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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 9794/2 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 \(B C\).
Pre-U Pre-U 9794/2 2013 November Q5
Moderate -0.8
5 Solve \(\sin \left( 2 \theta + 30 ^ { \circ } \right) = 0.1\) in the range \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
Pre-U Pre-U 9794/2 2013 November Q6
Moderate -0.3
6 The curve \(y = x ^ { 3 } + a x ^ { 2 } + b x + 1\) has a gradient of 11 at the point \(( 1,7 )\). Find the values of \(a\) and \(b\).
Pre-U Pre-U 9794/2 2013 November Q7
Moderate -0.3
7
  1. Differentiate \(3 \ln \left( x ^ { 2 } + 1 \right)\).
  2. Find \(\int \frac { x ^ { 2 } } { 3 - 4 x ^ { 3 } } \mathrm {~d} x\).
Pre-U Pre-U 9794/2 2013 November Q8
Moderate -0.3
8 Find the exact volume of the solid of revolution generated by rotating the graph of \(y = 3 \mathrm { e } ^ { x }\) between \(x = 0\) and \(x = 2\) through \(360 ^ { \circ }\) about the \(x\)-axis.
Pre-U Pre-U 9794/2 2013 November Q9
Moderate -0.3
9 Two straight lines have equations $$\mathbf { r } = \left( \begin{array} { r } 16 \\ 2 \\ 3 \end{array} \right) + \lambda \left( \begin{array} { r } 3 \\ 2 \\ - 1 \end{array} \right) \quad \text { and } \quad \mathbf { r } = \left( \begin{array} { r } - 3 \\ 8 \\ 12 \end{array} \right) + \mu \left( \begin{array} { r } 5 \\ - 6 \\ - 3 \end{array} \right) .$$ Show that the two lines intersect and find the coordinates of their point of intersection.
Pre-U Pre-U 9794/2 2013 November Q10
Standard +0.3
10
  1. Given that \(10 + 4 x - x ^ { 2 } \equiv p - ( x - q ) ^ { 2 }\), show that \(q = 2\) and find the value of \(p\).
  2. Hence find the coordinates of all the points of intersection of the curve \(y = 10 + 4 x - x ^ { 2 }\) and the circle \(( x - 2 ) ^ { 2 } + ( y - 1 ) ^ { 2 } = 25\).
Pre-U Pre-U 9794/2 2013 November Q11
Standard +0.3
11
  1. Expand \(( 1 + x ) ^ { - 1 }\) up to and including the term in \(x ^ { 2 }\).
  2. (a) Expand \(\sqrt { 2 + 3 x ^ { 2 } }\) up to and including the term in \(x ^ { 4 }\).
    (b) For what range of values of \(x\) is this expansion valid?
  3. Find the first three terms of the expansion of \(\frac { \sqrt { 2 + 3 x ^ { 2 } } } { 1 + x }\) in ascending powers of \(x\) and hence show that \(\int _ { 0 } ^ { 0.1 } \frac { \sqrt { 2 + 3 x ^ { 2 } } } { 1 + x } \mathrm {~d} x \approx 0.135\).
Pre-U Pre-U 9794/2 2013 November Q12
Standard +0.3
12 A curve \(C\) is given by the parametric equations \(x = 2 \tan \theta , y = 1 + \operatorname { cosec } \theta\) for \(0 < \theta < 2 \pi , \theta \neq \frac { 1 } { 2 } \pi , \pi , \frac { 3 } { 2 } \pi\).
  1. Show that the cartesian equation for \(C\) is \(\frac { 4 } { x ^ { 2 } } = y ^ { 2 } - 2 y\).
  2. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence show that \(C\) has no stationary points.
  3. \(P\) is the point on \(C\) where \(\theta = \frac { 1 } { 4 } \pi\). The tangent to \(C\) at \(P\) intersects the \(y\)-axis at \(Q\) and the \(x\)-axis at \(R\). Find the exact area of triangle \(O Q R\).
Pre-U Pre-U 9795/2 2013 November Q1
Moderate -0.3
1 The lifetime, \(T\) years, of a mortgage may be modelled by the random variable \(T\) with probability density function \(\mathrm { f } ( t )\), where $$\mathrm { f } ( t ) = \begin{cases} k \sin \left( \frac { 3 } { 32 } t \right) & 0 \leqslant t \leqslant 8 \pi \\ 0 & \text { otherwise } \end{cases}$$
  1. Show that \(k = \frac { 3 } { 32 } ( 2 - \sqrt { 2 } )\).
  2. Sketch the graph of \(\mathrm { f } ( t )\) and state the mode.
Pre-U Pre-U 9795/2 2013 November Q2
Standard +0.8
2
  1. The statistic \(T\) is derived from a random sample taken from a population which has an unknown parameter \(\theta\). \(T\) is an unbiased estimator of \(\theta\). What does the statement ' \(T\) is an unbiased estimator of \(\theta ^ { \prime }\) imply?
  2. A random sample of size \(n\) is taken from each of two independent populations. The first population has a non-zero mean \(\mu\) and variance \(\sigma ^ { 2 }\) and \(\bar { X } _ { 1 }\) denotes the sample mean. The second population has mean \(\frac { 1 } { 2 } \mu\) and variance \(b \sigma ^ { 2 }\), where \(b\) is a positive constant, and \(\bar { X } _ { 2 }\) denotes the sample mean. Two unbiased estimators for \(\mu\) are defined by $$T _ { 1 } = 3 \bar { X } _ { 1 } - a \bar { X } _ { 2 } \quad \text { and } \quad T _ { 2 } = \frac { 1 } { 5 } \left( 4 \bar { X } _ { 1 } + 2 \bar { X } _ { 2 } \right) .$$
    1. Determine the value of \(a\).
    2. Show that \(\operatorname { Var } \left( T _ { 1 } \right) = \frac { \sigma ^ { 2 } } { n } ( 9 + 16 b )\) and find a similar expression for \(\operatorname { Var } \left( T _ { 2 } \right)\).
    3. The estimator with the smaller variance is preferred. State which of \(T _ { 1 }\) and \(T _ { 2 }\) is the preferred estimator of \(\mu\).