Questions — OCR (4907 questions)

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OCR FP3 2006 June Q7
12 marks Challenging +1.3
The series \(C\) and \(S\) are defined for \(0 < \theta < \pi\) by \begin{align} C &= 1 + \cos \theta + \cos 2\theta + \cos 3\theta + \cos 4\theta + \cos 5\theta,
S &= \sin \theta + \sin 2\theta + \sin 3\theta + \sin 4\theta + \sin 5\theta. \end{align}
  1. Show that \(C + iS = \frac{e^{3i\theta} - e^{-3i\theta}}{e^{i\theta} - e^{-i\theta}} \cdot e^{i\theta}\). [4]
  2. Deduce that \(C = \sin 3\theta \cos \frac{5}{2}\theta \operatorname{cosec} \frac{1}{2}\theta\) and write down the corresponding expression for \(S\). [4]
  3. Hence find the values of \(\theta\), in the range \(0 < \theta < \pi\), for which \(C = S\). [4]
OCR FP3 2006 June Q8
14 marks Challenging +1.2
A group \(D\) of order 10 is generated by the elements \(a\) and \(r\), with the properties \(a^2 = e\), \(r^5 = e\) and \(r^4a = ar\), where \(e\) is the identity. Part of the operation table is shown below. \includegraphics{figure_1}
  1. Give a reason why \(D\) is not commutative. [1]
  2. Write down the orders of any possible proper subgroups of \(D\). [2]
  3. List the elements of a proper subgroup which contains
    1. the element \(a\), [1]
    2. the element \(r\). [1]
  4. Determine the order of each of the elements \(r^3\), \(ar\) and \(ar^2\). [4]
  5. Copy and complete the section of the table marked E, showing the products of the elements \(ar\), \(ar^2\), \(ar^3\) and \(ar^4\). [5]
OCR FP3 2010 June Q1
7 marks Standard +0.8
The line \(l_1\) passes through the points \((0, 0, 10)\) and \((7, 0, 0)\) and the line \(l_2\) passes through the points \((4, 6, 0)\) and \((3, 3, 1)\). Find the shortest distance between \(l_1\) and \(l_2\). [7]
OCR FP3 2010 June Q2
6 marks Challenging +1.2
A multiplicative group with identity \(e\) contains distinct elements \(a\) and \(r\), with the properties \(r^6 = e\) and \(ar = r^2a\).
  1. Prove that \(rar = a\). [2]
  2. Prove, by induction or otherwise, that \(r^n ar^n = a\) for all positive integers \(n\). [4]
OCR FP3 2010 June Q3
9 marks Standard +0.8
In this question, \(w\) denotes the complex number \(\cos \frac{2\pi}{5} + i \sin \frac{2\pi}{5}\).
  1. Express \(w^2\), \(w^3\) and \(w^4\) in polar form, with arguments in the interval \(0 \leq \theta < 2\pi\). [4]
  2. The points in an Argand diagram which represent the numbers $$1, \quad 1 + w, \quad 1 + w + w^2, \quad 1 + w + w^2 + w^3, \quad 1 + w + w^2 + w^3 + w^4$$ are denoted by \(A\), \(B\), \(C\), \(D\), \(E\) respectively. Sketch the Argand diagram to show these points and join them in the order stated. (Your diagram need not be exactly to scale, but it should show the important features.) [4]
  3. Write down a polynomial equation of degree 5 which is satisfied by \(w\). [1]
OCR FP3 2010 June Q4
8 marks Challenging +1.2
  1. Use the substitution \(y = xz\) to find the general solution of the differential equation $$x \frac{dy}{dx} - y = x \cos \left(\frac{y}{x}\right),$$ giving your answer in a form without logarithms. (You may quote an appropriate result given in the List of Formulae (MF1).) [6]
  2. Find the solution of the differential equation for which \(y = \pi\) when \(x = 4\). [2]
OCR FP3 2010 June Q5
8 marks Challenging +1.2
Convergent infinite series \(C\) and \(S\) are defined by \begin{align} C &= 1 + \frac{1}{4} \cos \theta + \frac{1}{4} \cos 2\theta + \frac{1}{8} \cos 3\theta + \ldots,
S &= \frac{1}{2} \sin \theta + \frac{1}{4} \sin 2\theta + \frac{1}{8} \sin 3\theta + \ldots. \end{align}
  1. Show that \(C + iS = \frac{2}{2 - e^{i\theta}}\). [4]
  2. Hence show that \(C = \frac{4 - 2\cos \theta}{5 - 4\cos \theta}\) and find a similar expression for \(S\). [4]
OCR FP3 2010 June Q6
9 marks Standard +0.8
  1. Find the general solution of the differential equation $$\frac{d^2y}{dx^2} + 2 \frac{dy}{dx} + 17y = 17x + 36.$$ [7]
  2. Show that, when \(x\) is large and positive, the solution approximates to a linear function, and state its equation. [2]
OCR FP3 2010 June Q7
12 marks Challenging +1.2
A line \(l\) has equation \(\mathbf{r} = \begin{pmatrix} -7 \\ -3 \\ 0 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ -2 \\ 3 \end{pmatrix}\). A plane \(\Pi\) passes through the points \((1, 3, 5)\) and \((5, 2, 5)\), and is parallel to \(l\).
