Questions — OCR MEI (4455 questions)

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OCR MEI FP1 2007 June Q6
6 marks Moderate -0.8
  1. Show that \(\frac{1}{r+2} - \frac{1}{r+3} = \frac{1}{(r+2)(r+3)}\). [2]
  2. Hence use the method of differences to find \(\frac{1}{3 \times 4} + \frac{1}{4 \times 5} + \frac{1}{5 \times 6} + \ldots + \frac{1}{52 \times 53}\). [4]
OCR MEI FP1 2007 June Q7
6 marks Moderate -0.3
Prove by induction that \(\sum_{r=1}^{n} 3^{r-1} = \frac{3^n - 1}{2}\). [6]
OCR MEI FP1 2007 June Q8
14 marks Standard +0.3
A curve has equation \(y = \frac{x^2 - 4}{(x-3)(x+1)(x-1)}\).
  1. Write down the coordinates of the points where the curve crosses the axes. [3]
  2. Write down the equations of the three vertical asymptotes and the one horizontal asymptote. [4]
  3. Determine whether the curve approaches the horizontal asymptote from above or below for
    1. large positive values of \(x\),
    2. large negative values of \(x\). [3]
  4. Sketch the curve. [4]
OCR MEI FP1 2007 June Q9
11 marks Standard +0.8
The cubic equation \(x^3 + Ax^2 + Bx + 15 = 0\), where \(A\) and \(B\) are real numbers, has a root \(x = 1 + 2\mathrm{j}\).
  1. Write down the other complex root. [1]
  2. Explain why the equation must have a real root. [1]
  3. Find the value of the real root and the values of \(A\) and \(B\). [9]
OCR MEI FP1 2007 June Q10
11 marks Standard +0.8
You are given that \(\mathbf{A} = \begin{pmatrix} 1 & -2 & k \\ 2 & 1 & 2 \\ 3 & 2 & -1 \end{pmatrix}\) and \(\mathbf{B} = \begin{pmatrix} -5 & -2+2k & -4-k \\ 8 & -1-3k & -2+2k \\ 1 & -8 & 5 \end{pmatrix}\) and that \(\mathbf{AB}\) is of the form \(\mathbf{AB} = \begin{pmatrix} k-n & 0 & 0 \\ 0 & k-n & 0 \\ 0 & 0 & k-n \end{pmatrix}\).
  1. Find the value of \(n\). [2]
  2. Write down the inverse matrix \(\mathbf{A}^{-1}\) and state the condition on \(k\) for this inverse to exist. [4]
  3. Using the result from part (ii), or otherwise, solve the following simultaneous equations. \begin{align} x - 2y + z &= 1
    2x + y + 2z &= 12
    3x + 2y - z &= 3 \end{align} [5]
OCR MEI FP2 2011 January Q1
19 marks Standard +0.3
  1. A curve has polar equation \(r = 2(\cos \theta + \sin \theta)\) for \(-\frac{1}{4}\pi \leq \theta \leq \frac{3}{4}\pi\).
    1. Show that a cartesian equation of the curve is \(x^2 + y^2 = 2x + 2y\). Hence or otherwise sketch the curve. [5]
    2. Find, by integration, the area of the region bounded by the curve and the lines \(\theta = 0\) and \(\theta = \frac{1}{2}\pi\). Give your answer in terms of \(\pi\). [7]
    1. Given that \(f(x) = \arctan(\frac{1}{2}x)\), find \(f'(x)\). [2]
    2. Expand \(f'(x)\) in ascending powers of \(x\) as far as the term in \(x^4\). Hence obtain an expression for \(f(x)\) in ascending powers of \(x\) as far as the term in \(x^5\). [5]
OCR MEI FP2 2011 January Q2
19 marks Standard +0.3
    1. Given that \(z = \cos \theta + j \sin \theta\), express \(z^n + z^{-n}\) and \(z^n - z^{-n}\) in simplified trigonometrical form. [2]
    2. By considering \((z + z^{-1})^6\), show that $$\cos^6 \theta = \frac{1}{32}(\cos 6\theta + 6 \cos 4\theta + 15 \cos 2\theta + 10).$$ [3]
    3. Obtain an expression for \(\cos^6 \theta - \sin^6 \theta\) in terms of \(\cos 2\theta\) and \(\cos 6\theta\). [5]
  2. The complex number \(w\) is \(8e^{i\pi/3}\). You are given that \(z_1\) is a square root of \(w\) and that \(z_2\) is a cube root of \(w\). The points representing \(z_1\) and \(z_2\) in the Argand diagram both lie in the third quadrant.
