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Pre-U Pre-U 9795/1 2018 June Q3
7 marks Standard +0.3
The complex numbers \(z_1\) and \(z_2\) are such that \(|z_1| = 2\), \(\arg(z_1) = \frac{7}{12}\pi\), \(|z_2| = \sqrt{2}\) and \(\arg(z_2) = -\frac{1}{8}\pi\).
  1. Find, in exact form, the modulus and argument of \(\frac{z_1}{z_2}\). [3]
  2. Let \(z_3 = \left(\frac{z_1}{z_2}\right)^n\). It is given that \(n\) is the least positive integer for which \(z_3\) is a positive real number. Find this value of \(n\) and the exact value of \(z_3\). [4]
Pre-U Pre-U 9795/1 2018 June Q4
7 marks Challenging +1.2
A curve has polar equation \(r = \frac{3}{10}e^{3\theta}\) for \(\theta \geq 0\). The length of the arc of this curve between \(\theta = 0\) and \(\theta = \alpha\) is denoted by \(L(\alpha)\).
  1. Show that \(L(\alpha) = \frac{1}{3}(e^{3\alpha} - 1)\). [5]
  2. The point \(P\) on the curve corresponding to \(\theta = \beta\) is such that \(L(\beta) = OP\), where \(O\) is the pole. Find the value of \(\beta\). [2]
Pre-U Pre-U 9795/1 2018 June Q5
8 marks Standard +0.8
Find, in the form \(y = f(x)\), the solution of the differential equation \(\frac{dy}{dx} + y\tanh x = 2\cosh x\), given that \(y = \frac{3}{4}\) when \(x = \ln 2\). [8]
Pre-U Pre-U 9795/1 2018 June Q6
8 marks Challenging +1.8
The cubic equation \(4x^3 - 12x^2 + 9x - 16 = 0\) has roots \(r_1\), \(r_2\) and \(r_3\). A second cubic equation, with integer coefficients, has roots \(R_1 = \frac{r_2 + r_3}{r_1}\), \(R_2 = \frac{r_3 + r_1}{r_2}\) and \(R_3 = \frac{r_1 + r_2}{r_3}\).
  1. Show that \(1 + R_1 = \frac{3}{r_1}\) and write down the corresponding results for the other roots. [2]
  2. Using a substitution based on this result, or otherwise, find this second cubic equation. [6]
Pre-U Pre-U 9795/1 2018 June Q7
6 marks Challenging +1.2
The function \(y\) satisfies \(\frac{d^2y}{dx^2} + x^2y = x\), and is such that \(y = 1\) and \(\frac{dy}{dx} = 1\) when \(x = 1\).
  1. Using the given differential equation
    1. state the value of \(\frac{d^2y}{dx^2}\) when \(x = 1\), [1]
    2. find, by differentiation, the value of \(\frac{d^3y}{dx^3}\) when \(x = 1\). [2]
  2. Hence determine the Taylor series for \(y\) about \(x = 1\) up to and including the term in \((x-1)^3\) and deduce, correct to 4 decimal places, an approximation for \(y\) when \(x = 1.1\). [3]
Pre-U Pre-U 9795/1 2018 June Q8
8 marks Challenging +1.2
  1. Write down the values of the constants \(a\) and \(b\) for which \(m^3 = \frac{1}{6}m^3(am^2 + 2) - \frac{1}{12}m^2(bm)\). [1]
  2. Prove by induction that \(\sum_{r=1}^{n} r^5 = \frac{1}{6}n^3(n+1)^3 - \frac{1}{12}n^2(n+1)^2\) for all positive integers \(n\). [7]
Pre-U Pre-U 9795/1 2018 June Q9
8 marks Standard +0.3
  1. Use de Moivre's theorem to prove that \(\cos 3\theta = 4c^3 - 3c\), where \(c = \cos\theta\). [3]
  2. Solve the equation \(2\cos 3\theta - \sqrt{3} = 0\) for \(0 < \theta < \pi\), giving each answer in an exact form. [2]
  3. Deduce, in trigonometric form, the three roots of the equation \(x^3 - 3x - \sqrt{3} = 0\). [3]
Pre-U Pre-U 9795/1 2018 June Q10
10 marks Challenging +1.8
  1. Let \(G\) be a group of order 10. Write down the possible orders of the elements of \(G\) and justify your answer. [2]
  2. Let \(G_1\) be the cyclic group of order 10 and let \(g\) be a generator of \(G_1\) (that is, an element of order 10). List the ten elements of \(G_1\) in terms of \(g\) and state the order of each element. [4]
  3. The group \(G_2\) is defined as the set of ordered pairs \((x, y)\), where \(x \in \{0, 1\}\) and \(y \in \{0, 1, 2, 3, 4\}\), together with the binary operation \(\oplus\) defined by $$(x_1, y_1) \oplus (x_2, y_2) = (x_3, y_3),$$ where \(x_3 = x_1 + x_2\) modulo 2 and \(y_3 = y_1 + y_2\) modulo 5.
