Questions — Pre-U (885 questions)

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Pre-U Pre-U 9794/3 2013 June Q4
10 marks Standard +0.3
4 A tomato grower grows just one variety of tomatoes. The weights of these tomatoes are found to be normally distributed with a mean of 85.1 grams and a standard deviation of 3.4 grams.
  1. Find the probability that a randomly chosen tomato of this variety weighs less than 80 grams.
  2. The grower puts the tomatoes in packs of 6 . Find the probability that, in a randomly chosen pack of 6 , at most one tomato weighs less than 80 grams.
  3. The grower supplies consignments of 250 packs of these tomatoes to a retailer. For a randomly chosen consignment, find the expected number of packs having more than one tomato weighing less than 80 grams.
Pre-U Pre-U 9794/3 2013 June Q5
10 marks Standard +0.3
5 A game is played with cards, each of which has a single digit printed on it. Eleanor has 7 cards with the digits \(1,1,2,3,4,5,6\) on them.
  1. How many different 7-digit numbers can be made by arranging Eleanor's cards?
  2. Eleanor is going to select 5 of the 7 cards and use them to form a 5 -digit number. How many different 5-digit numbers are possible?
Pre-U Pre-U 9794/3 2013 June Q6
13 marks Moderate -0.3
6 A particle travels along a straight line. Its velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) after \(t\) seconds is given by $$v = t ^ { 3 } - 6 t ^ { 2 } + 8 t \text { for } 0 \leqslant t \leqslant 4$$ When \(t = 0\) the particle is at rest at the point \(P\).
  1. Find the times (other than \(t = 0\) ) when the particle is at rest. Sketch the velocity-time graph for \(0 \leqslant t \leqslant 4\).
  2. Find the acceleration of the particle when \(t = 2\).
  3. Find an expression for the displacement of the particle from \(P\) after \(t\) seconds. Hence state its displacement from \(P\) when \(t = 2\) and find its average speed between \(t = 0\) and \(t = 2\).
Pre-U Pre-U 9794/3 2013 June Q7
8 marks Standard +0.3
7 A particle \(A\) of mass \(4 m\), on a smooth horizontal plane, is moving with speed \(u\) directly towards another particle \(B\), of mass \(2 m\), which is at rest. The coefficient of restitution between the two particles is \(e\).
  1. Show that, after the collision, the velocity of \(A\) is \(\frac { 1 } { 3 } ( 2 - e ) u\) and find the velocity of \(B\).
  2. Hence write down their velocities in the case when \(e = \frac { 1 } { 2 }\). Particle \(B\) now collides directly with a third particle \(C\), of mass \(m\), which is at rest. The coefficient of restitution in both collisions is \(\frac { 1 } { 2 }\).
  3. Use your answers to part (ii) to find the velocities of \(A , B\) and \(C\) after the second collision has taken place.
  4. Explain briefly whether any further collisions take place.
Pre-U Pre-U 9794/3 2013 June Q8
10 marks Standard +0.3
8 A particle is projected from a point \(O\) with initial speed \(U\) at an angle \(\theta\) above the horizontal. At time \(t\) after projection the position of the particle is \(( x , y )\) relative to horizontal and vertical axes through \(O\).
  1. Write down expressions for \(x\) and \(y\) at time \(t\). Hence derive the cartesian equation of the trajectory of the particle.
  2. A player in a cricket match throws the ball with speed \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to another player who is 45 metres away. Assume that the players throw and catch the ball at the same height above the ground. Show that there are two possible trajectories and find their respective angles of projection. [4]
  3. Describe briefly one advantage of each trajectory.
Pre-U Pre-U 9794/3 2013 June Q9
9 marks Standard +0.3
9 A particle of mass \(m \mathrm {~kg}\) rests in equilibrium on a rough horizontal table. There is a string attached to the particle. The tension in the string is \(T \mathrm {~N}\) at an angle of \(\theta\) to the horizontal, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{2e3f056c-58a2-4466-94ea-3fb873e54752-4_205_547_1027_799}
  1. Copy and complete the diagram to show all the forces acting on the particle.
  2. The coefficient of friction between the particle and the table is \(\mu\) and the particle is on the point of slipping. Show that \(T = \frac { \mu m g } { \cos \theta + \mu \sin \theta }\).
  3. Given that \(\mu = 0.75\), find the value of \(\theta\) for which \(T\) is a minimum.
Pre-U Pre-U 9794/1 2013 November Q1
Easy -1.8
1 Solve the simultaneous equations $$\begin{aligned} x ^ { 2 } + y ^ { 2 } & = 5 \\ y & = 2 x \end{aligned}$$
Pre-U Pre-U 9794/1 2013 November Q2
Easy -1.2
2 Find the equation of the line perpendicular to the line \(y = 5 x\) which passes through the point \(( 2,11 )\). Give your answer in the form \(a x + b y = c\) where \(a , b\) and \(c\) are integers to be found.
Pre-U Pre-U 9794/1 2013 November Q3
Easy -1.2
3 The first term of a geometric progression is 50 and the common ratio is 0.9 .
  1. Find the fifth term.
  2. Find the sum of the first thirty terms.
  3. Find the sum to infinity.
Pre-U Pre-U 9794/1 2013 November Q4
Moderate -0.8
4 Solve the equation \(x ^ { 2 } + ( \sqrt { 3 } ) x - 18 = 0\), giving each root in the form \(p \sqrt { q }\) where \(p\) and \(q\) are integers.
