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OCR Further Pure Core 1 2021 November Q11
5 marks Standard +0.3
The displacement of a door from its equilibrium (closed) position is measured by the angle, \(\theta\) radians, which the door makes with its closed position. The door can swing either side of the equilibrium position so that \(\theta\) can take positive and negative values. The door is released from rest from an open position at time \(t = 0\). A proposed differential equation to model the motion of the door for \(t \geqslant 0\) is $$\frac{\mathrm{d}^2\theta}{\mathrm{d}t^2} + \lambda \frac{\mathrm{d}\theta}{\mathrm{d}t} + 3\theta = 0$$ where \(\lambda\) is a constant and \(\lambda \geqslant 0\).
    1. According to the model, for what value of \(\lambda\) will the motion of the door be simple harmonic? [1]
    2. Explain briefly why modelling the motion of the door as simple harmonic is unlikely to be realistic. [1]
  2. Find the range of values of \(\lambda\) for which the model predicts that the door will never pass through the equilibrium position. [2]
  3. Sketch a possible graph of \(\theta\) against \(t\) when \(\lambda\) lies outside the range found in part (b) but the motion is not simple harmonic. [1]
OCR Further Pure Core 2 2024 June Q1
5 marks Moderate -0.8
  1. Use the method of differences to show that \(\sum_{r=1}^{n}\left(\frac{1}{r} - \frac{1}{r+1}\right) = 1 - \frac{1}{n+1}\). [1]
  2. Hence determine the following sums.
    1. \(\sum_{r=1}^{90}\frac{1}{r} - \frac{1}{r+1}\) [1]
    2. \(\sum_{r=100}^{\infty}\frac{1}{r} - \frac{1}{r+1}\) [3]
OCR Further Pure Core 2 2024 June Q2
6 marks Moderate -0.8
In this question you must show detailed reasoning.
  1. Solve the equation \(x^2 - 6x + 58 = 0\). Give your solutions in the form \(a + bi\) where \(a\) and \(b\) are real numbers. [3]
  2. Determine, in exact form, \(\arg(-10 + (5\sqrt{12})i)^3\). [3]
OCR Further Pure Core 2 2024 June Q3
7 marks Standard +0.3
Matrices \(\mathbf{A}\) and \(\mathbf{B}\) are given by \(\mathbf{A} = \begin{pmatrix} 4 & -3 \\ -2 & 2 \end{pmatrix}\) and \(\mathbf{B} = \begin{pmatrix} 3 & -5 \\ 0 & 1 \end{pmatrix}\).
  1. Find \(2\mathbf{A} - 4\mathbf{B}\). [2]
  2. Write down the matrix \(\mathbf{C}\) such that \(\mathbf{A}\mathbf{C} = 2\mathbf{A}\). [1]
  3. Find the value of \(\det \mathbf{A}\). [1]
  4. In this question you must show detailed reasoning. Use \(\mathbf{A}^{-1}\) to solve the equations \(4x - 3y = 7\) and \(-2x + 2y = 9\). [3]
OCR Further Pure Core 2 2024 June Q4
5 marks Challenging +1.2
In this question you must show detailed reasoning. The series \(S\) is defined as being the sum of the squares of all positive odd integers from \(1^2\) to \(779^2\). Determine the value of \(S\). [5]
OCR Further Pure Core 2 2024 June Q5
6 marks Challenging +1.2
Vectors \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\), are given by \(\mathbf{a} = \mathbf{i} + (1-p)\mathbf{j} + (p+2)\mathbf{k}\), \(\mathbf{b} = 2\mathbf{i} + \mathbf{j} + \mathbf{k}\) and \(\mathbf{c} = \mathbf{i} + 14\mathbf{j} + (p-3)\mathbf{k}\) where \(p\) is a constant. You are given that \(\mathbf{a} \times \mathbf{b}\) is perpendicular to \(\mathbf{c}\). Determine the possible values of \(p\). [6]
OCR Further Pure Core 2 2024 June Q6
11 marks Challenging +1.8
In polar coordinates, the equation of a curve, \(C\), is \(r = 6\sin(2\theta)\sinh\left(\frac{1}{3}\theta\right)\) for \(0 \leqslant \theta \leqslant \frac{1}{2}\pi\). The pole of the polar coordinate system corresponds to the origin of the cartesian system and the initial line corresponds to the positive \(x\)-axis.
