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OCR Further Statistics 2020 November Q5
11 marks Challenging +1.2
26 cards are each labelled with a different letter of the alphabet, A to Z. The letters A, E, I, O and U are vowels.
  1. Five cards are selected at random without replacement. Determine the probability that the letters on at least three of the cards are vowels. [4]
  2. All 26 cards are arranged in a line, in random order.
    1. Show that the probability that all the vowels are next to one another is \(\frac{1}{2990}\). [3]
    2. Determine the probability that three of the vowels are next to each other, and the other two vowels are next to each other, but the five vowels are not all next to each other. [4]
OCR Further Statistics 2020 November Q6
11 marks Standard +0.3
The numbers of CD players sold in a shop on three consecutive weekends were 7, 6 and 2. It may be assumed that sales of CD players occur randomly and that nobody buys more than one CD player at a time. The number of CD players sold on a randomly chosen weekend is denoted by \(X\).
  1. How appropriate is the Poisson distribution as a model for \(X\)? [2]
Now assume that a Poisson distribution with mean 5 is an appropriate model for \(X\).
  1. Find
    1. P\((X = 6)\), [2]
    2. P\((X \geqslant 8)\). [2]
The number of integrated sound systems sold in a weekend at the same shop can be assumed to have the distribution Po(7.2).
  1. Find the probability that on a randomly chosen weekend the total number of CD players and integrated sound systems sold is between 10 and 15 inclusive. [3]
  2. State an assumption needed for your answer to part (c) to be valid. [1]
  3. Give a reason why the assumption in part (d) may not be valid in practice. [1]
OCR Further Statistics 2020 November Q7
10 marks Standard +0.3
A biased spinner has five sides, numbered 1 to 5. Elmer spins the spinner repeatedly and counts the number of spins, \(X\), up to and including the first time that the number 2 appears. He carries out this experiment 100 times and records the frequency \(f\) with which each value of \(X\) is obtained. His results are shown in Table 1, together with the values of \(xf\).
\(x\)123456\(\geqslant 7\)Total
Frequency \(f\)2015913101023100
\(xf\)203027525060161400
Table 1
  1. State an appropriate distribution with which to model \(X\), determining the value(s) of any parameter(s). [3]
Elmer carries out a goodness-of-fit test, at the 5\% level, for the distribution in part (a). Table 2 gives some of his calculations, in which numbers that are not exact have been rounded to 3 decimal places.
\(x\)123456\(\geqslant 7\)
Observed frequency \(O\)2015913101023
Expected frequency \(E\)2518.7514.06310.5477.9105.93317.798
\((O - E)^2/E\)10.751.8230.5710.5522.7891.520
Table 2
  1. Show how the expected frequency corresponding to \(x \geqslant 7\) was obtained. [2]
  2. Carry out the test. [5]
OCR Further Statistics 2020 November Q8
15 marks Standard +0.8
The continuous random variable \(X\) has probability density function $$f(x) = \begin{cases} \frac{k}{x^n} & x \geqslant 1, \\ 0 & \text{otherwise}, \end{cases}$$ where \(n\) and \(k\) are constants and \(n\) is an integer greater than 1.
  1. Find \(k\) in terms of \(n\). [3]
    1. When \(n = 4\), find the cumulative distribution function of \(X\). [3]
    2. Hence determine P\((X > 7 | X > 5)\) when \(n = 4\). [4]
  3. Determine the values of \(n\) for which Var\((X)\) is not defined. [5]
OCR Further Mechanics 2023 June Q1
8 marks Standard +0.3
One end of a light inextensible string of length \(0.8\) m is attached to a particle \(P\) of mass \(m\) kg. The other end of the string is attached to a fixed point \(O\). Initially \(P\) hangs in equilibrium vertically below \(O\). It is then projected horizontally with a speed of \(5.3\) m s\(^{-1}\) so that it moves in a vertical circular path with centre \(O\) (see diagram). \includegraphics{figure_1} At a certain instant, \(P\) first reaches the point where the string makes an angle of \(\frac{1}{3}\pi\) radians with the downward vertical through \(O\).