  1. Find an equation of \(\Pi\), giving your answer in the form \(\mathbf{r} \cdot \mathbf{n} = p\). [4]
  2. Find the distance between \(l\) and \(\Pi\). [4]
  3. Find an equation of the line which is the reflection of \(l\) in \(\Pi\), giving your answer in the form \(\mathbf{r} = \mathbf{a} + t\mathbf{b}\). [4]
OCR FP3 2010 June Q8
13 marks Challenging +1.2
A set of matrices \(M\) is defined by $$A = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \quad B = \begin{pmatrix} \omega & 0 \\ 0 & \omega^2 \end{pmatrix}, \quad C = \begin{pmatrix} \omega^2 & 0 \\ 0 & \omega \end{pmatrix}, \quad D = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad E = \begin{pmatrix} 0 & \omega^2 \\ \omega & 0 \end{pmatrix}, \quad F = \begin{pmatrix} 0 & \omega \\ \omega^2 & 0 \end{pmatrix},$$ where \(\omega\) and \(\omega^2\) are the complex cube roots of 1. It is given that \(M\) is a group under matrix multiplication.
  1. Write down the elements of a subgroup of order 2. [1]
  2. Explain why there is no element \(X\) of the group, other than \(A\), which satisfies the equation \(X^2 = A\). [2]
  3. By finding \(BE\) and \(EB\), verify the closure property for the pair of elements \(B\) and \(E\). [4]
  4. Find the inverses of \(B\) and \(E\). [3]
  5. Determine whether the group \(M\) is isomorphic to the group \(N\) which is defined as the set of numbers \(\{1, 2, 4, 8, 7, 5\}\) under multiplication modulo 9. Justify your answer clearly. [3]
OCR FP3 2011 June Q1
6 marks Standard +0.3
A line \(l\) has equation \(\frac{x-1}{5} = \frac{y-6}{6} = \frac{z+3}{-7}\) and a plane \(p\) has equation \(x + 2y - z = 40\).
  1. Find the acute angle between \(l\) and \(p\). [4]
  2. Find the perpendicular distance from the point \((1, 6, -3)\) to \(p\). [2]
OCR FP3 2011 June Q2
6 marks Standard +0.8
It is given that \(z = e^{i\theta}\), where \(0 < \theta < 2\pi\), and \(w = \frac{1+z}{1-z}\).
  1. Prove that \(w = i \cot \frac{1}{2}\theta\). [3]
  2. Sketch separate Argand diagrams to show the locus of \(z\) and the locus of \(w\). You should show the direction in which each locus is described when \(\theta\) increases in the interval \(0 < \theta < 2\pi\). [3]
OCR FP3 2011 June Q3
11 marks Standard +0.8
The variables \(x\) and \(y\) satisfy the differential equation $$\frac{dy}{dx} + 4y = 5 \cos 3x.$$
  1. Find the complementary function. [2]
  2. Hence, or otherwise, find the general solution. [7]
  3. Find the approximate range of values of \(y\) when \(x\) is large and positive. [2]
OCR FP3 2011 June Q4
9 marks Challenging +1.3
A group \(G\), of order 8, is generated by the elements \(a\), \(b\), \(c\). \(G\) has the properties $$a^2 = b^2 = c^2 = e, \quad ab = ba, \quad bc = cb, \quad ca = ac,$$ where \(e\) is the identity.
