Questions Pre-U 9794/2 (176 questions)

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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 9794/2 2016 Specimen Q1
9 marks Easy -1.3
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.
  3. Show that \(\frac { 3 \sqrt { 3 } - 1 } { 2 \sqrt { 3 } - 3 } = \frac { 15 + 7 \sqrt { 3 } } { 3 }\).
Pre-U Pre-U 9794/2 2016 Specimen Q2
5 marks Moderate -0.5
2 \includegraphics[max width=\textwidth, alt={}, center]{ac5bf967-8b97-4bf3-991f-28c3ec7a25da-2_399_933_968_561} 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 2016 Specimen Q3
5 marks Easy -1.2
3
  1. The points \(A\) and \(B\) have coordinates ( \(- 4,4\) ) and ( 8,1 ) respectively. Find the equation of the line \(A B\). Give your answer in the form \(y = m x + c\).
  2. Determine, with a reason, whether the line \(y = 7 - 4 x\) is perpendicular to the line \(A B\).
Pre-U Pre-U 9794/2 2016 Specimen Q4
7 marks Moderate -0.3
4
  1. Show that \(2 x ^ { 2 } - 10 x - 3\) may be expressed in the form \(a ( x + b ) ^ { 2 } + c\) where \(a , b\) and \(c\) are real numbers to be found. Hence write down the co-ordinates of the minimum point on the curve.
  2. Solve the equation \(4 x ^ { 4 } - 13 x ^ { 2 } + 9 = 0\).
Pre-U Pre-U 9794/2 2016 Specimen Q5
9 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{ac5bf967-8b97-4bf3-991f-28c3ec7a25da-3_570_736_292_667} The diagram shows a sector of a circle, \(O M N\). The angle \(M O N\) is \(2 x\) radians, the radius of the circle is \(r\) and \(O\) is the centre.
  1. Find expressions, in terms of \(r\) and \(x\), for the area, \(A\), and the perimeter, \(P\), of the sector.
  2. Given that \(P = 20\), show that \(A = \frac { 100 x } { ( 1 + x ) ^ { 2 } }\).
  3. Find \(\frac { \mathrm { d } A } { \mathrm {~d} x }\), and hence find the value of \(x\) for which the area of the sector is a maximum.
Pre-U Pre-U 9794/2 2016 Specimen Q6
8 marks Moderate -0.3
6 Diane is given an injection that combines two drugs, Antiflu and Coldcure. At time \(t\) hours after the injection, the concentration of Antiflu in Diane's bloodstream is \(3 \mathrm { e } ^ { - 0.02 t }\) units and the concentration of Coldcure is \(5 \mathrm { e } ^ { - 0.07 t }\) units. Each drug becomes ineffective when its concentration falls below 1 unit.
  1. Show that Coldcure becomes ineffective before Antiflu.
  2. Sketch, on the same diagram, the graphs of concentration against time for each drug.
  3. 20 hours after the first injection, Diane is given a second injection. Determine the concentration of Coldcure 10 hours later.
Pre-U Pre-U 9794/2 2016 Specimen Q7
6 marks Standard +0.3
7 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 9794/2 2016 Specimen Q8
8 marks Moderate -0.3
8 The parametric equations of a curve are $$x = \mathrm { e } ^ { 2 t } - 5 t , \quad y = \mathrm { e } ^ { 2 t } - 3 t .$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
  2. Find the equation of the tangent to the curve at the point when \(t = 0\), giving your answer in the form \(a y + b x + c = 0\) where \(a , b\) and \(c\) are integers.
Pre-U Pre-U 9794/2 2016 Specimen Q9
7 marks Moderate -0.3
9 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 9794/2 2016 Specimen Q10
15 marks Standard +0.8
10 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, \(\tan b x = \frac { a } { b }\).
  2. \includegraphics[max width=\textwidth, alt={}, center]{ac5bf967-8b97-4bf3-991f-28c3ec7a25da-4_622_896_957_333} 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. In the first model the equation is \(y = \mathrm { e } ^ { - x } \cos 15 x\).
