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OCR Further Pure Core 2 2018 December Q10
14 marks Standard +0.8
A swing door is a door to a room which is closed when in equilibrium but which can be pushed open from either side and which can swing both ways, into or out of the room, and through the equilibrium position. The door is sprung so that when displaced from the equilibrium position it will swing back towards it. The extent to which the door is open at any time, \(t\) seconds, is measured by the angle at the hinge, \(\theta\), which the plane of the door makes with the plane of the equilibrium position. See the diagram below.
[diagram]
In an initial model of the motion of a certain swing door it is suggested that \(\theta\) satisfies the following differential equation. $$4\frac{\mathrm{d}^2\theta}{\mathrm{d}t^2} + 25\theta = 0 \quad (*)$$
    1. Write down the general solution to (*). [2]
    2. With reference to the behaviour of your solution in part (a)(i) explain briefly why the model using (*) is unlikely to be realistic. [1]
In an improved model of the motion of the door an extra term is introduced to the differential equation so that it becomes $$4\frac{\mathrm{d}^2\theta}{\mathrm{d}t^2} + \lambda\frac{\mathrm{d}\theta}{\mathrm{d}t} + 25\theta = 0 \quad (\dagger)$$ where \(\lambda\) is a positive constant.
  1. In the case where \(\lambda = 16\) the door is held open at an angle of \(0.9\) radians and then released from rest at time \(t = 0\).
    1. Find, in a real form, the general solution of (\(\dagger\)). [3]
    2. Find the particular solution of (\(\dagger\)). [4]
    3. With reference to the behaviour of your solution found in part (b)(ii) explain briefly how the extra term in (\(\dagger\)) improves the model. [2]
  2. Find the value of \(\lambda\) for which the door is critically damped. [2]
OCR H240/03 2018 March Q1
4 marks Easy -1.2
Show in a sketch the region of the \(x\)-\(y\) plane within which all three of the following inequalities are satisfied. $$3y \geqslant 4x \qquad y - x \leqslant 1 \qquad y \geqslant (x-1)^2$$ You should indicate the region for which the inequalities hold by labelling the region R. [4]
OCR H240/03 2018 March Q2
8 marks Moderate -0.3
The first term of a geometric progression is 12 and the second term is 9.
  1. Find the fifth term. [3]
The sum of the first \(N\) terms is denoted by \(S_N\) and the sum to infinity is denoted by \(S_\infty\). It is given that the difference between \(S_\infty\) and \(S_N\) is at most 0.0096.
  1. Show that \(\left(\frac{3}{4}\right)^N \leqslant 0.0002\). [3]
  2. Use logarithms to find the smallest possible value of \(N\). [2]
OCR H240/03 2018 March Q3
4 marks Hard +2.5
A sequence of three transformations maps the curve \(y = \ln x\) to the curve \(y = \mathrm{e}^{3x} - 5\). Give details of these transformations. [4]
OCR H240/03 2018 March Q4
11 marks Standard +0.3
A curve is defined, for \(t \geqslant 0\), by the parametric equations $$x = t^2, \quad y = t^3.$$
  1. Show that the equation of the tangent at the point with parameter \(t\) is $$2y = 3tx - t^3.$$ [4]
  1. In this question you must show detailed reasoning. It is given that this tangent passes through the point \(A\left(\frac{19}{2}, -\frac{15}{8}\right)\) and it meets the \(x\)-axis at the point \(B\). Find the area of triangle \(OAB\), where \(O\) is the origin. [7]
OCR H240/03 2018 March Q5
14 marks Standard +0.8
In this question you must show detailed reasoning. \includegraphics{figure_5} The function f is defined for the domain \(x \geqslant 0\) by $$\mathrm{f}(x) = (2x^2 - 3x)\mathrm{e}^{-x}.$$ The diagram shows the curve \(y = \mathrm{f}(x)\).
