Questions — Pre-U (885 questions)

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Pre-U Pre-U 9795/1 2010 June Q1
4 marks Standard +0.8
1 The equation \(x ^ { 3 } + 2 x ^ { 2 } + x - 7 = 0\) has roots \(\alpha , \beta\) and \(\gamma\). Use the substitution \(y = 1 + x ^ { 2 }\) to find an equation, with integer coefficients, whose roots are \(1 + \alpha ^ { 2 } , 1 + \beta ^ { 2 }\) and \(1 + \gamma ^ { 2 }\).
Pre-U Pre-U 9795/1 2010 June Q2
5 marks Standard +0.3
2 Use the method of differences to express \(\sum _ { r = 1 } ^ { n } \frac { 1 } { 4 r ^ { 2 } - 1 }\) in terms of \(n\), and hence deduce the sum of the infinite series $$\frac { 1 } { 3 } + \frac { 1 } { 15 } + \frac { 1 } { 35 } + \ldots + \frac { 1 } { 4 n ^ { 2 } - 1 } + \ldots$$
Pre-U Pre-U 9795/1 2010 June Q3
4 marks Moderate -0.8
3 The points \(A ( 1,3 ) , B ( 4,36 )\) and \(C ( 9,151 )\) lie on the curve with equation \(y = p + q x + r x ^ { 2 }\).
  1. Using this information, write down three simultaneous equations in \(p , q\) and \(r\).
  2. Re-write this system of equations in the matrix form \(\mathbf { C x } = \mathbf { a }\), where \(\mathbf { C }\) is a \(3 \times 3\) matrix, \(\mathbf { x }\) is an unknown vector, and \(\mathbf { a }\) is a fixed vector.
  3. By finding \(\mathbf { C } ^ { - 1 }\), determine the values of \(p , q\) and \(r\).
Pre-U Pre-U 9795/1 2010 June Q4
5 marks Standard +0.3
4
  1. Using the definitions of sinh and cosh in terms of exponentials, prove that $$\cosh A \cosh B + \sinh A \sinh B \equiv \cosh ( A + B )$$
  2. Solve the equation \(5 \cosh x + 3 \sinh x = 12\), giving your answers in the form \(\ln ( p \pm q \sqrt { 2 } )\) for rational numbers \(p\) and \(q\) to be determined.
Pre-U Pre-U 9795/1 2010 June Q5
8 marks Standard +0.8
5 A curve has equation \(y = \frac { x ^ { 2 } + 5 x - 6 } { x + 3 }\) for \(x \neq - 3\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } > 1\) at all points on the curve.
  2. Sketch the curve, justifying all significant features.
Pre-U Pre-U 9795/1 2010 June Q6
8 marks Challenging +1.2
6
  1. The set \(S\) consists of all \(2 \times 2\) matrices of the form \(\left( \begin{array} { l l } 1 & n \\ 0 & 1 \end{array} \right)\), where \(n \in \mathbb { Z }\).
    1. Show that \(S\), under the operation of matrix multiplication, forms a group \(G\). [You may assume that matrix multiplication is associative.]
    2. State, giving a reason, whether \(G\) is abelian.
    3. The group \(H\) is the set \(\mathbb { Z }\) together with the operation of addition. Explain why \(G\) is isomorphic to \(H\).
    4. The plane transformation \(T\) is given by the matrix \(\left( \begin{array} { l l } 1 & n \\ 0 & 1 \end{array} \right)\), where \(n\) is a non-zero integer. Describe \(T\) fully.
Pre-U Pre-U 9795/1 2010 June Q7
9 marks Challenging +1.2
7 A curve \(C\) has polar equation \(r = 2 + \cos \theta\) for \(- \pi < \theta \leqslant \pi\).
  1. The point \(P\) on \(C\) corresponds to \(\theta = \alpha\), and the point \(Q\) on \(C\) is such that \(P O Q\) is a straight line, where \(O\) is the pole. Show that the length \(P Q\) is independent of \(\alpha\).
  2. Find, in an exact form, the area of the region enclosed by \(C\).
  3. Show that \(\left( x ^ { 2 } + y ^ { 2 } - x \right) ^ { 2 } = 4 \left( x ^ { 2 } + y ^ { 2 } \right)\) is a cartesian equation for \(C\). Identify the coordinates of the point which is included in this cartesian equation but is not on \(C\).
