Questions — Pre-U (885 questions)

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Pre-U Pre-U 9795/1 2013 June Q1
4 marks Standard +0.3
1 By completing the square, or otherwise, find the exact value of \(\int _ { 2 } ^ { 6 } \frac { 1 } { x ^ { 2 } - 6 x + 12 } \mathrm {~d} x\).
Pre-U Pre-U 9795/1 2013 June Q2
4 marks Challenging +1.2
2 Use the standard Maclaurin series expansions given in the List of Formulae MF20 to show that $$\frac { 1 } { 2 } \ln \left( \frac { 1 + x } { 1 - x } \right) \equiv \tanh ^ { - 1 } x \text { for } - 1 < x < 1$$
Pre-U Pre-U 9795/1 2013 June Q3
2 marks Standard +0.3
3 The curve \(C\) has equation \(y = \frac { x + 1 } { x ^ { 2 } - 4 }\).
  1. Show that the gradient of \(C\) is always negative.
  2. Sketch \(C\), showing all significant features.
Pre-U Pre-U 9795/1 2013 June Q4
2 marks Standard +0.8
4
  1. Find a vector which is perpendicular to both of the vectors $$\mathbf { d } _ { 1 } = \mathbf { i } + 2 \mathbf { j } + 4 \mathbf { k } \quad \text { and } \quad \mathbf { d } _ { 2 } = 9 \mathbf { i } - 3 \mathbf { j } + \mathbf { k } .$$
  2. Determine the shortest distance between the skew lines with equations $$\mathbf { r } = 2 \mathbf { i } + 4 \mathbf { j } + 3 \mathbf { k } + \lambda ( \mathbf { i } + 2 \mathbf { j } + 4 \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = \mathbf { i } + \mathbf { j } + 10 \mathbf { k } + \mu ( 9 \mathbf { i } - 3 \mathbf { j } + \mathbf { k } ) .$$
Pre-U Pre-U 9795/1 2013 June Q5
5 marks Standard +0.8
5 Let \(z = \cos \theta + \mathrm { i } \sin \theta\).
  1. Prove the result \(z ^ { n } - \frac { 1 } { z ^ { n } } = 2 \mathrm { i } \sin n \theta\).
  2. Use this result to express \(\sin ^ { 5 } \theta\) in the form \(A \sin 5 \theta + B \sin 3 \theta + C \sin \theta\), for constants \(A , B\) and \(C\) to be determined.
Pre-U Pre-U 9795/1 2013 June Q6
8 marks Standard +0.8
6 The curve \(P\) has polar equation \(r = \frac { 1 } { 1 - \sin \theta }\) for \(0 \leqslant \theta < 2 \pi , \theta \neq \frac { 1 } { 2 } \pi\).
  1. Determine, in the form \(y = \mathrm { f } ( x )\), the cartesian equation of \(P\).
  2. Sketch \(P\).
  3. Evaluate \(\int _ { \pi } ^ { 2 \pi } \frac { 1 } { ( 1 - \sin \theta ) ^ { 2 } } \mathrm {~d} \theta\).
Pre-U Pre-U 9795/1 2013 June Q7
7 marks Standard +0.8
7
  1. Express \(x ^ { 3 } + y ^ { 3 }\) in terms of \(( x + y )\) and \(x y\).
  2. The equation \(t ^ { 2 } - 3 t + \frac { 8 } { 9 } = 0\) has roots \(\alpha\) and \(\beta\).
    1. Determine the value of \(\alpha ^ { 3 } + \beta ^ { 3 }\).
    2. Hence express 19 as the sum of the cubes of two positive rational numbers.
Pre-U Pre-U 9795/1 2013 June Q8
8 marks Challenging +1.8
8 Let \(G = \left\{ g _ { 1 } , g _ { 2 } , g _ { 3 } , \ldots , g _ { n } \right\}\) be a finite abelian group of order \(n\) under a multiplicative binary operation, where \(g _ { 1 } = e\) is the identity of \(G\).
  1. Let \(x \in G\). Justify the following statements:
    1. \(x g _ { i } = x g _ { j } \Leftrightarrow g _ { i } = g _ { j }\);
    2. \(\left\{ x g _ { 1 } , x g _ { 2 } , x g _ { 3 } , \ldots , x g _ { n } \right\} = G\).
