Questions — Pre-U (885 questions)

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Pre-U Pre-U 9794/2 2018 June Q6
12 marks Standard +0.8
6 \includegraphics[max width=\textwidth, alt={}, center]{f4b66aaa-16b9-4b15-b3f5-b9657fe98274-3_545_557_269_794} The diagram shows a sector \(A O B\) of a circle, centre \(O\) and radius \(r\). Angle \(A O B\) is \(\theta\) radians. The point \(C\) lies on \(O B\), and \(A C\) is perpendicular to \(O B\). The area of the triangle \(A O C\) is equal to the area of the segment bounded by the chord \(A B\) and the \(\operatorname { arc } A B\).
  1. Show that \(\theta = \sin \theta ( 1 + \cos \theta )\). The equation \(\theta = \sin \theta ( 1 + \cos \theta )\) has only one positive root.
  2. Use an iterative process based on this equation to find the value of the root correct to 3 significant figures. Use a starting value of 1 and show the result of each iteration. Use a change of sign to verify that the value you have found is correct to 3 significant figures.
Pre-U Pre-U 9794/2 2018 June Q7
10 marks Standard +0.8
7 A curve is given parametrically by \(x = t ^ { 2 } + 1 , y = t ^ { 3 } - 2 t\) where \(t\) is any real number.
  1. Show that the equation of the normal to the curve at the point where \(t = 2\) can be written in the form \(2 x + 5 y = 30\).
  2. Show that this normal does not meet the curve again.
Pre-U Pre-U 9794/2 2018 June Q8
8 marks Standard +0.3
8
  1. Use integration by parts twice to show that $$\int \mathrm { e } ^ { x } \sin x \mathrm {~d} x = \frac { 1 } { 2 } \mathrm { e } ^ { x } ( \sin x - \cos x ) + c .$$
  2. Hence find the equation of the curve which passes through the point \(( 0,2 )\) and for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { e } ^ { x } \sin x\).
Pre-U Pre-U 9794/2 2018 June Q9
13 marks Standard +0.8
9 In this question, \(x\) denotes an angle measured in degrees.
  1. Express \(4 \sin \left( 2 x + 30 ^ { \circ } \right) + 3 \cos 2 x\) in the form \(R \cos ( 2 x - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
  2. Give full details of the sequence of transformations which maps the graph of \(y = \cos x\) onto the graph of \(y = 4 \sin \left( 2 x + 30 ^ { \circ } \right) + 3 \cos 2 x\).
  3. Find the smallest positive value of \(x\) that satisfies the equation \(4 \sin \left( 2 x + 30 ^ { \circ } \right) + 3 \cos 2 x = 6\).
Pre-U Pre-U 9794/2 2018 June Q10
10 marks Challenging +1.2
10
  1. By using the substitution \(u = 3 - 2 x\), or otherwise, show that \(\int _ { 0 } ^ { 1 } \left( \frac { 4 x } { 3 - 2 x } \right) ^ { 2 } \mathrm {~d} x = 16 - 12 \ln 3\).
  2. \includegraphics[max width=\textwidth, alt={}, center]{f4b66aaa-16b9-4b15-b3f5-b9657fe98274-4_595_588_927_817} The diagram shows the region \(R\), which is bounded by the curve \(y = \frac { 4 x } { 3 - 2 x }\), the \(y\)-axis and the line \(y = 4\). Find the exact volume generated when the region \(R\) is rotated completely around the \(x\)-axis. {www.cie.org.uk} after the live examination series. }
Pre-U Pre-U 9794/3 2018 June Q2
9 marks Moderate -0.3
2 A teacher is monitoring the progress of students. The length of time, \(x\) hours, spent revising in a given week is compared to the score, \(y\), achieved in an assessment at the end of the week. The scatter diagram for a random sample of 8 students is shown below. \includegraphics[max width=\textwidth, alt={}, center]{35d24778-1203-4d5d-be4b-bb375344fe09-2_866_967_715_589} The data are summarised as \(\Sigma x = 24.6 , \Sigma y = 404 , \Sigma x ^ { 2 } = 105.56 , \Sigma y ^ { 2 } = 20820\) and \(\Sigma x y = 1350.2\).
  1. Find the equation of the least squares regression line of \(y\) on \(x\).
  2. Calculate the product moment correlation coefficient for the data.
  3. A ninth student, Jane, revises for 1.5 hours.
    1. Estimate her score in the assessment.
    2. Comment on the reliability of this estimate.
Pre-U Pre-U 9794/3 2018 June Q3
5 marks Easy -1.2
3 John plays a game with two unbiased coins. John tosses the coins. If he gets two heads he wins \(\pounds 1\). If he gets two tails he wins 20 p. If he gets one head and one tail he wins nothing. Let \(X\) be the random variable for the amount of money, in pence, John wins per game.
