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Pre-U Pre-U 9794/1 2016 Specimen Q4
6 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 2016 Specimen Q5
4 marks Moderate -0.3
5 The coefficient of \(x ^ { 3 }\) in the expansion of \(( 2 + a x ) ^ { 5 }\) is 10 times the coefficient of \(x ^ { 2 }\) in \(\left( 1 + \frac { a x } { 3 } \right) ^ { 4 }\). Find \(a\).
Pre-U Pre-U 9794/1 2016 Specimen Q6
6 marks Moderate -0.5
6 Solve the simultaneous equations $$x + y = 1 , \quad x ^ { 2 } - 2 x y + y ^ { 2 } = 9$$
Pre-U Pre-U 9794/1 2016 Specimen Q7
9 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 \(\int _ { 2 } ^ { 5 } \frac { 8 x - 1 } { ( 2 x - 1 ) ( x + 1 ) } \mathrm { d } x = \ln 24\).
Pre-U Pre-U 9794/1 2016 Specimen Q8
4 marks Moderate -0.3
8 Given that the equation \(x ^ { 3 } + 2 x - 7 = 0\) has a root between \(x = 1\) and \(x = 2\), use the Newton-Raphson formula with \(x _ { \mathrm { o } } = 1\) to find this root correct to 3 decimal places.
Pre-U Pre-U 9794/1 2016 Specimen Q9
6 marks Easy -1.3
9 The complex number \(3 - 4 \mathrm { i }\) 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 2016 Specimen Q10
8 marks Standard +0.3
10
  1. Prove that \(\cot \theta + \frac { \sin \theta } { 1 + \cos \theta } = \operatorname { cosec } \theta\).
  2. Hence solve the equation \(\cot \left( \theta + \frac { \pi } { 4 } \right) + \frac { \sin \left( \theta + \frac { \pi } { 4 } \right) } { 1 + \cos \left( \theta + \frac { \pi } { 4 } \right) } = \frac { 5 } { 2 }\) for \(0 \leqslant \theta \leqslant 2 \pi\).
Pre-U Pre-U 9794/1 2016 Specimen Q11
9 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 2016 Specimen Q12
11 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 2016 Specimen Q1
9 marks Easy -1.3
1
  1. Express each of the following as a single logarithm.
    1. \(\log _ { a } 5 + \log _ { a } 3\)
    2. \(5 \log _ { b } 2 - 3 \log _ { b } 4\)
  2. Express \(\left( 9 a ^ { 4 } \right) ^ { - \frac { 1 } { 2 } }\) as an algebraic fraction in its simplest form.
  3. Show that \(\frac { 3 \sqrt { 3 } - 1 } { 2 \sqrt { 3 } - 3 } = \frac { 15 + 7 \sqrt { 3 } } { 3 }\).
Pre-U Pre-U 9794/2 2016 Specimen Q2
5 marks Moderate -0.5
2 \includegraphics[max width=\textwidth, alt={}, center]{ac5bf967-8b97-4bf3-991f-28c3ec7a25da-2_399_933_968_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 2016 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 \(A B\).
Pre-U Pre-U 9794/2 2016 Specimen Q4
7 marks Moderate -0.3
4
  1. Show that \(2 x ^ { 2 } - 10 x - 3\) may be expressed in the form \(a ( x + b ) ^ { 2 } + c\) where \(a , b\) and \(c\) are real numbers to be found. Hence write down the co-ordinates of the minimum point on the curve.
  2. Solve the equation \(4 x ^ { 4 } - 13 x ^ { 2 } + 9 = 0\).
Pre-U Pre-U 9794/2 2016 Specimen Q5
9 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{ac5bf967-8b97-4bf3-991f-28c3ec7a25da-3_570_736_292_667} The diagram shows a sector of a circle, \(O M N\). The angle \(M O N\) is \(2 x\) radians, the radius of the circle is \(r\) and \(O\) is the centre.
