Questions — Pre-U (885 questions)

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Pre-U Pre-U 9795/2 2016 June Q10
8 marks Standard +0.8
10 A uniform ladder \(A B\) of length 5 m and mass 8 kg is placed at an angle \(\theta\) to the horizontal, with \(A\) on rough horizontal ground and \(B\) against a smooth vertical wall. The coefficient of friction between the ladder and the ground is 0.4 .
  1. By taking moments, find the smallest value of \(\theta\) for which the ladder is in equilibrium.
  2. A man of mass 75 kg stands on the ladder when \(\theta = 60 ^ { \circ }\). Find the greatest distance from \(A\) that he can stand without the ladder slipping.
Pre-U Pre-U 9795/2 2016 June Q11
6 marks Challenging +1.2
11 A car of mass 800 kg has a constant power output of 32 kW while travelling on a horizontal road. At time \(t \mathrm {~s}\) the car's speed is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the resistive force has magnitude \(20 v \mathrm {~N}\).
  1. Show that \(v\) satisfies the differential equation \(\frac { \mathrm { d } v } { \mathrm {~d} t } = \frac { 1600 - v ^ { 2 } } { 40 v }\).
  2. Given that \(v = 0\) when \(t = 0\), solve this differential equation to find \(v\) in terms of \(t\). State what the solution predicts as \(t\) becomes large.
Pre-U Pre-U 9795/2 2016 June Q12
8 marks Challenging +1.8
12 \includegraphics[max width=\textwidth, alt={}, center]{1a89caec-6da8-4b83-9ffa-efc209ecbc8d-5_205_200_264_497} \includegraphics[max width=\textwidth, alt={}, center]{1a89caec-6da8-4b83-9ffa-efc209ecbc8d-5_284_899_349_753} A white snooker ball of mass \(m\) moves with speed \(u\) towards a stationary black snooker ball of the same mass and radius. Taking the \(x\)-axis to be the line of centres of the two balls at the moment of collision, the direction of motion of the white ball before the collision makes an angle of \(30 ^ { \circ }\) with the positive \(x\)-axis (see diagram).
  1. Given that the coefficient of restitution is 0.9 , find the angle made with the \(x\)-axis by the velocity of the white ball after the collision.
  2. Show that after the collision the white ball cannot have a negative \(x\)-component of velocity whatever the value of the coefficient of restitution.
Pre-U Pre-U 9795/2 2016 June Q13
9 marks Challenging +1.2
13 A cricket ball is hit from a point \(P\) on a sloping field. The initial velocity of the ball is \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(40 ^ { \circ }\) above the field, which under the path of the ball slopes upwards at \(10 ^ { \circ }\) to the horizontal. Air resistance is to be ignored.
  1. Find the vertical height of the ball above the field after 2.5 seconds.
  2. The ball lands on the field at the point \(X\). Find the distance \(P X\).
Pre-U Pre-U 9795/2 2016 June Q14
14 marks Challenging +1.2
14 One end of a light elastic string of natural length 0.5 m and modulus of elasticity 3 N is attached to a ceiling at a point \(P\). A particle of mass 0.3 kg is attached to the other end of the string.
  1. Find the extension of the string when the particle hangs vertically in equilibrium. The particle is released from rest at \(P\) so that it falls vertically. Find
  2. the maximum extension of the string,
  3. the equation of motion for the particle when the string is stretched, in terms of the displacement \(x \mathrm {~m}\) below the equilibrium position,
  4. the time between the string first becoming stretched and next becoming unstretched again.
Pre-U Pre-U 9795/1 2016 Specimen Q6
7 marks Challenging +1.8
6 A group \(G\) has order 12.
  1. State, with a reason, the possible orders of the elements of \(G\). The identity element of \(G\) is \(e\), and \(x\) and \(y\) are distinct, non-identity elements of \(G\) satisfying the three conditions
    (1) \(x\) has order 6 ,
    (2) \(x ^ { 3 } = y ^ { 2 }\),
    (3) \(x y x = y\).
  2. Prove that \(y x ^ { 2 } y = x\).
  3. Prove that \(G\) is not a cyclic group.
Pre-U Pre-U 9795/1 2016 Specimen Q7
9 marks Challenging +1.2
7
  1. Use de Moivre's theorem to show that \(\tan 4 \theta = \frac { 4 t \left( 1 - t ^ { 2 } \right) } { 1 - 6 t ^ { 2 } + t ^ { 4 } }\), where \(\mathrm { t } = \tan \theta\).
  2. Given that \(\theta\) is the acute angle such that \(\tan \theta = \frac { 1 } { 5 }\), express \(\tan 4 \theta\) as a rational number in its simplest form, and verify that $$\frac { 1 } { 4 } \pi + \tan ^ { - 1 } \left( \frac { 1 } { 239 } \right) = 4 \tan ^ { - 1 } \left( \frac { 1 } { 5 } \right) .$$
Pre-U Pre-U 9795/1 2016 Specimen Q8
10 marks Standard +0.8
8 The function f satisfies the differential equation $$x ^ { 2 } \mathrm { f } ^ { \prime \prime } ( x ) + ( 2 x - 1 ) \mathrm { f } ^ { \prime } ( x ) - 2 \mathrm { f } ( x ) = 3 \mathrm { e } ^ { x - 1 } + 1 ,$$ and the conditions \(f ( 1 ) = 2 , f ^ { \prime } ( 1 ) = 3\).
