Questions - OCR MEI FP2 - A-Level Maths

OCR MEI FP2 2013 January Q3

18 marks Standard +0.3

3 You are given the matrix $\mathbf { M } = \left( \begin{array} { r r r } 1 & 3 & 0 \\ 3 & - 2 & - 1 \\ 0 & - 1 & 1 \end{array} \right)$.

Show that the characteristic equation of $\mathbf { M }$ is $$\lambda ^ { 3 } - 13 \lambda + 12 = 0 .$$
Find the eigenvalues and corresponding eigenvectors of $\mathbf { M }$.
Write down a matrix $\mathbf { P }$ and a diagonal matrix $\mathbf { D }$ such that $$\mathbf { M } ^ { n } = \mathbf { P D P } ^ { - 1 } .$$ (You are not required to calculate $\mathbf { P } ^ { - 1 }$.)

OCR MEI FP2 2013 January Q4

18 marks Challenging +1.8

4

Show that the curve with equation $$y = 3 \sinh x - 2 \cosh x$$ has no turning points.
Show that the curve crosses the $x$-axis at $x = \frac { 1 } { 2 } \ln 5$. Show that this is also the point at which the gradient of the curve has a stationary value.
Sketch the curve.
Express $( 3 \sinh x - 2 \cosh x ) ^ { 2 }$ in terms of $\sinh 2 x$ and $\cosh 2 x$. Hence or otherwise, show that the volume of the solid of revolution formed by rotating the region bounded by the curve and the axes through $360 ^ { \circ }$ about the $x$-axis is $$\pi \left( 3 - \frac { 5 } { 4 } \ln 5 \right) .$$ Option 2: Investigation of curves \section*{This question requires the use of a graphical calculator.}

OCR MEI FP2 2013 January Q5

18 marks Challenging +1.8

5 This question concerns the curves with polar equation $$r = \sec \theta + a \cos \theta ,$$ where $a$ is a constant which may take any real value, and $0 \leqslant \theta \leqslant 2 \pi$.

On a single diagram, sketch the curves for $a = 0 , a = 1 , a = 2$.
On a single diagram, sketch the curves for $a = 0 , a = - 1 , a = - 2$.
Identify a feature that the curves for $a = 1 , a = 2 , a = - 1 , a = - 2$ share.
Name a distinctive feature of the curve for $a = - 1$, and a different distinctive feature of the curve for $a = - 2$.
Show that, in cartesian coordinates, equation (*) may be written $$y ^ { 2 } = \frac { a x ^ { 2 } } { x - 1 } - x ^ { 2 }$$ Hence comment further on the feature you identified in part (iii).
Show algebraically that, when $a > 0$, the curve exists for $1 < x < 1 + a$. Find the set of values of $x$ for which the curve exists when $a < 0$.

OCR MEI FP2 2014 June Q1

19 marks Standard +0.8

1

Given that $\mathrm { f } ( x ) = \arccos x$,
1. sketch the graph of $y = \mathrm { f } ( x )$,
2. show that $\mathrm { f } ^ { \prime } ( x ) = - \frac { 1 } { \sqrt { 1 - x ^ { 2 } } }$,
3. obtain the Maclaurin series for $\mathrm { f } ( x )$ as far as the term in $x ^ { 3 }$.
A curve has polar equation $r = \theta + \sin \theta , \theta \geqslant 0$.
1. By considering $\frac { \mathrm { d } r } { \mathrm {~d} \theta }$ show that $r$ increases as $\theta$ increases. Sketch the curve for $0 \leqslant \theta \leqslant 4 \pi$.
2. You are given that $\sin \theta \approx \theta$ for small $\theta$. Find in terms of $\alpha$ the approximate area bounded by the curve and the lines $\theta = 0$ and $\theta = \alpha$, where $\alpha$ is small.

