Questions — OCR (4907 questions)

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OCR H240/03 2023 June Q6
6 marks Standard +0.8
The first, third and fourth terms of an arithmetic progression are \(u_1\), \(u_3\) and \(u_4\) respectively, where $$u_1 = 2 \sin \theta, \quad u_3 = -\sqrt{3} \cos \theta, \quad u_4 = \frac{7}{3} \sin \theta,$$ and \(\frac{1}{2}\pi < \theta < \pi\).
  1. Determine the exact value of \(\theta\). [3]
  2. Hence determine the value of \(\sum_{r=1}^{100} u_r\). [3]
OCR H240/03 2023 June Q7
12 marks Standard +0.8
A car \(C\) is moving horizontally in a straight line with velocity \(v \text{ms}^{-1}\) at time \(t\) seconds, where \(v > 0\) and \(t \geq 0\). The acceleration, \(a \text{ms}^{-2}\), of \(C\) is modelled by the equation $$a = v\left(\frac{8t}{7 + 4t^2} - \frac{1}{2}\right).$$
  1. In this question you must show detailed reasoning. Find the times when the acceleration of \(C\) is zero. [3] At \(t = 0\) the velocity of \(C\) is \(17.5 \text{ms}^{-1}\) and at \(t = T\) the velocity of \(C\) is \(5 \text{ms}^{-1}\).
  2. By setting up and solving a differential equation, show that \(T\) satisfies the equation $$T = 2 \ln\left(\frac{7 + 4T^2}{2}\right).$$ [6]
  3. Use an iterative formula, based on the equation in part (b), to find the value of \(T\), giving your answer correct to 4 significant figures. Use an initial value of 11.25 and show the result of each step of the iteration process. [2]
  4. The diagram below shows the velocity-time graph for the motion of \(C\). \includegraphics{figure_7d} Find the time taken for \(C\) to decelerate from travelling at its maximum speed until it is travelling at \(5 \text{ms}^{-1}\). [1]
OCR H240/03 2023 June Q8
4 marks Moderate -0.8
A particle \(P\) moves with constant acceleration \((3\mathbf{i} - 2\mathbf{j}) \text{ms}^{-2}\). At time \(t = 4\) seconds, \(P\) has velocity \(6\mathbf{i} \text{ms}^{-1}\). Determine the speed of \(P\) at time \(t = 0\) seconds. [4]
OCR H240/03 2023 June Q9
6 marks Challenging +1.2
\includegraphics{figure_9} A block \(B\) of weight \(10 \text{N}\) lies at rest in equilibrium on a rough plane inclined at \(\theta\) to the horizontal. A horizontal force of magnitude \(2 \text{N}\), acting above a line of greatest slope, is applied to \(B\) (see diagram).
  1. Complete the diagram in the Printed Answer Booklet to show all the forces acting on \(B\). [1]
It is given that \(B\) remains at rest and the coefficient of friction between \(B\) and the plane is 0.8.
  1. Determine the greatest possible value of \(\tan \theta\). [5]
OCR H240/03 2023 June Q10
7 marks Standard +0.3
A particle \(P\) of mass \(m \text{kg}\) is moving on a smooth horizontal surface under the action of two constant horizontal forces \((-4\mathbf{i} + 2\mathbf{j}) \text{N}\) and \((a\mathbf{i} + b\mathbf{j}) \text{N}\). The resultant of these two forces is \(\mathbf{R} \text{N}\). It is given that \(\mathbf{R}\) acts in a direction which is parallel to the vector \(-\mathbf{i} + 3\mathbf{j}\).
  1. Show that \(3a + b = 10\). [3]
It is given that \(a = 6\) and that \(P\) moves with an acceleration of magnitude \(5\sqrt{10} \text{ms}^{-2}\).
  1. Determine the value of \(m\). [4]
OCR H240/03 2023 June Q11
8 marks Standard +0.3
\includegraphics{figure_11} A uniform rod \(AB\), of weight \(20 \text{N}\) and length \(2.8 \text{m}\), rests in equilibrium with the end \(A\) in contact with rough horizontal ground and the end \(B\) resting against a smooth wall inclined at \(55°\) to the horizontal. The rod, which rests in a vertical plane that is perpendicular to the wall, is inclined at \(30°\) to the horizontal (see diagram).
  1. Show that the magnitude of the force acting on the rod at \(B\) is \(9.56 \text{N}\), correct to 3 significant figures. [3]
  2. Determine the magnitude of the contact force between the rod and the ground. Give your answer correct to 3 significant figures. [5]
OCR H240/03 2023 June Q12
13 marks Standard +0.8
In this question you should take the acceleration due to gravity to be \(10 \text{ms^{-2}\).} \includegraphics{figure_12} A small ball \(P\) is projected from a point \(A\) with speed \(39 \text{ms}^{-1}\) at an angle of elevation \(\theta\), where \(\sin \theta = \frac{5}{13}\) and \(\cos \theta = \frac{12}{13}\). Point \(A\) is \(20 \text{m}\) vertically above a point \(B\) on horizontal ground. The ball first lands at a point \(C\) on the horizontal ground (see diagram). The ball \(P\) is modelled as a particle moving freely under gravity.
