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Pre-U Pre-U 9795/1 2012 June Q4
9 marks Standard +0.8
4 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 \(C\).
Pre-U Pre-U 9795/1 2012 June Q5
6 marks Standard +0.8
5
  1. Write down the \(2 \times 2\) matrices which represent the following plane transformations:
    1. an anticlockwise rotation about the origin through an angle \(\alpha\);
    2. a reflection in the line \(y = x \tan \left( \frac { 1 } { 2 } \beta \right)\).
    3. A reflection in the \(x - y\) plane in the line \(y = x \tan \left( \frac { 1 } { 2 } \theta \right)\) is followed by a reflection in the line \(y = x \tan \left( \frac { 1 } { 2 } \phi \right)\). Show that the composition of these two reflections (in this order) is a rotation and describe this rotation fully.
Pre-U Pre-U 9795/1 2012 June 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) \(\quad x\) 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 2012 June 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 \(t = \tan \theta\).
  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 2012 June Q8
11 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 \(\mathrm { f } ( 1.1 )\).
Pre-U Pre-U 9795/1 2012 June Q9
9 marks Challenging +1.2
9
  1. Show that the substitution \(u = \frac { 1 } { y ^ { 3 } }\) transforms the differential equation \(\frac { \mathrm { d } y } { \mathrm {~d} x } + y = 3 x y ^ { 4 }\) into $$\frac { \mathrm { d } u } { \mathrm {~d} x } - 3 u = - 9 x$$
  2. Solve the differential equation \(\frac { \mathrm { d } y } { \mathrm {~d} x } + y = 3 x y ^ { 4 }\), given that \(y = \frac { 1 } { 2 }\) when \(x = 0\). Give your answer in the form \(y ^ { 3 } = \mathrm { f } ( x )\).
Pre-U Pre-U 9795/1 2012 June Q10
2 marks Standard +0.3
10 The line \(L\) has equation \(\mathbf { r } = \left( \begin{array} { r } 1 \\ - 3 \\ 2 \end{array} \right) + \lambda \left( \begin{array} { l } 3 \\ 4 \\ 6 \end{array} \right)\) and the plane \(\Pi\) has equation \(\mathbf { r } \cdot \left( \begin{array} { r } 2 \\ - 6 \\ 3 \end{array} \right) = k\).
  1. Given that \(L\) lies in \(\Pi\), determine the value of \(k\).
  2. Find the coordinates of the point, \(Q\), in \(\Pi\) which is closest to \(P ( 10,2 , - 43 )\). Deduce the shortest distance from \(P\) to \(\Pi\).
  3. Find, in the form \(a x + b y + c z = d\), where \(a , b , c\) and \(d\) are integers, an equation for the plane which contains both \(L\) and \(P\).
Pre-U Pre-U 9795/1 2012 June Q11
11 marks Standard +0.8
11 The complex number \(w = ( \sqrt { 3 } - 1 ) + \mathrm { i } ( \sqrt { 3 } + 1 )\).
  1. Determine, showing full working, the exact values of \(| w |\) and \(\arg w\).
    [0pt] [You may use the result that \(\tan \left( \frac { 5 } { 12 } \pi \right) = 2 + \sqrt { 3 }\).]
  2. (a) Find, in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), the three roots, \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\), of the equation \(z ^ { 3 } = w\).
    (b) Determine \(z _ { 1 } z _ { 2 } z _ { 3 }\) in the form \(a + \mathrm { i } b\).
    (c) Mark the points representing \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\) on a sketch of the Argand diagram. Show that they form an equilateral triangle, \(\Delta _ { 1 }\), and determine the side-length of \(\Delta _ { 1 }\).
    (d) The points representing \(k z _ { 1 } , k z _ { 2 }\) and \(k z _ { 3 }\) form \(\Delta _ { 2 }\), an equilateral triangle which is congruent to \(\Delta _ { 1 }\), and one of whose vertices lies on the positive real axis. Write down a suitable value for the complex constant \(k\).
Pre-U Pre-U 9795/1 2012 June Q12
15 marks 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 9795/1 2012 June Q13
6 marks Challenging +1.8
13 Define the repunit number, \(R _ { n }\), to be the positive integer which consists of a string of \(n 1\) 's. Thus, $$R _ { 1 } = 1 , \quad R _ { 2 } = 11 , \quad R _ { 3 } = 111 , \quad \ldots , \quad R _ { 7 } = 1111111 , \quad \ldots , \text { etc. }$$ Use induction to prove that, for all integers \(n \geqslant 5\), the number $$13579 \times R _ { n }$$ contains a string of ( \(n - 4\) ) consecutive 7's.