    1. Find \(z_1\) and \(z_2\) in the form \(re^{i\theta}\). Draw an Argand diagram showing \(w\), \(z_1\) and \(z_2\). [6]
    2. Find the product \(z_1z_2\), and determine the quadrant of the Argand diagram in which it lies. [3]
OCR MEI FP2 2011 January Q3
16 marks Standard +0.3
  1. Show that the characteristic equation of the matrix $$\mathbf{M} = \begin{pmatrix} 1 & -4 & 5 \\ 2 & 3 & -2 \\ -1 & 4 & 1 \end{pmatrix}$$ is \(\lambda^3 - 5\lambda^2 + 28\lambda - 66 = 0\). [4]
  2. Show that \(\lambda = 3\) is an eigenvalue of \(\mathbf{M}\), and determine whether or not \(\mathbf{M}\) has any other real eigenvalues. [4]
  3. Find an eigenvector, \(\mathbf{v}\), of unit length corresponding to \(\lambda = 3\). State the magnitude of the vector \(\mathbf{M}^n\mathbf{v}\), where \(n\) is an integer. [5]
  4. Using the Cayley-Hamilton theorem, obtain an equation for \(\mathbf{M}^{-1}\) in terms of \(\mathbf{M}^2\), \(\mathbf{M}\) and \(\mathbf{I}\). [3]
OCR MEI FP2 2011 January Q4
18 marks Standard +0.8
  1. Solve the equation $$\sinh t + 7 \cosh t = 8,$$ expressing your answer in exact logarithmic form. [6]
A curve has equation \(y = \cosh 2x + 7 \sinh 2x\).
  1. Using part (i), or otherwise, find, in an exact form, the coordinates of the points on the curve at which the gradient is 16. Show that there is no point on the curve at which the gradient is zero. Sketch the curve. [8]
  2. Find, in an exact form, the positive value of \(a\) for which the area of the region between the curve, the \(x\)-axis, the \(y\)-axis and the line \(x = a\) is \(\frac{1}{2}\). [4]
OCR MEI FP2 2011 January Q5
18 marks Challenging +1.2
A curve has parametric equations $$x = t + a \sin t, \quad y = 1 - a \cos t,$$ where \(a\) is a positive constant.
  1. Draw, on separate diagrams, sketches of the curve for \(-2\pi < t < 2\pi\) in the cases \(a = 1\), \(a = 2\) and \(a = 0.5\). By investigating other cases, state the value(s) of \(a\) for which the curve has
    1. loops,
    2. cusps. [7]
  2. Suppose that the point P\((x, y)\) lies on the curve. Show that the point P\('(-x, y)\) also lies on the curve. What does this indicate about the symmetry of the curve? [3]
  3. Find an expression in terms of \(a\) and \(t\) for the gradient of the curve. Hence find, in terms of \(a\), the coordinates of the turning points on the curve for \(-2\pi < t < 2\pi\) and \(a \neq 1\). [5]
  4. In the case \(a = \frac{1}{2}\pi\), show that \(t = \frac{1}{3}\pi\) and \(t = \frac{5}{3}\pi\) give the same point. Find the angle at which the curve crosses itself at this point. [3]
OCR MEI FP2 2009 June Q1
16 marks Standard +0.3
    1. Use the Maclaurin series for \(\ln(1 + x)\) and \(\ln(1 - x)\) to obtain the first three non-zero terms in the Maclaurin series for \(\ln\left(\frac{1 + x}{1 - x}\right)\). State the range of validity of this series. [4]
    2. Find the value of \(x\) for which \(\frac{1 + x}{1 - x} = 3\). Hence find an approximation to \(\ln 3\), giving your answer to three decimal places. [4]
  2. A curve has polar equation \(r = \frac{a}{1 + \sin \theta}\) for \(0 \leq \theta \leq \pi\), where \(a\) is a positive constant. The points on the curve have cartesian coordinates \(x\) and \(y\).