  1. Using these properties and basic group properties as necessary, prove that \(abc = cba\). [2]
The operation table for \(G\) is shown below.
\(e\)\(a\)\(b\)\(c\)\(bc\)\(ca\)\(ab\)\(abc\)
\(e\)\(e\)\(a\)\(b\)\(c\)\(bc\)\(ca\)\(ab\)\(abc\)
\(a\)\(a\)\(e\)\(ab\)\(ca\)\(abc\)\(c\)\(b\)\(bc\)
\(b\)\(b\)\(ab\)\(e\)\(bc\)\(c\)\(abc\)\(a\)\(ca\)
\(c\)\(c\)\(ca\)\(bc\)\(e\)\(b\)\(a\)\(abc\)\(ab\)
\(bc\)\(bc\)\(abc\)\(c\)\(b\)\(e\)\(ab\)\(ca\)\(a\)
\(ca\)\(ca\)\(c\)\(abc\)\(a\)\(ab\)\(e\)\(bc\)\(b\)
\(ab\)\(ab\)\(b\)\(a\)\(abc\)\(ca\)\(bc\)\(e\)\(c\)
\(abc\)\(abc\)\(bc\)\(ca\)\(ab\)\(a\)\(b\)\(c\)\(e\)
  1. List all the subgroups of order 2. [2]
  2. List five subgroups of order 4. [3]
  3. Determine whether all the subgroups of \(G\) which are of order 4 are isomorphic. [2]
OCR FP3 2011 June Q5
9 marks Standard +0.8
The substitution \(y = u^k\), where \(k\) is an integer, is to be used to solve the differential equation $$x \frac{dy}{dx} + 3y = x^2 y^2 \qquad (A)$$ by changing it into an equation (B) in the variables \(u\) and \(x\).
  1. Show that equation (B) may be written in the form $$\frac{du}{dx} + \frac{3}{kx} u = \frac{1}{k} x u^{k+1}.$$ [4]
  2. Write down the value of \(k\) for which the integrating factor method may be used to solve equation (B). [1]
  3. Using this value of \(k\), solve equation (B) and hence find the general solution of equation (A), giving your answer in the form \(y = f(x)\). [4]
OCR FP3 2011 June Q6
10 marks Challenging +1.2
  1. The set of polynomials \(\{ax + b\}\), where \(a, b \in \mathbb{R}\), is denoted by \(P\). Assuming that the associativity property holds, prove that \(P\), under addition, is a group. [4]
  2. The set of polynomials \(\{ax + b\}\), where \(a, b \in \{0, 1, 2\}\), is denoted by \(Q\). It is given that \(Q\), under addition modulo 3, is a group, denoted by \((Q, +(\text{mod}3))\).
    1. State the order of the group. [1]
    2. Write down the inverse of the element \(2x + 1\). [1]
    3. \(q(x) = ax + b\) is any element of \(Q\) other than the identity. Find the order of \(q(x)\) and hence determine whether \((Q, +(\text{mod}3))\) is a cyclic group. [4]
OCR FP3 2011 June Q7
10 marks Challenging +1.2
(In this question, the notation \(\Delta ABC\) denotes the area of the triangle \(ABC\).) The points \(P\), \(Q\) and \(R\) have position vectors \(p\mathbf{i}\), \(q\mathbf{j}\) and \(r\mathbf{k}\) respectively, relative to the origin \(O\), where \(p\), \(q\) and \(r\) are positive. The points \(O\), \(P\), \(Q\) and \(R\) are joined to form a tetrahedron.
  1. Draw a sketch of the tetrahedron and write down the values of \(\Delta OPQ\), \(\Delta OQR\) and \(\Delta ORP\). [3]
  2. Use the definition of the vector product to show that \(\frac{1}{2}|\overrightarrow{RP} \times \overrightarrow{RQ}| = \Delta PQR\). [1]
  3. Show that \((\Delta OPQ)^2 + (\Delta OQR)^2 + (\Delta ORP)^2 = (\Delta PQR)^2\). [6]
OCR FP3 2011 June Q8
11 marks Challenging +1.2
  1. Use de Moivre's theorem to express \(\cos 4\theta\) as a polynomial in \(\cos \theta\). [4]
  2. Hence prove that \(\cos 4\theta \cos 2\theta \equiv 16 \cos^6 \theta - 24 \cos^4 \theta + 10 \cos^2 \theta - 1\). [1]
  3. Use part (ii) to show that the only roots of the equation \(\cos 4\theta \cos 2\theta = 1\) are \(\theta = n\pi\), where \(n\) is an integer. [3]
  4. Show that \(\cos 4\theta \cos 2\theta = -1\) only when \(\cos \theta = 0\). [3]
OCR D1 2008 January Q1
6 marks Easy -1.8
Five boxes weigh 5 kg, 2 kg, 4 kg, 3 kg and 8 kg. They are stacked, in the order given, with the first box at the top of the stack. The boxes are to be packed into bins that can each hold up to 10 kg.
