Questions - AQA - A-Level Maths

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 $y ( 2.1 )$.
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 $y ( 2.2 )$.

AQA FP3 2008 January Q2

2 The diagram shows a sketch of part of the curve $C$ whose polar equation is $r = 1 + \tan \theta$. The point $O$ is the pole.
\includegraphics[max width=\textwidth, alt={}, center]{0c177d90-02ae-4e91-bc9d-d0c7051799b8-3_561_629_406_772} The points $P$ and $Q$ on the curve are given by $\theta = 0$ and $\theta = \frac { \pi } { 3 }$ respectively.

Show that the area of the region bounded by the curve $C$ and the lines $O P$ and $O Q$ is $$\frac { 1 } { 2 } \sqrt { 3 } + \ln 2$$ (6 marks)
Hence find the area of the shaded region bounded by the line $P Q$ and the arc $P Q$ of $C$.

AQA FP3 2008 January Q3

Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 5 y = 5$$
Hence express $y$ in terms of $x$, given that $y = 2$ and $\frac { \mathrm { d } y } { \mathrm {~d} x } = 3$ when $x = 0$.

AQA FP3 2008 January Q4

Explain why $\int _ { 1 } ^ { \infty } x \mathrm { e } ^ { - 3 x } \mathrm {~d} x$ is an improper integral.
Find $\int x \mathrm { e } ^ { - 3 x } \mathrm {~d} x$.
Hence evaluate $\int _ { 1 } ^ { \infty } x \mathrm { e } ^ { - 3 x } \mathrm {~d} x$, showing the limiting process used.

AQA FP3 2008 January Q5

5 By using an integrating factor, find the solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { 4 x } { x ^ { 2 } + 1 } y = x$$ given that $y = 1$ when $x = 0$. Give your answer in the form $y = \mathrm { f } ( x )$.

AQA FP3 2008 January Q6

6 A curve $C$ has polar equation $$r ^ { 2 } \sin 2 \theta = 8$$

Find the cartesian equation of $C$ in the form $y = \mathrm { f } ( x )$.
Sketch the curve $C$.
The line with polar equation $r = 2 \sec \theta$ intersects $C$ at the point $A$. Find the polar coordinates of $A$.

AQA FP3 2008 January Q7

1. Write down the expansion of $\ln ( 1 + 2 x )$ in ascending powers of $x$ up to and including the term in $x ^ { 3 }$.
2. State the range of values of $x$ for which this expansion is valid.
1. Given that $y = \ln \cos x$, find $\frac { \mathrm { d } y } { \mathrm {~d} x } , \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ and $\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }$.
2. Find the value of $\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } }$ when $x = 0$.
3. Hence, by using Maclaurin's theorem, show that the first two non-zero terms in the expansion, in ascending powers of $x$, of $\ln \cos x$ are $$- \frac { x ^ { 2 } } { 2 } - \frac { x ^ { 4 } } { 12 }$$
Find $$\lim _ { x \rightarrow 0 } \left[ \frac { x \ln ( 1 + 2 x ) } { x ^ { 2 } - \ln \cos x } \right]$$

AQA FP3 2008 January Q8

Given that $x = \mathrm { e } ^ { t }$ and that $y$ is a function of $x$, show that:
1. $x \frac { \mathrm {~d} y } { \mathrm {~d} x } = \frac { \mathrm { d } y } { \mathrm {~d} t }$;
2. $\quad x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } - \frac { \mathrm { d } y } { \mathrm {~d} t }$.
Hence find the general solution of the differential equation $$x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 6 x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 6 y = 0$$

AQA FP3 2009 January Q1

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

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 $y ( 1.2 )$.
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 $h = 0.2$, to obtain an approximation to $y ( 1.2 )$, giving your answer to four decimal places.

