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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 FM1 AS 2021 June Q1
5 marks Moderate -0.3
A car of mass 1200 kg is driven on a long straight horizontal road. There is a constant force of 250 N resisting the motion of the car. The engine of the car is working at a constant power of 10 kW.
  1. The car can travel at constant speed \(v \text{ ms}^{-1}\) along the road. Find \(v\). [2]
  2. Find the acceleration of the car at an instant when its speed is \(30 \text{ ms}^{-1}\). [3]
OCR FM1 AS 2021 June Q2
6 marks Standard +0.8
A particle \(P\) of mass 5.6 kg is attached to one end of a light rod of length 2.1 m. The other end of the rod is freely hinged to a fixed point \(O\). The particle is initially at rest directly below \(O\). It is then projected horizontally with speed \(5 \text{ ms}^{-1}\). In the subsequent motion, the angle between the rod and the downward vertical at \(O\) is denoted by \(\theta\) radians, as shown in the diagram. \includegraphics{figure_2}
  1. Find the speed of \(P\) when \(\theta = \frac{1}{4}\pi\). [4]
  2. Find the value of \(\theta\) when \(P\) first comes to instantaneous rest. [2]
OCR FM1 AS 2021 June Q3
9 marks Standard +0.3
A particle of mass \(m\) moves in a straight line with constant acceleration \(a\). Its initial and final velocities are \(u\) and \(v\) respectively and its final displacement from its starting position is \(s\). In order to model the motion of the particle it is suggested that the velocity is given by the equation $$v^2 = pu^{\alpha} + qa^{\beta}s^{\gamma}$$ where \(p\) and \(q\) are dimensionless constants.
  1. Explain why \(\alpha\) must equal 2 for the equation to be dimensionally consistent. [2]
  2. By using dimensional analysis, determine the values of \(\beta\) and \(\gamma\). [4]
  3. By considering the case where \(s = 0\), determine the value of \(p\). [1]
  4. By multiplying both sides of the equation by \(\frac{1}{2}m\), and using the numerical values of \(\alpha\), \(\beta\) and \(\gamma\), determine the value of \(q\). [2]
OCR FM1 AS 2021 June Q4
12 marks Standard +0.8
Three particles \(A\), \(B\) and \(C\) are free to move in the same straight line on a large smooth horizontal surface. Their masses are 3.3 kg, 2.2 kg and 1 kg respectively. The coefficient of restitution in collisions between any two of them is \(e\). Initially, \(B\) and \(C\) are at rest and \(A\) is moving towards \(B\) with speed \(u \text{ ms}^{-1}\) (see diagram). \(A\) collides directly with \(B\) and \(B\) then goes on to collide directly with \(C\). \includegraphics{figure_4}
  1. The velocities of \(A\) and \(B\) immediately after the first collision are denoted by \(v_A \text{ ms}^{-1}\) and \(v_B \text{ ms}^{-1}\) respectively. \(\bullet\) Show that \(v_A = \frac{u(3-2e)}{5}\). \(\bullet\) Find an expression for \(v_B\) in terms of \(u\) and \(e\). [4]
  2. Find an expression in terms of \(u\) and \(e\) for the velocity of \(B\) immediately after its collision with \(C\). [4]
After the collision between \(B\) and \(C\) there is a further collision between \(A\) and \(B\).
  1. Determine the range of possible values of \(e\). [4]
OCR FP1 AS 2021 June Q1
5 marks Moderate -0.8
  1. Find a vector which is perpendicular to both \(\begin{pmatrix} 1 \\ 3 \\ -2 \end{pmatrix}\) and \(\begin{pmatrix} -3 \\ -6 \\ 4 \end{pmatrix}\). [2]
  2. The cartesian equation of a line is \(\frac{x}{2} = y - 3 = \frac{z + 4}{4}\). Express the equation of this line in vector form. [3]
OCR FP1 AS 2021 June Q2
9 marks Standard +0.3
In this question you must show detailed reasoning. The complex numbers \(z_1\) and \(z_2\) are given by \(z_1 = 2 - 3i\) and \(z_2 = a + 4i\) where \(a\) is a real number.
  1. Express \(z_1\) in modulus-argument form, giving the modulus in exact form and the argument correct to 3 significant figures. [3]
  2. Find \(z_1z_2\) in terms of \(a\), writing your answer in the form \(c + id\). [2]
  3. The real and imaginary parts of a complex number on an Argand diagram are \(x\) and \(y\) respectively. Given that the point representing \(z_1z_2\) lies on the line \(y = x\), find the value of \(a\). [2]
  4. Given instead that \(z_1z_2 = (z_1z_2)^*\) find the value of \(a\). [2]
OCR FP1 AS 2021 June Q3
10 marks Moderate -0.3
In this question you must show detailed reasoning.
