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OCR MEI FP2 2006 June Q1
18 marks Standard +0.8
1
  1. A curve has polar equation \(r = a ( \sqrt { 2 } + 2 \cos \theta )\) for \(- \frac { 3 } { 4 } \pi \leqslant \theta \leqslant \frac { 3 } { 4 } \pi\), where \(a\) is a positive constant.
    1. Sketch the curve.
    2. Find, in an exact form, the area of the region enclosed by the curve.
    1. Find the Maclaurin series for the function \(\mathrm { f } ( x ) = \tan \left( \frac { 1 } { 4 } \pi + x \right)\), up to the term in \(x ^ { 2 }\).
    2. Use the Maclaurin series to show that, when \(h\) is small, $$\int _ { - h } ^ { h } x ^ { 2 } \tan \left( \frac { 1 } { 4 } \pi + x \right) \mathrm { d } x \approx \frac { 2 } { 3 } h ^ { 3 } + \frac { 4 } { 5 } h ^ { 5 }$$
OCR MEI FP2 2006 June Q2
18 marks Challenging +1.2
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. By considering \(\left( z - \frac { 1 } { z } \right) ^ { 4 } \left( z + \frac { 1 } { z } \right) ^ { 2 }\), find \(A , B , C\) and \(D\) such that $$\sin ^ { 4 } \theta \cos ^ { 2 } \theta = A \cos 6 \theta + B \cos 4 \theta + C \cos 2 \theta + D$$
    1. Find the modulus and argument of \(4 + 4 \mathrm { j }\).
    2. Find the fifth roots of \(4 + 4 \mathrm { j }\) in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\). Illustrate these fifth roots on an Argand diagram.
    3. Find integers \(p\) and \(q\) such that \(( p + q \mathrm { j } ) ^ { 5 } = 4 + 4 \mathrm { j }\).
OCR MEI FP2 2006 June Q3
18 marks Standard +0.8
3
  1. Find the inverse of the matrix \(\left( \begin{array} { r r r } 4 & 1 & k \\ 3 & 2 & 5 \\ 8 & 5 & 13 \end{array} \right)\), where \(k \neq 5\).
  2. Solve the simultaneous equations $$\begin{aligned} & 4 x + y + 7 z = 12 \\ & 3 x + 2 y + 5 z = m \\ & 8 x + 5 y + 13 z = 0 \end{aligned}$$ giving \(x , y\) and \(z\) in terms of \(m\).
  3. Find the value of \(p\) for which the simultaneous equations $$\begin{aligned} & 4 x + y + 5 z = 12 \\ & 3 x + 2 y + 5 z = p \\ & 8 x + 5 y + 13 z = 0 \end{aligned}$$ have solutions, and find the general solution in this case.
OCR MEI FP2 2006 June Q4
18 marks Challenging +1.2
4
  1. Starting from the definitions of \(\sinh x\) and \(\cosh x\) in terms of exponentials, prove that $$1 + 2 \sinh ^ { 2 } x = \cosh 2 x$$
  2. Solve the equation $$2 \cosh 2 x + \sinh x = 5 ,$$ giving the answers in an exact logarithmic form.
  3. Show that \(\int _ { 0 } ^ { \ln 3 } \sinh ^ { 2 } x \mathrm {~d} x = \frac { 10 } { 9 } - \frac { 1 } { 2 } \ln 3\).
  4. Find the exact value of \(\int _ { 3 } ^ { 5 } \sqrt { x ^ { 2 } - 9 } \mathrm {~d} x\).
OCR MEI FP2 2007 June Q1
18 marks Standard +0.8
1
  1. A curve has polar equation \(r = a ( 1 - \cos \theta )\), where \(a\) is a positive constant.
    1. Sketch the curve.
    2. Find the area of the region enclosed by the section of the curve for which \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\) and the line \(\theta = \frac { 1 } { 2 } \pi\).
