Questions Pre-U 9794/2 (176 questions)

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Pre-U Pre-U 9794/2 2012 Specimen Q1
7 marks Easy -1.8
1
  1. Express each of the following as a single logarithm.
    1. \(\log _ { a } 5 + \log _ { a } 3\)
    2. \(5 \log _ { b } 2 - 3 \log _ { b } 4\)
  2. Express \(\left( 9 a ^ { 4 } \right) ^ { - \frac { 1 } { 2 } }\) as an algebraic fraction in its simplest form.
Pre-U Pre-U 9794/2 2012 Specimen Q2
5 marks Moderate -0.8
2 The diagram shows a triangle \(A B C\) in which angle \(C = 30 ^ { \circ } , B C = x \mathrm {~cm}\) and \(A C = ( x + 2 ) \mathrm { cm }\). Given that the area of triangle \(A B C\) is \(12 \mathrm {~cm} ^ { 2 }\), calculate the value of \(x\).
Pre-U Pre-U 9794/2 2012 Specimen Q3
6 marks Standard +0.3
3 Solve the simultaneous equations $$x + y = 1 , \quad x ^ { 2 } - x y + y ^ { 2 } = 7 .$$
Pre-U Pre-U 9794/2 2012 Specimen Q4
5 marks Moderate -0.8
4 Find
  1. \(\quad \int ( 2 x + 3 ) ^ { 4 } \mathrm {~d} x\)
  2. \(\quad \int \left( 1 + \tan ^ { 2 } 2 x \right) \mathrm { d } x\)
Pre-U Pre-U 9794/2 2012 Specimen Q5
5 marks Standard +0.3
5 When \(x ^ { 4 } - 4 x ^ { 3 } + 5 x ^ { 2 } + x + a\) is divided by \(x ^ { 2 } - x + 1\), the quotient is \(x ^ { 2 } + b x + 1\) and the remainder is \(c x - 3\). Find the values of the constants \(a , b\) and \(c\).
Pre-U Pre-U 9794/2 2012 Specimen Q6
8 marks Moderate -0.8
6 The complex number \(5 - 3 \mathrm { i }\) is denoted by \(z\). Giving your answers in the form \(x + \mathrm { i } y\), and showing clearly how you obtain them, find
  1. \(\quad 6 z - z ^ { * }\),
  2. \(\quad ( z - \mathrm { i } ) ^ { 2 }\),
  3. \(\frac { 5 } { z }\).
Pre-U Pre-U 9794/2 2012 Specimen Q7
5 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{f8b66d63-96ce-43d2-bd28-c048070feac3-3_456_606_182_735} The diagram shows the region \(R\) bounded by the curve \(y = \frac { 1 } { \sqrt { 5 x + 3 } }\) and the lines \(x = 0\), \(x = 3\) and \(y = 0\). Find the exact volume of the solid formed when the region \(R\) is rotated completely about the \(x\)-axis, simplifying your answer.
Pre-U Pre-U 9794/2 2012 Specimen Q8
9 marks Moderate -0.3
8
  1. Express \(\frac { 3 x + 2 } { ( x - 2 ) ^ { 2 } }\) in the form \(\frac { A } { x - 2 } + \frac { B } { ( x - 2 ) ^ { 2 } }\) where \(A\) and \(B\) are constants.
  2. Hence find the exact value of \(\int _ { 6 } ^ { 10 } \frac { 3 x + 2 } { ( x - 2 ) ^ { 2 } } \mathrm {~d} x\), giving your answer in the form \(a + b \ln c\), where \(a , b\) and \(c\) are integers.
Pre-U Pre-U 9794/2 2012 Specimen Q9
8 marks Standard +0.3
9 The parametric equations of a curve are $$x = \mathrm { e } ^ { 2 t } - 5 t , \quad y = \mathrm { e } ^ { 2 t } - 2 t$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
  2. Find the exact value of \(t\) at the point on the curve where the gradient is 2 .
Pre-U Pre-U 9794/2 2012 Specimen Q10
7 marks Standard +0.3
10 Lines \(L _ { 1 } , L _ { 2 }\) and \(L _ { 3 }\) have vector equations $$\begin{aligned} & L _ { 1 } = ( 4 \mathbf { i } + \mathbf { j } + 3 \mathbf { k } ) + s ( 6 \mathbf { i } + 9 \mathbf { j } - 3 \mathbf { k } ) , \\ & L _ { 2 } = ( 2 \mathbf { i } + 3 \mathbf { j } ) + t ( - 3 \mathbf { i } - 8 \mathbf { j } + 6 \mathbf { k } ) , \\ & L _ { 3 } = ( 3 \mathbf { i } - \mathbf { j } + 4 \mathbf { k } ) + u ( - 2 \mathbf { i } + c \mathbf { j } + \mathbf { k } ) . \end{aligned}$$ In each of the following cases, find the value of \(c\).
  1. \(\quad L _ { 1 }\) and \(L _ { 3 }\) are parallel.
  2. \(\quad L _ { 2 }\) and \(L _ { 3 }\) intersect.
Pre-U Pre-U 9794/2 2012 Specimen Q11
16 marks Standard +0.3
11 A curve has equation $$y = \mathrm { e } ^ { a x } \cos b x$$ where \(a\) and \(b\) are constants.
  1. Show that, at any stationary points on the curve, \(\tan b x = \frac { a } { b }\). \includegraphics[max width=\textwidth, alt={}, center]{f8b66d63-96ce-43d2-bd28-c048070feac3-4_631_901_532_571} Values of related quantities \(x\) and \(y\) were measured in an experiment and plotted on a graph of \(y\) against \(x\), as shown in the diagram. Two of the points, labelled \(A\) and \(B\), have coordinates \(( 0,1 )\) and \(( 0.2 , - 0.8 )\) respectively. A third point labelled C has coordinates ( \(0.3,0.04\) ). Attempts were then made to find the equation of a curve which fitted closely to these three points, and two models were proposed.
  2. In the first model the equation is $$y = \mathrm { e } ^ { - x } \cos 12 x$$ Show that this model has a maximum point close to \(A\) and a minimum point close to \(B\), and state the coordinates of these maximum and minimum points and also the \(y\) value when \(x = 0.3\).
  3. In an alternative model the equation is $$y = f \cos ( \lambda x ) + g$$ where the constants \(f , \lambda\) and \(g\) are chosen to give a maximum precisely at the point \(A ( 0,1 )\) and a minimum precisely at the point \(B ( 0.2 , - 0.8 )\). Find suitable values for \(f , \lambda\) and \(g\).
  4. Using the alternative model, state the value of \(y\) when \(x = 0.3\) and hence comment on how accurate each model is in fitting the three given points.
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 9794/2 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 \(B C\).