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Pre-U Pre-U 9794/3 2016 Specimen Q7
6 marks Moderate -0.3
7 A particle travels along a straight line. Its velocity \(v \mathrm {~ms} ^ { - 1 }\) after \(t\) seconds is given by $$v = t ^ { 3 } - 9 t ^ { 2 } + 20 t$$ 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. Find the displacement of the particle from \(P\) when \(t = 2\).
Pre-U Pre-U 9794/3 2016 Specimen Q8
6 marks Standard +0.3
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[max width=\textwidth, alt={}, center]{01bd6354-3514-4dad-901b-7ecbe155b2c7-5_213_1095_429_479} The acceleration of the system is \(0.3 \mathrm {~ms} ^ { - 2 }\) in the direction of the pulling force of magnitude \(P\).
  1. Calculate the value of \(P\). Truck \(S\) is now subjected to an extra resistive force of 1800 N . The pulling force, \(P\), does not change.
  2. Calculate the new acceleration of the trucks.
  3. Calculate the force in the coupling between the trucks.
Pre-U Pre-U 9794/3 2016 Specimen Q9
10 marks Standard +0.8
9 \includegraphics[max width=\textwidth, alt={}, center]{01bd6354-3514-4dad-901b-7ecbe155b2c7-5_118_851_1265_607} Three particles \(A , B\) and \(C\), having masses of \(1 \mathrm {~kg} , 2 \mathrm {~kg}\) 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 \(14 \mathrm {~ms} ^ { - 1 }\). 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.
  2. Show that \(B\) reversed direction after the impact with \(C\).
  3. Find the distances between \(B\) and \(C\) at the instant that \(B\) collides with \(A\) for the second time.
Pre-U Pre-U 9794/3 2016 Specimen Q10
11 marks Challenging +1.2
10 \includegraphics[max width=\textwidth, alt={}, center]{01bd6354-3514-4dad-901b-7ecbe155b2c7-6_490_661_267_703} Particles \(A\) and \(B\) of masses \(2 m\) 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 \(\alpha \leqslant 45 ^ { \circ }\) and \(B\) is above the plane. The vertical plane defined by \(A P B\) contains a line of greatest slope of the plane, and \(P A\) is inclined at angle \(2 \alpha\) to the horizontal (see diagram).
  1. Show that the normal reaction \(R\) between \(A\) and the plane is \(m g ( 2 \cos \alpha - \sin \alpha )\).
  2. Show that \(R \geqslant \frac { 1 } { 2 } m g \sqrt { 2 }\). The coefficient of friction between \(A\) and the plane is \(\mu\). The particle is about to slip down the plane.
  3. Show that \(0.5 < \tan \alpha \leqslant 1\).
  4. Express \(\mu\) as a function of \(\tan \alpha\) and deduce its maximum value as \(\alpha\) varies.
Pre-U Pre-U 9794/2 2016 Specimen Q2
5 marks Moderate -0.5
2 \includegraphics[max width=\textwidth, alt={}, center]{1c957cfe-bead-41d9-8985-479e876e1616-2_403_938_964_559} The diagram shows a triangle \(A B C\) in which angle \(C = 30 ^ { \circ } , B C = x \mathrm {~cm}\) and \(A C = ( x + 2 ) \mathrm { cm }\). Given that the area of triangle \(A B C\) is \(12 \mathrm {~cm} ^ { 2 }\), calculate the value of \(x\).
Pre-U Pre-U 9794/2 2016 Specimen Q5
9 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{1c957cfe-bead-41d9-8985-479e876e1616-3_577_743_287_662} The diagram shows a sector of a circle, \(O M N\). The angle \(M O N\) is \(2 x\) radians, the radius of the circle is \(r\) and \(O\) is the centre.
  1. Find expressions, in terms of \(r\) and \(x\), for the area, \(A\), and the perimeter, \(P\), of the sector.
  2. Given that \(P = 20\), show that \(A = \frac { 100 x } { ( 1 + x ) ^ { 2 } }\).
  3. Find \(\frac { \mathrm { d } A } { \mathrm {~d} x }\), and hence find the value of \(x\) for which the area of the sector is a maximum.