    3. By considering the product of all \(G\) 's elements, and using the result of part (i)(b), prove that \(x ^ { n } = e\) for each \(x \in G\).
    4. Explain why
      (a) this does not imply that all elements of \(G\) have order \(n\),
      (b) this argument cannot be used to justify the same result for non-abelian groups.
Pre-U Pre-U 9795/1 2013 June Q9
8 marks Challenging +1.8
9 The plane transformation \(T\) is the composition (in this order) of
  • a reflection in the line \(y = x \tan \frac { 1 } { 8 } \pi\); followed by
  • a shear parallel to the \(y\)-axis, mapping \(( 1,0 )\) to \(( 1,2 )\); followed by
  • a clockwise rotation through \(\frac { 1 } { 4 } \pi\) radians about the origin; followed by
  • a shear parallel to the \(x\)-axis, mapping \(( 0,1 )\) to \(( - 2,1 )\).
Determine the matrix \(\mathbf { M }\) which represents \(T\), and hence give a full geometrical description of \(T\) as a single plane transformation.
Pre-U Pre-U 9795/1 2013 June Q10
18 marks Challenging +1.3
10
  1. Given that \(y = k x \cos x\) is a particular integral for the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + y = 4 \sin x$$ determine the value of \(k\) and find the general solution of this differential equation.
  2. The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + y ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + x y = 5 x - 19$$
    1. Given that \(y = 2\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1\) when \(x = 1\), find the value of \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\) when \(x = 1\).
    2. Deduce the Taylor series expansion for \(y\) in ascending powers of \(( x - 1 )\), up to and including the term in \(( x - 1 ) ^ { 3 }\), and use this series to find an approximation correct to 3 decimal places for the value of \(y\) when \(x = 1.1\).
Pre-U Pre-U 9795/1 2013 June Q11
13 marks Standard +0.3
11
  1. Determine \(p\) and \(q\) given that \(( p + \mathrm { i } q ) ^ { 2 } = 63 - 16 \mathrm { i }\) and that \(p\) and \(q\) are real.
  2. Let \(\mathrm { f } ( z ) = z ^ { 3 } - A z ^ { 2 } + B z - C\) for complex numbers \(A , B\) and \(C\).
    1. Given that the cubic equation \(\mathrm { f } ( z ) = 0\) has roots \(\alpha = - 7 \mathrm { i } , \beta = 3 \mathrm { i }\) and \(\gamma = 4\), determine each of \(A , B\) and \(C\).
    2. Find the roots of the equation \(\mathrm { f } ^ { \prime } ( z ) = 0\).
Pre-U Pre-U 9795/1 2013 June Q12
11 marks Challenging +1.2
12 Given \(y = x \mathrm { e } ^ { 2 x }\),
  1. find the first four derivatives of \(y\) with respect to \(x\),
  2. conjecture an expression for \(\frac { \mathrm { d } ^ { n } y } { \mathrm {~d} x ^ { n } }\) in the form \(( a x + b ) \mathrm { e } ^ { 2 x }\), where \(a\) and \(b\) are functions of \(n\),
  3. prove by induction that your result holds for all positive integers \(n\).
Pre-U Pre-U 9795/1 2013 June Q13
4 marks Challenging +1.8
13
  1. Use the definitions \(\tanh \theta = \frac { \mathrm { e } ^ { \theta } - \mathrm { e } ^ { - \theta } } { \mathrm { e } ^ { \theta } + \mathrm { e } ^ { - \theta } }\) and \(\operatorname { sech } \theta = \frac { 2 } { \mathrm { e } ^ { \theta } + \mathrm { e } ^ { - \theta } }\) to prove the results
    1. \(\tanh ^ { 2 } \theta \equiv 1 - \operatorname { sech } ^ { 2 } \theta\),
    2. \(\frac { \mathrm { d } } { \mathrm { d } \theta } ( \tanh \theta ) = \operatorname { sech } ^ { 2 } \theta\).
    3. Let \(I _ { n } = \int _ { 0 } ^ { \alpha } \tanh ^ { 2 n } \theta \mathrm {~d} \theta\) for \(n \geqslant 0\), where \(\alpha > 0\).
