Questions — Pre-U (885 questions)

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Pre-U Pre-U 9795/1 2019 Specimen Q12
6 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 9795/2 2019 Specimen Q1
4 marks Standard +0.3
1 The discrete random variable X has probability generating function \(\mathrm { G } _ { X } ( t )\) given by $$G _ { X } ( t ) = a t \left( t + \frac { 1 } { t } \right) ^ { 3 } ,$$ 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 2019 Specimen Q2
5 marks 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 2019 Specimen Q3
4 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 (a) and (b).
Pre-U Pre-U 9795/2 2019 Specimen Q4
3 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 \(\operatorname { Var } ( \overline { a X } + b \bar { Y } )\) 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 2019 Specimen Q5
3 marks Standard +0.3
5 The random variable \(X\) has probability density function \(\mathrm { f } ( \mathrm { x } )\), 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 2019 Specimen Q6
2 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 (b) in relation to the sketch in part (a).
  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.
Pre-U Pre-U 9795/2 2019 Specimen Q7
4 marks Moderate -0.5
7 A child of mass 20 kg slides down a rough slope of length 16 m against a constant frictional force \(F \mathrm {~N}\). Starting with an initial speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at a point 8 m above the ground, she reaches the ground with a speed of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the value of \(F\).
Pre-U Pre-U 9795/2 2019 Specimen Q8
2 marks Standard +0.3
8 \includegraphics[max width=\textwidth, alt={}, center]{adf5bd3c-5408-421d-b7d5-dea2d0f0185b-4_604_734_1512_667} 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. Find the tension in the string in terms of \(m , l\) and \(\omega\).
  2. Show that \(\omega ^ { 2 } h = g\).
  3. Deduce an expression in terms of \(g\) and \(h\) for the time taken for \(P\) to complete one full circle during its motion.
Pre-U Pre-U 9795/2 2019 Specimen Q9
6 marks Standard +0.3
9 The diagram shows a uniform \(\operatorname { rod } A B\) of length 40 cm and mass 2 kg placed with the end \(A\) resting against a smooth vertical wall and the end \(B\) on rough horizontal ground. The angle between \(A B\) and the horizontal is \(60 ^ { \circ }\). \includegraphics[max width=\textwidth, alt={}, center]{adf5bd3c-5408-421d-b7d5-dea2d0f0185b-5_661_655_390_705} Given that the value of the coefficient of friction between the rod and the ground is 0.2 , determine whether the rod slips.
Pre-U Pre-U 9795/2 2019 Specimen Q10
1 marks Standard +0.3
10 A cyclist and her bicycle 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 \(v \mathrm {~ms} ^ { - 1 }\) and the resistance to motion is \(k v \mathrm {~N}\), where \(k\) is a constant.
  1. Given that the steady speed at which the cyclist can move is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), show that \(k = \frac { 3 } { 4 }\).
  2. 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 \(3 \mathrm {~ms} ^ { - 1 }\) to a speed of \(7 \mathrm {~ms} ^ { - 1 }\).
Pre-U Pre-U 9795/2 2019 Specimen Q11
5 marks Challenging +1.8
11 \includegraphics[max width=\textwidth, alt={}, center]{adf5bd3c-5408-421d-b7d5-dea2d0f0185b-6_438_951_255_559} 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 2019 Specimen Q12
4 marks Challenging +1.2
12 A particle is projected from the origin with speed \(20 \mathrm {~ms} ^ { - 1 }\) at an angle \(\alpha\) above the horizontal.
  1. Prove that the equation of its trajectory is $$y = x \tan \alpha - \frac { x ^ { 2 } } { 80 } \left( l + \tan ^ { 2 } \alpha \right) .$$
  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 of \(30 ^ { \circ }\) to the horizontal. The line \(l\), with equation \(y = x \tan 30 ^ { \circ }\), is a line of greatest slope in the plane. The particle is projected from the origin with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point on the plane, in the vertical plane containing \(l\). By considering the intersection of \(l\) with the curve \(y = 20 - \frac { x ^ { 2 } } { 80 }\), find the maximum range up this inclined plane.
Pre-U Pre-U 9795/2 2019 Specimen Q13
2 marks Challenging +1.8
13 Two light strings, each of natural length \(l\) 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 \(6 l\) 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 the tension in each string is \(\frac { 6 m g } { l } \left( \sqrt { 9 l ^ { 2 } + x ^ { 2 } } - l \right)\).
  2. Show that $$\ddot { x } = - \frac { 12 g x } { l } \left( 1 - \frac { l } { \sqrt { 9 l ^ { 2 } + x ^ { 2 } } } \right)$$
  3. Given that throughout the motion \(\frac { x ^ { 2 } } { l ^ { 2 } }\) is small enough to be negligible, show that the equation of motion is approximately $$\ddot { x } = - \frac { 8 g x } { l } .$$
  4. Given that the initial speed of \(P\) is \(\sqrt { \frac { g l } { 200 } }\), find the time taken for the particle to travel a distance of \(\frac { 1 } { 80 } l\).