    In the second model the equation is \(y = f \cos ( \lambda x ) + \mathrm { g }\), 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 \(B ( 0.2 , - 0.8 )\). By calculating suitable values evaluate the suitability of the two models.
Pre-U Pre-U 9794/2 2016 Specimen Q2
5 marks Moderate -0.5
2 \includegraphics[max width=\textwidth, alt={}, center]{1c957cfe-bead-41d9-8985-479e876e1616-2_403_938_964_559} 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 2016 Specimen Q5
9 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{1c957cfe-bead-41d9-8985-479e876e1616-3_577_743_287_662} The diagram shows a sector of a circle, \(O M N\). The angle \(M O N\) is \(2 x\) radians, the radius of the circle is \(r\) and \(O\) is the centre.
  1. Find expressions, in terms of \(r\) and \(x\), for the area, \(A\), and the perimeter, \(P\), of the sector.
  2. Given that \(P = 20\), show that \(A = \frac { 100 x } { ( 1 + x ) ^ { 2 } }\).
  3. Find \(\frac { \mathrm { d } A } { \mathrm {~d} x }\), and hence find the value of \(x\) for which the area of the sector is a maximum.
Pre-U Pre-U 9794/2 2016 Specimen Q10
15 marks Standard +0.8
10 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, \(\tan b x = \frac { a } { b }\).
  2. \includegraphics[max width=\textwidth, alt={}, center]{1c957cfe-bead-41d9-8985-479e876e1616-4_620_896_959_333} 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. In the first model the equation is \(y = \mathrm { e } ^ { - x } \cos 15 x\).
    In the second model the equation is \(y = f \cos ( \lambda x ) + \mathrm { g }\), 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 \(B ( 0.2 , - 0.8 )\). By calculating suitable values evaluate the suitability of the two models.
Pre-U Pre-U 9794/2 2017 June Q1
4 marks Easy -1.8
1 Find the equation of the line which passes through the points \(( 2,5 )\) and \(( 8 , - 1 )\). Show that this line also passes through the point \(( - 2,9 )\).
Pre-U Pre-U 9794/2 2017 June Q2
6 marks Moderate -0.8
2
    1. Find the value of the discriminant of \(x ^ { 2 } + 3 x + 5\).
    2. Use your value from part (i) to determine the number of real roots of the equation \(x ^ { 2 } + 3 x + 5 = 0\).
  2. Find the non-zero value of \(k\) for which the equation \(k x ^ { 2 } + 3 x + 5 = 0\) has only one distinct real root.
Pre-U Pre-U 9794/2 2017 June Q3
4 marks Moderate -0.8
3 Solve the equation \(\tan \left( \theta + 10 ^ { \circ } \right) = 0.1\) in the range \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
Pre-U Pre-U 9794/2 2017 June Q4
4 marks Moderate -0.3
4 A sequence of complex numbers is defined by $$u _ { 1 } = 1 + \mathrm { i } \quad \text { and } \quad u _ { n + 1 } = \mathrm { i } u _ { n } ( n = 1,2,3 , \ldots )$$
  1. Find \(u _ { 2 } , u _ { 3 } , u _ { 4 } , u _ { 5 }\) and \(u _ { 6 }\).
  2. Describe the behaviour of the sequence.
  3. Hence evaluate \(\sum _ { n = 1 } ^ { 73 } u _ { n }\).
Pre-U Pre-U 9794/2 2017 June Q5
7 marks Moderate -0.3
5
  1. Differentiate \(\frac { x } { \sqrt { 1 + x ^ { 2 } } }\) with respect to \(x\).
  2. Hence show that \(\frac { x } { \sqrt { 1 + x ^ { 2 } } }\) is increasing for all \(x\).
Pre-U Pre-U 9794/2 2017 June Q6
7 marks Moderate -0.3
6 Find the solution of the differential equation $$x y ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } = x + 1$$ given that \(y = 3\) when \(x = 1\). Give your answer in the form \(y = \mathrm { f } ( x )\).