Questions — OCR (4907 questions)

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OCR H240/01 2018 December Q10
13 marks Standard +0.3
10 \includegraphics[max width=\textwidth, alt={}, center]{a16ab26f-21fb-4a73-8b94-c16bef611bcb-7_524_714_274_246} The diagram shows the graph of \(y = - \tan ^ { - 1 } \left( \frac { 1 } { 2 } x - \frac { 1 } { 3 } \pi \right)\), which crosses the \(x\)-axis at the point \(A\) and the \(y\)-axis at the point \(B\).
  1. Determine the coordinates of the points \(A\) and \(B\).
  2. Give full details of a sequence of three geometrical transformations which transform the graph of \(y = \tan ^ { - 1 } x\) to the graph of \(y = - \tan ^ { - 1 } \left( \frac { 1 } { 2 } x - \frac { 1 } { 3 } \pi \right)\). The equation \(x = - \tan ^ { - 1 } \left( \frac { 1 } { 2 } x - \frac { 1 } { 3 } \pi \right)\) has only one root.
  3. Show by calculation that this root lies between \(x = 0\) and \(x = 1\).
  4. Use the iterative formula \(x _ { n + 1 } = - \tan ^ { - 1 } \left( \frac { 1 } { 2 } x _ { n } - \frac { 1 } { 3 } \pi \right)\), with a suitable starting value, to find the root correct to 3 significant figures. Show the result of each iteration.
  5. Using the diagram in the Printed Answer Booklet, show how the iterative process converges to the root.
OCR H240/01 2018 December Q11
12 marks Standard +0.3
11 In this question you must show detailed reasoning. A function f is given by \(\mathrm { f } ( x ) = \frac { x - 4 } { ( x + 2 ) ( x - 1 ) } + \frac { 3 x + 1 } { ( x + 3 ) ( x - 1 ) }\).
  1. Show that \(\mathrm { f } ( x )\) can be written as \(\frac { 2 ( 2 x + 5 ) } { ( x + 2 ) ( x + 3 ) }\).
  2. Given that \(\int _ { a } ^ { a + 4 } \mathrm { f } ( x ) \mathrm { d } x = 2 \ln 3\), find the value of the positive constant \(a\).
OCR H240/01 2018 December Q12
9 marks Challenging +1.2
12
  1. By first writing \(\tan 3 \theta\) as \(\tan ( 2 \theta + \theta )\), show that \(\tan 3 \theta = \frac { 3 \tan \theta - \tan ^ { 3 } \theta } { 1 - 3 \tan ^ { 2 } \theta }\).
  2. Hence show that there are always exactly two different values of \(\theta\) between \(0 ^ { \circ }\) and \(180 ^ { \circ }\) which satisfy the equation \(3 \tan 3 \theta = \tan \theta + k\),
    where \(k\) is a non-zero constant. \section*{END OF QUESTION PAPER} \section*{OCR
    Oxford Cambridge and RSA}
OCR AS Pure 2017 Specimen Q1
4 marks Easy -1.3
1 Given that \(\mathrm { f } ( x ) = 6 x ^ { 3 } - 5 x\), find
  1. \(\mathrm { f } ^ { \prime } ( x )\),
  2. \(f ^ { \prime \prime } ( 2 )\).
OCR AS Pure 2017 Specimen Q2
5 marks Moderate -0.8
2 Points \(A\) and \(B\) have coordinates \(( 3,0 )\) and \(( 9,8 )\) respectively. The line \(A B\) is a diameter of a circle.
  1. Find the coordinates of the centre of the circle.
  2. Find the equation of the tangent to the circle at the point \(B\).
OCR AS Pure 2017 Specimen Q3
4 marks Moderate -0.3
3 The points \(P , Q\) and \(R\) have coordinates \(( - 1,6 ) , ( 2,10 )\) and \(( 11,1 )\) respectively. Find the angle \(P R Q\).
OCR AS Pure 2017 Specimen Q4
7 marks Moderate -0.8
4 The curve \(y = 2 x ^ { 3 } + 3 x ^ { 2 } - k x + 4\) has a stationary point where \(x = 2\).
  1. Determine the value of the constant \(k\).
  2. Determine whether this stationary point is a maximum or a minimum point.
OCR AS Pure 2017 Specimen Q5
9 marks Moderate -0.8
5
  1. Find \(\int \left( x ^ { 3 } - 6 x \right) \mathrm { d } x\).
    1. Find \(\int \left( \frac { 4 } { x ^ { 2 } } - 1 \right) \mathrm { d } x\).
    2. The diagram shows part of the curve \(y = \frac { 4 } { x ^ { 2 } } - 1\). \includegraphics[max width=\textwidth, alt={}, center]{35d8bb6d-ff0f-4590-b13d-46e4869e2587-04_707_1283_708_415} The curve crosses the \(x\)-axis at \(( 2,0 )\).
      The shaded region is bounded by the curve, the \(x\)-axis, and the lines \(x = 1\) and \(x = 5\). Calculate the area of the shaded region.