    1. By plotting suitable points, or otherwise, sketch the curve. [3]
    2. Show that, for this curve, \(r + y = a\) and hence find the cartesian equation of the curve. [5]
OCR MEI FP2 2009 June Q2
19 marks Standard +0.3
  1. Obtain the characteristic equation for the matrix \(\mathbf{M}\) where $$\mathbf{M} = \begin{pmatrix} 3 & 1 & -2 \\ 6 & -1 & 0 \\ 2 & 0 & 1 \end{pmatrix}.$$ Hence or otherwise obtain the value of \(\det(\mathbf{M})\). [3]
  2. Show that \(-1\) is an eigenvalue of \(\mathbf{M}\), and show that the other two eigenvalues are not real. Find an eigenvector corresponding to the eigenvalue \(-1\). Hence or otherwise write down the solution to the following system of equations. [9] $$3x + y - 2z = -0.1$$ $$-y = 0.6$$ $$2x + z = 0.1$$
  3. State the Cayley-Hamilton theorem and use it to show that $$\mathbf{M}^3 = 3\mathbf{M}^2 - 3\mathbf{M} - 7\mathbf{I}.$$ Obtain an expression for \(\mathbf{M}^{-1}\) in terms of \(\mathbf{M}^2\), \(\mathbf{M}\) and \(\mathbf{I}\). [4]
  4. Find the numerical values of the elements of \(\mathbf{M}^{-1}\), showing your working. [3]
OCR MEI FP2 2009 June Q3
19 marks Standard +0.8
    1. Sketch the graph of \(y = \arcsin x\) for \(-1 \leq x \leq 1\). [1] Find \(\frac{dy}{dx}\), justifying the sign of your answer by reference to your sketch. [4]
    2. Find the exact value of the integral \(\int_0^1 \frac{1}{\sqrt{2 - x^2}} dx\). [3]
  2. The infinite series \(C\) and \(S\) are defined as follows. $$C = \cos \theta + \frac{1}{3}\cos 3\theta + \frac{1}{5}\cos 5\theta + \ldots$$ $$S = \sin \theta + \frac{1}{3}\sin 3\theta + \frac{1}{5}\sin 5\theta + \ldots$$ By considering \(C + jS\), show that $$C = \frac{3\cos \theta}{5 - 3\cos 2\theta},$$ and find a similar expression for \(S\). [11]
OCR MEI FP2 2009 June Q4
18 marks Standard +0.8
  1. Prove, from definitions involving exponentials, that $$\cosh 2u = 2\cosh^2 u - 1.$$ [3]
  2. Prove that \(\arsinh y = \ln\left(y + \sqrt{y^2 + 1}\right)\). [4]
  3. Use the substitution \(x = 2\sinh u\) to show that $$\int \sqrt{x^2 + 4} dx = 2\arsinh \frac{x}{2} + \frac{x}{2}\sqrt{x^2 + 4} + c,$$ where \(c\) is an arbitrary constant. [6]
  4. By first expressing \(t^2 + 2t + 5\) in completed square form, show that $$\int_{-1}^1 \sqrt{t^2 + 2t + 5} dt = 2\left(\ln(1 + \sqrt{2}) + \sqrt{2}\right).$$ [5]
OCR MEI FP2 2009 June Q5
18 marks Challenging +1.3
Fig. 5 shows a circle with centre C \((a, 0)\) and radius \(a\). B is the point \((0, 1)\). The line BC intersects the circle at P and Q. P is above the \(x\)-axis and Q is below. \includegraphics{figure_5}
  1. Show that, in the case \(a = 1\), P has coordinates \(\left(1 - \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)\). Write down the coordinates of Q. [3]
  2. Show that, for all positive values of \(a\), the coordinates of P are $$x = a\left(1 - \frac{a}{\sqrt{a^2 + 1}}\right), \quad y = \frac{a}{\sqrt{a^2 + 1}} \quad (*)$$ Write down the coordinates of Q in a similar form. [4] Now let the variable point P be defined by the parametric equations (*) for all values of the parameter \(a\), positive, zero and negative. Let Q be defined for all \(a\) by your answer in part (ii).
