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Pre-U Pre-U 9794/1 2013 June Q10
6 marks Standard +0.3
10 Two intersecting straight lines have equations $$\frac { x - 5 } { 4 } = \frac { y - 11 } { 3 } = \frac { z - 7 } { - 5 } \quad \text { and } \quad \frac { x - 9 } { - 2 } = \frac { y - 4 } { 1 } = \frac { z + 4 } { 4 } .$$ Find the coordinates of their point of intersection.
Pre-U Pre-U 9794/1 2013 June Q11
10 marks Moderate -0.3
11 A curve has parametric equations given by $$x = 2 \sin \theta , \quad y = \cos 2 \theta$$
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - 2 \sin \theta\).
  2. Hence find the equation of the tangent to the curve at \(\theta = \frac { 1 } { 2 } \pi\).
  3. Find the cartesian equation of the curve.
Pre-U Pre-U 9794/1 2013 June Q12
6 marks Standard +0.3
12
  1. Prove the identity \(\frac { 1 } { ( x + h ) ^ { 2 } } - \frac { 1 } { x ^ { 2 } } \equiv \frac { - 2 h x - h ^ { 2 } } { x ^ { 2 } ( x + h ) ^ { 2 } }\).
  2. Given that \(\mathrm { f } ( x ) = x ^ { - 2 }\), use differentiation from first principles to find an expression for \(\mathrm { f } ^ { \prime } ( x )\).
Pre-U Pre-U 9794/1 2013 June Q13
12 marks Standard +0.8
13 By first factorising completely \(x ^ { 3 } + x ^ { 2 } - 5 x + 3\), find \(\int \frac { 2 x ^ { 2 } + x + 1 } { x ^ { 3 } + x ^ { 2 } - 5 x + 3 } \mathrm {~d} x\).
Pre-U Pre-U 9794/2 2013 June Q1
4 marks Easy -1.3
1 Vectors \(\mathbf { u }\) and \(\mathbf { v }\) are given by \(\mathbf { u } = \binom { 4 } { 6 }\) and \(\mathbf { v } = \binom { - 3 } { 2 }\).
  1. Find \(\mathbf { u } + \mathbf { v }\) and \(\mathbf { u } - \mathbf { v }\).
  2. Show that \(| \mathbf { u } + \mathbf { v } | = | \mathbf { u } - \mathbf { v } |\).
Pre-U Pre-U 9794/2 2013 June Q2
7 marks Easy -1.3
2
  1. An arithmetic sequence has first term 3 and common difference 2. Find the twenty-first term of this sequence.
  2. Find the sum to infinity of a geometric progression with first term 162 and second term 54.
  3. A sequence is given by the recurrence relation \(u _ { 1 } = 3 , u _ { n + 1 } = 2 - u _ { n } , n = 1,2,3 , \ldots\). Find \(u _ { 2 } , u _ { 3 }\), \(u _ { 4 } , u _ { 5 }\) and describe the behaviour of this sequence.
Pre-U Pre-U 9794/2 2013 June Q3
7 marks Easy -1.2
3
  1. Express \(x ^ { 2 } + 2 x - 3\) in the form \(( x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are integers to be found.
  2. Sketch the graph of \(y = x ^ { 2 } + 2 x - 3\) giving the coordinates of the vertex and of any intersections with the coordinate axes.
Pre-U Pre-U 9794/2 2013 June Q4
10 marks Moderate -0.3
4
  1. Verify that \(z = - 1\) is a root of the equation \(z ^ { 3 } + 5 z ^ { 2 } + 9 z + 5 = 0\).
  2. Find the two complex roots of the equation \(z ^ { 3 } + 5 z ^ { 2 } + 9 z + 5 = 0\).
  3. Show all three roots on an Argand diagram.
Pre-U Pre-U 9794/2 2013 June Q5
8 marks Moderate -0.3
5 The curve \(C\) has equation \(x ^ { 2 } + x y + y ^ { 2 } = 19\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { - 2 x - y } { x + 2 y }\).
  2. Hence find the equation of the normal to \(C\) at the point \(( 2,3 )\) in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
Pre-U Pre-U 9794/2 2013 June Q6
14 marks Moderate -0.3
6 The table below gives the population of breeding pairs of red kites in Yorkshire from 2001 to 2008.
Year20012002200320042005200620072008
Number of breeding pairs810162433404769
Source: \href{http://www.gigrin.co.uk}{www.gigrin.co.uk}
The following model for the population has been proposed: $$N = a \times b ^ { t } ,$$ where \(N\) is the number of breeding pairs \(t\) years after the year 2000, and \(a\) and \(b\) are constants.
