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Pre-U Pre-U 9795/1 2016 June Q6
16 marks Challenging +1.2
6 The equation \(\sinh x + \sin x = 3 x\) has one positive root \(\alpha\).
  1. Show that \(2.5 < \alpha < 3\).
  2. By using the first two non-zero terms in the Maclaurin series for \(\sinh x + \sin x\), show that \(\alpha \approx \sqrt [ 4 ] { 60 }\).
  3. By taking the third non-zero term in this series, find a second approximation to \(\alpha\), giving your answer correct to 4 decimal places.
Pre-U Pre-U 9795/1 2016 June Q7
9 marks Standard +0.3
7
  1. Find all values of \(z\) for which \(z ^ { 3 } = 2 + 2 \mathrm { i }\). Give your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(\theta\) is an exact multiple of \(\pi\) in the interval \(0 < \theta < 2 \pi\).
  2. The vertices of a triangle in the Argand diagram correspond to the three roots of the equation \(z ^ { 3 } = 2 + 2 \mathrm { i }\). Sketch the triangle and determine its area.
Pre-U Pre-U 9795/1 2016 June Q8
12 marks Challenging +1.2
8
  1. \(S\) is the set \(\{ 1,2,4,8,16,32 \}\) and \(\times _ { 63 }\) is the operation of multiplication modulo 63 .
    1. Construct the multiplication table for \(\left( S , \times _ { 63 } \right)\).
    2. Show that \(\left( S , \times _ { 63 } \right)\) forms a group, \(G\). (You may assume that \(\times _ { 63 }\) is associative.)
    3. The group \(H\), also of order 6, has identity element \(e\) and contains two further elements \(x\) and \(y\) with the properties $$x ^ { 2 } = y ^ { 3 } = e \quad \text { and } \quad x y x = y ^ { 2 } .$$ (a) Construct the group table of \(H\).
      (b) List all the proper subgroups of \(H\).
    4. State, with justification, whether \(G\) and \(H\) are isomorphic.
Pre-U Pre-U 9795/1 2016 June Q9
10 marks Challenging +1.2
9 The cubic equation \(x ^ { 3 } - a x ^ { 2 } + b x - c = 0\) has roots \(\alpha , \beta\) and \(\gamma\).
  1. State, in terms of \(a , b\) and \(c\), the values of \(\alpha + \beta + \gamma , \alpha \beta + \beta \gamma + \gamma \alpha\) and \(\alpha \beta \gamma\).
  2. Find, in terms of \(a , b\) and \(c\), the values of \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\) and \(\alpha ^ { 2 } \beta ^ { 2 } + \beta ^ { 2 } \gamma ^ { 2 } + \gamma ^ { 2 } \alpha ^ { 2 }\).
  3. Show that \(( \alpha - 2 \beta \gamma ) ( \beta - 2 \gamma \alpha ) ( \gamma - 2 \alpha \beta ) = c ( 2 a + 1 ) ^ { 2 } - 2 ( b + 2 c ) ^ { 2 }\).
  4. Deduce that one root of the equation \(x ^ { 3 } - a x ^ { 2 } + b x - c = 0\) is twice the product of the other two roots if and only if \(c ( 2 a + 1 ) ^ { 2 } = 2 ( b + 2 c ) ^ { 2 }\).
Pre-U Pre-U 9795/1 2016 June Q10
10 marks Challenging +1.2
10
  1. Sketch the curve with polar equation \(r = \left| \frac { 1 } { 2 } + \sin \theta \right|\), for \(0 \leqslant \theta < 2 \pi\).
  2. Find in an exact form the total area enclosed by the curve.
Pre-U Pre-U 9795/1 2016 June Q11
5 marks Challenging +1.8
11
  1. The sequence of Fibonacci Numbers \(\left\{ F _ { n } \right\}\) is given by $$F _ { 1 } = 1 , \quad F _ { 2 } = 1 \quad \text { and } \quad F _ { n + 1 } = F _ { n } + F _ { n - 1 } \text { for } n \geqslant 2 .$$ Write down the values of \(F _ { 3 }\) to \(F _ { 6 }\).
  2. The sequence of functions \(\left\{ \mathrm { p } _ { n } ( x ) \right\}\) is given by $$\mathrm { p } _ { 1 } ( x ) = x + 1 \quad \text { and } \quad \mathrm { p } _ { n + 1 } ( x ) = 1 + \frac { 1 } { \mathrm { p } _ { n } ( x ) } \text { for } n \geqslant 1$$
    1. Find \(\mathrm { p } _ { 2 } ( x )\) and \(\mathrm { p } _ { 3 } ( x )\), giving each answer as a single algebraic fraction, and show that \(\mathrm { p } _ { 4 } ( x ) = \frac { 3 x + 5 } { 2 x + 3 }\).
