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OCR FP1 AS 2017 December Q8
13 marks Standard +0.8
  1. Find, in terms of \(x\), a vector which is perpendicular to the vectors \(\begin{pmatrix} x-2 \\ 5 \\ 1 \end{pmatrix}\) and \(\begin{pmatrix} x \\ 6 \\ 2 \end{pmatrix}\). [2]
  2. Find the shortest possible vector of the form \(\begin{pmatrix} 1 \\ a \\ b \end{pmatrix}\) which is perpendicular to the vectors \(\begin{pmatrix} x-2 \\ 5 \\ 1 \end{pmatrix}\) and \(\begin{pmatrix} x \\ 6 \\ 2 \end{pmatrix}\). [5]
  1. Vector \(\mathbf{v}\) is perpendicular to both \(\begin{pmatrix} -1 \\ 1 \\ 1 \end{pmatrix}\) and \(\begin{pmatrix} 1 \\ p \\ p^2 \end{pmatrix}\) where \(p\) is a real number. Show that it is impossible for \(\mathbf{v}\) to be perpendicular to the vector \(\begin{pmatrix} 1 \\ 1 \\ p-1 \end{pmatrix}\). [6]
OCR Further Pure Core 2 2018 March Q1
8 marks Standard +0.3
Plane \(\Pi\) has equation \(3x - y + 2z = 33\). Line \(l\) has the following vector equation. $$l: \quad \mathbf{r} = \begin{pmatrix} 1 \\ 0 \\ 5 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 2 \\ 3 \end{pmatrix}$$
  1. Find the acute angle between \(\Pi\) and \(l\). [3]
  2. Find the coordinates of the point of intersection of \(\Pi\) and \(l\). [3]
  3. \(S\) is the point \((4, 5, -5)\). Find the shortest distance from \(S\) to \(\Pi\). [2]
OCR Further Pure Core 2 2018 March Q2
5 marks Moderate -0.8
The complex number \(2 + i\) is denoted by \(z\).
  1. Show that \(z^2 = 3 + 4i\). [2]
  2. Plot the following on the Argand diagram in the Printed Answer Booklet.
    [1]
  3. State the relationship between \(|z^2|\) and \(|z|\). [1]
  4. State the relationship between \(\arg(z^2)\) and \(\arg(z)\). [1]
OCR Further Pure Core 2 2018 March Q3
3 marks Standard +0.3
In this question you must show detailed reasoning. Use the formula \(\sum_{r=1}^n r^2 = \frac{1}{6}n(n+1)(2n+1)\) to evaluate \(121^2 + 122^2 + 123^2 + \ldots + 300^2\). [3]
OCR Further Pure Core 2 2018 March Q4
4 marks Standard +0.8
You are given that the cubic equation \(2x^3 - 3x^2 + x + 4 = 0\) has three roots, \(\alpha\), \(\beta\) and \(\gamma\). By making a suitable substitution to obtain a related cubic equation, determine the value of \(\frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma}\). [4]
OCR Further Pure Core 2 2018 March Q5
5 marks Standard +0.8
In this question you must show detailed reasoning. An ant starts from a fixed point \(O\) and walks in a straight line for \(1.5\) s. Its velocity, \(v\) cms\(^{-1}\), can be modelled by \(v = \frac{1}{\sqrt{9-t^2}}\). By finding the mean value of \(v\) in \(0 \leq t \leq 1.5\), deduce the average velocity of the ant. [5]
OCR Further Pure Core 2 2018 March Q6
12 marks Challenging +1.2
In this question you must show detailed reasoning.