  1. Show that the model can be transformed to a linear relationship between \(\log _ { 10 } N\) and \(t\).
  2. On graph paper, plot \(\log _ { 10 } N\) against \(t\) and draw by eye a line of best fit. Use your line to estimate the values of \(a\) and \(b\) in the equation for \(N\) in terms of \(t\).
  3. What values of \(N\) does the model give for the years 2008 and 2020?
  4. In which year will the number of breeding pairs first exceed 500 according to the model?
  5. Comment on the suitability of the model to predict the population of breeding pairs of red kites in Yorkshire.
Pre-U Pre-U 9794/2 2013 June Q7
7 marks Moderate -0.3
7 It is given that \(y = x ^ { 2 } \mathrm { e } ^ { - x }\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = x \mathrm { e } ^ { - x } ( 2 - x )\).
  2. Hence find the exact coordinates of the stationary points on the curve \(y = x ^ { 2 } \mathrm { e } ^ { - x }\).
Pre-U Pre-U 9794/2 2013 June Q8
4 marks Moderate -0.8
8 Evaluate the following, giving your answers in exact form.
  1. \(\sum _ { n = 1 } ^ { 30 } \frac { 1 } { n } - \sum _ { n = 2 } ^ { 29 } \frac { 1 } { n }\).
  2. \(\sum _ { n = 1 } ^ { 100 } n \times ( - 1 ) ^ { n }\).
Pre-U Pre-U 9794/2 2013 June Q9
12 marks Challenging +1.2
9
  1. Prove that \(\operatorname { cosec } 2 x - \cot 2 x \equiv \tan x\) and hence find an exact value for \(\tan \left( \frac { 3 } { 8 } \pi \right)\).
  2. Find the exact value of \(\int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 3 } { 8 } \pi } ( \operatorname { cosec } 2 x - \cot 2 x ) ^ { 2 } \mathrm {~d} x\).
Pre-U Pre-U 9794/2 2013 June Q10
11 marks Standard +0.3
10 A tank with vertical sides and rectangular cross-section is initially full of water. The water is leaking out of a hole in the base of the tank at a rate which is proportional to the square root of the depth of the water. \(V \mathrm {~m} ^ { 3 }\) is the volume of water in the tank at time \(t\) hours.
  1. Show that \(\frac { \mathrm { d } V } { \mathrm {~d} t } = a \sqrt { V }\), where \(a\) is a constant.
  2. Given that the tank is half full after one hour, show that \(V = V _ { 0 } \left( \left( \frac { 1 } { \sqrt { 2 } } - 1 \right) t + 1 \right) ^ { 2 }\), where \(V _ { 0 } \mathrm {~m} ^ { 3 }\) is the initial volume of water in the tank.
  3. Hence show that the tank will be empty after approximately 3 hours and 25 minutes.
Pre-U Pre-U 9794/3 2013 June Q1
4 marks Easy -1.3
1 Pupils at a certain school carried out a survey of traffic passing the school during a two-hour period one morning. One pupil recorded the number of people in each of the first 100 cars. Her results were as follows.
Number of people12345
Number of cars482614102
Find the mean and the standard deviation of the number of people per car in her sample.
Pre-U Pre-U 9794/3 2013 June Q2
4 marks Moderate -0.8
2 Events \(A\) and \(B\) are such that \(\mathrm { P } ( A ) = \frac { 1 } { 2 } , \mathrm { P } ( A \cup B ) = \frac { 5 } { 6 }\) and \(\mathrm { P } ( B \mid A ) = \frac { 1 } { 4 }\).
Find
  1. \(\mathrm { P } ( A \cap B )\),
  2. \(\mathrm { P } ( B )\).
Pre-U Pre-U 9794/3 2013 June Q3
12 marks Moderate -0.8
3 At a local athletics club, data on the ages of the members and their times to run a 10 km course are recorded. For a random sample of 25 club members aged between 20 and 60, their ages ( \(x\) years) and times ( \(y\) minutes) are summarised as follows. $$n = 25 \quad \Sigma x = 1002 \quad \Sigma x ^ { 2 } = 43508 \quad \Sigma y = 1865 \quad \Sigma y ^ { 2 } = 142749 \quad \Sigma x y = 77532$$
  1. Calculate the product moment correlation coefficient for these data.
  2. Show that the equation of the least squares regression line of \(y\) on \(x\) is \(y = 0.83 x + 41.28\), where the coefficients are given correct to 2 decimal places.
  3. Use the equation given in part (ii) to estimate the time taken by someone who is
    1. 50 years old,
    2. 65 years old. Comment on the validity of each of these estimates.
Pre-U Pre-U 9794/3 2013 June Q4
10 marks Standard +0.3
4 A tomato grower grows just one variety of tomatoes. The weights of these tomatoes are found to be normally distributed with a mean of 85.1 grams and a standard deviation of 3.4 grams.
