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OCR S4 2016 June Q4
9 marks Standard +0.8
4 The continuous random variable \(Y\) has a uniform (rectangular) distribution on \([ a , b ]\), where \(a\) and \(b\) are constants.
  1. Show that the moment generating function \(\mathrm { M } _ { Y } ( \mathrm { t } )\) of \(Y\) is \(\frac { \left( \mathrm { e } ^ { b t } - \mathrm { e } ^ { a t } \right) } { t ( b - a ) }\).
  2. Use the series expansion of \(\mathrm { e } ^ { x }\) to show that the mean and variance of \(Y\) are \(\frac { 1 } { 2 } ( a + b )\) and \(\frac { 1 } { 12 } ( b - a ) ^ { 2 }\), respectively.
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 FP3 2009 January Q1
5 marks Standard +0.8
1 In this question \(G\) is a group of order \(n\), where \(3 \leqslant n < 8\).
  1. In each case, write down the smallest possible value of \(n\) :
    1. if \(G\) is cyclic,
    2. if \(G\) has a proper subgroup of order 3,
    3. if \(G\) has at least two elements of order 2 .
    4. Another group has the same order as \(G\), but is not isomorphic to \(G\). Write down the possible value(s) of \(n\).
OCR FP3 2009 January Q2
5 marks Standard +0.3
2
  1. Express \(\frac { \sqrt { 3 } + \mathrm { i } } { \sqrt { 3 } - \mathrm { i } }\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(0 \leqslant \theta < 2 \pi\).
  2. Hence find the smallest positive value of \(n\) for which \(\left( \frac { \sqrt { 3 } + \mathrm { i } } { \sqrt { 3 } - \mathrm { i } } \right) ^ { n }\) is real and positive.
OCR FP3 2009 January Q3
6 marks Challenging +1.2
3 Two skew lines have equations $$\frac { x } { 2 } = \frac { y + 3 } { 1 } = \frac { z - 6 } { 3 } \quad \text { and } \quad \frac { x - 5 } { 3 } = \frac { y + 1 } { 1 } = \frac { z - 7 } { 5 } .$$
  1. Find the direction of the common perpendicular to the lines.
  2. Find the shortest distance between the lines.
OCR FP3 2009 January Q4
9 marks Standard +0.8
4 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 5 y = 65 \sin 2 x$$
OCR FP3 2009 January Q5
9 marks Standard +0.8
5 The variables \(x\) and \(y\) are related by the differential equation $$x ^ { 3 } \frac { \mathrm {~d} y } { \mathrm {~d} x } = x y + x + 1 .$$
  1. Use the substitution \(y = u - \frac { 1 } { x }\), where \(u\) is a function of \(x\), to show that the differential equation may be written as $$x ^ { 2 } \frac { \mathrm {~d} u } { \mathrm {~d} x } = u .$$
  2. Hence find the general solution of the differential equation (A), giving your answer in the form \(y = \mathrm { f } ( x )\).
OCR FP3 2009 January Q6
13 marks Standard +0.8
6
[diagram]
The cuboid \(O A B C D E F G\) shown in the diagram has \(\overrightarrow { O A } = 4 \mathbf { i } , \overrightarrow { O C } = 2 \mathbf { j } , \overrightarrow { O D } = 3 \mathbf { k }\), and \(M\) is the mid-point of \(G F\).
  1. Find the equation of the plane \(A C G E\), giving your answer in the form \(\mathbf{r} \cdot \mathbf{n} = p\).
  2. The plane \(O E F C\) has equation \(\mathbf { r } \cdot ( 3 \mathbf { i } - 4 \mathbf { k } ) = 0\). Find the acute angle between the planes \(O E F C\) and \(A C G E\).
  3. The line \(A M\) meets the plane \(O E F C\) at the point \(W\). Find the ratio \(A W : W M\).
OCR FP3 2009 January Q7
13 marks Standard +0.3
7
  1. The operation \(*\) is defined by \(x * y = x + y - a\), where \(x\) and \(y\) are real numbers and \(a\) is a real constant.
    1. Prove that the set of real numbers, together with the operation \(*\), forms a group.
    2. State, with a reason, whether the group is commutative.
    3. Prove that there are no elements of order 2.
    4. The operation \(\circ\) is defined by \(x \circ y = x + y - 5\), where \(x\) and \(y\) are positive real numbers. By giving a numerical example in each case, show that two of the basic group properties are not necessarily satisfied.
OCR FP3 2009 January Q8
12 marks Challenging +1.3
8
  1. By expressing \(\sin \theta\) in terms of \(\mathrm { e } ^ { \mathrm { i } \theta }\) and \(\mathrm { e } ^ { - \mathrm { i } \theta }\), show that $$\sin ^ { 6 } \theta \equiv - \frac { 1 } { 32 } ( \cos 6 \theta - 6 \cos 4 \theta + 15 \cos 2 \theta - 10 )$$
  2. Replace \(\theta\) by ( \(\frac { 1 } { 2 } \pi - \theta\) ) in the identity in part (i) to obtain a similar identity for \(\cos ^ { 6 } \theta\).
