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OCR S4 2016 June Q5
11 marks Standard +0.8
5 Events \(A\) and \(B\) are such that \(\mathrm { P } ( A ) = 0.5 , \mathrm { P } ( B ) = 0.6\) and \(\mathrm { P } \left( A \mid B ^ { \prime } \right) = 0.75\).
  1. Find \(\mathrm { P } ( A \cap B )\) and \(\mathrm { P } ( A \cup B )\).
  2. Determine, giving a reason in each case,
    1. whether \(A\) and \(B\) are mutually exclusive,
    2. whether \(A\) and \(B\) are independent.
    3. A further event \(C\) is such that \(\mathrm { P } ( A \cup B \cup C ) = 1\) and \(\mathrm { P } ( A \cap B \cap C ) = 0.05\). It is also given that \(\mathrm { P } \left( A \cap B ^ { \prime } \cap C \right) = \mathrm { P } \left( A ^ { \prime } \cap B \cap C \right) = x\) and \(\mathrm { P } \left( A \cap B ^ { \prime } \cap C ^ { \prime } \right) = 2 x\).
      Find \(\mathrm { P } ( C )\).
OCR S4 2016 June Q6
13 marks Standard +0.3
6 Andrew has five coins. Three of them are unbiased. The other two are biased such that the probability of obtaining a head when one of them is tossed is \(\frac { 3 } { 5 }\). Andrew tosses all five coins. It is given that the probability generating function of \(X\), the number of heads obtained on the unbiased coins, is \(\mathrm { G } _ { X } ( t )\), where $$\mathrm { G } _ { X } ( t ) = \frac { 1 } { 8 } + \frac { 3 } { 8 } t + \frac { 3 } { 8 } t ^ { 2 } + \frac { 1 } { 8 } t ^ { 3 }$$
  1. Find \(G _ { Y } ( \mathrm { t } )\), the probability generating function of \(Y\), the number of heads on the biased coins.
  2. The random variable \(Z\) is the total number of heads obtained when Andrew tosses all five coins. Find the probability generating function of \(Z\), giving your answer as a polynomial.
  3. Find \(\mathrm { E } ( Z )\) and \(\operatorname { Var } ( Z )\).
  4. Write down the value of \(\mathrm { P } ( Z = 3 )\).
OCR S4 2016 June Q7
14 marks Challenging +1.8
7 A continuous random variable \(Y\) has cumulative distribution function $$\mathrm { F } ( y ) = \left\{ \begin{array} { c c } 0 & y < a \\ 1 - \frac { a ^ { 5 } } { y ^ { 5 } } & y \geqslant a \end{array} \right.$$ where \(a\) is a parameter.
Two independent observations of \(Y\) are denoted by \(Y _ { 1 }\) and \(Y _ { 2 }\). The smaller of them is denoted by S .
  1. Show that \(P ( S > \mathrm { s } ) = \frac { a ^ { 10 } } { s ^ { 10 } }\) and hence find the probability density function of \(S\).
  2. Show that \(S\) is not an unbiased estimator of \(a\), and construct an unbiased estimator of \(a , T _ { 1 }\) based on \(S\).
  3. Construct another unbiased estimator of \(a , T _ { 2 }\), of the form \(k \left( Y _ { 1 } + Y _ { 2 } \right)\), where \(k\) is a constant to be found.
  4. Without further calculation, explain how you would decide which of \(T _ { 1 }\) and \(T _ { 2 }\) is the more efficient estimator.
OCR S4 2017 June Q1
4 marks Standard +0.3
1 A meteorologist claims that the median daily rainfall in London is 2.2 mm . A single sample sign test is to be used to test the claim, using the following hypotheses: \(\mathrm { H } _ { 0 }\) : a sample comes from a population with median 2.2, \(\mathrm { H } _ { 1 }\) : the sample does not come from a population with median 2.2.
30 randomly selected observations of daily rainfall in London are compared with 2.2, and given a '+' sign if greater than 2.2 and a '-' sign if less than 2.2. (You may assume that no data values are exactly equal to 2.2.) The test is to be carried out at the \(5 \%\) level of significance. Let the number of ' + ' signs be \(k\). Find, in terms of \(k\), the critical region for the test showing the values of any relevant probabilities.
