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OCR Further Discrete 2018 December Q5
12 marks Moderate -0.3
5 A rapid transport system connects 8 stations using three railway lines.
The blue line connects A to B to C to D .
FromtoTravel time
AB5
BC3
CD9
The red line connects \(B\) to \(F\) to \(E\) to \(D\).
FromtoTravel time
BF2
FE3
ED2
The green line connects E to G to H to A .
FromtoTravel time
EG5
GH6
HA4
  • The travel times for the return journeys are the same as for the outward journeys (so, for example, the travel time from B to A is 5 minutes, the same as the time from A to B ).
  • All travel times include time spent stopped at stations.
  • No trains are delayed so the travel times are all correct.
  • Give a reason why the quickest journey from A to D may take longer than 12 minutes.
OCR Further Discrete 2018 December Q6
22 marks Standard +0.3
6 Jack is making pizzas for a party. He can make three types of pizza:
Suitable for vegansSuitable for vegetariansSuitable for meat eaters
Type X✓
Type Y✓✓
Type Z✓✓✓
  • There is enough dough to make 30 pizzas.
  • Type Z pizzas use vegan cheese. Jack only has enough vegan cheese to make 2 type Z pizzas.
  • At least half the pizzas made must be suitable for vegetarians.
  • Jack has enough onions to make 50 type X pizzas or 20 type Y pizzas or 20 type Z pizzas or some mixture of the three types.
Suppose that Jack makes \(x\) type X pizzas, \(y\) type Y pizzas and \(z\) type Z pizzas.
  1. Formulate the constraints above in terms of the non-negative, integer valued variables \(x , y\) and \(z\), together with non-negative slack variables \(s , t , u\) and \(v\). Jack wants to find out the maximum total number of pizzas that he can make.
    1. Set up an initial simplex tableau for Jack's problem.
    2. Carry out one iteration of the simplex algorithm, choosing your pivot so that \(x\) becomes a basic variable. When Jack carries out the simplex algorithm his final tableau is:
      \(P\)\(x\)\(y\)\(z\)\(s\)\(t\)\(u\)\(v\)RHS
      100000\(\frac { 3 } { 7 }\)\(\frac { 2 } { 7 }\)\(28 \frac { 4 } { 7 }\)
      000010\(- \frac { 3 } { 7 }\)\(- \frac { 2 } { 7 }\)\(1 \frac { 3 } { 7 }\)
      000101002
      010000\(\frac { 5 } { 7 }\)\(\frac { 1 } { 7 }\)\(14 \frac { 2 } { 7 }\)
      001100\(- \frac { 2 } { 7 }\)\(\frac { 1 } { 7 }\)\(14 \frac { 2 } { 7 }\)
  3. Use this final tableau to deduce how many pizzas of each type Jack should make. Jack knows that some of the guests are vegans. He decides to make 2 pizzas of type \(Z\).
    1. Plot the feasible region for \(x\) and \(y\).
    2. Complete the branch-and-bound formulation in the Printed Answer Booklet to find the number of pizzas of each type that Jack should make.
      You should branch on \(x\) first. \section*{END OF QUESTION PAPER}
OCR Further Additional Pure 2018 December Q1
7 marks Standard +0.8
1 A surface has equation \(z = x \tan y\) for \(- \frac { 1 } { 2 } \pi < y < \frac { 1 } { 2 } \pi\).
  1. Find
OCR Further Additional Pure 2018 December Q2
13 marks Challenging +1.2
2 A sequence \(\left\{ u _ { n } \right\}\) is given by \(u _ { n + 1 } = 4 u _ { n } + 1\) for \(n \geqslant 1\) and \(u _ { 1 } = 3\).
  1. Find the values of \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\).
  2. Solve the recurrence system (*).
    1. Prove by induction that each term of the sequence can be written in the form \(( 10 m + 3 )\) where \(m\) is an integer.
    2. Show that no term of the sequence is a square number.
OCR Further Additional Pure 2018 December Q3
9 marks Challenging +1.8
3
  1. Show that \(10 ^ { 2 } \equiv 6 ( \bmod 47 )\).
  2. Determine the integer \(r\), with \(0 < r < 47\), such that \(6 r \equiv 1 ( \bmod 47 )\).
  3. Determine the least positive integer \(n\) for which \(10 ^ { n } \equiv 1\) or \(- 1 ( \bmod 47 )\).
OCR Further Additional Pure 2018 December Q4
12 marks Challenging +1.2
4 The set \(L\) consists of all points \(( x , y )\) in the cartesian plane, with \(x \neq 0\). The operation ◇ is defined by \(( a , b ) \diamond ( c , d ) = ( a c , b + a d )\) for \(( a , b ) , ( c , d ) \in L\).
