Questions — OCR MEI FP2 (82 questions)

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OCR MEI FP2 2007 January Q1
1
  1. A curve has polar equation \(r = a \mathrm { e } ^ { - k \theta }\) for \(0 \leqslant \theta \leqslant \pi\), where \(a\) and \(k\) are positive constants. The points A and B on the curve correspond to \(\theta = 0\) and \(\theta = \pi\) respectively.
    1. Sketch the curve.
    2. Find the area of the region enclosed by the curve and the line AB .
  2. Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } } \frac { 1 } { 3 + 4 x ^ { 2 } } \mathrm {~d} x\).
    1. Find the Maclaurin series for \(\tan x\), up to the term in \(x ^ { 3 }\).
    2. Use this Maclaurin series to show that, when \(h\) is small, \(\int _ { h } ^ { 4 h } \frac { \tan x } { x } \mathrm {~d} x \approx 3 h + 7 h ^ { 3 }\).
OCR MEI FP2 2007 January Q2
2
  1. You are given the complex numbers \(w = 3 \mathrm { e } ^ { - \frac { 1 } { 12 } \pi \mathrm { j } }\) and \(z = 1 - \sqrt { 3 } \mathrm { j }\).
    1. Find the modulus and argument of each of the complex numbers \(w , z\) and \(\frac { w } { z }\).
    2. Hence write \(\frac { w } { z }\) in the form \(a + b \mathrm { j }\), giving the exact values of \(a\) and \(b\).
  2. In this part of the question, \(n\) is a positive integer and \(\theta\) is a real number with \(0 < \theta < \frac { \pi } { n }\).
    1. Express \(\mathrm { e } ^ { - \frac { 1 } { 2 } \mathrm { j } \theta } + \mathrm { e } ^ { \frac { 1 } { 2 } \mathrm { j } \theta }\) in simplified trigonometric form, and hence, or otherwise, show that $$1 + \mathrm { e } ^ { \mathrm { j } \theta } = 2 \mathrm { e } ^ { \frac { 1 } { 2 } \mathrm { j } \theta } \cos \frac { 1 } { 2 } \theta$$ Series \(C\) and \(S\) are defined by $$\begin{aligned} & C = 1 + \binom { n } { 1 } \cos \theta + \binom { n } { 2 } \cos 2 \theta + \binom { n } { 3 } \cos 3 \theta + \ldots + \binom { n } { n } \cos n \theta
      & S = \binom { n } { 1 } \sin \theta + \binom { n } { 2 } \sin 2 \theta + \binom { n } { 3 } \sin 3 \theta + \ldots + \binom { n } { n } \sin n \theta \end{aligned}$$
    2. Find \(C\) and \(S\), and show that \(\frac { S } { C } = \tan \frac { 1 } { 2 } n \theta\).
OCR MEI FP2 2007 January Q3
3 Let \(\mathbf { P } = \left( \begin{array} { r r r } 4 & 2 & k
1 & 1 & 3
1 & 0 & - 1 \end{array} \right) (\) where \(k \neq 4 )\) and \(\mathbf { M } = \left( \begin{array} { r r r } 2 & - 2 & - 6
- 1 & 3 & 1
1 & - 2 & - 2 \end{array} \right)\).
  1. Find \(\mathbf { P } ^ { - 1 }\) in terms of \(k\), and show that, when \(k = 2 , \mathbf { P } ^ { - 1 } = \frac { 1 } { 2 } \left( \begin{array} { r r r } - 1 & 2 & 4
    4 & - 6 & - 10
    - 1 & 2 & 2 \end{array} \right)\).
  2. Verify that \(\left( \begin{array} { l } 4
    1
    1 \end{array} \right) , \left( \begin{array} { l } 2
    1
    0 \end{array} \right)\) and \(\left( \begin{array} { r } 2
    3
    - 1 \end{array} \right)\) are eigenvectors of \(\mathbf { M }\), and find the corresponding eigenvalues.