OCR S4 2017 June Q2
11 marks Challenging +1.2
2 The independent discrete random variables \(X\) and \(Y\) can take the values 0,1 and 2 with probabilities as given in the tables.
\(x\)012
\(\mathrm { P } ( X = x )\)0.50.30.2
\(\quad\)
\(y\)012
\(\mathrm { P } ( Y = y )\)0.50.30.2
The random variables \(U\) and \(V\) are defined as follows: $$U = X Y , V = | X - Y | .$$
  1. In the Printed Answer Book complete the table giving the joint distribution of \(U\) and \(V\).
  2. Find \(\operatorname { Cov } ( U , V )\).
  3. Find \(\mathrm { P } ( U V = 0 \mid V = 2 )\).
OCR S4 2017 June Q3
10 marks Standard +0.8
3 For events \(A , B\) and \(C\) it is given that \(\mathrm { P } ( A ) = 0.6 , \mathrm { P } ( B ) = 0.5 , \mathrm { P } ( C ) = 0.4\) and \(\mathrm { P } ( A \cap B \cap C ) = 0.1\). It is also given that events \(A\) and \(B\) are independent and that events \(A\) and \(C\) are independent.
  1. Find \(\mathrm { P } ( B \mid A )\).
  2. Given also that events \(B\) and \(C\) are independent, find \(\mathrm { P } \left( A ^ { \prime } \cap B ^ { \prime } \cap C ^ { \prime } \right)\).
  3. Given instead that events \(B\) and \(C\) are not independent, find the greatest and least possible values of \(\mathrm { P } \left( A ^ { \prime } \cap B ^ { \prime } \cap C ^ { \prime } \right)\).
OCR S4 2017 June Q4
12 marks Standard +0.3
4 The heights of eleven randomly selected primary school children are measured. The results, in metres, are
Girls1.481.311.631.381.561.57
Boys1.441.351.321.281.27
  1. Use a Wilcoxon rank-sum test, at the \(1 \%\) significance level, to test whether primary school girls are taller than primary school boys.
  2. It is decided to repeat the test, using larger random samples. The heights of twenty girls and eighteen boys are measured. Find the greatest value of the test statistic \(W\) which will result in the conclusion that there is evidence, at the \(1 \%\) level of significance, that primary school girls are taller than primary school boys.
OCR S4 2017 June Q5
11 marks Standard +0.3
5 The discrete random variable \(X\) is such that \(\mathrm { P } ( X = x ) = \frac { 3 } { 4 } \left( \frac { 1 } { 4 } \right) ^ { x } , x = 0,1,2 , \ldots\).
  1. Show that the moment generating function of \(X , \mathrm { M } _ { X } ( t )\), can be written as \(\mathrm { M } _ { X } ( t ) = \frac { 3 } { 4 - \mathrm { e } ^ { t } }\).
  2. Find the range of values of \(t\) for which the formula for \(\mathrm { M } _ { X } ( t )\) in part (i) is valid.
  3. Use \(\mathrm { M } _ { X } ( t )\) to find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
OCR S4 2017 June Q6
15 marks Standard +0.3
6 The continuous random variable \(Z\) has probability density function $$f ( z ) = \left\{ \begin{array} { c c } \frac { 4 z ^ { 3 } } { k ^ { 4 } } & 0 \leqslant z \leqslant k \\ 0 & \text { otherwise } \end{array} \right.$$ where \(k\) is a parameter whose value is to be estimated.
  1. Show that \(\frac { 5 Z } { 4 }\) is an unbiased estimator of \(k\).
  2. Find the variance of \(\frac { 5 Z } { 4 }\). The parameter \(k\) can also be estimated by making observations of a random variable \(X\) which has mean \(\frac { 1 } { 2 } k\) and variance \(\frac { 1 } { 12 } k ^ { 2 }\). Let \(Y = X _ { 1 } + X _ { 2 } + X _ { 3 }\) where \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\) are independent observations of \(X\).
  3. \(c Y\) is also an unbiased estimator of \(k\). Find the value of \(c\).
  4. For the value of \(c\) found in part (iii), determine which of \(\frac { 5 Z } { 4 }\) and \(c Y\) is the more efficient estimator of \(k\).