OCR MEI FP2 2014 June Q2

17 marks Standard +0.8

2

The infinite series $C$ and $S$ are defined as follows. $$\begin{gathered} C = a \cos \theta + a ^ { 2 } \cos 2 \theta + a ^ { 3 } \cos 3 \theta + \ldots \\ S = a \sin \theta + a ^ { 2 } \sin 2 \theta + a ^ { 3 } \sin 3 \theta + \ldots \end{gathered}$$ where $a$ is a real number and $| a | < 1$.
By considering $C + \mathrm { j } S$, show that $$S = \frac { a \sin \theta } { 1 - 2 a \cos \theta + a ^ { 2 } }$$ Find a corresponding expression for $C$.
P is one vertex of a regular hexagon in an Argand diagram. The centre of the hexagon is at the origin. P corresponds to the complex number $\sqrt { 3 } + \mathrm { j }$.
1. Find, in the form $x + \mathrm { j } y$, the complex numbers corresponding to the other vertices of the hexagon.
2. The six complex numbers corresponding to the vertices of the hexagon are squared to form the vertices of a new figure. Find, in the form $x + \mathrm { j } y$, the vertices of the new figure. Find the area of the new figure.

OCR MEI FP2 2014 June Q3

18 marks Standard +0.3

3

1. Find the eigenvalues and corresponding eigenvectors for the matrix $\mathbf { A }$, where $$\mathbf { A } = \left( \begin{array} { l l } 6 & - 3 \\ 4 & - 1 \end{array} \right)$$
2. Write down a matrix $\mathbf { P }$ and a diagonal matrix $\mathbf { D }$ such that $\mathbf { A } = \mathbf { P D P } ^ { - 1 }$.
1. The $3 \times 3$ matrix $\mathbf { B }$ has characteristic equation $$\lambda ^ { 3 } - 4 \lambda ^ { 2 } - 3 \lambda - 10 = 0$$ Show that 5 is an eigenvalue of $\mathbf { B }$. Show that $\mathbf { B }$ has no other real eigenvalues.
2. An eigenvector corresponding to the eigenvalue 5 is $\left( \begin{array} { r } - 2 \\ 1 \\ 4 \end{array} \right)$. Evaluate $\mathbf { B } \left( \begin{array} { r } - 2 \\ 1 \\ 4 \end{array} \right)$ and $\mathbf { B } ^ { 2 } \left( \begin{array} { r } 4 \\ - 2 \\ - 8 \end{array} \right)$.
  Solve the equation $\mathbf { B } \left( \begin{array} { l } x \\ y \\ z \end{array} \right) = \left( \begin{array} { r } - 20 \\ 10 \\ 40 \end{array} \right)$ for $x , y , z$.
3. Show that $\mathbf { B } ^ { 4 } = 19 \mathbf { B } ^ { 2 } + 22 \mathbf { B } + 40 \mathbf { I }$.

OCR MEI FP2 2014 June Q4

18 marks Challenging +1.2

4

Given that $\sinh y = x$, show that $$y = \ln \left( x + \sqrt { 1 + x ^ { 2 } } \right)$$ Differentiate (*) to show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sqrt { 1 + x ^ { 2 } } }$$
Find $\int \frac { 1 } { \sqrt { 25 + 4 x ^ { 2 } } } \mathrm {~d} x$, expressing your answer in logarithmic form.
Use integration by substitution with $2 x = 5 \sinh u$ to show that $$\int \sqrt { 25 + 4 x ^ { 2 } } \mathrm {~d} x = \frac { 25 } { 4 } \left( \ln \left( \frac { 2 x } { 5 } + \sqrt { 1 + \frac { 4 x ^ { 2 } } { 25 } } \right) + \frac { 2 x } { 5 } \sqrt { 1 + \frac { 4 x ^ { 2 } } { 25 } } \right) + c$$ where $c$ is an arbitrary constant. \section*{OCR}

OCR MEI FP2 2015 June Q3

18 marks Standard +0.8

3 This question concerns the matrix $\mathbf { M }$ where $\mathbf { M } = \left( \begin{array} { r r r } 5 & - 1 & 3 \\ 4 & - 3 & - 2 \\ 2 & 1 & 4 \end{array} \right)$.

