Questions — OCR MEI FP2 (82 questions)

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OCR MEI FP2 2012 January Q3
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
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
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
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
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
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
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
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
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
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
4
  1. Given that \(\sinh y = x\), show that $$y = \ln \left( x + \sqrt { 1 + x ^ { 2 } } \right)$$ Differentiate (*) to show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sqrt { 1 + x ^ { 2 } } }$$
  2. Find \(\int \frac { 1 } { \sqrt { 25 + 4 x ^ { 2 } } } \mathrm {~d} x\), expressing your answer in logarithmic form.
  3. 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
3 This question concerns the matrix \(\mathbf { M }\) where \(\mathbf { M } = \left( \begin{array} { r r r } 5 & - 1 & 3
4 & - 3 & - 2
2 & 1 & 4 \end{array} \right)\).
  1. 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 \(\lambda _ { 1 } < \lambda _ { 2 } < \lambda _ { 3 }\).
  2. 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 \(\lambda _ { 3 }\).
  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.
  4. 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
4
  1. 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 \(y = \operatorname { artanh } x\).
  2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 1 - x ^ { 2 } }\).
  3. Show, by integrating the result in part (ii), that \(y = \frac { 1 } { 2 } \ln \left( \frac { 1 + x } { 1 - x } \right)\).
  4. 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.
  5. 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
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.
OCR MEI FP2 2012 June Q2
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
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.
OCR MEI FP2 2012 June Q4
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.
OCR MEI FP2 2012 June Q5
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 \(\left( x ^ { 2 } + y ^ { 2 } \right) ( x - 1 ) ^ { 2 } = a ^ { 2 } x ^ { 2 }\).
OCR MEI FP2 2013 June Q3
3 You are given the matrix \(\mathbf { A } = \left( \begin{array} { r r r } k & - 7 & 4
2 & - 2 & 3
1 & - 3 & - 2 \end{array} \right)\).
  1. Show that when \(k = 5\) the determinant of \(\mathbf { A }\) is zero. Obtain an expression for the inverse of \(\mathbf { A }\) when \(k \neq 5\).
  2. Solve the matrix equation $$\left( \begin{array} { r r r } 4 & - 7 & 4
    2 & - 2 & 3
    1 & - 3 & - 2 \end{array} \right) \left( \begin{array} { l } x
    y
    z \end{array} \right) = \left( \begin{array} { c } p
    1
    2 \end{array} \right)$$ giving your answer in terms of \(p\).
  3. Find the value of \(p\) for which the matrix equation $$\left( \begin{array} { r r r } 5 & - 7 & 4
    2 & - 2 & 3
    1 & - 3 & - 2 \end{array} \right) \left( \begin{array} { c } x
    y
    z \end{array} \right) = \left( \begin{array} { c } p
    1
    2 \end{array} \right)$$ has a solution. Give the general solution in this case and describe it geometrically.
OCR MEI FP2 2013 June Q4
4
  1. Prove, using exponential functions, that \(\cosh ^ { 2 } u - \sinh ^ { 2 } u = 1\).
  2. Given that \(y = \operatorname { arsinh } x\), show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sqrt { 1 + x ^ { 2 } } }$$ and that $$y = \ln \left( x + \sqrt { 1 + x ^ { 2 } } \right)$$
  3. Show that $$\int _ { 0 } ^ { 2 } \frac { 1 } { \sqrt { 4 + 9 x ^ { 2 } } } \mathrm {~d} x = \frac { 1 } { 3 } \ln ( 3 + \sqrt { 10 } )$$
  4. Find, in exact logarithmic form, $$\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 1 + x ^ { 2 } } } \operatorname { arsinh } x \mathrm {~d} x$$
OCR MEI FP2 2009 June Q1
1
    1. Use the Maclaurin series for \(\ln ( 1 + x )\) and \(\ln ( 1 - x )\) to obtain the first three non-zero terms in the Maclaurin series for \(\ln \left( \frac { 1 + x } { 1 - x } \right)\). State the range of validity of this series.
    2. Find the value of \(x\) for which \(\frac { 1 + x } { 1 - x } = 3\). Hence find an approximation to \(\ln 3\), giving your answer to three decimal places.
  2. A curve has polar equation \(r = \frac { a } { 1 + \sin \theta }\) for \(0 \leqslant \theta \leqslant \pi\), where \(a\) is a positive constant. The points on the curve have cartesian coordinates \(x\) and \(y\).
    1. By plotting suitable points, or otherwise, sketch the curve.
    2. Show that, for this curve, \(r + y = a\) and hence find the cartesian equation of the curve.
    3. Obtain the characteristic equation for the matrix \(\mathbf { M }\) where $$\mathbf { M } = \left( \begin{array} { r r r } 3 & 1 & - 2
      0 & - 1 & 0
      2 & 0 & 1 \end{array} \right)$$ Hence or otherwise obtain the value of \(\operatorname { det } ( \mathbf { M } )\).
