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OCR MEI C4 Q12
Standard +0.3
12
9 \end{array} \right) + \lambda \left( \begin{array} { l } 1
3
2 \end{array} \right)$$ \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ff20b83a-5e38-437e-8115-5b0a6a54fa9d-2_745_1300_256_399} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure} Fig. 7 illustrates a house. All units are in metres. The coordinates of A, B, C and E are as shown. BD is horizontal and parallel to AE .
  1. Find the length AE .
  2. Find a vector equation of the line BD . Given that the length of BD is 15 metres, find the coordinates of D.
  3. Verify that the equation of the plane ABC is $$- 3 x + 4 y + 5 z = 30 .$$ Write down a vector normal to this plane.
  4. Show that the vector \(\left( \begin{array} { l } 4 \\ 3 \\ 5 \end{array} \right)\) is normal to the plane ABDE . Hence find the equation of the plane ABDE .
  5. Find the angle between the planes ABC and ABDE .
OCR MEI C4 Q1
6 marks Moderate -0.5
1 The points \(\mathrm { A } , \mathrm { B }\) and C have coordinates \(\mathrm { A } ( 3,2 , - 1 ) , \mathrm { B } ( - 1,1,2 )\) and \(\mathrm { C } ( 10,5 , - 5 )\), relative to the origin O . Show that \(\overrightarrow { \mathrm { OC } }\) can be written in the form \(\lambda \overrightarrow { \mathrm { OA } } + \mu \overrightarrow { \mathrm { OB } }\), where \(\lambda\) and \(\mu\) are to be determined. What can you deduce about the points \(\mathrm { O } , \mathrm { A } , \mathrm { B }\) and C from the fact that \(\overrightarrow { \mathrm { OC } }\) can be expressed as a combination of \(\overrightarrow { \mathrm { OA } }\) and \(\overrightarrow { \mathrm { OB } }\) ?
OCR MEI C4 Q2
5 marks Moderate -0.8
2 Vectors \(\mathbf { a }\) and \(\mathbf { b }\) are given by \(\mathbf { a } = 2 \mathbf { i } + \mathbf { j } - \mathbf { k }\) and \(\mathbf { b } = 4 \mathbf { i } - 2 \mathbf { j } + \mathbf { k }\).
Find constants \(\lambda\) and \(\mu\) such that \(\lambda \mathbf { a } + \mu \mathbf { b } = 4 \mathbf { j } - 3 \mathbf { k }\).
OCR MEI C4 Q3
6 marks Standard +0.3
3 A triangle ABC has vertices \(\mathrm { A } ( - 2,4,1 ) , \mathrm { B } ( 2,3,4 )\) and \(\mathrm { C } ( 4,8,3 )\). By calculating a suitable scalar product, show that angle ABC is a right angle. Hence calculate the area of the triangle. [6]
OCR MEI FP2 Q5
18 marks Challenging +1.2
5 A curve has equation \(y = \frac { x ^ { 3 } - k ^ { 3 } } { x ^ { 2 } - 4 }\), where \(k\) is a positive constant and \(k \neq 2\).
  1. Find the equations of the three asymptotes.
  2. Use your graphical calculator to obtain rough sketches of the curve in the two separate cases \(k < 2\) and \(k > 2\).
  3. In the case \(k < 2\), your sketch may not show clearly the shape of the curve near \(x = 0\). Use calculus to show that the curve has a minimum point when \(x = 0\).
  4. In the case \(k > 2\), your sketch may not show clearly how the curve approaches its asymptote as \(x \rightarrow + \infty\). Show algebraically that the curve crosses this asymptote.
  5. Use the results of parts (iii) and (iv) to produce more accurate sketches of the curve in the two separate cases \(k < 2\) and \(k > 2\). These sketches should indicate where the curve crosses the axes, and should show clearly how the curve approaches its asymptotes. The presence of stationary points should be clearly shown, but there is no need to find their coordinates. RECOGNISING ACHIEVEMENT \section*{OXFORD CAMBRIDGE AND RSA EXAMINATIONS} \section*{MEI STRUCTURED MATHEMATICS} Further Methods for Advanced Mathematics (FP2)
    Tuesday
OCR MEI FP2 2006 January Q1
18 marks Standard +0.8
1
  1. A curve has polar equation \(r = a \cos 3 \theta\) for \(- \frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\), where \(a\) is a positive constant.
    1. Sketch the curve, using a continuous line for sections where \(r > 0\) and a broken line for sections where \(r < 0\).
    2. Find the area enclosed by one of the loops.
  2. Find the exact value of \(\int _ { 0 } ^ { \frac { 3 } { 4 } } \frac { 1 } { \sqrt { 3 - 4 x ^ { 2 } } } \mathrm {~d} x\).
  3. Use a trigonometric substitution to find \(\int _ { 0 } ^ { 1 } \frac { 1 } { \left( 1 + 3 x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x\).
OCR MEI FP2 2006 January Q2
18 marks Challenging +1.2
2 In this question, \(\theta\) is a real number with \(0 < \theta < \frac { 1 } { 6 } \pi\), and \(w = \frac { 1 } { 2 } \mathrm { e } ^ { 3 \mathrm { j } \theta }\).
  1. State the modulus and argument of each of the complex numbers $$w , \quad w ^ { * } \quad \text { and } \quad \mathrm { j } w .$$ Illustrate these three complex numbers on an Argand diagram.
