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Pre-U Pre-U 9794/1 Specimen Q5
4 marks Moderate -0.3
5
  1. Show that the equation \(4 - \frac { 4 } { \operatorname { cosec } x } = 3 \cos ^ { 2 } x\) can be expressed in the form $$3 \sin ^ { 2 } x - 4 \sin x + 1 = 0$$
  2. Hence find all values of \(x\) for which \(0 < x < \pi\) that satisfy the equation $$4 - \frac { 4 } { \operatorname { cosec } x } = 3 \cos ^ { 2 } x$$
Pre-U Pre-U 9794/1 Specimen Q6
6 marks Moderate -0.3
6 The equation \(x ^ { 3 } - x - 1 = 0\) has exactly one real root in the interval \(0 \leq x \leq 3\).
  1. Denoting this root by \(\alpha\), find the integer \(n\) such that \(n < \alpha < n + 1\).
  2. Taking \(n\) as a first approximation, use the Newton-Raphson method to find \(\alpha\), correct to 2 decimal places. You must show the result of each iteration correct to an appropriate degree of accuracy.
Pre-U Pre-U 9794/1 Specimen Q7
8 marks Standard +0.3
7 Express \(\frac { 1 - 6 x - 2 x ^ { 2 } } { ( x + 2 ) \left( x ^ { 2 } + 1 \right) }\) in the form \(\frac { A } { x + 2 } + \frac { B x + C } { x ^ { 2 } + 1 }\) where the numerical values of \(A , B\) and \(C\) are to be found. Hence show that \(\int _ { 0 } ^ { 1 } \frac { 1 - 6 x - 2 x ^ { 2 } } { ( x + 2 ) \left( x ^ { 2 } + 1 \right) } \mathrm { d } x = \ln 3 - \frac { 5 } { 2 } \ln 2\).
Pre-U Pre-U 9794/1 Specimen Q8
9 marks Standard +0.3
8
  1. Show that the lines $$\mathbf { r } = - 3 \mathbf { i } + \mathbf { j } - 5 \mathbf { k } + \lambda ( \mathbf { i } + 2 \mathbf { j } + \mathbf { 6 } \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = 4 \mathbf { i } + \mathbf { j } + 5 \mathbf { k } + \mu ( - 3 \mathbf { i } + \mathbf { j } - 2 \mathbf { k } )$$ intersect and write down the coordinates of their point of intersection.
  2. Find in degrees the obtuse angle between the two lines.
Pre-U Pre-U 9794/1 Specimen Q9
5 marks Moderate -0.3
9
  1. Show that \(z = ( 1 + \mathrm { i } )\) is a root of the cubic equation \(3 z ^ { 3 } - 8 z ^ { 2 } + 10 z - 4 = 0\).
  2. Show that the equation \(3 z ^ { 3 } - 8 z ^ { 2 } + 10 z - 4 = 0\) has a quadratic factor with real coefficients and hence solve this equation completely.
Pre-U Pre-U 9794/1 Specimen Q10
7 marks Standard +0.8
10
  1. Show that \(\frac { \mathrm { d } } { \mathrm { d } x } ( \sin 3 x - 3 x \cos 3 x ) = 9 x \sin 3 x\). The curve shown in the figure below is part of the graph of the function \(y = x \sin 3 x\). \includegraphics[max width=\textwidth, alt={}, center]{3e4281d1-dbad-46a2-bbb7-97706bda2dfa-3_508_1136_1939_466}
  2. Show that \(\int _ { 0 } ^ { \frac { 2 \pi } { 3 } } | x \sin 3 x | \mathrm { d } x = \frac { 4 \pi } { 9 }\).
Pre-U Pre-U 9794/1 Specimen Q11
11 marks Challenging +1.8
11 A sequence of terms \(x _ { n }\) generated by a recurrence relation is said to be strictly increasing if, for each \(x _ { n } , x _ { n + 1 } > x _ { n }\).
  1. Let a recurrence relation be defined by $$x _ { n + 1 } = \frac { x _ { n } ^ { 2 } + 2 } { 3 } \quad \text { and } \quad x _ { 0 } = \frac { 1 } { 2 } \quad \text { for } n \geq 0$$ Calculate \(x _ { 1 } , x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\) correct to 3 significant figures where appropriate.
