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OCR MEI FP3 2009 June Q2
24 marks Challenging +1.8
2 A surface has equation \(z = 3 x ( x + y ) ^ { 3 } - 2 x ^ { 3 } + 24 x\).
  1. Find \(\frac { \partial z } { \partial x }\) and \(\frac { \partial z } { \partial y }\).
  2. Find the coordinates of the three stationary points on the surface.
  3. Find the equation of the normal line at the point \(\mathrm { P } ( 1 , - 2,19 )\) on the surface.
  4. The point \(\mathrm { Q } ( 1 + k , - 2 + h , 19 + 3 h )\) is on the surface and is close to P . Find an approximate expression for \(k\) in terms of \(h\).
  5. Show that there is only one point on the surface at which the tangent plane has an equation of the form \(27 x - z = d\). Find the coordinates of this point and the corresponding value of \(d\).
OCR MEI FP3 2009 June Q3
24 marks Challenging +1.8
3 A curve has parametric equations \(x = a ( \theta + \sin \theta ) , y = a ( 1 - \cos \theta )\), for \(0 \leqslant \theta \leqslant \pi\), where \(a\) is a positive constant.
  1. Show that the arc length \(s\) from the origin to a general point on the curve is given by \(s = 4 a \sin \frac { 1 } { 2 } \theta\).
  2. Find the intrinsic equation of the curve giving \(s\) in terms of \(a\) and \(\psi\), where \(\tan \psi = \frac { \mathrm { d } y } { \mathrm {~d} x }\).
  3. Hence, or otherwise, show that the radius of curvature at a point on the curve is \(4 a \cos \frac { 1 } { 2 } \theta\).
  4. Find the coordinates of the centre of curvature corresponding to the point on the curve where \(\theta = \frac { 2 } { 3 } \pi\).
  5. Find the area of the surface generated when the curve is rotated through \(2 \pi\) radians about the \(x\)-axis.
OCR MEI FP3 2009 June Q4
24 marks Challenging +1.2
4 The group \(G = \{ 1,2,3,4,5,6 \}\) has multiplication modulo 7 as its operation. The group \(H = \{ 1,5,7,11,13,17 \}\) has multiplication modulo 18 as its operation.
  1. Show that the groups \(G\) and \(H\) are both cyclic.
  2. List all the proper subgroups of \(G\).
  3. Specify an isomorphism between \(G\) and \(H\). The group \(S = \{ \mathrm { a } , \mathrm { b } , \mathrm { c } , \mathrm { d } , \mathrm { e } , \mathrm { f } \}\) consists of functions with domain \(\{ 1,2,3 \}\) given by $$\begin{array} { l l l } \mathrm { a } ( 1 ) = 2 & \mathrm { a } ( 2 ) = 3 & \mathrm { a } ( 3 ) = 1 \\ \mathrm {~b} ( 1 ) = 3 & \mathrm {~b} ( 2 ) = 1 & \mathrm {~b} ( 3 ) = 2 \\ \mathrm { c } ( 1 ) = 1 & \mathrm { c } ( 2 ) = 3 & \mathrm { c } ( 3 ) = 2 \\ \mathrm {~d} ( 1 ) = 3 & \mathrm {~d} ( 2 ) = 2 & \mathrm {~d} ( 3 ) = 1 \\ \mathrm { e } ( 1 ) = 1 & \mathrm { e } ( 2 ) = 2 & \mathrm { e } ( 3 ) = 3 \\ \mathrm { f } ( 1 ) = 2 & \mathrm { f } ( 2 ) = 1 & \mathrm { f } ( 3 ) = 3 \end{array}$$ and the group operation is composition of functions.
  4. Show that ad \(= \mathrm { c }\) and find da.
  5. Give a reason why \(S\) is not isomorphic to \(G\).
  6. Find the order of each element of \(S\).
  7. List all the proper subgroups of \(S\).
OCR MEI FP3 2009 June Q5
24 marks Moderate -0.5
5 Each level of a fantasy computer game is set in a single location, Alphaworld, Betaworld, Chiworld or Deltaworld. After completing a level, a player goes on to the next level, which could be set in the same location as the previous level, or in a different location. In the first version of the game, the initial and transition probabilities are as follows.