Obtain the characteristic equation of $\mathbf { M }$. Find the eigenvalues of $\mathbf { M }$. These eigenvalues are denoted by $\lambda _ { 1 } , \lambda _ { 2 } , \lambda _ { 3 }$, where $\lambda _ { 1 } < \lambda _ { 2 } < \lambda _ { 3 }$.
Verify that an eigenvector corresponding to $\lambda _ { 1 }$ is $\left( \begin{array} { r } 1 \\ 3 \\ - 1 \end{array} \right)$ and that an eigenvector corresponding to $\lambda _ { 2 }$ is $\left( \begin{array} { r } 1 \\ 2 \\ - 1 \end{array} \right)$. Find an eigenvector of the form $\left( \begin{array} { l } a \\ 1 \\ c \end{array} \right)$ corresponding to $\lambda _ { 3 }$.
Write down a matrix $\mathbf { P }$ and a diagonal matrix $\mathbf { D }$ such that $\mathbf { M } = \mathbf { P D P } ^ { - 1 }$. (You are not required to calculate $\mathbf { P } ^ { - 1 }$.) Hence write down an expression for $\mathbf { M } ^ { 4 }$ in terms of $\mathbf { P }$ and a diagonal matrix. You should give the elements of the diagonal matrix explicitly.
Use the Cayley-Hamilton theorem to obtain an expression for $\mathbf { M } ^ { 4 }$ as a linear combination of $\mathbf { M }$ and $\mathbf { M } ^ { 2 }$.

OCR MEI FP2 2015 June Q4

18 marks Standard +0.8

4

Starting with the relationship $\cosh ^ { 2 } t - \sinh ^ { 2 } t = 1$, deduce a relationship between $\tanh ^ { 2 } t$ and $\operatorname { sech } ^ { 2 } t$. You are given that $y = \operatorname { artanh } x$.
Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 1 - x ^ { 2 } }$.
Show, by integrating the result in part (ii), that $y = \frac { 1 } { 2 } \ln \left( \frac { 1 + x } { 1 - x } \right)$.
Show that $\int _ { 0 } ^ { \frac { \sqrt { 3 } } { 6 } } \frac { 1 } { 1 - 3 x ^ { 2 } } \mathrm {~d} x = \frac { 1 } { \sqrt { 3 } } \operatorname { artanh } \frac { 1 } { 2 }$. Express this answer in logarithmic form.
Use integration by parts to find $\int \operatorname { artanh } x \mathrm {~d} x$, giving your answer in terms of logarithms. \section*{END OF QUESTION PAPER}

OCR MEI FP2 2012 June Q1

18 marks Standard +0.3

1

1. Differentiate the equation $\sin y = x$ with respect to $x$, and hence show that the derivative of $\arcsin x$ is $\frac { 1 } { \sqrt { 1 - x ^ { 2 } } }$.
2. Evaluate the following integrals, giving your answers in exact form.
  (A) $\int _ { - 1 } ^ { 1 } \frac { 1 } { \sqrt { 2 - x ^ { 2 } } } \mathrm {~d} x$ (B) $\int _ { - \frac { 1 } { 2 } } ^ { \frac { 1 } { 2 } } \frac { 1 } { \sqrt { 1 - 2 x ^ { 2 } } } \mathrm {~d} x$
A curve has polar equation $r = \tan \theta , 0 \leqslant \theta < \frac { 1 } { 2 } \pi$. The points on the curve have cartesian coordinates $( x , y )$. A sketch of the curve is given in Fig. 1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{99f0c663-bb5b-4456-854c-df177f5d8349-2_493_796_1123_605} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} Show that $x = \sin \theta$ and that $r ^ { 2 } = \frac { x ^ { 2 } } { 1 - x ^ { 2 } }$.
Hence show that the cartesian equation of the curve is $$y = \frac { x ^ { 2 } } { \sqrt { 1 - x ^ { 2 } } } .$$ Give the cartesian equation of the asymptote of the curve.