Pre-U Pre-U 9795/2 2012 June Q1
5 marks Standard +0.8
1 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 \(\operatorname { Var } ( X )\).
Pre-U Pre-U 9795/2 2012 June Q2
9 marks Standard +0.3
2 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 2012 June Q3
10 marks Standard +0.3
3 Small amounts of a potentially hazardous chemical are discharged into a river from a nearby industrial site. A random sample of size 6 was taken from the river and the concentration of the chemical present in each item was measured in grams per litre. The results are shown below. $$\begin{array} { l l l l l l } 1.64 & 1.53 & 1.78 & 1.60 & 1.73 & 1.77 \end{array}$$
  1. Assuming that the sample was taken from a normal distribution with known variance 0.01 , construct a \(99 \%\) confidence interval for the mean concentration of the chemical present in the river.
  2. If instead the sample was taken from a normal distribution, but with unknown variance, construct a revised \(99 \%\) confidence interval for the mean concentration of the chemical present in the river.
  3. If the mean concentration of the chemical in the river exceeds 1.8 grams per litre, then remedial action needs to be taken. Comment briefly on the need for remedial action in the light of the results in parts (i) and (ii).
Pre-U Pre-U 9795/2 2012 June Q4
10 marks Challenging +1.3
4
  1. The random variable \(X\) has the distribution \(\operatorname { Po } ( \lambda )\). Prove that the probability generating function, \(\mathrm { G } _ { X } ( t )\), is given by $$\mathrm { G } _ { X } ( t ) = \mathrm { e } ^ { \lambda ( t - 1 ) } .$$
  2. The independent random variables \(X\) and \(Y\) have distributions \(\operatorname { Po } ( \lambda )\) and \(\operatorname { Po } ( \mu )\) respectively. Use probability generating functions to show that the distribution of \(X + Y\) is \(\operatorname { Po } ( \lambda + \mu )\).
  3. Given that \(X \sim \operatorname { Po } ( 1.5 )\) and \(Y \sim \operatorname { Po } ( 2.5 )\), find \(\mathrm { P } ( X \leqslant 2 \mid X + Y = 4 )\).
Pre-U Pre-U 9795/2 2012 June Q5
11 marks Standard +0.3
5
  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
    1. at least 15 times,
    2. no more than 12 times.
    3. 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 2012 June Q6
12 marks Standard +0.3
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 $$f ( x ) = \begin{cases} \frac { 4 } { 27 } x ^ { 2 } ( 3 - x ) & 0 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$
  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. Given that \(\sigma ^ { 2 } = 0.36\), find \(\mathrm { P } ( | X - \mu | < \sigma )\), where \(\mu\) and \(\sigma ^ { 2 }\) denote the mean and the variance of \(X\) respectively.
Pre-U Pre-U 9795/2 2012 June Q7
8 marks Standard +0.3
7 A cyclist and her machine have a combined mass of 90 kg and she is riding along a straight horizontal road. She is working at a constant power of 75 W . At time \(t\) seconds her speed is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the resistance to motion is \(k v \mathrm {~N}\), where \(k\) is a constant.
  1. If the cyclist's maximum steady speed is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), show that \(k = \frac { 3 } { 4 }\).
  2. Use Newton's second law to show that $$\frac { 25 } { v } - \frac { v } { 4 } = 30 \frac { \mathrm {~d} v } { \mathrm {~d} t } .$$
  3. Find the time taken for the cyclist to accelerate from a speed of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to a speed of \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
Pre-U Pre-U 9795/2 2012 June Q8
8 marks Challenging +1.2
8 \includegraphics[max width=\textwidth, alt={}, center]{d8ca5464-435f-45e0-8e19-1830415a7c60-4_757_729_260_708} An aircraft carrier, \(A\), is heading due north at \(40 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). A destroyer, \(D\), which is 8 km south-west of \(A\), is ordered to take up a position 3 km east of \(A\) as quickly as possible. The speed of \(D\) is \(60 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) (see diagram). Find the bearing, \(\theta\), of the course that \(D\) should take, giving your answer to the nearest degree.
Pre-U Pre-U 9795/2 2012 June Q9
9 marks Standard +0.3
9 \includegraphics[max width=\textwidth, alt={}, center]{d8ca5464-435f-45e0-8e19-1830415a7c60-4_666_816_1384_662} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(l\). The other end of the string is attached to a fixed point \(A\). The particle moves with constant angular speed \(\omega\) in a horizontal circle whose centre is at a distance \(h\) vertically below \(A\) (see diagram).