      (a) Show that \(I _ { n - 1 } - I _ { n } = \frac { \tanh ^ { 2 n - 1 } \alpha } { 2 n - 1 }\) for \(n \geqslant 1\). Given that \(\alpha = \frac { 1 } { 2 } \ln 3\),
      (b) evaluate \(I _ { 0 }\),
    4. use the method of differences to show that \(I _ { n } = \frac { 1 } { 2 } \ln 3 - \sum _ { r = 1 } ^ { n } \frac { \left( \frac { 1 } { 2 } \right) ^ { 2 r - 1 } } { 2 r - 1 }\) and deduce the sum of the infinite series \(\sum _ { r = 0 } ^ { \infty } \frac { 1 } { ( 2 r + 1 ) 4 ^ { r } }\).
Pre-U Pre-U 9795/2 2013 June Q1
3 marks Moderate -0.3
1 A company hires out narrowboats on a canal. It may be assumed that demands to hire a narrowboat occur independently and randomly at a constant mean rate of 25 per week. Using a suitable normal approximation, find
  1. the probability that 15 or fewer narrowboats are hired out during a certain week,
  2. the number of narrowboats that the company needs to have available for a week in order that the probability of running out of boats is 0.05 or less.
Pre-U Pre-U 9795/2 2013 June Q2
9 marks Moderate -0.3
2
  1. The heights of boys in Year 9 are normally distributed with mean 156 cm and standard deviation 8 cm . The heights of girls in Year 10 are, independently, normally distributed with mean 160 cm and standard deviation 7 cm . Find the probability that the mean height of a random sample of 9 boys in Year 9 exceeds the mean height of a random sample of 16 girls in Year 10.
  2. State why the distributions of the sample means are normally distributed.
Pre-U Pre-U 9795/2 2013 June Q3
9 marks Standard +0.3
3
  1. Given that \(X \sim \operatorname { Po } ( 5 )\), find \(\mathrm { P } ( X > 6 \mid X > 3 )\).
  2. Given that \(Y \sim \operatorname { Po } ( \lambda )\) and \(\mathrm { P } ( Y \leqslant 1 ) = \frac { 1 } { 2 }\), show that \(\lambda\) satisfies the equation \(\lambda = \ln \{ 2 ( 1 + \lambda ) \}\).
  3. Starting with a suitable approximation from the table of cumulative Poisson probabilities, use iteration to find \(\lambda\) correct to 3 decimal places.
Pre-U Pre-U 9795/2 2013 June Q4
10 marks Standard +0.8
4 The broadband speed in village \(P\) was measured on 8 randomly selected occasions and the broadband speed in village \(Q\) was measured on 6 randomly selected occasions. The results, measured in megabits per second, are shown below.
Village \(P :\)4.83.52.93.74.24.65.13.3
Village \(Q :\)2.41.92.33.12.72.9
  1. Calculate a \(90 \%\) confidence interval for the difference in mean broadband speed in these two villages.
  2. State two assumptions that you have made in carrying out the calculation.
Pre-U Pre-U 9795/2 2013 June Q5
8 marks Standard +0.3
5 The discrete random variable \(X\) has probability generating function given by $$\mathrm { G } _ { X } ( t ) = k \left( 5 t ^ { - 1 } + 3 + 2 t ^ { 2 } \right) ,$$ where \(k\) is a constant.
  1. Find
    1. the value of \(k\),
    2. the modal value of \(X\).
    3. The random variables \(X _ { 1 }\) and \(X _ { 2 }\) are independent observations of \(X\).
      (a) Write down the probability generating function of \(Y\), where \(Y = X _ { 1 } + X _ { 2 }\).
      (b) Use your answer to part (ii)(a) to find \(\mathrm { E } ( Y )\) and \(\operatorname { Var } ( Y )\).