  1. Find the coordinates of all stationary points on the graph of \(y = 6\sinh^2 x - 13\cosh x\), giving your answers in an exact, simplified form. [9]
  2. By finding the second derivative, classify the stationary points found in part (i). [3]
OCR Further Pure Core 2 2018 March Q7
12 marks Challenging +1.2
In the following set of simultaneous equations, \(a\) and \(b\) are constants. \begin{align} 3x + 2y - z &= 5
2x - 4y + 7z &= 60
ax + 20y - 25z &= b \end{align}
  1. In the case where \(a = 10\), solve the simultaneous equations, giving your solution in terms of \(b\). [3]
  2. Determine the value of \(a\) for which there is no unique solution for \(x\), \(y\) and \(z\). [3]
    1. Find the values of \(\alpha\) and \(\beta\) for which \(\alpha(2y - z) + \beta(-4y + 7z) = 20y - 25z\) for any \(y\) and \(z\). [3]
    2. Hence, for the case where there is no unique solution for \(x\), \(y\) and \(z\), determine the value of \(b\) for which there is an infinite number of solutions. [2]
    3. When \(a\) takes the value in part (ii) and \(b\) takes the value in part (iii)(b) describe the geometrical arrangement of the planes represented by the three equations. [1]
OCR Further Pure Core 2 2018 March Q8
12 marks Challenging +1.8
In this question you must show detailed reasoning. Show that \(\int_0^2 \frac{2x^2 + 3x - 1}{x^3 - 3x^2 + 4x - 12} dx = \frac{3}{8}\pi - \ln 9\). [12]
OCR Further Pure Core 2 2018 March Q9
14 marks Challenging +1.2
In this question you must show detailed reasoning.
  1. Show that \(e^{i\theta} - e^{-i\theta} = 2i\sin\theta\). [1]
  2. Hence, show that \(\frac{2}{e^{2i\theta} - 1} = -(1 + i\cot\theta)\). [3]
  3. Two series, \(C\) and \(S\), are defined as follows. $$C = 2 + 2\cos\frac{\pi}{10} + 2\cos\frac{\pi}{5} + 2\cos\frac{3\pi}{10} + 2\cos\frac{2\pi}{5}$$ $$S = 2\sin\frac{\pi}{10} + 2\sin\frac{\pi}{5} + 2\sin\frac{3\pi}{10} + 2\sin\frac{2\pi}{5}$$ By considering \(C + iS\), find a simplified expression for \(C\) in terms of only integers and \(\cot\frac{\pi}{10}\). [8]
  4. Verify that \(S = C - 2\) and, by considering the series in their original form, explain why this is so. [2]
OCR Further Discrete 2018 March Q1
6 marks Easy -1.2
The masses, in kg, of ten bags are given below. 8 \quad 10 \quad 10 \quad 12 \quad 12 \quad 12 \quad 13 \quad 15 \quad 18 \quad 18
  1. Use first-fit decreasing to pack the bags into crates that can hold a maximum of 50 kg each. [3]
Only two crates are available, so only some of the bags will be packed. Each bag is given a value.
BagABCDEFGHIJ
Mass (kg)8101012121213151818
Value6332454644
  1. Find a packing into two crates so that the total value of the bags in the crates is at least 32. [3]
OCR Further Discrete 2018 March Q2
14 marks Challenging +1.2
A linear programming problem is \begin{align} \text{Maximise } P &= 4x - y - 2z
\text{subject to } x + 5y + 3z &\leq 60
2x - 5y &\leq 80
2y + z &\leq 10
x \geq 0, y &\geq 0, z \geq 0 \end{align}
  1. Use the simplex algorithm to solve the problem. [7]
In the case when \(z = 0\) the feasible region can be represented graphically. \includegraphics{figure_1} The vertices of the feasible region are \((0, 0)\), \((40, 0)\), \((46.67, 2.67)\), \((35, 5)\) and \((0, 5)\), where non-integer values are given to 2 decimal places. The linear programming problem is given the additional constraint that \(x\) and \(y\) are integers.
  1. Use branch-and-bound, branching on \(x\) first, to show that the optimum solution with this additional constraint is \(x = 45, y = 2\). [7]
OCR Further Discrete 2018 March Q3
8 marks Standard +0.8
50 people are at a TV game show. 21 of the 50 are there to take part in the game show and the others are friends who are in the audience, 22 are women and 20 are from London, 2 are women from London who are there to take part in the game show and 15 are men who are not from London and are friends who are in the audience.
  1. Deduce how many of the 50 people are in two of the categories 'there to take part in the game show', 'is a woman' and 'is from London', but are not in all three categories. [3]
The 21 people who are there to take part in the game show are moved to the stage where they are seated in two rows of seats with 20 seats in each row. Some of the seats are empty.
  1. Show how the pigeonhole principle can be used to show that there must be at least one pair of these 21 people with no empty chair between them. [2]
The 21 people are split into three sets of 7. In each round of the game show, three of the people are chosen. The three people must all be from the same set of 7 but once two people have played in the same round they cannot play together in another round. For example, if A plays with B and C in round 1 then A cannot play with B or with C in any other round.
