Questions FP2 - A-Level Maths

Sketch the curve $C$.
Find the polar coordinates of the points where tangents to $C$ are parallel to the initial line.
Find the area of the region bounded by $C$.

Edexcel FP2 Specimen Q7

7. (a) Given that $x = e ^ { t }$, show that

$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { e } ^ { - t } \frac { \mathrm {~d} y } { \mathrm {~d} t }$$
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \mathrm { e } ^ { - 2 t } \left( \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } - \frac { \mathrm { d } y } { \mathrm {~d} t } \right)$$ (b) Use you answers to part (a) to show that the substitution $x = \mathrm { e } ^ { t }$ transforms the differential equation $$x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y = x ^ { 3 }$$ into $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } - 3 \frac { \mathrm {~d} y } { \mathrm {~d} t } + 2 y = \mathrm { e } ^ { 3 t }$$ (c) Hence find the general solution of $$x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y = x ^ { 3 }$$

Edexcel FP2 Specimen Q8

(a) Given that $z = e ^ { i \theta }$, show that

$$z ^ { p } + \frac { 1 } { z ^ { p } } = 2 \cos p \theta$$ where $p$ is a positive integer.
(b) Given that $$\cos ^ { 4 } \theta = A \cos 4 \theta + B \cos 2 \theta + C$$ find the values of the constants $A , B$ and $C$. The region $R$ bounded by the curve with equation $y = \cos ^ { 2 } x , - \frac { \pi } { 2 } \leq x \leq \frac { \pi } { 2 }$, and the $x$-axis is rotated through $2 \pi$ about the $x$-axis.
(c) Find the volume of the solid generated.

Edexcel FP2 2006 January Q2

Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 5 x = 0$$
Given that $x = 1$ and $\frac { \mathrm { d } x } { \mathrm {~d} t } = 1$ at $t = 0$, find the particular solution of the differential equation, giving your answer in the form $x = \mathrm { f } ( t )$.
Sketch the curve with equation $x = \mathrm { f } ( t ) , 0 \leq t \leq \pi$, showing the coordinates, as multiples of $\pi$, of the points where the curve cuts the $x$-axis.
(4)(Total 13 marks)

Edexcel FP2 2002 June Q3

Show that $y = \frac { 1 } { 2 } x ^ { 2 } \mathrm { e } ^ { x }$ is a solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = \mathrm { e } ^ { x }$$
Solve the differential equation $\quad \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = \mathrm { e } ^ { x }$.
given that at $x = 0 , y = 1$ and $\frac { \mathrm { d } y } { \mathrm {~d} x } = 2$.

Edexcel FP2 2004 June Q2

$$\frac { \mathrm { d } y } { \mathrm {~d} x } + y \left( 1 + \frac { 3 } { x } \right) = \frac { 1 } { x ^ { 2 } } , \quad x > 0$$

Verify that $x ^ { 3 } \mathrm { e } ^ { x }$ is an integrating factor for the differential equation.
Find the general solution of the differential equation.
Given that $y = 1$ at $x = 1$, find $y$ at $x = 2$.
(3)(Total 10 marks)

Edexcel FP2 2005 June Q2

Find the general solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + 2 y \cot 2 x = \sin x , \quad 0 < x < \frac { \pi } { 2 }$$ giving your answer in the form $y = \mathrm { f } ( x )$.
(Total 7 marks)

Edexcel FP2 2009 June Q2

Solve the equation $$z ^ { 3 } = 4 \sqrt { } 2 - 4 \sqrt { } 2 i$$ giving your answers in the form $r ( \cos \theta + \mathrm { i } \sin \theta )$, where $- \pi < \theta \leqslant \pi$.

