Questions — Pre-U (885 questions)

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Pre-U Pre-U 9795/2 2012 June Q8
8 marks Challenging +1.2
8 \includegraphics[max width=\textwidth, alt={}, center]{d8ca5464-435f-45e0-8e19-1830415a7c60-4_757_729_260_708} An aircraft carrier, \(A\), is heading due north at \(40 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). A destroyer, \(D\), which is 8 km south-west of \(A\), is ordered to take up a position 3 km east of \(A\) as quickly as possible. The speed of \(D\) is \(60 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) (see diagram). Find the bearing, \(\theta\), of the course that \(D\) should take, giving your answer to the nearest degree.
Pre-U Pre-U 9795/2 2012 June Q9
9 marks Standard +0.3
9 \includegraphics[max width=\textwidth, alt={}, center]{d8ca5464-435f-45e0-8e19-1830415a7c60-4_666_816_1384_662} 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. Show that however fast the particle travels \(A P\) will never become horizontal, and that the tension in the string is always greater than the weight of the particle.
  2. Find the tension in the string in terms of \(m , l\) and \(\omega\).
  3. Show that \(\omega ^ { 2 } h = g\) and calculate \(\omega\) when \(h\) is 0.5 m .
Pre-U Pre-U 9795/2 2012 June Q10
12 marks Challenging +1.8
10 \includegraphics[max width=\textwidth, alt={}, center]{d8ca5464-435f-45e0-8e19-1830415a7c60-5_432_949_258_598} 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 2012 June Q11
11 marks Challenging +1.2
11 Two light strings, each of natural length \(a\) 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 \(6 a\) 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 $$\ddot { x } = - \frac { 12 g x } { a } \left( 1 - \frac { a } { \sqrt { 9 a ^ { 2 } + x ^ { 2 } } } \right) .$$
  2. Given that \(\frac { x } { a }\) is small throughout the motion, show that the equation of motion is approximately $$\ddot { x } = - \frac { 8 g x } { a }$$ and state the period of the simple harmonic motion that this equation represents.
  3. Given that the initial speed of \(P\) is \(\sqrt { \frac { g a } { 200 } }\), show that the amplitude of the simple harmonic motion is \(\frac { 1 } { 40 } a\).
Pre-U Pre-U 9795/2 2012 June Q12
12 marks Challenging +1.8
12 A projectile is launched from the origin with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\alpha\) above the horizontal.
  1. Prove that the equation of its trajectory is $$y = x \tan \alpha - \frac { x ^ { 2 } } { 80 } \left( 1 + \tan ^ { 2 } \alpha \right)$$
  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 \(\beta\) to the horizontal. The line \(l\), with equation \(y = x \tan \beta\), is a line of greatest slope in the plane. A particle is projected from a point on the plane, in the vertical plane containing \(l\). By considering the intersection of \(l\) with the bounding parabola \(y = 20 - \frac { x ^ { 2 } } { 80 }\), deduce that the maximum range up, or down, this inclined plane is \(\frac { 40 } { 1 + \sin \beta }\), or \(\frac { 40 } { 1 - \sin \beta }\), respectively.
Pre-U Pre-U 9794/1 2012 June Q1
5 marks Easy -1.2
1 The first term of a geometric progression is 16 and the common ratio is 0.8 .
  1. Calculate the sum of the first 12 terms.
  2. Find the sum to infinity.
Pre-U Pre-U 9794/1 2012 June Q2
6 marks Moderate -0.8
2 Let \(\mathrm { f } ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 13 x + 15\).
  1. Show that \(\mathrm { f } ( 1 ) = 0\) and hence factorise \(x ^ { 3 } - 3 x ^ { 2 } - 13 x + 15\) completely.
  2. Hence solve the equation \(x ^ { 3 } - 3 x ^ { 2 } - 13 x + 15 = 0\).
Pre-U Pre-U 9794/1 2012 June Q3
6 marks Moderate -0.8
3 The equation of a curve is \(y = x ^ { 3 } + x ^ { 2 } - x + 3\).
  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 2012 June Q4
5 marks Moderate -0.8
4
  1. Show that the equation \(x ^ { 3 } - 6 x + 2 = 0\) has a root between \(x = 0\) and \(x = 1\).
  2. Use the iterative formula \(x _ { n + 1 } = \frac { 2 + x _ { n } ^ { 3 } } { 6 }\) with \(x _ { 0 } = 0.5\) to find this root correct to 4 decimal places, showing the result of each iteration.