  2. Use a trigonometric substitution to show that \(\int _ { 0 } ^ { 1 } \frac { 1 } { \left( 4 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x = \frac { 1 } { 4 \sqrt { 3 } }\).
  3. In this part of the question, \(\mathrm { f } ( x ) = \arccos ( 2 x )\).
    1. Find \(\mathrm { f } ^ { \prime } ( x )\).
    2. Use a standard series to expand \(\mathrm { f } ^ { \prime } ( x )\), and hence find the series for \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to the term in \(x ^ { 5 }\).
OCR MEI FP2 2007 June Q2
18 marks Standard +0.8
2
  1. Use de Moivre's theorem to show that \(\sin 5 \theta = 5 \sin \theta - 20 \sin ^ { 3 } \theta + 16 \sin ^ { 5 } \theta\).
    1. Find the cube roots of \(- 2 + 2 \mathrm { j }\) in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\) where \(r > 0\) and \(- \pi < \theta \leqslant \pi\). These cube roots are represented by points \(\mathrm { A } , \mathrm { B }\) and C in the Argand diagram, with A in the first quadrant and ABC going anticlockwise. The midpoint of AB is M , and M represents the complex number \(w\).
    2. Draw an Argand diagram, showing the points \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and M .
    3. Find the modulus and argument of \(w\).
    4. Find \(w ^ { 6 }\) in the form \(a + b \mathrm { j }\).
OCR MEI FP2 2007 June Q3
18 marks Challenging +1.2
3 Let \(\mathbf { M } = \left( \begin{array} { r r r } 3 & 5 & 2 \\ 5 & 3 & - 2 \\ 2 & - 2 & - 4 \end{array} \right)\).
  1. Show that the characteristic equation for \(\mathbf { M }\) is \(\lambda ^ { 3 } - 2 \lambda ^ { 2 } - 48 \lambda = 0\). You are given that \(\left( \begin{array} { r } 1 \\ - 1 \\ 1 \end{array} \right)\) is an eigenvector of \(\mathbf { M }\) corresponding to the eigenvalue 0 .
  2. Find the other two eigenvalues of \(\mathbf { M }\), and corresponding eigenvectors.
  3. Write down a matrix \(\mathbf { P }\), and a diagonal matrix \(\mathbf { D }\), such that \(\mathbf { P } ^ { - 1 } \mathbf { M } ^ { 2 } \mathbf { P } = \mathbf { D }\).
  4. Use the Cayley-Hamilton theorem to find integers \(a\) and \(b\) such that \(\mathbf { M } ^ { 4 } = a \mathbf { M } ^ { 2 } + b \mathbf { M }\). Section B (18 marks)
OCR MEI FP2 2007 June Q4
18 marks Standard +0.8
4
  1. Find \(\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 9 x ^ { 2 } + 16 } } \mathrm {~d} x\), giving your answer in an exact logarithmic form.
    1. Starting from the definitions of \(\sinh x\) and \(\cosh x\) in terms of exponentials, prove that \(\sinh 2 x = 2 \sinh x \cosh x\).
    2. Show that one of the stationary points on the curve $$y = 20 \cosh x - 3 \cosh 2 x$$ has coordinates \(\left( \ln 3 , \frac { 59 } { 3 } \right)\), and find the coordinates of the other two stationary points.
    3. Show that \(\int _ { - \ln 3 } ^ { \ln 3 } ( 20 \cosh x - 3 \cosh 2 x ) \mathrm { d } x = 40\).
OCR MEI FP2 2007 June Q5
18 marks Standard +0.8
5 The curve with equation \(y = \frac { x ^ { 2 } - k x + 2 k } { x + k }\) is to be investigated for different values of \(k\).
  1. Use your graphical calculator to obtain rough sketches of the curve in the cases \(k = - 2\), \(k = - 0.5\) and \(k = 1\).
  2. Show that the equation of the curve may be written as \(y = x - 2 k + \frac { 2 k ( k + 1 ) } { x + k }\). Hence find the two values of \(k\) for which the curve is a straight line.