OCR MEI FP2 2006 June Q1

A curve has polar equation $r = a ( \sqrt { 2 } + 2 \cos \theta )$ for $- \frac { 3 } { 4 } \pi \leqslant \theta \leqslant \frac { 3 } { 4 } \pi$, where $a$ is a positive constant.
1. Sketch the curve.
2. Find, in an exact form, the area of the region enclosed by the curve.
1. Find the Maclaurin series for the function $\mathrm { f } ( x ) = \tan \left( \frac { 1 } { 4 } \pi + x \right)$, up to the term in $x ^ { 2 }$.
2. Use the Maclaurin series to show that, when $h$ is small, $$\int _ { - h } ^ { h } x ^ { 2 } \tan \left( \frac { 1 } { 4 } \pi + x \right) \mathrm { d } x \approx \frac { 2 } { 3 } h ^ { 3 } + \frac { 4 } { 5 } h ^ { 5 }$$

OCR MEI FP2 2006 June Q2

1. Given that $z = \cos \theta + \mathrm { j } \sin \theta$, express $z ^ { n } + \frac { 1 } { z ^ { n } }$ and $z ^ { n } - \frac { 1 } { z ^ { n } }$ in simplified trigonometric form.
2. By considering $\left( z - \frac { 1 } { z } \right) ^ { 4 } \left( z + \frac { 1 } { z } \right) ^ { 2 }$, find $A , B , C$ and $D$ such that $$\sin ^ { 4 } \theta \cos ^ { 2 } \theta = A \cos 6 \theta + B \cos 4 \theta + C \cos 2 \theta + D$$
1. Find the modulus and argument of $4 + 4 \mathrm { j }$.
2. Find the fifth roots of $4 + 4 \mathrm { j }$ in the form $r \mathrm { e } ^ { \mathrm { j } \theta }$, where $r > 0$ and $- \pi < \theta \leqslant \pi$. Illustrate these fifth roots on an Argand diagram.
3. Find integers $p$ and $q$ such that $( p + q \mathrm { j } ) ^ { 5 } = 4 + 4 \mathrm { j }$.

OCR MEI FP2 2006 June Q3

Find the inverse of the matrix $\left( \begin{array} { r r r } 4 & 1 & k
3 & 2 & 5
8 & 5 & 13 \end{array} \right)$, where $k \neq 5$.
Solve the simultaneous equations $$\begin{aligned} & 4 x + y + 7 z = 12
& 3 x + 2 y + 5 z = m
& 8 x + 5 y + 13 z = 0 \end{aligned}$$ giving $x , y$ and $z$ in terms of $m$.
Find the value of $p$ for which the simultaneous equations $$\begin{aligned} & 4 x + y + 5 z = 12
& 3 x + 2 y + 5 z = p
& 8 x + 5 y + 13 z = 0 \end{aligned}$$ have solutions, and find the general solution in this case.

OCR MEI FP2 2006 June Q4

Starting from the definitions of $\sinh x$ and $\cosh x$ in terms of exponentials, prove that $$1 + 2 \sinh ^ { 2 } x = \cosh 2 x$$
Solve the equation $$2 \cosh 2 x + \sinh x = 5 ,$$ giving the answers in an exact logarithmic form.
Show that $\int _ { 0 } ^ { \ln 3 } \sinh ^ { 2 } x \mathrm {~d} x = \frac { 10 } { 9 } - \frac { 1 } { 2 } \ln 3$.
Find the exact value of $\int _ { 3 } ^ { 5 } \sqrt { x ^ { 2 } - 9 } \mathrm {~d} x$.

OCR MEI FP2 2007 June Q1

A curve has polar equation $r = a ( 1 - \cos \theta )$, where $a$ is a positive constant.
1. Sketch the curve.
2. Find the area of the region enclosed by the section of the curve for which $0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi$ and the line $\theta = \frac { 1 } { 2 } \pi$.
Use a trigonometric substitution to show that $\int _ { 0 } ^ { 1 } \frac { 1 } { \left( 4 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x = \frac { 1 } { 4 \sqrt { 3 } }$.
In this part of the question, $\mathrm { f } ( x ) = \arccos ( 2 x )$.
1. Find $\mathrm { f } ^ { \prime } ( x )$.
2. Use a standard series to expand $\mathrm { f } ^ { \prime } ( x )$, and hence find the series for $\mathrm { f } ( x )$ in ascending powers of $x$, up to the term in $x ^ { 5 }$.