    1. Show that \(L\) is closed under ◇.
    2. Prove that \(\diamond\) is associative on \(L\).
    3. Find the identity element of \(L\) under ◇ .
    4. Find the inverse element of \(( a , b )\) under ◇.
  2. Find a subgroup of \(( L , \diamond )\) of order 2.
OCR Further Additional Pure 2018 December Q5
10 marks Standard +0.3
5 Torque is a vector quantity that measures how much a force acting on an object causes that object to rotate. The torque (about the origin), \(\mathbf { T }\), exerted on an object is given by \(\mathbf { T } = \mathbf { p } \times \mathbf { F }\), where \(\mathbf { F }\) is the force acting on the object and \(\mathbf { p }\) is the position vector of the point at which \(\mathbf { F }\) is applied to the object. The points \(A\) and \(B\), with position vectors \(\mathbf { a } = 3 \mathbf { i } + 4 \mathbf { j } + \mathbf { k }\) and \(\mathbf { b } = 3 \mathbf { i } + 5 \mathbf { j } + \mathbf { k }\) are on the surface of a rock. The force \(\mathbf { F } _ { 1 } = 6 \mathbf { i } + 7 \mathbf { j } - 3 \mathbf { k }\) is applied to the rock at \(A\) while the force \(\mathbf { F } _ { 2 } = - 7 \mathbf { i } - 10 \mathbf { j } + 2 \mathbf { k }\) is applied to the rock at \(B\).
  1. Find the torque (about the origin) exerted on the rock by \(\mathbf { F } _ { 1 }\).
  2. Determine which of the two forces \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) exerts a torque (about the origin) of greater magnitude on the rock.
  3. Show that the torque (about the origin) is the same as your answer to part (a) when \(\mathbf { F } _ { 1 }\) acts on the rock at any point on the line \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { p }\), where \(\mathbf { p }\) is a vector in the same direction as \(\mathbf { F } _ { 1 }\). A third force \(\mathbf { F } _ { 3 }\) is now applied to the rock at \(A\), which exerts zero torque (about the origin).
  4. Show that \(\mathbf { F } _ { 3 }\) must act in the direction of the line through \(A\) and the origin.
OCR Further Additional Pure 2018 December Q6
13 marks Challenging +1.8
6 For positive integers \(n\), the integrals \(I _ { n }\) are given by \(I _ { n } = \int _ { 1 } ^ { 5 } x ^ { n } \sqrt { 2 + x ^ { 2 } } \mathrm {~d} x\).
  1. Show that \(I _ { 1 } = 26 \sqrt { 3 }\).
  2. Prove that, for \(n \geqslant 3 , ( n + 2 ) I _ { n } = 3 \sqrt { 3 } \left( 27 \times 5 ^ { n - 1 } - 1 \right) - 2 ( n - 1 ) I _ { n - 2 }\).
  3. Determine the exact value of \(I _ { 5 }\) as a rational multiple of \(\sqrt { 3 }\).
OCR Further Additional Pure 2018 December Q7
11 marks Challenging +1.2
7 For each value of \(t\), the surface \(S _ { t }\) has equation \(z = t x ^ { 2 } + y ^ { 2 } + 3 x y - y\).
  1. Verify that there are no stationary points on \(S _ { t }\) when \(t = \frac { 9 } { 4 }\).
  2. Determine, as \(t\) varies, the nature of any stationary point(s) of \(S _ { t }\).
    (You do not have to find the coordinates of the stationary points.) \section*{OCR} Oxford Cambridge and RSA
OCR H240/01 2018 March Q1
4 marks Easy -1.2
1 A circle with equation \(x ^ { 2 } + y ^ { 2 } + 6 x - 4 y = k\) has a radius of 4 .
  1. Find the coordinates of the centre of the circle.
  2. Find the value of the constant \(k\).
OCR H240/01 2018 March Q2
6 marks Moderate -0.8
2
  1. Given that \(| n | = 5\), find the greatest value of \(| 2 n - 3 |\), justifying your answer.
  2. Solve the equation \(| 3 x - 6 | = | x - 6 |\).
OCR H240/01 2018 March Q3
6 marks Moderate -0.3
3 The equation \(k x ^ { 2 } + ( k - 6 ) x + 2 = 0\) has two distinct real roots. Find the set of possible values of the constant \(k\), giving your answer in set notation.