  3. When the curve is not a straight line, it is a conic.
    (A) Name the type of conic.
    (B) Write down the equations of the asymptotes.
  4. Draw a sketch to show the shape of the curve when \(1 < k < 8\). This sketch should show where the curve crosses the axes and how it approaches its asymptotes. Indicate the points A and B on the curve where \(x = 1\) and \(x = k\) respectively.
OCR MEI FP2 2008 June Q1
18 marks Standard +0.8
1
  1. A curve has cartesian equation \(\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 3 x y ^ { 2 }\).
    1. Show that the polar equation of the curve is \(r = 3 \cos \theta \sin ^ { 2 } \theta\).
    2. Hence sketch the curve.
  2. Find the exact value of \(\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 4 - 3 x ^ { 2 } } } \mathrm {~d} x\).
    1. Write down the series for \(\ln ( 1 + x )\) and the series for \(\ln ( 1 - x )\), both as far as the term in \(x ^ { 5 }\).
    2. Hence find the first three non-zero terms in the series for \(\ln \left( \frac { 1 + x } { 1 - x } \right)\).
    3. Use the series in part (ii) to show that \(\sum _ { r = 0 } ^ { \infty } \frac { 1 } { ( 2 r + 1 ) 4 ^ { r } } = \ln 3\).
OCR MEI FP2 2008 June Q2
18 marks Standard +0.8
2 You are given the complex numbers \(z = \sqrt { 32 } ( 1 + \mathrm { j } )\) and \(w = 8 \left( \cos \frac { 7 } { 12 } \pi + \mathrm { j } \sin \frac { 7 } { 12 } \pi \right)\).
  1. Find the modulus and argument of each of the complex numbers \(z , z ^ { * } , z w\) and \(\frac { z } { w }\).
  2. Express \(\frac { z } { w }\) in the form \(a + b \mathrm { j }\), giving the exact values of \(a\) and \(b\).
  3. Find the cube roots of \(z\), in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
  4. Show that the cube roots of \(z\) can be written as $$k _ { 1 } w ^ { * } , \quad k _ { 2 } z ^ { * } \quad \text { and } \quad k _ { 3 } \mathrm { j } w ,$$ where \(k _ { 1 } , k _ { 2 }\) and \(k _ { 3 }\) are real numbers. State the values of \(k _ { 1 } , k _ { 2 }\) and \(k _ { 3 }\).
OCR MEI FP2 2008 June Q3
18 marks Standard +0.8
3
  1. Given the matrix \(\mathbf { Q } = \left( \begin{array} { r r r } 2 & - 1 & k \\ 1 & 0 & 1 \\ 3 & 1 & 2 \end{array} \right)\) (where \(k \neq 3\) ), find \(\mathbf { Q } ^ { - 1 }\) in terms of \(k\).
    Show that, when \(k = 4 , \mathbf { Q } ^ { - 1 } = \left( \begin{array} { r r r } - 1 & 6 & - 1 \\ 1 & - 8 & 2 \\ 1 & - 5 & 1 \end{array} \right)\). The matrix \(\mathbf { M }\) has eigenvectors \(\left( \begin{array} { l } 2 \\ 1 \\ 3 \end{array} \right) , \left( \begin{array} { r } - 1 \\ 0 \\ 1 \end{array} \right)\) and \(\left( \begin{array} { l } 4 \\ 1 \\ 2 \end{array} \right)\), with corresponding eigenvalues \(1 , - 1\) and 3 respectively.
  2. Write down a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { P } ^ { - 1 } \mathbf { M P } = \mathbf { D }\), and hence find the matrix \(\mathbf { M }\).
