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OCR Further Pure Core 2 2021 June Q2
5 marks Standard +0.3
In this question you must show detailed reasoning. Show that \(\int_5^{\infty} (x-1)^{-2} dx = 1\). [5]
OCR Further Pure Core 2 2021 June Q3
6 marks Standard +0.3
\(A\) is a fixed point on a smooth horizontal surface. A particle \(P\) is initially held at \(A\) and released from rest. It subsequently performs simple harmonic motion in a straight line on the surface. After its release it is next at rest after \(0.2\) seconds at point \(B\) whose displacement is \(0.2\) m from \(A\). The point \(M\) is halfway between \(A\) and \(B\). The displacement of \(P\) from \(M\) at time \(t\) seconds after release is denoted by \(x\) m.
  1. Sketch a graph of \(x\) against \(t\) for \(0 \leq t \leq 0.4\). [4]
  2. Find the displacement of \(P\) from \(M\) at \(0.75\) seconds after release. [2]
OCR Further Pure Core 2 2021 June Q4
7 marks Standard +0.8
In an Argand diagram the points representing the numbers \(2 + 3i\) and \(1 - i\) are two adjacent vertices of a square \(S\).
  1. Find the area of \(S\). [3]
  2. Find all the possible pairs of numbers represented by the other two vertices of \(S\). [4]
OCR Further Pure Core 2 2021 June Q5
11 marks Challenging +1.3
In this question you must show detailed reasoning. The diagram below shows the curve \(r = \sqrt{\sin\theta}e^{\cos\theta}\) for \(0 \leq \theta < \pi\). \includegraphics{figure_5}
  1. Find the exact area enclosed by the curve. [4]
  2. Show that the greatest value of \(r\) on the curve is \(\sqrt{\frac{3}{2}}e^{\frac{1}{2}}\). [7]
OCR Further Pure Core 2 2021 June Q1
5 marks Moderate -0.5
In this question you must show detailed reasoning. Solve the equation \(4z^2 - 20z + 169 = 0\). Give your answers in modulus-argument form. [5]
OCR Further Pure Core 2 2021 June Q2
9 marks Standard +0.3
The equations of two intersecting lines \(l_1\) and \(l_2\) are $$l_1: \mathbf{r} = \begin{pmatrix} 1 \\ 0 \\ a \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 1 \\ -3 \end{pmatrix}$$ $$l_2: \mathbf{r} = \begin{pmatrix} 7 \\ 9 \\ -2 \end{pmatrix} + \mu \begin{pmatrix} -1 \\ 1 \\ 2 \end{pmatrix}$$ where \(a\) is a constant. The equation of the plane \(\Pi\) is $$\mathbf{r} \cdot \begin{pmatrix} 1 \\ 5 \\ 3 \end{pmatrix} = -14.$$ \(l_1\) and \(\Pi\) intersect at \(Q\). \(l_2\) and \(\Pi\) intersect at \(R\).
  1. Verify that the coordinates of \(R\) are \((13, 3, -14)\). [2]
  2. Determine the exact value of the length of \(QR\). [7]
OCR Further Pure Core 2 2021 June Q3
7 marks Standard +0.3
A capacitor is an electrical component which stores charge. The value of the charge stored by the capacitor, in suitable units, is denoted by \(Q\). The capacitor is placed in an electrical circuit. At any time \(t\) seconds, where \(t \geq 0\), \(Q\) can be modelled by the differential equation $$\frac{d^2Q}{dt^2} - 2\frac{dQ}{dt} - 15Q = 0.$$ Initially the charge is 100 units and it is given that \(Q\) tends to a finite limit as \(t\) tends to infinity.
  1. Determine the charge on the capacitor when \(t = 0.5\). [6]
  2. Determine the finite limit of \(Q\) as \(t\) tends to infinity. [1]
OCR Further Pure Core 2 2021 June Q4
6 marks Standard +0.8
The matrix \(\mathbf{A}\) is given by \(\mathbf{A} = \begin{pmatrix} 0.6 & 2.4 \\ -0.8 & 1.8 \end{pmatrix}\).
  1. Find \(\det \mathbf{A}\). [1]
The matrix \(\mathbf{A}\) represents a stretch parallel to one of the coordinate axes followed by a rotation about the origin.