Level 1 is set in Alphaworld or Betaworld, with probabilities 0.6, 0.4 respectively.
After a level set in Alphaworld, the next level will be set in Betaworld, Chiworld or Deltaworld, with probabilities \(0.7,0.1,0.2\) respectively.
After a level set in Betaworld, the next level will be set in Alphaworld, Betaworld or Deltaworld, with probabilities \(0.1,0.8,0.1\) respectively.
After a level set in Chiworld, the next level will also be set in Chiworld.
After a level set in Deltaworld, the next level will be set in Alphaworld, Betaworld or Chiworld, with probabilities \(0.3,0.6,0.1\) respectively. The situation is modelled as a Markov chain with four states.
  1. Write down the transition matrix.
  2. Find the probabilities that level 14 is set in each location.
  3. Find the probability that level 15 is set in the same location as level 14 .
  4. Find the level at which the probability of being set in Chiworld first exceeds 0.5.
  5. Following a level set in Betaworld, find the expected number of further levels which will be set in Betaworld before changing to a different location. In the second version of the game, the initial probabilities and the transition probabilities after Alphaworld, Betaworld and Deltaworld are all the same as in the first version; but after a level set in Chiworld, the next level will be set in Chiworld or Deltaworld, with probabilities \(0.9,0.1\) respectively.
  6. By considering powers of the new transition matrix, or otherwise, find the equilibrium probabilities for the four locations. In the third version of the game, the initial probabilities and the transition probabilities after Alphaworld, Betaworld and Deltaworld are again all the same as in the first version; but the transition probabilities after Chiworld have changed again. The equilibrium probabilities for Alphaworld, Betaworld, Chiworld and Deltaworld are now 0.11, 0.75, 0.04, 0.1 respectively.
  7. Find the new transition probabilities after a level set in Chiworld. OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements Booklet. This is produced for each series of examinations, is given to all schools that receive assessment material and is freely available to download from our public website (\href{http://www.ocr.org.uk}{www.ocr.org.uk}) after the live examination series.
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OCR MEI FP2 2010 January Q1
18 marks Standard +0.8
1
  1. Given that \(y = \arctan \sqrt { x }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), giving your answer in terms of \(x\). Hence show that $$\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { x } ( x + 1 ) } \mathrm { d } x = \frac { \pi } { 2 }$$
  2. A curve has cartesian equation $$x ^ { 2 } + y ^ { 2 } = x y + 1$$
    1. Show that the polar equation of the curve is $$r ^ { 2 } = \frac { 2 } { 2 - \sin 2 \theta }$$
    2. Determine the greatest and least positive values of \(r\) and the values of \(\theta\) between 0 and \(2 \pi\) for which they occur.
    3. Sketch the curve.
OCR MEI FP2 2010 January Q2
18 marks Standard +0.8
2
  1. Use de Moivre's theorem to find the constants \(a , b , c\) in the identity $$\cos 5 \theta \equiv a \cos ^ { 5 } \theta + b \cos ^ { 3 } \theta + c \cos \theta$$
  2. Let $$\begin{aligned} C & = \cos \theta + \cos \left( \theta + \frac { 2 \pi } { n } \right) + \cos \left( \theta + \frac { 4 \pi } { n } \right) + \ldots + \cos \left( \theta + \frac { ( 2 n - 2 ) \pi } { n } \right) \\ \text { and } S & = \sin \theta + \sin \left( \theta + \frac { 2 \pi } { n } \right) + \sin \left( \theta + \frac { 4 \pi } { n } \right) + \ldots + \sin \left( \theta + \frac { ( 2 n - 2 ) \pi } { n } \right) \end{aligned}$$ where \(n\) is an integer greater than 1 .
    By considering \(C + \mathrm { j } S\), show that \(C = 0\) and \(S = 0\).
  3. Write down the Maclaurin series for \(\mathrm { e } ^ { t }\) as far as the term in \(t ^ { 2 }\). Hence show that, for \(t\) close to zero, $$\frac { t } { \mathrm { e } ^ { t } - 1 } \approx 1 - \frac { 1 } { 2 } t$$
OCR MEI FP2 2010 January Q3
18 marks Standard +0.3
3
  1. Find the inverse of the matrix $$\left( \begin{array} { r r r } 1 & 1 & a \\ 2 & - 1 & 2 \\ 3 & - 2 & 2 \end{array} \right)$$ where \(a \neq 4\).