  3. Write down the characteristic equation for \(\mathbf { M }\), and use the Cayley-Hamilton theorem to find integers \(a , b\) and \(c\) such that \(\mathbf { M } ^ { 4 } = a \mathbf { M } ^ { 2 } + b \mathbf { M } + c \mathbf { I }\). Section B (18 marks)
OCR MEI FP2 2008 June Q4
18 marks Standard +0.3
4
  1. Starting from the definitions of \(\sinh x\) and \(\cosh x\) in terms of exponentials, prove that $$\cosh ^ { 2 } x - \sinh ^ { 2 } x = 1$$
  2. Solve the equation \(4 \cosh ^ { 2 } x + 9 \sinh x = 13\), giving the answers in exact logarithmic form.
  3. Show that there is only one stationary point on the curve $$y = 4 \cosh ^ { 2 } x + 9 \sinh x$$ and find the \(y\)-coordinate of the stationary point.
  4. Show that \(\int _ { 0 } ^ { \ln 2 } \left( 4 \cosh ^ { 2 } x + 9 \sinh x \right) \mathrm { d } x = 2 \ln 2 + \frac { 33 } { 8 }\).
OCR MEI FP2 2008 June Q5
18 marks Challenging +1.2
5 A curve has parametric equations \(x = \lambda \cos \theta - \frac { 1 } { \lambda } \sin \theta , y = \cos \theta + \sin \theta\), where \(\lambda\) is a positive constant.
  1. Use your calculator to obtain a sketch of the curve in each of the cases $$\lambda = 0.5 , \quad \lambda = 3 \quad \text { and } \quad \lambda = 5 .$$
  2. Given that the curve is a conic, name the type of conic.
  3. Show that \(y\) has a maximum value of \(\sqrt { 2 }\) when \(\theta = \frac { 1 } { 4 } \pi\).
  4. Show that \(x ^ { 2 } + y ^ { 2 } = \left( 1 + \lambda ^ { 2 } \right) + \left( \frac { 1 } { \lambda ^ { 2 } } - \lambda ^ { 2 } \right) \sin ^ { 2 } \theta\), and deduce that the distance from the origin of any point on the curve is between \(\sqrt { 1 + \frac { 1 } { \lambda ^ { 2 } } }\) and \(\sqrt { 1 + \lambda ^ { 2 } }\).
  5. For the case \(\lambda = 1\), show that the curve is a circle, and find its radius.
  6. For the case \(\lambda = 2\), draw a sketch of the curve, and label the points \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E } , \mathrm { F } , \mathrm { G } , \mathrm { H }\) on the curve corresponding to \(\theta = 0 , \frac { 1 } { 4 } \pi , \frac { 1 } { 2 } \pi , \frac { 3 } { 4 } \pi , \pi , \frac { 5 } { 4 } \pi , \frac { 3 } { 2 } \pi , \frac { 7 } { 4 } \pi\) respectively. You should make clear what is special about each of these points.
OCR MEI FP2 2010 June Q1
19 marks Standard +0.8
1
    1. Given that \(\mathrm { f } ( t ) = \arcsin t\), write down an expression for \(\mathrm { f } ^ { \prime } ( t )\) and show that $$\mathrm { f } ^ { \prime \prime } ( t ) = \frac { t } { \left( 1 - t ^ { 2 } \right) ^ { \frac { 3 } { 2 } } }$$
    2. Show that the Maclaurin expansion of the function \(\arcsin \left( x + \frac { 1 } { 2 } \right)\) begins $$\frac { \pi } { 6 } + \frac { 2 } { \sqrt { 3 } } x$$ and find the term in \(x ^ { 2 }\).
  2. Sketch the curve with polar equation \(r = \frac { \pi a } { \pi + \theta }\), where \(a > 0\), for \(0 \leqslant \theta < 2 \pi\). Find, in terms of \(a\), the area of the region bounded by the part of the curve for which \(0 \leqslant \theta \leqslant \pi\) and the lines \(\theta = 0\) and \(\theta = \pi\).
  3. Find the exact value of the integral $$\int _ { 0 } ^ { \frac { 3 } { 2 } } \frac { 1 } { 9 + 4 x ^ { 2 } } \mathrm {~d} x$$
OCR MEI FP2 2010 June Q2
16 marks Challenging +1.2
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.
