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OCR Further Pure Core AS 2020 November Q8
8 marks Challenging +1.8
Two loci, \(C_1\) and \(C_2\), are defined by $$C_1 = \{z:|z| = |z - 4d^2 - 36|\}$$ $$C_2 = \left\{z:\arg(z - 12d - 3\text{i}) = \frac{1}{4}\pi\right\}$$ where \(d\) is a real number.
  1. Find, in terms of \(d\), the complex number which is represented on an Argand diagram by the point of intersection of \(C_1\) and \(C_2\). [You may assume that \(C_1 \cap C_2 \neq \emptyset\).] [6]
  2. Explain why the solution found in part (a) is not valid when \(d = 3\). [2]
OCR Further Statistics AS Specimen Q1
5 marks Moderate -0.8
Two music critics, \(P\) and \(Q\), give scores to seven concerts as follows.
Concert1234567
Score by critic \(P\)1211613171614
Score by critic \(Q\)913814181620
  1. Calculate Spearman's rank correlation coefficient, \(r_s\), for these scores. [4]
  2. Without carrying out a hypothesis test, state what your answer tells you about the views of the two critics. [1]
OCR Further Statistics AS Specimen Q2
7 marks Standard +0.8
The probability distribution of a discrete random variable \(W\) is given in the table.
\(w\)0123
\(\mathrm{P}(W = w)\)0.190.18\(x\)\(y\)
Given that \(\mathrm{E}(W) = 1.61\), find the value of \(\mathrm{Var}(3W + 2)\). [7]
OCR Further Statistics AS Specimen Q3
8 marks Standard +0.3
Carl believes that the proportions of men and women who own black cars are different. He obtained a random sample of people who each owned exactly one car. The results are summarised in the table below.
BlackNon-black
Men6971
Women3055
Test at the 5\% significance level whether Carl's belief is justified. [8]
OCR Further Statistics AS Specimen Q4
6 marks Challenging +1.2
  1. Four men and four women stand in a random order in a straight line. Determine the probability that no one is standing next to a person of the same gender. [3]
  2. \(x\) men, including Mr Adam, and \(x\) women, including Mrs Adam, are arranged at random in a straight line. Show that the probability that Mr Adam is standing next to Mrs Adam is \(\frac{1}{x}\). [3]
OCR Further Statistics AS Specimen Q5
7 marks Standard +0.3
  1. The random variable \(X\) has the distribution \(\mathrm{Geo}(0.6)\).
    1. Find \(\mathrm{P}(X \geq 8)\). [2]
    2. Find the value of \(\mathrm{E}(X)\). [1]
    3. Find the value of \(\mathrm{Var}(X)\). [1]
  2. The random variable \(Y\) has the distribution \(\mathrm{Geo}(p)\). It is given that \(\mathrm{P}(Y < 4) = 0.986\) correct to 3 significant figures. Use an algebraic method to find the value of \(p\). [3]
OCR Further Statistics AS Specimen Q6
13 marks Moderate -0.3
Sabrina counts the number of cars passing her house during randomly chosen one minute intervals. Two assumptions are needed for the number of cars passing her house in a fixed time interval to be well modelled by a Poisson distribution.
  1. State these two assumptions. [2]
  2. For each assumption in part (i) give a reason why it might not be a reasonable assumption for this context. [2]
Assume now that the number of cars that pass Sabrina's house in one minute can be well modelled by the distribution \(\mathrm{Po}(0.8)\).
    1. Write down an expression for the probability that, in a given one minute period, exactly \(r\) cars pass Sabrina's house. [1]
    2. Hence find the probability that, in a given one minute period, exactly 2 cars pass Sabrina's house. [1]
  2. Find the probability that, in a given 30 minute period, at least 28 cars pass Sabrina's house. [3]
  3. The number of bicycles that pass Sabrina's house in a 5 minute period is a random variable with the distribution \(\mathrm{Po}(1.5)\). Find the probability that, in a given 10 minute period, the total number of cars and bicycles that pass Sabrina's house is between 12 and 15 inclusive. State a necessary condition. [4]
OCR Further Statistics AS Specimen Q7
4 marks Standard +0.3
The discrete random variable \(X\) is equally likely to take values 0, 1 and 2. \(3N\) observations of \(X\) are obtained, and the observed frequencies corresponding to \(X = 0\), \(X = 1\) and \(X = 2\) are given in the following table.
