Questions — Pre-U (885 questions)

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Pre-U Pre-U 9794/2 2015 June Q6
11 marks Moderate -0.3
6 A cup of tea is served at \(80 ^ { \circ } \mathrm { C }\) in a room which is kept at a constant \(20 ^ { \circ } \mathrm { C }\). The temperature, \(T ^ { \circ } \mathrm { C }\), of the tea after \(t\) minutes can be modelled by assuming that the rate of change of \(T\) is proportional to the difference in temperature between the tea and the room.
  1. Explain why the rate of change of the temperature in this model is given by \(\frac { \mathrm { d } T } { \mathrm {~d} t } = - k ( T - 20 )\), where \(k\) is a positive constant.
  2. Show by integration that the temperature of the tea after \(t\) minutes is given by \(T = 20 + 60 \mathrm { e } ^ { - k t }\).
  3. After 2 minutes the tea has cooled to \(60 ^ { \circ } \mathrm { C }\). Find the value of \(k\).
Pre-U Pre-U 9794/2 2015 June Q7
6 marks Standard +0.3
7 A curve is given parametrically by \(x = 3 t , y = 1 + t ^ { 3 }\) where \(t\) is any real number.
  1. Show that a cartesian equation for this curve is given by \(y = 1 + \frac { 1 } { 27 } x ^ { 3 }\). A second curve is given by \(y = x ^ { 2 } + 4 x - 19\).
  2. Given that the curves intersect at the point \(( 3,2 )\), find the coordinates of all the other points of intersection between the two curves.
Pre-U Pre-U 9794/2 2015 June Q8
5 marks Moderate -0.3
8 The function f is given by \(\mathrm { f } ( x ) = \frac { x ^ { 2 } } { 3 x ^ { 2 } - 1 }\), for \(x > 1\). Show that f is a decreasing function.
Pre-U Pre-U 9794/2 2015 June Q9
8 marks Standard +0.8
9 Find the equations of all the horizontal tangents to the curve with equation \(y ^ { 2 } = x ^ { 4 } - 4 x ^ { 3 } + 36\).
Pre-U Pre-U 9794/2 2015 June Q10
14 marks Challenging +1.2
10
  1. Show that \(\sin \left( 2 \theta + \frac { 1 } { 2 } \pi \right) = \cos 2 \theta\).
  2. Hence solve the equation \(\sin 3 \theta = \cos 2 \theta\) for \(0 \leqslant \theta \leqslant 2 \pi\).
  3. Show that \(\sin 3 \theta = 3 \sin \theta - 4 \sin ^ { 3 } \theta\). Hence, by writing \(\cos 2 \theta - \sin 3 \theta\) in terms of \(\sin \theta\), use your answer to part (ii) to determine the solutions of \(4 x ^ { 3 } - 2 x ^ { 2 } - 3 x + 1 = 0\).
Pre-U Pre-U 9794/2 2015 June Q11
11 marks Standard +0.3
11 \includegraphics[max width=\textwidth, alt={}, center]{2f48a6ee-e8ce-47e4-a07f-2c55a6904e7d-3_661_953_767_596} The diagram shows a circle, centre \(O\), radius \(r\). The points \(R\) and \(S\) lie on the circumference of the circle, and the line \(R T\) is a tangent to the circle at \(R\). The angle \(R O S\) is \(\theta\) radians where \(0 < \theta < \frac { 1 } { 2 } \pi\).
  1. Find expressions for the perimeter, \(P\), and the area, \(A\), of the shaded region in terms of \(r\) and \(\theta\).
  2. Hence show that \(A \neq r P\).
Pre-U Pre-U 9794/3 2015 June Q1
5 marks Moderate -0.8
1 The information below summarises the percentages of males unemployed ( \(x\) ) and the percentages of females unemployed ( \(y\) ) in 10 different locations in the UK. $$n = 10 \quad \Sigma x = 87.6 \quad \Sigma x ^ { 2 } = 804.34 \quad \Sigma y = 76.4 \quad \Sigma y ^ { 2 } = 596 \quad \Sigma x y = 684.02$$ Find the product-moment correlation coefficient for these data.