    Hence find the constants \(A , B , C\) in the identity $$\sin ^ { 5 } \theta \equiv A \sin \theta + B \sin 3 \theta + C \sin 5 \theta$$
    1. Find the 4th roots of - 9 j in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\), where \(r > 0\) and \(0 < \theta < 2 \pi\). Illustrate the roots on an Argand diagram.
    2. Let the points representing these roots, taken in order of increasing \(\theta\), be \(\mathrm { P } , \mathrm { Q } , \mathrm { R } , \mathrm { S }\). The mid-points of the sides of PQRS represent the 4th roots of a complex number \(w\). Find the modulus and argument of \(w\). Mark the point representing \(w\) on your Argand diagram.
OCR MEI FP2 2010 June Q3
19 marks Standard +0.8
3
    1. A \(3 \times 3\) matrix \(\mathbf { M }\) has characteristic equation $$2 \lambda ^ { 3 } + \lambda ^ { 2 } - 13 \lambda + 6 = 0$$ Show that \(\lambda = 2\) is an eigenvalue of \(\mathbf { M }\). Find the other eigenvalues.
    2. An eigenvector corresponding to \(\lambda = 2\) is \(\left( \begin{array} { r } 3 \\ - 3 \\ 1 \end{array} \right)\). Evaluate \(\mathbf { M } \left( \begin{array} { r } 3 \\ - 3 \\ 1 \end{array} \right)\) and \(\mathbf { M } ^ { 2 } \left( \begin{array} { r } 1 \\ - 1 \\ \frac { 1 } { 3 } \end{array} \right)\).
      Solve the equation \(\mathbf { M } \left( \begin{array} { l } x \\ y \\ z \end{array} \right) = \left( \begin{array} { r } 3 \\ - 3 \\ 1 \end{array} \right)\).
    3. Find constants \(A , B , C\) such that $$\mathbf { M } ^ { 4 } = A \mathbf { M } ^ { 2 } + B \mathbf { M } + C \mathbf { I }$$
  2. A \(2 \times 2\) matrix \(\mathbf { N }\) has eigenvalues -1 and 2, with eigenvectors \(\binom { 1 } { 2 }\) and \(\binom { - 1 } { 1 }\) respectively. Find \(\mathbf { N }\). Section B (18 marks)
OCR MEI FP2 2010 June Q4
18 marks Challenging +1.2
4
  1. Prove, using exponential functions, that $$\sinh 2 x = 2 \sinh x \cosh x$$ Differentiate this result to obtain a formula for \(\cosh 2 x\).
  2. Sketch the curve with equation \(y = \cosh x - 1\). The region bounded by this curve, the \(x\)-axis, and the line \(x = 2\) is rotated through \(2 \pi\) radians about the \(x\)-axis. Find, correct to 3 decimal places, the volume generated. (You must show your working; numerical integration by calculator will receive no credit.)
  3. Show that the curve with equation $$y = \cosh 2 x + \sinh x$$ has exactly one stationary point.
    Determine, in exact logarithmic form, the \(x\)-coordinate of the stationary point.
OCR MEI FP2 2010 June Q5
18 marks Challenging +1.2
5 In parts (i), (ii), (iii) of this question you are required to investigate curves with the equation $$x ^ { k } + y ^ { k } = 1$$ for various positive values of \(k\).
  1. Firstly consider cases in which \(k\) is a positive even integer.
    (A) State the shape of the curve when \(k = 2\).
    (B) Sketch, on the same axes, the curves for \(k = 2\) and \(k = 4\).
    (C) Describe the shape that the curve tends to as \(k\) becomes very large.
    (D) State the range of possible values of \(x\) and \(y\).
  2. Now consider cases in which \(k\) is a positive odd integer.
    (A) Explain why \(x\) and \(y\) may take any value.
    (B) State the shape of the curve when \(k = 1\).
    (C) Sketch the curve for \(k = 3\). State the equation of the asymptote of this curve.
    (D) Sketch the shape that the curve tends to as \(k\) becomes very large.