  1. Show that at this instant the speed of \(P\) is \(4.5\) m s\(^{-1}\). [3]
  2. Find the magnitude and direction of the radial acceleration of \(P\) at this instant. [3]
  3. Find the magnitude of the tangential acceleration of \(P\) at this instant. [2]
OCR Further Mechanics 2023 June Q2
11 marks Standard +0.3
Materials have a measurable property known as the Young's Modulus, \(E\). If a force is applied to one face of a block of the material then the material is stretched by a distance called the extension. Young's modulus is defined as the ratio \(\frac{\text{Stress}}{\text{Strain}}\) where Stress is defined as the force per unit area and Strain is the ratio of the extension of the block to the length of the block.
  1. Show that Strain is a dimensionless quantity. [1]
  2. By considering the dimensions of both Stress and Strain determine the dimensions of \(E\). [2]
It is suggested that the speed of sound in a material, \(c\), depends only upon the value of Young's modulus for the material, \(E\), the volume of the material, \(V\), and the density (or mass per unit volume) of the material, \(\rho\).
  1. Use dimensional analysis to suggest a formula for \(c\) in terms of \(E\), \(V\) and \(\rho\). [5]
  2. The speed of sound in a certain material is \(500\) m s\(^{-1}\).
    1. Use your formula from part (c) to predict the speed of sound in the material if the value of Young's modulus is doubled but all other conditions are unchanged. [1]
    2. With reference to your formula from part (c), comment on the effect on the speed of sound in the material if the volume is doubled but all other conditions are unchanged. [1]
  3. Suggest one possible limitation caused by using dimensional analysis to set up the model in part (c). [1]
OCR Further Mechanics 2023 June Q3
7 marks Challenging +1.2
Two smooth circular discs \(A\) and \(B\) are moving on a smooth horizontal plane when they collide. The mass of \(A\) is \(5\) kg and the mass of \(B\) is \(3\) kg. At the instant before they collide, • the velocity of \(A\) is \(4\) m s\(^{-1}\) at an angle of \(60°\) to the line of centres, • the velocity of \(B\) is \(6\) m s\(^{-1}\) along the line of centres (see diagram). \includegraphics{figure_3} The coefficient of restitution for collisions between the two discs is \(\frac{3}{4}\). Determine the angle that the velocity of \(A\) makes with the line of centres after the collision. [7]
OCR Further Mechanics 2023 June Q4
9 marks Standard +0.3
\(ABCD\) is a uniform lamina in the shape of a kite with \(BA = BC = 0.37\) m, \(DA = DC = 0.91\) m and \(AC = 0.7\) m (see diagram). The centre of mass of \(ABCD\) is \(G\). \includegraphics{figure_4}
  1. Explain why \(G\) lies on \(BD\). [1]
  2. Show that the distance of \(G\) from \(B\) is \(0.36\) m. [4]
The lamina \(ABCD\) is freely suspended from the point \(A\).
  1. Determine the acute angle that \(CD\) makes with the horizontal, stating which of \(C\) or \(D\) is higher. [4]
OCR Further Mechanics 2023 June Q5
13 marks Challenging +1.3
A particle \(P\) of mass \(2\) kg moves along the \(x\)-axis. At time \(t = 0\), \(P\) passes through the origin \(O\) with speed \(3\) m s\(^{-1}\). At time \(t\) seconds the displacement of \(P\) from \(O\) is \(x\) m and the velocity of \(P\) is \(v\) m s\(^{-1}\), where \(t \geqslant 0\), \(x \geqslant 0\) and \(v \geqslant 0\). While \(P\) is in motion the only force acting on \(P\) is a resistive force \(F\) of magnitude \((v^2 + 1)\) N acting in the negative \(x\)-direction.
  1. Find an expression for \(v\) in terms of \(x\). [5]
  2. Determine the distance travelled by \(P\) while its speed drops from \(3\) m s\(^{-1}\) to \(2\) m s\(^{-1}\). [2]
Particle \(Q\) is identical to particle \(P\). At a different time, \(Q\) is moving along the \(x\)-axis under the influence of a single constant resistive force of magnitude \(1\) N. When \(t' = 0\), \(Q\) is at the origin and its speed is \(3\) m s\(^{-1}\).