Pre-U Pre-U 9794/1 2019 Specimen Q1
2 marks Easy -1.8
1 A circle has equation \(( x - 4 ) ^ { 2 } + ( y + 7 ) ^ { 2 } = 64\).
  1. Write down the coordinates of the centre and the radius of the circle. Two points, \(A\) and \(B\), lie on the circle and have coordinates \(( 4,1 )\) and \(( 12 , - 7 )\) respectively.
  2. Find the coordinates of the midpoint of the chord \(A B\).
Pre-U Pre-U 9794/1 2019 Specimen Q2
2 marks Moderate -0.8
2 The equation of a curve is \(y = x ^ { 3 } - 2 x ^ { 2 } - 4 x + 3\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Hence find the coordinates of the stationary points on the curve.
Pre-U Pre-U 9794/1 2019 Specimen Q3
1 marks Easy -1.3
3 Let \(\mathrm { f } ( x ) = x ^ { 2 }\) and \(\mathrm { g } ( x ) = 7 x - 2\) for all real values of \(x\).
  1. Give a reason why f has no inverse function.
  2. Write down an expression for \(\operatorname { gf } ( x )\).
  3. Find \(\mathrm { g } ^ { - 1 } ( x )\).
  4. Explain the relationship between the graph of \(y = \mathrm { g } ( x )\) and \(y = \mathrm { g } ^ { - 1 } ( x )\).
Pre-U Pre-U 9794/1 2019 Specimen Q4
1 marks Moderate -0.8
4
  1. Show that \(x = 2\) is a root of the equation \(2 x ^ { 3 } - x ^ { 2 } - 15 x + 18 = 0\).
  2. Hence solve the equation \(2 x ^ { 3 } - x ^ { 2 } - 15 x + 18 = 0\).
Pre-U Pre-U 9794/1 2019 Specimen Q7
4 marks Moderate -0.3
7
  1. Express \(\frac { 8 x - 1 } { ( 2 x - 1 ) ( x - 1 ) }\) in the form \(\frac { A } { 2 x - 1 } + \frac { B } { x + 1 }\) where \(A\) and \(B\) are constants.
  2. Hence show that \(\equiv \frac { 5 x - 1 } { \overline { 2 } } \frac { 8 x - 1 ) ( x + 1 ) } { ( 2 x - \ln 24 \text {. } }\)
Pre-U Pre-U 9794/1 2019 Specimen Q9
2 marks Easy -1.2
9 The complex number 3-4i is denoted by \(z\). Giving your answers in the form \(x + \mathrm { i } y\), and showing clearly how you obtain them, find
  1. \(2 z + z ^ { * }\),
  2. \(\frac { 5 } { z }\).
  3. Show \(z\) and \(z ^ { * }\) on an Argand diagram.
Pre-U Pre-U 9794/1 2019 Specimen Q10
4 marks Standard +0.3
10
  1. Prove that \(\cot \theta + \frac { \sin \theta } { 1 + \cos \theta } = \operatorname { cosec } \theta\).
  2. Hence solve the equation \(\cot \left( \theta + \frac { \neq } { 4 } \right) + \frac { \sin \left( \theta + \frac { \neq } { 4 } \right) } { 1 + \cos \left( \theta + \frac { \neq } { 4 } \right) } = \frac { 5 } { 2 }\) for \(0 \leqslant \theta \leqslant 2 \pi\).
Pre-U Pre-U 9794/1 2019 Specimen Q11
7 marks Standard +0.8
11 An arithmetic progression has first term \(a\) and common difference \(d\). The first, ninth and fourteenth terms are, respectively, the first three terms of a geometric progression with common ratio \(r\), where \(r \neq 1\).
  1. Find \(d\) in terms of \(a\) and show that \(r = \frac { 5 } { 8 }\).
  2. Find the sum to infinity of the geometric progression in terms of \(a\).
Pre-U Pre-U 9794/1 2019 Specimen Q12
2 marks Standard +0.8
12
  1. Use integration by parts to show that \(\int \ln x \mathrm {~d} x = x \ln x - x + c\).
  2. Find
    1. \(\quad \int ( \ln x ) ^ { 2 } \mathrm {~d} x\),
    2. \(\quad \int \frac { \ln ( \ln x ) } { x } \mathrm {~d} x\).
Pre-U Pre-U 9794/2 2019 Specimen Q2
5 marks Moderate -0.5
2 \includegraphics[max width=\textwidth, alt={}, center]{48b63de9-f022-4881-a187-f08e3c7d9f1a-2_399_940_952_561} 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 2019 Specimen Q3
5 marks Easy -1.2
3
  1. The points \(A\) and \(B\) have coordinates \(( - 4,4 )\) and \(( 8,1 )\) respectively. Find the equation of the line \(A B\). Give your answer in the form \(y = m x + c\).
  2. Determine, with a reason, whether the line \(y = 7 - 4 x\) is perpendicular to the line \(A B\).