  1. Find expressions, in terms of \(r\) and \(x\), for the area, \(A\), and the perimeter, \(P\), of the sector.
  2. Given that \(P = 20\), show that \(A = \frac { 100 x } { ( 1 + x ) ^ { 2 } }\).
  3. Find \(\frac { \mathrm { d } A } { \mathrm {~d} x }\), and hence find the value of \(x\) for which the area of the sector is a maximum.
Pre-U Pre-U 9794/2 2016 Specimen Q6
8 marks Moderate -0.3
6 Diane is given an injection that combines two drugs, Antiflu and Coldcure. At time \(t\) hours after the injection, the concentration of Antiflu in Diane's bloodstream is \(3 \mathrm { e } ^ { - 0.02 t }\) units and the concentration of Coldcure is \(5 \mathrm { e } ^ { - 0.07 t }\) units. Each drug becomes ineffective when its concentration falls below 1 unit.
  1. Show that Coldcure becomes ineffective before Antiflu.
  2. Sketch, on the same diagram, the graphs of concentration against time for each drug.
  3. 20 hours after the first injection, Diane is given a second injection. Determine the concentration of Coldcure 10 hours later.
Pre-U Pre-U 9794/2 2016 Specimen Q7
6 marks Standard +0.3
7 Solve the differential equation \(x ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } = \sec y\) given that \(y = \frac { \pi } { 6 }\) when \(x = 4\) giving your answer in the form \(y = \mathrm { f } ( x )\).
Pre-U Pre-U 9794/2 2016 Specimen Q8
8 marks Moderate -0.3
8 The parametric equations of a curve are $$x = \mathrm { e } ^ { 2 t } - 5 t , \quad y = \mathrm { e } ^ { 2 t } - 3 t .$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
  2. Find the equation of the tangent to the curve at the point when \(t = 0\), giving your answer in the form \(a y + b x + c = 0\) where \(a , b\) and \(c\) are integers.
Pre-U Pre-U 9794/2 2016 Specimen Q9
7 marks Moderate -0.3
9 The points \(A\) and \(B\) have position vectors \(\mathbf { a }\) and \(\mathbf { b }\) relative to an origin \(O\), where \(\mathbf { a } = 5 \mathbf { i } + 4 \mathbf { j } - 2 \mathbf { k }\) and \(\mathbf { b } = - 7 \mathbf { i } + 3 \mathbf { j } + \mathbf { k }\).
  1. Find the length of \(A B\).
  2. Use a scalar product to find angle \(O A B\).
Pre-U Pre-U 9794/2 2016 Specimen Q10
15 marks Standard +0.8
10 A curve has equation $$y = \mathrm { e } ^ { a x } \cos b x$$ where \(a\) and \(b\) are constants.
  1. Show that, at any stationary points on the curve, \(\tan b x = \frac { a } { b }\).
  2. \includegraphics[max width=\textwidth, alt={}, center]{ac5bf967-8b97-4bf3-991f-28c3ec7a25da-4_622_896_957_333} Values of related quantities \(x\) and \(y\) were measured in an experiment and plotted on a graph of \(y\) against \(x\), as shown in the diagram. Two of the points, labelled \(A\) and \(B\), have coordinates \(( 0,1 )\) and \(( 0.2 , - 0.8 )\) respectively. A third point labelled C has coordinates ( \(0.3,0.04\) ). Attempts were then made to find the equation of a curve which fitted closely to these three points, and two models were proposed. In the first model the equation is \(y = \mathrm { e } ^ { - x } \cos 15 x\).
    In the second model the equation is \(y = f \cos ( \lambda x ) + \mathrm { g }\), where the constants \(f , \lambda\), and \(g\) are chosen to give a maximum precisely at the point \(A ( 0,1 )\) and a minimum precisely at the point \(B ( 0.2 , - 0.8 )\). By calculating suitable values evaluate the suitability of the two models.