  1. By comparing the motion of \(P\) with the motion of \(Q\) explain why \(P\) must come to rest at some finite time when \(t < 6\) with \(x < 9\). [3]
  2. Sketch the velocity-time graph for \(P\). You do not need to indicate any values on your sketch. [1]
  3. Determine the maximum displacement of \(P\) from \(O\) during \(P\)'s motion. [2]
OCR Further Mechanics 2023 June Q6
12 marks Challenging +1.2
A particle \(P\) of mass \(3\) kg is moving on a smooth horizontal surface under the influence of a variable horizontal force \(\mathbf{F}\) N. At time \(t\) seconds, where \(t \geqslant 0\), the velocity of \(P\), \(\mathbf{v}\) m s\(^{-1}\), is given by $$\mathbf{v} = (32\sinh(2t))\mathbf{i} + (32\cosh(2t) - 257)\mathbf{j}.$$
    1. By considering kinetic energy, determine the work done by \(\mathbf{F}\) over the interval \(0 \leqslant t \leqslant \ln 2\). [5]
    2. Explain the significance of the sign of the answer to part (a)(i). [1]
  2. Determine the rate at which \(\mathbf{F}\) is working at the instant when \(P\) is moving parallel to the \(\mathbf{i}\)-direction. [6]
OCR Further Mechanics 2023 June Q7
7 marks Challenging +1.2
Two particles \(A\) and \(B\) are connected by a light inextensible string of length \(1.26\) m. Particle \(A\) has a mass of \(1.25\) kg and moves on a smooth horizontal table in a circular path of radius \(0.9\) m and centre \(O\). The string passes through a small smooth hole at \(O\). Particle \(B\) has a mass of \(2\) kg and moves in a horizontal circle as shown in the diagram. The angle that the portion of string below the table makes with the downwards vertical through \(O\) is \(\theta\), where \(\cos\theta = \frac{4}{5}\) (see diagram). \includegraphics{figure_7}
  1. Determine the angular speed of \(A\) and the angular speed of \(B\). [5]
At the start of the motion, \(A\), \(O\) and \(B\) all lie in the same vertical plane.
  1. Find the first subsequent time when \(A\), \(O\) and \(B\) all lie in the same vertical plane. [2]
OCR Further Mechanics 2023 June Q8
8 marks Challenging +1.2
One end of a light elastic string of natural length \(2.1\) m and modulus of elasticity \(4.8\) N is attached to a particle, \(P\), of mass \(1.75\) kg. The other end of the string is attached to a fixed point, \(O\), which is on a rough inclined plane. The angle between the plane and the horizontal is \(\theta\) where \(\sin\theta = \frac{3}{5}\). The coefficient of friction between \(P\) and the plane is \(0.732\). Particle \(P\) is placed on the plane at \(O\) and then projected down a line of greatest slope of the plane with an initial speed of \(2.4\) m s\(^{-1}\). Determine the distance that \(P\) has travelled from \(O\) at the instant when it first comes to rest. You can assume that during its motion \(P\) does not reach the bottom of the inclined plane. [8]
OCR MEI Further Pure Core AS 2018 June Q1
4 marks Moderate -0.8
The matrices \(\mathbf{A}\), \(\mathbf{B}\) and \(\mathbf{C}\) are defined as follows: $$\mathbf{A} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}, \quad \mathbf{B} = \begin{pmatrix} 2 & 0 & 3 \\ 1 & -1 & 3 \end{pmatrix}, \quad \mathbf{C} = \begin{pmatrix} 1 & 3 \end{pmatrix}.$$ Calculate all possible products formed from two of these three matrices. [4]
OCR MEI Further Pure Core AS 2018 June Q2
3 marks Moderate -0.8
Find, to the nearest degree, the angle between the vectors \(\begin{pmatrix} 1 \\ 0 \\ -2 \end{pmatrix}\) and \(\begin{pmatrix} -2 \\ 3 \\ -3 \end{pmatrix}\). [3]
OCR MEI Further Pure Core AS 2018 June Q3
5 marks Moderate -0.8
Find real numbers \(a\) and \(b\) such that \((a - 3i)(5 - i) = b - 17i\). [5]
OCR MEI Further Pure Core AS 2018 June Q4
5 marks Moderate -0.3
Find a cubic equation with real coefficients, two of whose roots are \(2 - i\) and \(3\). [5]
OCR MEI Further Pure Core AS 2018 June Q5
7 marks Standard +0.3
A transformation of the \(x\)-\(y\) plane is represented by the matrix \(\begin{pmatrix} \cos \theta & 2 \sin \theta \\ 2 \sin \theta & -\cos \theta \end{pmatrix}\), where \(\theta\) is a positive acute angle.
