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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 \(P\).
Pre-U Pre-U 9794/3 2012 June Q10
10 marks Challenging +1.2
10 \includegraphics[max width=\textwidth, alt={}, center]{f0c32e07-f3a0-4d58-bd00-c266177ceaac-4_81_949_1283_598} Three particles \(A , B\) and \(C\), having masses \(1 \mathrm {~kg} , 2 \mathrm {~kg}\) and 5 kg , respectively, are placed 1 metre apart in a straight line on a smooth horizontal plane (see diagram). The particles \(B\) and \(C\) are initially at rest and \(A\) is moving towards \(B\) with speed \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The coefficient of restitution between each pair of particles is 0.5 .
  1. Find the velocity of \(B\) immediately after the first impact and show that \(A\) comes to rest.
  2. Show that \(B\) reverses direction after an impact with \(C\).
  3. Find the distance between \(B\) and \(C\) at the instant that \(B\) collides with \(A\) for the second time.
Pre-U Pre-U 9794/3 2012 June Q11
13 marks Standard +0.3
11 A particle \(P\) of mass 2 kg can move along a line of greatest slope on the smooth surface of a wedge which is fixed to the ground. The sloping face \(O A\) of the wedge has length 10 metres and is inclined at \(30 ^ { \circ }\) to the horizontal (see Fig. 1). \(P\) is fired up the slope from the lowest point \(O\), with an initial speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f0c32e07-f3a0-4d58-bd00-c266177ceaac-5_295_1529_484_310} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure}
  1. Find the time taken for \(P\) to reach \(A\) and show that the speed of \(P\) at \(A\) is \(10 \sqrt { 3 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). After \(P\) has reached \(A\) it becomes a projectile (see Fig. 2). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{f0c32e07-f3a0-4d58-bd00-c266177ceaac-5_424_1533_1123_306} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure}
  2. Find the total horizontal distance travelled by \(P\) from \(O\) when it hits the ground.
Pre-U Pre-U 9794/1 2012 Specimen Q1
2 marks Easy -1.8
1 Write down the coordinates of the centre and the radius of the circle with equation $$( x + 5 ) ^ { 2 } + ( y - 3 ) ^ { 2 } = 36 .$$
Pre-U Pre-U 9794/1 2012 Specimen Q2
5 marks Moderate -0.8
2
  1. Show that \(x = 2\) is a root of the equation \(2 x ^ { 3 } - x ^ { 2 } - 15 x + 18 = 0\).
  2. Hence solve the equation \(2 x ^ { 3 } - x ^ { 2 } - 15 x + 18 = 0\).
Pre-U Pre-U 9794/1 2012 Specimen Q3
6 marks Easy -1.2
3
  1. In an arithmetic progression, the first term is 7 and the sum of the first 40 terms is 4960. Find the common difference.
  2. A geometric progression has first term 14 and common ratio 0.3. Find the sum to infinity.
Pre-U Pre-U 9794/1 2012 Specimen Q4
7 marks Moderate -0.3
4 A sector \(A O B\) of a circle has radius \(r \mathrm {~cm}\) and the angle \(A O B\) is \(\theta\) radians. The perimeter of the sector is 40 cm and its area is \(100 \mathrm {~cm} ^ { 2 }\).
  1. Write down equations for the perimeter and area of the sector in terms of \(r\) and \(\theta\).
  2. Use your equations to show that \(r ^ { 2 } - 20 r + 100 = 0\) and hence find the value of \(r\) and of \(\theta\).
Pre-U Pre-U 9794/1 2012 Specimen Q5
8 marks Moderate -0.3
5
  1. Find \(\int \left( \frac { 1 } { x - 2 } - \frac { 2 } { 2 x + 3 } \right) \mathrm { d } x\) giving your answer in its simplest form.
  2. Use integration by parts to find \(\int x ^ { 2 } \ln x \mathrm {~d} x\).
Pre-U Pre-U 9794/1 2012 Specimen Q6
7 marks Moderate -0.3
6
  1. Find and simplify the first four terms in the expansion of \(( 1 - 2 x ) ^ { 9 }\) in ascending powers of \(x\).
  2. In the expansion of $$( 2 + a x ) ( 1 - 2 x ) ^ { 9 }$$ the coefficient of \(x ^ { 2 }\) is 66 . Find the value of \(a\).