  1. Find the range of f. [6]
  1. The function g is defined for the domain \(x \geqslant k\) by $$\mathrm{g}(x) = (2x^2 - 3x)\mathrm{e}^{-x}.$$ Given that g is a one-one function, state the least possible value of \(k\). [1]
  1. Find the exact area of the shaded region enclosed by the curve and the \(x\)-axis. [7]
OCR H240/03 2018 March Q6
10 marks Standard +0.3
  1. Determine the values of \(p\) and \(q\) for which $$x^2 - 6x + 10 \equiv (x - p)^2 + q.$$ [2]
  1. Use the substitution \(x - p = \tan u\), where \(p\) takes the value found in part (i), to evaluate $$\int_3^4 \frac{1}{x^2 - 6x + 10} \, dx.$$ [3]
  1. Determine the value of $$\int_3^4 \frac{x}{x^2 - 6x + 10} \, dx,$$ giving your answer in the form \(a + \ln b\), where \(a\) and \(b\) are constants to be determined. [5]
OCR H240/03 2018 March Q7
3 marks Moderate -0.8
Three forces \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\) acting on a particle are given by $$\mathbf{F}_1 = (3\mathbf{i} - 2a\mathbf{j})\text{N}, \quad \mathbf{F}_2 = (2b\mathbf{i} + 3a\mathbf{j})\text{N} \quad \text{and} \quad \mathbf{F}_3 = (-2\mathbf{i} + b\mathbf{j})\text{N}.$$ The particle is in equilibrium under the action of these three forces. Find the value of \(a\) and the value of \(b\). [3]
OCR H240/03 2018 March Q8
11 marks Standard +0.3
A jogger is running along a straight horizontal road. The jogger starts from rest and accelerates at a constant rate of \(0.4\,\text{m}\,\text{s}^{-2}\) until reaching a velocity of \(V\,\text{m}\,\text{s}^{-1}\). The jogger then runs at a constant velocity of \(V\,\text{m}\,\text{s}^{-1}\) before decelerating at a constant rate of \(0.08\,\text{m}\,\text{s}^{-2}\) back to rest. The jogger runs a total distance of \(880\,\text{m}\) in \(250\,\text{s}\).
  1. Sketch the velocity-time graph for the jogger's journey. [2]
  2. Show that \(3V^2 - 100V + 352 = 0\). [6]
  3. Hence find the value of \(V\), giving a reason for your answer. [3]
OCR H240/03 2018 March Q9
14 marks Standard +0.8
Two particles \(A\) and \(B\) have position vectors \(\mathbf{r}_A\) metres and \(\mathbf{r}_B\) metres at time \(t\) seconds, where $$\mathbf{r}_A = t^2\mathbf{i} + (3t - 1)\mathbf{j} \quad \text{and} \quad \mathbf{r}_B = (1 - 2t^2)\mathbf{i} + (3t - 2t^2)\mathbf{j}, \quad \text{for } t \geqslant 0.$$
  1. Find the values of \(t\) when \(A\) and \(B\) are moving with the same speed. [5]
  2. Show that the distance, \(d\) metres, between \(A\) and \(B\) at time \(t\) satisfies $$d^2 = 13t^4 - 10t^2 + 2.$$ [3]
  3. Hence find the shortest distance between \(A\) and \(B\) in the subsequent motion. [6]
OCR H240/03 2018 March Q10
9 marks Standard +0.3
\includegraphics{figure_10} A uniform rod \(AB\), of weight \(W\) N and length \(2a\) m, rests with the end \(A\) on a rough horizontal table. A small object of weight \(2W\) N is attached to the rod at \(B\). The rod is maintained in equilibrium at an angle of \(30°\) to the horizontal by a force acting at \(B\) in a direction perpendicular to the rod in the same vertical plane as the rod (see diagram).