    Show that when \(a = - 1\) the inverse is $$\frac { 1 } { 5 } \left( \begin{array} { r r r } 2 & 0 & 1 \\ 2 & 5 & - 4 \\ - 1 & 5 & - 3 \end{array} \right)$$
  2. Solve, in terms of \(b\), the following system of equations. $$\begin{aligned} x + y - z & = - 2 \\ 2 x - y + 2 z & = b \\ 3 x - 2 y + 2 z & = 1 \end{aligned}$$
  3. Find the value of \(b\) for which the equations $$\begin{aligned} x + y + 4 z & = - 2 \\ 2 x - y + 2 z & = b \\ 3 x - 2 y + 2 z & = 1 \end{aligned}$$ have solutions. Give a geometrical interpretation of the solutions in this case. Section B (18 marks)
OCR MEI FP2 2010 January Q4
18 marks Standard +0.8
4
  1. Prove, using exponential functions, that $$\cosh 2 x = 1 + 2 \sinh ^ { 2 } x$$ Differentiate this result to obtain a formula for \(\sinh 2 x\).
  2. Solve the equation $$2 \cosh 2 x + 3 \sinh x = 3$$ expressing your answers in exact logarithmic form.
  3. Given that \(\cosh t = \frac { 5 } { 4 }\), show by using exponential functions that \(t = \pm \ln 2\). Find the exact value of the integral $$\int _ { 4 } ^ { 5 } \frac { 1 } { \sqrt { x ^ { 2 } - 16 } } \mathrm {~d} x$$
OCR MEI FP2 2010 January Q5
18 marks Challenging +1.8
5 A line PQ is of length \(k\) (where \(k > 1\) ) and it passes through the point ( 1,0 ). PQ is inclined at angle \(\theta\) to the positive \(x\)-axis. The end Q moves along the \(y\)-axis. See Fig. 5. The end P traces out a locus. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d43d1e11-3173-47c4-88c9-0397c8630a39-4_639_977_552_584} \captionsetup{labelformat=empty} \caption{Fig. 5}
\end{figure}
  1. Show that the locus of P may be expressed parametrically as follows. $$x = k \cos \theta \quad y = k \sin \theta - \tan \theta$$ You are now required to investigate curves with these parametric equations, where \(k\) may take any non-zero value and \(- \frac { 1 } { 2 } \pi < \theta < \frac { 1 } { 2 } \pi\).
  2. Use your calculator to sketch the curve in each of the cases \(k = 2 , k = 1 , k = \frac { 1 } { 2 }\) and \(k = - 1\).
  3. For what value(s) of \(k\) does the curve have
    (A) an asymptote (you should state what the asymptote is),
    (B) a cusp,
    (C) a loop?
  4. For the case \(k = 2\), find the angle at which the curve crosses itself.
  5. For the case \(k = 8\), find in an exact form the coordinates of the highest point on the loop.
  6. Verify that the cartesian equation of the curve is $$y ^ { 2 } = \frac { ( x - 1 ) ^ { 2 } } { x ^ { 2 } } \left( k ^ { 2 } - x ^ { 2 } \right) .$$
OCR MEI FP2 2011 January Q1
19 marks Standard +0.8
1
  1. A curve has polar equation \(r = 2 ( \cos \theta + \sin \theta )\) for \(- \frac { 1 } { 4 } \pi \leqslant \theta \leqslant \frac { 3 } { 4 } \pi\).
    1. Show that a cartesian equation of the curve is \(x ^ { 2 } + y ^ { 2 } = 2 x + 2 y\). Hence or otherwise sketch the curve.
    2. Find, by integration, the area of the region bounded by the curve and the lines \(\theta = 0\) and \(\theta = \frac { 1 } { 2 } \pi\). Give your answer in terms of \(\pi\).
    1. Given that \(\mathrm { f } ( x ) = \arctan \left( \frac { 1 } { 2 } x \right)\), find \(\mathrm { f } ^ { \prime } ( x )\).