OCR S4 2017 June Q7
9 marks Challenging +1.2
7 The discrete random variable \(Y\) has probability generating function \(\mathrm { G } _ { Y } ( t ) = \frac { 1 } { 126 } t \left( 64 - t ^ { 6 } \right) \left( 1 - \frac { t } { 2 } \right) ^ { - 1 }\).
  1. Find \(\mathrm { P } ( Y = 3 )\).
  2. Find \(\mathrm { E } ( Y )\).
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 .
OCR MEI FP3 2009 June Q2
24 marks Challenging +1.8
2 A surface has equation \(z = 3 x ( x + y ) ^ { 3 } - 2 x ^ { 3 } + 24 x\).
  1. Find \(\frac { \partial z } { \partial x }\) and \(\frac { \partial z } { \partial y }\).
  2. Find the coordinates of the three stationary points on the surface.
  3. Find the equation of the normal line at the point \(\mathrm { P } ( 1 , - 2,19 )\) on the surface.
  4. The point \(\mathrm { Q } ( 1 + k , - 2 + h , 19 + 3 h )\) is on the surface and is close to P . Find an approximate expression for \(k\) in terms of \(h\).
  5. Show that there is only one point on the surface at which the tangent plane has an equation of the form \(27 x - z = d\). Find the coordinates of this point and the corresponding value of \(d\).
OCR MEI FP3 2009 June Q3
24 marks Challenging +1.8
3 A curve has parametric equations \(x = a ( \theta + \sin \theta ) , y = a ( 1 - \cos \theta )\), for \(0 \leqslant \theta \leqslant \pi\), where \(a\) is a positive constant.
  1. Show that the arc length \(s\) from the origin to a general point on the curve is given by \(s = 4 a \sin \frac { 1 } { 2 } \theta\).
  2. Find the intrinsic equation of the curve giving \(s\) in terms of \(a\) and \(\psi\), where \(\tan \psi = \frac { \mathrm { d } y } { \mathrm {~d} x }\).
  3. Hence, or otherwise, show that the radius of curvature at a point on the curve is \(4 a \cos \frac { 1 } { 2 } \theta\).
  4. Find the coordinates of the centre of curvature corresponding to the point on the curve where \(\theta = \frac { 2 } { 3 } \pi\).
  5. Find the area of the surface generated when the curve is rotated through \(2 \pi\) radians about the \(x\)-axis.
OCR MEI FP3 2009 June Q4
24 marks Challenging +1.2
4 The group \(G = \{ 1,2,3,4,5,6 \}\) has multiplication modulo 7 as its operation. The group \(H = \{ 1,5,7,11,13,17 \}\) has multiplication modulo 18 as its operation.
  1. Show that the groups \(G\) and \(H\) are both cyclic.
  2. List all the proper subgroups of \(G\).
  3. Specify an isomorphism between \(G\) and \(H\). The group \(S = \{ \mathrm { a } , \mathrm { b } , \mathrm { c } , \mathrm { d } , \mathrm { e } , \mathrm { f } \}\) consists of functions with domain \(\{ 1,2,3 \}\) given by $$\begin{array} { l l l } \mathrm { a } ( 1 ) = 2 & \mathrm { a } ( 2 ) = 3 & \mathrm { a } ( 3 ) = 1 \\ \mathrm {~b} ( 1 ) = 3 & \mathrm {~b} ( 2 ) = 1 & \mathrm {~b} ( 3 ) = 2 \\ \mathrm { c } ( 1 ) = 1 & \mathrm { c } ( 2 ) = 3 & \mathrm { c } ( 3 ) = 2 \\ \mathrm {~d} ( 1 ) = 3 & \mathrm {~d} ( 2 ) = 2 & \mathrm {~d} ( 3 ) = 1 \\ \mathrm { e } ( 1 ) = 1 & \mathrm { e } ( 2 ) = 2 & \mathrm { e } ( 3 ) = 3 \\ \mathrm { f } ( 1 ) = 2 & \mathrm { f } ( 2 ) = 1 & \mathrm { f } ( 3 ) = 3 \end{array}$$ and the group operation is composition of functions.