OCR MEI FP2 2007 June Q2

Use de Moivre's theorem to show that $\sin 5 \theta = 5 \sin \theta - 20 \sin ^ { 3 } \theta + 16 \sin ^ { 5 } \theta$.
1. Find the cube roots of $- 2 + 2 \mathrm { j }$ in the form $r \mathrm { e } ^ { \mathrm { j } \theta }$ where $r > 0$ and $- \pi < \theta \leqslant \pi$. These cube roots are represented by points $\mathrm { A } , \mathrm { B }$ and C in the Argand diagram, with A in the first quadrant and ABC going anticlockwise. The midpoint of AB is M , and M represents the complex number $w$.
2. Draw an Argand diagram, showing the points $\mathrm { A } , \mathrm { B } , \mathrm { C }$ and M .
3. Find the modulus and argument of $w$.
4. Find $w ^ { 6 }$ in the form $a + b \mathrm { j }$.

OCR MEI FP2 2007 June Q3

3 Let $\mathbf { M } = \left( \begin{array} { r r r } 3 & 5 & 2
5 & 3 & - 2
2 & - 2 & - 4 \end{array} \right)$.

Show that the characteristic equation for $\mathbf { M }$ is $\lambda ^ { 3 } - 2 \lambda ^ { 2 } - 48 \lambda = 0$. You are given that $\left( \begin{array} { r } 1
- 1
1 \end{array} \right)$ is an eigenvector of $\mathbf { M }$ corresponding to the eigenvalue 0 .
Find the other two eigenvalues of $\mathbf { M }$, and corresponding eigenvectors.
Write down a matrix $\mathbf { P }$, and a diagonal matrix $\mathbf { D }$, such that $\mathbf { P } ^ { - 1 } \mathbf { M } ^ { 2 } \mathbf { P } = \mathbf { D }$.
Use the Cayley-Hamilton theorem to find integers $a$ and $b$ such that $\mathbf { M } ^ { 4 } = a \mathbf { M } ^ { 2 } + b \mathbf { M }$. Section B (18 marks)

OCR MEI FP2 2007 June Q4

Find $\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 9 x ^ { 2 } + 16 } } \mathrm {~d} x$, giving your answer in an exact logarithmic form.
1. Starting from the definitions of $\sinh x$ and $\cosh x$ in terms of exponentials, prove that $\sinh 2 x = 2 \sinh x \cosh x$.
2. Show that one of the stationary points on the curve $$y = 20 \cosh x - 3 \cosh 2 x$$ has coordinates $\left( \ln 3 , \frac { 59 } { 3 } \right)$, and find the coordinates of the other two stationary points.
3. Show that $\int _ { - \ln 3 } ^ { \ln 3 } ( 20 \cosh x - 3 \cosh 2 x ) \mathrm { d } x = 40$.

OCR MEI FP2 2007 June Q5

5 The curve with equation $y = \frac { x ^ { 2 } - k x + 2 k } { x + k }$ is to be investigated for different values of $k$.

Use your graphical calculator to obtain rough sketches of the curve in the cases $k = - 2$, $k = - 0.5$ and $k = 1$.
Show that the equation of the curve may be written as $y = x - 2 k + \frac { 2 k ( k + 1 ) } { x + k }$. Hence find the two values of $k$ for which the curve is a straight line.
When the curve is not a straight line, it is a conic.
(A) Name the type of conic.
(B) Write down the equations of the asymptotes.
Draw a sketch to show the shape of the curve when $1 < k < 8$. This sketch should show where the curve crosses the axes and how it approaches its asymptotes. Indicate the points A and B on the curve where $x = 1$ and $x = k$ respectively.

OCR MEI FP2 2008 June Q1

A curve has cartesian equation $\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 3 x y ^ { 2 }$.
1. Show that the polar equation of the curve is $r = 3 \cos \theta \sin ^ { 2 } \theta$.
2. Hence sketch the curve.
Find the exact value of $\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 4 - 3 x ^ { 2 } } } \mathrm {~d} x$.
1. Write down the series for $\ln ( 1 + x )$ and the series for $\ln ( 1 - x )$, both as far as the term in $x ^ { 5 }$.
2. Hence find the first three non-zero terms in the series for $\ln \left( \frac { 1 + x } { 1 - x } \right)$.
3. Use the series in part (ii) to show that $\sum _ { r = 0 } ^ { \infty } \frac { 1 } { ( 2 r + 1 ) 4 ^ { r } } = \ln 3$.

OCR MEI FP2 2008 June Q2

2 You are given the complex numbers $z = \sqrt { 32 } ( 1 + \mathrm { j } )$ and $w = 8 \left( \cos \frac { 7 } { 12 } \pi + \mathrm { j } \sin \frac { 7 } { 12 } \pi \right)$.