OCR MEI FP2 2012 June Q2

18 marks Standard +0.8

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. Beginning with an expression for $\left( z + \frac { 1 } { z } \right) ^ { 4 }$, find the constants $A , B , C$ in the identity $$\cos ^ { 4 } \theta \equiv A + B \cos 2 \theta + C \cos 4 \theta$$
3. Use the identity in part (ii) to obtain an expression for $\cos 4 \theta$ as a polynomial in $\cos \theta$.
1. Given that $z = 4 \mathrm { e } ^ { \mathrm { j } \pi / 3 }$ and that $w ^ { 2 } = z$, write down the possible values of $w$ in the form $r \mathrm { e } ^ { \mathrm { j } \theta }$, where $r > 0$. Show $z$ and the possible values of $w$ in an Argand diagram.
2. Find the least positive integer $n$ for which $z ^ { n }$ is real. Show that there is no positive integer $n$ for which $z ^ { n }$ is imaginary.
  For each possible value of $w$, find the value of $w ^ { 3 }$ in the form $a + \mathrm { j } b$ where $a$ and $b$ are real.

OCR MEI FP2 2012 June Q3

18 marks Challenging +1.2

3

Find the value of $a$ for which the matrix $$\mathbf { M } = \left( \begin{array} { r r r } 1 & 2 & 3 \\ - 1 & a & 4 \\ 3 & - 2 & 2 \end{array} \right)$$ does not have an inverse.
Assuming that $a$ does not have this value, find the inverse of $\mathbf { M }$ in terms of $a$.
Hence solve the following system of equations. $$\begin{aligned} x + 2 y + 3 z & = 1 \\ - x + 4 z & = - 2 \\ 3 x - 2 y + 2 z & = 1 \end{aligned}$$
Find the value of $b$ for which the following system of equations has a solution. $$\begin{aligned} x + 2 y + 3 z & = 1 \\ - x + 6 y + 4 z & = - 2 \\ 3 x - 2 y + 2 z & = b \end{aligned}$$ Find the general solution in this case and describe the solution geometrically.

OCR MEI FP2 2012 June Q4

18 marks Challenging +1.2

4

Prove, from definitions involving exponential functions, that $$\cosh 2 u = 2 \sinh ^ { 2 } u + 1$$
Prove that, if $y \geqslant 0$ and $\cosh y = u$, then $y = \ln \left( u + \sqrt { } \left( u ^ { 2 } - 1 \right) \right)$.
Using the substitution $2 x = \cosh u$, show that $$\int \sqrt { 4 x ^ { 2 } - 1 } \mathrm {~d} x = a x \sqrt { 4 x ^ { 2 } - 1 } - b \operatorname { arcosh } 2 x + c$$ where $a$ and $b$ are constants to be determined and $c$ is an arbitrary constant.
Find $\int _ { \frac { 1 } { 2 } } ^ { 1 } \sqrt { 4 x ^ { 2 } - 1 } \mathrm {~d} x$, expressing your answer in an exact form involving logarithms.

OCR MEI FP2 2012 June Q5

18 marks Challenging +1.2

5 This question concerns curves with polar equation $r = \sec \theta + a$, where $a$ is a constant.

State the set of values of $\theta$ between 0 and $2 \pi$ for which $r$ is undefined. For the rest of the question you should assume that $\theta$ takes all values between 0 and $2 \pi$ for which $r$ is defined.
Use your graphical calculator to obtain a sketch of the curve in the case $a = 0$. Confirm the shape of the curve by writing the equation in cartesian form.
Sketch the curve in the case $a = 1$. Now consider the curve in the case $a = - 1$. What do you notice?
By considering both curves for $0 < \theta < \pi$ and $\pi < \theta < 2 \pi$ separately, describe the relationship between the cases $a = 1$ and $a = - 1$.
What feature does the curve exhibit for values of $a$ greater than 1 ? Sketch a typical case.
Show that a cartesian equation of the curve $r = \sec \theta + a$ is $\left( x ^ { 2 } + y ^ { 2 } \right) ( x - 1 ) ^ { 2 } = a ^ { 2 } x ^ { 2 }$.