  4. Show that ad \(= \mathrm { c }\) and find da.
  5. Give a reason why \(S\) is not isomorphic to \(G\).
  6. Find the order of each element of \(S\).
  7. List all the proper subgroups of \(S\).
OCR MEI FP3 2009 June Q5
24 marks Moderate -0.5
5 Each level of a fantasy computer game is set in a single location, Alphaworld, Betaworld, Chiworld or Deltaworld. After completing a level, a player goes on to the next level, which could be set in the same location as the previous level, or in a different location. In the first version of the game, the initial and transition probabilities are as follows.
Level 1 is set in Alphaworld or Betaworld, with probabilities 0.6, 0.4 respectively.
After a level set in Alphaworld, the next level will be set in Betaworld, Chiworld or Deltaworld, with probabilities \(0.7,0.1,0.2\) respectively.
After a level set in Betaworld, the next level will be set in Alphaworld, Betaworld or Deltaworld, with probabilities \(0.1,0.8,0.1\) respectively.
After a level set in Chiworld, the next level will also be set in Chiworld.
After a level set in Deltaworld, the next level will be set in Alphaworld, Betaworld or Chiworld, with probabilities \(0.3,0.6,0.1\) respectively. The situation is modelled as a Markov chain with four states.
  1. Write down the transition matrix.
  2. Find the probabilities that level 14 is set in each location.
  3. Find the probability that level 15 is set in the same location as level 14 .
  4. Find the level at which the probability of being set in Chiworld first exceeds 0.5.
  5. Following a level set in Betaworld, find the expected number of further levels which will be set in Betaworld before changing to a different location. In the second version of the game, the initial probabilities and the transition probabilities after Alphaworld, Betaworld and Deltaworld are all the same as in the first version; but after a level set in Chiworld, the next level will be set in Chiworld or Deltaworld, with probabilities \(0.9,0.1\) respectively.
  6. By considering powers of the new transition matrix, or otherwise, find the equilibrium probabilities for the four locations. In the third version of the game, the initial probabilities and the transition probabilities after Alphaworld, Betaworld and Deltaworld are again all the same as in the first version; but the transition probabilities after Chiworld have changed again. The equilibrium probabilities for Alphaworld, Betaworld, Chiworld and Deltaworld are now 0.11, 0.75, 0.04, 0.1 respectively.
  7. Find the new transition probabilities after a level set in Chiworld. }{www.ocr.org.uk}) after the live examination series.
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OCR MEI FP2 2010 January Q1
18 marks Standard +0.8
1
  1. Given that \(y = \arctan \sqrt { x }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), giving your answer in terms of \(x\). Hence show that $$\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { x } ( x + 1 ) } \mathrm { d } x = \frac { \pi } { 2 }$$
  2. A curve has cartesian equation $$x ^ { 2 } + y ^ { 2 } = x y + 1$$
    1. Show that the polar equation of the curve is $$r ^ { 2 } = \frac { 2 } { 2 - \sin 2 \theta }$$
    2. Determine the greatest and least positive values of \(r\) and the values of \(\theta\) between 0 and \(2 \pi\) for which they occur.
    3. Sketch the curve.
OCR MEI FP2 2010 January Q2
18 marks Standard +0.8
2
  1. Use de Moivre's theorem to find the constants \(a , b , c\) in the identity $$\cos 5 \theta \equiv a \cos ^ { 5 } \theta + b \cos ^ { 3 } \theta + c \cos \theta$$
  2. Let $$\begin{aligned} C & = \cos \theta + \cos \left( \theta + \frac { 2 \pi } { n } \right) + \cos \left( \theta + \frac { 4 \pi } { n } \right) + \ldots + \cos \left( \theta + \frac { ( 2 n - 2 ) \pi } { n } \right) \\ \text { and } S & = \sin \theta + \sin \left( \theta + \frac { 2 \pi } { n } \right) + \sin \left( \theta + \frac { 4 \pi } { n } \right) + \ldots + \sin \left( \theta + \frac { ( 2 n - 2 ) \pi } { n } \right) \end{aligned}$$ where \(n\) is an integer greater than 1 .
    By considering \(C + \mathrm { j } S\), show that \(C = 0\) and \(S = 0\).