    4. Show that - 1 is an eigenvalue of \(\mathbf { M }\), and show that the other two eigenvalues are not real. Find an eigenvector corresponding to the eigenvalue - 1 .
      Hence or otherwise write down the solution to the following system of equations. $$\begin{aligned} 3 x + y - 2 z & = - 0.1
      - y & = 0.6
      2 x + z & = 0.1 \end{aligned}$$
    5. State the Cayley-Hamilton theorem and use it to show that $$\mathbf { M } ^ { 3 } = 3 \mathbf { M } ^ { 2 } - 3 \mathbf { M } - 7 \mathbf { I }$$ Obtain an expression for \(\mathbf { M } ^ { - 1 }\) in terms of \(\mathbf { M } ^ { 2 } , \mathbf { M }\) and \(\mathbf { I }\).
    6. Find the numerical values of the elements of \(\mathbf { M } ^ { - 1 }\), showing your working.
OCR MEI FP2 2009 June Q3
3
    1. Sketch the graph of \(y = \arcsin x\) for \(- 1 \leqslant x \leqslant 1\). Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), justifying the sign of your answer by reference to your sketch.
    2. Find the exact value of the integral \(\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 2 - x ^ { 2 } } } \mathrm {~d} x\).
  2. The infinite series \(C\) and \(S\) are defined as follows. $$\begin{gathered} C = \cos \theta + \frac { 1 } { 3 } \cos 3 \theta + \frac { 1 } { 9 } \cos 5 \theta + \ldots
    S = \sin \theta + \frac { 1 } { 3 } \sin 3 \theta + \frac { 1 } { 9 } \sin 5 \theta + \ldots \end{gathered}$$ By considering \(C + \mathrm { j } S\), show that $$C = \frac { 3 \cos \theta } { 5 - 3 \cos 2 \theta }$$ and find a similar expression for \(S\). Section B (18 marks)
OCR MEI FP2 2009 June Q4
4
  1. Prove, from definitions involving exponentials, that $$\cosh 2 u = 2 \cosh ^ { 2 } u - 1$$
  2. Prove that \(\operatorname { arsinh } y = \ln \left( y + \sqrt { y ^ { 2 } + 1 } \right)\).
  3. Use the substitution \(x = 2 \sinh u\) to show that $$\int \sqrt { x ^ { 2 } + 4 } \mathrm {~d} x = 2 \operatorname { arsinh } \frac { 1 } { 2 } x + \frac { 1 } { 2 } x \sqrt { x ^ { 2 } + 4 } + c$$ where \(c\) is an arbitrary constant.
  4. By first expressing \(t ^ { 2 } + 2 t + 5\) in completed square form, show that $$\int _ { - 1 } ^ { 1 } \sqrt { t ^ { 2 } + 2 t + 5 } \mathrm {~d} t = 2 ( \ln ( 1 + \sqrt { 2 } ) + \sqrt { 2 } )$$ \section*{[Question 5 is printed overleaf.]}
OCR MEI FP2 2009 June Q5
5 Fig. 5 shows a circle with centre \(\mathrm { C } ( a , 0 )\) and radius \(a\). B is the point \(( 0,1 )\). The line BC intersects the circle at P and \(\mathrm { Q } ; \mathrm { P }\) is above the \(x\)-axis and Q is below. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{66ca36f1-099c-44ce-a6e2-027172e44fd8-4_556_659_539_742} \captionsetup{labelformat=empty} \caption{Fig. 5}
\end{figure}
  1. Show that, in the case \(a = 1 , \mathrm { P }\) has coordinates \(\left( 1 - \frac { 1 } { \sqrt { 2 } } , \frac { 1 } { \sqrt { 2 } } \right)\). Write down the coordinates of Q .
  2. Show that, for all positive values of \(a\), the coordinates of P are $$x = a \left( 1 - \frac { a } { \sqrt { a ^ { 2 } + 1 } } \right) , \quad y = \frac { a } { \sqrt { a ^ { 2 } + 1 } } .$$ Write down the coordinates of Q in a similar form. Now let the variable point P be defined by the parametric equations \(( * )\) for all values of the parameter \(a\), positive, zero and negative. Let Q be defined for all \(a\) by your answer in part (ii).
  3. Using your calculator, sketch the locus of P as \(a\) varies. State what happens to P as \(a \rightarrow \infty\) and as \(a \rightarrow - \infty\). Show algebraically that this locus has an asymptote at \(y = - 1\).
    On the same axes, sketch, as a dotted line, the locus of Q as \(a\) varies.
    (The single curve made up of these two loci and including the point B is called a right strophoid.)
  4. State, with a reason, the size of the angle POQ in Fig. 5. What does this indicate about the angle at which a right strophoid crosses itself? \section*{OCR
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OCR MEI FP2 2011 June Q1
1
  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 \(\int _ { - \frac { 1 } { 2 } } ^ { \frac { 1 } { 2 } } \frac { 1 } { \left( 1 + 4 x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x\).