Pre-U Pre-U 9794/3 2015 June Q2
6 marks Standard +0.3
2 Jill is collecting picture cards given away in packets of a particular brand of breakfast cereal. There are five different cards in the complete set. Each packet contains one card which is equally likely to be any of the five cards in the set.
  1. Find the probability that Jill has a complete set of cards from the first five packets that she buys.
  2. At some point Jill needs just one more card to complete the set. Let \(X\) be the random variable that represents the number of additional packets that Jill will need to buy in order to complete the set.
    1. Write down the distribution of \(X\).
    2. State the expected number of additional packets that Jill will need to buy.
    3. Find the probability that Jill will need to buy at least 3 additional packets in order to complete the set.
Pre-U Pre-U 9794/3 2015 June Q3
4 marks Moderate -0.5
3 Jack's journey time, in minutes, to work each morning is modelled by the normal distribution \(\mathrm { N } \left( 43.2,6.3 ^ { 2 } \right)\).
  1. If Jack leaves home at 0810 , find the probability that he arrives at work by 0900 .
  2. Find the time by which Jack should leave home in order to be at least \(95 \%\) certain that he arrives at work by 0900 .
Pre-U Pre-U 9794/3 2015 June Q4
9 marks Moderate -0.3
4 At a sixth form college, the student council has 16 members made up as follows. There are 3 male and 3 female students from Year 12, and 6 male and 4 female students from Year 13. Two members of the council are chosen at random to represent the college at conference. Find the probability that the 2 members chosen are
  1. the same sex,
  2. the same sex and from the same year,
  3. from the same year given that they are the same sex.
Pre-U Pre-U 9794/3 2015 June Q5
12 marks Standard +0.3
5 A garden centre grows a particular variety of plant for sale. They sow 3 seeds in each pot and there are 6 pots in a tray. The probability that a seed germinates is 0.7 , independently of any other seeds.
  1. State the probability distribution of the number of seeds in a pot that germinate.
  2. Find the probability that, in a randomly chosen pot,
    1. exactly 2 seeds germinate,
    2. at least 1 seed germinates. After the seeds have germinated and become seedlings, some are removed (and discarded) so that there remains at most 1 seedling per pot.
    3. Write out the probability distribution of the number of seedlings per pot that remain.
    4. Find the probability that there is a seedling in every one of the 6 pots in a randomly chosen tray.
Pre-U Pre-U 9794/3 2015 June Q6
4 marks Moderate -0.8
6 \includegraphics[max width=\textwidth, alt={}, center]{9ddae838-2639-4952-bbc0-3944a81e5762-3_401_1224_1315_456} The diagram shows a barge being towed along a canal by a force of 240 N at an angle of \(25 ^ { \circ }\) to its direction of motion. A force, \(F \mathrm {~N}\), perpendicular to the direction of motion, is applied to the barge to keep it moving in the direction shown.
  1. Find the magnitude of \(F\).
  2. The mass of the barge is 1100 kg and there is a resistance force of 100 N parallel to the direction of motion. Find the acceleration of the barge.
Pre-U Pre-U 9794/3 2015 June Q7
8 marks Standard +0.3
7 A particle is projected from the origin with initial speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\theta\) above the horizontal. After 2 seconds the particle is at a point which is 18 m horizontally from the origin and 4 m above it.
  1. Show that \(\tan \theta = \frac { 4 } { 3 }\) and find \(u\).
  2. Find the horizontal range of the particle.
Pre-U Pre-U 9794/3 2015 June Q8
5 marks Moderate -0.8
8 A tram travels from stop \(A\) to stop \(B\), a distance of 300 m . First the tram starts from rest at \(A\) and accelerates uniformly at \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for 16 seconds. Then it travels at a constant speed and finally it slows down uniformly at \(1 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) coming to rest at \(B\).