Pre-U Pre-U 9795/1 2010 June Q8
10 marks Challenging +1.8
8 For the differential equation \(t ^ { 2 } \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } - 4 t \frac { \mathrm {~d} x } { \mathrm {~d} t } + \left( 6 - 4 t ^ { 2 } \right) x = 0\), use the substitution \(x = t ^ { 2 } u\) to find a differential equation involving \(t\) and \(u\) only. Hence solve the above differential equation, given that \(x = \mathrm { e } - 1\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 4 \mathrm { e }\) when \(t = 1\).
Pre-U Pre-U 9795/1 2010 June Q9
10 marks Challenging +1.2
9 Three non-collinear points \(A , B\) and \(C\) have position vectors \(\mathbf { a } , \mathbf { b }\) and \(\mathbf { c }\) respectively, relative to the origin \(O\). The plane through \(A , B\) and \(C\) is denoted by \(\Pi\).
  1. (a) Prove that the area of triangle \(A B C\) is \(\frac { 1 } { 2 } | \mathbf { a } \times \mathbf { b } + \mathbf { b } \times \mathbf { c } + \mathbf { c } \times \mathbf { a } |\).
    (b) Describe the significance of the vector \(\mathbf { a } \times \mathbf { b } + \mathbf { b } \times \mathbf { c } + \mathbf { c } \times \mathbf { a }\) in relation to \(\Pi\).
  2. (a) In the case when \(\mathbf { a } = a \mathbf { i } , \mathbf { b } = b \mathbf { j }\) and \(\mathbf { c } = c \mathbf { k }\), where \(a , b\) and \(c\) are positive scalar constants, determine the equation of \(\Pi\) in the form r.n \(= d\), where the components of \(\mathbf { n }\) and the value of the scalar constant \(d\) are to be given in terms of \(a , b\) and \(c\).
    (b) Deduce the shortest distance from the origin \(O\) to \(\Pi\) in this case.
Pre-U Pre-U 9795/1 2010 June Q10
11 marks Challenging +1.2
10 One root of the equation \(z ^ { 5 } - 1 = 0\) is the complex number \(\omega = \mathrm { e } ^ { \frac { 2 } { 5 } \pi \mathrm { i } }\).
  1. Show that
    1. \(\quad \omega ^ { 5 } = 1\),
    2. \(\quad \omega + \omega ^ { 2 } + \omega ^ { 3 } + \omega ^ { 4 } = - 1\),
    3. \(\quad \omega + \omega ^ { 4 } = 2 \cos \frac { 2 } { 5 } \pi\), and write down a similar expression for \(\omega ^ { 2 } + \omega ^ { 3 }\).
    4. Using these results, find the values of \(\cos \frac { 2 } { 5 } \pi + \cos \frac { 4 } { 5 } \pi\) and \(\cos \frac { 2 } { 5 } \pi \times \cos \frac { 4 } { 5 } \pi\), and deduce a quadratic equation, with integer coefficients, which has roots $$\cos \frac { 2 } { 5 } \pi \quad \text { and } \quad \cos \frac { 4 } { 5 } \pi$$
Pre-U Pre-U 9795/1 2010 June Q11
18 marks Challenging +1.8
11
  1. At all points \(( x , y )\) on the curve \(C , \frac { \mathrm {~d} y } { \mathrm {~d} x } + x y = 0\).
    1. Prove by induction that, for all integers \(n \geqslant 1\), $$\frac { \mathrm { d } ^ { n + 1 } y } { \mathrm {~d} x ^ { n + 1 } } + x \frac { \mathrm {~d} ^ { n } y } { \mathrm {~d} x ^ { n } } + n \frac { \mathrm {~d} ^ { n - 1 } y } { \mathrm {~d} x ^ { n - 1 } } = 0$$ where \(\frac { \mathrm { d } ^ { 0 } y } { \mathrm {~d} x ^ { 0 } } = y\).
    2. Given that \(y = 1\) when \(x = 0\), determine the Maclaurin expansion of \(y\) in ascending powers of \(x\), up to and including the term in \(x ^ { 6 }\).
    3. Solve the differential equation \(\frac { \mathrm { d } y } { \mathrm {~d} x } + x y = 0\) given that \(y = 1\) when \(x = 0\).
