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Pre-U Pre-U 9794/1 2014 June Q6
7 marks Moderate -0.8
6 The diagram shows the curve with equation \(y = 7 x - 10 - x ^ { 2 }\) and the tangent to the curve at the point where \(x = 3\). \includegraphics[max width=\textwidth, alt={}, center]{0eb5bd24-e656-40f0-ad85-f21d3e2edf77-3_648_684_342_731}
  1. Show that the curve crosses the \(x\)-axis at \(x = 2\).
  2. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence find the equation of the tangent to the curve at \(x = 3\). Show that the tangent crosses the \(x\)-axis at \(x = 1\).
  3. Evaluate \(\int _ { 2 } ^ { 3 } \left( 7 x - 10 - x ^ { 2 } \right) \mathrm { d } x\) and hence find the exact area of the shaded region bounded by the curve, the tangent and the \(x\)-axis.
Pre-U Pre-U 9794/1 2015 June Q1
3 marks Easy -1.2
1 Find the set of values of \(x\) for which \(x ^ { 2 } - x - 12 < 0\).
Pre-U Pre-U 9794/1 2015 June Q2
5 marks Moderate -0.8
2 Solve the following simultaneous equations. $$x ^ { 2 } + 2 y ^ { 2 } = 36 \quad x + 2 y = 10$$
Pre-U Pre-U 9794/1 2015 June Q3
3 marks Standard +0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{816a16df-e3a5-48ae-84c6-7f6f5bbba9ca-2_305_825_630_660} The diagram shows a triangle \(A B C\) in which angle \(B = 39 ^ { \circ }\), angle \(C = 28 ^ { \circ } , A B = x \mathrm {~cm}\) and \(A C = ( 2 x - 1 ) \mathrm { cm }\). Find the value of \(x\).
Pre-U Pre-U 9794/1 2015 June Q4
6 marks Moderate -0.8
4 A population, \(P\), is modelled by the equation \(P = a \mathrm { e } ^ { b t }\) where \(t\) is time in years, and \(a\) and \(b\) are constants.
  1. By considering logarithms, show that a graph of \(\ln P\) against \(t\) is a straight line. State the intercept on the vertical axis and the gradient.
  2. Use the graph below to obtain values for \(a\) and \(b\). \includegraphics[max width=\textwidth, alt={}, center]{816a16df-e3a5-48ae-84c6-7f6f5bbba9ca-2_657_750_1530_740}
Pre-U Pre-U 9794/1 2015 June Q5
9 marks Moderate -0.8
5 A circle has equation \(x ^ { 2 } - 6 x + y ^ { 2 } - 4 y = 12\).
  1. Show that the centre of the circle is at the point \(( 3,2 )\) and find the radius.
  2. \(P Q\) is a diameter of the circle where \(P\) has coordinates \(( - 1 , - 1 )\). Find the equation of \(P Q\), giving your answer in the form \(a x + b y = c\) where \(a , b\) and \(c\) are integers.
  3. Another diameter of the circle passes through the point \(( 0,6 )\). Show that this diameter is perpendicular to \(P Q\).
Pre-U Pre-U 9794/1 2015 June Q6
6 marks Moderate -0.3
6 The functions f and g are given by \(\mathrm { f } ( x ) = \frac { 3 } { x - 1 }\) for all \(x \neq 1\) and \(\mathrm { g } ( x ) = x + 2\) for all real \(x\).
  1. Find gf, stating its domain and range.
  2. Find \(( \mathrm { gf } ) ^ { - 1 }\), stating any values of \(x\) for which \(( \mathrm { gf } ) ^ { - 1 }\) is not defined.
Pre-U Pre-U 9794/1 2015 June Q7
9 marks Standard +0.3
7 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have the following vector equations. $$\begin{aligned} & l _ { 1 } : \mathbf { r } = 3 \mathbf { i } + 2 \mathbf { j } + \mathbf { k } + \lambda ( \mathbf { i } - 6 \mathbf { j } - 2 \mathbf { k } ) \\ & l _ { 2 } : \mathbf { r } = \mathbf { i } + 5 \mathbf { j } + 2 \mathbf { k } + \mu ( 3 \mathbf { j } + \mathbf { k } ) \end{aligned}$$
  1. Show that the lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect and find the coordinates of their point of intersection.
  2. Find the acute angle between the lines \(l _ { 1 }\) and \(l _ { 2 }\).
Pre-U Pre-U 9794/1 2015 June Q8
11 marks Moderate -0.3
8 The complex numbers \(w\) and \(z\) are given by \(w = 3 - \mathrm { i }\) and \(z = 1 + \mathrm { i }\).
  1. Express \(\frac { z } { w }\) in the form \(p + \mathrm { i } q\) where \(p\) and \(q\) are real numbers.
  2. On the same Argand diagram, mark the points representing \(z , w\) and \(\frac { z } { w }\).
  3. Find the value in radians of \(\arg w\).
  4. Show that \(z + \frac { 2 } { z }\) is a real number.
Pre-U Pre-U 9794/1 2015 June Q9
7 marks Standard +0.3
9 A curve has equation \(y = \left( x ^ { 2 } - 3 \right) \mathrm { e } ^ { - x }\). Find the exact coordinates of the stationary points of the curve.
Pre-U Pre-U 9794/1 2015 June Q10
11 marks Standard +0.3
10 A curve has parametric equations given by $$x = - \sqrt { ( 1 - t ) ^ { 3 } } \quad y = \sqrt { ( 1 + t ) ^ { 3 } } \quad \text { for } - 1 < t < 1$$
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 + t } { \sqrt { 1 - t ^ { 2 } } }\).