  1. Express \((2 + 3i)^3\) in the form \(a + ib\). [3]
  2. Hence verify that \(2 + 3i\) is a root of the equation \(3z^3 - 8z^2 + 23z + 52 = 0\). [3]
  3. Express \(3z^3 - 8z^2 + 23z + 52\) as the product of a linear factor and a quadratic factor with real coefficients. [4]
OCR FP1 AS 2021 June Q4
6 marks Standard +0.3
Prove by induction that \(2^{n+1} + 5 \times 9^n\) is divisible by 7 for all integers \(n \geq 1\). [6]
OCR FS1 AS 2021 June Q1
8 marks Moderate -0.8
A book reviewer estimates that the probability that he receives a delivery of books to review on any one weekday is \(0.1\). The first weekday in September on which he receives a delivery of books to review is the \(X\)th weekday of September.
  1. State an assumption needed for \(X\) to be well modelled by a geometric distribution. [1]
  2. Find \(P(X = 11)\). [2]
  3. Find \(P(X \leq 8)\). [2]
  4. Find \(\text{Var}(X)\). [2]
  5. Give a reason why a geometric distribution might not be an appropriate model for the first weekday in a calendar year on which the reviewer receives a delivery of books to review. [1]
OCR FS1 AS 2021 June Q2
8 marks Standard +0.3
The probability distribution for the discrete random variable \(W\) is given in the table.
\(w\)1234
\(P(W = w)\)0.250.36\(x\)\(x^2\)
  1. Show that \(\text{Var}(W) = 0.8571\). [7]
  2. Find \(\text{Var}(3W + 6)\). [1]
OCR FS1 AS 2021 June Q3
5 marks Standard +0.8
In this question you must show detailed reasoning. The random variable \(T\) has a binomial distribution. It is known that \(E(T) = 5.625\) and the standard deviation of \(T\) is \(1.875\). Find the values of the parameters of the distribution. [5]
OCR FS1 AS 2021 June Q4
9 marks Challenging +1.2
The table shows the results of a random sample drawn from a population which is thought to have the distribution \(U(20)\).
Range\(1 \leq x \leq 8\)\(9 \leq x \leq 12\)\(13 \leq x \leq 20\)
Observed frequency12\(y\)\(28 - y\)
Find the range of values of \(y\) for which the data are not consistent with the distribution at the \(5\%\) significance level. [9]
OCR Further Pure Core 1 2021 June Q1
9 marks Standard +0.3
The equation of the curve shown on the graph is, in polar coordinates, \(r = 3\sin 2\theta\) for \(0 \leqslant \theta \leqslant \frac{1}{2}\pi\). \includegraphics{figure_1}
  1. The greatest value of \(r\) on the curve occurs at the point \(P\).
    1. Show that \(\theta = \frac{1}{4}\pi\) at the point \(P\). [2]
    2. Find the value of \(r\) at the point \(P\). [1]
    3. Mark the point \(P\) on a copy of the graph. [1]
  2. In this question you must show detailed reasoning. Find the exact area of the region enclosed by the curve. [5]
OCR Further Pure Core 1 2021 June Q2
7 marks Standard +0.3
You are given that \(f(x) = \ln(2 + x)\).
  1. Determine the exact value of \(f'(0)\). [2]
  2. Show that \(f''(0) = -\frac{1}{4}\). [2]
  3. Hence write down the first three terms of the Maclaurin series for \(f(x)\). [3]
OCR Further Pure Core 1 2021 June Q3
6 marks Standard +0.3
You are given that \(\mathbf{A} = \begin{pmatrix} 1 & 2 & 1 \\ 2 & 5 & 2 \\ 3 & -2 & -1 \end{pmatrix}\) and \(\mathbf{B} = \begin{pmatrix} 1 & 0 & 1 \\ -8 & 4 & 0 \\ 19 & -8 & -1 \end{pmatrix}\).