  1. Sketch the velocity-time graph for the journey of the tram from \(A\) to \(B\).
  2. Find the speed of the tram and the distance travelled at the end of the first 16 seconds.
  3. Show that the journey from \(A\) to \(B\) takes 49.5 seconds.
Pre-U Pre-U 9794/3 2015 June Q9
7 marks Moderate -0.3
9 A particle of mass 0.5 kg moving on a smooth horizontal plane with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) collides directly with another particle of mass \(k \mathrm {~kg}\) (where \(k\) is a constant) which is at rest. After the collision the first particle comes to rest but the second particle moves off with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find \(v\) in terms of \(k\) and \(u\).
  2. The coefficient of restitution between the two particles is \(e\). Find \(e\) in terms of \(k\) only.
  3. Show that \(k \geqslant \frac { 1 } { 2 }\).
Pre-U Pre-U 9794/3 2015 June Q10
10 marks Standard +0.3
10 A particle is projected up a long smooth slope at a speed of \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The slope is at an angle \(\theta\) to the horizontal where \(\sin \theta = \frac { 1 } { 25 }\). After 2 seconds it passes a mark on the slope. Find the total time taken from the moment of projection until it passes the mark again and the total distance travelled in that time. {www.cie.org.uk} after the live examination series. }
Pre-U Pre-U 9795/2 2015 June Q1
4 marks Standard +0.8
1 The independent random variables \(X\) and \(Y\) are such that $$X \sim \mathrm {~N} ( \mu , 11 ) , \quad Y \sim \mathrm {~N} \left( 10 , \sigma ^ { 2 } \right) \quad \text { and } \quad 2 X - 5 Y \sim \mathrm {~N} ( 0,144 ) .$$ Find
  1. the values of \(\mu\) and \(\sigma ^ { 2 }\),
  2. \(\mathrm { P } ( X - Y > 10 )\).
Pre-U Pre-U 9795/2 2015 June Q2
8 marks Standard +0.3
2 The pH value, \(X\), which is a measure of acidity, was measured for soil taken from a random sample of 20 villages in which rhododendrons grow well. The results are summarised below, where \(\bar { x }\) denotes the sample mean. You may assume that the sample is selected from a normal population. $$\Sigma x = 114 \quad \Sigma ( x - \bar { x } ) ^ { 2 } = 2.382$$
  1. Calculate a \(98 \%\) confidence interval for the mean pH value in villages where rhododendrons grow well, giving 3 decimal places in your answer.
  2. Comment, justifying your answer, on a suggestion that the average pH value in villages where rhododendrons grow well is 5.5.
Pre-U Pre-U 9795/2 2015 June Q3
14 marks Challenging +1.2
3 The probability generating function of the random variable \(X\) is \(\frac { 1 } { 81 } \left( t + \frac { 2 } { t } \right) ^ { 4 }\).
  1. Use the probability generating function to find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
  2. The random variable \(Y\) is defined by \(Y = \frac { 1 } { 2 } ( X + 4 )\). By finding the probability distribution of \(X\), or otherwise, show that \(Y \sim \mathrm {~B} ( n , p )\), stating the values of \(n\) and \(p\).
Pre-U Pre-U 9795/2 2015 June Q4
11 marks Challenging +1.2
4
  1. (a) Derive the moment generating function for a Poisson distribution with mean \(\lambda\).
    (b) The independent random variables \(X\) and \(Y\) are such that \(X \sim \operatorname { Po } ( \mu )\) and \(Y \sim \operatorname { Po } ( v )\). Use moment generating functions to show that \(( X + Y ) \sim \operatorname { Po } ( \mu + v )\).