Find the modulus and argument of each of the complex numbers $z , z ^ { * } , z w$ and $\frac { z } { w }$.
Express $\frac { z } { w }$ in the form $a + b \mathrm { j }$, giving the exact values of $a$ and $b$.
Find the cube roots of $z$, in the form $r \mathrm { e } ^ { \mathrm { j } \theta }$, where $r > 0$ and $- \pi < \theta \leqslant \pi$.
Show that the cube roots of $z$ can be written as $$k _ { 1 } w ^ { * } , \quad k _ { 2 } z ^ { * } \quad \text { and } \quad k _ { 3 } \mathrm { j } w ,$$ where $k _ { 1 } , k _ { 2 }$ and $k _ { 3 }$ are real numbers. State the values of $k _ { 1 } , k _ { 2 }$ and $k _ { 3 }$.

OCR MEI FP2 2008 June Q3

Given the matrix $\mathbf { Q } = \left( \begin{array} { r r r } 2 & - 1 & k
1 & 0 & 1
3 & 1 & 2 \end{array} \right)$ (where $k \neq 3$ ), find $\mathbf { Q } ^ { - 1 }$ in terms of $k$.
Show that, when $k = 4 , \mathbf { Q } ^ { - 1 } = \left( \begin{array} { r r r } - 1 & 6 & - 1
1 & - 8 & 2
1 & - 5 & 1 \end{array} \right)$. The matrix $\mathbf { M }$ has eigenvectors $\left( \begin{array} { l } 2
1
3 \end{array} \right) , \left( \begin{array} { r } - 1
0
1 \end{array} \right)$ and $\left( \begin{array} { l } 4
1
2 \end{array} \right)$, with corresponding eigenvalues $1 , - 1$ and 3 respectively.
Write down a matrix $\mathbf { P }$ and a diagonal matrix $\mathbf { D }$ such that $\mathbf { P } ^ { - 1 } \mathbf { M P } = \mathbf { D }$, and hence find the matrix $\mathbf { M }$.
Write down the characteristic equation for $\mathbf { M }$, and use the Cayley-Hamilton theorem to find integers $a , b$ and $c$ such that $\mathbf { M } ^ { 4 } = a \mathbf { M } ^ { 2 } + b \mathbf { M } + c \mathbf { I }$. Section B (18 marks)

OCR MEI FP2 2008 June Q4

Starting from the definitions of $\sinh x$ and $\cosh x$ in terms of exponentials, prove that $$\cosh ^ { 2 } x - \sinh ^ { 2 } x = 1$$
Solve the equation $4 \cosh ^ { 2 } x + 9 \sinh x = 13$, giving the answers in exact logarithmic form.
Show that there is only one stationary point on the curve $$y = 4 \cosh ^ { 2 } x + 9 \sinh x$$ and find the $y$-coordinate of the stationary point.
Show that $\int _ { 0 } ^ { \ln 2 } \left( 4 \cosh ^ { 2 } x + 9 \sinh x \right) \mathrm { d } x = 2 \ln 2 + \frac { 33 } { 8 }$.

OCR MEI FP2 2008 June Q5

5 A curve has parametric equations $x = \lambda \cos \theta - \frac { 1 } { \lambda } \sin \theta , y = \cos \theta + \sin \theta$, where $\lambda$ is a positive constant.

Use your calculator to obtain a sketch of the curve in each of the cases $$\lambda = 0.5 , \quad \lambda = 3 \quad \text { and } \quad \lambda = 5 .$$
Given that the curve is a conic, name the type of conic.
Show that $y$ has a maximum value of $\sqrt { 2 }$ when $\theta = \frac { 1 } { 4 } \pi$.
Show that $x ^ { 2 } + y ^ { 2 } = \left( 1 + \lambda ^ { 2 } \right) + \left( \frac { 1 } { \lambda ^ { 2 } } - \lambda ^ { 2 } \right) \sin ^ { 2 } \theta$, and deduce that the distance from the origin of any point on the curve is between $\sqrt { 1 + \frac { 1 } { \lambda ^ { 2 } } }$ and $\sqrt { 1 + \lambda ^ { 2 } }$.
For the case $\lambda = 1$, show that the curve is a circle, and find its radius.
For the case $\lambda = 2$, draw a sketch of the curve, and label the points $\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E } , \mathrm { F } , \mathrm { G } , \mathrm { H }$ on the curve corresponding to $\theta = 0 , \frac { 1 } { 4 } \pi , \frac { 1 } { 2 } \pi , \frac { 3 } { 4 } \pi , \pi , \frac { 5 } { 4 } \pi , \frac { 3 } { 2 } \pi , \frac { 7 } { 4 } \pi$ respectively. You should make clear what is special about each of these points. \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (OCR) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. OCR is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }

OCR MEI FP2 2010 June Q1

1. Given that $\mathrm { f } ( t ) = \arcsin t$, write down an expression for $\mathrm { f } ^ { \prime } ( t )$ and show that $$\mathrm { f } ^ { \prime \prime } ( t ) = \frac { t } { \left( 1 - t ^ { 2 } \right) ^ { \frac { 3 } { 2 } } }$$
2. Show that the Maclaurin expansion of the function $\arcsin \left( x + \frac { 1 } { 2 } \right)$ begins $$\frac { \pi } { 6 } + \frac { 2 } { \sqrt { 3 } } x$$ and find the term in $x ^ { 2 }$.
Sketch the curve with polar equation $r = \frac { \pi a } { \pi + \theta }$, where $a > 0$, for $0 \leqslant \theta < 2 \pi$. Find, in terms of $a$, the area of the region bounded by the part of the curve for which $0 \leqslant \theta \leqslant \pi$ and the lines $\theta = 0$ and $\theta = \pi$.
Find the exact value of the integral $$\int _ { 0 } ^ { \frac { 3 } { 2 } } \frac { 1 } { 9 + 4 x ^ { 2 } } \mathrm {~d} x$$

OCR MEI FP2 2010 June Q2

Given that $z = \cos \theta + \mathrm { j } \sin \theta$, express $z ^ { n } + \frac { 1 } { z ^ { n } }$ and $z ^ { n } - \frac { 1 } { z ^ { n } }$ in simplified trigonometric form.
Hence find the constants $A , B , C$ in the identity $$\sin ^ { 5 } \theta \equiv A \sin \theta + B \sin 3 \theta + C \sin 5 \theta$$
1. Find the 4th roots of - 9 j in the form $r \mathrm { e } ^ { \mathrm { j } \theta }$, where $r > 0$ and $0 < \theta < 2 \pi$. Illustrate the roots on an Argand diagram.
2. Let the points representing these roots, taken in order of increasing $\theta$, be $\mathrm { P } , \mathrm { Q } , \mathrm { R } , \mathrm { S }$. The mid-points of the sides of PQRS represent the 4th roots of a complex number $w$. Find the modulus and argument of $w$. Mark the point representing $w$ on your Argand diagram.

OCR MEI FP2 2010 June Q3

1. A $3 \times 3$ matrix $\mathbf { M }$ has characteristic equation $$2 \lambda ^ { 3 } + \lambda ^ { 2 } - 13 \lambda + 6 = 0$$ Show that $\lambda = 2$ is an eigenvalue of $\mathbf { M }$. Find the other eigenvalues.
2. An eigenvector corresponding to $\lambda = 2$ is $\left( \begin{array} { r } 3
  - 3
  1 \end{array} \right)$. Evaluate $\mathbf { M } \left( \begin{array} { r } 3
  - 3
  1 \end{array} \right)$ and $\mathbf { M } ^ { 2 } \left( \begin{array} { r } 1
  - 1
  \frac { 1 } { 3 } \end{array} \right)$.
  Solve the equation $\mathbf { M } \left( \begin{array} { l } x
  y
  z \end{array} \right) = \left( \begin{array} { r } 3
  - 3
  1 \end{array} \right)$.
3. Find constants $A , B , C$ such that $$\mathbf { M } ^ { 4 } = A \mathbf { M } ^ { 2 } + B \mathbf { M } + C \mathbf { I }$$
A $2 \times 2$ matrix $\mathbf { N }$ has eigenvalues -1 and 2, with eigenvectors $\binom { 1 } { 2 }$ and $\binom { - 1 } { 1 }$ respectively. Find $\mathbf { N }$. Section B (18 marks)

Questions FP2 (1157 questions)

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