    4. Given that \(Z \sim \mathrm {~N} ( 0,1 )\), use your answers to parts (i) and (ii) to find an approximation, to 4 decimal places, to the probability \(\mathrm { P } ( Z \leqslant 1 )\).
      [0pt] [Note that the probability density function of the standard normal distribution is \(\mathrm { f } ( z ) = \frac { 1 } { \sqrt { 2 \pi } } \mathrm { e } ^ { - \frac { 1 } { 2 } z ^ { 2 } }\).]
Pre-U Pre-U 9795/1 2010 June Q12
22 marks Challenging +1.8
12
  1. Let \(I _ { n } = \int \frac { x ^ { n } } { \sqrt { x ^ { 2 } + 1 } } \mathrm {~d} x\), for integers \(n \geqslant 0\).
    By writing \(\frac { x ^ { n } } { \sqrt { x ^ { 2 } + 1 } }\) as \(x ^ { n - 1 } \times \frac { x } { \sqrt { x ^ { 2 } + 1 } }\), or otherwise, show that, for \(n \geqslant 2\), $$n I _ { n } = x ^ { n - 1 } \sqrt { x ^ { 2 } + 1 } - ( n - 1 ) I _ { n - 2 } .$$
  2. The diagram shows a sketch of the hyperbola \(H\) with equation \(\frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 16 } = 1\). \includegraphics[max width=\textwidth, alt={}, center]{32ed7cc8-3456-4cf0-952a-ee04eada1298-6_593_666_776_776}
    1. Find the coordinates of the points where \(H\) crosses the \(x\)-axis.
    2. The curve \(J\) has parametric equations \(x = 2 \cosh \theta , y = 4 \sinh \theta\), for \(\theta \geqslant 0\). Show that these parametric equations satisfy the cartesian equation of \(H\), and indicate on a copy of the above diagram which part of \(H\) is \(J\).
    3. The arc of the curve \(J\) between the points where \(x = 2\) and \(x = 34\) is rotated once completely about the \(x\)-axis to form a surface of revolution with area \(S\). Show that $$S = 16 \pi \int _ { \alpha } ^ { \beta } \sinh \theta \sqrt { 5 \cosh ^ { 2 } \theta - 1 } \mathrm {~d} \theta$$ for suitable constants \(\alpha\) and \(\beta\).
    4. Use the substitution \(u ^ { 2 } = 5 \cosh ^ { 2 } \theta - 1\) to show that $$S = \frac { 8 \pi } { \sqrt { 5 } } ( 644 \sqrt { 5 } - \ln ( 9 + 4 \sqrt { 5 } ) )$$
Pre-U Pre-U 9795/2 2010 June Q1
7 marks Standard +0.8
1 A lorry moves along a straight horizontal road. The engine of the lorry produces a constant power of 80 kW . The mass of the lorry is 10 tonnes and the resistance to motion is constant at 4000 N .
  1. Express the driving force of the lorry in terms of its velocity and hence, using Newton's second law, write down a differential equation which connects the velocity of the lorry and the time for which it has been moving.
  2. Hence find the time taken, in seconds, for the lorry to accelerate from \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
Pre-U Pre-U 9795/2 2010 June Q2
9 marks Standard +0.8
2 At 1200 hours an aircraft, \(A\), sets out to intercept a second aircraft, \(B\), which is 200 km away on a bearing of \(300 ^ { \circ }\) and is flying due east at \(600 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). Both aircraft are at the same altitude and continue to fly horizontally.
  1. (a) Find the bearing on which \(A\) should fly when travelling at \(800 \mathrm {~km} \mathrm {~h} ^ { - 1 }\).
    (b) Find the time at which \(A\) intercepts \(B\) in this case.
  2. Find the least steady speed at which \(A\) can fly to intercept \(B\).
Pre-U Pre-U 9795/2 2010 June Q3
11 marks Challenging +1.8
3 A particle is projected at an angle \(\theta\) above the horizontal from the foot of a plane which is inclined at \(45 ^ { \circ }\) to the horizontal. Subsequently the particle impacts on the plane at a higher point.
  1. Prove that the angle at which the particle strikes the plane is \(\phi\), where $$\tan \phi = \frac { \tan \theta - 1 } { 3 - \tan \theta }$$
  2. Find the angle to the horizontal at which the particle would have to be projected if it were to strike the plane horizontally.