  3. Show that \(\mathbf { M } ^ { n } = \left( \begin{array} { r r r } 4 & - 6 & - 10
    2 & - 3 & - 5
    0 & 0 & 0 \end{array} \right) + 2 ^ { n - 1 } \left( \begin{array} { r r r } - 2 & 4 & 4
    - 3 & 6 & 6
    1 & - 2 & - 2 \end{array} \right)\). Section B (18 marks)
OCR MEI FP2 2007 January Q4
4
  1. Show that \(\operatorname { arcosh } x = \ln \left( x + \sqrt { x ^ { 2 } - 1 } \right)\).
  2. Find \(\int _ { 2.5 } ^ { 3.9 } \frac { 1 } { \sqrt { 4 x ^ { 2 } - 9 } } \mathrm {~d} x\), giving your answer in the form \(a \ln b\), where \(a\) and \(b\) are rational numbers.
  3. There are two points on the curve \(y = \frac { \cosh x } { 2 + \sinh x }\) at which the gradient is \(\frac { 1 } { 9 }\). Show that one of these points is \(\left( \ln ( 1 + \sqrt { 2 } ) , \frac { 1 } { 3 } \sqrt { 2 } \right)\), and find the coordinates of the other point, in a similar form.
OCR MEI FP2 2007 January Q5
5 Cartesian coordinates \(( x , y )\) and polar coordinates \(( r , \theta )\) are set up in the usual way, with the pole at the origin and the initial line along the positive \(x\)-axis, so that \(x = r \cos \theta\) and \(y = r \sin \theta\). A curve has polar equation \(r = k + \cos \theta\), where \(k\) is a constant with \(k \geqslant 1\).
  1. Use your graphical calculator to obtain sketches of the curve in the three cases $$k = 1 , k = 1.5 \text { and } k = 4$$
  2. Name the special feature which the curve has when \(k = 1\).
  3. For each of the three cases, state the number of points on the curve at which the tangent is parallel to the \(y\)-axis.
  4. Express \(x\) in terms of \(k\) and \(\theta\), and find \(\frac { \mathrm { d } x } { \mathrm {~d} \theta }\). Hence find the range of values of \(k\) for which there are just two points on the curve where the tangent is parallel to the \(y\)-axis. The distance between the point ( \(r , \theta\) ) on the curve and the point ( 1,0 ) on the \(x\)-axis is \(d\).
  5. Use the cosine rule to express \(d ^ { 2 }\) in terms of \(k\) and \(\theta\), and deduce that \(k ^ { 2 } \leqslant d ^ { 2 } \leqslant k ^ { 2 } + 1\).
  6. Hence show that, when \(k\) is large, the shape of the curve is very nearly circular.
OCR MEI FP2 2008 January Q1
1
  1. Fig. 1 shows the curve with polar equation \(r = a ( 1 - \cos 2 \theta )\) for \(0 \leqslant \theta \leqslant \pi\), where \(a\) is a positive constant. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{43b4c7ed-3556-4d87-8aef-0111fe747982-2_529_620_577_799} \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{figure} Find the area of the region enclosed by the curve.
    1. Given that \(\mathrm { f } ( x ) = \arctan ( \sqrt { 3 } + x )\), find \(\mathrm { f } ^ { \prime } ( x )\) and \(\mathrm { f } ^ { \prime \prime } ( x )\).
    2. Hence find the Maclaurin series for \(\arctan ( \sqrt { 3 } + x )\), as far as the term in \(x ^ { 2 }\).
    3. Hence show that, if \(h\) is small, \(\int _ { - h } ^ { h } x \arctan ( \sqrt { 3 } + x ) \mathrm { d } x \approx \frac { 1 } { 6 } h ^ { 3 }\).
OCR MEI FP2 2008 January Q2
2
  1. Find the 4th roots of 16j, in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\) where \(r > 0\) and \(- \pi < \theta \leqslant \pi\). Illustrate the 4th roots on an Argand diagram.
