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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 \(n\) :
    (a) if \(G\) is cyclic,
    (b) if \(G\) has a proper subgroup of order 3,
    (c) if \(G\) has at least two elements of order 2 .
  2. 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
\includegraphics[max width=\textwidth, alt={}, center]{bc975428-c594-427b-a32e-268412b3cd26-3_554_825_264_660} 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 r.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.
    (a) Prove that the set of real numbers, together with the operation \(*\), forms a group.
    (b) State, with a reason, whether the group is commutative.
    (c) Prove that there are no elements of order 2.
  2. 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 2010 January Q1
5 marks Standard +0.3
1 Determine whether the lines $$\frac { x - 1 } { 1 } = \frac { y + 2 } { - 1 } = \frac { z + 4 } { 2 } \quad \text { and } \quad \frac { x + 3 } { 2 } = \frac { y - 1 } { 3 } = \frac { z - 5 } { 4 }$$ intersect or are skew.
\(2 \quad H\) denotes the set of numbers of the form \(a + b \sqrt { 5 }\), where \(a\) and \(b\) are rational. The numbers are combined under multiplication.
  1. Show that the product of any two members of \(H\) is a member of \(H\). It is now given that, for \(a\) and \(b\) not both zero, \(H\) forms a group under multiplication.
  2. State the identity element of the group.
  3. Find the inverse of \(a + b \sqrt { 5 }\).
  4. With reference to your answer to part (iii), state a property of the number 5 which ensures that every number in the group has an inverse.
OCR FP3 2010 January Q3
6 marks Moderate -0.3
3 Use the integrating factor method to find the solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + 2 y = \mathrm { e } ^ { - 3 x }$$ for which \(y = 1\) when \(x = 0\). Express your answer in the form \(y = \mathrm { f } ( x )\).
OCR FP3 2010 January Q4
7 marks Standard +0.8
4
  1. Write down, in cartesian form, the roots of the equation \(z ^ { 4 } = 16\).
  2. Hence solve the equation \(w ^ { 4 } = 16 ( 1 - w ) ^ { 4 }\), giving your answers in cartesian form.
OCR FP3 2010 January Q5
11 marks Challenging +1.2
5 A regular tetrahedron has vertices at the points $$A \left( 0,0 , \frac { 2 } { 3 } \sqrt { 6 } \right) , \quad B \left( \frac { 2 } { 3 } \sqrt { 3 } , 0,0 \right) , \quad C \left( - \frac { 1 } { 3 } \sqrt { 3 } , 1,0 \right) , \quad D \left( - \frac { 1 } { 3 } \sqrt { 3 } , - 1,0 \right) .$$
  1. Obtain the equation of the face \(A B C\) in the form $$x + \sqrt { 3 } y + \left( \frac { 1 } { 2 } \sqrt { 2 } \right) z = \frac { 2 } { 3 } \sqrt { 3 }$$ (Answers which only verify the given equation will not receive full credit.)
  2. Give a geometrical reason why the equation of the face \(A B D\) can be expressed as $$x - \sqrt { 3 } y + \left( \frac { 1 } { 2 } \sqrt { 2 } \right) z = \frac { 2 } { 3 } \sqrt { 3 }$$
  3. Hence find the cosine of the angle between two faces of the tetrahedron.
OCR FP3 2010 January Q6
12 marks Challenging +1.3
6 The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 16 y = 8 \cos 4 x$$
  1. Find the complementary function of the differential equation.
  2. Given that there is a particular integral of the form \(y = p x \sin 4 x\), where \(p\) is a constant, find the general solution of the equation.
  3. Find the solution of the equation for which \(y = 2\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) when \(x = 0\).
OCR FP3 2010 January Q7
13 marks
7
  1. Solve the equation \(\cos 6 \theta = 0\), for \(0 < \theta < \pi\).
  2. By using de Moivre's theorem, show that $$\cos 6 \theta \equiv \left( 2 \cos ^ { 2 } \theta - 1 \right) \left( 16 \cos ^ { 4 } \theta - 16 \cos ^ { 2 } \theta + 1 \right)$$
  3. Hence find the exact value of $$\cos \left( \frac { 1 } { 12 } \pi \right) \cos \left( \frac { 5 } { 12 } \pi \right) \cos \left( \frac { 7 } { 12 } \pi \right) \cos \left( \frac { 11 } { 12 } \pi \right)$$ justifying your answer.
