Questions Pre-U 9794/1 (194 questions)

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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 9794/1 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 9794/1 2014 June Q1
5 marks Easy -1.2
1
  1. Express \(x ^ { 2 } - 8 x + 10\) in the form \(( x - a ) ^ { 2 } + b\) where \(a\) and \(b\) are integers to be found.
  2. Hence write down the minimum value of \(x ^ { 2 } - 8 x + 10\) and the corresponding value of \(x\).
Pre-U Pre-U 9794/1 2014 June Q2
3 marks Moderate -0.8
2 Sketch the curve with equation \(y = \tan x\) for \(- \frac { 1 } { 2 } \pi < x < \frac { 1 } { 2 } \pi\).
On the same diagram, sketch the curve with equation \(y = \tan ^ { - 1 } x\) for all \(x\).
State the geometrical relationship between the curves.