  2. Given the recurrence relation $$x _ { n + 1 } = \frac { x _ { n } ^ { 2 } + 2 } { 3 }$$ show that the sequence is strictly increasing when \(x _ { n } > 2\) or \(x _ { n } < 1\).
  3. If \(- 1 < x _ { 0 } < 1\), then the sequence \(x _ { n } ( n \geq 0 )\) converges to a limit. Explain briefly why this limit is 1 .
  4. Given the recurrence relation $$x _ { n + 1 } = \frac { x _ { n } ^ { 2 } + k } { m } \text { with } m > 0$$ prove that \(x _ { n }\) is a strictly increasing sequence for all \(x _ { n }\) if \(m ^ { 2 } < 4 k\).
Pre-U Pre-U 9794/1 Specimen Q12
6 marks Moderate -0.8
12 A set of data is shown in the table below.
\(x\)012345678
frequency3104320001
  1. Calculate the mean and standard deviation of the data. The value 8 may be regarded as an outlier.
  2. Explain how you would treat this outlier if the data represents
    1. the difference of the scores obtained when throwing a pair of ordinary dice,
    2. the number of thunderstorms per year in Cambridgeshire over a 23-year period.
    3. Without doing any further calculations state what effect, if any, removing the outlier would have on the mean and standard deviation.
Pre-U Pre-U 9794/1 Specimen Q13
9 marks Moderate -0.3
13 A seed company investigated how well African Marigold seeds germinated when the seeds were past their sell-by date. The table shows the average number of seeds which germinated per packet, \(y\), and the number of months past their sell-by date, \(t\).
\(t\)1020304050
\(y\)24.524.021.718.612.4
The summary data for the investigation were as follows. $$\Sigma t = 150 \quad \Sigma t ^ { 2 } = 5500 \quad \Sigma y = 101.2 \quad \Sigma y ^ { 2 } = 2146.86 \quad \Sigma t y = 2740$$
  1. Calculate the equation of the regression line of \(y\) on \(t\).
  2. Use your regression line to calculate \(y\) when \(t = 10\). Compare your answer with the value of \(y\) when \(t = 10\) in the table and comment on the result.
  3. Use your regression line to calculate \(y\) when \(t = 100\). Comment on the validity of this result.
  4. Suggest with reasons whether the regression line provides a good model for predicting the germination of seeds past their sell-by date.
Pre-U Pre-U 9794/1 Specimen Q14
14 marks Moderate -0.3
14 The maximum pressure exerted by the blood on the arteries in a population of elderly male patients may be modelled by a random variable having a normal distribution with a mean of 150 and standard deviation 15, measured in suitable units.
  1. Find the probability that the maximum pressure for a randomly chosen patient is more than 160.
  2. If the maximum pressure is found to be \(t\) or more, the patient must be referred to a consultant. If \(5 \%\) of the patients are referred to a consultant, find the value of \(t\).
  3. Find the percentage of patients whose maximum pressure is between 130 and 160 . The probability that a randomly chosen patient attending a doctor's surgery has their blood pressure measured is 0.4 .
  4. Find the probability that of 18 people attending a doctor's surgery more than 8 have their blood pressure measured, assuming that each measurement is random and independent of any other.
  5. If 450 patients visited the surgery in a week, find the expected number of patients whose blood pressure would be measured.
Pre-U Pre-U 9794/1 Specimen Q15
12 marks Standard +0.8
15 In order to be accepted on a university course, a student needs to pass three exams.
The probability that the student passes the first exam is \(\frac { 3 } { 4 }\).
For each of the second and third exams, the probability of passing the exam is
  • the same as the probability of passing the preceding exam if the student passed the preceding exam,
  • half of the probability of passing the preceding exam if the student failed the preceding exam.
    1. Draw a tree diagram to represent the above information.
    2. Find the probability that the student passes all three exams.
    3. Find the probability that the student passes at least two of the exams.
    4. Find the probability that the student passes the third exam given that exactly two of the three exams are passed.
Pre-U Pre-U 9794/2 Specimen Q1
3 marks Moderate -0.3
1 Solve the equation $$x \sqrt { 32 } - \sqrt { 24 } = ( 3 \sqrt { 3 } - 5 ) ( \sqrt { 6 } + x \sqrt { 2 } )$$
Pre-U Pre-U 9794/2 Specimen Q2
4 marks Easy -1.2
2 You are given that \(\ln ( 12 ) = 2.484907\) and \(\ln ( 3 ) = 1.098612\), correct to 6 decimal places. Use the laws of logarithms to obtain the values of \(\ln ( 36 )\) and \(\ln ( 0.5 )\), correct to 4 decimal places. You must show your numerical working.