  1. Explain how you can tell that \(C\) comprises a single loop in the first quadrant, passing through the pole. [3]
The incomplete table below shows values of \(r\) for various values of \(\theta\).
\(\theta\)0\(\frac{1}{12}\pi\)\(\frac{1}{6}\pi\)\(\frac{1}{4}\pi\)\(\frac{1}{3}\pi\)\(\frac{5}{12}\pi\)\(\frac{1}{2}\pi\)
\(r\)00.2621.851
  1. Use the copy of the table and the polar coordinate system diagram given in the Printed Answer Booklet to complete the table and sketch \(C\). [3]
The point on \(C\) which is furthest away from the pole is denoted by \(A\) and the value of \(\theta\) at \(A\) is denoted by \(\phi\).
  1. Show that \(\phi\) satisfies the equation \(\phi = \frac{3}{4}\ln\left(\frac{6-\tan 2\phi}{6+\tan 2\phi}\right)\) [4]
  2. You are given that the relevant solution of the equation given in part (c) is \(\phi = 1.0207\) correct to 5 significant figures. Find the distance from \(A\) to the pole. Give your answer correct to 3 significant figures. [1]
OCR Further Pure Core 2 2024 June Q7
10 marks Challenging +1.8
  1. Express \(17\cosh x - 15\sinh x\) in the form \(e^{-x}(ae^{bx} + c)\) where \(a\), \(b\) and \(c\) are integers to be determined. [3]
A function is defined by \(f(x) = \frac{1}{\sqrt{17\cosh x - 15\sinh x}}\). The region bounded by the curve \(y = f(x)\), the \(x\)-axis, the \(y\)-axis and the line \(x = \ln 3\) is rotated by \(2\pi\) radians about the \(x\)-axis to form a solid of revolution \(S\).
  1. In this question you must show detailed reasoning. Use a suitable substitution, together with known results from the formula book, to show that the volume of \(S\) is given by \(k\pi\tan^{-1} q\) where \(k\) and \(q\) are rational numbers to be determined. [7]
OCR Further Pure Core 2 2024 June Q8
13 marks Standard +0.8
A children's play centre has two rooms, a room full of bouncy castles and a room full of ball pits. At any given instant, each child in the centre is playing either on the bouncy castles or in the ball pits. Each child can see one room from the other room and can decide to change freely between the two rooms. It is assumed that such changes happen instantaneously. The number of children playing on the bouncy castles at time \(t\) hours, is denoted by \(C\) and the corresponding number of children playing in the ball pits is \(P\). Because the number of children is large for most of the time, \(C\) and \(P\) are modelled as being continuous. When there is a different number of children in each room, some children will move from the room with more children to the room with fewer children. A researcher therefore decides to model \(C\) and \(P\) with the following coupled differential equations. $$\frac{dP}{dt} = \alpha(P-C) + \gamma t$$ $$\frac{dC}{dt} = \alpha(C-P)$$
  1. Explain why \(\alpha\) must be negative. [1]
After examining data, the researcher chooses \(\alpha = -2\) and \(\gamma = 32\).
  1. Show that \(P\) satisfies the second order differential equation \(\frac{d^2P}{dt^2} + 4\frac{dP}{dt} = 64t + 32\). [2]
    1. Find the complementary function for the differential equation from part (b). [1]
    2. Explain why a particular integral of the form \(P = at + b\) will not work in this situation. [1]
    3. Using a particular integral of the form \(P = at^2 + bt\), find the general solution of the differential equation from part (b). [3]
At a certain time there are 55 children playing in the ball pits and 24 children per hour are arriving at the ball pits.
  1. Use the model, starting from this time, to estimate the number of children in the ball pits 30 minutes later. [4]
  2. Explain why the model becomes unreliable as \(t\) gets very large. [1]
OCR Further Pure Core 2 2024 June Q9
12 marks Challenging +1.2
In this question, the argument of a complex number is defined as being in the range \([0, 2\pi)\). You are given that \(\omega_k\), where \(k = 0, 1, 2, ..., n-1\), are the \(n\) \(n\)th roots of unity for some integer \(n\), \(n \geqslant 3\), and that these are given in order of increasing argument (so that \(\omega_0 = 1\)).