Pre-U Pre-U 9794/1 2012 June Q5
5 marks Easy -1.3
5 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 \(\mathrm { gf } ( x )\).
  3. Find \(\mathrm { g } ^ { - 1 } ( x )\).
Pre-U Pre-U 9794/1 2012 June Q6
5 marks Moderate -0.8
6 The roots of the equation \(z ^ { 2 } - 6 z + 10 = 0\) are \(z _ { 1 }\) and \(z _ { 2 }\), where \(z _ { 1 } = 3 + \mathrm { i }\).
  1. Write down the value of \(z _ { 2 }\).
  2. Show \(z _ { 1 }\) and \(z _ { 2 }\) on an Argand diagram.
  3. Show that \(z _ { 1 } ^ { 2 } = 8 + 6 \mathrm { i }\).
Pre-U Pre-U 9794/1 2012 June Q7
9 marks Moderate -0.3
7
  1. Show that the first three terms in the expansion of \(( 1 - 2 x ) ^ { \frac { 1 } { 2 } }\) are \(1 - x - \frac { 1 } { 2 } x ^ { 2 }\) and find the next term.
  2. State the range of values of \(x\) for which this expansion is valid.
  3. Hence show that the first four terms in the expansion of \(( 2 + x ) ( 1 - 2 x ) ^ { \frac { 1 } { 2 } }\) are \(2 - x + a x ^ { 2 } + b x ^ { 3 }\) and state the values of \(a\) and \(b\).
Pre-U Pre-U 9794/1 2012 June Q8
9 marks Moderate -0.3
8
  1. Given that \(\frac { 2 x + 11 } { ( 2 x + 1 ) ( x + 3 ) } \equiv \frac { A } { 2 x + 1 } + \frac { B } { x + 3 }\), find the values of the constants \(A\) and \(B\).
  2. Hence show that \(\int _ { 0 } ^ { 2 } \frac { 2 x + 11 } { ( 2 x + 1 ) ( x + 3 ) } \mathrm { d } x = \ln 15\).
Pre-U Pre-U 9794/1 2012 June Q9
10 marks Standard +0.3
9 Three points \(A , B\) and \(C\) have coordinates \(( 1,0,7 ) , ( 13,9,1 )\) and \(( 2 , - 1 , - 7 )\) respectively.
  1. Use a scalar product to find angle \(A C B\).
  2. Hence find the area of triangle \(A C B\).
  3. Show that a vector equation of the line \(A B\) is given by \(\mathbf { r } = \mathbf { i } + 7 \mathbf { k } + \lambda ( 4 \mathbf { i } + 3 \mathbf { j } - 2 \mathbf { k } )\), where \(\lambda\) is a scalar parameter.
Pre-U Pre-U 9794/1 2012 June Q10
9 marks Standard +0.3
10
  1. Prove that $$\sin ^ { 2 } 2 \theta \left( \cot ^ { 2 } \theta - \tan ^ { 2 } \theta \right) = 4 \left( \cos ^ { 4 } \theta - \sin ^ { 4 } \theta \right)$$ and hence show that $$\sin ^ { 2 } 2 \theta \left( \cot ^ { 2 } \theta - \tan ^ { 2 } \theta \right) = 4 \cos 2 \theta$$
  2. Hence solve the equation \(\sin ^ { 2 } 2 \theta \left( \cot ^ { 2 } \theta - \tan ^ { 2 } \theta \right) = 2\) for \(0 ^ { \circ } \leqslant \theta < 360 ^ { \circ }\).
Pre-U Pre-U 9794/1 2012 June Q11
11 marks Standard +0.3
11
  1. Use integration by parts to show that \(\int \ln x \mathrm {~d} x = x \ln x - x + c\).
  2. Find
    1. \(\int ( \ln x ) ^ { 2 } \mathrm {~d} x\),
    2. \(\quad \int \frac { \ln ( \ln x ) } { x } \mathrm {~d} x\).