    2. Expand \(\mathrm { f } ^ { \prime } ( x )\) in ascending powers of \(x\) as far as the term in \(x ^ { 4 }\). Hence obtain an expression for \(\mathrm { f } ( x )\) in ascending powers of \(x\) as far as the term in \(x ^ { 5 }\).
OCR MEI FP2 2011 January Q2
19 marks
2
    1. Given that \(z = \cos \theta + \mathrm { j } \sin \theta\), express \(z ^ { n } + z ^ { - n }\) and \(z ^ { n } - z ^ { - n }\) in simplified trigonometrical form.
    2. By considering \(\left( z + z ^ { - 1 } \right) ^ { 6 }\), show that $$\cos ^ { 6 } \theta = \frac { 1 } { 32 } ( \cos 6 \theta + 6 \cos 4 \theta + 15 \cos 2 \theta + 10 )$$
    3. Obtain an expression for \(\cos ^ { 6 } \theta - \sin ^ { 6 } \theta\) in terms of \(\cos 2 \theta\) and \(\cos 6 \theta\).
  2. The complex number \(w\) is \(8 \mathrm { e } ^ { \mathrm { j } \pi / 3 }\). You are given that \(z _ { 1 }\) is a square root of \(w\) and that \(z _ { 2 }\) is a cube root of \(w\). The points representing \(z _ { 1 }\) and \(z _ { 2 }\) in the Argand diagram both lie in the third quadrant.
    1. Find \(z _ { 1 }\) and \(z _ { 2 }\) in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\). Draw an Argand diagram showing \(w , z _ { 1 }\) and \(z _ { 2 }\).
    2. Find the product \(z _ { 1 } z _ { 2 }\), and determine the quadrant of the Argand diagram in which it lies.
    3. Show that the characteristic equation of the matrix $$\mathbf { M } = \left( \begin{array} { r r r } 1 & - 4 & 5 \\ 2 & 3 & - 2 \\ - 1 & 4 & 1 \end{array} \right)$$ is \(\lambda ^ { 3 } - 5 \lambda ^ { 2 } + 28 \lambda - 66 = 0\).
    4. Show that \(\lambda = 3\) is an eigenvalue of \(\mathbf { M }\), and determine whether or not \(\mathbf { M }\) has any other real eigenvalues.
    5. Find an eigenvector, \(\mathbf { v }\), of unit length corresponding to \(\lambda = 3\). State the magnitude of the vector \(\mathbf { M } ^ { n } \mathbf { v }\), where \(n\) is an integer.
    6. Using the Cayley-Hamilton theorem, obtain an equation for \(\mathbf { M } ^ { - 1 }\) in terms of \(\mathbf { M } ^ { 2 } , \mathbf { M }\) and \(\mathbf { I }\).
OCR MEI FP2 2011 January Q4
18 marks Challenging +1.2
4
  1. Solve the equation $$\sinh t + 7 \cosh t = 8$$ expressing your answer in exact logarithmic form. A curve has equation \(y = \cosh 2 x + 7 \sinh 2 x\).
  2. Using part (i), or otherwise, find, in an exact form, the coordinates of the points on the curve at which the gradient is 16 . Show that there is no point on the curve at which the gradient is zero.
    Sketch the curve.
  3. Find, in an exact form, the positive value of \(a\) for which the area of the region between the curve, the \(x\)-axis, the \(y\)-axis and the line \(x = a\) is \(\frac { 1 } { 2 }\).
OCR MEI FP2 2011 January Q5
18 marks Challenging +1.8
5 A curve has parametric equations $$x = t + a \sin t , \quad y = 1 - a \cos t$$ where \(a\) is a positive constant.
  1. Draw, on separate diagrams, sketches of the curve for \(- 2 \pi < t < 2 \pi\) in the cases \(a = 1 , a = 2\) and \(a = 0.5\). By investigating other cases, state the value(s) of \(a\) for which the curve has
    (A) loops,
    (B) cusps.
  2. Suppose that the point \(\mathrm { P } ( x , y )\) lies on the curve. Show that the point \(\mathrm { P } ^ { \prime } ( - x , y )\) also lies on the curve. What does this indicate about the symmetry of the curve?