  3. Now let \(k = \frac { 1 } { 2 }\). Sketch the curve, indicating the range of possible values of \(x\) and \(y\).
  4. Now consider the modified equation \(| x | ^ { k } + | y | ^ { k } = 1\).
    (A) Sketch the curve for \(k = \frac { 1 } { 2 }\).
    (B) Investigate the shape of the curve for \(k = \frac { 1 } { n }\) as the positive integer \(n\) becomes very large.
OCR S1 2005 January Q1
4 marks Easy -1.3
1 The scatter diagrams below illustrate three sets of bivariate data, \(A , B\) and \(C\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f0c0a4ca-da0a-4c74-b8b1-bac4fd3f2487-2_440_428_360_317} \captionsetup{labelformat=empty} \caption{Set \(A\)}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f0c0a4ca-da0a-4c74-b8b1-bac4fd3f2487-2_440_426_360_858} \captionsetup{labelformat=empty} \caption{Set \(B\)}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f0c0a4ca-da0a-4c74-b8b1-bac4fd3f2487-2_435_424_365_1402} \captionsetup{labelformat=empty} \caption{Set \(C\)}
\end{figure} State, with an explanation in each case, which of the three sets of data has
  1. the largest,
  2. the smallest,
    value of the product moment correlation coefficient.
OCR S1 2005 January Q2
6 marks Easy -1.8
2 The back-to-back stem-and-leaf diagram below shows the number of hours of television watched per week by each of 15 boys and 15 girls. $$\begin{aligned} & \text { Boys Girls } \\ & \left. \begin{array} { r r r r r r r r | r r r r r r r r r r r r r } & 677664 & 4 & 3 & 0 & 0 & 5 & 5 & 6 & 677888 \end{array} \right\} \end{aligned}$$ Key: 4 | 2 | 2 means a boy who watched 24 hours and a girl who watched 22 hours of television per week.
  1. Find the median and the quartiles of the results for the boys.
  2. Give a reason why the median might be preferred to the mean in using an average to compare the two data sets.
  3. State one advantage, and one disadvantage, of using stem-and-leaf diagrams rather than box-andwhisker plots to represent the data.
OCR S1 2005 January Q3
6 marks Moderate -0.8
3 Two commentators gave ratings out of 100 for seven sports personalities. The ratings are shown in the table below.
Personality\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)
Commentator I73767865868291
Commentator II77787980868995
  1. Calculate Spearman's rank correlation coefficient for these ratings.
  2. State what your answer tells you about the ratings given by the two commentators.
OCR S1 2005 January Q4
7 marks Easy -1.3
4 The table below shows the probability distribution of the random variable \(X\).
\(x\)- 2- 1012
\(\mathrm { P } ( X = x )\)\(\frac { 1 } { 4 }\)\(\frac { 1 } { 5 }\)\(k\)\(\frac { 2 } { 5 }\)\(\frac { 1 } { 10 }\)
  1. Find the value of the constant \(k\).
  2. Calculate the values of \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
OCR S1 2005 January Q5
8 marks Moderate -0.8
5 On average 1 in 20 members of the population of this country has a particular DNA feature. Members of the population are selected at random until one is found who has this feature.
  1. Find the probability that the first person to have this feature is
    1. the sixth person selected,
    2. not among the first 10 people selected.
    3. Find the expected number of people selected.
OCR S1 2005 January Q6
7 marks Moderate -0.3
6 Louise and Marie play a series of tennis matches. It is given that, in any match, the probability that Louise wins the first two sets is \(\frac { 3 } { 8 }\).
  1. Find the probability that, in 5 randomly chosen matches, Louise wins the first two sets in exactly 2 of the matches. It is also given that Louise and Marie are equally likely to win the first set.
  2. Show that P (Louise wins the second set, given that she won the first set) \(= \frac { 3 } { 4 }\).
  3. The probability that Marie wins the first two sets is \(\frac { 1 } { 3 }\). Find P(Marie wins the second set, given that she won the first set).