  1. Construct a probability distribution table for \(X\).
  2. Calculate \(\mathrm { E } ( X )\).
  3. John pays \(s\) pence to play the game. State the values of \(s\) for which John should expect to make a loss.
Pre-U Pre-U 9794/3 2018 June Q4
6 marks Moderate -0.3
4 On a particular day at a busy international airport, 75\% of the scheduled flights depart on time. A random sample of 16 flights is chosen.
  1. Find the expected number of flights that depart on time.
  2. For these 16 flights, find the probability that fewer than 14 flights depart on time.
  3. For these 16 flights, the probability that at least \(k\) flights depart on time is greater than 0.9 . Find the largest possible value of \(k\).
Pre-U Pre-U 9794/3 2018 June Q5
9 marks Standard +0.3
5 A soft drinks company has an automated bottling machine that fills 500 ml bottles with soft drink. The contents of the bottles are measured during a check on the machine. In the check, \(5 \%\) of the bottles contain more than 500 ml and \(2.5 \%\) contain less than 495 ml . It is given that the amount of drink dispensed per bottle is normally distributed.
  1. Find the mean and standard deviation of the amount of drink dispensed per bottle, giving your answers to 4 significant figures.
  2. It is subsequently found that the measurements of volume made in the checking process are all 3 ml below their true value. Using a corrected distribution, find the probability that a bottle chosen at random contains more than 500 ml of the drink.
Pre-U Pre-U 9794/3 2018 June Q6
12 marks Moderate -0.3
6 A volleyball squad has 11 players. A volleyball team consists of 6 players.
  1. Find the total number of different teams that could be chosen from the squad. The squad has 5 women and 6 men.
  2. Find the total number of different teams that contain at least 3 women. The squad includes a man and a woman who are married to one another.
  3. It is given that the team chosen has exactly 3 women and all such teams are equally likely to be chosen. Calculate the probability that a team chosen includes the married couple.
Pre-U Pre-U 9794/3 2018 June Q7
5 marks Moderate -0.8
7 A particle is projected with a speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), at an angle of \(40 ^ { \circ }\) above the horizontal. Find the speed and direction of motion of the particle 0.4 s after projection.
Pre-U Pre-U 9794/3 2018 June Q8
7 marks Easy -1.2
8 A small ball is thrown vertically upwards with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), from a point 5 m above the ground. Assuming air resistance is negligible, find
  1. the greatest height above the ground reached by the ball,
  2. the time taken for the ball to reach the ground.
Pre-U Pre-U 9794/3 2018 June Q9
5 marks Moderate -0.8
9 A particle \(P\) of mass \(m \mathrm {~kg}\) is moving with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a straight line on a smooth horizontal table. \(P\) collides directly with a stationary particle \(Q\) of mass 0.5 kg . This collision reverses the direction of motion of \(P\). Immediately after the collision the speed of \(P\) is \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(Q\) is \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find
  1. the value of \(m\),
  2. the coefficient of restitution between the two particles.
Pre-U Pre-U 9794/3 2018 June Q10
7 marks Standard +0.3
10 A particle \(P\) moves in a straight line starting from \(O\). At time \(t\) seconds after leaving \(O\), the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = 5 + 1.5 t - 0.125 t ^ { 3 }\).
  1. Find the displacement of \(P\) between the times \(t = 1\) and \(t = 4\).
  2. Find the time at which the velocity of \(P\) is a maximum, justifying your answer.
Pre-U Pre-U 9794/3 2018 June Q11
13 marks Standard +0.3
11 \includegraphics[max width=\textwidth, alt={}, center]{35d24778-1203-4d5d-be4b-bb375344fe09-4_285_700_1043_721} Three forces are acting on a particle \(A\) as shown in the diagram. The forces act in the same plane and the particle is in equilibrium.