  1. By considering the determinants of these transformations, determine the scale factor of the stretch. [2]
  2. Explain whether the stretch is parallel to the \(x\)-axis or the \(y\)-axis, justifying your answer. [1]
  3. Find the angle of rotation. [2]
OCR Further Pure Core 2 2021 June Q5
11 marks Challenging +1.2
Two thin poles, \(OA\) and \(BC\), are fixed vertically on horizontal ground. A chain is fixed at \(A\) and \(C\) such that it touches the ground at point \(D\) as shown in the diagram. On a coordinate system the coordinates of \(A\), \(B\) and \(D\) are \((0, 3)\), \((5, 0)\) and \((2, 0)\). \includegraphics{figure_5} It is required to find the height of pole \(BC\) by modelling the shape of the curve that the chain forms. Jofra models the curve using the equation \(y = k \cosh(ax - b) - 1\) where \(k\), \(a\) and \(b\) are positive constants.
  1. Determine the value of \(k\). [2]
  2. Find the exact value of \(a\) and the exact value of \(b\), giving your answers in logarithmic form. [5]
Holly models the curve using the equation \(y = \frac{1}{4}x^2 - 3x + 3\).
  1. Write down the coordinates of the point, \((u, v)\) where \(u\) and \(v\) are both non-zero, at which the two models will agree. [1]
  2. Show that Jofra's model and Holly's model disagree in their predictions of the height of pole \(BC\) by \(3.32\)m to 3 significant figures. [3]
OCR Further Statistics 2021 June Q1
5 marks Moderate -0.3
A set of bivariate data \((X, Y)\) is summarised as follows. \(n = 25\), \(\Sigma x = 9.975\), \(\Sigma y = 11.175\), \(\Sigma x^2 = 5.725\), \(\Sigma y^2 = 46.200\), \(\Sigma xy = 11.575\)
  1. Calculate the value of Pearson's product-moment correlation coefficient. [1]
  2. Calculate the equation of the regression line of \(y\) on \(x\). [2]
It is desired to know whether the regression line of \(y\) on \(x\) will provide a reliable estimate of \(y\) when \(x = 0.75\).
  1. State one reason for believing that the estimate will be reliable. [1]
  2. State what further information is needed in order to determine whether the estimate is reliable. [1]
OCR Further Statistics 2021 June Q2
4 marks Standard +0.3
The average numbers of cars, lorries and buses passing a point on a busy road in a period of 30 minutes are 400, 80 and 17 respectively.
  1. Assuming that the numbers of each type of vehicle passing the point in a period of 30 minutes have independent Poisson distributions, calculate the probability that the total number of vehicles passing the point in a randomly chosen period of 30 minutes is at least 520. [3]
  2. Buses are known to run in approximate accordance with a fixed timetable. Explain why this casts doubt on the use of a Poisson distribution to model the number of buses passing the point in a fixed time interval. [1]
OCR Further Statistics 2021 June Q3
9 marks Standard +0.3
The greatest weight \(W\) N that can be supported by a shelving bracket of traditional design is a normally distributed random variable with mean 500 and standard deviation 80. A sample of 40 shelving brackets of a new design are tested and it is found that the mean of the greatest weights that the brackets in the sample can support is 473.0 N.
  1. Test at the 1% significance level whether the mean of the greatest weight that a bracket of the new design can support is less than the mean of the greatest weight that a bracket of the traditional design can support. [7]
  2. State an assumption needed in carrying out the test in part (a). [1]
  3. Explain whether it is necessary to use the central limit theorem in carrying out the test. [1]
OCR Further Statistics 2021 June Q4
10 marks Standard +0.3
The random variable \(D\) has the distribution Geo\((p)\). It is given that Var\((D) = \frac{40}{9}\). Determine
  1. Var\((3D + 5)\). [1]
  2. E\((3D + 5)\). [6]
  3. \(\text{P}(D > \text{E}(D))\). [3]
OCR Further Statistics 2021 June Q5
10 marks Standard +0.8
A university course was taught by two different professors. Students could choose whether to attend the lectures given by Professor \(Q\) or the lectures given by Professor \(R\). At the end of the course all the students took the same examination. The examination marks of a random sample of 30 students taught by Professor \(Q\) and a random sample of 24 students taught by Professor \(R\) were ranked. The sum of the ranks of the students taught by Professor \(Q\) was 726. Test at the 5% significance level whether there is a difference in the ranks of the students taught by the two professors. [10]
OCR Further Statistics 2021 June Q1
9 marks Standard +0.8
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 2021 June Q2
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 E\((3X) = 30\) and Var\((3X) = 36\). Determine the value of \(m\) and the value of \(n\). [7]
OCR Further Statistics 2021 June Q3
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 2021 June Q4
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\(\geq 7\)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\(\geq 7\)
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 \(x \geq 7\) was obtained. [2]
  2. Carry out the test. [5]
OCR FP1 AS 2017 December Q1
4 marks Standard +0.3
The matrix \(\mathbf{A}\) is given by \(\mathbf{A} = \begin{pmatrix} -3 & 3 & 2 \\ 5 & -4 & -3 \\ -1 & 1 & 1 \end{pmatrix}\).
