1. Given that $y = \ln \left( 2 - \mathrm { e } ^ { x } \right)$, find $\frac { \mathrm { d } y } { \mathrm {~d} x } , \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ and $\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }$.
  (You may leave your expression for $\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }$ unsimplified.)
2. Hence, by using Maclaurin's theorem, show that the first three non-zero terms in the expansion, in ascending powers of $x$, of $\ln \left( 2 - \mathrm { e } ^ { x } \right)$ are $$- x - x ^ { 2 } - x ^ { 3 }$$
Find $$\lim _ { x \rightarrow 0 } \left[ \frac { x \ln \left( 2 - \mathrm { e } ^ { x } \right) } { 1 - \cos 4 x } \right]$$

Questions — AQA FP3 (144 questions)

Browse by module

Browse by board

AQA FP3 2006 June Q4

AQA FP3 2006 June Q5

AQA FP3 2006 June Q6

AQA FP3 2006 June Q7

AQA FP3 2008 June Q1

AQA FP3 2008 June Q2

AQA FP3 2008 June Q3

AQA FP3 2008 June Q4

AQA FP3 2008 June Q5

AQA FP3 2008 June Q6

AQA FP3 2008 June Q7

AQA FP3 2008 June Q8

AQA FP3 2009 June Q1

AQA FP3 2009 June Q2

AQA FP3 2009 June Q3

AQA FP3 2009 June Q4

AQA FP3 2009 June Q5

AQA FP3 2009 June Q6

AQA FP3 2009 June Q7

AQA FP3 2009 June Q8

AQA FP3 2010 June Q1

AQA FP3 2010 June Q2

AQA FP3 2010 June Q3

AQA FP3 2010 June Q4

AQA FP3 2010 June Q5