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OCR FP2 2016 June Q3
5 marks Challenging +1.2
3 The diagram shows the curve \(y = \mathrm { f } ( x )\). Points \(A , B , C\) and \(D\) on the curve have coordinates ( \(- 1,0 ) , ( 2,0 )\), \(( 5,0 )\) and \(( 0,2 )\) respectively. \includegraphics[max width=\textwidth, alt={}, center]{a31997f4-7890-42c1-9725-1b7058e8741f-2_593_1221_1041_406} On the copy of this diagram in the Printed Answer Book, sketch the curve \(y ^ { 2 } = \mathrm { f } ( x )\), giving the coordinates of the points where the curve crosses the axes.
OCR FP2 2016 June Q4
12 marks Standard +0.8
4 You are given the equation \(( 2 x - 1 ) ^ { 2 } - \mathrm { e } ^ { x } = 0\).
  1. Verify that 0 is a root of the equation. There are also two other roots, \(\alpha\) and \(\beta\), where \(0 < \alpha < \beta\).
  2. The iterative formula \(x _ { r + 1 } = \ln \left( 2 x _ { r } - 1 \right) ^ { 2 }\) is to be used to find a root of the equation.
    1. Sketch the line \(y = x\) and the curve \(y = \ln ( 2 x - 1 ) ^ { 2 }\) on the same axes, showing the roots \(0 , \alpha\) and \(\beta\).
    2. By drawing a 'staircase' diagram on your sketch, starting with a value of \(x\) that is between \(\alpha\) and \(\beta\), show that this iteration does not converge to \(\alpha\).
    3. Using this iterative formula with \(x _ { 1 } = 3.75\), find the value of \(\beta\) correct to 3 decimal places.
    4. Using the Newton-Raphson method with \(x _ { 1 } = 1.6\), find the root \(\alpha\) of the equation \(( 2 x - 1 ) ^ { 2 } - \mathrm { e } ^ { x } = 0\) correct to 5 significant figures. Show the result of each iteration.
OCR FP2 2016 June Q5
9 marks Standard +0.8
5 It is given that \(y = \tan ^ { - 1 } 2 x\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 x \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } = 0\).
  2. Find the Maclaurin series for \(y\) up to and including the term in \(x ^ { 3 }\). Show all your working.
  3. The result in part (ii), together with the value \(x = \frac { 1 } { 2 }\), is used to find an estimate for \(\pi\). Show that this estimate is only correct to 1 significant figure.
OCR FP2 2016 June Q6
10 marks Standard +0.8
6 The equation of a curve in polar coordinates is \(r = \sin 5 \theta\) for \(0 \leqslant \theta \leqslant \frac { 1 } { 5 } \pi\).
  1. Sketch the curve and write down the equations of the tangents at the pole.
  2. The line of symmetry meets the curve at the pole and at one other point \(A\). Find the equation of the line of symmetry and the cartesian coordinates of \(A\).
  3. Find the area of the region enclosed by this curve.
OCR FP2 2016 June Q7
9 marks Challenging +1.2
7
  1. By using a set of rectangles of unit width to approximate an area under the curve \(y = \frac { 1 } { x }\), show that \(\sum _ { x = 1 } ^ { \infty } \frac { 1 } { x }\) is infinite.
  2. By using a set of rectangles of unit width to approximate an area under the curve \(y = \frac { 1 } { x ^ { 2 } }\), find an upper limit for the series \(\sum _ { x = 1 } ^ { \infty } \frac { 1 } { x ^ { 2 } }\).
OCR FP2 2016 June Q8
12 marks Challenging +1.8
8 It is given that \(I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sec ^ { n } x \mathrm {~d} x\) where \(n\) is a positive integer.