OCR H240/01 2018 March Q4
9 marks Moderate -0.3
4
  1. Sketch the curves \(y = \frac { 3 } { x ^ { 2 } }\) and \(y = x ^ { 2 } - 2\) on the axes provided in the Printed Answer Booklet.
  2. In this question you must show detailed reasoning. Find the exact coordinates of the points of intersection of the curves \(y = \frac { 3 } { x ^ { 2 } }\) and \(y = x ^ { 2 } - 2\).
OCR H240/01 2018 March Q5
10 marks Moderate -0.8
5 An ice cream seller expects that the number of sales will increase by the same amount every week from May onwards. 150 ice creams are sold in Week 1 and 166 ice creams are sold in Week 2. The ice cream seller makes a profit of \(\pounds 1.25\) for each ice cream sold.
  1. Find the expected profit in Week 10.
  2. In which week will the total expected profits first exceed \(\pounds 5000\) ?
  3. Give two reasons why this model may not be appropriate.
OCR H240/01 2018 March Q6
5 marks Standard +0.5
6 Prove by contradiction that \(\sqrt { 7 }\) is irrational.
OCR H240/01 2018 March Q7
7 marks Standard +0.3
7 Two lifeboat stations, \(P\) and \(Q\), are situated on the coastline with \(Q\) being due south of \(P\). A stationary ship is at sea, at a distance of 4.8 km from \(P\) and a distance of 2.2 km from \(Q\). The ship is on a bearing of \(155 ^ { \circ }\) from \(P\).
  1. Find any possible bearings of the ship from \(Q\).
  2. Find the shortest distance from the ship to the line \(P Q\).
  3. Give a reason why the actual distance from the ship to the coastline may be different to your answer to part (ii).
OCR H240/01 2018 March Q8
9 marks Standard +0.8
8
  1. Given that \(y = \sec x\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sec x \tan x\).
  2. In this question you must show detailed reasoning. Find the exact value of \(\int _ { \frac { 1 } { 12 } \pi } ^ { \frac { 1 } { 6 } \pi } ( \sec 2 x + \tan 2 x ) ^ { 2 } \mathrm {~d} x\).
OCR H240/01 2018 March Q9
11 marks Standard +0.8
9
  1. Express \(\frac { 5 + 4 x - 3 x ^ { 2 } } { ( 1 - 2 x ) ( 2 + x ) ^ { 2 } }\) in three partial fractions.
  2. Hence find the first three terms in the expansion of \(\frac { 5 + 4 x - 3 x ^ { 2 } } { ( 1 - 2 x ) ( 2 + x ) ^ { 2 } }\) in ascending powers of \(x\).
  3. State the set of values for which the expansion in part (ii) is valid.
OCR H240/01 2018 March Q10
10 marks Challenging +1.2
10 In this question you must show detailed reasoning.
Show that the curve with equation \(x ^ { 2 } - 4 x y + 8 y ^ { 3 } - 4 = 0\) has exactly one stationary point.
OCR H240/01 2018 March Q11
12 marks Moderate -0.3
11 The height, in metres, of the sea at a coastal town during a day may be modelled by the function $$\mathrm { f } ( t ) = 1.7 + 0.8 \sin ( 30 t ) ^ { \circ }$$ where \(t\) is the number of hours after midnight.
  1. (a) Find the maximum height of the sea as given by this model.
    (b) Find the time of day at which this maximum height first occurs.
  2. Determine the time when, according to this model, the height of the sea will first be 1.2 m . The height, in metres, at a different coastal town during a day may be modelled by the function $$\mathrm { g } ( t ) = a + b \sin ( c t + d ) ^ { \circ }$$ where \(t\) is the number of hours after midnight.
  3. It is given that at this different coastal town the maximum height of the sea is 3.1 m , and this height occurs at 0500 and 1700. The minimum height of the sea is 0.7 m , and this height occurs at 1100 and 2300 . Find the values of the constants \(a , b , c\) and \(d\).