Pre-U Pre-U 9794/2 Specimen Q3
5 marks Standard +0.3
3 Show that $$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } ( \pi - x ) \cos 2 x \mathrm {~d} x = \frac { 1 } { 4 } + \frac { 3 } { 8 } \pi$$
Pre-U Pre-U 9794/2 Specimen Q4
6 marks Standard +0.3
4 The complex number \(p\) satisfies the equation $$p + \mathrm { i } p ^ { * } = 2 \left( p - \mathrm { i } p ^ { * } \right) - 8$$ Determine the exact values of the modulus and argument of \(p\).
Pre-U Pre-U 9794/2 Specimen Q5
9 marks Standard +0.3
5 A circle \(S\) has centre at the point \(( 3,1 )\) and passes through the point \(( 0,5 )\).
  1. Find the radius of \(S\) and hence write down its cartesian equation.
  2. (a) Determine the two points on \(S\) where the \(y\)-coordinate is twice the \(x\)-coordinate.
    (b) Calculate the length of the minor arc joining these two points.
Pre-U Pre-U 9794/2 Specimen Q6
5 marks Standard +0.8
6
  1. Given that the numbers \(a , b\) and \(c\) are in arithmetic progression, show that \(a + c = 2 b\).
  2. Find an analogous result for three numbers in geometric progression.
  3. The numbers \(2 - 3 x , 2 x , 3 - 2 x\) are the first three terms of a convergent geometric progression. Find \(x\) and hence calculate the sum to infinity.
Pre-U Pre-U 9794/2 Specimen Q7
12 marks Standard +0.3
7 A cubic polynomial is given by $$\mathrm { P } ( x ) = x ^ { 3 } - 3 x ^ { 2 } + a x + b$$ where \(a\) and \(b\) are constants.
  1. If \(\mathrm { P } ( x )\) is exactly divisible by \(x - 1\), and has a local maximum at \(x = - 1\), determine the values of \(a\) and \(b\).
  2. Sketch the curve \(y = \mathrm { P } ( x )\), marking the intercepts and the \(x\)-coordinates of the stationary points.
  3. Expand and simplify \(\mathrm { P } ( 1 + x )\), and deduce that \(\mathrm { P } ( 1 + x ) = - \mathrm { P } ( 1 - x )\). Interpret this result graphically.
Pre-U Pre-U 9794/2 Specimen Q8
6 marks Standard +0.8
8
  1. Show that $$\tan x = \frac { 2 t } { 1 - t ^ { 2 } } \text { for } 0 \leq t < 1 , \text { where } t = \tan \frac { 1 } { 2 } x$$ and deduce that $$\sin x = \frac { 2 t } { 1 + t ^ { 2 } }$$
  2. Using the substitution \(t = \tan \frac { 1 } { 2 } x\), show that $$\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \frac { 1 } { 1 + \sin x } \mathrm {~d} x = \sqrt { 3 } - 1$$
Pre-U Pre-U 9794/2 Specimen Q9
11 marks Standard +0.3
9 A curve has equation $$y = \mathrm { e } ^ { 3 x } - 5 \mathrm { e } ^ { 2 x } + 8 \mathrm { e } ^ { x }$$
  1. Find the exact coordinates of the stationary points of \(y\).
  2. Determine the range of values of \(x\) for which $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } > 0$$
  3. Determine the nature of the stationary points on the curve.
Pre-U Pre-U 9794/2 Specimen Q10
12 marks Challenging +1.2
10
    1. By writing \(\sec x = \frac { 1 } { \cos x }\), prove that $$\frac { \mathrm { d } } { \mathrm {~d} x } ( \sec x ) = \sec x \tan x .$$
    2. Deduce that \(y = \sec x\) satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = y \sqrt { y ^ { 2 } - 1 } , \quad 0 \leq x < \frac { 1 } { 2 } \pi .$$
  2. A curve lies in the first quadrant of the cartesian plane with origin \(O\) as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{85043199-527d-4105-aa0b-c913dec0e35b-4_707_698_845_685} The normal to the curve at the point \(P ( x , y )\) meets the \(x\)-axis at the point \(Q\). The angle between \(O P\) and the \(x\)-axis is \(u\), and the angle between \(Q P\) and the \(x\)-axis is \(v\).