    1. List the elements of \(G_2\) and state the order of each element. [3]
    2. State, with justification, whether \(G_1\) and \(G_2\) are isomorphic. [1]
Pre-U Pre-U 9795/1 2018 June Q11
10 marks Challenging +1.3
Let \(\mathbf{A}\) be the matrix \(\begin{pmatrix} 17 & 12 \\ 12 & 10 \end{pmatrix}\).
    1. Determine the integer \(n\) for which \(27\mathbf{A} - \mathbf{A}^2 = n\mathbf{I}\), where \(\mathbf{I}\) is the \(2 \times 2\) identity matrix. [2]
    2. Hence find \(\mathbf{A}^{-1}\) in the form \(p\mathbf{A} + q\mathbf{I}\) for rational numbers \(p\) and \(q\). [2]
  2. The plane transformation \(T\) is defined by \(T: \begin{pmatrix} x \\ y \end{pmatrix} \mapsto \mathbf{A} \begin{pmatrix} x \\ y \end{pmatrix}\). It is given that \(T\) is a stretch, with scale factor \(k\), parallel to the line \(y = mx\), where \(m > 0\).
    1. Find the value of \(k\). [2]
    2. By considering \(\mathbf{A} \begin{pmatrix} x \\ mx \end{pmatrix}\), or otherwise, determine the value of \(m\). [4]
Pre-U Pre-U 9795/1 2018 June Q12
15 marks Challenging +1.8
The curve \(C\) is given by \(y = \frac{1}{4}x^2 - \frac{1}{2}\ln x\) for \(2 \leq x \leq 8\).
  1. Find, in its simplest exact form, the length of \(C\). [5]
  2. When \(C\) is rotated through \(2\pi\) radians about the \(x\)-axis, a surface of revolution is formed. Show that the area of this surface is \(\pi(270 - 47\ln 2 - 2(\ln 2)^2)\). [10]
Pre-U Pre-U 9795/1 2018 June Q13
18 marks Challenging +1.2
The planes \(\Pi_1\) and \(\Pi_2\) are both perpendicular to \(\mathbf{n}\), where \(\mathbf{n} = \begin{pmatrix} 1 \\ 2 \\ -2 \end{pmatrix}\). The points \(A(0, -9, 13)\) and \(B(8, 7, -3)\) lie in \(\Pi_1\) and \(\Pi_2\) respectively.
  1. Find the equations of \(\Pi_1\) and \(\Pi_2\) in the form \(\mathbf{r} \cdot \mathbf{n} = d\) and show that \(\overrightarrow{AB}\) is parallel to \(\mathbf{n}\). [4]
  2. Calculate the perpendicular distance between \(\Pi_1\) and \(\Pi_2\). [2]
  3. Write down two vectors which are perpendicular to \(\mathbf{n}\) and hence find, in the form $$\mathbf{r} = \mathbf{u} + \lambda\mathbf{v} + \mu\mathbf{w},$$ an equation for the plane \(\Pi_3\) which is parallel to \(\Pi_1\) and \(\Pi_2\) and exactly half-way between them. [4]
  4. The locus of all points \(P\) such that \(AP = BP = 12\sqrt{2}\) is denoted by \(L\).
    1. Give a full geometrical description of \(L\). [4]
    2. Using the result of part (iii), or otherwise, find a point on \(L\) which has integer coordinates. [4]
Pre-U Pre-U 9794/3 2019 Specimen Q1
6 marks Easy -1.2
The times for a motorist to travel from home to work are normally distributed with a mean of 24 minutes and a standard deviation of 4 minutes. Find the probability that a particular trip from home to work takes
  1. more than 27 minutes, [3]
  2. between 20 and 25 minutes. [3]
Pre-U Pre-U 9794/3 2019 Specimen Q2
12 marks Moderate -0.8
  1. A music club has 200 members. 75 members play the piano, 130 members like Elgar, and 30 members do not play the piano, nor do they like Elgar.