Pre-U Pre-U 9794/3 2012 June Q1
4 marks Easy -1.8
1 The heights in centimetres of 10 young women were measured and are given below. $$\begin{array} { l l l l l l l l l l } 140 & 145 & 162 & 174 & 153 & 167 & 147 & 151 & 148 & 156 \end{array}$$ Calculate the mean height of these women and show that the standard deviation is approximately 10 cm .
Pre-U Pre-U 9794/3 2012 June Q2
5 marks Moderate -0.8
2 A bag contains four black balls and one white ball. A man chooses a ball at random. If it is a black ball, he replaces it and chooses another at random. If he chooses the white ball, he stops.
  1. Name the probability distribution which models this situation.
  2. Calculate the probability that he will make exactly three attempts before he stops.
  3. Calculate the probability that he will make fewer than three attempts before he stops.
Pre-U Pre-U 9794/3 2012 June Q3
4 marks Easy -1.8
3 The lengths of snakes on a tropical island were measured and found to be normally distributed with a mean of 160 cm and a standard deviation of 6 cm . Find the probability that a randomly selected snake has a length of less than 170 cm .
Pre-U Pre-U 9794/3 2012 June Q4
6 marks Easy -1.2
4 In one department of a firm, four employees are selected for promotion from a staff of eighteen.
  1. In how many ways can four employees be selected? It is known that throughout the firm 5\% of those selected for promotion decline it.
  2. If 100 employees are randomly selected for promotion in the firm, calculate the number expected to decline promotion.
  3. If 20 employees are selected at random for promotion, use the binomial distribution to find the probability that fewer than five employees will decline promotion.
Pre-U Pre-U 9794/3 2012 June Q5
10 marks Moderate -0.8
5 In an archery competition, competitors are allowed up to three attempts to hit the bulls-eye. No one who succeeds may try again. \(45 \%\) of those entering the competition hit the bulls-eye first time. For those who fail to hit it the first time, \(60 \%\) of those attempting it for the second time succeed in hitting it. For those who fail twice, only \(15 \%\) of those attempting it for the third time succeed in hitting it. By drawing a tree diagram, or otherwise,
  1. find the probability that a randomly chosen competitor fails at all three attempts,
  2. find the probability that a randomly chosen competitor fails at the first attempt but succeeds at either the second or third attempt,
  3. find the probability that a randomly chosen competitor succeeds in hitting the bulls-eye,
  4. find the probability that a randomly chosen competitor requires exactly two attempts given that the competitor is successful.
Pre-U Pre-U 9794/3 2012 June Q6
11 marks Moderate -0.3
6 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 2012 June Q7
7 marks Moderate -0.8
7 \includegraphics[max width=\textwidth, alt={}, center]{f0c32e07-f3a0-4d58-bd00-c266177ceaac-3_343_401_1439_872} The diagram shows two forces of magnitudes 10 N and 15 N acting in a horizontal plane on a particle \(P\).
  1. Find the component of the 15 N force which is parallel to the 10 N force.
  2. Write down the component of the 15 N force which is perpendicular to the 10 N force.
  3. Hence, or otherwise, calculate the magnitude and direction of the resultant force on \(P\).
Pre-U Pre-U 9794/3 2012 June Q8
4 marks Moderate -0.8
8 A crane lifts a crate of mass 20 kg using a light inextensible cable. The crate starts from rest and ascends 10 metres in 4 seconds during which time a constant tension of \(T \mathrm {~N}\) is applied in the cable. Find the value of \(T\).
Pre-U Pre-U 9794/3 2012 June Q9
6 marks Moderate -0.3
9 \includegraphics[max width=\textwidth, alt={}, center]{f0c32e07-f3a0-4d58-bd00-c266177ceaac-4_430_565_260_790} The diagram shows a block of wood, weighing 100 N , at rest on a rough plane inclined at \(35 ^ { \circ }\) to the horizontal. The coefficient of friction between the block and the plane is 0.2 . A force of \(P \mathrm {~N}\) acts on the block up the slope.
  1. Find the maximum possible value of the friction acting on the block.
  2. Given that the block is on the point of moving up the slope, find \(P\).
  3. Given that the block is on the point of moving down the slope, find \(P\).