  1. Show that however fast the particle travels \(A P\) will never become horizontal, and that the tension in the string is always greater than the weight of the particle.
  2. Find the tension in the string in terms of \(m , l\) and \(\omega\).
  3. Show that \(\omega ^ { 2 } h = g\) and calculate \(\omega\) when \(h\) is 0.5 m .
Pre-U Pre-U 9795/2 2012 June Q10
12 marks Challenging +1.8
10 \includegraphics[max width=\textwidth, alt={}, center]{d8ca5464-435f-45e0-8e19-1830415a7c60-5_432_949_258_598} A smooth sphere \(P\) of mass \(3 m\) is at rest on a smooth horizontal table. A second smooth sphere \(Q\) of mass \(m\) and the same radius as \(P\) is moving along the table towards \(P\) and strikes it obliquely (see diagram). After the collision, the directions of motion of the two spheres are perpendicular.
  1. Find the coefficient of restitution.
  2. Given that one sixth of the original kinetic energy is lost as a result of the collision, find the angle between the initial direction of motion of \(Q\) and the line of centres.
Pre-U Pre-U 9795/2 2012 June Q11
11 marks Challenging +1.2
11 Two light strings, each of natural length \(a\) and modulus of elasticity \(6 m g\), are attached at their ends to a particle \(P\) of mass \(m\). The other ends of the strings are attached to two fixed points \(A\) and \(B\), which are at a distance \(6 a\) apart on a smooth horizontal table. Initially \(P\) is at rest at the mid-point of \(A B\). The particle is now given a horizontal impulse in the direction perpendicular to \(A B\). At time \(t\) the displacement of \(P\) from the line \(A B\) is \(x\).
  1. Show that $$\ddot { x } = - \frac { 12 g x } { a } \left( 1 - \frac { a } { \sqrt { 9 a ^ { 2 } + x ^ { 2 } } } \right) .$$
  2. Given that \(\frac { x } { a }\) is small throughout the motion, show that the equation of motion is approximately $$\ddot { x } = - \frac { 8 g x } { a }$$ and state the period of the simple harmonic motion that this equation represents.
  3. Given that the initial speed of \(P\) is \(\sqrt { \frac { g a } { 200 } }\), show that the amplitude of the simple harmonic motion is \(\frac { 1 } { 40 } a\).
Pre-U Pre-U 9795/2 2012 June Q12
12 marks Challenging +1.8
12 A projectile is launched from the origin with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\alpha\) above the horizontal.
  1. Prove that the equation of its trajectory is $$y = x \tan \alpha - \frac { x ^ { 2 } } { 80 } \left( 1 + \tan ^ { 2 } \alpha \right)$$
  2. Regarding the equation of the trajectory as a quadratic equation in \(\tan \alpha\), show that \(\tan \alpha\) has real values provided that $$y \leqslant 20 - \frac { x ^ { 2 } } { 80 }$$
  3. A plane is inclined at an angle \(\beta\) to the horizontal. The line \(l\), with equation \(y = x \tan \beta\), is a line of greatest slope in the plane. A particle is projected from a point on the plane, in the vertical plane containing \(l\). By considering the intersection of \(l\) with the bounding parabola \(y = 20 - \frac { x ^ { 2 } } { 80 }\), deduce that the maximum range up, or down, this inclined plane is \(\frac { 40 } { 1 + \sin \beta }\), or \(\frac { 40 } { 1 - \sin \beta }\), respectively.
Pre-U Pre-U 9794/1 2012 June Q1
5 marks Easy -1.2
1 The first term of a geometric progression is 16 and the common ratio is 0.8 .
  1. Calculate the sum of the first 12 terms.
  2. Find the sum to infinity.
Pre-U Pre-U 9794/1 2012 June Q2
6 marks Moderate -0.8
2 Let \(\mathrm { f } ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 13 x + 15\).
  1. Show that \(\mathrm { f } ( 1 ) = 0\) and hence factorise \(x ^ { 3 } - 3 x ^ { 2 } - 13 x + 15\) completely.
  2. Hence solve the equation \(x ^ { 3 } - 3 x ^ { 2 } - 13 x + 15 = 0\).
Pre-U Pre-U 9794/1 2012 June Q3
6 marks Moderate -0.8
3 The equation of a curve is \(y = x ^ { 3 } + x ^ { 2 } - x + 3\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Hence find the coordinates of the stationary points on the curve.