  1. Find the least possible value of the coefficient of friction between the rod and the table. [7]
  2. Given that the magnitude of the contact force at \(A\) is \(\sqrt{39}\) N, find the value of \(W\). [2]
OCR H240/03 2018 March Q11
12 marks Challenging +1.8
In this question you must show detailed reasoning. \includegraphics{figure_11} A football \(P\) is kicked with speed \(25\,\text{m}\,\text{s}^{-1}\) at an angle of elevation \(\alpha\) from a point \(A\) on horizontal ground. At the same instant a second football \(Q\) is kicked with speed \(15\,\text{m}\,\text{s}^{-1}\) at an angle of elevation \(2\alpha\) from a point \(B\) on the same horizontal ground, where \(AB = 72\) m. The footballs are modelled as particles moving freely under gravity in the same vertical plane and they collide with each other at the point \(C\) (see diagram).
  1. Calculate the height of \(C\) above the ground. [7]
  2. Find the direction of motion of \(P\) at the moment of impact. [4]
  3. Suggest one improvement that could be made to the model. [1]
OCR H240/02 2018 December Q1
4 marks Standard +0.3
\includegraphics{figure_1} The diagram shows the curve \(y = \sqrt{x - 3}\). The shaded region is bounded by the curve and the two axes. Find the exact area of the shaded region. [4]
OCR H240/02 2018 December Q2
5 marks Moderate -0.3
\(\text{f}(x)\) is a cubic polynomial in which the coefficient of \(x^3\) is 1. The equation \(\text{f}(x) = 0\) has exactly two roots.
  1. Sketch a possible graph of \(y = \text{f}(x)\). [2]
It is now given that the two roots are \(x = 2\) and \(x = 3\).
  1. Find, in expanded form, the two possible polynomials \(\text{f}(x)\). [3]
OCR H240/02 2018 December Q3
4 marks Easy -1.8
\includegraphics{figure_3} The diagram shows the graph of \(y = \text{g}(x)\). In the printed answer booklet, using the same scale as in this diagram, sketch the curves
  1. \(y = \frac{3}{2}\text{g}(x)\), [2]
  2. \(y = \text{g}\left(\frac{1}{2}x\right)\). [2]
OCR H240/02 2018 December Q4
10 marks Standard +0.3
In this question you must show detailed reasoning.
  1. Show that \(\cos A + \sin A \tan A = \sec A\). [3]
  2. Solve the equation \(\tan 2\theta = 3 \tan \theta\) for \(0° \leqslant \theta \leqslant 180°\). [7]
OCR H240/02 2018 December Q5
8 marks Moderate -0.3
Points \(A\) and \(B\) have position vectors \(\mathbf{a}\) and \(\mathbf{b}\). Point \(C\) lies on \(AB\) such that \(AC : CB = p : 1\).
  1. Show that the position vector of \(C\) is \(\frac{1}{p+1}(\mathbf{a} + p\mathbf{b})\). [3]
It is now given that \(\mathbf{a} = 2\mathbf{i} + 3\mathbf{j} - 4\mathbf{k}\) and \(\mathbf{b} = -6\mathbf{i} + 4\mathbf{j} + 12\mathbf{k}\), and that \(C\) lies on the \(y\)-axis.
  1. Find the value of \(p\). [4]
  2. Write down the position vector of \(C\). [1]
OCR H240/02 2018 December Q6
8 marks Moderate -0.8
The table shows information about three geometric series. The three geometric series have different common ratios.