Pre-U Pre-U 9794/2 2016 Specimen Q10
15 marks Standard +0.8
10 A curve has equation $$y = \mathrm { e } ^ { a x } \cos b x$$ where \(a\) and \(b\) are constants.
  1. Show that, at any stationary points on the curve, \(\tan b x = \frac { a } { b }\).
  2. \includegraphics[max width=\textwidth, alt={}, center]{1c957cfe-bead-41d9-8985-479e876e1616-4_620_896_959_333} Values of related quantities \(x\) and \(y\) were measured in an experiment and plotted on a graph of \(y\) against \(x\), as shown in the diagram. Two of the points, labelled \(A\) and \(B\), have coordinates \(( 0,1 )\) and \(( 0.2 , - 0.8 )\) respectively. A third point labelled C has coordinates ( \(0.3,0.04\) ). Attempts were then made to find the equation of a curve which fitted closely to these three points, and two models were proposed. In the first model the equation is \(y = \mathrm { e } ^ { - x } \cos 15 x\).
    In the second model the equation is \(y = f \cos ( \lambda x ) + \mathrm { g }\), where the constants \(f , \lambda\), and \(g\) are chosen to give a maximum precisely at the point \(A ( 0,1 )\) and a minimum precisely at the point \(B ( 0.2 , - 0.8 )\). By calculating suitable values evaluate the suitability of the two models.
Pre-U Pre-U 9794/3 2016 Specimen Q6
7 marks Easy -1.2
6 \includegraphics[max width=\textwidth, alt={}, center]{b18b1bc5-bf26-4161-b5a5-764b00e97bea-4_572_672_456_701} The diagram shows two horizontal forces \(\mathbf { P }\) and \(\mathbf { Q }\) acting at the origin \(O\) of rectangular coordinates \(O x y\). 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. Hence, or otherwise, calculate the magnitude of this resultant and the angle the resultant makes with the positive \(x\)-axis.
Pre-U Pre-U 9794/3 2016 Specimen Q8
6 marks Moderate -0.3
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[max width=\textwidth, alt={}, center]{b18b1bc5-bf26-4161-b5a5-764b00e97bea-5_215_1095_427_479} The acceleration of the system is \(0.3 \mathrm {~ms} ^ { - 2 }\) in the direction of the pulling force of magnitude \(P\).
  1. Calculate the value of \(P\). Truck \(S\) is now subjected to an extra resistive force of 1800 N . The pulling force, \(P\), does not change.
  2. Calculate the new acceleration of the trucks.
  3. Calculate the force in the coupling between the trucks.
Pre-U Pre-U 9794/3 2016 Specimen Q9
10 marks Standard +0.8
9 \includegraphics[max width=\textwidth, alt={}, center]{b18b1bc5-bf26-4161-b5a5-764b00e97bea-5_118_851_1265_607} Three particles \(A , B\) and \(C\), having masses of \(1 \mathrm {~kg} , 2 \mathrm {~kg}\) 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 \(14 \mathrm {~ms} ^ { - 1 }\). 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.
  2. Show that \(B\) reversed direction after the impact with \(C\).
  3. Find the distances between \(B\) and \(C\) at the instant that \(B\) collides with \(A\) for the second time.
Pre-U Pre-U 9794/3 2016 Specimen Q10
12 marks Challenging +1.2
10 \includegraphics[max width=\textwidth, alt={}, center]{b18b1bc5-bf26-4161-b5a5-764b00e97bea-6_490_661_267_703} Particles \(A\) and \(B\) of masses \(2 m\) 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 \(\alpha \leqslant 45 ^ { \circ }\) and \(B\) is above the plane. The vertical plane defined by \(A P B\) contains a line of greatest slope of the plane, and \(P A\) is inclined at angle \(2 \alpha\) to the horizontal (see diagram).
  1. Show that the normal reaction \(R\) between \(A\) and the plane is \(m g ( 2 \cos \alpha - \sin \alpha )\).
  2. Show that \(R \geqslant \frac { 1 } { 2 } m g \sqrt { 2 }\). The coefficient of friction between \(A\) and the plane is \(\mu\). The particle is about to slip down the plane.
  3. Show that \(0.5 < \tan \alpha \leqslant 1\).
  4. Express \(\mu\) as a function of \(\tan \alpha\) and deduce its maximum value as \(\alpha\) varies.
Pre-U Pre-U 9795/1 2016 Specimen Q2
4 marks Standard +0.3
2 A curve has polar equation \(r = \sin \theta + \cos \theta\). Find the area enclosed by the curve and the lines \(\theta = 0\) and \(\theta = \frac { 1 } { 2 } \pi\).
Pre-U Pre-U 9795/1 2016 Specimen Q3
Standard +0.3
3
  1. Evaluate, in terms of \(k\), the determinant of the matrix \(\left( \begin{array} { c c c } 1 & 2 & 1 \\ - 3 & 5 & 8 \\ 6 & 12 & k \end{array} \right)\). Three planes have equations \(x + 2 y + z = 4 , - 3 x + 5 y + 8 z = 21\) and \(6 x + 12 y + k z = 31\).
