Questions — Pre-U (885 questions)

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Pre-U Pre-U 9794/1 2013 June Q2
4 marks Easy -1.2
2 Find the coefficient of \(x ^ { 3 }\) in the expansion of \(( 1 - 2 x ) ^ { 5 }\).
Pre-U Pre-U 9794/1 2013 June Q3
5 marks Moderate -0.8
3 A sector, \(P O Q\), of a circle centre \(O\) has radius 7 cm and angle 1.7 radians (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{41d9ff74-82de-4ac5-928f-f6ab008319d2-2_469_723_662_712}
  1. Find the length of the line \(P Q\).
  2. Hence find the perimeter of the shaded area.
Pre-U Pre-U 9794/1 2013 June Q4
4 marks Easy -1.2
4 Solve the equation \(2 ^ { 5 x } = 15\), giving the value of \(x\) correct to 3 significant figures.
Pre-U Pre-U 9794/1 2013 June Q5
5 marks Easy -1.3
5
  1. Find \(\int \left( 3 x ^ { 2 } - 4 x + 8 \right) \mathrm { d } x\).
  2. Hence find \(\int _ { 1 } ^ { 3 } \left( 3 x ^ { 2 } - 4 x + 8 \right) \mathrm { d } x\).
Pre-U Pre-U 9794/1 2013 June Q6
3 marks Easy -1.2
6
  1. Sketch the graph of \(y = \cos 2 x\) for \(0 \leqslant x \leqslant 2 \pi\).
  2. Describe the transformation which maps the graph of \(y = \cos x\) onto the graph of \(y = \cos 2 x\).
Pre-U Pre-U 9794/1 2013 June Q7
6 marks Moderate -0.8
7 The complex number \(z\) is given by \(- 20 + 21 \mathrm { i }\). Showing all your working,
  1. find the value of \(| z |\),
  2. calculate the value of \(\arg z\) correct to 3 significant figures,
  3. express \(\frac { 1 } { z }\) in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real numbers.
Pre-U Pre-U 9794/1 2013 June Q8
6 marks Moderate -0.3
8
  1. Let \(\mathrm { f } ( x ) = x ^ { 3 } - x - 1\). Use a sign change method to show that the equation \(x ^ { 3 } - x - 1 = 0\) has a root between \(x = 1\) and \(x = 2\).
  2. By taking \(x = 1\) as a first approximation to this root, use the Newton-Raphson formula to find this root correct to 3 decimal places.
Pre-U Pre-U 9794/1 2013 June Q9
8 marks Moderate -0.3
9
  1. Show that \(\sin \theta + \sqrt { 3 } \cos \theta\) can be expressed in the form \(R \sin ( \theta + \alpha )\) where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\). State the values of \(R\) and \(\alpha\).
  2. Hence find the value of \(\theta\), where \(0 < \theta < \pi\), such that \(\sin \theta + \sqrt { 3 } \cos \theta = 0.8\).
Pre-U Pre-U 9794/1 2013 June Q10
6 marks Standard +0.3
10 Two intersecting straight lines have equations $$\frac { x - 5 } { 4 } = \frac { y - 11 } { 3 } = \frac { z - 7 } { - 5 } \quad \text { and } \quad \frac { x - 9 } { - 2 } = \frac { y - 4 } { 1 } = \frac { z + 4 } { 4 } .$$ Find the coordinates of their point of intersection.
Pre-U Pre-U 9794/1 2013 June Q11
10 marks Moderate -0.3
11 A curve has parametric equations given by $$x = 2 \sin \theta , \quad y = \cos 2 \theta$$
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - 2 \sin \theta\).
  2. Hence find the equation of the tangent to the curve at \(\theta = \frac { 1 } { 2 } \pi\).
  3. Find the cartesian equation of the curve.
Pre-U Pre-U 9794/1 2013 June Q12
6 marks Standard +0.3
12
  1. Prove the identity \(\frac { 1 } { ( x + h ) ^ { 2 } } - \frac { 1 } { x ^ { 2 } } \equiv \frac { - 2 h x - h ^ { 2 } } { x ^ { 2 } ( x + h ) ^ { 2 } }\).