\(x\)012
Observed frequency\(N - 1\)\(N - 1\)\(N + 2\)
The test statistic for a chi-squared goodness of fit test for the data is 0.3. Find the value of \(N\). [4]
OCR Further Statistics AS Specimen Q8
10 marks Standard +0.3
The following table gives the mean per capita consumption of mozzarella cheese per annum, \(x\) pounds, and the number of civil engineering doctorates awarded, \(y\), in the United States in each of 10 years.
\(x\)9.39.79.79.79.910.210.511.010.610.6
\(y\)480501540552547622655701712708
source: www.tylervigen.com
  1. Find the equation of the regression line of \(y\) on \(x\). [2]
You are given that the product moment correlation coefficient is 0.959.
  1. Explain whether this value would be different if \(x\) is measured in kilograms instead of pounds. [1]
It is desired to carry out a hypothesis test to investigate whether there is correlation between these two variables.
  1. Assume that the data is a random sample of all years.
    1. Carry out the test at the 10\% significance level. [6]
    2. Explain whether your conclusion suggests that manufacturers of mozzarella cheese could increase consumption by sponsoring doctoral candidates in civil engineering. [1]
OCR Further Mechanics AS Specimen Q1
6 marks Moderate -0.8
A roundabout in a playground can be modeled as a horizontal circular platform with centre \(O\). The roundabout is free to rotate about a vertical axis through \(O\). A child sits without slipping on the roundabout at a horizontal distance of 1.5 m from \(O\) and completes one revolution in 2.4 seconds.
  1. Calculate the speed of the child. [3]
  2. Find the magnitude and direction of the acceleration of the child. [3]
OCR Further Mechanics AS Specimen Q2
7 marks Standard +0.3
\includegraphics{figure_2} A smooth wire is shaped into a circle of centre \(O\) and radius 0.8 m. The wire is fixed in a vertical plane. A small bead \(P\) of mass 0.03 kg is threaded on the wire and is projected along the wire from the highest point with a speed of \(4.2 \, \text{m s}^{-1}\). When \(OP\) makes an angle \(\theta\) with the upward vertical the speed of \(P\) is \(v \, \text{m s}^{-1}\) (see diagram).
  1. Show that \(v^2 = 33.32 - 15.68\cos\theta\). [4]
  2. Prove that the bead is never at rest. [1]
  3. Find the maximum value of \(v\). [2]
OCR Further Mechanics AS Specimen Q3
9 marks Standard +0.3
  1. Write down the dimension of density. [1]
The workings of an oil pump consist of a right, solid cylinder which is partially submerged in oil. The cylinder is free to oscillate along its central axis which is vertical. If the base area of the pump is \(0.4 \, \text{m}^2\) and the density of the oil is \(920 \, \text{kg m}^{-3}\) then the period of oscillation of the pump is 0.7 s. A student assumes that the period of oscillation of the pump is dependent only on the density of the oil, \(\rho\), the acceleration due to gravity, \(g\), and the surface area, \(A\), of the circular base of the pump. The student attempts to test this assumption by stating that the period of oscillation, \(T\), is given by \(T = C\rho^{\alpha} g^{\beta} A^{\gamma}\) where \(C\) is a dimensionless constant.
  1. Use dimensional analysis to find the values of \(\alpha\), \(\beta\) and \(\gamma\). [4]
  2. Hence give the value of \(C\) to 3 significant figures. [2]
  3. Comment, with justification, on the assumption made by the student that the formula for the period of oscillation of the pump was dependent on only \(\rho\), \(g\) and \(A\). [2]
OCR Further Mechanics AS Specimen Q4
10 marks Standard +0.8
A car of mass 1250 kg experiences a resistance to its motion of magnitude \(kv^2\) N, where \(k\) is a constant and \(v \, \text{m s}^{-1}\) is the car's speed. The car travels in a straight line along a horizontal road with its engine working at a constant rate of \(P\) W. At a point \(A\) on the road the car's speed is \(15 \, \text{m s}^{-1}\) and it has an acceleration of magnitude \(0.54 \, \text{m s}^{-2}\). At a point \(B\) on the road the car's speed is \(20 \, \text{m s}^{-1}\) and it has an acceleration of magnitude \(0.3 \, \text{m s}^{-2}\).