Pre-U Pre-U 9794/3 2016 Specimen Q1
5 marks Moderate -0.8
1 The times for a motorist to travel from home to work are normally distributed with a mean of 24 minutes and a standard deviation of 4 minutes. Find the probability that a particular trip from home to work takes
  1. more than 27 minutes,
  2. between 20 and 25 minutes.
Pre-U Pre-U 9794/3 2016 Specimen Q2
11 marks Moderate -0.8
2
  1. A music club has 200 members. 75 members play the piano, 130 members like Elgar, and 30 members do not play the piano, nor do they like Elgar.
    1. Calculate the probability that a member chosen at random plays the piano but does not like Elgar.
    2. Calculate the probability that a member chosen at random plays the piano given that this member likes Elgar.
  2. The music club is organising a concert. The programme is to consist of 7 pieces of music which are to be selected from 9 classical pieces and 6 modern pieces. Find the number of different concert programmes than can be produced if
    1. there are no restrictions,
    2. the programme must consist of 5 classical pieces and 2 modern pieces,
    3. there are to be more modern pieces than classical pieces.
Pre-U Pre-U 9794/3 2016 Specimen Q3
5 marks Easy -1.2
3 The table shows fuel economy figures in miles per gallon (mpg) for some new cars.
CarABCDEFGHIJKLMNO
Mpg574034331117302731203524262332
  1. Find the median and quartiles for the mpg of these fifteen cars.
  2. Use the values in part (i) to identify any cars for which the mpg is an outlier.
Pre-U Pre-U 9794/3 2016 Specimen Q4
6 marks Moderate -0.8
4 A survey into left-handedness found that 13\% of the population of the world are left-handed.
  1. State the assumptions necessary for it to be appropriate to model the number of left-handed children in a class of 20 children using the binomial distribution \(\mathrm { B } ( 20,0.13 )\).
  2. Assuming that this binomial model is appropriate, calculate the probability that fewer than \(13 \%\) of the 20 children are left-handed.
Pre-U Pre-U 9794/3 2016 Specimen Q5
11 marks Moderate -0.8
5 James plays an arcade game. Each time he plays, he puts a \(\pounds 1\) coin in the slot to start the game. The possible outcomes of each game are as follows: James loses the game with a probability of 0.7 and the machine pays out nothing, James draws the game with a probability of 0.25 and the machine pays out a \(\pounds 1\) coin, James wins the game with a probability of 0.05 and the machine pays out ten \(\pounds 1\) coins. The outcomes can be modelled by a random variable \(X\) representing the number of \(\pounds 1\) coins gained at the end of a game.
  1. Construct a probability distribution table for \(X\).
  2. Show that \(\mathrm { E } ( X ) = - 0.25\) and find \(\operatorname { Var } ( X )\). James starts off with \(10 \pounds 1\) coins and decides to play exactly 10 games.
  3. Find the expected number of \(\pounds 1\) coins that James will have at the end of his 10 games.
  4. Find the probability that after his 10 games James will have at least \(10 \pounds 1\) coins left.
Pre-U Pre-U 9794/3 2016 Specimen Q6
6 marks Easy -1.3
6 \includegraphics[max width=\textwidth, alt={}, center]{01bd6354-3514-4dad-901b-7ecbe155b2c7-4_572_672_456_701} The diagram shows two horizontal forces \(\mathbf { P }\) and \(\mathbf { Q }\) acting at the origin \(O\) of rectangular coordinates \(O x y\). The components of \(\mathbf { P }\) in the \(x\) - and \(y\)-directions are 12 N and 17 N respectively. The components of \(\mathbf { Q }\) in the \(x\) - and \(y\)-directions are - 5 N and 7 N respectively.
  1. Write down the components, in the \(x\) - and \(y\)-directions, of the resultant of \(\mathbf { P }\) and \(\mathbf { Q }\).
  2. Hence, or otherwise, calculate the magnitude of this resultant and the angle the resultant makes with the positive \(x\)-axis.