Questions — OCR (4907 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 PURE S1 S2 S3 S4 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
Sort by: Default | Easiest first | Hardest first
OCR Further Discrete AS 2020 November Q4
10 marks Moderate -0.8
4 Bob is extending his attic with the help of some friends, including his architect friend Archie. The activities involved, their durations (in days) and Bob's notes are given below.
ActivityDuration (days)Notes
AArchie takes measurements1
BArchie draws up plans3Must come after A
CPlans are approved21Must come after B
DBob orders materials2Must come after B
EMaterials delivered10Must come after D
FWork area cleared5Must come after A
GPlumbing and electrics3Must come after C, E and F
HFloors, walls and ceilings24Must come after G
IStaircase2Must come after H
JWindows1Must come after H
KDecorating6Must come after I and J
Archie has started to construct an activity network to represent the project. \includegraphics[max width=\textwidth, alt={}, center]{c2deec7d-0617-4eb0-a47e-5b42ba55b753-5_401_1253_1475_406}
  1. Complete the activity network in the Printed Answer Booklet and use it to determine
OCR Further Discrete AS 2020 November Q5
9 marks Standard +0.8
5 The number of points won by player 1 in a zero-sum game is shown in the pay-off matrix below, where \(k\) is a constant. \begin{table}[h]
\captionsetup{labelformat=empty} \caption{Player 2}
Strategy EStrategy FStrategy GStrategy H
Strategy A\(2 k\)-2\(1 - k\)4
Strategy B-334-5
Strategy C14-42
Strategy D4-2-56
\end{table}
  1. In one game, player 2 chooses strategy H. Write down the greatest number of points that player 2 could win. You are given that strategy A is a play-safe strategy for player 1.
  2. Determine the range of possible values for \(k\).
  3. Determine the column minimax value.
OCR Further Discrete AS 2020 November Q6
15 marks Moderate -0.3
6 Tamsin is planning how to spend a day off. She will divide her time between walking the coast path, visiting a bird sanctuary and visiting the garden centre. Tamsin has given a value to each hour spent doing each activity. She wants to decide how much time to spend on each activity to maximise the total value of the activities.
ActivityWalking coast pathVisiting bird sanctuaryVisiting garden centre
Value5 points per hour3 points per hour2 points per hour
Tamsin's requirements are that she will spend:
  • a total of exactly 6 hours on the three activities
  • at most 3.5 hours walking the coast path
  • at least as long at the bird sanctuary as at the garden centre
  • at least 1 hour at the garden centre.
      1. Explain why the maximum total value of the activities done is achieved when \(3 x + y\) is maximised.
      2. Show how the requirement that she spends at least as long at the bird sanctuary as at the garden centre leads to the constraint \(x + 2 y \geqslant 6\).
      3. Explain why there is no need to require that \(y \geqslant 0\).
    2. Represent the constraints graphically and hence find a solution to Tamsin's problem.
OCR Further Discrete AS Specimen Q1
2 marks Easy -2.5
1 Hussain wants to travel by train from Edinburgh to Southampton, leaving Edinburgh after 9 am and arriving in Southampton by 4 pm . He wants to leave Edinburgh as late as possible.
Hussain rings the train company to find out about the train times. Write down a question he might ask that leads to
(A) an existence problem,
(B) an optimisation problem.
OCR Further Discrete AS Specimen Q2
7 marks Standard +0.3
2 Some of the activities that may be involved in making a cup of tea are listed below. A: Boil water.
B: Put teabag in teapot, pour on boiled water and let tea brew.
C: Get cup from cupboard.
D: Pour tea into cup.
E: Add milk to cup.
F: Add sugar to cup. Activity A must happen before activity B.
Activities B and C must happen before activity D .
Activities E and F cannot happen until after activity C.
Other than that, the activities can happen in any order.
  1. Lisa does not take milk or sugar in her tea, so she only needs to use activities \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D . In how many different orders can activities \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D be arranged, subject to the restrictions above?
  2. Mick takes milk but no sugar, so he needs to use activities \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D }\) and E . Explain carefully why there are exactly nine different orders for these activities, subject to the restrictions above.
  3. Find the number of different orders for all six activities, subject to the restrictions above. Explain your reasoning carefully.