  1. Write down the image of the point \((2, 3)\) under this transformation. [2]
  2. You are given that this image is the point \((a, 0)\). Find the value of \(a\). [5]
OCR MEI Further Pure Core AS 2018 June Q6
4 marks Standard +0.3
Find the invariant line of the transformation of the \(x\)-\(y\) plane represented by the matrix \(\begin{pmatrix} 2 & 0 \\ 4 & -1 \end{pmatrix}\). [4]
OCR MEI Further Pure Core AS 2018 June Q7
9 marks Standard +0.8
  1. Express \(\frac{1}{2r-1} - \frac{1}{2r+1}\) as a single fraction. [2]
  2. Find how many terms of the series $$\frac{2}{1 \times 3} + \frac{2}{3 \times 5} + \frac{2}{5 \times 7} + \ldots + \frac{2}{(2r-1)(2r+1)} + \ldots$$ are needed for the sum to exceed \(0.999999\). [7]
OCR MEI Further Pure Core AS 2018 June Q8
6 marks Standard +0.3
Prove by induction that \(\begin{pmatrix} 1 & 1 \\ 0 & 2 \end{pmatrix}^n = \begin{pmatrix} 1 & 2^n - 1 \\ 0 & 2^n \end{pmatrix}\) for all positive integers \(n\). [6]
OCR MEI Further Pure Core AS 2018 June Q9
9 marks Standard +0.3
Fig. 9 shows a sketch of the region OPQ of the Argand diagram defined by $$\left\{z : |z| \leq 4\sqrt{2}\right\} \cap \left\{z : -\frac{1}{4}\pi \leq \arg z \leq \frac{1}{4}\pi\right\}.$$ \includegraphics{figure_9}
  1. Find, in modulus-argument form, the complex number represented by the point P. [2]
  2. Find, in the form \(a + ib\), where \(a\) and \(b\) are exact real numbers, the complex number represented by the point Q. [3]
  3. In this question you must show detailed reasoning. Determine whether the points representing the complex numbers
    lie within this region. [4]
OCR MEI Further Pure Core AS 2018 June Q10
8 marks Standard +0.3
Three planes have equations \begin{align} -x + 2y + z &= 0
2x - y - z &= 0
x + y &= a \end{align} where \(a\) is a constant.
  1. Investigate the arrangement of the planes:
    [6]
  2. Chris claims that the position vectors \(-\mathbf{i} + 2\mathbf{j} + \mathbf{k}\), \(2\mathbf{i} - \mathbf{j} - \mathbf{k}\) and \(\mathbf{i} + \mathbf{j}\) lie in a plane. Determine whether or not Chris is correct. [2]
OCR MEI Further Pure Core AS Specimen Q1
4 marks Moderate -0.8
The complex number \(z_1\) is \(1+ i\) and the complex number \(z_2\) has modulus 4 and argument \(\frac{\pi}{3}\).
  1. Express \(z_2\) in the form \(a + bi\), giving \(a\) and \(b\) in exact form. [2]
  2. Express \(\frac{z_2}{z_1}\) in the form \(c + di\), giving \(c\) and \(d\) in exact form. [2]
OCR MEI Further Pure Core AS Specimen Q2
4 marks Moderate -0.8
  1. Describe fully the transformation represented by the matrix \(\begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}\). [2]
  2. A triangle of area 5 square units undergoes the transformation represented by the matrix \(\begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}\). Explaining your reasoning, find the area of the image of the triangle following this transformation. [2]
OCR MEI Further Pure Core AS Specimen Q3
4 marks Moderate -0.3
  1. Write down, in complex form, the equation of the locus represented by the circle in the Argand diagram shown in Fig. 3. [2] \includegraphics{figure_3}
  2. On the copy of Fig. 3 in the Printed Answer Booklet mark with a cross any point(s) on the circle for which \(\arg(z - 2i) = \frac{\pi}{4}\). [2]