  1. Determine \(f ^ { \prime \prime } ( 1 )\).
  2. Differentiate (*) with respect to \(x\) and hence evaluate \(\mathrm { f } ^ { \prime \prime \prime } ( 1 )\).
  3. Hence determine the Taylor series approximation for \(\mathrm { f } ( x )\) about \(x = 1\), up to and including the term in \(( x - 1 ) ^ { 3 }\).
  4. Deduce, to 3 decimal places, an approximation for \(\mathrm { f } ( 1.1 )\).
Pre-U Pre-U 9795/1 2016 Specimen Q9
13 marks Challenging +1.2
9
  1. Show that the substitution \(u = \frac { 1 } { y ^ { 3 } }\) transforms the differential equation \(\frac { \mathrm { d } y } { \mathrm {~d} x } + y = 3 x y ^ { 4 }\) into $$\frac { \mathrm { d } u } { \mathrm {~d} x } - 3 u = - 9 x .$$
  2. Solve the differential equation \(\frac { \mathrm { d } y } { \mathrm {~d} x } + y = 3 x y ^ { 4 }\), given that \(y = \frac { 1 } { 2 }\) when \(x = 0\). Give your answer in the form \(y ^ { 3 } = \mathrm { f } ( x )\).
Pre-U Pre-U 9795/1 2016 Specimen Q10
12 marks Standard +0.3
10 The line \(L\) has equation \(\mathbf { r } = \left( \begin{array} { c } 1 \\ - 3 \\ 2 \end{array} \right) + \lambda \left( \begin{array} { l } 3 \\ 4 \\ 6 \end{array} \right)\) and the plane \(\Pi\) has equation \(\mathbf { r } \cdot \left( \begin{array} { c } 2 \\ - 6 \\ 3 \end{array} \right) = k\).
  1. Given that \(L\) lies in \(\Pi\), determine the value of \(k\).
  2. Find the coordinates of the point, \(Q\), in \(\Pi\) which is closest to \(P ( 10,2 , - 43 )\). Deduce the shortest distance from \(P\) to \(\Pi\).
  3. Find, in the form \(a x + b y + c z = d\), where \(a , b , c\) and \(d\) are integers, an equation for the plane which contains both \(L\) and \(P\).
Pre-U Pre-U 9795/1 2016 Specimen Q13
6 marks Challenging +1.8
13 Define the repunit number, \(R _ { n }\), to be the positive integer which consists of a string of \(n 1 \mathrm {~s}\). Thus, $$R _ { 1 } = 1 , \quad R _ { 2 } = 11 , \quad R _ { 3 } = 111 , \quad \ldots , \quad R _ { 7 } = 1111111 , \quad \ldots , \text { etc. }$$ Use induction to prove that, for all integers \(n \geqslant 5\), the number $$13579 \times R _ { n }$$ contains a string of ( \(n - 4\) ) consecutive 7s.
Pre-U Pre-U 9795/2 2016 Specimen Q1
9 marks Standard +0.3
1 The random variable \(X\) has probability density function \(\mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = \begin{cases} k \mathrm { e } ^ { - k x } & x \geqslant 0 \\ 0 & x < 0 \end{cases}$$ and \(k\) is a positive constant.
  1. Show that the moment generating function of \(X\) is \(\mathrm { M } _ { X } ( t ) = k ( k - t ) ^ { - 1 } , t < k\).
  2. Use the moment generating function to find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
Pre-U Pre-U 9795/2 2016 Specimen Q4
10 marks Challenging +1.2
4
  1. The random variable \(X\) has the distribution \(\operatorname { Po } ( \lambda )\). Prove that the probability generating function, \(\mathrm { G } _ { X } ( t )\), is given by $$\mathrm { G } _ { X } ( t ) = \mathrm { e } ^ { \lambda ( t - 1 ) }$$
  2. The independent random variables \(X\) and \(Y\) have distributions \(\operatorname { Po } ( \lambda )\) and \(\operatorname { Po } ( \mu )\) respectively. Use probability generating functions to show that the distribution of \(X + Y\) is \(\operatorname { Po } ( \lambda + \mu )\).
  3. Given that \(X \sim \operatorname { Po } ( 1.5 )\) and \(Y \sim \operatorname { Po } ( 2.5 )\), find \(\mathrm { P } ( X \leqslant 2 \mid X + Y = 4 )\).