  1. By first considering how many different rounds can be formed using the first set of seven people, deduce how many rounds there can be altogether. [3]
OCR Further Discrete 2018 March Q4
12 marks Challenging +1.3
The graph below connects nine vertices A, B, \(\ldots\), H, I. \includegraphics{figure_2}
    1. Show that the minimum sum of the degrees of each pair of non-adjacent vertices is 9. [2]
    2. Explain what you can deduce from the result in part (a). [1]
  2. Use Kuratowski's theorem to prove that the graph is non-planar. [3]
  3. Prove that there is no subgraph of the graph that is isomorphic to \(K_4\), without using subdivision or contraction. [6]
OCR Further Discrete 2018 March Q5
12 marks Standard +0.3
The diagram represents a map of seven locations (A to G) and the direct road distances (km) between some of them. \includegraphics{figure_3} A delivery driver needs to start from his depot at D, make deliveries at each of A, B, F and G, and finish at D.
  1. Write down a route from A to G of length 70 km. [1]
The table shows the length of the shortest path between some pairs of places.
DABFG
D-
A-70
B-84
F84-
G70-
    1. Complete the table.
    2. Use the nearest neighbour method on the table, starting at D, to find the length of a cycle through D, A, B, F and G, ignoring possibly repeating E and C. [4]
  2. By first considering the table with the row and column for D removed, find a lower bound for the distance that the driver must travel. [2]
  3. What can you conclude from your previous answers about the distance that the driver must travel? [1]
A new road is constructed between D and F. Using this road the driver starts from D, makes the deliveries and returns to D having travelled just 172 km.
  1. Find the length of the new road if
    1. the driver does not return to D until all the deliveries have been made,
    2. the driver uses the new road twice in making the deliveries. [4]
OCR Further Discrete 2018 March Q6
15 marks Standard +0.3
The activities involved in a project, their durations, immediate predecessors and the number of workers required for each activity are shown in the table.
ActivityDuration (hours)Immediate predecessorsNumber of workers
A6-2
B4-1
C4-1
D2A2
E3A, B1
F4C1
G3D1
H3E, F2
  1. Model the project using an activity network.
  2. Draw a cascade chart for the project, showing each activity starting at its earliest possible start time. [3]
  3. Construct a schedule to show how three workers can complete the project in the minimum possible time. [4]
OCR Further Discrete 2018 March Q7
8 marks Challenging +1.2
Each day Alix and Ben play a game. They each choose a card and use the table below to find the number of points they win. The table shows the cards available to each player. The entries in the cells are of the form \((a, b)\), where \(a =\) points won by Alix and \(b =\) points won by Ben. Each is trying to maximise the points they win.
Ben
\cline{2-4} \multicolumn{1}{c}{}Card XCard YCard Z
\cline{2-4} \multirow{3}{*}{Alix}
Card P(4, 4)(5, 9)(1, 7)
\cline{2-4} Card Q(3, 5)(4, 1)(8, 2)
\cline{2-4} Card R\((x, y)\)(2, 2)(9, 4)
\cline{2-4}
  1. Explain why the table cannot be reduced through dominance no matter what values \(x\) and \(y\) have. [2]
  2. Show that the game is not stable no matter what values \(x\) and \(y\) have. [2]
  3. Find the Nash equilibrium solutions for the various values that \(x\) and \(y\) can have. [4]
OCR Further Pure Core 2 2018 September Q1
8 marks Standard +0.3
Line \(l_1\) has Cartesian equation $$l_1: \frac{-x}{2} = \frac{y-5}{2} = \frac{-z-6}{7}.$$
  1. Find a vector equation for \(l_1\). [2]
Line \(l_2\) has vector equation $$l_2: \mathbf{r} = \begin{pmatrix} 2 \\ 7 \\ -1 \end{pmatrix} + \mu \begin{pmatrix} 1 \\ -2 \\ 4 \end{pmatrix}.$$
  1. Find the point of intersection of \(l_1\) and \(l_2\). [3]
  2. Find the acute angle between \(l_1\) and \(l_2\). [3]
OCR Further Pure Core 2 2018 September Q2
5 marks Challenging +1.2
In this question you must show detailed reasoning.