  1. Find the probability that a randomly chosen tomato of this variety weighs less than 80 grams.
  2. The grower puts the tomatoes in packs of 6 . Find the probability that, in a randomly chosen pack of 6 , at most one tomato weighs less than 80 grams.
  3. The grower supplies consignments of 250 packs of these tomatoes to a retailer. For a randomly chosen consignment, find the expected number of packs having more than one tomato weighing less than 80 grams.
Pre-U Pre-U 9794/3 2013 June Q5
10 marks Standard +0.3
5 A game is played with cards, each of which has a single digit printed on it. Eleanor has 7 cards with the digits \(1,1,2,3,4,5,6\) on them.
  1. How many different 7-digit numbers can be made by arranging Eleanor's cards?
  2. Eleanor is going to select 5 of the 7 cards and use them to form a 5 -digit number. How many different 5-digit numbers are possible?
Pre-U Pre-U 9794/3 2013 June Q6
13 marks Moderate -0.3
6 A particle travels along a straight line. Its velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) after \(t\) seconds is given by $$v = t ^ { 3 } - 6 t ^ { 2 } + 8 t \text { for } 0 \leqslant t \leqslant 4$$ When \(t = 0\) the particle is at rest at the point \(P\).
  1. Find the times (other than \(t = 0\) ) when the particle is at rest. Sketch the velocity-time graph for \(0 \leqslant t \leqslant 4\).
  2. Find the acceleration of the particle when \(t = 2\).
  3. Find an expression for the displacement of the particle from \(P\) after \(t\) seconds. Hence state its displacement from \(P\) when \(t = 2\) and find its average speed between \(t = 0\) and \(t = 2\).
Pre-U Pre-U 9794/3 2013 June Q7
8 marks Standard +0.3
7 A particle \(A\) of mass \(4 m\), on a smooth horizontal plane, is moving with speed \(u\) directly towards another particle \(B\), of mass \(2 m\), which is at rest. The coefficient of restitution between the two particles is \(e\).
  1. Show that, after the collision, the velocity of \(A\) is \(\frac { 1 } { 3 } ( 2 - e ) u\) and find the velocity of \(B\).
  2. Hence write down their velocities in the case when \(e = \frac { 1 } { 2 }\). Particle \(B\) now collides directly with a third particle \(C\), of mass \(m\), which is at rest. The coefficient of restitution in both collisions is \(\frac { 1 } { 2 }\).
  3. Use your answers to part (ii) to find the velocities of \(A , B\) and \(C\) after the second collision has taken place.
  4. Explain briefly whether any further collisions take place.
Pre-U Pre-U 9794/3 2013 June Q8
10 marks Standard +0.3
8 A particle is projected from a point \(O\) with initial speed \(U\) at an angle \(\theta\) above the horizontal. At time \(t\) after projection the position of the particle is \(( x , y )\) relative to horizontal and vertical axes through \(O\).
  1. Write down expressions for \(x\) and \(y\) at time \(t\). Hence derive the cartesian equation of the trajectory of the particle.
  2. A player in a cricket match throws the ball with speed \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to another player who is 45 metres away. Assume that the players throw and catch the ball at the same height above the ground. Show that there are two possible trajectories and find their respective angles of projection. [4]
  3. Describe briefly one advantage of each trajectory.
Pre-U Pre-U 9794/3 2013 June Q9
9 marks Standard +0.3
9 A particle of mass \(m \mathrm {~kg}\) rests in equilibrium on a rough horizontal table. There is a string attached to the particle. The tension in the string is \(T \mathrm {~N}\) at an angle of \(\theta\) to the horizontal, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{2e3f056c-58a2-4466-94ea-3fb873e54752-4_205_547_1027_799}
  1. Copy and complete the diagram to show all the forces acting on the particle.
  2. The coefficient of friction between the particle and the table is \(\mu\) and the particle is on the point of slipping. Show that \(T = \frac { \mu m g } { \cos \theta + \mu \sin \theta }\).
  3. Given that \(\mu = 0.75\), find the value of \(\theta\) for which \(T\) is a minimum.
Pre-U Pre-U 9794/1 2013 November Q1
Easy -1.8
1 Solve the simultaneous equations $$\begin{aligned} x ^ { 2 } + y ^ { 2 } & = 5 \\ y & = 2 x \end{aligned}$$
Pre-U Pre-U 9794/1 2013 November Q2
Easy -1.2
2 Find the equation of the line perpendicular to the line \(y = 5 x\) which passes through the point \(( 2,11 )\). Give your answer in the form \(a x + b y = c\) where \(a , b\) and \(c\) are integers to be found.