Questions — Pre-U Pre-U 9794/2 (176 questions)

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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 9794/2 2014 June Q1
2 marks Easy -1.2
1 The diagram shows the triangle \(A B C\). \(A B = 10 \mathrm {~cm} , A C = 7 \mathrm {~cm}\) and angle \(B A C = 100 ^ { \circ }\).
  1. Find the length \(B C\).
  2. Find the area of the triangle \(A B C\).
Pre-U Pre-U 9794/2 2014 June Q2
3 marks Moderate -0.8
2 Let \(\mathrm { f } ( x ) = x ^ { 2 } + k x + 4\), where \(k\) is a constant.
  1. Find an expression for the discriminant of f in terms of \(k\).
  2. Hence find the range of values of \(k\) for which the equation \(\mathrm { f } ( x ) = 0\) has two distinct real roots.
Pre-U Pre-U 9794/2 2014 June Q3
4 marks Moderate -0.8
3 Given that \(\mathrm { f } ( x ) = x ^ { 3 }\), use differentiation from first principles to prove that \(\mathrm { f } ^ { \prime } ( x ) = 3 x ^ { 2 }\).
Pre-U Pre-U 9794/2 2014 June Q4
7 marks Standard +0.3
4 The points \(A , B , C\) and \(D\) have coordinates \(( 2 , - 1,0 ) , ( 3,2,5 ) , ( 4,2,3 )\) and \(( - 1 , a , b )\) respectively, where \(a\) and \(b\) are constants.
  1. Find the angle \(A B C\).
  2. Given that the lines \(A B\) and \(C D\) are parallel, find the values of \(a\) and \(b\).
Pre-U Pre-U 9794/2 2014 June Q5
3 marks Easy -1.3
5 An arithmetic progression has first term 5 and common difference 7.
  1. Find the value of the 10th term.
  2. Find the sum of the first 15 terms. The terms of the progression are given by \(x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots\).
  3. Evaluate \(\sum _ { n = 1 } ^ { 15 } \left( 2 x _ { n } + 1 \right)\).
Pre-U Pre-U 9794/2 2014 June Q6
5 marks Moderate -0.8
6 Given that the angle \(\theta\) is acute and \(\cos \theta = \frac { 3 } { 4 }\) find, without using a calculator, the exact value of \(\sin 2 \theta\) and of \(\cot \theta\).
Pre-U Pre-U 9794/2 2014 June Q7
2 marks Moderate -0.8
7
  1. Express \(z ^ { 4 } + 3 z ^ { 2 } - 4\) in the form \(\left( z ^ { 2 } + a \right) \left( z ^ { 2 } + b \right)\) where \(a\) and \(b\) are real constants to be found.
  2. Hence draw an Argand diagram showing the points that represent the roots of the equation \(z ^ { 4 } + 3 z ^ { 2 } - 4 = 0\).
Pre-U Pre-U 9794/2 2014 June Q8
6 marks Standard +0.3
8 Show that the graph of \(y = x ^ { 2 } - \ln x\) has only one stationary point and give the coordinates of that point in exact form.
Pre-U Pre-U 9794/2 2014 June Q9
7 marks Challenging +1.2
9 A new lake is stocked with fish. Let \(P _ { t }\) be the population of fish in the lake after \(t\) years. Two models using recurrence relations are proposed for \(P _ { t }\), with \(P _ { 0 } = 550\). $$\begin{aligned} & \text { Model } 1 : P _ { t } = 2 P _ { t - 1 } \mathrm { e } ^ { - 0.001 P _ { t - 1 } } \\ & \text { Model } 2 : P _ { t } = \frac { 1 } { 2 } P _ { t - 1 } \left( 7 - \frac { 1 } { 160 } P _ { t - 1 } \right) \end{aligned}$$