OCR MEI FP2 2013 June Q3

18 marks Standard +0.8

3 You are given the matrix $\mathbf { A } = \left( \begin{array} { r r r } k & - 7 & 4 \\ 2 & - 2 & 3 \\ 1 & - 3 & - 2 \end{array} \right)$.

Show that when $k = 5$ the determinant of $\mathbf { A }$ is zero. Obtain an expression for the inverse of $\mathbf { A }$ when $k \neq 5$.
Solve the matrix equation $$\left( \begin{array} { r r r }

OCR MEI FP2 2013 June Q4

18 marks Challenging +1.2

4

Prove, using exponential functions, that $\cosh ^ { 2 } u - \sinh ^ { 2 } u = 1$.
Given that $y = \operatorname { arsinh } x$, show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sqrt { 1 + x ^ { 2 } } }$$ and that $$y = \ln \left( x + \sqrt { 1 + x ^ { 2 } } \right)$$
Show that $$\int _ { 0 } ^ { 2 } \frac { 1 } { \sqrt { 4 + 9 x ^ { 2 } } } \mathrm {~d} x = \frac { 1 } { 3 } \ln ( 3 + \sqrt { 10 } )$$
Find, in exact logarithmic form, $$\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 1 + x ^ { 2 } } } \operatorname { arsinh } x \mathrm {~d} x$$

OCR MEI FP2 2011 June Q1

18 marks Standard +0.8

1

A curve has polar equation $r = a ( 1 - \sin \theta )$, where $a > 0$ and $0 \leqslant \theta < 2 \pi$.
1. Sketch the curve.
2. Find, in an exact form, the area of the region enclosed by the curve.
1. Find, in an exact form, the value of the integral $\int _ { - \frac { 1 } { 2 } } ^ { \frac { 1 } { 2 } } \frac { 1 } { 1 + 4 x ^ { 2 } } \mathrm {~d} x$.
2. Find, in an exact form, the value of the integral $\int _ { - \frac { 1 } { 2 } } ^ { \frac { 1 } { 2 } } \frac { 1 } { \left( 1 + 4 x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x$.

OCR MEI FP2 2011 June Q2

18 marks Challenging +1.2

2

Use de Moivre's theorem to find expressions for $\sin 5 \theta$ and $\cos 5 \theta$ in terms of $\sin \theta$ and $\cos \theta$.
Hence show that, if $t = \tan \theta$, then $$\tan 5 \theta = \frac { t \left( t ^ { 4 } - 10 t ^ { 2 } + 5 \right) } { 5 t ^ { 4 } - 10 t ^ { 2 } + 1 }$$
1. Find the 5th roots of $- 4 \sqrt { 2 }$ in the form $r \mathrm { e } ^ { \mathrm { j } \theta }$, where $r > 0$ and $0 \leqslant \theta < 2 \pi$. These 5th roots are represented in the Argand diagram, in order of increasing $\theta$, by the points A , $\mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E }$.
2. Draw the Argand diagram, making clear which point is which. The mid-point of AB is the point P which represents the complex number $w$.
3. Find, in exact form, the modulus and argument of $w$.
4. $w$ is an $n$th root of a real number $a$, where $n$ is a positive integer. State the least possible value of $n$ and find the corresponding value of $a$.

OCR MEI FP2 2011 June Q3

18 marks Challenging +1.2

3

Find the value of $k$ for which the matrix $$\mathbf { M } = \left( \begin{array} { r r r } 1 & - 1 & k \\ 5 & 4 & 6 \\ 3 & 2 & 4 \end{array} \right)$$ does not have an inverse.
Assuming that $k$ does not take this value, find the inverse of $\mathbf { M }$ in terms of $k$.
In the case $k = 3$, evaluate $$\mathbf { M } \left( \begin{array} { r } - 3 \\ 3 \\ 1 \end{array} \right)$$
State the significance of what you have found in part (ii).
Find the value of $t$ for which the system of equations $$\begin{array} { r } x - y + 3 z = t \\ 5 x + 4 y + 6 z = 1 \\ 3 x + 2 y + 4 z = 0 \end{array}$$ has solutions. Find the general solution in this case and describe the solution geometrically.