  1. Use the first-fit method to put the boxes into bins. Show clearly which boxes are packed in which bins. [2]
  2. Use the first-fit decreasing method to put the boxes into bins. You do not need to use an algorithm for sorting. Show clearly which boxes are packed in which bins. [2]
  3. Why might the first-fit decreasing method not be practical? [1]
  4. Show that if the bins can only hold up to 8 kg each it is still possible to pack the boxes into three bins. [1]
OCR D1 2008 January Q2
5 marks Moderate -0.8
A puzzle involves a 3 by 3 grid of squares, numbered 1 to 9, as shown in Fig. 1a below. Eight of the squares are covered by blank tiles. Fig. 1b shows the puzzle with all of the squares covered except for square 4. This arrangement of tiles will be called position 4. \includegraphics{figure_1} A move consists of sliding a tile into the empty space. From position 4, the next move will result in position 1, position 5 or position 7.
  1. Draw a graph with nine vertices to represent the nine positions and arcs that show which positions can be reached from one another in one move. What is the least number of moves needed to get from position 1 to position 9? [3]
  2. State whether the graph from part (i) is Eulerian, semi-Eulerian or neither. Explain how you know which it is. [2]
OCR D1 2008 January Q3
11 marks Moderate -0.8
Answer this question on the insert provided. \includegraphics{figure_2}
  1. This diagram shows a network. The insert has a copy of this network together with a list of the arcs, sorted into increasing order of weight. Use Kruskal's algorithm on the insert to find a minimum spanning tree for this network. Draw your tree and give its total weight. [5]
  2. Use your answer to part (i) to find the weight of a minimum spanning tree for the network with vertex \(G\), and all the arcs joined to \(G\), removed. Hence find a lower bound for the travelling salesperson problem on the original network. [3]
  3. Apply the nearest neighbour method, starting from vertex \(A\), to find an upper bound for the travelling salesperson problem on the original network. [3]
OCR D1 2008 January Q4
12 marks Moderate -0.8
Answer this question on the insert provided. Jenny needs to travel to London to arrive in time for a morning meeting. The graph below represents the various travel options that are available to her. \includegraphics{figure_3} It takes Jenny 120 minutes to drive from her home to the local airport and check in (arc \(JA\)). The journey from the local airport to Gatwick takes 80 minutes. From Gatwick to the underground station takes 60 minutes, and walking from the underground station to the meeting venue takes 15 minutes. Alternatively, Jenny could get a taxi from Gatwick to the meeting venue; this takes 80 minutes. It takes Jenny 15 minutes to drive from her house to the train station. Alternatively, she can walk to the bus stop, which takes 5 minutes, and then get a bus to the train station, taking another 20 minutes. From the train station to Paddington takes 300 minutes, and from Paddington to the underground station takes a further 20 minutes. Alternatively, Jenny could walk from Paddington to the meeting venue, taking 30 minutes. Jenny can catch a coach from her local bus stop to Victoria, taking 400 minutes. From Victoria she can either travel to the underground station, which takes 10 minutes, or she can walk to the meeting venue, which takes 15 minutes. The final option available to Jenny is to drive to a friend's house, taking 240 minutes, and then continue the journey into London by train. The journey from her friend's house to Waterloo takes Jenny 30 minutes. From here she can either go to the underground station, which takes 20 minutes, or walk to the meeting venue, which takes 40 minutes.