  1. Evaluate \(P\) and \(\theta\). The 8 N force is removed.
  2. State the direction of the instantaneous acceleration of \(A\).
Pre-U Pre-U 9794/3 2018 June Q12
12 marks Challenging +1.8
12 \includegraphics[max width=\textwidth, alt={}, center]{35d24778-1203-4d5d-be4b-bb375344fe09-5_429_873_264_635} The diagram shows a block \(B\) of mass 2 kg and a particle \(A\) of mass 3 kg attached to opposite ends of a light inextensible string. The block is held at rest on a rough plane inclined at \(20 ^ { \circ }\) to the horizontal, and the coefficient of friction between the block and the plane is 0.4 . The string passes over a small smooth pulley \(C\) at the edge of the plane and \(A\) hangs in equilibrium 1.2 m above horizontal ground. The part of the string between \(B\) and \(C\) is parallel to a line of greatest slope of the plane. \(B\) is released and begins to move up the plane.
  1. Show that the acceleration of \(A\) is \(3.13 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), correct to 3 significant figures, and find the tension in the string.
  2. When \(A\) reaches the ground it remains there. Given that \(B\) does not reach \(C\) in the subsequent motion, find the total time that \(B\) is moving up the plane.
Pre-U Pre-U 9794/1 2018 June Q5
10 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{69214874-18a7-495d-892d-2a0a7019cbe9-2_746_1182_1304_479} The diagram shows the curve with equation \(y = x ^ { 3 } + 2 x ^ { 2 } - 13 x + 10\) and the tangent to the curve at the point ( 2,0 ).
  1. Find the equation of this tangent and verify that the tangent intersects the curve when \(x = - 6\).
  2. Calculate the exact area of the region bounded by the curve and the tangent.
Pre-U Pre-U 9795/1 2019 Specimen Q3
2 marks Standard +0.3
3
  1. Evaluate, in terms of \(k\), the determinant of the matrix \(\left( \begin{array} { c c c } 1 & 2 & 1 \\ - 3 & 5 & 8 \\ 6 & 12 & k \end{array} \right)\). Three planes have equations \(x + 2 y + z = 4 , - 3 x + 5 y + 8 z = 21\) and \(6 x + 12 y + k z = 31\).
  2. State the value of \(k\) for which these three planes do not meet at a single point.
  3. Find the coordinates of the point of intersection of the three planes when \(k = 7\).
Pre-U Pre-U 9795/1 2019 Specimen Q4
3 marks Challenging +1.2
4
  1. Given that \(y = \sqrt { \sinh x }\) for \(x \geqslant 0\), express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(y\) only.
  2. Hence or otherwise find \(\int \frac { 2 t } { \sqrt { 1 + t ^ { 4 } } } \mathrm {~d} t\).
Pre-U Pre-U 9795/1 2019 Specimen Q6
5 marks Standard +0.8
6 The curve \(C\) has equation \(y = \frac { x + 1 } { x ^ { 2 } + 3 }\).
  1. By considering a suitable quadratic equation in \(x\), find the set of possible values of \(y\) for points on \(C\).
  2. Deduce the coordinates of the turning points on \(C\).
  3. Sketch \(C\).
Pre-U Pre-U 9795/1 2019 Specimen Q7
2 marks Challenging +1.2
7 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 \(\mathrm { f } ^ { \prime \prime } ( 1 )\).
  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 2019 Specimen Q8
5 marks Challenging +1.8
8 Consider the set \(S\) of all matrices of the form \(\left( \begin{array} { l l } p & p \\ p & p \end{array} \right)\), where p is a non-zero rational number.
  1. Show that \(S\), under the operation of matrix multiplication, forms a group, \(G\). (You may assume that matrix multiplication is associative.)
  2. Find a subgroup of \(G\) of order 2 and show that \(G\) contains no subgroups of order 3 .
Pre-U Pre-U 9795/1 2019 Specimen Q9
3 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 2019 Specimen Q10
8 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 2019 Specimen Q11
8 marks Challenging +1.8
11
  1. Use de Moivre's theorem to prove that \(\sin 5 \theta \equiv s \left( 16 s ^ { 4 } - 20 s ^ { 2 } + 5 \right)\), where \(s = \sin \theta\), and deduce that $$\sin \frac { 2 \pi } { 5 } = \sqrt { \frac { 5 + \sqrt { 5 } } { 8 } } .$$ The complex number \(\omega = 16 ( - 1 + \mathrm { i } \sqrt { 3 } )\).
  2. State the value of \(| \omega |\) and find \(\arg \omega\) as a rational multiple of \(\pi\).
    1. Determine the five roots of the equation \(z ^ { 5 } = \omega\), giving your answers in the form ( \(\mathrm { r } , \theta\) ), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
    2. These five roots are represented in the complex plane by the points \(A , B , C , D\) and \(E\). Show these points on an Argand diagram, and find the area of the pentagon \(A B C D E\) in an exact surd form.[3]