    2. Conjecture an expression for \(\mathrm { p } _ { n } ( x )\) as a single algebraic fraction involving Fibonacci numbers, and prove it by induction for all integers \(n \geqslant 2\).
Pre-U Pre-U 9795/1 2016 June Q12
10 marks Challenging +1.8
12 The curve \(C\) has equation \(y = \ln \left( \tanh \frac { 1 } { 2 } x \right)\), for \(x > 0\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \operatorname { cosech } x\).
  2. For positive integers \(n\), the length of the arc of \(C\) between \(x = n\) and \(x = 2 n\) is \(L _ { n }\).
    1. Show by calculus that, when \(n\) is large, \(L _ { n } \approx n\).
    2. Explain how this result corresponds to the shape of \(C\).
Pre-U Pre-U 9795/1 2016 June Q13
17 marks Challenging +1.8
13
  1. (a) Given that \(x \geqslant 1\), show that \(\sec ^ { - 1 } x = \cos ^ { - 1 } \left( \frac { 1 } { x } \right)\), and deduce that \(\frac { \mathrm { d } } { \mathrm { d } x } \left( \sec ^ { - 1 } x \right) = \frac { 1 } { x \sqrt { x ^ { 2 } - 1 } }\).
    (b) Use integration by parts to determine \(\int \sec ^ { - 1 } x \mathrm {~d} x\).
  2. \includegraphics[max width=\textwidth, alt={}, center]{5d526fd9-72f8-42b1-b156-fd4a0c764c82-4_670_1029_1073_596} The diagram shows the curve \(S\) with equation \(y = \sec ^ { - 1 } x\) for \(x \geqslant 1\). The line \(L\), with gradient \(\frac { 1 } { \sqrt { 2 } }\), is the tangent to \(S\) at the point \(P\) and cuts the \(x\)-axis at the point \(Q\). The point \(I\) has coordinates \(( 1,0 )\).
    (a) Determine the exact coordinates of \(P\) and \(Q\).
    (b) The region \(R\), shaded on the diagram, is bounded by the line segments \(P Q\) and \(Q I\) and the \(\operatorname { arc } I P\) of \(S\). Show that \(R\) has area $$\ln ( 1 + \sqrt { 2 } ) - \frac { \pi ( 8 - \pi ) \sqrt { 2 } } { 32 } .$$ {www.cie.org.uk} after the live examination series. }
Pre-U Pre-U 9795/2 2016 June Q1
5 marks Moderate -0.5
1 An investigation was carried out of the lengths of commuters' journeys. For a random sample of 500 commuters, the mean journey time was 75 minutes, and the standard deviation was 40 minutes.
  1. Calculate a 95\% confidence interval for the mean journey time.
  2. Explain whether you need to assume that journey times are normally distributed.
Pre-U Pre-U 9795/2 2016 June Q2
4 marks Standard +0.8
2 The mass in grams of a pre-cut piece of Brie cheese is a random variable with the distribution \(\mathrm { N } ( 150,1200 )\). Brie costs 80 p per 100 g .
  1. Find the probability that a randomly chosen piece of Brie costs more than \(\pounds 1.40\). The mass in grams of a pre-cut piece of Stilton cheese is an independent random variable with the distribution \(\mathrm { N } ( 180,1500 )\).
  2. Find the probability that the total mass of four randomly chosen pieces of Brie is less than the total mass of three randomly chosen pieces of Stilton.
Pre-U Pre-U 9795/2 2016 June Q3
5 marks Standard +0.8
3
  1. Show that the probability generating function of a random variable with the distribution \(\mathrm { B } ( n , p )\) is \(( 1 - p + p t ) ^ { n }\).
  2. \(R\) and \(S\) are independent random variables with the distributions \(\mathrm { B } \left( 8 , \frac { 1 } { 4 } \right)\) and \(\mathrm { B } \left( 8 , \frac { 3 } { 4 } \right)\) respectively. Show that the probability generating function of \(R + S\) can be expressed as $$\left( \frac { 3 } { 16 } + \frac { 1 } { 16 } t ( 10 + 3 t ) \right) ^ { 8 }$$ and use this result to find \(\mathrm { P } ( R + S = 1 )\).
Pre-U Pre-U 9795/2 2016 June Q4
7 marks Standard +0.3
4 In a Football League match, the number of goals scored by the home team can be modelled by the distribution \(\mathrm { Po } ( 2.4 )\). The number of goals scored by the away team can be modelled by the distribution Po(1.8).