  2. State the value of \(k\) for which these three planes do not meet at a single point.
  3. Find the coordinates of the point of intersection of the three planes when \(k = 7\).
Pre-U Pre-U 9795/1 2016 Specimen Q4
6 marks Challenging +1.2
4
  1. Given that \(y = \sqrt { \sinh x }\) for \(x \geqslant 0\), express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(y\) only.
  2. Hence or otherwise find \(\int \frac { 2 t } { \sqrt { 1 + t ^ { 4 } } } \mathrm {~d} t\).
Pre-U Pre-U 9795/1 2016 Specimen Q5
Standard +0.3
5 Use induction to prove that \(\sum _ { r = 1 } ^ { n } \left( \frac { 2 } { 4 r - 1 } \right) \left( \frac { 2 } { 4 r + 3 } \right) = \frac { 1 } { 3 } - \frac { 1 } { 4 n + 3 }\) for all positive integers \(n\).
Pre-U Pre-U 9795/1 2016 Specimen Q6
9 marks Standard +0.8
6 The curve \(C\) has equation \(y = \frac { x + 1 } { x ^ { 2 } + 3 }\).
  1. By considering a suitable quadratic equation in \(x\), find the set of possible values of \(y\) for points on \(C\).
  2. Deduce the coordinates of the turning points on \(C\).
  3. Sketch \(C\).
Pre-U Pre-U 9795/1 2016 Specimen Q7
Standard +0.8
7 The function \(f\) satisfies the differential equation $$x ^ { 2 } \mathrm { f } ^ { \prime \prime } ( x ) + ( 2 x - 1 ) \mathrm { f } ^ { \prime } ( x ) - 2 \mathrm { f } ( x ) = 3 \mathrm { e } ^ { x - 1 } + 1 ,$$ and the conditions \(f ( 1 ) = 2 , f ^ { \prime } ( 1 ) = 3\).
  1. Determine \(f ^ { \prime \prime } ( 1 )\).
  2. Differentiate (*) with respect to \(x\) and hence evaluate \(\mathrm { f } ^ { \prime \prime \prime } ( 1 )\).
  3. Hence determine the Taylor series approximation for \(\mathrm { f } ( x )\) about \(x = 1\), up to and including the term in \(( x - 1 ) ^ { 3 }\).
  4. Deduce, to 3 decimal places, an approximation for \(\mathrm { f } ( 1.1 )\).
Pre-U Pre-U 9795/1 2016 Specimen Q8
Challenging +1.2
8 Consider the set \(S\) of all matrices of the form \(\left( \begin{array} { l l } p & p \\ p & p \end{array} \right)\), where \(p\) is a non-zero rational number.
  1. Show that \(S\), under the operation of matrix multiplication, forms a group, \(G\). (You may assume that matrix multiplication is associative.)
  2. Find a subgroup of \(G\) of order 2 and show that \(G\) contains no subgroups of order 3.
Pre-U Pre-U 9795/1 2016 Specimen Q11
11 marks Challenging +1.8
11
  1. Use de Moivre's theorem to prove that \(\sin 5 \theta \equiv s \left( 16 s ^ { 4 } - 20 s ^ { 2 } + 5 \right)\), where \(s = \sin \theta\), and deduce that $$\sin \frac { 2 \pi } { 5 } = \sqrt { \frac { 5 + \sqrt { 5 } } { 8 } }$$ The complex number \(\omega = 16 ( - 1 + \mathrm { i } \sqrt { 3 } )\).
  2. State the value of \(| \omega |\) and find \(\arg \omega\) as a rational multiple of \(\pi\).
  3. (a) Determine the five roots of the equation \(z ^ { 5 } = \omega\), giving your answers in the form ( \(\mathrm { r } , \theta\) ), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
    (b) These five roots are represented in the complex plane by the points \(A , B , C , D\) and \(E\). Show these points on an Argand diagram, and find the area of the pentagon \(A B C D E\) in an exact surd form.
Pre-U Pre-U 9795/1 2016 Specimen Q12
Challenging +1.8
12
  1. Let \(I _ { n } = \int _ { 0 } ^ { 3 } x ^ { n } \sqrt { 16 + x ^ { 2 } } \mathrm {~d} x\), for \(n \geqslant 0\). Show that, for \(n \geqslant 2\), $$( n + 2 ) I _ { n } = 125 \times 3 ^ { n - 1 } - 16 ( n - 1 ) I _ { n - 2 }$$
  2. A curve has polar equation \(r = \frac { 1 } { 4 } \theta ^ { 4 }\) for \(0 \leqslant \theta \leqslant 3\).
    1. Sketch this curve.
    2. Find the exact length of the curve.
Pre-U Pre-U 9795/2 2016 Specimen Q1
Standard +0.8
1 The discrete random variable \(X\) has probability generating function \(\mathrm { G } _ { X } ( t )\) given by $$\mathrm { G } _ { X } ( t ) = a t \left( t + \frac { 1 } { t } \right) ^ { 3 } ,$$ where \(a\) is a constant.