  3. Using your calculator, sketch the locus of P as \(a\) varies. State what happens to P as \(a \to \infty\) and as \(a \to -\infty\). Show algebraically that this locus has an asymptote at \(y = -1\). On the same axes, sketch, as a dotted line, the locus of Q as \(a\) varies. [8] (The single curve made up of these two loci and including the point B is called a right strophoid.)
  4. State, with a reason, the size of the angle POQ in Fig. 5. What does this indicate about the angle at which a right strophoid crosses itself? [3]
OCR MEI D1 2007 January Q1
8 marks Easy -1.3
Each of the following symbols consists of boundaries enclosing regions. \includegraphics{figure_1} The symbol representing zero has three regions, the outside, that between the two boundaries and the inside. To classify the symbols a graph is produced for each one. The graph has a vertex for each region, with arcs connecting regions which share a boundary. Thus the graph for \includegraphics{figure_2} is \(\bullet \longrightarrow \bullet \longrightarrow \bullet\).
  1. Produce the graph for the symbol \includegraphics{figure_3}. [1]
  2. Give two symbols each having the graph \(\bullet \longrightarrow \bullet\). [2]
  3. Produce the graph for the symbol \includegraphics{figure_4}. [2]
  4. Produce a single graph for the composite symbol \includegraphics{figure_5}. [2]
  5. Give the name of a connected graph with \(n\) nodes and \(n - 1\) arcs. [1]
OCR MEI D1 2007 January Q2
8 marks Easy -1.2
The following algorithm is a version of bubble sort. Step 1 \quad Store the values to be sorted in locations L(1), L(2), \(\ldots\) , L(n) and set i to be the number, n, of values to be sorted. Step 2 \quad Set j = 1. Step 3 \quad Compare the values in locations L(j) and L(j+1) and swap them if that in L(j) is larger than that in L(j+1). Step 4 \quad Add 1 to j. Step 5 \quad If j is less than i then go to step 3. Step 5 \quad Write out the current list, L(1), L(2), \(\ldots\) , L(n). Step 6 \quad Subtract 1 from i. Step 7 \quad If i is larger than 1 then go to step 2. Step 8 \quad Stop.
  1. Apply this algorithm to sort the following list. 109 \quad 32 \quad 3 \quad 523 \quad 58. Count the number of comparisons and the number of swaps which you make in applying the algorithm. [4]
  2. Put the five values into the order which maximises the number of swaps made in applying the algorithm, and give that number. [2]
  3. Bubble sort has quadratic complexity. Using bubble sort it takes a computer 1.5 seconds to sort a list of 1000 values. Approximately how long would it take to sort a list of 100 000 values? (Give your answer in hours and minutes.) [2]
OCR MEI D1 2007 January Q3
8 marks Easy -1.2
A bag contains five pieces of paper labelled A, B, C, D and E. One piece is drawn at random from the bag. If the piece is labelled with a vowel (A or E) then the process stops. Otherwise the piece of paper is replaced, the bag is shaken, and the process is repeated. You are to simulate this process to estimate the mean number of draws needed to get a vowel.
  1. Show how to use single digit random numbers to simulate the process efficiently. You need to describe exactly how your simulation will work. [3]
  2. Use the random numbers in your answer book to run your simulation 5 times, recording your results. [2]
  3. From your results compute an estimate of the mean number of draws needed to get a vowel. [2]
  4. State how you could produce a more accurate estimate. [1]
OCR MEI D1 2007 January Q4
16 marks Moderate -0.8
Cassi is managing the building of a house. The table shows the major activities that are involved, their durations and their precedences.
ActivityDuration (days)Immediate predecessors
ABuild concrete frame10\(-\)
BLay bricks7A
CLay roof tiles10A
DFirst fit electrics5B
EFirst fit plumbing4B
FPlastering6C, D, E
GSecond fit electrics3F
HSecond fit plumbing2F
ITiling10G, H
JFit sanitary ware2H
KFit windows and doors5I
  1. Draw an activity-on-arc network to represent this information. [5]
  2. Find the early time and the late time for each event. Give the project duration and list the critical activities. [6]
  3. Calculate total and independent floats for each non-critical activity. [2]
Cassi's clients wish to take delivery in 42 days. Some durations can be reduced, at extra cost, to achieve this.