  3. Hence find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \left( \sin ^ { 6 } \theta - \cos ^ { 6 } \theta \right) \mathrm { d } \theta\).
OCR FP3 2012 January Q1
7 marks Standard +0.3
1 The variables \(x\) and \(y\) are related by the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 x ^ { 2 } + y ^ { 2 } } { x y } .$$
  1. Use the substitution \(y = u x\), where \(u\) is a function of \(x\), to obtain the differential equation $$x \frac { \mathrm {~d} u } { \mathrm {~d} x } = \frac { 2 } { u } .$$
  2. Hence find the general solution of the differential equation (A), giving your answer in the form \(y ^ { 2 } = \mathrm { f } ( x )\).
OCR FP3 2012 January Q2
7 marks Standard +0.8
2
  1. Show that \(\left( z ^ { n } - \mathrm { e } ^ { \mathrm { i } \theta } \right) \left( z ^ { n } - \mathrm { e } ^ { - \mathrm { i } \theta } \right) \equiv z ^ { 2 n } - ( 2 \cos \theta ) z ^ { n } + 1\).
  2. Express \(z ^ { 4 } - z ^ { 2 } + 1\) as the product of four factors of the form \(\left( z - e ^ { \mathrm { i } \alpha } \right)\), where \(0 \leqslant \alpha < 2 \pi\).
OCR FP3 2012 January Q3
7 marks Challenging +1.2
3 A multiplicative group contains the distinct elements \(e , x\) and \(y\), where \(e\) is the identity.
  1. Prove that \(x ^ { - 1 } y ^ { - 1 } = ( y x ) ^ { - 1 }\).
  2. Given that \(x ^ { n } y ^ { n } = ( x y ) ^ { n }\) for some integer \(n \geqslant 2\), prove that \(x ^ { n - 1 } y ^ { n - 1 } = ( y x ) ^ { n - 1 }\).
  3. If \(x ^ { n - 1 } y ^ { n - 1 } = ( y x ) ^ { n - 1 }\), does it follow that \(x ^ { n } y ^ { n } = ( x y ) ^ { n }\) ? Give a reason for your answer.
OCR FP3 2012 January Q4
10 marks Standard +0.3
4 The line \(l\) has equations \(\frac { x - 1 } { 2 } = \frac { y - 1 } { 3 } = \frac { z + 1 } { 2 }\) and the point \(A\) is ( \(7,3,7\) ). \(M\) is the point where the perpendicular from \(A\) meets \(l\).
  1. Find, in either order, the coordinates of \(M\) and the perpendicular distance from \(A\) to \(l\).
  2. Find the coordinates of the point \(B\) on \(A M\) such that \(\overrightarrow { A B } = 3 \overrightarrow { B M }\).
OCR FP3 2012 January Q5
11 marks Challenging +1.3
5 The variables \(x\) and \(y\) satisfy the differential equation $$2 \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 3 \frac { \mathrm {~d} y } { \mathrm {~d} x } - 2 y = 5 \mathrm { e } ^ { - 2 x }$$
  1. Find the complementary function of the differential equation.
  2. Given that there is a particular integral of the form \(y = p x \mathrm { e } ^ { - 2 x }\), find the constant \(p\).
  3. Find the solution of the equation for which \(y = 0\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 4\) when \(x = 0\).
OCR FP3 2012 January Q6
9 marks Standard +0.3
6 The plane \(\Pi\) has equation \(\mathbf { r } = \left( \begin{array} { l } 1 \\ 6 \\ 7 \end{array} \right) + \lambda \left( \begin{array} { r } 2 \\ - 1 \\ - 1 \end{array} \right) + \mu \left( \begin{array} { r } 2 \\ - 3 \\ - 5 \end{array} \right)\) and the line \(l\) has equation \(\mathbf { r } = \left( \begin{array} { l } 7 \\ 4 \\ 1 \end{array} \right) + t \left( \begin{array} { r } 3 \\ 0 \\ - 1 \end{array} \right)\).
  1. Express the equation of \(\Pi\) in the form r.n \(= p\).
  2. Find the point of intersection of \(l\) and \(\Pi\).
  3. The equation of \(\Pi\) may be expressed in the form \(\mathbf { r } = \left( \begin{array} { l } 1 \\ 6 \\ 7 \end{array} \right) + \lambda \left( \begin{array} { r } 2 \\ - 1 \\ - 1 \end{array} \right) + \mu \mathbf { c }\), where \(\mathbf { c }\) is perpendicular to \(\left( \begin{array} { r } 2 \\ - 1 \\ - 1 \end{array} \right)\). Find \(\mathbf { c }\).