    1. Calculate the probability that a member chosen at random plays the piano but does not like Elgar. [3]
    2. Calculate the probability that a member chosen at random plays the piano given that this member likes Elgar. [2]
  2. The music club is organising a concert. The programme is to consist of 7 pieces of music which are to be selected from 9 classical pieces and 6 modern pieces. Find the number of different concert programmes that can be produced if
    1. there are no restrictions, [2]
    2. the programme must consist of 5 classical pieces and 2 modern pieces, [2]
    3. there are to be more modern pieces than classical pieces. [3]
Pre-U Pre-U 9794/3 2019 Specimen Q3
5 marks Easy -1.8
The table shows fuel economy figures in miles per gallon (mpg) for some new cars.
CarABCDEFGHIJKLMNO
Mpg574034331117302731203524262332
  1. Find the median and quartiles for the mpg of these 15 cars. [2]
  2. Use the values in part (a) to identify any cars for which the mpg is an outlier. [3]
Pre-U Pre-U 9794/3 2019 Specimen Q4
6 marks Moderate -0.3
A survey into left-handedness found that 13% of the population of the world are left-handed.
  1. State the assumptions necessary for it to be appropriate to model the number of left-handed children in a class of 20 children using the binomial distribution B(20, 0.13). [2]
  2. Assuming that this binomial model is appropriate, calculate the probability that fewer than 13% of the 20 children are left-handed. [4]
Pre-U Pre-U 9794/3 2019 Specimen Q5
11 marks Moderate -0.3
James plays an arcade game. Each time he plays, he puts a £1 coin in the slot to start the game. The possible outcomes of each game are as follows: James loses the game with a probability of 0.7 and the machine pays out nothing, James draws the game with a probability of 0.25 and the machine pays out a £1 coin, James wins the game with a probability of 0.05 and the machine pays out ten £1 coins. The outcomes can be modelled by a random variable \(X\) representing the number of £1 coins gained at the end of a game.
  1. Construct a probability distribution table for \(X\). [2]
  2. Show that E(\(X\)) = \(-0.25\) and find Var(\(X\)). [4]
James starts off with 10 £1 coins and decides to play exactly 10 games.
  1. Find the expected number of £1 coins that James will have at the end of his 10 games. [2]
  2. Find the probability that after his 10 games James will have at least 10 £1 coins left. [3]
Pre-U Pre-U 9794/3 2019 Specimen Q6
6 marks Easy -1.2
\includegraphics{figure_6} The diagram shows two horizontal forces \(\mathbf{P}\) and \(\mathbf{Q}\) acting at the origin \(O\) of rectangular coordinates \(Oxy\). The components of \(\mathbf{P}\) in the \(x\)- and \(y\)-directions are 12 N and 17 N respectively. The components of \(\mathbf{Q}\) in the \(x\)- and \(y\)-directions are \(-5\) N and 7 N respectively.
  1. Write down the components, in the \(x\)- and \(y\)-directions, of the resultant of \(\mathbf{P}\) and \(\mathbf{Q}\). [2]
  2. Hence, or otherwise, calculate the magnitude of this resultant and the angle the resultant makes with the positive \(x\)-axis. [4]
Pre-U Pre-U 9794/3 2019 Specimen Q7
6 marks Moderate -0.3
A particle travels along a straight line. Its velocity \(v\) m s\(^{-1}\) after \(t\) seconds is given by $$v = t^3 - 9t^2 + 20t$$ When \(t = 0\), the particle is at rest at \(P\).
  1. Find the times, other than \(t = 0\), at which the particle is at rest. [2]
  2. Find the displacement of the particle from \(P\) when \(t = 2\). [4]
Pre-U Pre-U 9794/3 2019 Specimen Q8
6 marks Moderate -0.8
Two trucks, \(S\) and \(T\), of masses 8000 kg and 10000 kg respectively, are pulled along a straight, horizontal track by a constant, horizontal force of \(P\) N. A resistive force of 600 N acts on \(S\) and a resistive force of 450 N acts on \(T\). The coupling between the trucks is light and horizontal (see diagram). \includegraphics{figure_8} The acceleration of the system is 0.3 m s\(^{-2}\) in the direction of the pulling force of magnitude \(P\).