Pre-U Pre-U 9794/1 2013 November Q5
Easy -1.2
5 Express \(\frac { 7 - x } { ( x - 1 ) ( x + 2 ) }\) in partial fractions.
Pre-U Pre-U 9794/1 2013 November Q6
Easy -1.3
6 Describe fully the transformations which, when applied to the graph of \(y = \mathrm { f } ( x )\), will produce the graphs with equations given by
  1. \(y = \mathrm { f } ( - x )\),
  2. \(y = \mathrm { f } ( x - 3 )\),
  3. \(y = \mathrm { f } \left( \frac { x } { 2 } \right)\).
Pre-U Pre-U 9794/1 2013 November Q7
Easy -1.2
7 Given that \(z\) is a complex number, prove that \(z z ^ { * } = | z | ^ { 2 }\).
Pre-U Pre-U 9794/1 2013 November Q8
Moderate -0.3
8
  1. Express \(\sin x - \sqrt { 8 } \cos x\) in the form \(R \sin ( x - \alpha )\) where \(R \geqslant 0\) and \(0 \leqslant \alpha \leqslant 90 ^ { \circ }\).
  2. Hence write down the maximum value of \(\sin x - \sqrt { 8 } \cos x\) and find the smallest positive value of \(x\) for which it occurs.
Pre-U Pre-U 9794/1 2013 November Q9
Moderate -0.5
9 Find \(\int x \sin 2 x \mathrm {~d} x\).
Pre-U Pre-U 9794/1 2013 November Q10
Moderate -0.3
10 A curve has equation \(y = \frac { \mathrm { e } ^ { x } } { x ^ { 2 } }\). Show that
  1. the gradient of the curve at \(x = 1\) is - e ,
  2. there is a stationary point at \(x = 2\) and determine its nature.
Pre-U Pre-U 9794/1 2013 November Q11
Standard +0.3
11 The functions f and g are defined by \(\mathrm { f } ( x ) = \frac { 1 } { 2 + x } + 5 , x > - 2\) and \(\mathrm { g } ( x ) = | x | , x \in \mathbb { R }\).
  1. Given that the range of f is of the form \(\mathrm { f } ( x ) > a\), find \(a\).
  2. Find an expression for \(\mathrm { f } ^ { - 1 }\), stating its domain and range.
  3. Show that \(\mathrm { gf } ( x ) = \mathrm { f } ( x )\).
  4. Find an expression for \(\mathrm { fg } ( x )\). Determine whether fg has an inverse.
Pre-U Pre-U 9794/1 2013 November Q12
Standard +0.3
12 The diagram shows the curve \(y = \frac { x ^ { 2 } - 3 } { x + 1 }\) for \(x > - 1\). \includegraphics[max width=\textwidth, alt={}, center]{806dc286-416e-4785-8d13-0d524f808cb0-3_435_874_897_639}
  1. Find the coordinates of the points where the curve crosses the axes.
  2. Express \(\frac { x ^ { 2 } - 3 } { x + 1 }\) in the form \(A x + B + \frac { C } { x + 1 }\), where \(A , B\) and \(C\) are constants, and hence show that the exact area enclosed by the \(x\)-axis, the curve \(y = \frac { x ^ { 2 } - 3 } { x + 1 }\) and the lines \(x = 2\) and \(x = 4\) is \(4 + \ln \frac { 9 } { 25 }\).
Pre-U Pre-U 9794/1 2013 November Q13
10 marks Standard +0.3
13 Solve the differential equation \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - k ( y - 10 )\), where \(k\) is a constant, given that \(y = 70\) when \(x = 0\) and \(y = 40\) when \(x = 1\). Express your answer in the form \(y = a + b \left( \frac { 1 } { 2 } \right) ^ { x }\) where \(a\) and \(b\) are constants to be found.
[0pt] [10]
Pre-U Pre-U 9794/2 2013 November Q1
Moderate -0.8
1 The diagram shows a sector \(O A B\) of a circle with centre \(O\) and radius 9 cm . The angle \(A O B\) is \(100 ^ { \circ }\).
  1. Express \(100 ^ { \circ }\) in radians, giving your answer in exact form.
  2. Find the perimeter of the sector \(O A B\).
  3. Find the area of the sector \(O A B\).
Pre-U Pre-U 9794/2 2013 November Q2
Easy -1.8
2 Solve the equation \(| x + 3 | = 5\).
Pre-U Pre-U 9794/2 2013 November Q3
Moderate -0.3
3
  1. Show that the equation \(x ^ { 2 } - \ln x - 2 = 0\) has a solution between \(x = 1\) and \(x = 2\).
  2. Find an approximation to that solution using the iteration \(x _ { n + 1 } = \sqrt { 2 + \ln x _ { n } }\), giving your answer correct to 2 decimal places.
Pre-U Pre-U 9794/2 2013 November Q4
Standard +0.3
4 The diagram shows a triangle \(A B C\) in which \(A B = 5 \mathrm {~cm} , B C = 10 \mathrm {~cm}\) and angle \(B C A = 20 ^ { \circ }\). \includegraphics[max width=\textwidth, alt={}, center]{f4e774e5-76fd-48ff-9bce-a995b3ba517b-2_355_767_1695_689}
  1. Find angle \(B A C\), given that it is obtuse.
  2. Find the shortest distance from \(A\) to \(B C\).
Pre-U Pre-U 9794/2 2013 November Q5
Moderate -0.8
5 Solve \(\sin \left( 2 \theta + 30 ^ { \circ } \right) = 0.1\) in the range \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
Pre-U Pre-U 9794/2 2013 November Q6
Moderate -0.3
6 The curve \(y = x ^ { 3 } + a x ^ { 2 } + b x + 1\) has a gradient of 11 at the point \(( 1,7 )\). Find the values of \(a\) and \(b\).