AQA FP3 2009 January Q2

Show that $\frac { 1 } { x ^ { 2 } }$ is an integrating factor for the first-order differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } - \frac { 2 } { x } y = x$$
Hence find the general solution of this differential equation, giving your answer in the form $y = \mathrm { f } ( x )$.

AQA FP3 2009 January Q3

3 The diagram shows a sketch of a loop, the pole $O$ and the initial line.
\includegraphics[max width=\textwidth, alt={}, center]{f4fdffc7-5647-4462-a983-1564d4e76a4d-3_305_553_383_740} The polar equation of the loop is $$r = ( 2 + \cos \theta ) \sqrt { \sin \theta } , \quad 0 \leqslant \theta \leqslant \pi$$ Find the area enclosed by the loop.

AQA FP3 2009 January Q4

Use integration by parts to show that $\int \ln x \mathrm {~d} x = x \ln x - x + c$, where $c$ is an arbitrary constant.
Hence evaluate $\int _ { 0 } ^ { 1 } \ln x \mathrm {~d} x$, showing the limiting process used.

AQA FP3 2009 January Q5

5 The diagram shows a sketch of a curve $C$, the pole $O$ and the initial line.
\includegraphics[max width=\textwidth, alt={}, center]{f4fdffc7-5647-4462-a983-1564d4e76a4d-3_301_668_1644_689} The curve $C$ has polar equation $$r = \frac { 2 } { 3 + 2 \cos \theta } , \quad 0 \leqslant \theta \leqslant 2 \pi$$

Verify that the point $L$ with polar coordinates ( $2 , \pi$ ) lies on $C$.
The circle with polar equation $r = 1$ intersects $C$ at the points $M$ and $N$.
1. Find the polar coordinates of $M$ and $N$.
2. Find the area of triangle $L M N$.
Find a cartesian equation of $C$, giving your answer in the form $9 y ^ { 2 } = \mathrm { f } ( x )$.

AQA FP3 2009 January Q6

6 The function f is defined by $\mathrm { f } ( x ) = \mathrm { e } ^ { 2 x } ( 1 + 3 x ) ^ { - \frac { 2 } { 3 } }$.

1. Use the series expansion for $\mathrm { e } ^ { x }$ to write down the first four terms in the series expansion of $\mathrm { e } ^ { 2 x }$.
2. Use the binomial series expansion of $( 1 + 3 x ) ^ { - \frac { 2 } { 3 } }$ and your answer to part (a)(i) to show that the first three non-zero terms in the series expansion of $\mathrm { f } ( x )$ are $1 + 3 x ^ { 2 } - 6 x ^ { 3 }$.
1. Given that $y = \ln ( 1 + 2 \sin x )$, find $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$.
2. By using Maclaurin's theorem, show that, for small values of $x$, $$\ln ( 1 + 2 \sin x ) \approx 2 x - 2 x ^ { 2 }$$
Find $$\lim _ { x \rightarrow 0 } \frac { 1 - \mathrm { f } ( x ) } { x \ln ( 1 + 2 \sin x ) }$$

AQA FP3 2009 January Q7

Given that $x = \mathrm { e } ^ { t }$ and that $y$ is a function of $x$, show that $$x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } - \frac { \mathrm { d } y } { \mathrm {~d} t }$$
Hence show that the substitution $x = \mathrm { e } ^ { t }$ transforms the differential equation $$x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 4 x \frac { \mathrm {~d} y } { \mathrm {~d} x } = 10$$ into $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } - 5 \frac { \mathrm {~d} y } { \mathrm {~d} t } = 10$$
Find the general solution of the differential equation $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } - 5 \frac { \mathrm {~d} y } { \mathrm {~d} t } = 10$.
Hence solve the differential equation $x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 4 x \frac { \mathrm {~d} y } { \mathrm {~d} x } = 10$, given that $y = 0$ and $\frac { \mathrm { d } y } { \mathrm {~d} x } = 8$ when $x = 1$.