  2. Given that \(\mathrm { f } ( x ) = x ^ { - 2 }\), use differentiation from first principles to find an expression for \(\mathrm { f } ^ { \prime } ( x )\).
Pre-U Pre-U 9794/1 2013 June Q13
12 marks Standard +0.8
13 By first factorising completely \(x ^ { 3 } + x ^ { 2 } - 5 x + 3\), find \(\int \frac { 2 x ^ { 2 } + x + 1 } { x ^ { 3 } + x ^ { 2 } - 5 x + 3 } \mathrm {~d} x\).
Pre-U Pre-U 9794/2 2013 June Q1
4 marks Easy -1.3
1 Vectors \(\mathbf { u }\) and \(\mathbf { v }\) are given by \(\mathbf { u } = \binom { 4 } { 6 }\) and \(\mathbf { v } = \binom { - 3 } { 2 }\).
  1. Find \(\mathbf { u } + \mathbf { v }\) and \(\mathbf { u } - \mathbf { v }\).
  2. Show that \(| \mathbf { u } + \mathbf { v } | = | \mathbf { u } - \mathbf { v } |\).
Pre-U Pre-U 9794/2 2013 June Q2
7 marks Easy -1.3
2
  1. An arithmetic sequence has first term 3 and common difference 2. Find the twenty-first term of this sequence.
  2. Find the sum to infinity of a geometric progression with first term 162 and second term 54.
  3. A sequence is given by the recurrence relation \(u _ { 1 } = 3 , u _ { n + 1 } = 2 - u _ { n } , n = 1,2,3 , \ldots\). Find \(u _ { 2 } , u _ { 3 }\), \(u _ { 4 } , u _ { 5 }\) and describe the behaviour of this sequence.
Pre-U Pre-U 9794/2 2013 June Q3
7 marks Easy -1.2
3
  1. Express \(x ^ { 2 } + 2 x - 3\) in the form \(( x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are integers to be found.
  2. Sketch the graph of \(y = x ^ { 2 } + 2 x - 3\) giving the coordinates of the vertex and of any intersections with the coordinate axes.
Pre-U Pre-U 9794/2 2013 June Q4
10 marks Moderate -0.3
4
  1. Verify that \(z = - 1\) is a root of the equation \(z ^ { 3 } + 5 z ^ { 2 } + 9 z + 5 = 0\).
  2. Find the two complex roots of the equation \(z ^ { 3 } + 5 z ^ { 2 } + 9 z + 5 = 0\).
  3. Show all three roots on an Argand diagram.
Pre-U Pre-U 9794/2 2013 June Q5
8 marks Moderate -0.3
5 The curve \(C\) has equation \(x ^ { 2 } + x y + y ^ { 2 } = 19\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { - 2 x - y } { x + 2 y }\).
  2. Hence find the equation of the normal to \(C\) at the point \(( 2,3 )\) in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
Pre-U Pre-U 9794/2 2013 June Q6
14 marks Moderate -0.3
6 The table below gives the population of breeding pairs of red kites in Yorkshire from 2001 to 2008.
Year20012002200320042005200620072008
Number of breeding pairs810162433404769
Source: \href{http://www.gigrin.co.uk}{www.gigrin.co.uk}
The following model for the population has been proposed: $$N = a \times b ^ { t } ,$$ where \(N\) is the number of breeding pairs \(t\) years after the year 2000, and \(a\) and \(b\) are constants.
  1. Show that the model can be transformed to a linear relationship between \(\log _ { 10 } N\) and \(t\).
  2. On graph paper, plot \(\log _ { 10 } N\) against \(t\) and draw by eye a line of best fit. Use your line to estimate the values of \(a\) and \(b\) in the equation for \(N\) in terms of \(t\).