  • The tiler will finish activity I in 9 days for an extra £250, or in 8 days for an extra £500.
  • The bricklayer will cut his total of 7 days on activity B by up to 3 days at an extra cost of £350 per day.
  • The electrician could be paid £300 more to cut a day off activity D, or £600 more to cut two days.
  1. What is the cheapest way in which Cassi can get the house built in 42 days? [3]
OCR MEI D1 2007 January Q5
16 marks Moderate -0.8
Leone is designing her new garden. She wants to have at least 1000 m\(^2\), split between lawn and flower beds. Initial costs are £0.80 per m\(^2\) for lawn and £0.40 per m\(^2\) for flowerbeds. Leone's budget is £500. Leone prefers flower beds to lawn, and she wants the area for flower beds to be at least twice the area for lawn. However, she wants to have at least 200 m\(^2\) of lawn. Maintenance costs each year are £0.15 per m\(^2\) for lawn and £0.25 per m\(^2\) for flower beds. Leone wants to minimize the maintenance costs of her garden.
  1. Formulate Leone's problem as a linear programming problem. [7]
  2. Produce a graph to illustrate the inequalities. [6]
  3. Solve Leone's problem. [2]
  4. If Leone had more than £500 available initially, how much extra could she spend to minimize maintenance costs? [1]
OCR MEI D1 2007 January Q6
16 marks Moderate -0.3
In a factory a network of pipes connects 6 vats, A, B, C, D, E and F. Two separate connectors need to be chosen from the network The table shows the lengths of pipes (metres) connecting the 6 vats.
ABCDEF
A\(-\)7\(-\)\(-\)12\(-\)
B7\(-\)5366
C\(-\)5\(-\)847
D\(-\)38\(-\)15
E12641\(-\)7
F\(-\)6757\(-\)
  1. Use Kruskal's algorithm to find a minimum connector. Show the order in which you select pipes, draw your connector and give its total length. [5]
  2. Produce a new table excluding the pipes which you selected in part (i). Use the tabular form of Prim's algorithm to find a second minimum connector from this reduced set of pipes. Show your working, draw your connector and give its total length. [7]
  3. The factory manager prefers the following pair of connectors: \(\{\)AB, BC, BD, BE, BF\(\}\) and \(\{\)AE, BF, CE, DE, DF\(\}\). Give two possible reasons for this preference. [4]
OCR MEI D2 Q1
16 marks Easy -1.2
The switching circuit in Fig. 1.1 shows switches, s for a car's sidelights, h for its dipped headlights and f for its high-intensity rear foglights. It also shows the three sets of lights. \includegraphics{figure_1} (Note: s and h are each "ganged" switches. A ganged switch consists of two connected switches sharing a single switch control, so that both are either on or off together.)
    1. Describe in words the conditions under which the foglights will come on. [2]
    Fig. 1.2 shows a combinatorial circuit. \includegraphics{figure_2}
    1. Write the output in terms of a Boolean expression involving s, h and f. [2]
    2. Use a truth table to prove that \(s \wedge h \wedge f = \sim (\sim s \vee \sim h) \wedge f\). [3]
  2. A car's first gear can be engaged (g) if either both the road speed is low (r) and the clutch is depressed (d), or if both the road speed is low (r) and the engine speed is the correct multiple of the road speed (m).
    1. Draw a switching circuit to represent the conditions under which first gear can be engaged. Use two ganged switches to represent r, and single switches to represent each of d, m and g. [2]
    2. Draw a combinatorial circuit to represent the Boolean expression \(r \wedge (d \vee m) \wedge g\). [4]
    3. Use Boolean algebra to prove that \(r \wedge (d \vee m) \wedge g = ((r \wedge d) \vee (r \wedge m)) \wedge g\). [2]
    4. Draw another switching circuit to represent the conditions under which first gear can be selected, but without using a ganged switch. [1]
OCR MEI D2 Q2
16 marks Moderate -0.8
Karl is considering investing in a villa in Greece. It will cost him 56000 euros (€ 56000). His alternative is to invest his money, £35000, in the United Kingdom. He is concerned with what will happen over the next 5 years. He estimates that there is a 60% chance that a house currently worth € 56000 will appreciate to be worth € 75000 in that time, but that there is a 40% chance that it will be worth only € 55000. If he invests in the United Kingdom then there is a 50% chance that there will be 20% growth over the 5 years, and a 50% chance that there will be 10% growth.