  1. Find \(\mathbf{A}^{-1}\). [1]
  2. Solve the simultaneous equations $$-3x + 3y + 2z = 12a$$ $$5x - 4y - 3z = -6$$ $$-x + y + z = 7$$ giving your solution in terms of \(a\). [3]
OCR FP1 AS 2017 December Q2
9 marks Standard +0.3
The loci \(C_1\) and \(C_2\) are given by \(|z - (3 + 2i)| = 2\) and \(\arg(z - (3 + 2i)) = \frac{5\pi}{6}\) respectively.
  1. Sketch \(C_1\) and \(C_2\) on a single Argand diagram. [4]
  2. Find, in surd form, the number represented by the point of intersection of \(C_1\) and \(C_2\). [3]
  3. Indicate, by shading, the region of the Argand diagram for which $$|z - (3 + 2i)| \leq 2 \text{ and } \frac{5\pi}{6} \leq \arg(z - (3 + 2i)) \leq \pi.$$ [2]
OCR FP1 AS 2017 December Q3
8 marks Standard +0.3
Two lines, \(l_1\) and \(l_2\), have the following equations. $$l_1: \mathbf{r} = \begin{pmatrix} -11 \\ 10 \\ 3 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ -2 \\ 1 \end{pmatrix}$$ $$l_2: \mathbf{r} = \begin{pmatrix} 5 \\ 2 \\ 4 \end{pmatrix} + \mu \begin{pmatrix} 3 \\ 1 \\ -2 \end{pmatrix}$$ \(P\) is the point of intersection of \(l_1\) and \(l_2\).
  1. Find the position vector of \(P\). [3]
  2. Find, correct to 1 decimal place, the acute angle between \(l_1\) and \(l_2\). [3]
\(Q\) is a point on \(l_1\) which is 12 metres away from \(P\). \(R\) is the point on \(l_2\) such that \(QR\) is perpendicular to \(l_1\).
  1. Determine the length \(QR\). [2]
OCR FP1 AS 2017 December Q4
7 marks Standard +0.8
In this question you must show detailed reasoning. The distinct numbers \(\omega_1\) and \(\omega_2\) both satisfy the quadratic equation \(4x^2 + 4x + 17 = 0\).
  1. Write down the value of \(\omega_1 \omega_2\). [1]
  2. \(A\), \(B\) and \(C\) are the points on an Argand diagram which represent \(\omega_1\), \(\omega_2\) and \(\omega_1 \omega_2\). Find the area of triangle \(ABC\). [6]
OCR FP1 AS 2017 December Q5
7 marks Challenging +1.3
In this question you must show detailed reasoning. The equation \(x^3 + 3x^2 - 2x + 4 = 0\) has roots \(\alpha\), \(\beta\) and \(\gamma\).
  1. Using the identity \(\alpha^3 + \beta^3 + \gamma^3 \equiv (\alpha + \beta + \gamma)^3 - 3(\alpha\beta + \beta\gamma + \gamma\alpha)(\alpha + \beta + \gamma) + 3\alpha\beta\gamma\) find the value of \(\alpha^3 + \beta^3 + \gamma^3\). [3]
  2. Given that \(\alpha^2\beta^3 + \beta^3\gamma^3 + \gamma^3\alpha^3 = 112\) find a cubic equation whose roots are \(\alpha^2\), \(\beta^3\) and \(\gamma^3\). [4]
OCR FP1 AS 2017 December Q6
5 marks Standard +0.3
Prove by induction that \(n! \geq 6n\) for \(n \geq 4\). [5]
OCR FP1 AS 2017 December Q7
7 marks Standard +0.8
A transformation is equivalent to a shear parallel to the \(x\)-axis followed by a shear parallel to the \(y\)-axis and is represented by the matrix \(\begin{pmatrix} 1 & s \\ t & 0 \end{pmatrix}\). Find in terms of \(s\) the matrices which represent each of the shears. [7]