Pre-U Pre-U 9795/2 2010 June Q4
11 marks Challenging +1.2
4 One end of a light elastic string of natural length 0.2 m and modulus of elasticity 100 N is attached to a fixed point \(A\). The other end is attached to a particle of mass 5 kg . The particle moves with angular speed \(\omega\) radians per second in a horizontal circle with the centre vertically below \(A\). The string makes an angle \(\theta\) with the vertical.
  1. By considering the horizontal component of the tension in the string, show that the tension in the string is \(( 1 + 5 x ) \omega ^ { 2 } \mathrm {~N}\), where \(x\) is the extension, in metres, of the string.
  2. (a) By considering vertical forces and also Hooke's law, deduce that \(\cos \theta = \frac { 1 } { 10 x }\).
    (b) Show that \(\omega > \frac { 10 \sqrt { 3 } } { 3 }\).
  3. When the value of \(\omega\) is \(5 \sqrt { 2 }\), find the radius of the circular motion.
Pre-U Pre-U 9795/2 2010 June Q5
11 marks Challenging +1.8
5 A particle of mass \(m\) is attached by a light elastic string of natural length \(l\) and modulus of elasticity \(\lambda\) to a fixed point \(A\), from which it is allowed to fall freely. The particle first comes to rest, instantaneously, at \(B\), where \(A B = 2 l\). Prove that
  1. \(\lambda = 4 m g\),
  2. while the string is taut, \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - \frac { 4 g } { l } x\), where \(x\) is the displacement from the equilibrium position at time \(t\),
  3. the time taken between the first occasion when the string becomes taut and the next occasion when it becomes slack is $$\left[ \frac { 1 } { 2 } \pi + \sin ^ { - 1 } \left( \frac { 1 } { 3 } \right) \right] \sqrt { \frac { l } { g } }$$
Pre-U Pre-U 9795/2 2010 June Q6
11 marks Challenging +1.8
6 Two smooth spheres, \(A\) and \(B\), have masses \(m\) and \(2 m\) respectively and equal radii. Sphere \(B\) is at rest on a smooth horizontal floor. Sphere \(A\) is projected with speed \(u\) along the floor in a direction parallel to a smooth vertical wall and strikes \(B\) obliquely. Subsequently \(B\) strikes the wall at an angle \(\alpha\) with the wall. The coefficient of restitution between \(A\) and \(B\) and between \(B\) and the wall is 0.5. After \(B\) has struck the wall, \(A\) and \(B\) are moving parallel to each other.
  1. Write down a momentum equation and a restitution equation along the line of centres for the impact between \(A\) and \(B\). Hence find the components of velocity of \(A\) and \(B\) in this direction after this first impact.
  2. Find the value of \(\alpha\), giving your answer in degrees.
Pre-U Pre-U 9795/2 2010 June Q7
8 marks Standard +0.3
7 The number of goals scored by a hockey team in an interval of time of length \(t\) minutes follows a Poisson distribution with mean \(\frac { 1 } { 24 } t\). The random variable \(T\) is defined as the length of time, in minutes, between successive goals.
  1. (a) Show that \(\mathrm { P } ( T < t ) = 1 - \mathrm { e } ^ { - \frac { 1 } { 24 } t }\) for \(t \geqslant 0\).
    (b) Hence find the probability density function of \(T\).
  2. Find the exact value of the interquartile range of \(T\).
Pre-U Pre-U 9795/2 2010 June Q8
8 marks Standard +0.3
8 Two groups of Year 12 pupils, one at each of schools \(A\) and \(B\), are given the same mathematics test. The scores, \(x\) and \(y\), of pupils at schools \(A\) and \(B\) respectively are summarised as follows.
School \(A\)\(n _ { A } = 15\)\(\bar { x } = 53\)\(\Sigma ( x - \bar { x } ) ^ { 2 } = 925\)
School \(B\)\(n _ { B } = 12\)\(\bar { y } = 47\)\(\Sigma ( y - \bar { y } ) ^ { 2 } = 850\)
  1. Assuming that the two groups are random samples from independent normal populations with means \(\mu _ { A }\) and \(\mu _ { B }\) respectively and a common, but unknown, variance, construct a \(98 \%\) confidence interval for \(\mu _ { A } - \mu _ { B }\).