  1. Find, in either order, the value of \(a\) and the set of values that \(X\) can take.
  2. Find the value of \(\mathrm { E } ( X )\).
Pre-U Pre-U 9795/2 2016 Specimen Q2
Standard +0.3
2
  1. The probability that a shopper obtains a parking space on the river embankment on any given Saturday morning is 0.2 . Using a suitable normal approximation, find the probability that, over a period of 100 Saturday mornings, the shopper finds a parking space at least 15 times. Justify the use of the normal approximation in this case.
  2. The number of parking tickets that a traffic warden issues on the river embankment during the course of a week has a Poisson distribution with mean 36 . The probability that the traffic warden issues more than \(N\) parking tickets is less than 0.05 . Using a suitable normal approximation, find the least possible value of \(N\).
Pre-U Pre-U 9795/2 2016 Specimen Q4
9 marks Standard +0.3
4 The independent random variables \(X\) and \(Y\) have normal distributions where \(X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\) and \(Y \sim \mathrm {~N} \left( 3 \mu , 4 \sigma ^ { 2 } \right)\). Two random samples each of size \(n\) are taken, one from each of these normal populations.
  1. Show that \(a \bar { X } + b \bar { Y }\) is an unbiased estimator of \(\mu\) provided that \(a + 3 b = 1\), where \(a\) and \(b\) are constants and \(\bar { X }\) and \(\bar { Y }\) are the respective sample means. In the remainder of the question assume that \(a \bar { X } + b \bar { Y }\) is an unbiased estimator of \(\mu\).
  2. Show that \(\operatorname { Var } ( a \bar { X } + b \bar { Y } )\) can be written as \(\frac { \sigma ^ { 2 } } { n } \left( 1 - 6 b + 13 b ^ { 2 } \right)\).
  3. The value of the constant \(b\) can be varied. Find the value of \(b\) that gives the minimum of \(\operatorname { Var } ( a \bar { X } + b \bar { Y } )\), and hence find the minimum of \(\operatorname { Var } ( a \bar { X } + b \bar { Y } )\) in terms of \(\sigma\) and \(n\).
Pre-U Pre-U 9795/2 2016 Specimen Q5
9 marks Standard +0.8
5 The random variable \(X\) has probability density function \(\mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = \begin{cases} k \mathrm { e } ^ { - k x } & x \geqslant 0 \\ 0 & x < 0 \end{cases}$$ and \(k\) is a positive constant.
  1. Show that the moment generating function of \(X\) is \(\mathrm { M } _ { X } ( t ) = k ( k - t ) ^ { - 1 } , t < k\).
  2. Use the moment generating function to find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
  3. Show that the moment generating function of \(- X\) is \(k ( k + t ) ^ { - 1 }\).
  4. \(X _ { 1 }\) and \(X _ { 2 }\) are two independent observations of \(X\). Use the moment generating function of \(X _ { 1 } - X _ { 2 }\) to find the value of \(\mathrm { E } \left[ \left( X _ { 1 } - X _ { 2 } \right) ^ { 2 } \right]\).
Pre-U Pre-U 9795/2 2016 Specimen Q6
12 marks Standard +0.8
6 The lengths of time, in years, that sales representatives for a certain company keep their company cars may be modelled by the distribution with probability density function \(\mathrm { f } ( x )\), where $$f ( x ) = \left\{ \begin{array} { c c } \frac { 4 } { 27 } x ^ { 2 } ( 3 - x ) & 0 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{array} \right.$$
  1. Draw a sketch of this probability density function.
  2. Calculate the mean and the mode of \(X\).
  3. Comment briefly on the values obtained in part (ii) in relation to the sketch in part (i).
  4. Show that the lower quartile \(\mathrm { Q } _ { 1 }\) of \(X\) satisfies the equation \(\mathrm { Q } _ { 1 } { } ^ { 4 } - 4 \mathrm { Q } _ { 1 } { } ^ { 3 } + 6.75 = 0\), and use an appropriate numerical method to find the value of \(\mathrm { Q } _ { 1 }\) correct to 2 decimal places, showing full details of your method.