Pre-U Pre-U 9795/2 2013 June Q6
14 marks Challenging +1.2
6 A rectangle of area \(Y \mathrm {~m} ^ { 2 }\) has a perimeter of 16 m and a side of length \(X \mathrm {~m}\), where \(X\) is a random variable with probability density function, f, given by $$f ( x ) = \begin{cases} \frac { 1 } { 2 } & 0 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$
  1. Obtain the cumulative distribution function, F , of \(X\).
  2. Show that $$16 - Y = ( 4 - X ) ^ { 2 }$$ and deduce that the probability density function of the random variable \(Y\) is $$g ( y ) = \begin{cases} \frac { 1 } { 4 \sqrt { 16 - y } } & 0 \leqslant y \leqslant 12 \\ 0 & \text { otherwise } \end{cases}$$
  3. Find the median of \(Y\).
  4. Find \(\mathrm { E } ( Y )\).
Pre-U Pre-U 9795/2 2013 June Q7
8 marks Standard +0.3
7 Find the power required to pump \(3 \mathrm {~m} ^ { 3 }\) of water per minute from a depth of 25 m and deliver it through a circular pipe of diameter 10 cm . Assume that friction may be neglected and that the density of water is \(1000 \mathrm {~kg} \mathrm {~m} ^ { - 3 }\).
Pre-U Pre-U 9795/2 2013 June Q8
9 marks Challenging +1.2
8 A particle, \(P\), is moving in a straight line with simple harmonic motion about a centre \(O\). When \(P\) is at the point \(A , 2 \mathrm {~m}\) from \(O\), it has speed \(4 \sqrt { 5 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When \(P\) is at the point \(B , \sqrt { 5 } \mathrm {~m}\) from \(O\), it has speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the amplitude and period of the motion.
  2. Given that \(A\) and \(B\) are on opposite sides of \(O\), find the time taken for \(P\) to travel directly from \(A\) to \(B\).
Pre-U Pre-U 9795/2 2013 June Q9
10 marks Challenging +1.2
9 A particle of mass 2 kg is moving along the \(x\)-axis, which is horizontal, against a resistive force which is proportional to the cube of the speed of the particle at any instant. At time \(t\) seconds the particle's velocity is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its displacement is \(x \mathrm {~m}\). When \(t = 0 , x = 0 , v = 4\) and the retardation is \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  1. Show that $$\frac { 1 } { v } = \frac { x + 8 } { 32 } .$$
  2. Find the time taken to cover the first 8 metres.
Pre-U Pre-U 9795/2 2013 June Q10
6 marks Standard +0.8
10 Ship \(A\) is 15 km due south of ship \(B\). Ship \(B\) is travelling at \(20 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) on a bearing of \(300 ^ { \circ }\). Ship \(A\) is travelling at \(16 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). Find
  1. the bearing, to the nearest degree, that \(A\) must take in order to get as close as possible to \(B\), [4]
  2. the time, in minutes, that it takes for the ships to be as close as possible.
Pre-U Pre-U 9795/2 2013 June Q11
11 marks Standard +0.3
11 \includegraphics[max width=\textwidth, alt={}, center]{742ef62b-bd72-45b4-88e3-70399632e9d6-4_384_524_587_808} One end of a light inextensible string of length 10 m is attached to a fixed point \(O\). A particle of mass 5 kg is attached to the other end of the string. The particle rests in equilibrium below \(O\). The particle is pulled aside until the string makes an angle of \(60 ^ { \circ }\) with the downward vertical and released from rest (see diagram). At the instant when the string makes an angle \(\cos ^ { - 1 } \left( \frac { 4 } { 5 } \right)\) with the downward vertical,
  1. find the speed of the particle and the tension in the string,
  2. show that the magnitude of the acceleration of the particle is \(6 \sqrt { 2 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
Pre-U Pre-U 9795/2 2013 June Q12
6 marks Challenging +1.2
12 \includegraphics[max width=\textwidth, alt={}, center]{742ef62b-bd72-45b4-88e3-70399632e9d6-4_247_801_1535_671} A small smooth sphere is projected from a point \(A\) across a smooth horizontal surface. The sphere strikes a smooth vertical wall at the point \(P\). The acute angle between the direction of motion of the sphere and the wall is \(\theta\). After the impact, the sphere passes through the point \(B\), where angle \(A P B = \phi\) (see diagram). The coefficient of restitution between the sphere and the wall is \(e\).
  1. Given that \(\theta = \tan ^ { - 1 } 3\) and \(\phi = 90 ^ { \circ }\), find the exact value of \(e\).
  2. Given instead that \(e = \frac { 2 } { 3 }\) and \(\phi = 45 ^ { \circ }\), show that \(\theta = \tan ^ { - 1 } 3\).