  1. Find the maximum height of \(P\) above the ground during its motion. [3]
The time taken for \(P\) to travel from \(A\) to \(C\) is \(7\) seconds.
  1. Determine the value of \(T\). [3]
  2. State one limitation of the model, other than air resistance or the wind, that could affect the answer to part (b). [1]
At the instant that \(P\) is projected, a second small ball \(Q\) is released from rest at \(B\) and moves towards \(C\) along the horizontal ground. At time \(t\) seconds, where \(t \geq 0\), the velocity \(v \text{ms}^{-1}\) of \(Q\) is given by $$v = kt^3 + 6t^2 + \frac{3}{2}t,$$ where \(k\) is a positive constant.
  1. Given that \(P\) and \(Q\) collide at \(C\), determine the acceleration of \(Q\) immediately before this collision. [6]
OCR H240/03 2023 June Q13
12 marks Challenging +1.2
\includegraphics{figure_13} The diagram shows a small block \(B\), of mass \(2 \text{kg}\), and a particle \(P\), of mass \(4 \text{kg}\), which are attached to the ends of a light inextensible string. The string is taut and passes over a small smooth pulley fixed at the intersection of a horizontal surface and an inclined plane. The particle can move on the inclined plane, which is rough, and which makes an angle of \(60°\) with the horizontal. The block can move on the horizontal surface, which is also rough. The system is released from rest, and in the subsequent motion \(P\) moves down the plane and \(B\) does not reach the pulley. It is given that the coefficient of friction between \(P\) and the inclined plane is twice the coefficient of friction between \(B\) and the horizontal surface.
  1. Determine, in terms of \(g\), the tension in the string. [7]
When \(P\) is moving at \(2 \text{ms}^{-1}\) the string breaks. In the \(0.5\) seconds after the string breaks \(P\) moves \(1.9 \text{m}\) down the plane.
  1. Determine the deceleration of \(B\) after the string breaks. Give your answer correct to 3 significant figures. [5]
OCR PURE Q1
5 marks Easy -1.3
In this question you must show detailed reasoning.
  1. Express \(3^{\frac{1}{2}}\) in the form \(a\sqrt{b}\), where \(a\) is an integer and \(b\) is a prime number. [2]
  2. Express \(\frac{\sqrt{2}}{1-\sqrt{2}}\) in the form \(c + d\sqrt{e}\), where \(c\) and \(d\) are integers and \(e\) is a prime number. [3]
OCR PURE Q2
4 marks Easy -1.2
  1. The equation \(x^2 + 3x + k = 0\) has repeated roots. Find the value of the constant \(k\). [2]
  2. Solve the inequality \(6 + x - x^2 > 0\). [2]
OCR PURE Q3
6 marks Moderate -0.8
  1. Solve the equation \(\sin^2\theta = 0.25\) for \(0° \leq \theta < 360°\). [3]
  2. In this question you must show detailed reasoning. Solve the equation \(\tan 3\phi = \sqrt{3}\) for \(0° \leq \phi < 90°\). [3]
OCR PURE Q4
9 marks Moderate -0.8
  1. It is given that \(y = x^2 + 3x\).
    1. Find \(\frac{dy}{dx}\). [2]
    2. Find the values of \(x\) for which \(y\) is increasing. [2]
  2. Find \(\int(3 - 4\sqrt{x})dx\). [5]
OCR PURE Q5
5 marks Standard +0.8
\(N\) is an integer that is not divisible by 3. Prove that \(N^2\) is of the form \(3p + 1\), where \(p\) is an integer. [5]
OCR PURE Q6
7 marks Easy -1.2
Sketch the following curves.
  1. \(y = \frac{2}{x}\) [2]
  2. \(y = x^3 - 6x^2 + 9x\) [5]
OCR PURE Q7
7 marks Moderate -0.8
\(OABC\) is a parallelogram with \(\overrightarrow{OA} = \mathbf{a}\) and \(\overrightarrow{OC} = \mathbf{c}\). \(P\) is the midpoint of \(AC\). \includegraphics{figure_7}
  1. Find the following in terms of \(\mathbf{a}\) and \(\mathbf{c}\), simplifying your answers.
    1. \(\overrightarrow{AC}\) [1]
    2. \(\overrightarrow{OP}\) [2]
  2. Hence prove that the diagonals of a parallelogram bisect one another. [4]
OCR PURE Q8
6 marks Standard +0.8
In this question you must show detailed reasoning. The lines \(y = \frac{1}{2}x\) and \(y = -\frac{1}{2}x\) are tangents to a circle at \((2, 1)\) and \((-2, 1)\) respectively. Find the equation of the circle in the form \(x^2 + y^2 + ax + by + c = 0\), where \(a\), \(b\) and \(c\) are constants. [6]
OCR PURE Q9
2 marks Easy -2.0
Jo is investigating the popularity of a certain band amongst students at her school. She decides to survey a sample of 100 students.