Pre-U Pre-U 9795/2 2016 Specimen Q7
9 marks Standard +0.3
7 A cyclist and her machine have a combined mass of 90 kg and she is riding along a straight horizontal road. She is working at a constant power of 75 W . At time \(t\) seconds her speed is \(v \mathrm {~ms} ^ { - 1 }\) and the resistance to motion is \(k v \mathrm {~N}\), where \(k\) is a constant.
  1. If the cyclist's maximum steady speed is \(10 \mathrm {~ms} ^ { - 1 }\), show that \(k = \frac { 3 } { 4 }\).
  2. Use Newton's second law to show that $$\frac { 25 } { v } - \frac { v } { 4 } = 30 \frac { \mathrm {~d} v } { \mathrm {~d} t } .$$
  3. Find the time taken for the cyclist to accelerate from a speed of \(3 \mathrm {~ms} ^ { - 1 }\) to a speed of \(7 \mathrm {~ms} ^ { - 1 }\).
Pre-U Pre-U 9795/2 2016 Specimen Q8
8 marks Standard +0.3
8 The diagram shows a uniform rod \(A B\) of length 40 cm and mass 2 kg placed with the end \(A\) resting against a smooth vertical wall and the end \(B\) on rough horizontal ground. The angle between \(A B\) and the horizontal is \(60 ^ { \circ }\). \includegraphics[max width=\textwidth, alt={}, center]{a19fab61-da1c-4803-9dbc-38d618a0c58e-4_657_655_1128_705}
  1. Given that the value of the coefficient of friction between the rod and the ground is 0.2 , determine whether the rod slips.
  2. Explain why it is impossible for the rod to be in equilibrium with one end on smooth horizontal ground and the other against a rough vertical wall.
Pre-U Pre-U 9795/2 2016 Specimen Q9
9 marks Standard +0.3
9 \includegraphics[max width=\textwidth, alt={}, center]{a19fab61-da1c-4803-9dbc-38d618a0c58e-5_671_817_255_623} 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 2016 Specimen Q10
12 marks Challenging +1.8
10 \includegraphics[max width=\textwidth, alt={}, center]{a19fab61-da1c-4803-9dbc-38d618a0c58e-5_435_951_1528_557} 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 2016 Specimen Q12
12 marks Challenging +1.8
12 A projectile is launched from the origin with speed \(20 \mathrm {~ms} ^ { - 1 }\) 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 2016 June Q1
3 marks Easy -1.3
1 Find the equation of the line perpendicular to the line \(y = 5 x + 6\) which passes through the point \(( 1,11 )\). Give your answer in the form \(y = m x + c\).
Pre-U Pre-U 9794/1 2016 June Q2
2 marks Easy -1.8
2 Without using a calculator, simplify the following, giving each answer in the form \(a \sqrt { 5 }\) where \(a\) is an integer. Show all your working.
  1. \(4 \sqrt { 10 } \times \sqrt { 2 }\)
  2. \(\sqrt { 500 } + \sqrt { 125 }\)
Pre-U Pre-U 9794/1 2016 June Q3
4 marks Easy -1.2
3 Solve \(3 x ^ { 2 } + 11 x - 20 > 0\).
Pre-U Pre-U 9794/1 2016 June Q4
3 marks Easy -1.3
4 A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\), is defined by \(u _ { n } = 3 n + 5\).
  1. State the values of \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\).
  2. Find the value of \(n\) such that \(u _ { n } = 254\).
  3. Evaluate \(\sum _ { n = 1 } ^ { 500 } u _ { n }\).
Pre-U Pre-U 9794/1 2016 June Q5
4 marks Moderate -0.8
5 The circle with equation \(x ^ { 2 } + y ^ { 2 } - 6 x - k = 0\) has radius 5 . Find the coordinates of the centre of the circle and the value of \(k\).
Pre-U Pre-U 9794/1 2016 June Q6
9 marks Moderate -0.3
6
  1. Find the coordinates of the stationary points of the curve with equation $$y = 3 x ^ { 4 } - 20 x ^ { 3 } + 36 x ^ { 2 }$$ and determine their nature.
  2. Sketch the graph of \(y = 3 x ^ { 4 } - 20 x ^ { 3 } + 36 x ^ { 2 }\) and hence state the set of values of \(k\) for which the equation \(3 x ^ { 4 } - 20 x ^ { 3 } + 36 x ^ { 2 } = k\) has exactly four distinct real roots.
Pre-U Pre-U 9794/1 2016 June Q7
8 marks Moderate -0.8
7 The functions f and g are defined for all real numbers by $$\mathrm { f } ( x ) = x ^ { 2 } + 2 \quad \text { and } \quad \mathrm { g } ( x ) = 4 x + 3$$
  1. State the range of each of the functions f and g .
  2. Find the values of \(x\) for which \(\mathrm { fg } ( x ) = \mathrm { gf } ( x )\).
  3. The function h , given by \(\mathrm { h } ( x ) = x ^ { 2 } + 2\), has the same range as f but is such that \(\mathrm { h } ^ { - 1 } ( x )\) exists. State a possible domain for h and find an expression for \(\mathrm { h } ^ { - 1 } ( x )\).