  2. Comment, with a reason, on any difference in ability between the two schools.
Pre-U Pre-U 9795/2 2010 June Q9
10 marks Challenging +1.2
9
  1. Two independent discrete random variables \(X\) and \(Y\) follow Poisson distributions with means \(\lambda\) and \(\mu\) respectively. Prove that the discrete random variable \(Z = X + Y\) follows a Poisson distribution with mean \(\lambda + \mu\). A garage has a white limousine and a green limousine for hire. Demands to hire the white limousine occur at a constant mean rate of 3 per week and demands to hire the green limousine occur at a constant mean rate of 2 per week. Demands for hire are received independently and randomly.
  2. Calculate the probability that in a period of two weeks
    1. no demands for hire are received, giving your answer to 3 significant figures,
    2. seven demands for hire are received.
    3. Find the least value of \(n\) such that the probability of at least \(n\) demands for hire in a period of three weeks is less than 0.005 .
Pre-U Pre-U 9795/2 2010 June Q10
11 marks Challenging +1.2
10 A box contains a large number, \(n\), of identical dice, which are thought to be biased. The probability that one of these dice will show a six in a single roll is \(p\). The \(n\) dice are rolled many times and the number of sixes obtained in each trial is recorded. In \(4.01 \%\) of these trials 56 or more dice showed a six. In \(10.56 \%\) of these trials 37 or fewer dice showed a six. Using a suitable normal approximation, find the values of \(n\) and \(p\).
Pre-U Pre-U 9795/2 2010 June Q11
12 marks Standard +0.8
11 The thickness of a randomly chosen paperback book is \(P \mathrm {~cm}\) and the thickness of a randomly chosen hardback is \(H \mathrm {~cm}\), where \(P\) and \(H\) have distributions \(\mathrm { N } ( 2.0,0.75 )\) and \(\mathrm { N } ( 5.0,2.25 )\) respectively. When more than one book is selected, any book is selected independently of all other books.
  1. Calculate the probability that a randomly chosen hardback is more than 1 cm thicker than a randomly chosen paperback.
  2. Calculate the probability that 2 paperbacks and 4 hardbacks, randomly chosen, have a combined thickness of less than 20 cm .
  3. Find the probability that a randomly chosen hardback is more than twice the thickness of a randomly chosen paperback.
Pre-U Pre-U 9795/2 2010 June Q12
11 marks Challenging +1.2
12 Two players, \(A\) and \(B\), are taking turns to shoot at a basket with a basketball. The winner of this game is the first player to score a basket. The probability that \(A\) scores a basket with any shot is \(\frac { 1 } { 4 }\) and the probability that \(B\) scores a basket with any shot is \(\frac { 1 } { 5 }\). Each shot is independent of all other shots. \(A\) shoots first.
  1. Find
    1. the probability that \(B\) wins with his first shot,
    2. the probability that \(A\) wins with his second shot,
    3. the probability that \(A\) wins the game.
    4. \(R\) is the total number of shots taken by \(A\) and \(B\) up to and including the shot that scores a basket.
      (a) Show that the probability generating function of \(R\) is given by $$\mathrm { G } ( t ) = \frac { 5 t + 3 t ^ { 2 } } { 4 \left( 5 - 3 t ^ { 2 } \right) }$$ (b) Hence find \(\mathrm { E } ( R )\).
Pre-U Pre-U 9795/2 2011 June Q1
3 marks Standard +0.3
1 The independent random variables \(X\) and \(Y\) have distributions \(\mathrm { N } ( 30,9 )\) and \(\mathrm { N } ( 20,4 )\) respectively.
  1. Give the distribution of $$\left( X _ { 1 } + X _ { 2 } + X _ { 3 } \right) - \left( Y _ { 1 } + Y _ { 2 } + Y _ { 3 } + Y _ { 4 } \right)$$ where \(X _ { i } , i = 1,2,3\), and \(Y _ { j } , j = 1,2,3,4\), are independent observations of \(X\) and \(Y\) respectively. The time for female cadets to complete an assault course is \(X\) minutes and the time for male cadets to complete the same assault course is \(Y\) minutes.
  2. Find the probability that the total time for three randomly chosen female cadets to complete the assault course is greater than the total time for four randomly chosen male cadets to complete the assault course.