  1. Given that £1 is worth € 1.60, draw a decision tree for Karl, and advise him what to do, using the EMV of his investment (in thousands of euros) as his criterion. [4]
In fact the £/€ exchange rate is not fixed. It is estimated that at the end of 5 years, if there has been 20% growth in the UK then there is a 70% chance that the exchange rate will stand at 1.70 euros per pound, and a 30% chance that it will be 1.50. If growth has been 10% then there is a 40% chance that the exchange rate will stand at 1.70 and a 60% chance that it will be 1.50.
  1. Produce a revised decision tree incorporating this information, and give appropriate advice. [3]
A financial analyst asks Karl a number of questions to determine his utility function. He estimates that for x in cash (in thousands of euros) Karl's utility is \(x^{0.5}\), and that for y in property (in thousands of euros), Karl's utility is \(y^{0.75}\).
  1. Repeat your computations from part (ii) using utility instead of the EMV of his investment. Does this change your advice? [3]
  2. Using EMVs, find the exchange rate (number of euros per pound) which will make Karl indifferent between investing in the UK and investing in a villa in Greece. [2]
  3. Show that, using Karl's utility function, the exchange rate would have to drop to 1.277 euros per pound to make Karl indifferent between investing in the UK and investing in a villa in Greece. [4]
OCR MEI D2 Q3
20 marks Standard +0.3
The distance and route matrices shown in Fig. 3.1 are the result of applying Floyd's algorithm to the incomplete network on 4 vertices shown in Fig. 3.2. \includegraphics{figure_3} \includegraphics{figure_4}
  1. Draw the complete network of shortest distances. [2]
  2. Explain how to use the route matrix to find the shortest route from vertex 4 to vertex 1 in the original incomplete network. [2]
A new vertex, vertex 5, is added to the original network. Its distances from vertices to which it is connected are shown in Fig. 3.3. \includegraphics{figure_5}
  1. Draw the extended network and the complete 5-node network of shortest distances. (You are not required to use an algorithm to find the shortest distances.) [3]
  2. Produce the shortest distance matrix and the route matrix for the extended 5-node network. [3]
  3. Apply the nearest neighbour algorithm to your \(5 \times 5\) distance matrix, starting at vertex 1. Give the length of the cycle produced, together with the actual cycle in the original 5-node network. [3]
  4. By deleting vertex 1 and its arcs, and by using Prim's algorithm on the reduced distance matrix, produce a lower bound for the solution to the practical travelling salesperson problem in the original 5-node network. Show clearly your use of the matrix form of Prim's algorithm. [4]
  5. In the original 5-node network find a shortest route starting at vertex 1 and using each of the 6 arcs at least once. Give the length of your route. [3]
OCR MEI D2 Q4
20 marks Standard +0.8
Kassi and Theo are discussing how much oil and how much vinegar to use to dress their salad. They agree to use between 5 and 10ml of oil and between 3 and 6ml of vinegar and that the amount of oil should not exceed twice the amount of vinegar. Theo prefers to have more oil than vinegar. He formulates the following problem to maximise the proportion of oil: Maximise \(\frac{x}{x + y}\) subject to \(0 \leq x \leq 10\), \(0 \leq y \leq 6\), \(x - 2y \leq 0\).
  1. Explain why this problem is not an LP. [1]
  2. Use the simplex method to solve the following LP. Maximise \(x - y\) subject to \(0 \leq x \leq 10\), \(0 \leq y \leq 6\), \(x - 2y \leq 0\). [7]
  3. Kassi prefers to have more vinegar than oil. She formulates the following LP. Maximise \(y - x\) subject to \(5 \leq x \leq 10\), \(3 \leq y \leq 6\), \(x - 2y \leq 0\). Draw separate graphs to show the feasible regions for this problem and for the problem in part (ii). [5]
  4. Explain why the formulation in part (ii) produced a solution for Theo's problem, and why it is more difficult to use the simplex method to solve Kassi's problem in part (iii). [2]
  5. Produce an initial tableau for using the two-stage simplex method to solve Kassi's problem. Explain briefly how to proceed. [5]