Questions - AQA - 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 $$\lim _ { x \rightarrow 0 } \left[ \frac { x \ln \left( 2 - \mathrm { e } ^ { x } \right) } { 1 - \cos 4 x } \right]$$

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