    1. Show that \(\left( 1 - 2 \mathrm { e } ^ { \mathrm { j } \theta } \right) \left( 1 - 2 \mathrm { e } ^ { - \mathrm { j } \theta } \right) = 5 - 4 \cos \theta\). Series \(C\) and \(S\) are defined by $$\begin{aligned} & C = 2 \cos \theta + 4 \cos 2 \theta + 8 \cos 3 \theta + \ldots + 2 ^ { n } \cos n \theta
      & S = 2 \sin \theta + 4 \sin 2 \theta + 8 \sin 3 \theta + \ldots + 2 ^ { n } \sin n \theta \end{aligned}$$
    2. Show that \(C = \frac { 2 \cos \theta - 4 - 2 ^ { n + 1 } \cos ( n + 1 ) \theta + 2 ^ { n + 2 } \cos n \theta } { 5 - 4 \cos \theta }\), and find a similar expression for \(S\).
OCR MEI FP2 2008 January Q3
3 You are given the matrix \(\mathbf { M } = \left( \begin{array} { r r } 7 & 3
- 4 & - 1 \end{array} \right)\).
  1. Find the eigenvalues, and corresponding eigenvectors, of the matrix \(\mathbf { M }\).
  2. Write down a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { P } ^ { - 1 } \mathbf { M P } = \mathbf { D }\).
  3. Given that \(\mathbf { M } ^ { n } = \left( \begin{array} { l l } a & b
    c & d \end{array} \right)\), show that \(a = - \frac { 1 } { 2 } + \frac { 3 } { 2 } \times 5 ^ { n }\), and find similar expressions for \(b , c\) and \(d\). Section B (18 marks)
OCR MEI FP2 2008 January Q4
4
  1. Given that \(k \geqslant 1\) and \(\cosh x = k\), show that \(x = \pm \ln \left( k + \sqrt { k ^ { 2 } - 1 } \right)\).
  2. Find \(\int _ { 1 } ^ { 2 } \frac { 1 } { \sqrt { 4 x ^ { 2 } - 1 } } \mathrm {~d} x\), giving the answer in an exact logarithmic form.
  3. Solve the equation \(6 \sinh x - \sinh 2 x = 0\), giving the answers in an exact form, using logarithms where appropriate.
  4. Show that there is no point on the curve \(y = 6 \sinh x - \sinh 2 x\) at which the gradient is 5 .
OCR MEI FP2 2008 January Q5
5 A curve has parametric equations \(x = \frac { t ^ { 2 } } { 1 + t ^ { 2 } } , y = t ^ { 3 } - \lambda t\), where \(\lambda\) is a constant.
  1. Use your calculator to obtain a sketch of the curve in each of the cases $$\lambda = - 1 , \quad \lambda = 0 \quad \text { and } \quad \lambda = 1 .$$ Name any special features of these curves.
  2. By considering the value of \(x\) when \(t\) is large, write down the equation of the asymptote. For the remainder of this question, assume that \(\lambda\) is positive.
  3. Find, in terms of \(\lambda\), the coordinates of the point where the curve intersects itself.
  4. Show that the two points on the curve where the tangent is parallel to the \(x\)-axis have coordinates $$\left( \frac { \lambda } { 3 + \lambda } , \pm \sqrt { \frac { 4 \lambda ^ { 3 } } { 27 } } \right)$$ Fig. 5 shows a curve which intersects itself at the point ( 2,0 ) and has asymptote \(x = 8\). The stationary points A and B have \(y\)-coordinates 2 and - 2 . \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{43b4c7ed-3556-4d87-8aef-0111fe747982-4_791_609_1482_769} \captionsetup{labelformat=empty} \caption{Fig. 5}
    \end{figure}
  5. For the curve sketched in Fig. 5, find parametric equations of the form \(x = \frac { a t ^ { 2 } } { 1 + t ^ { 2 } } , y = b \left( t ^ { 3 } - \lambda t \right)\), where \(a , \lambda\) and \(b\) are to be determined.
OCR MEI FP2 2009 January Q1
1
    1. By considering the derivatives of \(\cos x\), show that the Maclaurin expansion of \(\cos x\) begins $$1 - \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 24 } x ^ { 4 }$$
    2. The Maclaurin expansion of \(\sec x\) begins $$1 + a x ^ { 2 } + b x ^ { 4 }$$ where \(a\) and \(b\) are constants. Explain why, for sufficiently small \(x\), $$\left( 1 - \frac { 1 } { 2 } x ^ { 2 } + \frac { 1 } { 24 } x ^ { 4 } \right) \left( 1 + a x ^ { 2 } + b x ^ { 4 } \right) \approx 1$$ Hence find the values of \(a\) and \(b\).
