OCR MEI FP2 (Further Pure Mathematics 2) 2012 June

Question 1
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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) \(\int _ { - \frac { 1 } { 2 } } ^ { \frac { 1 } { 2 } } \frac { 1 } { \sqrt { 1 - 2 x ^ { 2 } } } \mathrm {~d} x\)
  2. 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.
Question 2
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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.
Question 3
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3
  1. 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 \(a\).
  2. Hence solve the following system of equations. $$\begin{aligned} x + 2 y + 3 z & = 1
    - x + 4 z & = - 2
    3 x - 2 y + 2 z & = 1 \end{aligned}$$
  3. 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.
Question 4
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4
  1. Prove, from definitions involving exponential functions, that $$\cosh 2 u = 2 \sinh ^ { 2 } u + 1$$
  2. Prove that, if \(y \geqslant 0\) and \(\cosh y = u\), then \(y = \ln \left( u + \sqrt { } \left( u ^ { 2 } - 1 \right) \right)\).
  3. 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.
  4. Find \(\int _ { \frac { 1 } { 2 } } ^ { 1 } \sqrt { 4 x ^ { 2 } - 1 } \mathrm {~d} x\), expressing your answer in an exact form involving logarithms.
Question 5
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5 This question concerns curves with polar equation \(r = \sec \theta + a\), where \(a\) is a constant.
  1. 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.
  2. 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.
  3. 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 \(a = - 1\).
  4. What feature does the curve exhibit for values of \(a\) greater than 1 ? Sketch a typical case.
  5. Show that a cartesian equation of the curve \(r = \sec \theta + a\) is \(\left( x ^ { 2 } + y ^ { 2 } \right) ( x - 1 ) ^ { 2 } = a ^ { 2 } x ^ { 2 }\).