  3. Find an expression in terms of \(a\) and \(t\) for the gradient of the curve. Hence find, in terms of \(a\), the coordinates of the turning points on the curve for \(- 2 \pi < t < 2 \pi\) and \(a \neq 1\).
  4. In the case \(a = \frac { 1 } { 2 } \pi\), show that \(t = \frac { 1 } { 2 } \pi\) and \(t = \frac { 3 } { 2 } \pi\) give the same point. Find the angle at which the curve crosses itself at this point.
OCR MEI FP2 2012 January Q1
18 marks Standard +0.3
1
  1. A curve has polar equation \(r = 1 + \cos \theta\) for \(0 \leqslant \theta < 2 \pi\).
    1. Sketch the curve.
    2. Find the area of the region enclosed by the curve, giving your answer in exact form.
  2. Assuming that \(x ^ { 4 }\) and higher powers may be neglected, write down the Maclaurin series approximations for \(\sin x\) and \(\cos x\) (where \(x\) is in radians). Hence or otherwise obtain an approximation for \(\tan x\) in the form \(a x + b x ^ { 3 }\).
  3. Find \(\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 1 - \frac { 1 } { 4 } X ^ { 2 } } } \mathrm {~d} x\), giving your answer in exact form.
OCR MEI FP2 2012 January Q2
18 marks Standard +0.3
2
  1. The infinite series \(C\) and \(S\) are defined as follows. $$\begin{aligned} & C = 1 + a \cos \theta + a ^ { 2 } \cos 2 \theta + \ldots \\ & S = a \sin \theta + a ^ { 2 } \sin 2 \theta + a ^ { 3 } \sin 3 \theta + \ldots \end{aligned}$$ where \(a\) is a real number and \(| a | < 1\).
    By considering \(C + \mathrm { j } S\), show that \(C = \frac { 1 - a \cos \theta } { 1 + a ^ { 2 } - 2 a \cos \theta }\) and find a corresponding expression for \(S\).
  2. Express the complex number \(z = - 1 + \mathrm { j } \sqrt { 3 }\) in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\). Find the 4th roots of \(z\) in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\).
    Show \(z\) and its 4th roots in an Argand diagram.
    Find the product of the 4th roots and mark this as a point on your Argand diagram.
OCR MEI FP2 2012 January Q3
18 marks Standard +0.3
3
  1. Show that the characteristic equation of the matrix $$\mathbf { M } = \left( \begin{array} { r r r } 3 & - 1 & 2 \\ - 4 & 3 & 2 \\ 2 & 1 & - 1 \end{array} \right)$$ is \(\lambda ^ { 3 } - 5 \lambda ^ { 2 } - 7 \lambda + 35 = 0\).
  2. Show that \(\lambda = 5\) is an eigenvalue of \(\mathbf { M }\), and find its other eigenvalues.
  3. Find an eigenvector, \(\mathbf { v }\), of unit length corresponding to \(\lambda = 5\). State the magnitudes and directions of the vectors \(\mathbf { M } ^ { 2 } \mathbf { v }\) and \(\mathbf { M } ^ { - 1 } \mathbf { v }\).
  4. Use the Cayley-Hamilton theorem to find the constants \(a , b , c\) such that $$\mathbf { M } ^ { 4 } = a \mathbf { M } ^ { 2 } + b \mathbf { M } + c \mathbf { I } .$$ Section B (18 marks)
OCR MEI FP2 2012 January Q4
18 marks Standard +0.8
4
  1. Define tanh \(t\) in terms of exponential functions. Sketch the graph of \(\tanh t\).
  2. Show that \(\operatorname { artanh } x = \frac { 1 } { 2 } \ln \left( \frac { 1 + x } { 1 - x } \right)\). State the set of values of \(x\) for which this equation is valid.
  3. Differentiate the equation \(\tanh y = x\) with respect to \(x\) and hence show that the derivative of \(\operatorname { artanh } x\) is \(\frac { 1 } { 1 - x ^ { 2 } }\). Show that this result may also be obtained by differentiating the equation in part (ii).