OCR AS Pure 2017 Specimen Q6
9 marks Standard +0.3
6 In this question you must show detailed reasoning. The cubic polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 4 x ^ { 3 } + 4 x ^ { 2 } + 7 x - 5\).
  1. Show that \(( 2 x - 1 )\) is a factor of \(\mathrm { f } ( x )\).
  2. Hence solve the equation \(4 \sin ^ { 3 } \theta + 4 \sin ^ { 2 } \theta + 7 \sin \theta - 5 = 0\) for \(0 ^ { \circ } \leq \theta \leq 360 ^ { \circ }\).
OCR AS Pure 2017 Specimen Q7
8 marks Moderate -0.8
7
  1. Sketch the curve \(y = 2 x ^ { 2 } - x - 3\).
  2. Hence, or otherwise, solve \(2 x ^ { 2 } - x - 3 < 0\).
  3. Given that the equation \(2 x ^ { 2 } - x - 3 = k\) has no real roots, find the set of possible values of k .
OCR AS Pure 2017 Specimen Q8
3 marks Easy -1.8
8 A club secretary wishes to survey a sample of members of his club. He uses all members present at a particular meeting as his sample.
  1. Explain why this sample is likely to be biased. Later the secretary decides to choose a random sample of members.
    The club has 253 members and the secretary numbers the members from 1 to 253 . He then generates random 3-digit numbers on his calculator. The first six random numbers generated are 156, 965, 248, 156, 073 and 181. The secretary uses each number, where possible, as the number of a member in the sample.
  2. Find possible numbers for the first four members in the sample.
OCR AS Pure 2017 Specimen Q9
3 marks Standard +0.3
9 The probability distribution of a random variable \(X\) is given in the table.
\(x\)123
\(\mathrm { P } ( X = x )\)0.60.30.1
Two values of \(X\) are chosen at random. Find the probability that the second value is greater than the first.
OCR AS Pure 2017 Specimen Q10
7 marks Moderate -0.8
10
  1. Write down and simplify the first four terms in the expansion of \(( x + y ) ^ { 7 }\).
    Give your answer in ascending powers of \(x\).
  2. Given that the terms in \(x ^ { 2 } y ^ { 5 }\) and \(x ^ { 3 } y ^ { 4 }\) in this expansion are equal, find the value of \(\frac { x } { y }\).
  3. A hospital consultant has seven appointments every day. The number of these appointments which start late on a randomly chosen day is denoted by \(L\).
    The variable \(L\) is modelled by the distribution \(\mathrm { B } \left( 7 , \frac { 3 } { 8 } \right)\). Show that, in this model, the hospital consultant is equally likely to have two appointments start late or three appointments start late.
OCR AS Pure 2017 Specimen Q11
4 marks Easy -1.2
11 The scatter diagram below shows data taken from the 2011 UK census for each of the Local Authorities in the North East and North West regions.
The scatter diagram shows the total population of the Local Authority and the proportion of its workforce that travel to work by bus, minibus or coach. \includegraphics[max width=\textwidth, alt={}, center]{35d8bb6d-ff0f-4590-b13d-46e4869e2587-07_938_1136_664_260}
  1. Samuel suggests that, with a few exceptions, the data points in the diagram show that Local Authorities with larger populations generally have higher proportions of workers travelling by bus, minibus or coach. On the diagram in the Printed Answer Booklet draw a ring around each of the data points that Samuel might regard as an exception.
  2. Jasper suggests that it is possible to separate these Local Authorities into more than one group with different relationships between population and proportion travelling to work by bus, minibus or coach. Discuss Jasper's suggestion, referring to the data and to how differences between the Local Authorities could explain the patterns seen in the diagram.
    [0pt] [3]
OCR AS Pure 2017 Specimen Q12
8 marks Standard +0.3
12 It is known that under the standard treatment for a certain disease, \(9.7 \%\) of patients with the disease experience side effects within one year. In a trial of a new treatment, 450 patients with this disease were selected and the number, \(X\), that experienced side effects within one year was noted. It was found that 51 of the 450 patients experienced side effects within one year.
  1. Test, at the \(10 \%\) significance level, whether the proportion of patients experiencing side effects within one year is greater under the new treatment than under the standard treatment.
  2. It was later discovered that all 450 patients selected for the trial were treated in the same hospital. Comment on the validity of the model used in part (a).
OCR Further Pure Core 1 2017 Specimen Q8
8 marks Standard +0.8
8
  1. Find the solution to the following simultaneous equations. $$\begin{array} { r r r } x + y + & z = & 3 \\ 2 x + 4 y + 5 z = & 9 \\ 7 x + 11 y + 12 z = & 20 \end{array}$$
  2. Determine the values of \(p\) and \(k\) for which there are an infinity of solutions to the following simultaneous equations. $$\begin{array} { r r r r } x + & y + & z = & 3 \\ 2 x + & 4 y + & 5 z = & 9 \\ 7 x + & 11 y + & p z = & k \end{array}$$
OCR Further Pure Core 1 2017 Specimen Q10
10 marks Standard +0.3
10 The Argand diagram below shows the origin \(O\) and pentagon \(A B C D E\), where \(A , B , C , D\) and \(E\) are the points that represent the complex numbers \(a , b , c , d\) and \(e\), and where \(a\) is a positive real number. You are given that these five complex numbers are the roots of the equation \(z ^ { 5 } - a ^ { 5 } = 0\). \includegraphics[max width=\textwidth, alt={}, center]{bc258133-b0d6-49bb-96a7-a5ef7f9c31fc-04_885_851_482_516}
  1. Justify each of the following statements.
    1. \(A , B , C , D\) and \(E\) lie on a circle with centre \(O\).