    1. If $$\tan v = \tan ^ { 2 } u$$ obtain a differential equation satisfied by the curve.
    2. The curve passes through the point \(( 2,1 )\). By solving the differential equation, find an equation for the curve in the implicit form $$\mathrm { F } ( x , y ) = C ,$$ where \(C\) is a constant that should be determined.
Pre-U Pre-U 9794/2 Specimen Q11
4 marks Challenging +1.2
11 Three light inextensible strings \(A C , C D\) and \(D B\), each of length 10 cm , are joined as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{85043199-527d-4105-aa0b-c913dec0e35b-5_300_670_475_699} The ends \(A\) and \(B\) are fixed to points 20 cm apart on the same horizontal level. Two heavy particles, each of mass 2 kg , are attached at \(C\) and \(D\). The system remains in a vertical plane.
  1. Determine the tension in each string.
  2. The string \(C D\) is replaced by one of length \(L \mathrm {~cm}\), made of the same material. If the tension in \(A C\) is 50 N , show that \(L = 20 - 4 \sqrt { 21 }\).
Pre-U Pre-U 9794/2 Specimen Q12
5 marks Standard +0.3
12
  1. Whilst a helicopter is hovering, the floor of its cargo hold maintains an angle of \(30 ^ { \circ }\) to the horizontal. There is a box of mass 20 kg on the floor. If the box is just on the point of sliding, show by resolving forces that the coefficient of friction between the box and the floor is \(\frac { 1 } { \sqrt { 3 } }\).
  2. The helicopter ascends at a constant acceleration 0.5 g . If the cargo hold is now maintained at \(10 ^ { \circ }\) to the horizontal, determine the frictional force and the normal reaction between the box and the floor.
Pre-U Pre-U 9794/2 Specimen Q13
10 marks Standard +0.3
13 Professor Oldham wishes to illustrate and test Newton's experimental law of impacts. A ball is dropped from rest from a height \(h\) above a rigid horizontal board and rebounds to a height \(H\). The time taken to reach the height \(H\) after the first impact is \(T\). These quantities are recorded using very accurate measuring devices.
  1. Show that $$H = e ^ { 2 } h \quad \text { and } \quad T = e \sqrt { \frac { 2 h } { g } }$$ are predicted by Newton's law, where \(e\) is the coefficient of restitution between the ball and the board.
  2. If \(h = 180 \mathrm {~cm}\) and \(H = 45 \mathrm {~cm}\), determine \(T\) from these formulae. The experiment is repeated for initial heights \(h , 2 h , 3 h , \ldots , 15 h\) where \(h = 180 \mathrm {~cm}\). The corresponding rebound heights and times taken to reach that height after the first impact are recorded. The mean of the 15 rebound heights is found to be 3.3 m .
  3. Find the mean of the rebound heights predicted by Newton's law and give one reason why this differs from the experimental value. Professor Oldham is able to repeat the experiment on the surface of the moon using the same experimental set-up inside a laboratory.
  4. The mean of the rebound heights is unchanged, but the mean of the rebound times is substantially increased. Comment on these findings.
Pre-U Pre-U 9794/2 Specimen Q14
13 marks Standard +0.8
14 A particle \(P\) is projected from the point \(O\), at the top of a vertical wall of height \(H\) above a horizontal plane, with initial speed \(V\) at an angle \(\alpha\) above the horizontal. At time \(t\) the coordinates of the particle are \(( x , y )\) referred to horizontal and vertical axes at \(O\).
  1. Express \(x\) and \(y\) as functions of \(t\). Let \(\theta\) be the angle \(O P\) makes with the horizontal at time \(t\).
  2. (a) Show that $$\tan \theta = \tan \alpha - \frac { g } { 2 V \cos \alpha } t$$ (b) Show that when the particle attains its greatest height above the point of projection, where \(O P\) makes an angle \(\beta\) with the horizontal, $$\tan \beta = \frac { 1 } { 2 } \tan \alpha .$$ (c) If the particle strikes the ground where \(O P\) makes an angle \(\beta\) below the horizontal, show that $$H = \frac { 3 V ^ { 2 } \sin ^ { 2 } \alpha } { 2 g }$$