AQA FP3 2010 January Q1

1 The function $y ( x )$ satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { f } ( x , y )$$ where $$\mathrm { f } ( x , y ) = x \ln ( 2 x + y )$$ 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 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 $y ( 3.1 )$, giving your answer to four decimal places.

AQA FP3 2010 January Q2

Given that $y = \ln ( 4 + 3 x )$, find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ and $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$.
Hence, by using Maclaurin's theorem, find the first three terms in the expansion, in ascending powers of $x$, of $\ln ( 4 + 3 x )$.
Write down the first three terms in the expansion, in ascending powers of $x$, of $\ln ( 4 - 3 x )$.
Show that, for small values of $x$, $$\ln \left( \frac { 4 + 3 x } { 4 - 3 x } \right) \approx \frac { 3 } { 2 } x$$

AQA FP3 2010 January Q3

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

AQA FP3 2010 January Q4

Write down the expansion of $\sin 3 x$ in ascending powers of $x$ up to and including the term in $x ^ { 3 }$.
Find $$\lim _ { x \rightarrow 0 } \left[ \frac { 3 x \cos 2 x - \sin 3 x } { 5 x ^ { 3 } } \right]$$

AQA FP3 2010 January Q5

5 It is given that $y$ satisfies the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 3 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y = 2 \mathrm { e } ^ { - 2 x }$$

Find the value of the constant $p$ for which $y = p x \mathrm { e } ^ { - 2 x }$ is a particular integral of the given differential equation.
Solve the differential equation, expressing $y$ in terms of $x$, given that $y = 2$ and $\frac { \mathrm { d } y } { \mathrm {~d} x } = 0$ when $x = 0$.

AQA FP3 2010 January Q6

Explain why $\int _ { 1 } ^ { \infty } \frac { \ln x ^ { 2 } } { x ^ { 3 } } \mathrm {~d} x$ is an improper integral.
1. Show that the substitution $y = \frac { 1 } { x }$ transforms $\int \frac { \ln x ^ { 2 } } { x ^ { 3 } } \mathrm {~d} x$ into $\int 2 y \ln y \mathrm {~d} y$.
2. Evaluate $\int _ { 0 } ^ { 1 } 2 y \ln y \mathrm {~d} y$, showing the limiting process used.
3. Hence write down the value of $\int _ { 1 } ^ { \infty } \frac { \ln x ^ { 2 } } { x ^ { 3 } } \mathrm {~d} x$.

AQA FP3 2010 January Q7

7 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 y = 8 x ^ { 2 } + 9 \sin x$$ (8 marks)

AQA FP3 2010 January Q8

8 The diagram shows a sketch of a curve $C$ and a line $L$, which is parallel to the initial line and touches the curve at the points $P$ and $Q$.
\includegraphics[max width=\textwidth, alt={}, center]{32de7ef6-b7aa-4bfd-a73a-e12bfc0147e2-5_506_762_447_639} The polar equation of the curve $C$ is $$r = 4 ( 1 - \sin \theta ) , \quad 0 \leqslant \theta < 2 \pi$$ and the polar equation of the line $L$ is $$r \sin \theta = 1$$

Show that the polar coordinates of $P$ are $\left( 2 , \frac { \pi } { 6 } \right)$ and find the polar coordinates of $Q$.
Find the area of the shaded region $R$ bounded by the line $L$ and the curve $C$. Give your answer in the form $m \sqrt { 3 } + n \pi$, where $m$ and $n$ are integers.

AQA FP3 2011 January Q1

1 The function $y ( x )$ satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { f } ( x , y )$$ where $$\mathrm { f } ( x , y ) = x + \sqrt { y }$$ and $$y ( 3 ) = 4$$ 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 $y ( 3.1 )$, giving your answer to three decimal places.

AQA FP3 2011 January Q2

Find the values of the constants $p$ and $q$ for which $p \sin x + q \cos x$ is a particular integral of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + 5 y = 13 \cos x$$
Hence find the general solution of this differential equation.

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