First termCommon ratioNumber of termsLast term
Series 112\(n_1\)1024
Series 21\(r_2\)\(n_2\)1024
Series 31\(r_3\)\(n_3\)1024
  1. Find \(n_1\). [2]
  2. Given that \(r_2\) is an integer less than 10, find the value of \(r_2\) and the value of \(n_2\). [2]
  3. Given that \(r_3\) is not an integer, find a possible value for the sum of all the terms in Series 3. [4]
OCR H240/02 2018 December Q7
5 marks Standard +0.8
  1. Show that, if \(n\) is a positive integer, then \((x^n - 1)\) is divisible by \((x - 1)\). [1]
  2. Hence show that, if \(k\) is a positive integer, then \(2^{8k} - 1\) is divisible by 17. [4]
OCR H240/02 2018 December Q8
7 marks Challenging +1.8
Use a suitable trigonometric substitution to find \(\int \frac{x^2}{\sqrt{1-x^2}} \text{d}x\). [7]
OCR H240/02 2018 December Q9
7 marks Standard +0.3
Research has shown that drug A is effective in 32% of patients with a certain disease. In a trial, drug B is given to a random sample of 1000 patients with the disease, and it is found that the drug is effective in 290 of these patients. Test at the 2.5% significance level whether there is evidence that drug B is effective in a lower proportion of patients than drug A. [7]
OCR H240/02 2018 December Q10
6 marks Moderate -0.8
Using the 2001 UK census results and some software, Javid intended to calculate the mean number of people who travelled to work by underground, metro, light rail or tram (UMLT) for all 348 Local Authorities. However, Javid noticed that for one LA the entry in the UMLT column is a dash, rather than a 0. See the extract below.
Data extract for one LA in 2001
Work mainly at or from homeUMLTTrainBus, minibus or coach
295–44
Javid felt that it was not clear how this LA was to be treated so he decided to omit it from his calculation.
  1. Explain how the omission of this LA affects Javid's calculation of the mean. [1]
The value of the mean that Javid obtained was 2046.3.
  1. Calculate the value of the mean when this LA is not removed. [2]
Javid finds that the corresponding mean for all Local Authorities for 2011 is 2860.8. In order to compare the means for the two years, Javid also finds the total number of employees in each of these years. His results are given below.
Year20012011
Total number of employees23 627 75326 526 336
  1. Show that a higher proportion of employees used the metro to travel to work in 2011 than in 2001. [2]
  2. Suggest a reason for this increase. [1]
OCR H240/02 2018 December Q11
6 marks Moderate -0.8
Laxmi wishes to test whether there is linear correlation between the mass and the height of adult males.
  1. State, with a reason, whether Laxmi should use a 1-tail or a 2-tail test. [1]
Laxmi chooses a random sample of 40 adult males and calculates Pearson's product-moment correlation coefficient, \(r\). She finds that \(r = 0.2705\).
  1. Use the table below to carry out the test at the 5% significance level. [5]
Critical values of Pearson's product-moment correlation coefficient.
1-tail test2-tail test
5%2.5%1%0.5%
10%5%2.5%1%
380.27090.32020.37600.4128
390.26730.31600.37120.4076
\(n\) 400.26380.31200.36650.4026
410.26050.30810.36210.3978
OCR H240/02 2018 December Q12
7 marks Moderate -0.8
Paul drew a cumulative frequency graph showing information about the numbers of people in various age-groups in a certain region X. He forgot to include the scale on the cumulative frequency axis, as shown below. \includegraphics{figure_12}
  1. Find an estimate of the median age of the population of region X. [1]
  2. Find an estimate of the proportion of people aged over 60 in region X. [2]
Sonika drew similar cumulative graphs for another two regions, Y and Z, but she included the scales on the cumulative frequency axes, as shown below. \includegraphics{figure_12b}
  1. Find an age group, of width 20 years, in which region Z has approximately 3 times as many people as region Y. [1]
  2. State one advantage and one disadvantage of using Sonika's two diagrams to compare the populations in Regions Y and Z. [2]
  3. Without calculation state, with a reason, which of regions Y or Z has the greater proportion of people aged under 40. [1]
OCR H240/02 2018 December Q13
3 marks Moderate -0.8
The marks of 24 students in a test had mean \(m\) and standard deviation \(\sqrt{6}\). Two new students took the same test. Their marks were \(m - 4\) and \(m + 4\). Show that the standard deviation of the marks of all 26 students is 2.60, correct to 3 significant figures. [3]