  1. State a necessary assumption for the total number of goals scored in one match to be modelled by the distribution \(\operatorname { Po } ( 4.2 )\).
  2. Assume now that this assumption holds.
    1. Write down an expression for the probability that the total number of goals scored in \(n\) randomly chosen games is less than 4 .
    2. Find the probability that the result of a randomly chosen game is either 0-0 or 1-1.
Pre-U Pre-U 9795/2 2016 June Q5
10 marks Challenging +1.2
5 The random variable \(R\) has the distribution \(\mathrm { B } ( n , p )\).
  1. State two conditions that \(n\) and \(p\) must satisfy if the distribution of \(R\) can be well approximated by a normal distribution. Assume now that these conditions hold. Using the normal approximation, it is given that \(\mathrm { P } ( R < 25 ) = 0.8282\) and \(\mathrm { P } ( R \geqslant 28 ) = 0.0393\), correct to 4 decimal places.
  2. Find the mean and standard deviation of the approximating normal distribution.
  3. Hence find the value of \(p\) and the value of \(n\).
Pre-U Pre-U 9795/2 2016 June Q6
6 marks Standard +0.3
6 A continuous random variable \(X\) has probability density function $$\mathrm { f } ( x ) = \begin{cases} 4 x \mathrm { e } ^ { - 2 x } & x \geqslant 0 \\ 0 & \text { otherwise } . \end{cases}$$
  1. Show that the moment generating function \(\mathrm { M } _ { X } ( t )\) of \(X\) is \(\frac { 4 } { ( 2 - t ) ^ { 2 } }\). You may assume that \(x \mathrm { e } ^ { - k x } \rightarrow 0\) as \(x \rightarrow + \infty\).
  2. What condition on \(t\) is needed in finding \(\mathrm { M } _ { X } ( t )\) ?
  3. \(Y\) is the sum of three independent observations of \(X\). Find the moment generating function of \(Y\), and use your answer to find \(\operatorname { Var } ( Y )\).
Pre-U Pre-U 9795/2 2016 June Q7
10 marks Challenging +1.8
7 A continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} \frac { 3 x ^ { 2 } } { k ^ { 3 } } & 0 \leqslant x \leqslant k \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a parameter.
  1. Find \(\mathrm { E } ( X )\). Hence show that \(\frac { 4 } { 3 } X\) is an unbiased estimator of \(k\). Three independent observations of \(X\) are denoted by \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\), and the largest value of \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\) is denoted by \(M\).
  2. Write down an expression for \(\mathrm { P } ( M \leqslant x )\), and hence show that the probability density function of \(M\) is $$f _ { M } ( x ) = \begin{cases} \frac { 9 x ^ { 8 } } { k ^ { 9 } } & 0 \leqslant x \leqslant k \\ 0 & \text { otherwise } . \end{cases}$$
  3. Find \(\mathrm { E } ( M )\) and use your answer to construct an unbiased estimator of \(k\) based on \(M\).
Pre-U Pre-U 9795/2 2016 June Q8
5 marks Moderate -0.3
8 A rough plane is inclined at \(20 ^ { \circ }\) to the horizontal. A particle of mass 0.4 kg is projected down the plane, along a line of greatest slope, at \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). After it has travelled 3 m down the plane its speed is \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). By considering the change in energy, find the magnitude of the frictional force, assumed constant.
Pre-U Pre-U 9795/2 2016 June Q9
4 marks Standard +0.3
9 \includegraphics[max width=\textwidth, alt={}, center]{1a89caec-6da8-4b83-9ffa-efc209ecbc8d-4_506_730_625_712} Particles \(P\) and \(Q\), of masses 1.2 kg and 1.5 kg respectively, are attached to the ends of a light inextensible string. The string passes through a small smooth ring which is attached to the ceiling but which is free to rotate. \(P\) rotates at \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\) in a horizontal circle of radius 0.12 m , and \(Q\) hangs vertically in equilibrium (see diagram). Determine
  1. the vertical distance below the ring at which \(P\) rotates,
  2. the value of \(\omega\).
Pre-U Pre-U 9795/2 2016 June Q10
8 marks Standard +0.8
10 A uniform ladder \(A B\) of length 5 m and mass 8 kg is placed at an angle \(\theta\) to the horizontal, with \(A\) on rough horizontal ground and \(B\) against a smooth vertical wall. The coefficient of friction between the ladder and the ground is 0.4 .