OCR FP3 2010 January Q8
12 marks Challenging +1.3
8 The function f is defined by \(\mathrm { f } : x \mapsto \frac { 1 } { 2 - 2 x }\) for \(x \in \mathbb { R } , x \neq 0 , x \neq \frac { 1 } { 2 } , x \neq 1\). The function g is defined by \(\mathrm { g } ( x ) = \mathrm { ff } ( x )\).
  1. Show that \(\mathrm { g } ( x ) = \frac { 1 - x } { 1 - 2 x }\) and that \(\operatorname { gg } ( x ) = x\). It is given that f and g are elements of a group \(K\) under the operation of composition of functions. The element e is the identity, where e : \(x \mapsto x\) for \(x \in \mathbb { R } , x \neq 0 , x \neq \frac { 1 } { 2 } , x \neq 1\).
  2. State the orders of the elements f and g .
  3. The inverse of the element f is denoted by h . Find \(\mathrm { h } ( x )\).
  4. Construct the operation table for the elements e, f, g, h of the group \(K\).
OCR FP3 2011 January Q1
6 marks Standard +0.3
1
  1. Find the general solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + x y = x \mathrm { e } ^ { \frac { 1 } { 2 } x ^ { 2 } }$$ giving your answer in the form \(y = \mathrm { f } ( x )\).
  2. Find the particular solution for which \(y = 1\) when \(x = 0\).
OCR FP3 2011 January Q2
6 marks Standard +0.8
2 Two intersecting lines, lying in a plane \(p\), have equations $$\frac { x - 1 } { 2 } = \frac { y - 3 } { 1 } = \frac { z - 4 } { - 3 } \quad \text { and } \quad \frac { x - 1 } { - 1 } = \frac { y - 3 } { 2 } = \frac { z - 4 } { 4 } .$$
  1. Obtain the equation of \(p\) in the form \(2 x - y + z = 3\).
  2. Plane \(q\) has equation \(2 x - y + z = 21\). Find the distance between \(p\) and \(q\).
OCR FP3 2011 January Q3
8 marks Standard +0.8
3
  1. Express \(\sin \theta\) in terms of \(\mathrm { e } ^ { \mathrm { i } \theta }\) and \(\mathrm { e } ^ { - \mathrm { i } \theta }\) and show that $$\sin ^ { 4 } \theta \equiv \frac { 1 } { 8 } ( \cos 4 \theta - 4 \cos 2 \theta + 3 )$$
  2. Hence find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } \sin ^ { 4 } \theta \mathrm {~d} \theta\).
OCR FP3 2011 January Q4
8 marks Standard +0.8
4 The cube roots of 1 are denoted by \(1 , \omega\) and \(\omega ^ { 2 }\), where the imaginary part of \(\omega\) is positive.
  1. Show that \(1 + \omega + \omega ^ { 2 } = 0\).
    \includegraphics[max width=\textwidth, alt={}, center]{d12573dd-c0c2-4f0d-8e49-8fdf8d5864a5-2_616_748_1676_699} In the diagram, \(A B C\) is an equilateral triangle, labelled anticlockwise. The points \(A , B\) and \(C\) represent the complex numbers \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\) respectively.
  2. State the geometrical effect of multiplication by \(\omega\) and hence explain why \(z _ { 1 } - z _ { 3 } = \omega \left( z _ { 3 } - z _ { 2 } \right)\).
  3. Hence show that \(z _ { 1 } + \omega z _ { 2 } + \omega ^ { 2 } z _ { 3 } = 0\).
OCR FP3 2011 January Q7
10 marks Challenging +1.2
7 Three planes \(\Pi _ { 1 } , \Pi _ { 2 }\) and \(\Pi _ { 3 }\) have equations $$\mathbf { r } . ( \mathbf { i } + \mathbf { j } - 2 \mathbf { k } ) = 5 , \quad \mathbf { r } . ( \mathbf { i } - \mathbf { j } + 3 \mathbf { k } ) = 6 , \quad \mathbf { r } . ( \mathbf { i } + 5 \mathbf { j } - 12 \mathbf { k } ) = 12 ,$$ respectively. Planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) intersect in a line \(l\); planes \(\Pi _ { 2 }\) and \(\Pi _ { 3 }\) intersect in a line \(m\).
  1. Show that \(l\) and \(m\) are in the same direction.
  2. Write down what you can deduce about the line of intersection of planes \(\Pi _ { 1 }\) and \(\Pi _ { 3 }\).
  3. By considering the cartesian equations of \(\Pi _ { 1 } , \Pi _ { 2 }\) and \(\Pi _ { 3 }\), or otherwise, determine whether or not the three planes have a common line of intersection.