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Pre-U Pre-U 9794/1 2012 Specimen Q7
4 marks Standard +0.3
7 Given that the equation \(x = 2 - \frac { 1 } { ( x + 1 ) ^ { 2 } }\) has a root between \(x = 1\) and \(x = 2\), use the Newton-Raphson formula with \(x _ { 0 } = 2\) to find this root correct to 3 decimal places.
Pre-U Pre-U 9794/1 2012 Specimen Q8
7 marks Moderate -0.8
8 A curve has equation \(y = 2 x ^ { 3 } - 5 x ^ { 2 } - 4 x + 1\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Hence find the \(x\)-coordinates of the stationary points of the curve.
  3. By using the second derivative, determine whether each of the stationary points is a maximum or a minimum.
Pre-U Pre-U 9794/1 2012 Specimen Q9
10 marks Moderate -0.3
9
  1. On the same axes, sketch the curves \(y = 3 + 2 x - x ^ { 2 }\) and \(y = x + 1\).
  2. Find the exact area of the region contained between the curves \(y = 3 + 2 x - x ^ { 2 }\) and \(y = x + 1\).
Pre-U Pre-U 9794/1 2012 Specimen Q10
6 marks Moderate -0.3
10 The points \(A\) and \(B\) have position vectors \(\mathbf { a }\) and \(\mathbf { b }\) relative to an origin \(O\), where \(\mathbf { a } = 5 \mathbf { i } + 4 \mathbf { j } - 2 \mathbf { k }\) and \(\mathbf { b } = - 7 \mathbf { i } + 3 \mathbf { j } + \mathbf { k }\).
  1. Find the length of \(A B\).
  2. Use a scalar product to find angle \(O A B\).
Pre-U Pre-U 9794/1 2012 Specimen Q11
6 marks Standard +0.3
11 Solve the differential equation \(x ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } = \sec y\) given that \(y = \frac { \pi } { 6 }\) when \(x = 4\) giving your answer in the form \(y = \mathrm { f } ( x )\).
Pre-U Pre-U 9794/1 2012 Specimen Q12
11 marks Challenging +1.2
12 Calculate the maximum and minimum values of \(\frac { 1 } { 2 + \cos \theta + \sqrt { 2 } \sin \theta }\) and the smallest positive values of \(\theta\) for which they occur.
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 9795/1 2013 June Q1
4 marks Standard +0.3
1 By completing the square, or otherwise, find the exact value of \(\int _ { 2 } ^ { 6 } \frac { 1 } { x ^ { 2 } - 6 x + 12 } \mathrm {~d} x\).
Pre-U Pre-U 9795/1 2013 June Q2
4 marks Challenging +1.2
2 Use the standard Maclaurin series expansions given in the List of Formulae MF20 to show that $$\frac { 1 } { 2 } \ln \left( \frac { 1 + x } { 1 - x } \right) \equiv \tanh ^ { - 1 } x \text { for } - 1 < x < 1$$
Pre-U Pre-U 9795/1 2013 June Q3
2 marks Standard +0.3
3 The curve \(C\) has equation \(y = \frac { x + 1 } { x ^ { 2 } - 4 }\).
  1. Show that the gradient of \(C\) is always negative.
  2. Sketch \(C\), showing all significant features.
Pre-U Pre-U 9795/1 2013 June Q4
2 marks Standard +0.8
4
  1. Find a vector which is perpendicular to both of the vectors $$\mathbf { d } _ { 1 } = \mathbf { i } + 2 \mathbf { j } + 4 \mathbf { k } \quad \text { and } \quad \mathbf { d } _ { 2 } = 9 \mathbf { i } - 3 \mathbf { j } + \mathbf { k } .$$
  2. Determine the shortest distance between the skew lines with equations $$\mathbf { r } = 2 \mathbf { i } + 4 \mathbf { j } + 3 \mathbf { k } + \lambda ( \mathbf { i } + 2 \mathbf { j } + 4 \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = \mathbf { i } + \mathbf { j } + 10 \mathbf { k } + \mu ( 9 \mathbf { i } - 3 \mathbf { j } + \mathbf { k } ) .$$
Pre-U Pre-U 9795/1 2013 June Q5
5 marks Standard +0.8
5 Let \(z = \cos \theta + \mathrm { i } \sin \theta\).
  1. Prove the result \(z ^ { n } - \frac { 1 } { z ^ { n } } = 2 \mathrm { i } \sin n \theta\).
  2. Use this result to express \(\sin ^ { 5 } \theta\) in the form \(A \sin 5 \theta + B \sin 3 \theta + C \sin \theta\), for constants \(A , B\) and \(C\) to be determined.
Pre-U Pre-U 9795/1 2013 June Q6
8 marks Standard +0.8
6 The curve \(P\) has polar equation \(r = \frac { 1 } { 1 - \sin \theta }\) for \(0 \leqslant \theta < 2 \pi , \theta \neq \frac { 1 } { 2 } \pi\).
  1. Determine, in the form \(y = \mathrm { f } ( x )\), the cartesian equation of \(P\).
  2. Sketch \(P\).
  3. Evaluate \(\int _ { \pi } ^ { 2 \pi } \frac { 1 } { ( 1 - \sin \theta ) ^ { 2 } } \mathrm {~d} \theta\).
Pre-U Pre-U 9795/1 2013 June Q7
7 marks Standard +0.8
7
  1. Express \(x ^ { 3 } + y ^ { 3 }\) in terms of \(( x + y )\) and \(x y\).
  2. The equation \(t ^ { 2 } - 3 t + \frac { 8 } { 9 } = 0\) has roots \(\alpha\) and \(\beta\).
    1. Determine the value of \(\alpha ^ { 3 } + \beta ^ { 3 }\).
    2. Hence express 19 as the sum of the cubes of two positive rational numbers.
Pre-U Pre-U 9795/1 2013 June Q8
8 marks Challenging +1.8
8 Let \(G = \left\{ g _ { 1 } , g _ { 2 } , g _ { 3 } , \ldots , g _ { n } \right\}\) be a finite abelian group of order \(n\) under a multiplicative binary operation, where \(g _ { 1 } = e\) is the identity of \(G\).
  1. Let \(x \in G\). Justify the following statements:
    1. \(x g _ { i } = x g _ { j } \Leftrightarrow g _ { i } = g _ { j }\);
    2. \(\left\{ x g _ { 1 } , x g _ { 2 } , x g _ { 3 } , \ldots , x g _ { n } \right\} = G\).
    3. By considering the product of all \(G\) 's elements, and using the result of part (i)(b), prove that \(x ^ { n } = e\) for each \(x \in G\).
    4. Explain why
      (a) this does not imply that all elements of \(G\) have order \(n\),
      (b) this argument cannot be used to justify the same result for non-abelian groups.