  1. Find the values of \(k\) and \(P\). [7]
The power is increased to 15 kW.
  1. Calculate the maximum steady speed of the car on a straight horizontal road. [3]
OCR Further Mechanics AS Specimen Q5
15 marks Standard +0.8
\includegraphics{figure_5} The masses of two spheres \(A\) and \(B\) are \(3m\) kg and \(m\) kg respectively. The spheres are moving towards each other with constant speeds \(2u \, \text{m s}^{-1}\) and \(u \, \text{m s}^{-1}\) respectively along the same straight line towards each other on a smooth horizontal surface (see diagram). The two spheres collide and the coefficient of restitution between the spheres is \(e\). After colliding, \(A\) and \(B\) both move in the same direction with speeds \(v \, \text{m s}^{-1}\) and \(w \, \text{m s}^{-1}\), respectively.
  1. Find an expression for \(v\) in terms of \(e\) and \(u\). [6]
  2. Write down unsimplified expressions in terms of \(e\) and \(u\) for
    1. the total kinetic energy of the spheres before the collision, [1]
    2. the total kinetic energy of the spheres after the collision. [2]
  3. Given that the total kinetic energy of the spheres after the collision is \(\lambda\) times the total kinetic energy before the collision, show that $$\lambda = \frac{27e^2 + 25}{52}.$$ [3]
  4. Comment on the cases when
    1. \(\lambda = 1\),
    2. \(\lambda = \frac{25}{52}\). [3]
OCR Further Mechanics AS Specimen Q6
13 marks Challenging +1.2
\includegraphics{figure_6} The fixed points \(A\), \(B\) and \(C\) are in a vertical line with \(A\) above \(B\) and \(B\) above \(C\). A particle \(P\) of mass 2.5 kg is joined to \(A\), to \(B\) and to a particle \(Q\) of mass 2 kg, by three light rods where the length of rod \(AP\) is 1.5 m and the length of rod \(PQ\) is 0.75 m. Particle \(P\) moves in a horizontal circle with centre \(B\). Particle \(Q\) moves in a horizontal circle with centre \(C\) at the same constant angular speed \(\omega\) as \(P\), in such a way that \(A\), \(B\), \(P\) and \(Q\) are coplanar. The rod \(AP\) makes an angle of \(60°\) with the downward vertical, rod \(PQ\) makes an angle of \(30°\) with the downward vertical and rod \(BP\) is horizontal (see diagram).
  1. Find the tension in the rod \(PQ\). [2]
  2. Find \(\omega\). [3]
  3. Find the speed of \(P\). [1]
  4. Find the tension in the rod \(AP\). [3]
  5. Hence find the magnitude of the force in rod \(BP\). Decide whether this rod is under tension or compression. [4]
OCR Further Pure Core 1 2021 November Q1
6 marks Moderate -0.8
  1. Sketch on a single Argand diagram the loci given by
    1. \(|z - 1 + 2\mathrm{i}| = 3\), [2]
    2. \(|z + 1| = |z - 2|\). [2]
  2. Indicate, by shading, the region of the Argand diagram for which \(|z - 1 + 2\mathrm{i}| \leqslant 3\) and \(|z + 1| \leqslant |z - 2|\). [2]
OCR Further Pure Core 1 2021 November Q2
8 marks Standard +0.3
You are given that \(\mathrm{f}(x) = \tan^{-1}(1 + x)\).