OCR MEI FP2 2011 June Q4

18 marks Challenging +1.2

4

Given that $\cosh y = x$, show that $y = \pm \ln \left( x + \sqrt { x ^ { 2 } - 1 } \right)$ and that $\operatorname { arcosh } x = \ln \left( x + \sqrt { x ^ { 2 } - 1 } \right)$.
Find $\int _ { \frac { 4 } { 5 } } ^ { 1 } \frac { 1 } { \sqrt { 25 x ^ { 2 } - 16 } } \mathrm {~d} x$, expressing your answer in an exact logarithmic form.
Solve the equation $$5 \cosh x - \cosh 2 x = 3$$ giving your answers in an exact logarithmic form.

OCR MEI FP2 2011 June Q5

18 marks Standard +0.8

5 In this question, you are required to investigate the curve with equation $$y = x ^ { m } ( 1 - x ) ^ { n } , \quad 0 \leqslant x \leqslant 1 ,$$ for various positive values of $m$ and $n$.

On separate diagrams, sketch the curve in each of the following cases.
(A) $m = 1 , n = 1$,
(B) $m = 2 , n = 2$,
(C) $m = 2 , n = 4$,
(D) $m = 4 , n = 2$.
What feature does the curve have when $m = n$ ? What is the effect on the curve of interchanging $m$ and $n$ when $m \neq n$ ?
Describe how the $x$-coordinate of the maximum on the curve varies as $m$ and $n$ vary. Use calculus to determine the $x$-coordinate of the maximum.
Find the condition on $m$ for the gradient to be zero when $x = 0$. State a corresponding result for the gradient to be zero when $x = 1$.
Use your calculator to investigate the shape of the curve for large values of $m$ and $n$. Hence conjecture what happens to the value of the integral $\int _ { 0 } ^ { 1 } x ^ { m } ( 1 - x ) ^ { n } \mathrm {~d} x$ as $m$ and $n$ tend to infinity.
Use your calculator to investigate the shape of the curve for small values of $m$ and $n$. Hence conjecture what happens to the shape of the curve as $m$ and $n$ tend to zero. }{www.ocr.org.uk}) after the live examination series.
If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity. For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1 GE.
OCR is part of the

OCR MEI FP2 2009 January Q2

18 marks Standard +0.3

Write down the modulus and argument of the complex number $\mathrm { e } ^ { \mathrm { j } \pi / 3 }$.
The triangle OAB in an Argand diagram is equilateral. O is the origin; A corresponds to the complex number $a = \sqrt { 2 } ( 1 + \mathrm { j } ) ; \mathrm { B }$ corresponds to the complex number $b$. Show A and the two possible positions for B in a sketch. Express $a$ in the form $r \mathrm { e } ^ { \mathrm { j } \theta }$. Find the two possibilities for $b$ in the form $r \mathrm { e } ^ { \mathrm { j } \theta }$.
Given that $z _ { 1 } = \sqrt { 2 } \mathrm { e } ^ { \mathrm { j } \pi / 3 }$, show that $z _ { 1 } ^ { 6 } = 8$. Write down, in the form $r \mathrm { e } ^ { \mathrm { j } \theta }$, the other five complex numbers $z$ such that $z ^ { 6 } = 8$. Sketch all six complex numbers in a new Argand diagram. Let $w = z _ { 1 } \mathrm { e } ^ { - \mathrm { j } \pi / 12 }$.
Find $w$ in the form $x + \mathrm { j } y$, and mark this complex number on your Argand diagram.
Find $w ^ { 6 }$, expressing your answer in as simple a form as possible.