  1. Evaluate the population predicted by each model when \(t = 3\).
  2. Identify, with evidence, which one of the models predicts a stable population in the long term.
  3. Describe the long term behaviour of the population for the other model.
Pre-U Pre-U 9794/2 2014 June Q10
11 marks Standard +0.3
10 Let \(\mathrm { f } ( x ) = x ^ { 4 } - 4 x ^ { 3 } - 10 x ^ { 2 } + 28 x - 15\).
  1. Show that \(x = 1\) is a root of the equation \(\mathrm { f } ( x ) = 0\).
  2. Find the quotient and remainder when \(\mathrm { f } ( x )\) is divided by \(x - 5\).
  3. Factorise \(\mathrm { f } ( x )\) fully and hence sketch the graph of \(y = \mathrm { f } ( x )\).
Pre-U Pre-U 9794/2 2014 June Q11
12 marks Challenging +1.2
11 The cubic equation \(x ^ { 3 } - 2 x ^ { 2 } + 4 x - 7 = 0\) has a single root \(\alpha\), close to 1.9 , which can be found using an iteration of the form \(x _ { n + 1 } = \mathrm { F } \left( x _ { n } \right)\). Three possible functions that can be used for such an iteration are $$\mathrm { F } _ { 1 } ( x ) = \frac { 7 } { 4 } + \frac { 1 } { 2 } x ^ { 2 } - \frac { 1 } { 4 } x ^ { 3 } , \quad \mathrm {~F} _ { 2 } ( x ) = \sqrt [ 3 ] { 2 x ^ { 2 } - 4 x + 7 } , \quad \mathrm {~F} _ { 3 } ( x ) = \frac { 7 - 4 x } { x ^ { 2 } - 2 x }$$
  1. Differentiate each of these functions with respect to \(x\).
  2. Without performing any iterations, and using \(x = 1.9\), show that an iterative process based on only two of the given functions will converge. Determine which one will do so more rapidly. The sequence of errors, \(e _ { n }\), is such that \(e _ { n + 1 } \approx \mathrm {~F} ^ { \prime } ( \alpha ) e _ { n }\).
  3. Using the iteration from part (ii) with the most rapid convergence, estimate the number of iterations required to reduce the magnitude of the error from \(\left| e _ { 1 } \right|\) in the first term to less than \(10 ^ { - 10 } \left| e _ { 1 } \right|\).
Pre-U Pre-U 9794/2 2014 June Q12
9 marks Standard +0.8
12 A curve \(C\) is defined parametrically by $$x = \cos t ( 1 - 2 \sin t ) , \quad y = \sin t ( 1 - 3 \sin t ) , \quad 0 \leqslant t < 2 \pi$$
  1. Show that \(C\) intersects the \(y\)-axis at exactly three points, and state the values of \(t\) and \(y\) at these points.
  2. Find the range of values of \(t\) for which \(C\) lies above the \(x\)-axis.
Pre-U Pre-U 9794/2 2015 June Q1
3 marks Easy -1.8
1 Show that \(\frac { 31 } { 6 - \sqrt { 5 } } = 6 + \sqrt { 5 }\).
Pre-U Pre-U 9794/2 2015 June Q2
4 marks Easy -1.2
2 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 x ^ { 2 } + 2\). The curve passes through the point \(( 1,3 )\). Find the equation of the curve.
Pre-U Pre-U 9794/2 2015 June Q3
4 marks Easy -1.2
3 The function f is given by \(\mathrm { f } ( x ) = | x - 2 | + 3\) for \(- 5 \leqslant x \leqslant 5\).
  1. Sketch the graph of \(y = \mathrm { f } ( x )\).
  2. Explain why f is not one-one.
Pre-U Pre-U 9794/2 2015 June Q4
4 marks Moderate -0.8
4 Find the volume of the solid generated when the region bounded by the \(x\)-axis, \(x = 1 , x = 2\) and the curve given by \(y = x ^ { 3 }\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
Pre-U Pre-U 9794/2 2015 June Q5
8 marks Moderate -0.3
5
  1. Show that the equation \(\sin x - x + 1 = 0\) has a root between 1.5 and 2 .
  2. Use the iteration \(x _ { n + 1 } = 1 + \sin x _ { n }\), with a suitable starting value, to find that root correct to 2 decimal places.
  3. Sketch the graphs of \(y = \sin x\) and \(y = x - 1\), on the same set of axes, for \(0 \leqslant x \leqslant \pi\).
  4. Explain why the equation \(\sin x - x + 1 = 0\) has no root other than the one found in part (ii). [1]