  1. Find \(\mathbf{AB}\). [1]
  2. Hence write down \(\mathbf{A}^{-1}\). [1]
  3. You are given three simultaneous equations $$x + 2y + z = 0$$ $$2x + 5y + 2z = 1$$ $$3x - 2y - z = 4$$
    1. Explain how you can tell, without solving them, that there is a unique solution to these equations. [2]
    2. Find this unique solution. [2]
OCR Further Pure Core 1 2021 June Q4
9 marks Standard +0.3
  1. Given that \(u = \tanh x\), use the definition of \(\tanh x\) in terms of exponentials to show that $$x = \frac{1}{2}\ln\left(\frac{1+u}{1-u}\right).$$ [4]
  2. Solve the equation \(4\tanh^2 x + \tanh x - 3 = 0\), giving the solution in the form \(a\ln b\) where \(a\) and \(b\) are rational numbers to be determined. [4]
  3. Explain why the equation in part (b) has only one root. [1]
OCR Further Pure Core 1 2021 June Q5
7 marks Challenging +1.2
In this question you must show detailed reasoning. Find \(\int_{-1}^{11} \frac{1}{\sqrt{x^2 + 6x + 13}} dx\) giving your answer in the form \(\ln(p + q\sqrt{2})\) where \(p\) and \(q\) are integers to be determined. [7]
OCR Further Pure Core 2 2021 June Q1
8 marks Moderate -0.8
In this question you must show detailed reasoning. S is the 2-D transformation which is a stretch of scale factor 3 parallel to the x-axis. A is the matrix which represents S.
  1. Write down A. [1]
  2. By considering the transformation represented by \(\mathbf{A}^{-1}\), determine the matrix \(\mathbf{A}^{-1}\). [2]
Matrix B is given by \(\mathbf{B} = \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}\). T is the transformation represented by B.
  1. Describe T. [1]
  2. Determine the matrix which represents the transformation S followed by T. [2]
  3. Demonstrate, by direct calculation, that \((\mathbf{BA})^{-1} = \mathbf{A}^{-1}\mathbf{B}^{-1}\). [2]
OCR Further Pure Core 2 2021 June Q2
7 marks Standard +0.8
  1. Find the shortest distance between the point \((-6, 4)\) and the line \(y = -0.75x + 7\). [2]
Two lines, \(l_1\) and \(l_2\), are given by $$l_1: \mathbf{r} = \begin{pmatrix} 4 \\ 3 \\ -2 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 1 \\ -4 \end{pmatrix} \text{ and } l_2: \mathbf{r} = \begin{pmatrix} 11 \\ -1 \\ 5 \end{pmatrix} + \mu \begin{pmatrix} 3 \\ -1 \\ 1 \end{pmatrix}$$
  1. Find the shortest distance between \(l_1\) and \(l_2\). [3]
  2. Hence determine the geometrical arrangement of \(l_1\) and \(l_2\). [2]
OCR Further Pure Core 2 2021 June Q3
9 marks Standard +0.3
Three matrices, A, B and C, are given by \(\mathbf{A} = \begin{pmatrix} 1 & 2 \\ a & -1 \end{pmatrix}\), \(\mathbf{B} = \begin{pmatrix} 2 & -1 \\ 4 & 1 \end{pmatrix}\) and \(\mathbf{C} = \begin{pmatrix} 5 & 0 \\ -2 & 2 \end{pmatrix}\) where \(a\) is a constant.
  1. Using A, B and C in that order demonstrate explicitly the associativity property of matrix multiplication. [4]
  2. Use A and C to disprove by counterexample the proposition 'Matrix multiplication is commutative'. [2]
For a certain value of \(a\), \(\mathbf{A}\begin{pmatrix} x \\ y \end{pmatrix} = 3\begin{pmatrix} x \\ y \end{pmatrix}\)
  1. Find
OCR Further Pure Core 2 2021 June Q4
7 marks Challenging +1.8
\includegraphics{figure_4} The figure shows part of the graph of \(y = (x - 3)\sqrt{\ln x}\). The portion of the graph below the x-axis is rotated by \(2\pi\) radians around the x-axis to form a solid of revolution, S. Determine the exact volume of S. [7]
OCR Further Pure Core 2 2021 June Q5
7 marks Challenging +1.8
\(C\) is the locus of numbers, \(z\), for which \(\ln\left(\frac{z + 7i}{z - 24}\right) = \frac{1}{4}\). By writing \(z = x + iy\) give a complete description of the shape of \(C\) on an Argand diagram. [7]
OCR Further Pure Core 2 2021 June Q1
8 marks Standard +0.3
  1. A plane \(\Pi\) has the equation \(\mathbf{r} \cdot \begin{pmatrix} 3 \\ 6 \\ -2 \end{pmatrix} = 15\). \(C\) is the point \((4, -5, 1)\). Find the shortest distance between \(\Pi\) and \(C\). [3]
  2. Lines \(l_1\) and \(l_2\) have the following equations. \(l_1: \mathbf{r} = \begin{pmatrix} 4 \\ 3 \\ 1 \end{pmatrix} + \lambda \begin{pmatrix} -2 \\ 4 \\ -2 \end{pmatrix}\) \(l_2: \mathbf{r} = \begin{pmatrix} 5 \\ 2 \\ 4 \end{pmatrix} + \mu \begin{pmatrix} 1 \\ -2 \\ 1 \end{pmatrix}\) Find, in exact form, the distance between \(l_1\) and \(l_2\). [5]