OCR Further Discrete AS Specimen Q3
6 marks Challenging +1.2
3 A zero-sum game is being played between two players, \(X\) and \(Y\). The pay-off matrix for \(X\) is given below. \section*{Player X}
Player \(\boldsymbol { Y }\)
Strategy \(\boldsymbol { R }\)Strategy \(\boldsymbol { S }\)
Strategy \(\boldsymbol { P }\)4- 2
Strategy \(\boldsymbol { Q }\)- 31
  1. Find an optimal mixed strategy for player \(X\).
  2. Give one assumption that must be made about the behaviour of \(Y\) in order to make the mixed strategy of Player \(X\) valid.
OCR Further Discrete AS Specimen Q4
6 marks Moderate -0.3
4 Two graphs are shown below. Each has exactly five vertices with vertex orders 2, 3, 3, 4, 4 . \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a6800c9f-583b-493a-906c-015df63b842f-3_605_616_360_278} \captionsetup{labelformat=empty} \caption{Graph 1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a6800c9f-583b-493a-906c-015df63b842f-3_420_501_497_1169} \captionsetup{labelformat=empty} \caption{Graph 2}
\end{figure}
  1. Write down a semi-Eulerian route for graph 1 .
  2. Explain how the vertex orders show that graph 2 is also semi-Eulerian.
  3. By referring to specific vertices, explain how you know that these graphs are not simple.
  4. By referring to specific vertices, explain how you know that these graphs are not isomorphic.
OCR Further Discrete AS Specimen Q5
8 marks Standard +0.8
5 There are three non-isomorphic trees on five vertices.
  1. Draw an example of each of these trees.
  2. State three properties that must be satisfied by the vertex orders of a tree on six vertices.
  3. List the five different sets of possible vertex orders for trees on six vertices.
  4. Draw an example of each type listed in part (iii).
OCR Further Discrete AS Specimen Q6
8 marks Standard +0.8
6 The following masses, in kg, are to be packed into bins. $$\begin{array} { l l l l l l l l l l } 8 & 5 & 9 & 7 & 7 & 9 & 1 & 3 & 3 & 8 \end{array}$$
  1. Chloe says that first-fit decreasing gives a packing that requires 4 bins, but first-fit only requires 3 bins. Find the maximum capacity of the bins. First-fit requires one pass through the list and the time taken may be regarded as being proportional to the length of the list. Suppose that shuttle sort was used to sort the list into decreasing order.
  2. What can be deduced, in this case, about the order of the time complexity, \(\mathrm { T } ( n )\), for first-fit decreasing?
OCR Further Discrete AS Specimen Q7
11 marks Moderate -0.8
7 A complete graph on five vertices is weighted to form a network, as given in the weighted matrix below.
ABCDE
\cline { 2 - 6 } A-9542
\cline { 2 - 6 } B9-757
\cline { 2 - 6 } C57-68
\cline { 2 - 6 } D456-5
\cline { 2 - 6 } E2785-
\cline { 2 - 6 }
\cline { 2 - 6 }
  1. Apply Prim's algorithm to the copy of this weighted matrix in the Printed Answer Booklet to construct a minimum spanning tree for the five vertices.
    Draw your minimum spanning tree, stating the order in which you built the tree and giving its total weight.
  2. (a) Using only the arcs in the minimum spanning tree, which vertex should be chosen to find the smallest total of the weights of the paths from that vertex to each of the other vertices?
    (b) State the minimum total for this vertex.
  3. Show that the total number of comparisons needed to find a minimum spanning tree for a \(5 \times 5\) matrix is 16 .
  4. If a computer takes 4 seconds to find a minimum spanning tree for a network with 100 vertices, how long would it take to find a minimum spanning tree for a network with 500 vertices?
OCR Further Discrete AS Specimen Q8
12 marks Standard +0.8
8 A sweet shop sells three different types of boxes of chocolate truffles. The cost of each type of box and the number of truffles of each variety in each type of box are given in the table below.
TypeCost (£)Milk chocolatePlain chocolateWhite chocolateNutty chocolate
Assorted2.005555
No Nuts1.005870
Speciality2.505492
Narendra wants to buy some boxes of truffles so that in total he has at least 20 milk chocolate, 10 plain chocolate, 16 white chocolate and 12 nutty chocolate truffles.
  1. Explain why Narendra needs to buy at least four boxes of truffles.
  2. Narendra decides that he will buy exactly four boxes. Determine the minimum number of Assorted boxes that Narendra must buy.
  3. For your answer in part (ii),
    Narendra finds that the sweet shop has sold out of Assorted boxes, but he then spots that it also sells small boxes of milk chocolate truffles and small boxes of nutty chocolate truffles. Each small box contains 4 truffles (all of one variety) and costs \(\pounds 0.50\). He decides to buy \(x\) boxes of No Nuts and \(y\) boxes of Speciality, where \(x + y < 4\), so that he has at least 10 plain chocolate and 16 white chocolate truffles. He will then buy as many small boxes as he needs to give a total of at least 20 milk chocolate and 12 nutty chocolate truffles.