OCR MEI FP2 2013 January Q2

18 marks Challenging +1.3

1. Show that $$1 + \mathrm { e } ^ { \mathrm { j } 2 \theta } = 2 \cos \theta ( \cos \theta + \mathrm { j } \sin \theta )$$
2. The series $C$ and $S$ are defined as follows. $$\begin{aligned} & C = 1 + \binom { n } { 1 } \cos 2 \theta + \binom { n } { 2 } \cos 4 \theta + \ldots + \cos 2 n \theta \\ & S = \binom { n } { 1 } \sin 2 \theta + \binom { n } { 2 } \sin 4 \theta + \ldots + \sin 2 n \theta \end{aligned}$$ By considering $C + \mathrm { j } S$, show that $$C = 2 ^ { n } \cos ^ { n } \theta \cos n \theta$$ and find a corresponding expression for $S$.
1. Express $\mathrm { e } ^ { \mathrm { j } 2 \pi / 3 }$ in the form $x + \mathrm { j } y$, where the real numbers $x$ and $y$ should be given exactly.
2. An equilateral triangle in the Argand diagram has its centre at the origin. One vertex of the triangle is at the point representing $2 + 4 \mathrm { j }$. Obtain the complex numbers representing the other two vertices, giving your answers in the form $x + \mathrm { j } y$, where the real numbers $x$ and $y$ should be given exactly.
3. Show that the length of a side of the triangle is $2 \sqrt { 15 }$.

OCR MEI FP2 2006 June Q5

18 marks Challenging +1.2

5 A curve has parametric equations $$x = \theta - k \sin \theta , \quad y = 1 - \cos \theta ,$$ where $k$ is a positive constant.

For the case $k = 1$, use your graphical calculator to sketch the curve. Describe its main features.
Sketch the curve for a value of $k$ between 0 and 1 . Describe briefly how the main features differ from those for the case $k = 1$.
For the case $k = 2$ :
(A) sketch the curve;
(B) find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $\theta$;
(C) show that the width of each loop, measured parallel to the $x$-axis, is $$2 \sqrt { 3 } - \frac { 2 \pi } { 3 }$$
Use your calculator to find, correct to one decimal place, the value of $k$ for which successive loops just touch each other.