  1. With the help of a diagram explain why \(\omega_k = (\omega_1)^k\) for \(k = 2, ..., n-1\). [3]
  2. Using the identity given in part (a), show that \(\sum_{k=0}^{n-1}\omega_k = 0\). [2]
  3. Show that if \(z\) is a complex number then \(z + z^* = 2\text{Re}(z)\). [1]
  4. Using the results from parts (b) and (c) show that \(\sum_{k=0}^{n-1}\text{Re}(\omega_k) = 0\). [1]
  5. With the help of a diagram explain why \(\text{Re}(\omega_k) = \text{Re}(\omega_{n-k})\) for \(k = 1, 2, ..., n-1\). [1]
You should now consider the case when \(n = 5\).
    1. Use parts (d) and (e) to deduce that \(\cos\frac{4\pi}{5} = a + b\cos\frac{2\pi}{5}\), for some rational constants \(a\) and \(b\). [2]
    2. Hence determine the exact value of \(\cos\frac{2\pi}{5}\). [2]
OCR Further Pure Core 2 Specimen Q1
4 marks Standard +0.3
Find \(\sum_{r=1}^{n}(r+1)(r+5)\). Give your answer in a fully factorised form. [4]
OCR Further Pure Core 2 Specimen Q2
4 marks Standard +0.8
In this question you must show detailed reasoning. The finite region \(R\) is enclosed by the curve with equation \(y = \frac{8}{\sqrt{16+x^3}}\), the \(x\)-axis and the lines \(x=0\) and \(x=4\). Region \(R\) is rotated through \(360°\) about the \(x\)-axis. Find the exact value of the volume generated. [4]
OCR Further Pure Core 2 Specimen Q3
4 marks Standard +0.3
\begin{enumerate}[label=(\roman*)] \item Find \(\sum_{r=1}^{n}\left(\frac{1}{r}-\frac{1}{r+2}\right)\). [3] \item What does the sum in part (i) tend to as \(n \to \infty\)? Justify your answer. [1]
OCR Further Pure Core 2 Specimen Q4
5 marks Challenging +1.2
It is given that \(\frac{5x^2+x+12}{x^2+kx} = \frac{A}{x} + \frac{Bx+C}{x^2+k}\) where \(k\), \(A\), \(B\) and \(C\) are positive integers. Determine the set of possible values of \(k\). [5]
OCR Further Pure Core 2 Specimen Q5
4 marks Standard +0.8
In this question you must show detailed reasoning. Evaluate \(\int_0^{\infty} 2xe^{-x} dx\). [You may use the result \(\lim_{x \to \infty} xe^{-x} = 0\).] [4]
OCR Further Pure Core 2 Specimen Q6
8 marks Standard +0.3
The equation of a plane \(\Pi\) is \(x-2y-z=30\). \begin{enumerate}[label=(\roman*)] \item Find the acute angle between the line \(\mathbf{r} = \begin{pmatrix} 3 \\ 2 \\ -5 \end{pmatrix} + \lambda \begin{pmatrix} -5 \\ 3 \\ 2 \end{pmatrix}\) and \(\Pi\). [4] \item Determine the geometrical relationship between the line \(\mathbf{r} = \begin{pmatrix} 1 \\ 4 \\ 2 \end{pmatrix} + \mu \begin{pmatrix} 3 \\ -1 \\ 5 \end{pmatrix}\) and \(\Pi\). [4]
OCR Further Pure Core 2 Specimen Q7
7 marks Challenging +1.8
\begin{enumerate}[label=(\roman*)] \item Use the Maclaurin series for \(\sin x\) to work out the series expansion of \(\sin x \sin 2x \sin 4x\) up to and including the term in \(x^3\). [4] \item Hence find, in exact surd form, an approximation to the least positive root of the equation \(2\sin x \sin 2x \sin 4x = x\). [3]
OCR Further Pure Core 2 Specimen Q8
8 marks Challenging +1.2
The equation of a curve is \(y = \cosh^2 x - 3\sinh x\). Show that \(\left(\ln\left(\frac{3+\sqrt{13}}{2}\right), -\frac{5}{4}\right)\) is the only stationary point on the curve. [8]
OCR Further Pure Core 2 Specimen Q9
6 marks Standard +0.8
A curve has equation \(x^4 + y^4 = x^2 + y^2\), where \(x\) and \(y\) are not both zero. \begin{enumerate}[label=(\roman*)] \item Show that the equation of the curve in polar coordinates is \(r^2 = \frac{2}{2-\sin^2 2\theta}\). [4] \item Deduce that no point on the curve \(x^4 + y^4 = x^2 + y^2\) is further than \(\sqrt{2}\) from the origin. [2]
OCR Further Pure Core 2 Specimen Q10
8 marks Hard +2.3