  4. (a) Set up constraints on the values of \(x\) and \(y\).
    (b) Represent the feasible region graphically.
    (c) Hence determine the cheapest cost for Narendra. www.ocr.org.uk after the live examination series. If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity. For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
    OCR is part of the
OCR Further Additional Pure AS 2018 June Q1
8 marks Standard +0.3
1 The points \(A , B\) and \(C\) have position vectors \(6 \mathbf { i } + 2 \mathbf { j } + 4 \mathbf { k } , 13 \mathbf { i } + 2 \mathbf { j } + 5 \mathbf { k }\) and \(16 \mathbf { i } + 6 \mathbf { j } + 3 \mathbf { k }\) respectively.
  1. Using the vector product, calculate the area of triangle \(A B C\).
  2. Hence find, in simplest surd form, the perpendicular distance from \(C\) to the line through \(A\) and \(B\).
OCR Further Additional Pure AS 2018 June Q2
9 marks Standard +0.8
2 The surface with equation \(z = 6 x ^ { 3 } + \frac { 1 } { 9 } y ^ { 2 } + x ^ { 2 } y\) has two stationary points.
  1. Verify that one of these stationary points is at the origin.
  2. Find the coordinates of the second stationary point.
OCR Further Additional Pure AS 2018 June Q3
3 marks Challenging +1.2
3 Given that \(n\) is a positive integer, show that the numbers ( \(4 n + 1\) ) and ( \(6 n + 1\) ) are co-prime.
OCR Further Additional Pure AS 2018 June Q4
11 marks Challenging +1.2
4 The group \(G\) consists of a set of six matrices under matrix multiplication. Two of the elements of \(G\) are \(\mathbf { A } = \left( \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { l l } 1 & - 1 \\ 0 & - 1 \end{array} \right)\).
  1. Determine each of the following:
OCR Further Additional Pure AS 2018 June Q5
8 marks Challenging +1.2
5 For integers \(a\) and \(b\), with \(a \geqslant 0\) and \(0 \leqslant b \leqslant 99\), the numbers \(M\) and \(N\) are such that $$M = 100 a + b \text { and } N = a - 9 b .$$
  1. By considering the number \(M + 2 N\), show that \(17 \mid M\) if and only if \(17 \mid N\).
  2. Demonstrate step-by-step how an algorithm based on the result of part (i) can be used to show that 2058376813901 is a multiple of 17 .
OCR Further Additional Pure AS 2018 June Q6
8 marks Challenging +1.8
6 The Fibonacci sequence \(\left\{ F _ { n } \right\}\) is defined by \(F _ { 0 } = 0 , F _ { 1 } = 1\) and \(F _ { n } = F _ { n - 1 } + F _ { n - 2 }\) for all \(n \geqslant 2\).
  1. Show that \(F _ { n + 5 } = 5 F _ { n + 1 } + 3 F _ { n }\)
  2. Prove that \(F _ { n }\) is a multiple of 5 when \(n\) is a multiple of 5 .
OCR Further Additional Pure AS 2018 June Q7
13 marks Challenging +1.8
7 The 'parabolic' TV satellite dish in the diagram can be modelled by the surface generated by the rotation of part of a parabola around a vertical \(z\)-axis. The model is represented by part of the surface with equation \(z = \mathrm { f } ( x , y )\) and \(O\) is on the surface. The point \(P\) is on the rim of the dish and directly above the \(x\)-axis.
The object, \(B\), modelled as a point on the \(z\)-axis is the receiving box which collects the TV signals reflected by the dish. \includegraphics[max width=\textwidth, alt={}, center]{f2166e0a-cd4c-40af-b4b4-04ef4919d996-3_753_995_584_525}
  1. The horizontal plane \(\Pi _ { 1 }\), containing the point \(P\), intersects the surface of the model in a contour of the surface.
    1. Sketch this contour in the Printed Answer Booklet.
    2. State a suitable equation for this contour.
    3. A second plane, \(\Pi _ { 2 }\), containing both \(P\) and the \(z\)-axis, intersects the surface of the model in a section of the surface.
      (a) Sketch this section in the Printed Answer Booklet.
      (b) State a suitable equation for this section.
    4. A proposed equation for the surface is \(z = a x ^ { 2 } + b y ^ { 2 }\). What can you say about the constants \(a\) and \(b\) within this equation? Justify your answers.