  1. By taking moments, find the smallest value of \(\theta\) for which the ladder is in equilibrium.
  2. A man of mass 75 kg stands on the ladder when \(\theta = 60 ^ { \circ }\). Find the greatest distance from \(A\) that he can stand without the ladder slipping.
Pre-U Pre-U 9795/2 2016 June Q11
6 marks Challenging +1.2
11 A car of mass 800 kg has a constant power output of 32 kW while travelling on a horizontal road. At time \(t \mathrm {~s}\) the car's speed is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the resistive force has magnitude \(20 v \mathrm {~N}\).
  1. Show that \(v\) satisfies the differential equation \(\frac { \mathrm { d } v } { \mathrm {~d} t } = \frac { 1600 - v ^ { 2 } } { 40 v }\).
  2. Given that \(v = 0\) when \(t = 0\), solve this differential equation to find \(v\) in terms of \(t\). State what the solution predicts as \(t\) becomes large.
Pre-U Pre-U 9795/2 2016 June Q12
8 marks Challenging +1.8
12 \includegraphics[max width=\textwidth, alt={}, center]{1a89caec-6da8-4b83-9ffa-efc209ecbc8d-5_205_200_264_497} \includegraphics[max width=\textwidth, alt={}, center]{1a89caec-6da8-4b83-9ffa-efc209ecbc8d-5_284_899_349_753} A white snooker ball of mass \(m\) moves with speed \(u\) towards a stationary black snooker ball of the same mass and radius. Taking the \(x\)-axis to be the line of centres of the two balls at the moment of collision, the direction of motion of the white ball before the collision makes an angle of \(30 ^ { \circ }\) with the positive \(x\)-axis (see diagram).
  1. Given that the coefficient of restitution is 0.9 , find the angle made with the \(x\)-axis by the velocity of the white ball after the collision.
  2. Show that after the collision the white ball cannot have a negative \(x\)-component of velocity whatever the value of the coefficient of restitution.
Pre-U Pre-U 9795/2 2016 June Q13
9 marks Challenging +1.2
13 A cricket ball is hit from a point \(P\) on a sloping field. The initial velocity of the ball is \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(40 ^ { \circ }\) above the field, which under the path of the ball slopes upwards at \(10 ^ { \circ }\) to the horizontal. Air resistance is to be ignored.
  1. Find the vertical height of the ball above the field after 2.5 seconds.
  2. The ball lands on the field at the point \(X\). Find the distance \(P X\).
Pre-U Pre-U 9795/2 2016 June Q14
14 marks Challenging +1.2
14 One end of a light elastic string of natural length 0.5 m and modulus of elasticity 3 N is attached to a ceiling at a point \(P\). A particle of mass 0.3 kg is attached to the other end of the string.
  1. Find the extension of the string when the particle hangs vertically in equilibrium. The particle is released from rest at \(P\) so that it falls vertically. Find
  2. the maximum extension of the string,
  3. the equation of motion for the particle when the string is stretched, in terms of the displacement \(x \mathrm {~m}\) below the equilibrium position,
  4. the time between the string first becoming stretched and next becoming unstretched again.
Pre-U Pre-U 9795/1 2016 Specimen Q6
7 marks Challenging +1.8
6 A group \(G\) has order 12.
  1. State, with a reason, the possible orders of the elements of \(G\). The identity element of \(G\) is \(e\), and \(x\) and \(y\) are distinct, non-identity elements of \(G\) satisfying the three conditions
    (1) \(x\) has order 6 ,
    (2) \(x ^ { 3 } = y ^ { 2 }\),
    (3) \(x y x = y\).
  2. Prove that \(y x ^ { 2 } y = x\).
  3. Prove that \(G\) is not a cyclic group.
Pre-U Pre-U 9795/1 2016 Specimen Q7
9 marks Challenging +1.2
7
  1. Use de Moivre's theorem to show that \(\tan 4 \theta = \frac { 4 t \left( 1 - t ^ { 2 } \right) } { 1 - 6 t ^ { 2 } + t ^ { 4 } }\), where \(\mathrm { t } = \tan \theta\).
  2. Given that \(\theta\) is the acute angle such that \(\tan \theta = \frac { 1 } { 5 }\), express \(\tan 4 \theta\) as a rational number in its simplest form, and verify that $$\frac { 1 } { 4 } \pi + \tan ^ { - 1 } \left( \frac { 1 } { 239 } \right) = 4 \tan ^ { - 1 } \left( \frac { 1 } { 5 } \right) .$$
Pre-U Pre-U 9795/1 2016 Specimen Q8
10 marks Standard +0.8
8 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 \(\mathrm { f } ( 1.1 )\).