  3. Write down the Maclaurin series for \(\mathrm { e } ^ { t }\) as far as the term in \(t ^ { 2 }\). Hence show that, for \(t\) close to zero, $$\frac { t } { \mathrm { e } ^ { t } - 1 } \approx 1 - \frac { 1 } { 2 } t$$
OCR MEI FP2 2010 January Q3
18 marks Standard +0.3
3
  1. Find the inverse of the matrix $$\left( \begin{array} { r r r } 1 & 1 & a \\ 2 & - 1 & 2 \\ 3 & - 2 & 2 \end{array} \right)$$ where \(a \neq 4\).
    Show that when \(a = - 1\) the inverse is $$\frac { 1 } { 5 } \left( \begin{array} { r r r } 2 & 0 & 1 \\ 2 & 5 & - 4 \\ - 1 & 5 & - 3 \end{array} \right)$$
  2. Solve, in terms of \(b\), the following system of equations. $$\begin{aligned} x + y - z & = - 2 \\ 2 x - y + 2 z & = b \\ 3 x - 2 y + 2 z & = 1 \end{aligned}$$
  3. Find the value of \(b\) for which the equations $$\begin{aligned} x + y + 4 z & = - 2 \\ 2 x - y + 2 z & = b \\ 3 x - 2 y + 2 z & = 1 \end{aligned}$$ have solutions. Give a geometrical interpretation of the solutions in this case. Section B (18 marks)
OCR MEI FP2 2010 January Q4
18 marks Standard +0.8
4
  1. Prove, using exponential functions, that $$\cosh 2 x = 1 + 2 \sinh ^ { 2 } x$$ Differentiate this result to obtain a formula for \(\sinh 2 x\).
  2. Solve the equation $$2 \cosh 2 x + 3 \sinh x = 3$$ expressing your answers in exact logarithmic form.
  3. Given that \(\cosh t = \frac { 5 } { 4 }\), show by using exponential functions that \(t = \pm \ln 2\). Find the exact value of the integral $$\int _ { 4 } ^ { 5 } \frac { 1 } { \sqrt { x ^ { 2 } - 16 } } \mathrm {~d} x$$
OCR MEI FP2 2010 January Q5
18 marks Challenging +1.8
5 A line PQ is of length \(k\) (where \(k > 1\) ) and it passes through the point ( 1,0 ). PQ is inclined at angle \(\theta\) to the positive \(x\)-axis. The end Q moves along the \(y\)-axis. See Fig. 5. The end P traces out a locus. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d43d1e11-3173-47c4-88c9-0397c8630a39-4_639_977_552_584} \captionsetup{labelformat=empty} \caption{Fig. 5}
\end{figure}
  1. Show that the locus of P may be expressed parametrically as follows. $$x = k \cos \theta \quad y = k \sin \theta - \tan \theta$$ You are now required to investigate curves with these parametric equations, where \(k\) may take any non-zero value and \(- \frac { 1 } { 2 } \pi < \theta < \frac { 1 } { 2 } \pi\).
  2. Use your calculator to sketch the curve in each of the cases \(k = 2 , k = 1 , k = \frac { 1 } { 2 }\) and \(k = - 1\).
  3. For what value(s) of \(k\) does the curve have
    (A) an asymptote (you should state what the asymptote is),
    (B) a cusp,
    (C) a loop?
  4. For the case \(k = 2\), find the angle at which the curve crosses itself.
  5. For the case \(k = 8\), find in an exact form the coordinates of the highest point on the loop.
  6. Verify that the cartesian equation of the curve is $$y ^ { 2 } = \frac { ( x - 1 ) ^ { 2 } } { x ^ { 2 } } \left( k ^ { 2 } - x ^ { 2 } \right) .$$
OCR MEI FP2 2012 January Q1
18 marks Standard +0.3
1
  1. A curve has polar equation \(r = 1 + \cos \theta\) for \(0 \leqslant \theta < 2 \pi\).
    1. Sketch the curve.
    2. Find the area of the region enclosed by the curve, giving your answer in exact form.
  2. Assuming that \(x ^ { 4 }\) and higher powers may be neglected, write down the Maclaurin series approximations for \(\sin x\) and \(\cos x\) (where \(x\) is in radians). Hence or otherwise obtain an approximation for \(\tan x\) in the form \(a x + b x ^ { 3 }\).
  3. Find \(\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 1 - \frac { 1 } { 4 } X ^ { 2 } } } \mathrm {~d} x\), giving your answer in exact form.