  1. Find \(\int_{-\frac{3\pi}{4}}^{\frac{3\pi}{4}} 2\tan x \, dx\) giving your answer in the form \(\ln p\). [3]
  2. Show that \(\int_0^{\frac{3\pi}{4}} 2\tan x \, dx\) is undefined explaining your reasoning. [2]
OCR Further Pure Core 2 2018 September Q3
6 marks Standard +0.3
The equation of a plane, \(\Pi\), is $$\Pi: \mathbf{r} = \begin{pmatrix} 2 \\ -3 \\ 5 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 1 \\ 3 \end{pmatrix} + \mu \begin{pmatrix} -1 \\ 2 \\ 1 \end{pmatrix}.$$
  1. Find a vector which is perpendicular to \(\Pi\). [2]
  2. Hence find an equation for \(\Pi\) in the form \(\mathbf{r} \cdot \mathbf{n} = p\). [2]
  3. Find in the form \(\sqrt{q}\) the shortest distance between \(\Pi\) and the origin, where \(q\) is a rational number. [2]
OCR Further Pure Core 2 2018 September Q4
10 marks Challenging +1.2
The matrix \(\mathbf{A}\) is given by \(\mathbf{A} = \begin{pmatrix} a & 2 & 3 \\ 4 & 4 & 6 \\ -2 & 2 & 9 \end{pmatrix}\) where \(a\) is a constant. It is given that if \(\mathbf{A}\) is not singular then $$\mathbf{A}^{-1} = \frac{1}{24a-48} \begin{pmatrix} 24 & -12 & 0 \\ -48 & 9a+6 & 12-6a \\ 16 & -2a-4 & 4a-8 \end{pmatrix}.$$
  1. Use \(\mathbf{A}^{-1}\) to solve the simultaneous equations below, giving your answer in terms of \(k\). \begin{align} x + 2y + 3z &= 6
    4x + 4y + 6z &= 8
    -2x + 2y + 9z &= k \end{align} [3]
  2. Consider the equations below where \(a\) takes the value which makes \(\mathbf{A}\) singular. \begin{align} ax + 2y + 3z &= b
    4x + 4y + 6z &= 10
    -2x + 2y + 9z &= -13 \end{align} \(b\) takes the value for which the equations have an infinite number of solutions.
  3. For the equations in part (ii) with the values of \(a\) and \(b\) found in part (ii) describe fully the geometrical arrangement of the planes represented by the equations. [2]
OCR Further Pure Core 2 2018 September Q5
8 marks Challenging +1.2
The region \(R\) between the \(x\)-axis, the curve \(y = \frac{1}{\sqrt{p+x^3}}\) and the lines \(x = \sqrt{p}\) and \(x = \sqrt{3p}\), where \(p\) is a positive parameter, is rotated by \(2\pi\) radians about the \(x\)-axis to form a solid of revolution \(S\).
  1. Find and simplify an algebraic expression, in terms of \(p\), for the exact volume of \(S\). [5]
  2. Given that \(R\) must lie entirely between the lines \(x = 1\) and \(x = \sqrt{48}\) find in exact form
OCR Further Pure Core 2 2018 September Q6
8 marks Challenging +1.2
  1. By considering \(\sum_{r=1}^n ((r+1)^5 - r^5)\) show that \(\sum_{r=1}^n r^4 = \frac{1}{30}n(n+1)(2n+1)(3n^2+3n-1)\). [6]
  2. Use the formula given in part (i) to find \(50^4 + 51^4 + \ldots + 80^4\). [2]
OCR Further Pure Core 2 2018 September Q7
9 marks Challenging +1.2
The roots of the equation \(ax^2 + bx + c = 0\), where \(a\), \(b\) and \(c\) are positive integers, are \(\alpha\) and \(\beta\).
  1. Find a quadratic equation with integer coefficients whose roots are \(\alpha + \beta\) and \(\alpha\beta\). [4]
  2. Show that it is not possible for the original equation and the equation found in part (i) both to have repeated roots. [2]
  3. Show that the discriminant of the equation found in part (i) is always positive. [3]
OCR Further Pure Core 2 2018 September Q8
6 marks Challenging +1.2
In this question you must show detailed reasoning.
  1. Express \((6+5i)(7+5i)\) in the form \(a+bi\). [2]
  2. You are given that \(17^2 + 65^2 = 4514\). Using the result in part (i) and by considering \((6-5i)(7-5i)\) express \(4514\) as a product of its prime factors. [4]