Let \(C = \sum_{r=0}^{20} \binom{20}{r} \cos r\theta\). Show that \(C = 2^{20} \cos^{20}\left(\frac{1}{2}\theta\right) \cos 10\theta\). [8]
OCR Further Pure Core 2 Specimen Q11
17 marks Challenging +1.2
During an industrial process substance \(X\) is converted into substance \(Z\). Some of the substance \(X\) goes through an intermediate phase, and is converted to substance \(Y\), before being converted to substance \(Z\). The situation is modelled by $$\frac{dy}{dt} = 0.3x + 0.2y \text{ and } \frac{dz}{dt} = 0.2y + 0.1x$$ where \(x\), \(y\) and \(z\) are the amounts in kg of \(X\), \(Y\) and \(Z\) at time \(t\) hours after the process starts. Initially there is 10 kg of substance \(X\) and nothing of substances \(Y\) and \(Z\). The amount of substance \(X\) decreases exponentially. The initial rate of decrease is 4 kg per hour.
  1. Show that \(x = Ae^{-0.4t}\), stating the value of \(A\). [3]
    1. Show that \(\frac{dx}{dt} + \frac{dy}{dt} + \frac{dz}{dt} = 0\). [2]
    2. Comment on this result in the context of the industrial process. [2]
  3. Express \(y\) in terms of \(t\). [5]
  4. Determine the maximum amount of substance \(Y\) present during the process. [3]
  5. How long does it take to produce 9 kg of substance \(Z\)? [2]
OCR Further Statistics 2020 November Q1
4 marks Moderate -0.8
The continuous random variable \(X\) has the distribution \(\text{N}(\mu, 30)\). The mean of a random sample of 8 observations of \(X\) is 53.1. Determine a 95\% confidence interval for \(\mu\). You should give the end points of the interval correct to 4 significant figures. [4]
OCR Further Statistics 2020 November Q2
8 marks Standard +0.3
A book collector compared the prices of some books, \(£x\), when new in 1972 and the prices of copies of the same books, \(£y\), on a second-hand website in 2018. The results are shown in Table 1 and are summarised below the table.
BookABCDEFGHIJKL
\(x\)0.950.650.700.900.551.401.500.501.150.350.200.35
\(y\)6.067.002.005.874.005.367.192.503.008.291.372.00
Table 1 \(n = 12, \Sigma x = 9.20, \Sigma y = 54.64, \Sigma x^2 = 8.9950, \Sigma y^2 = 310.4572, \Sigma xy = 46.0545\)
  1. It is given that the value of Pearson's product-moment correlation coefficient for the data is 0.381, correct to 3 significant figures.
    1. State what this information tells you about a scatter diagram illustrating the data. [1]
    2. Test at the 5\% significance level whether there is evidence of positive correlation between prices in 1972 and prices in 2018. [5]
  2. The collector noticed that the second-hand copy of book J was unusually expensive and he decided to ignore the data for book J. Calculate the value of Pearson's product-moment correlation coefficient for the other 11 books. [2]
OCR Further Statistics 2020 November Q3
9 marks Challenging +1.2
Jo can use either of two different routes, A or B, for her journey to school. She believes that route A has shorter journey times. She measures how long her journey takes for 17 journeys by route A and 12 journeys by route B. She ranks the 29 journeys in increasing order of time taken, and she finds that the sum of the ranks of the journeys by route B is 219.
  1. Test at the 10\% significance level whether route A has shorter journey times than route B. [8]
  2. State an assumption about the 29 journeys which is necessary for the conclusion of the test to be valid. [1]
OCR Further Statistics 2020 November Q4
7 marks Standard +0.8
The random variable \(X\) is equally likely to take any of the \(n\) integer values from \(m + 1\) to \(m + n\) inclusive. It is given that \(\text{E}(3X) = 30\) and \(\text{Var}(3X) = 36\). Determine the value of \(m\) and the value of \(n\). [7]