  1. By writing \(\sec ^ { n } x = \sec ^ { n - 2 } x \sec ^ { 2 } x\), or otherwise, show that $$( n - 1 ) I _ { n } = ( \sqrt { 2 } ) ^ { n - 2 } + ( n - 2 ) I _ { n - 2 } \text { for } n > 1 .$$
  2. Show that \(I _ { 8 } = \frac { 96 } { 35 }\).
  3. Prove by induction that \(I _ { 2 n }\) is rational for all values of \(n > 1\). \section*{END OF QUESTION PAPER}
OCR FP2 Specimen Q1
6 marks Standard +0.3
1
  1. Starting from the definition of \(\cosh x\) in terms of \(\mathrm { e } ^ { x }\), show that \(\cosh 2 x = 2 \cosh ^ { 2 } x - 1\).
  2. Given that \(\cosh 2 x = k\), where \(k > 1\), express each of \(\cosh x\) and \(\sinh x\) in terms of \(k\).
OCR FP2 Specimen Q2
7 marks Challenging +1.2
2 \includegraphics[max width=\textwidth, alt={}, center]{e4e1c424-8dd5-4d18-9950-e902de0301b0-2_728_951_486_534} The diagram shows the graph of $$y = \frac { 2 x ^ { 2 } + 3 x + 3 } { x + 1 }$$
  1. Find the equations of the asymptotes of the curve.
  2. Prove that the values of \(y\) between which there are no points on the curve are - 5 and 3 .
OCR FP2 Specimen Q3
7 marks Standard +0.3
3
  1. Find the first three terms of the Maclaurin series for \(\ln ( 2 + x )\).
  2. Write down the first three terms of the series for \(\ln ( 2 - x )\), and hence show that, if \(x\) is small, then $$\ln \left( \frac { 2 + x } { 2 - x } \right) \approx x$$
OCR FP2 Specimen Q4
8 marks Standard +0.8
4 The equation of a curve, in polar coordinates, is $$r = 2 \cos 2 \theta \quad ( - \pi < \theta \leqslant \pi ) .$$
  1. Find the values of \(\theta\) which give the directions of the tangents at the pole. One loop of the curve is shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{e4e1c424-8dd5-4d18-9950-e902de0301b0-3_362_720_653_708}
  2. Find the exact value of the area of the region enclosed by the loop.
OCR FP2 Specimen Q5
8 marks Challenging +1.2
5 \includegraphics[max width=\textwidth, alt={}, center]{e4e1c424-8dd5-4d18-9950-e902de0301b0-3_444_999_1258_539} The diagram shows the curve \(y = \frac { 1 } { x + 1 }\) together with four rectangles of unit width.
  1. Explain how the diagram shows that $$\frac { 1 } { 2 } + \frac { 1 } { 3 } + \frac { 1 } { 4 } + \frac { 1 } { 5 } < \int _ { 0 } ^ { 4 } \frac { 1 } { x + 1 } \mathrm {~d} x$$ The curve \(y = \frac { 1 } { x + 2 }\) passes through the top left-hand corner of each of the four rectangles shown.
  2. By considering the rectangles in relation to this curve, write down a second inequality involving \(\frac { 1 } { 2 } + \frac { 1 } { 3 } + \frac { 1 } { 4 } + \frac { 1 } { 5 }\) and a definite integral.
  3. By considering a suitable range of integration and corresponding rectangles, show that $$\ln ( 500.5 ) < \sum _ { r = 2 } ^ { 1000 } \frac { 1 } { r } < \ln ( 1000 ) .$$
OCR FP2 Specimen Q6
10 marks Challenging +1.2
6
  1. Given that \(I _ { n } = \int _ { 0 } ^ { 1 } x ^ { n } \sqrt { } ( 1 - x ) \mathrm { d } x\), prove that, for \(n \geqslant 1\), $$( 2 n + 3 ) I _ { n } = 2 n I _ { n - 1 } .$$
  2. Hence find the exact value of \(I _ { 2 }\).
OCR FP2 Specimen Q7
13 marks Standard +0.8
7 The curve with equation $$y = \frac { x } { \cosh x }$$ has one stationary point for \(x > 0\).
  1. Show that the \(x\)-coordinate of this stationary point satisfies the equation \(x \tanh x - 1 = 0\). The positive root of the equation \(x \tanh x - 1 = 0\) is denoted by \(\alpha\).
  2. Draw a sketch showing (for positive values of \(x\) ) the graph of \(y = \tanh x\) and its asymptote, and the graph of \(y = \frac { 1 } { x }\). Explain how you can deduce from your sketch that \(\alpha > 1\).