  2. Write \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) as a series of ascending powers of \(t\) up to and including the term in \(t ^ { 4 }\), and hence estimate the gradient of the curve when \(t = 0.5\).
Pre-U Pre-U 9794/1 2015 June Q11
10 marks Standard +0.8
11 Using the substitution \(x = u ^ { 2 } - 1\), or otherwise, show that $$\int \frac { 1 } { 2 x \sqrt { x + 1 } } \mathrm {~d} x = \ln \left( A \sqrt { \frac { \sqrt { x + 1 } - 1 } { \sqrt { x + 1 } + 1 } } \right)$$ where \(A\) is an arbitrary constant and \(x > 0\).
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]
Pre-U Pre-U 9794/2 2015 June Q6
11 marks Moderate -0.3
6 A cup of tea is served at \(80 ^ { \circ } \mathrm { C }\) in a room which is kept at a constant \(20 ^ { \circ } \mathrm { C }\). The temperature, \(T ^ { \circ } \mathrm { C }\), of the tea after \(t\) minutes can be modelled by assuming that the rate of change of \(T\) is proportional to the difference in temperature between the tea and the room.
  1. Explain why the rate of change of the temperature in this model is given by \(\frac { \mathrm { d } T } { \mathrm {~d} t } = - k ( T - 20 )\), where \(k\) is a positive constant.
  2. Show by integration that the temperature of the tea after \(t\) minutes is given by \(T = 20 + 60 \mathrm { e } ^ { - k t }\).
  3. After 2 minutes the tea has cooled to \(60 ^ { \circ } \mathrm { C }\). Find the value of \(k\).
Pre-U Pre-U 9794/2 2015 June Q7
6 marks Standard +0.3
7 A curve is given parametrically by \(x = 3 t , y = 1 + t ^ { 3 }\) where \(t\) is any real number.
  1. Show that a cartesian equation for this curve is given by \(y = 1 + \frac { 1 } { 27 } x ^ { 3 }\). A second curve is given by \(y = x ^ { 2 } + 4 x - 19\).
  2. Given that the curves intersect at the point \(( 3,2 )\), find the coordinates of all the other points of intersection between the two curves.
Pre-U Pre-U 9794/2 2015 June Q8
5 marks Moderate -0.3
8 The function f is given by \(\mathrm { f } ( x ) = \frac { x ^ { 2 } } { 3 x ^ { 2 } - 1 }\), for \(x > 1\). Show that f is a decreasing function.
Pre-U Pre-U 9794/2 2015 June Q9
8 marks Standard +0.8
9 Find the equations of all the horizontal tangents to the curve with equation \(y ^ { 2 } = x ^ { 4 } - 4 x ^ { 3 } + 36\).
Pre-U Pre-U 9794/2 2015 June Q10
14 marks Challenging +1.2
10
  1. Show that \(\sin \left( 2 \theta + \frac { 1 } { 2 } \pi \right) = \cos 2 \theta\).
  2. Hence solve the equation \(\sin 3 \theta = \cos 2 \theta\) for \(0 \leqslant \theta \leqslant 2 \pi\).
  3. Show that \(\sin 3 \theta = 3 \sin \theta - 4 \sin ^ { 3 } \theta\). Hence, by writing \(\cos 2 \theta - \sin 3 \theta\) in terms of \(\sin \theta\), use your answer to part (ii) to determine the solutions of \(4 x ^ { 3 } - 2 x ^ { 2 } - 3 x + 1 = 0\).
Pre-U Pre-U 9794/2 2015 June Q11
11 marks Standard +0.3
11 \includegraphics[max width=\textwidth, alt={}, center]{2f48a6ee-e8ce-47e4-a07f-2c55a6904e7d-3_661_953_767_596} The diagram shows a circle, centre \(O\), radius \(r\). The points \(R\) and \(S\) lie on the circumference of the circle, and the line \(R T\) is a tangent to the circle at \(R\). The angle \(R O S\) is \(\theta\) radians where \(0 < \theta < \frac { 1 } { 2 } \pi\).
  1. Find expressions for the perimeter, \(P\), and the area, \(A\), of the shaded region in terms of \(r\) and \(\theta\).
  2. Hence show that \(A \neq r P\).
Pre-U Pre-U 9794/3 2015 June Q1
5 marks Moderate -0.8
1 The information below summarises the percentages of males unemployed ( \(x\) ) and the percentages of females unemployed ( \(y\) ) in 10 different locations in the UK. $$n = 10 \quad \Sigma x = 87.6 \quad \Sigma x ^ { 2 } = 804.34 \quad \Sigma y = 76.4 \quad \Sigma y ^ { 2 } = 596 \quad \Sigma x y = 684.02$$ Find the product-moment correlation coefficient for these data.
Pre-U Pre-U 9794/3 2015 June Q2
6 marks Standard +0.3
2 Jill is collecting picture cards given away in packets of a particular brand of breakfast cereal. There are five different cards in the complete set. Each packet contains one card which is equally likely to be any of the five cards in the set.
  1. Find the probability that Jill has a complete set of cards from the first five packets that she buys.
  2. At some point Jill needs just one more card to complete the set. Let \(X\) be the random variable that represents the number of additional packets that Jill will need to buy in order to complete the set.
    1. Write down the distribution of \(X\).
    2. State the expected number of additional packets that Jill will need to buy.
    3. Find the probability that Jill will need to buy at least 3 additional packets in order to complete the set.