  4. It is instead given that the maximum height of the sea actually occurs at 0500 and 1709 . State, with a reason, how this will affect the value of \(c\) found in part (iii). \includegraphics[max width=\textwidth, alt={}, center]{74a37bca-0b28-4c48-bd21-a9304f31b8f8-6_563_568_322_751} The diagram shows the curve \(y = \mathrm { e } ^ { \sqrt { x + 1 } }\) for \(x \geqslant 0\).
  5. Use the substitution \(u ^ { 2 } = x + 1\) to find \(\int \mathrm { e } ^ { \sqrt { x + 1 } } \mathrm {~d} x\).
  6. Make \(x\) the subject of the equation \(y = \mathrm { e } ^ { \sqrt { x + 1 } }\).
  7. Hence show that \(\int _ { \mathrm { e } } ^ { \mathrm { e } ^ { 4 } } \left( ( \ln y ) ^ { 2 } - 1 \right) \mathrm { d } y = 9 \mathrm { e } ^ { 4 }\). \section*{END OF QUESTION PAPER} \section*{OCR} Oxford Cambridge and RSA
OCR H240/02 2018 March Q1
6 marks Moderate -0.8
1 Part of the graph of \(y = \mathrm { f } ( x )\) is shown below, where \(\mathrm { f } ( x )\) is a cubic polynomial. \includegraphics[max width=\textwidth, alt={}, center]{6a6316e4-7b2d-4533-988a-4863d79ce668-04_681_679_475_694}
  1. Find \(\mathrm { f } ( - 1 )\).
  2. Write down three linear factors of \(\mathrm { f } ( x )\). It is given that \(\mathrm { f } ( x ) \equiv a x ^ { 3 } + b x ^ { 2 } + c x + d\).
  3. Show that \(a = - 2\).
  4. Find \(b , c\) and \(d\).
OCR H240/02 2018 March Q2
4 marks Moderate -0.8
2 Angela makes the following claim. \begin{displayquote} " \(n\) is an odd positive integer greater than \(1 \Rightarrow 2 ^ { n } - 1\) is prime" \end{displayquote} Prove that Angela's claim is false.
OCR H240/02 2018 March Q3
7 marks Standard +0.3
3 On a particular voyage, a ship sails 500 km at a constant speed of \(v \mathrm {~km} / \mathrm { h }\). The cost for the voyage is \(\pounds R\) per hour. The total cost of the voyage is \(\pounds T\).
  1. Show that \(T = \frac { 500 R } { v }\). The running cost is modelled by the following formula. $$R = 270 + \frac { v ^ { 3 } } { 200 }$$ The ship's owner wishes to sail at a speed that will minimise the total cost for the voyage. It is given that the graph of \(T\) against \(v\) has exactly one stationary point, which is a minimum.
  2. Find the speed that gives the minimum value of \(T\).
  3. Find the minimum value of the total cost.
OCR H240/02 2018 March Q4
5 marks Moderate -0.5
4 The diagram shows part of the graph of \(y = \cos x\), where \(x\) is measured in radians. \includegraphics[max width=\textwidth, alt={}, center]{6a6316e4-7b2d-4533-988a-4863d79ce668-05_609_846_294_607}
  1. Use the copy of this diagram in the Printed Answer Booklet to find an approximate solution to the equation \(x = \cos x\).
  2. Use an iterative method to find the solution to the equation \(x = \cos x\) correct to 3 significant figures. You should show your first, second and last two iterations, writing down all the figures on your calculator.
OCR H240/02 2018 March Q5
8 marks Moderate -0.3
5 Points \(A , B\) and \(C\) have position vectors \(\left( \begin{array} { l } 1 \\ 2 \\ 3 \end{array} \right) , \left( \begin{array} { c } 2 \\ - 1 \\ 5 \end{array} \right)\) and \(\left( \begin{array} { c } - 4 \\ 0 \\ 3 \end{array} \right)\) respectively.
  1. Find the exact distance between the midpoint of \(A B\) and the midpoint of \(B C\). Point \(D\) has position vector \(\left( \begin{array} { c } x \\ - 6 \\ z \end{array} \right)\) and the line \(C D\) is parallel to the line \(A B\).
  2. Find all the possible pairs of \(x\) and \(z\).