  4. By considering \(\operatorname { artanh } x\) as \(1 \times \operatorname { artanh } x\) and using integration by parts, show that $$\int _ { 0 } ^ { \frac { 1 } { 2 } } \operatorname { artanh } x \mathrm {~d} x = \frac { 1 } { 4 } \ln \frac { 27 } { 16 }$$
OCR MEI FP2 2012 January Q5
18 marks Challenging +1.2
5 The points \(\mathrm { A } ( - 1,0 ) , \mathrm { B } ( 1,0 )\) and \(\mathrm { P } ( x , y )\) are such that the product of the distances PA and PB is 1 . You are given that the cartesian equation of the locus of P is $$\left( ( x + 1 ) ^ { 2 } + y ^ { 2 } \right) \left( ( x - 1 ) ^ { 2 } + y ^ { 2 } \right) = 1 .$$
  1. Show that this equation may be written in polar form as $$r ^ { 4 } + 2 r ^ { 2 } = 4 r ^ { 2 } \cos ^ { 2 } \theta$$ Show that the polar equation simplifies to $$r ^ { 2 } = 2 \cos 2 \theta$$
  2. Give a sketch of the curve, stating the values of \(\theta\) for which the curve is defined.
  3. The equation in part (i) is now to be generalised to $$r ^ { 2 } = 2 \cos 2 \theta + k$$ where \(k\) is a constant.
    (A) Give sketches of the curve in the cases \(k = 1 , k = 2\). Describe how these two curves differ at the pole.
    (B) Give a sketch of the curve in the case \(k = 4\). What happens to the shape of the curve as \(k\) tends to infinity?
  4. Sketch the curve for the case \(k = - 1\). What happens to the curve as \(k \rightarrow - 2\) ? \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE.}
OCR MEI FP2 2013 January Q1
18 marks Standard +0.3
1
    1. Differentiate with respect to \(x\) the equation \(a \tan y = x\) (where \(a\) is a constant), and hence show that the derivative of \(\arctan \frac { x } { a }\) is \(\frac { a } { a ^ { 2 } + x ^ { 2 } }\).
    2. By first expressing \(x ^ { 2 } - 4 x + 8\) in completed square form, evaluate the integral \(\int _ { 0 } ^ { 4 } \frac { 1 } { x ^ { 2 } - 4 x + 8 } \mathrm {~d} x\), giving your answer exactly.
    3. Use integration by parts to find \(\int \arctan x \mathrm {~d} x\).
    1. A curve has polar equation \(r = 2 \cos \theta\), for \(- \frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\). Show, by considering its cartesian equation, that the curve is a circle. State the centre and radius of the circle.
    2. Another circle has radius 2 and its centre, in cartesian coordinates, is ( 0,2 ). Find the polar equation of this circle.
OCR MEI FP2 2013 January Q3
18 marks Standard +0.3
3 You are given the matrix \(\mathbf { M } = \left( \begin{array} { r r r } 1 & 3 & 0 \\ 3 & - 2 & - 1 \\ 0 & - 1 & 1 \end{array} \right)\).
  1. Show that the characteristic equation of \(\mathbf { M }\) is $$\lambda ^ { 3 } - 13 \lambda + 12 = 0 .$$
  2. Find the eigenvalues and corresponding eigenvectors of \(\mathbf { M }\).
  3. 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
  1. 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.
  2. Sketch the curve.
  3. 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 \(0 \leqslant \theta \leqslant 2 \pi\).
  1. On a single diagram, sketch the curves for \(a = 0 , a = 1 , a = 2\).
  2. On a single diagram, sketch the curves for \(a = 0 , a = - 1 , a = - 2\).
  3. Identify a feature that the curves for \(a = 1 , a = 2 , a = - 1 , a = - 2\) share.
  4. Name a distinctive feature of the curve for \(a = - 1\), and a different distinctive feature of the curve for \(a = - 2\).
  5. 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).
  6. 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
  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 \(x ^ { 3 }\).
  2. 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
  1. 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 \(C\).
  2. 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 \(\mathbf { B } ^ { 4 } = 19 \mathbf { B } ^ { 2 } + 22 \mathbf { B } + 40 \mathbf { I }\).