OCR MEI FP2 2016 June Q1

Standard +0.3

1

1. Given that $\mathrm { f } ( x ) = \arctan x$, write down an expression for $\mathrm { f } ^ { \prime } ( x )$. Assuming that $x$ is small, use a binomial expansion to express $\mathrm { f } ^ { \prime } ( x )$ in ascending powers of $x$ as far as the term in $x ^ { 4 }$.
2. Hence express $\arctan x$ in ascending powers of $x$ as far as the term in $x ^ { 5 }$.
Find, in exact form, the value of the following integral. $$\int _ { 0 } ^ { \frac { 3 } { 4 } } \frac { 1 } { \sqrt { 3 - 4 x ^ { 2 } } } \mathrm {~d} x$$
A curve has polar equation $r = \frac { a } { \sqrt { \theta } }$ where $a > 0$.
1. Sketch the curve for $\frac { \pi } { 4 } \leqslant \theta \leqslant 2 \pi$.
2. State what happens to $r$ as $\theta$ tends to zero.
3. Find the area of the region enclosed by the part of the curve sketched in part (i) and the lines $\theta = \frac { \pi } { 4 }$ and $\theta = 2 \pi$. Give your answer in an exact simplified form.
  1. (i) Express $2 \sin \frac { 1 } { 2 } \theta \left( \sin \frac { 1 } { 2 } \theta - \mathrm { j } \cos \frac { 1 } { 2 } \theta \right)$ in terms of $z$ where $z = \cos \theta + \mathrm { j } \sin \theta$.
    (ii) The series $C$ and $S$ are defined as follows. $$\begin{aligned} C & = 1 - \binom { n } { 1 } \cos \theta + \binom { n } { 2 } \cos 2 \theta - \ldots + ( - 1 ) ^ { n } \binom { n } { n } \cos n \theta \\ S & = - \binom { n } { 1 } \sin \theta + \binom { n } { 2 } \sin 2 \theta - \ldots + ( - 1 ) ^ { n } \binom { n } { n } \sin n \theta \end{aligned}$$ Show that $$C + \mathrm { j } S = \left\{ - 2 \mathrm { j } \sin \frac { 1 } { 2 } \theta \left( \cos \frac { 1 } { 2 } \theta + \mathrm { j } \sin \frac { 1 } { 2 } \theta \right) \right\} ^ { n } .$$ Hence show that, for even values of $n$, $$\frac { C } { S } = \cot \left( \frac { 1 } { 2 } n \theta \right)$$
  2. Write the complex number $z = \sqrt { 6 } + \mathrm { j } \sqrt { 2 }$ in the form $r \mathrm { e } ^ { \mathrm { j } \theta }$, expressing $r$ and $\theta$ as simply as possible. Hence find the cube roots of $z$ in the form $r \mathrm { e } ^ { \mathrm { j } \theta }$. Show the points representing $z$ and its cube roots on an Argand diagram.
    1. Find the eigenvalues and eigenvectors of the matrix $\mathbf { M }$, where $$\mathbf { M } = \left( \begin{array} { l l } \frac { 1 } { 2 } & \frac { 1 } { 2 } \\ \frac { 2 } { 3 } & \frac { 1 } { 3 } \end{array} \right)$$ Hence express $\mathbf { M }$ in the form $\mathbf { P D P } ^ { - 1 }$ where $\mathbf { D }$ is a diagonal matrix.
    2. Write down an equation for $\mathbf { M } ^ { n }$ in terms of the matrices $\mathbf { P }$ and $\mathbf { D }$. Hence obtain expressions for the elements of $\mathbf { M } ^ { n }$.
      Show that $\mathbf { M } ^ { n }$ tends to a limit as $n$ tends to infinity. Find that limit.
    3. Express $\mathbf { M } ^ { - 1 }$ in terms of the matrices $\mathbf { P }$ and $\mathbf { D }$. Hence determine whether or not $\left( \mathbf { M } ^ { - 1 } \right) ^ { n }$ tends to a limit as $n$ tends to infinity. Section B (18 marks)
      1. Given that $y = \cosh x$, use the definition of $\cosh x$ in terms of exponential functions to prove that $$x = \pm \ln \left( y + \sqrt { y ^ { 2 } - 1 } \right) .$$
      2. Solve the equation $$\cosh x + \cosh 2 x = 5$$ giving the roots in an exact logarithmic form.
      3. Sketch the curve with equation $y = \cosh x + \cosh 2 x$. Show on your sketch the line $y = 5$. Find the area of the finite region bounded by the curve and the line $y = 5$. Give your answer in an exact form that does not involve hyperbolic functions. \section*{END OF QUESTION PAPER}

Questions — OCR MEI FP2 (86 questions)

Browse by module

Browse by board

OCR MEI FP2 2013 January Q3

OCR MEI FP2 2013 January Q4

OCR MEI FP2 2013 January Q5

OCR MEI FP2 2014 June Q1

OCR MEI FP2 2014 June Q2

OCR MEI FP2 2014 June Q3

OCR MEI FP2 2014 June Q4

OCR MEI FP2 2015 June Q3

OCR MEI FP2 2015 June Q4

OCR MEI FP2 2012 June Q1

OCR MEI FP2 2012 June Q2

OCR MEI FP2 2012 June Q3

OCR MEI FP2 2012 June Q4

OCR MEI FP2 2012 June Q5

OCR MEI FP2 2013 June Q3

OCR MEI FP2 2013 June Q4

OCR MEI FP2 2011 June Q1

OCR MEI FP2 2011 June Q2

OCR MEI FP2 2011 June Q3

OCR MEI FP2 2011 June Q4

OCR MEI FP2 2011 June Q5

OCR MEI FP2 2009 January Q2

OCR MEI FP2 2013 January Q2

OCR MEI FP2 2006 June Q5

OCR MEI FP2 2016 June Q1