    1. Given that \(y = \arctan \left( \frac { x } { a } \right)\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { a } { a ^ { 2 } + x ^ { 2 } }\).
    2. Find the exact values of the following integrals. $$\begin{aligned} & \text { (A) } \int _ { - 2 } ^ { 2 } \frac { 1 } { 4 + x ^ { 2 } } \mathrm {~d} x
      & \text { (B) } \int _ { - \frac { 1 } { 2 } } ^ { \frac { 1 } { 2 } } \frac { 4 } { 1 + 4 x ^ { 2 } } \mathrm {~d} x \end{aligned}$$
OCR MEI FP2 2009 January Q3
3
  1. A curve has polar equation \(r = a \tan \theta\) for \(0 \leqslant \theta \leqslant \frac { 1 } { 3 } \pi\), where \(a\) is a positive constant.
    1. Sketch the curve.
    2. Find the area of the region between the curve and the line \(\theta = \frac { 1 } { 4 } \pi\). Indicate this region on your sketch.
    1. Find the eigenvalues and corresponding eigenvectors for the matrix \(\mathbf { M }\) where $$\mathbf { M } = \left( \begin{array} { l l } 0.2 & 0.8
      0.3 & 0.7 \end{array} \right)$$
    2. Give a matrix \(\mathbf { Q }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { M } = \mathbf { Q D } \mathbf { Q } ^ { - 1 }\). Section B (18 marks)
OCR MEI FP2 2009 January Q4
4
    1. Prove, from definitions involving exponentials, that $$\cosh ^ { 2 } x - \sinh ^ { 2 } x = 1$$
    2. Given that \(\sinh x = \tan y\), where \(- \frac { 1 } { 2 } \pi < y < \frac { 1 } { 2 } \pi\), show that
      (A) \(\tanh x = \sin y\),
      (B) \(x = \ln ( \tan y + \sec y )\).
    1. Given that \(y = \operatorname { artanh } x\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\). Hence show that \(\int _ { - \frac { 1 } { 2 } } ^ { \frac { 1 } { 2 } } \frac { 1 } { 1 - x ^ { 2 } } \mathrm {~d} x = 2 \operatorname { artanh } \frac { 1 } { 2 }\).
    2. Express \(\frac { 1 } { 1 - x ^ { 2 } }\) in partial fractions and hence find an expression for \(\int \frac { 1 } { 1 - x ^ { 2 } } \mathrm {~d} x\) in terms of logarithms.
    3. Use the results in parts (i) and (ii) to show that \(\operatorname { artanh } \frac { 1 } { 2 } = \frac { 1 } { 2 } \ln 3\).
OCR MEI FP2 2009 January Q5
5 The limaçon of Pascal has polar equation \(r = 1 + 2 a \cos \theta\), where \(a\) is a constant.
  1. Use your calculator to sketch the curve when \(a = 1\). (You need not distinguish between parts of the curve where \(r\) is positive and negative.)
  2. By using your calculator to investigate the shape of the curve for different values of \(a\), positive and negative,
    (A) state the set of values of \(a\) for which the curve has a loop within a loop,
    (B) state, with a reason, the shape of the curve when \(a = 0\),
    (C) state what happens to the shape of the curve as \(a \rightarrow \pm \infty\),
    (D) name the feature of the curve that is evident when \(a = 0.5\), and find another value of \(a\) for which the curve has this feature.
  3. Given that \(a > 0\) and that \(a\) is such that the curve has a loop within a loop, write down an equation for the values of \(\theta\) at which \(r = 0\). Hence show that the angle at which the curve crosses itself is \(2 \arccos \left( \frac { 1 } { 2 a } \right)\). Obtain the cartesian equations of the tangents at the point where the curve crosses itself. Explain briefly how these equations relate to the answer to part (ii)(A).
OCR MEI FP2 2010 January Q1
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
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
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
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
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
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
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
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
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
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
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.