    2. \(A B C D E\) is a regular pentagon.
    3. \(b \times \mathrm { e } ^ { \frac { 2 \mathrm { i } \pi } { 5 } } = c\)
    4. \(b ^ { * } = e\)
    5. \(a + b + c + d + e = 0\)
    6. The midpoints of sides \(A B , B C , C D , D E\) and \(E A\) represent the complex numbers \(p , q , r , s\) and \(t\). Determine a polynomial equation, with real coefficients, that has roots \(p , q , r , s\) and \(t\).
OCR Further Pure Core 2 2017 Specimen Q2
4 marks Standard +0.3
2 In this question you must show detailed reasoning. The finite region \(R\) is enclosed by the curve with equation \(y = \frac { 8 } { \sqrt { 16 + x ^ { 2 } } }\), the \(x\)-axis and the lines \(x = 0\) and \(x = 4\). Region \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Find the exact value of the volume generated. [4]
OCR Further Pure Core 2 2017 Specimen Q3
4 marks Standard +0.3
3
  1. Find \(\sum _ { r = 1 } ^ { n } \left( \frac { 1 } { r } - \frac { 1 } { r + 2 } \right)\).
  2. What does the sum in part (i) tend to as \(n \rightarrow \infty\) ? Justify your answer.
OCR Further Pure Core 2 2017 Specimen Q4
5 marks Standard +0.3
4 Express \(\frac { 5 x ^ { 2 } + x + 12 } { x ^ { 3 } + 4 x }\) in partial fractions.
OCR Further Pure Core 2 2017 Specimen Q5
4 marks Standard +0.8
5 In this question you must show detailed reasoning. Evaluate \(\int _ { 0 } ^ { \infty } 2 x \mathrm { e } ^ { - x } \mathrm {~d} x\).
[0pt] [You may use the result \(\lim _ { x \rightarrow \infty } x \mathrm { e } ^ { - x } = 0\).]
OCR Further Pure Core 2 2017 Specimen Q6
8 marks Standard +0.3
6 The equation of a plane \(\Pi\) is \(x - 2 y - z = 30\).
  1. Find the acute angle between the line \(\mathbf { r } = \left( \begin{array} { c } 3 \\ 2 \\ - 5 \end{array} \right) + \lambda \left( \begin{array} { r } - 5 \\ 3 \\ 2 \end{array} \right)\) and \(\Pi\).
  2. Determine the geometrical relationship between the line \(\mathbf { r } = \left( \begin{array} { l } 1 \\ 4 \\ 2 \end{array} \right) + \mu \left( \begin{array} { r } 3 \\ - 1 \\ 5 \end{array} \right)\) and \(\Pi\).
OCR Further Pure Core 2 2017 Specimen Q7
7 marks Challenging +1.2
7
  1. Use the Maclaurin series for \(\sin x\) to work out the series expansion of \(\sin x \sin 2 x \sin 4 x\) up to and including the term in \(x ^ { 3 }\).
  2. Hence find, in exact surd form, an approximation to the least positive root of the equation \(2 \sin x \sin 2 x \sin 4 x = x\).
OCR Further Mechanics 2017 Specimen Q1
9 marks Standard +0.8
1 A body, \(P\), of mass 2 kg moves under the action of a single force \(\mathbf { F } \mathrm { N }\). At time \(t \mathrm {~s}\), the velocity of the body is \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\), where $$\mathbf { v } = \left( t ^ { 2 } - 3 \right) \mathbf { i } + \frac { 5 } { 2 t + 1 } \mathbf { j } \text { for } t \geq 2 .$$
  1. Obtain \(\mathbf { F }\) in terms of \(t\).
  2. Calculate the rate at which the force \(\mathbf { F }\) is working at \(t = 4\).
  3. By considering the change in kinetic energy of \(P\), calculate the work done by the force \(\mathbf { F }\) during the time interval \(2 \leq t \leq 4\).
OCR Further Mechanics 2017 Specimen Q3
5 marks Standard +0.3
3 A body, \(Q\), of mass 2 kg moves in a straight line under the action of a single force which acts in the direction of motion of \(Q\). Initially the speed of \(Q\) is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At time \(t \mathrm {~s}\), the magnitude \(F N\) of the force is given by $$F = t ^ { 2 } + 3 \mathrm { e } ^ { t } , \quad 0 \leq t \leq 4 .$$
  1. Calculate the impulse of the force over the time interval.
  2. Hence find the speed of \(Q\) when \(t = 4\).