  2. Show that \(( 1 + w ) \left( 1 + w ^ { * } \right) = \frac { 5 } { 4 } + \cos 3 \theta\). Infinite series \(C\) and \(S\) are defined by $$\begin{aligned} & C = \cos 2 \theta - \frac { 1 } { 2 } \cos 5 \theta + \frac { 1 } { 4 } \cos 8 \theta - \frac { 1 } { 8 } \cos 11 \theta + \ldots \\ & S = \sin 2 \theta - \frac { 1 } { 2 } \sin 5 \theta + \frac { 1 } { 4 } \sin 8 \theta - \frac { 1 } { 8 } \sin 11 \theta + \ldots \end{aligned}$$
  3. Show that \(C = \frac { 4 \cos 2 \theta + 2 \cos \theta } { 5 + 4 \cos 3 \theta }\), and find a similar expression for \(S\).
OCR MEI FP2 2006 January Q3
18 marks Challenging +1.2
3 The matrix \(\mathbf { M } = \left( \begin{array} { r r r } 1 & 2 & 3 \\ - 2 & - 3 & 6 \\ 2 & 2 & - 4 \end{array} \right)\).
  1. Show that the characteristic equation for \(\mathbf { M }\) is \(\lambda ^ { 3 } + 6 \lambda ^ { 2 } - 9 \lambda - 14 = 0\).
  2. Show that - 1 is an eigenvalue of \(\mathbf { M }\), and find the other two eigenvalues.
  3. Find an eigenvector corresponding to the eigenvalue - 1 .
  4. Verify that \(\left( \begin{array} { l } 3 \\ 0 \\ 1 \end{array} \right)\) and \(\left( \begin{array} { r } 0 \\ 3 \\ - 2 \end{array} \right)\) are eigenvectors of \(\mathbf { M }\).
  5. Write down a matrix \(\mathbf { P }\), and a diagonal matrix \(\mathbf { D }\), such that \(\mathbf { M } ^ { 3 } = \mathbf { P D P } \mathbf { P } ^ { - 1 }\).
  6. Use the Cayley-Hamilton theorem to express \(\mathbf { M } ^ { - 1 }\) in the form \(a \mathbf { M } ^ { 2 } + b \mathbf { M } + c \mathbf { I }\). Section B (18 marks)
OCR MEI FP2 2006 January Q4
18 marks Standard +0.8
4
  1. Solve the equation $$\sinh x + 4 \cosh x = 8$$ giving the answers in an exact logarithmic form.
  2. Find the exact value of \(\int _ { 0 } ^ { 2 } \mathrm { e } ^ { x } \sinh x \mathrm {~d} x\).
    1. Differentiate \(\operatorname { arsinh } \left( \frac { 2 } { 3 } x \right)\) with respect to \(x\).
    2. Use integration by parts to show that \(\int _ { 0 } ^ { 2 } \operatorname { arsinh } \left( \frac { 2 } { 3 } x \right) \mathrm { d } x = 2 \ln 3 - 1\).
OCR MEI FP2 2006 January Q5
18 marks Challenging +1.2
5 A curve has equation \(y = \frac { x ^ { 3 } - k ^ { 3 } } { x ^ { 2 } - 4 }\), where \(k\) is a positive constant and \(k \neq 2\).
  1. Find the equations of the three asymptotes.
  2. Use your graphical calculator to obtain rough sketches of the curve in the two separate cases \(k < 2\) and \(k > 2\).
  3. In the case \(k < 2\), your sketch may not show clearly the shape of the curve near \(x = 0\). Use calculus to show that the curve has a minimum point when \(x = 0\).
  4. In the case \(k > 2\), your sketch may not show clearly how the curve approaches its asymptote as \(x \rightarrow + \infty\). Show algebraically that the curve crosses this asymptote.
  5. Use the results of parts (iii) and (iv) to produce more accurate sketches of the curve in the two separate cases \(k < 2\) and \(k > 2\). These sketches should indicate where the curve crosses the axes, and should show clearly how the curve approaches its asymptotes. The presence of stationary points should be clearly shown, but there is no need to find their coordinates.
OCR MEI FP2 2007 January Q1
18 marks Challenging +1.2
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
18 marks Challenging +1.8
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
18 marks Challenging +1.2
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
18 marks Challenging +1.2
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
18 marks Challenging +1.2
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
18 marks Standard +0.8
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
18 marks Challenging +1.2
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
18 marks Standard +0.3
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
18 marks Standard +0.8
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
18 marks Challenging +1.2
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
19 marks Standard +0.3
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
17 marks Standard +0.8
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
18 marks Standard +0.8
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
18 marks Challenging +1.8
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 FP3 2009 June Q1
24 marks Challenging +1.8
1 The point \(\mathrm { A } ( - 1,12,5 )\) lies on the plane \(P\) with equation \(8 x - 3 y + 10 z = 6\). The point \(\mathrm { B } ( 6 , - 2,9 )\) lies on the plane \(Q\) with equation \(3 x - 4 y - 2 z = 8\). The planes \(P\) and \(Q\) intersect in the line \(L\).
  1. Find an equation for the line \(L\).
  2. Find the shortest distance between \(L\) and the line AB . The lines \(M\) and \(N\) are both parallel to \(L\), with \(M\) passing through A and \(N\) passing through B .
  3. Find the distance between the parallel lines \(M\) and \(N\). The point C has coordinates \(( k , 0,2 )\), and the line AC intersects the line \(N\) at the point D .
  4. Find the value of \(k\), and the coordinates of D .