  3. Use the Newton-Raphson method, taking first approximation \(x _ { 1 } = 1\), to find further approximations \(x _ { 2 }\) and \(x _ { 3 }\) for \(\alpha\).
  4. By considering the approximate errors in \(x _ { 1 }\) and \(x _ { 2 }\), estimate the error in \(x _ { 3 }\).
OCR FP2 Specimen Q8
13 marks Challenging +1.8
8
  1. Use the substitution \(t = \tan \frac { 1 } { 2 } x\) to show that $$\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sqrt { \frac { 1 - \cos x } { 1 + \sin x } } \mathrm {~d} x = 2 \sqrt { } 2 \int _ { 0 } ^ { 1 } \frac { t } { ( 1 + t ) \left( 1 + t ^ { 2 } \right) } \mathrm { d } t$$
  2. Express \(\frac { t } { ( 1 + t ) \left( 1 + t ^ { 2 } \right) }\) in partial fractions.
  3. Hence find \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sqrt { \frac { 1 - \cos x } { 1 + \sin x } } \mathrm {~d} x\), expressing your answer in an exact form.
OCR S2 2013 January Q1
4 marks Standard +0.8
1 A random variable has the distribution \(\mathrm { B } ( n , p )\). It is required to test \(\mathrm { H } _ { 0 } : p = \frac { 2 } { 3 }\) against \(\mathrm { H } _ { 1 } : p < \frac { 2 } { 3 }\) at a significance level as close to \(1 \%\) as possible, using a sample of size \(n = 8,9\) or 10 . Use tables to find which value of \(n\) gives such a test, stating the critical region for the test and the corresponding significance level.
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OCR S2 2013 January Q2
6 marks Moderate -0.3
2 A random variable \(C\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). A random sample of 10 observations of \(C\) is obtained, and the results are summarised as $$n = 10 , \Sigma c = 380 , \Sigma c ^ { 2 } = 14602 .$$
  1. Calculate unbiased estimates of \(\mu\) and \(\sigma ^ { 2 }\).
  2. Hence calculate an estimate of the probability that \(C > 40\).
OCR S2 2013 January Q3
8 marks Moderate -0.8
3 A factory produces 9000 music DVDs each day. A random sample of 100 such DVDs is obtained.
  1. Explain how to obtain this sample using random numbers.
  2. Given that \(24 \%\) of the DVDs produced by the factory are classical, use a suitable approximation to find the probability that, in the sample of 100 DVDs, fewer than 20 are classical.
OCR S2 2013 January Q4
9 marks Moderate -0.5
4 A continuous random variable \(X\) has probability density function $$\mathrm { f } ( x ) = \left\{ \begin{array} { c l } k x & 0 \leqslant x \leqslant a \\ 0 & \text { otherwise } \end{array} \right.$$ where \(k\) and \(a\) are constants.
  1. State what the letter \(x\) represents.
  2. Find \(k\) in terms of \(a\).
  3. Find \(\operatorname { Var } ( X )\) in terms of \(a\).
OCR S2 2013 January Q5
8 marks Standard +0.3
5 In a mine, a deposit of the substance pitchblende emits radioactive particles. The number of particles emitted has a Poisson distribution with mean 70 particles per second. The warning level is reached if the total number of particles emitted in one minute is more than 4350.
  1. A one-minute period is chosen at random. Use a suitable approximation to show that the probability that the warning level is reached during this period is 0.01 , correct to 2 decimal places. You should calculate the answer correct to 4 decimal places.
  2. Use a suitable approximation to find the probability that in 30 one-minute periods the warning level is reached on at least 4 occasions. (You should use the given rounded value of 0.01 from part (i) in your calculation.)
OCR S2 2013 January Q6
10 marks Standard +0.3
6 Gordon is a cricketer. Over a long period he knows that his population mean score, in number of runs per innings, is 28 , and the population standard deviation is 12 . In a new season he adopts a different batting style and he finds that in 30 innings using this style his mean score is 28.98 .