  3. What values of \(N\) does the model give for the years 2008 and 2020?
  4. In which year will the number of breeding pairs first exceed 500 according to the model?
  5. Comment on the suitability of the model to predict the population of breeding pairs of red kites in Yorkshire.
Pre-U Pre-U 9794/2 2013 June Q7
7 marks Moderate -0.3
7 It is given that \(y = x ^ { 2 } \mathrm { e } ^ { - x }\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = x \mathrm { e } ^ { - x } ( 2 - x )\).
  2. Hence find the exact coordinates of the stationary points on the curve \(y = x ^ { 2 } \mathrm { e } ^ { - x }\).
Pre-U Pre-U 9794/2 2013 June Q8
4 marks Moderate -0.8
8 Evaluate the following, giving your answers in exact form.
  1. \(\sum _ { n = 1 } ^ { 30 } \frac { 1 } { n } - \sum _ { n = 2 } ^ { 29 } \frac { 1 } { n }\).
  2. \(\sum _ { n = 1 } ^ { 100 } n \times ( - 1 ) ^ { n }\).
Pre-U Pre-U 9794/2 2013 June Q9
12 marks Challenging +1.2
9
  1. Prove that \(\operatorname { cosec } 2 x - \cot 2 x \equiv \tan x\) and hence find an exact value for \(\tan \left( \frac { 3 } { 8 } \pi \right)\).
  2. Find the exact value of \(\int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 3 } { 8 } \pi } ( \operatorname { cosec } 2 x - \cot 2 x ) ^ { 2 } \mathrm {~d} x\).
Pre-U Pre-U 9794/2 2013 June Q10
11 marks Standard +0.3
10 A tank with vertical sides and rectangular cross-section is initially full of water. The water is leaking out of a hole in the base of the tank at a rate which is proportional to the square root of the depth of the water. \(V \mathrm {~m} ^ { 3 }\) is the volume of water in the tank at time \(t\) hours.
  1. Show that \(\frac { \mathrm { d } V } { \mathrm {~d} t } = a \sqrt { V }\), where \(a\) is a constant.
  2. Given that the tank is half full after one hour, show that \(V = V _ { 0 } \left( \left( \frac { 1 } { \sqrt { 2 } } - 1 \right) t + 1 \right) ^ { 2 }\), where \(V _ { 0 } \mathrm {~m} ^ { 3 }\) is the initial volume of water in the tank.
  3. Hence show that the tank will be empty after approximately 3 hours and 25 minutes.
Pre-U Pre-U 9794/3 2013 June Q1
4 marks Easy -1.3
1 Pupils at a certain school carried out a survey of traffic passing the school during a two-hour period one morning. One pupil recorded the number of people in each of the first 100 cars. Her results were as follows.
Number of people12345
Number of cars482614102
Find the mean and the standard deviation of the number of people per car in her sample.
Pre-U Pre-U 9794/3 2013 June Q2
4 marks Moderate -0.8
2 Events \(A\) and \(B\) are such that \(\mathrm { P } ( A ) = \frac { 1 } { 2 } , \mathrm { P } ( A \cup B ) = \frac { 5 } { 6 }\) and \(\mathrm { P } ( B \mid A ) = \frac { 1 } { 4 }\).
Find
  1. \(\mathrm { P } ( A \cap B )\),
  2. \(\mathrm { P } ( B )\).
Pre-U Pre-U 9794/3 2013 June Q3
12 marks Moderate -0.8
3 At a local athletics club, data on the ages of the members and their times to run a 10 km course are recorded. For a random sample of 25 club members aged between 20 and 60, their ages ( \(x\) years) and times ( \(y\) minutes) are summarised as follows. $$n = 25 \quad \Sigma x = 1002 \quad \Sigma x ^ { 2 } = 43508 \quad \Sigma y = 1865 \quad \Sigma y ^ { 2 } = 142749 \quad \Sigma x y = 77532$$
  1. Calculate the product moment correlation coefficient for these data.
  2. Show that the equation of the least squares regression line of \(y\) on \(x\) is \(y = 0.83 x + 41.28\), where the coefficients are given correct to 2 decimal places.
  3. Use the equation given in part (ii) to estimate the time taken by someone who is
    1. 50 years old,
    2. 65 years old. Comment on the validity of each of these estimates.