  1. State an advantage of using a stratified sample rather than a simple random sample. [1]
  2. Explain whether it would be reasonable for Jo to use her results to draw conclusions about all students in the UK. [1]
OCR PURE Q10
5 marks Moderate -0.3
The probability distribution of a random variable \(X\) is given in the table.
\(x\)0246
P\((X = x)\)\(\frac{3}{8}\)\(\frac{5}{16}\)\(4p\)\(p\)
  1. Find the value of \(p\). [2]
  2. Two values of \(X\) are chosen at random. Find the probability that the product of these values is 0. [3]
OCR PURE Q11
5 marks Moderate -0.3
The probability that Janice sees a kingfisher on any particular day is 0.3. She notes the number, \(X\), of days in a week on which she sees a kingfisher.
  1. State one necessary condition for \(X\) to have a binomial distribution. [1]
Assume now that \(X\) has a binomial distribution.
  1. Find the probability that, in a week, Janice sees a kingfisher on exactly 2 days. [1]
Each week Janice notes the number of days on which she sees a kingfisher.
  1. Find the probability that Janice sees a kingfisher on exactly 2 days in a week during at least 4 of 6 randomly chosen weeks. [3]
OCR PURE Q12
7 marks Standard +0.3
It is known that 20% of plants of a certain type suffer from a fungal disease, when grown under normal conditions. Some plants of this type are grown using a new method. A random sample of 250 of these plants is chosen, and it is found that 36 suffer from the disease. Test, at the 2% significance level, whether there is evidence that the new method reduces the proportion of plants which suffer from the disease. [7]
OCR PURE Q13
7 marks Easy -2.5
The radar diagrams illustrate some population figures from the 2011 census results. \includegraphics{figure_13} Each radius represents an age group, as follows:
Radius123456
Age group0-1718-2930-4445-5960-7475+
The distance of each dot from the centre represents the number of people in the relevant age group.
  1. The scales on the two diagrams are different. State an advantage and a disadvantage of using different scales in order to make comparisons between the ages of people in these two Local Authorities. [2]
  2. Approximately how many people aged 45 to 59 were there in Liverpool? [1]
  3. State the main two differences between the age profiles of the two Local Authorities. [2]
  4. James makes the following claim. "Assuming that there are no significant movements of population either into or out of the two regions, the 2021 census results are likely to show an increase in the number of children in Liverpool and a decrease in the number of children in Rutland." Use the radar diagrams to give a justification for this claim. [2]
OCR PURE Q1
5 marks Moderate -0.8
  1. Prove that \(\cos x + \sin x \tan x \equiv \frac{1}{\cos x}\) (where \(x \neq \frac{1}{2}n\pi\) for any odd integer \(n\)). [3]
  2. Solve the equation \(2\sin^2 x = \cos^2 x\) for \(0° \leqslant x \leqslant 180°\). [2]
OCR PURE Q2
8 marks Moderate -0.3
  1. The points \(A\), \(B\) and \(C\) have position vectors \(\begin{pmatrix} -4 \\ 3 \end{pmatrix}\), \(\begin{pmatrix} -3 \\ 6 \end{pmatrix}\) and \(\begin{pmatrix} -1 \\ 12 \end{pmatrix}\) respectively.
    1. Show that \(B\) lies on \(AC\). [2]
    2. Find the ratio \(AB : BC\). [1]
  2. The diagram shows the line \(x + y = 6\) and the point \(P(2, 4)\) that lies on the line. A copy of the diagram is given in the Printed Answer Booklet. \includegraphics{figure_1} The distinct point \(Q\) also lies on the line \(x + y = 6\) and is such that \(|\overrightarrow{OQ}| = |\overrightarrow{OP}|\), where \(O\) is the origin. Find the magnitude and direction of the vector \(\overrightarrow{PQ}\). [3]
  3. The point \(R\) is such that \(\overrightarrow{PR}\) is perpendicular to \(\overrightarrow{OP}\) and \(|\overrightarrow{PR}| = \frac{1}{2}|\overrightarrow{OP}|\). Write down the position vectors of the two possible positions of the point \(R\). [2]
OCR PURE Q3
4 marks Moderate -0.8
The diagram shows the graph of \(y = f(x)\), where \(f(x)\) is a quadratic function of \(x\). A copy of the diagram is given in the Printed Answer Booklet. \includegraphics{figure_2}
  1. On the copy of the diagram in the Printed Answer Booklet, draw a possible graph of the gradient function \(y = f'(x)\). [3]
  2. State the gradient of the graph of \(y = f''(x)\). [1]
OCR PURE Q4
4 marks Easy -1.2
A curve has equation \(y = e^{3x}\).
  1. Determine the value of \(x\) when \(y = 10\). [2]
  2. Determine the gradient of the tangent to the curve at the point where \(x = 2\). [2]