    1. Find the value of \(\mathrm{f}(0)\). [1]
    2. Determine the value of \(\mathrm{f}'(0)\). [2]
    3. Show that \(\mathrm{f}''(0) = -\frac{1}{2}\). [3]
  2. Hence find the Maclaurin series for \(\mathrm{f}(x)\) up to and including the term in \(x^2\). [2]
OCR Further Pure Core 1 2021 November Q3
8 marks Standard +0.8
A function \(\mathrm{f}(z)\) is defined on all complex numbers \(z\) by \(\mathrm{f}(z) = z^3 - 3z^2 + kz - 5\) where \(k\) is a real constant. The roots of the equation \(\mathrm{f}(z) = 0\) are \(\alpha\), \(\beta\) and \(\gamma\). You are given that \(\alpha^2 + \beta^2 + \gamma^2 = -5\).
  1. Explain why \(\mathrm{f}(z) = 0\) has only one real root. [3]
  2. Find the value of \(k\). [3]
  3. Find a cubic equation with integer coefficients that has roots \(\frac{1}{\alpha}\), \(\frac{1}{\beta}\) and \(\frac{1}{\gamma}\). [2]
OCR Further Pure Core 1 2021 November Q4
11 marks Standard +0.3
Points \(A\), \(B\) and \(C\) have coordinates \((4, 2, 0)\), \((1, 5, 3)\) and \((1, 4, -2)\) respectively. The line \(l\) passes through \(A\) and \(B\).
  1. Find a cartesian equation for \(l\). [3]
\(M\) is the point on \(l\) that is closest to \(C\).
  1. Find the coordinates of \(M\). [4]
  2. Find the exact area of the triangle \(ABC\). [4]
OCR Further Pure Core 1 2021 November Q5
4 marks Standard +0.8
Use de Moivre's theorem to find the constants \(A\), \(B\) and \(C\) in the identity \(\sin^3 \theta \equiv A \sin \theta + B \sin 3\theta + C \sin 5\theta\). [4]
OCR Further Pure Core 1 2021 November Q6
3 marks Standard +0.8
\(O\) is the origin of a coordinate system whose units are cm. The points \(A\), \(B\), \(C\) and \(D\) have coordinates \((1, 0)\), \((1, 4)\), \((6, 9)\) and \((0, 9)\) respectively. The arc \(BC\) is part of the curve with equation \(x^2 + (y - 10)^2 = 37\). The closed shape \(OABCD\) is formed, in turn, from the line segments \(OA\) and \(AB\), the arc \(BC\) and the line segments \(CD\) and \(DO\) (see diagram). A funnel can be modelled by rotating \(OABCD\) by \(2\pi\) radians about the \(y\)-axis. \includegraphics{figure_6} Find the volume of the funnel according to the model. [3]
OCR Further Pure Core 1 2021 November Q7
9 marks Standard +0.8
The diagram below shows the curve with polar equation \(r = \sin 3\theta\) for \(0 \leqslant \theta \leqslant \frac{1}{3}\pi\). \includegraphics{figure_7}
  1. Find the values of \(\theta\) at the pole. [1]
  2. Find the polar coordinates of the point on the curve where \(r\) takes its maximum value. [2]
  3. In this question you must show detailed reasoning. Find the exact area enclosed by the curve. [4]
  4. Given that \(\sin 3\theta = 3 \sin \theta - 4 \sin^3 \theta\), find a cartesian equation for the curve. [2]
OCR Further Pure Core 1 2021 November Q8
8 marks Standard +0.3
You are given that \(\mathrm{f}(x) = 4 \sinh x + 3 \cosh x\).
  1. Show that the curve \(y = \mathrm{f}(x)\) has no turning points. [3]
  2. Determine the exact solution of the equation \(\mathrm{f}(x) = 5\). [5]
OCR Further Pure Core 1 2021 November Q9
5 marks Standard +0.3
You are given that the matrix \(\begin{pmatrix} 2 & 1 \\ -1 & 0 \end{pmatrix}\) represents a transformation T.
  1. You are given that the line with equation \(y = kx\) is invariant under T. Determine the value of \(k\). [4]
  2. Determine whether the line with equation \(y = kx\) in part (a) is a line of invariant points under T. [1]
OCR Further Pure Core 1 2021 November Q10
8 marks Challenging +1.2
Using an algebraic method, determine the least value of \(n\) for which \(\sum_{r=1}^{n} \frac{1}{(2r-1)(2r+1)} \geqslant 0.49\). [8]