    5. The real TV satellite dish has the following measurements (in metres): the height of \(P\) above \(O\) is 0.065 and the perimeter of the rim is 2.652 . Using this information, calculate correct to three decimal places the values of
OCR Further Additional Pure AS 2019 June Q1
3 marks Moderate -0.5
1 In decimal (base 10) form, the number \(N\) is 15260.
  1. Express \(N\) in binary (base 2) form.
  2. Using the binary form of \(N\), show that \(N\) is divisible by 7 .
OCR Further Additional Pure AS 2019 June Q2
4 marks Challenging +1.2
2
  1. The convergent sequence \(\left\{ \mathrm { a } _ { \mathrm { n } } \right\}\) is defined by \(a _ { 0 } = 1\) and \(\mathrm { a } _ { \mathrm { n } + 1 } = \sqrt { \mathrm { a } _ { \mathrm { n } } } + \frac { 4 } { \sqrt { \mathrm { a } _ { \mathrm { n } } } }\) for \(n \geqslant 0\). Calculate the limit of the sequence.
  2. The convergent sequence \(\left\{ \mathrm { b } _ { \mathrm { n } } \right\}\) is defined by \(\mathrm { b } _ { 0 } = 1\) and \(\mathrm { b } _ { \mathrm { n } + 1 } = \sqrt { \mathrm { b } _ { \mathrm { n } } } + \frac { \mathrm { k } } { \sqrt { \mathrm { b } _ { \mathrm { n } } } }\) for \(n \geqslant 0\), where \(k\) is a constant. Determine the value of \(k\) for which the limit of the sequence is 9 .
OCR Further Additional Pure AS 2019 June Q3
4 marks Standard +0.3
3 The non-zero vectors \(\mathbf { x }\) and \(\mathbf { y }\) are such that \(\mathbf { x } \times \mathbf { y } = \mathbf { 0 }\).
  1. Explain the geometrical significance of this statement.
  2. Use your answer to part (a) to explain how the line equation \(\mathbf { r } = \mathbf { a } + t \mathbf { d }\) can be written in the form \(( \mathbf { r } - \mathbf { a } ) \times \mathbf { d } = \mathbf { 0 }\).
OCR Further Additional Pure AS 2019 June Q4
8 marks Challenging +1.2
4 The sequence \(\left\{ \mathrm { u } _ { \mathrm { n } } \right\}\) is defined by \(u _ { 1 } = 1\) and \(\mathrm { u } _ { \mathrm { n } + 1 } = 2 \mathrm { u } _ { \mathrm { n } } + \mathrm { n } ^ { 2 }\) for \(\mathrm { n } \geqslant 1\).
Determine \(u _ { n }\) as a function of \(n\).
OCR Further Additional Pure AS 2019 June Q5
8 marks Challenging +1.2
5 The tetrahedron \(T\), shown below, has vertices at \(O ( 0,0,0 ) , A ( 1,2,2 ) , B ( 2,1,2 )\) and \(C ( 2,2,1 )\). \includegraphics[max width=\textwidth, alt={}, center]{59fa1650-a296-471e-93b9-0988177cd89d-3_360_464_319_555} Diagram not drawn to scale Show that the surface area of \(T\) is \(\frac { 1 } { 2 } \sqrt { 3 } ( 1 + \sqrt { 51 } )\).
OCR Further Additional Pure AS 2019 June Q6
8 marks Standard +0.8
6
  1. Determine all values of \(x\) for which \(16 x \equiv 5 ( \bmod 101 )\).
  2. Solve
    1. \(95 x \equiv 6 ( \bmod 101 )\),
    2. \(95 x \equiv 5 ( \bmod 101 )\).
OCR Further Additional Pure AS 2019 June Q7
12 marks Standard +0.8
7 You are given the set \(S = \{ 1,5,7,11,13,17 \}\) together with \(\times _ { 18 }\), the operation of multiplication modulo 18.
  1. Complete the Cayley table for \(\left( S , \times _ { 18 } \right)\) given in the Printed Answer Booklet.
  2. Prove that ( \(S , \times _ { 18 }\) ) is a group. (You may assume that \(\times _ { 18 }\) is associative.)
  3. Write down the order of each element of the group.
  4. Show that \(\left( S , \times _ { 18 } \right)\) is a cyclic group.
    1. Give an example of a non-cyclic group of order 6 .
    2. Give one reason why your example is structurally different to \(\left( S , { } _ { 18 } \right)\).