  1. Weight the arcs on the graph in the insert to show these times. Apply Dijkstra's algorithm, starting from \(J\), to give a permanent label and order of becoming permanent at each vertex. Stop when you have assigned a permanent label to vertex \(M\). Write down the route of the shortest path from \(J\) to \(M\). [9]
  2. What does the value of the permanent label at \(M\) represent? [1]
  3. Give two reasons why Jenny might choose to use a different route from \(J\) to \(M\). [2]
OCR D1 2008 January Q5
12 marks Moderate -0.8
Mark wants to decorate the walls of his study. The total wall area is 24 m\(^2\). Mark can cover the walls using any combination of three materials: panelling, paint and pinboard. He wants at least 2 m\(^2\) of pinboard and at least 10 m\(^2\) of panelling. Panelling costs £8 per m\(^2\) and it will take Mark 15 minutes to put up 1 m\(^2\) of panelling. Paint costs £4 per m\(^2\) and it will take Mark 30 minutes to paint 1 m\(^2\). Pinboard costs £10 per m\(^2\) and it will take Mark 20 minutes to put up 1 m\(^2\) of pinboard. He has all the equipment that he will need for the decorating jobs. Mark is able to spend up to £150 on the materials for the decorating. He wants to know what area should be covered with each material to enable him to complete the whole job in the shortest time possible. Mark models the problem as an LP with five constraints. His constraints are: $$x + y + z = 24,$$ $$4x + 2y + 5z \leqslant 75,$$ $$x \geqslant 10,$$ $$y \geqslant 0,$$ $$z \geqslant 2.$$
  1. Identify the meaning of each of the variables \(x\), \(y\) and \(z\). [2]
  2. Show how the constraint \(4x + 2y + 5z \leqslant 75\) was formed. [2]
  3. Write down an objective function, to be minimised. [1]
Mark rewrites the first constraint as \(z = 24 - x - y\) and uses this to eliminate \(z\) from the problem.
  1. Rewrite and simplify the objective and the remaining four constraints as functions of \(x\) and \(y\) only. [3]
  2. Represent your constraints from part (iv) graphically and identify the feasible region. Your graph should show \(x\) and \(y\) values from 0 to 15 only. [4]
OCR D1 2008 January Q6
13 marks Moderate -0.3
  1. Represent the linear programming problem below by an initial Simplex tableau. [2] Maximise \quad \(P = 25x + 14y - 32z\), subject to \quad \(6x - 4y + 3z \leqslant 24\), \qquad\qquad\quad \(5x - 3y + 10z \leqslant 15\), and \qquad\qquad \(x \geqslant 0\), \(y \geqslant 0\), \(z \geqslant 0\).
  2. Explain how you know that the first iteration will use a pivot from the \(x\) column. Show the calculations used to find the pivot element. [3]
  3. Perform one iteration of the Simplex algorithm. Show how each row was calculated and write down the values of \(x\), \(y\), \(z\) and \(P\) that result from this iteration. [7]
  4. Explain why the Simplex algorithm cannot be used to find the optimal value of \(P\) for this problem. [1]
OCR D1 2008 January Q7
13 marks Moderate -0.8
In this question, the function INT(\(X\)) is the largest integer less than or equal to \(X\). For example, $$\text{INT}(3.6) = 3,$$ $$\text{INT}(3) = 3,$$ $$\text{INT}(-3.6) = -4.$$ Consider the following algorithm. \begin{align} \text{Step 1} \quad & \text{Input } B
\text{Step 2} \quad & \text{Input } N
\text{Step 3} \quad & \text{Calculate } F = N \div B
\text{Step 4} \quad & \text{Let } G = \text{INT}(F)
\text{Step 5} \quad & \text{Calculate } H = B \times G
\text{Step 6} \quad & \text{Calculate } C = N - H
\text{Step 7} \quad & \text{Output } C
\text{Step 8} \quad & \text{Replace } N \text{ by the value of } G
\text{Step 9} \quad & \text{If } N = 0 \text{ then stop, otherwise go back to Step 3} \end{align}
  1. Apply the algorithm with the inputs \(B = 2\) and \(N = 5\). Record the values of \(F\), \(G\), \(H\), \(C\) and \(N\) each time Step 9 is reached. [5]
  2. Explain what happens when the algorithm is applied with the inputs \(B = 2\) and \(N = -5\). [4]
  3. Apply the algorithm with the inputs \(B = 10\) and \(N = 37\). Record the values of \(F\), \(G\), \(H\), \(C\) and \(N\) each time Step 9 is reached. What are the output values when \(B = 10\) and \(N\) is any positive integer? [4]