  1. Stating a necessary assumption, test at the \(5 \%\) significance level whether his population mean score has increased.
  2. Explain whether it was necessary to use the Central Limit Theorem in part (i).
OCR S2 2013 January Q7
9 marks Standard +0.8
7 The continuous random variable \(X\) has the distribution \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). The mean of a random sample of \(n\) observations of \(X\) is denoted by \(\bar { X }\). It is given that \(\mathrm { P } ( \bar { X } < 35.0 ) = 0.9772\) and \(\mathrm { P } ( \bar { X } < 20.0 ) = 0.1587\).
  1. Obtain a formula for \(\sigma\) in terms of \(n\). Two students are discussing this question. Aidan says "If you were told another probability, for instance \(\mathrm { P } ( \bar { X } > 32 ) = 0.1\), you could work out the value of \(\sigma\)." Binya says, "No, the value of \(\mathrm { P } ( \bar { X } > 32 )\) is fixed by the information you know already."
  2. State which of Aidan and Binya is right. If you think that Aidan is right, calculate the value of \(\sigma\) given that \(\mathrm { P } ( \bar { X } > 32 ) = 0.1\). If you think that Binya is right, calculate the value of \(\mathrm { P } ( \bar { X } > 32 )\).
OCR S2 2013 January Q8
10 marks Standard +0.3
8 In a large city the number of traffic lights that fail in one day of 24 hours is denoted by \(Y\). It may be assumed that failures occur randomly.
  1. Explain what the statement "failures occur randomly" means.
  2. State, in context, two different conditions that must be satisfied if \(Y\) is to be modelled by a Poisson distribution, and for each condition explain whether you think it is likely to be met in this context.
  3. For this part you may assume that \(Y\) is well modelled by the distribution \(\operatorname { Po } ( \lambda )\). It is given that \(\mathrm { P } ( Y = 7 ) = \mathrm { P } ( Y = 8 )\). Use an algebraic method to calculate the value of \(\lambda\) and hence calculate the corresponding value of \(\mathrm { P } ( Y = 7 )\).
OCR S2 2013 January Q9
8 marks Standard +0.8
9 The random variable \(A\) has the distribution \(\mathrm { B } ( 30 , p )\). A test is carried out of the hypotheses \(\mathrm { H } _ { 0 } : p = 0.6\) against \(\mathrm { H } _ { 1 } : p < 0.6\). The critical region is \(A \leqslant 13\).
  1. State the probability that \(\mathrm { H } _ { 0 }\) is rejected when \(p = 0.6\).
  2. Find the probability that a Type II error occurs when \(p = 0.5\).
  3. It is known that on average \(p = 0.5\) on one day in five, and on other days the value of \(p\) is 0.6 . On each day two tests are carried out. If the result of the first test is that \(\mathrm { H } _ { 0 }\) is rejected, the value of \(p\) is adjusted if necessary, to ensure that \(p = 0.6\) for the rest of the day. Otherwise the value of \(p\) remains the same as for the first test. Calculate the probability that the result of the second test is to reject \(\mathrm { H } _ { 0 }\). \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE.}
OCR S2 2015 June Q1
6 marks Standard +0.3
1 The random variable \(Y\) is normally distributed with mean \(\mu\) and variance \(\sigma ^ { 2 }\). It is found that \(\mathrm { P } ( Y > 150.0 ) = 0.0228\) and \(\mathrm { P } ( Y > 143.0 ) = 0.9332\). Find the values of \(\mu\) and \(\sigma\).
OCR S2 2015 June Q2
6 marks Moderate -0.8
2 A class investigated the number of dead rabbits found along a particular stretch of road.
  1. The class agrees that dead rabbits occur randomly along the road. Explain what this statement means.
  2. State, in this context, an assumption needed for the number of dead rabbits in a fixed length of road to be modelled by a Poisson distribution, and explain what your statement means. Assume now that the number of dead rabbits in a fixed length of road can be well modelled by a Poisson distribution with mean 1 per 600 m of road.
  3. Use an appropriate formula, showing your working, to find the probability that in a road of length 1650 m there are exactly 3 dead rabbits.