Questions — OCR MEI FP2 (82 questions)

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OCR MEI FP2 2011 June Q2
2
  1. 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
3
  1. 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 \(k\).
  2. In the case \(k = 3\), evaluate $$\mathbf { M } \left( \begin{array} { r } - 3
    3
    1 \end{array} \right)$$
  3. State the significance of what you have found in part (ii).
  4. 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
4
  1. Given that \(\cosh y = x\), show that \(y = \pm \ln \left( x + \sqrt { x ^ { 2 } - 1 } \right)\) and that \(\operatorname { arcosh } x = \ln \left( x + \sqrt { x ^ { 2 } - 1 } \right)\).
  2. Find \(\int _ { \frac { 4 } { 5 } } ^ { 1 } \frac { 1 } { \sqrt { 25 x ^ { 2 } - 16 } } \mathrm {~d} x\), expressing your answer in an exact logarithmic form.
  3. Solve the equation $$5 \cosh x - \cosh 2 x = 3$$ giving your answers in an exact logarithmic form.
OCR MEI FP2 2011 June Q5
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 \(n\).
  1. 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) \(m = 4 , n = 2\).
  2. What feature does the curve have when \(m = n\) ? What is the effect on the curve of interchanging \(m\) and \(n\) when \(m \neq n\) ?
  3. 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.
  4. 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 \(x = 1\).
  5. 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.
  6. 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. 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 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 2009 January Q2
  1. Write down the modulus and argument of the complex number \(\mathrm { e } ^ { \mathrm { j } \pi / 3 }\).
  2. 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 \(r \mathrm { e } ^ { \mathrm { j } \theta }\).
  3. 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 \(w = z _ { 1 } \mathrm { e } ^ { - \mathrm { j } \pi / 12 }\).
  4. Find \(w\) in the form \(x + \mathrm { j } y\), and mark this complex number on your Argand diagram.
  5. Find \(w ^ { 6 }\), expressing your answer in as simple a form as possible.
OCR MEI FP2 2013 January Q2
    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
5 A curve has parametric equations $$x = \theta - k \sin \theta , \quad y = 1 - \cos \theta ,$$ where \(k\) is a positive constant.
  1. For the case \(k = 1\), use your graphical calculator to sketch the curve. Describe its main features.
  2. Sketch the curve for a value of \(k\) between 0 and 1 . Describe briefly how the main features differ from those for the case \(k = 1\).
  3. 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 $$2 \sqrt { 3 } - \frac { 2 \pi } { 3 }$$
  4. Use your calculator to find, correct to one decimal place, the value of \(k\) for which successive loops just touch each other.