Questions - OCR MEI - A-Level Maths

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OCR MEI FP2 2006 June Q1

18 marks Standard +0.8

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

18 marks Challenging +1.2

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

18 marks Standard +0.8

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

18 marks Challenging +1.2

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

18 marks Standard +0.8

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

18 marks Standard +0.8

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

18 marks Challenging +1.2

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

18 marks Standard +0.8

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

18 marks Standard +0.8

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

18 marks Standard +0.8

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

18 marks Standard +0.8

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

18 marks Standard +0.8

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

18 marks Standard +0.3

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

18 marks Challenging +1.2

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.

OCR MEI FP2 2010 June Q1

19 marks Standard +0.8

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

16 marks Challenging +1.2

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

19 marks Standard +0.8

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)

OCR MEI FP2 2010 June Q4

18 marks Challenging +1.2

Prove, using exponential functions, that $$\sinh 2 x = 2 \sinh x \cosh x$$ Differentiate this result to obtain a formula for $\cosh 2 x$.
Sketch the curve with equation $y = \cosh x - 1$. The region bounded by this curve, the $x$-axis, and the line $x = 2$ is rotated through $2 \pi$ radians about the $x$-axis. Find, correct to 3 decimal places, the volume generated. (You must show your working; numerical integration by calculator will receive no credit.)
Show that the curve with equation $$y = \cosh 2 x + \sinh x$$ has exactly one stationary point.
Determine, in exact logarithmic form, the $x$-coordinate of the stationary point.

OCR MEI FP2 2010 June Q5

18 marks Challenging +1.2

5 In parts (i), (ii), (iii) of this question you are required to investigate curves with the equation $$x ^ { k } + y ^ { k } = 1$$ for various positive values of $k$.

Firstly consider cases in which $k$ is a positive even integer.
(A) State the shape of the curve when $k = 2$.
(B) Sketch, on the same axes, the curves for $k = 2$ and $k = 4$.
(C) Describe the shape that the curve tends to as $k$ becomes very large.
(D) State the range of possible values of $x$ and $y$.
Now consider cases in which $k$ is a positive odd integer.
(A) Explain why $x$ and $y$ may take any value.
(B) State the shape of the curve when $k = 1$.
(C) Sketch the curve for $k = 3$. State the equation of the asymptote of this curve.
(D) Sketch the shape that the curve tends to as $k$ becomes very large.
Now let $k = \frac { 1 } { 2 }$. Sketch the curve, indicating the range of possible values of $x$ and $y$.
Now consider the modified equation $| x | ^ { k } + | y | ^ { k } = 1$.
(A) Sketch the curve for $k = \frac { 1 } { 2 }$.
(B) Investigate the shape of the curve for $k = \frac { 1 } { n }$ as the positive integer $n$ becomes very large.

OCR MEI S1 2005 January Q1

7 marks Easy -1.8

1 The number of minutes of recorded music on a sample of 100 CDs is summarised below.

Time ( $t$ minutes)	$40 \leqslant t < 45$	$45 \leqslant t < 50$	$50 \leqslant t < 60$	$60 \leqslant t < 70$	$70 \leqslant t < 90$
Number of CDs	26	18	31	16	9

Illustrate the data by means of a histogram.
Identify two features of the distribution.

OCR MEI S1 2005 January Q2

7 marks Moderate -0.8

2 A sprinter runs many 100 -metre trials, and the time, $x$ seconds, for each is recorded. A sample of eight of these times is taken, as follows. $$\begin{array} { l l l l l l l l } 10.53 & 10.61 & 10.04 & 10.49 & 10.63 & 10.55 & 10.47 & 10.63 \end{array}$$

Calculate the sample mean, $\bar { x }$, and sample standard deviation, $s$, of these times.
Show that the time of 10.04 seconds may be regarded as an outlier.
Discuss briefly whether or not the time of 10.04 seconds should be discarded.

OCR MEI S1 2005 January Q3

3 marks Moderate -0.3

3 The Venn diagram illustrates the occurrence of two events $A$ and $B$. \includegraphics[max width=\textwidth, alt={}, center]{b35b2b3b-0d26-4a35-b4d2-110bf270d5dc-2_513_826_1713_658} You are given that $\mathrm { P } ( A \cap B ) = 0.3$ and that the probability that neither $A$ nor $B$ occurs is 0.1 . You are also given that $\mathrm { P } ( A ) = 2 \mathrm { P } ( B )$. Find $\mathrm { P } ( B )$.

OCR MEI S1 2005 January Q4

6 marks Moderate -0.8

4 The number, $X$, of children per family in a certain city is modelled by the probability distribution $\mathrm { P } ( X = r ) = k ( 6 - r ) ( 1 + r )$ for $r = 0,1,2,3,4$.

Copy and complete the following table and hence show that the value of $k$ is $\frac { 1 } { 50 }$.
$r$ 0 1 2 3 4
$\mathrm { P } ( X = r )$ $6 k$ $10 k$
Calculate $\mathrm { E } ( X )$.
Hence write down the probability that a randomly selected family in this city has more than the mean number of children.

OCR MEI S1 2005 January Q5

5 marks Easy -1.2

5 A rugby union team consists of 15 players made up of 8 forwards and 7 backs. A manager has to select his team from a squad of 12 forwards and 11 backs.

In how many ways can the manager select the forwards?
In how many ways can the manager select the team?

OCR MEI S1 2005 January Q6

8 marks Easy -1.2

6 An amateur weather forecaster describes each day as either sunny, cloudy or wet. He keeps a record each day of his forecast and of the actual weather. His results for one particular year are given in the table.

Weather Forecast

\multirow{2}{*}{Total}

\cline { 3 - 6 } \multicolumn{2}{|c|}{}

A day is selected at random from that year.

Show that the probability that the forecast is correct is $\frac { 264 } { 365 }$. Find the probability that
the forecast is correct, given that the forecast is sunny,
the forecast is correct, given that the weather is wet,
the weather is cloudy, given that the forecast is correct.

Time ( \(t\) minutes)	\(40 \leqslant t < 45\)	\(45 \leqslant t < 50\)	\(50 \leqslant t < 60\)	\(60 \leqslant t < 70\)	\(70 \leqslant t < 90\)
Number of CDs	26	18	31	16	9

\(r\)	0	1	2	3	4
\(\mathrm { P } ( X = r )\)	\(6 k\)	\(10 k\)

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