  1. Calculate the value of \(P\). [2]
Truck \(S\) is now subjected to an extra resistive force of 1800 N. The pulling force, \(P\), does not change.
  1. Calculate the new acceleration of the trucks. [2]
  2. Calculate the force in the coupling between the trucks. [2]
Pre-U Pre-U 9794/3 2019 Specimen Q9
10 marks Challenging +1.2
\includegraphics{figure_9} Three particles \(A\), \(B\) and \(C\), having masses of 1 kg, 2 kg and 5 kg respectively, are placed 1 metre apart in a straight line on a smooth horizontal plane (see diagram). The particles \(B\) and \(C\) are initially at rest and \(A\) is moving towards \(B\) with speed 14 m s\(^{-1}\). The coefficient of restitution between each pair of particles is 0.5.
  1. Find the velocity of \(B\) immediately after the first impact and show that \(A\) comes to rest. [4]
  2. Show that \(B\) reversed direction after the impact with \(C\). [3]
  3. Find the distances between \(B\) and \(C\) at the instant that \(B\) collides with \(A\) for the second time. [3]
Pre-U Pre-U 9794/3 2019 Specimen Q10
12 marks Challenging +1.8
\includegraphics{figure_10} Particles \(A\) and \(B\) of masses \(2m\) and \(m\), respectively, are attached to the ends of a light inextensible string. The string passes over a smooth fixed pulley \(P\). The particle \(A\) rests in equilibrium on a rough plane inclined at an angle \(\alpha\) to the horizontal, where \(\alpha \leqslant 45°\) and \(B\) is above the plane. The vertical plane defined by \(APB\) contains a line of greatest slope of the plane, and \(PA\) is inclined at angle \(2\alpha\) to the horizontal (see diagram).
  1. Show that the normal reaction R between A and the plane is mg(2\(\cos\alpha - \sin\alpha\)). [3]
  2. Show that R \(\geqslant \frac{1}{2}mg\sqrt{2}\). [3]
The coefficient of friction between \(A\) and the plane is \(\mu\). The particle is about to slip down the plane.
  1. Show that \(0.5 < \tan\alpha \leqslant 1\). [3]
  2. Express \(\mu\) as a function of \(\tan\alpha\) and deduce its maximum value as \(\alpha\) varies. [3]
Pre-U Pre-U 9794/3 2020 Specimen Q1
6 marks Easy -1.2
The times for a motorist to travel from home to work are normally distributed with a mean of 24 minutes and a standard deviation of 4 minutes. Find the probability that a particular trip from home to work takes
  1. more than 27 minutes, [3]
  2. between 20 and 25 minutes. [3]
Pre-U Pre-U 9794/3 2020 Specimen Q2
12 marks Moderate -0.8
  1. A music club has 200 members. 75 members play the piano, 130 members like Elgar, and 30 members do not play the piano, nor do they like Elgar.
    1. Calculate the probability that a member chosen at random plays the piano but does not like Elgar. [3]
    2. Calculate the probability that a member chosen at random plays the piano given that this member likes Elgar. [2]
  2. The music club is organising a concert. The programme is to consist of 7 pieces of music which are to be selected from 9 classical pieces and 6 modern pieces. Find the number of different concert programmes that can be produced if
    1. there are no restrictions, [2]
    2. the programme must consist of 5 classical pieces and 2 modern pieces, [2]
    3. there are to be more modern pieces than classical pieces. [3]
Pre-U Pre-U 9794/3 2020 Specimen Q3
5 marks Easy -1.3
The table shows fuel economy figures in miles per gallon (mpg) for some new cars.
CarABCDEFGHIJKLMNO
Mpg574034331117302731203524262332
  1. Find the median and quartiles for the mpg of these 15 cars. [2]
  2. Use the values in part (a) to identify any cars for which the mpg is an outlier. [3]
Pre-U Pre-U 9794/3 2020 Specimen Q4
6 marks Moderate -0.3
A survey into left-handedness found that 13% of the population of the world are left-handed.
  1. State the assumptions necessary for it to be appropriate to model the number of left-handed children in a class of 20 children using the binomial distribution B(20, 0.13). [2]
  2. Assuming that this binomial model is appropriate, calculate the probability that fewer than 13% of the 20 children are left-handed. [4]