  2. The number of goals scored per match by Camford Academicals FC may be modelled by a Poisson distribution with mean 2. The number of goals scored against Camford during a match may be modelled by an independent Poisson distribution with mean \(k\). The probability that no goals are scored, by either side, in a match involving Camford is 0.045 . Find
    (a) the value of \(k\),
    (b) the probability that exactly 3 goals are scored against Camford in a match,
    (c) the probability that the total number of goals scored, in a match involving Camford, is between 2 and 5 inclusive.
Pre-U Pre-U 9795/2 2015 June Q5
11 marks Challenging +1.8
5 Each year a college has a large fixed number, \(n\), of places to fill. The probability, \(p\), that a randomly chosen student comes from abroad is constant. Using a suitable normal approximation and applying a continuity correction, it is calculated that the probability of more than 60 students coming from abroad is 0.0187 and the probability of fewer than 40 students coming from abroad is 0.0783 . Find the values of \(n\) and \(p\).
Pre-U Pre-U 9795/2 2015 June Q6
18 marks Challenging +1.8
6 The object distance, \(U \mathrm {~cm}\), and the image distance, \(V \mathrm {~cm}\), for a convex lens of focal length 40 cm are related by the lens law $$\frac { 1 } { U } + \frac { 1 } { V } = \frac { 1 } { 40 } .$$ The random variable \(U\) is uniformly distributed over the interval \(80 \leqslant u \leqslant 120\).
  1. Show that the probability density function of \(V\) is given by $$f ( v ) = \begin{cases} \frac { 40 } { ( v - 40 ) ^ { 2 } } & 60 \leqslant v \leqslant 80 \\ 0 & \text { otherwise } \end{cases}$$
  2. Find
    1. the median value of \(V\),
    2. the expected value of \(V\).
Pre-U Pre-U 9795/2 2015 June Q7
6 marks Challenging +1.2
7 \includegraphics[max width=\textwidth, alt={}, center]{86cc07e7-ea69-4480-96c8-82b818445199-3_599_499_1279_822} A light inextensible string of length \(4 a\) has one end fixed at a point \(P\) and the other end fixed at a point \(Q\), which is vertically below \(P\) and at a distance \(3 a\) from \(P\). A small smooth ring \(R\) of mass \(m\) is threaded on the string. \(R\) moves in a horizontal circle with centre \(Q\) and with the string taut (see diagram).
  1. Show that \(Q R = \frac { 7 } { 8 } a\).
  2. Find the speed of \(R\) in terms of \(a\) and \(g\).
Pre-U Pre-U 9795/2 2015 June Q8
4 marks Standard +0.3
8 \includegraphics[max width=\textwidth, alt={}, center]{86cc07e7-ea69-4480-96c8-82b818445199-4_182_803_264_671} A light spring of modulus of elasticity 8 N and natural length 0.4 m has one end fixed to a smooth horizontal table at a fixed point \(L\). A particle of mass 0.2 kg is attached to the other end of the spring and pulled out horizontally to a point \(M\) on the table, so that the spring is extended by 0.2 m . The particle is then released from rest. The mid-point of \(L M\) is \(N\) and the point \(O\) is on \(L M\) such that \(L O = 0.4 \mathrm {~m}\) (see diagram).
  1. Show that the particle moves in simple harmonic motion with centre \(O\) and state the exact period of its motion.
  2. Find the exact time taken for the particle to move directly from \(M\) to \(N\).
Pre-U Pre-U 9795/2 2015 June Q9
6 marks Standard +0.3
9 A car of mass 800 kg is descending a straight hill which is inclined at \(2 ^ { \circ }\) to the horizontal. The car passes through the points \(A\) and \(B\) with speeds \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. The distance \(A B\) is 400 m .
  1. Assuming that resistances to motion are negligible, calculate the work done by the car's engine over the distance from \(A\) to \(B\).
  2. Assuming also that the driving force produced by the car's engine remains constant, calculate the power of the car's engine at the mid-point of \(A B\).