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

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Pre-U Pre-U 9794/2 2010 June Q11
10 marks Standard +0.3
Two forces \(\mathbf{F}_1\) and \(\mathbf{F}_2\) are given by $$\mathbf{F}_1 = 13\mathbf{i} + 4\mathbf{j} - 3\mathbf{k}, \quad \mathbf{F}_2 = -2\mathbf{i} + 6\mathbf{j} + \mathbf{k},$$ in which the units of the components are newtons. A third force, \(\mathbf{F}_3\), of magnitude 6 N acts parallel to the vector \(2\mathbf{i} - 2\mathbf{j} + \mathbf{k}\).
  1. Find the two possible resultants of \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\), and show that they have the same magnitude. [5]
A particle, \(P\), of mass 2 kg is initially at rest at the origin. The only forces acting on \(P\) are \(\mathbf{F}_1\) and \(\mathbf{F}_2\).
  1. Find the magnitude of the acceleration of \(P\). [3]
  2. Find the time taken for \(P\) to travel 60 m. [2]
Pre-U Pre-U 9794/2 2010 June Q12
13 marks Standard +0.3
A particle moves along a straight line under the action of a variable force. The acceleration is given by $$a = \begin{cases} 30 - 6t, & \text{for } 0 \leqslant t \leqslant 10 \\ 6t - 90, & \text{for } 10 \leqslant t \leqslant 20 \end{cases}$$ where time \(t\) is measured in seconds and \(a\) in m s\(^{-2}\). The particle is at rest at the origin at \(t = 0\).
    1. Find the velocity \(v\) of the particle in terms of \(t\). Verify that \(v = 0\) when \(t = 10\) and \(t = 20\). [7]
    2. Sketch the velocity-time graph for the motion. [2]
  2. Calculate the total distance travelled by the particle. [4]
Pre-U Pre-U 9794/2 2010 June Q13
8 marks Standard +0.3
A light inextensible string passes over a fixed smooth light pulley. Particles \(A\) and \(B\), of masses 2 kg and 3 kg respectively, are attached to the ends so that the portions of the string below the axis of the pulley are vertical (see diagram). The centre of the horizontal axis of the pulley is 4 m above ground level. \includegraphics{figure_13} The particles are released from rest 1 m above ground level with the string taut.
  1. Determine the acceleration of both particles prior to the impact of \(B\) with the ground. [3]
  2. Determine the greatest height attained by \(A\) above ground level. [3]
  3. If \(B\) rebounds after impact to a first maximum height of 0.05 m, determine the coefficient of restitution between \(B\) and the ground. [2]
Pre-U Pre-U 9794/2 2011 June Q1
5 marks Easy -1.3
  1. Show that \(x = 4\) is a root of \(x^3 - 12x - 16 = 0\). [2]
  2. Hence completely factorise the expression \(x^3 - 12x - 16\). [3]
Pre-U Pre-U 9794/2 2011 June Q2
6 marks Easy -1.2
  1. Expand and simplify \((7 - 2\sqrt{3})^2\). [2]
  2. Show that $$\frac{\sqrt{125}}{2 + \sqrt{5}} = 25 - 10\sqrt{5}.$$ [4]
Pre-U Pre-U 9794/2 2011 June Q3
5 marks Moderate -0.8
Use integration by parts to find \(\int x \sin 3x \, dx\). [5]
Pre-U Pre-U 9794/2 2011 June Q4
9 marks Standard +0.3
  1. On the same diagram, sketch the graphs of \(y = 2 \sec x\) and \(y = 1 + 3 \cos x\), for \(0 \leqslant x \leqslant \pi\). [4]
  2. Solve the equation \(2 \sec x = 1 + 3 \cos x\), where \(0 \leqslant x \leqslant \pi\). [5]
Pre-U Pre-U 9794/2 2011 June Q5
7 marks Moderate -0.8
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 \(3e^{-0.02t}\) units and the concentration of Coldcure is \(5e^{-0.07t}\) units. Each drug becomes ineffective when its concentration falls below 1 unit.
  1. Show that Coldcure becomes ineffective before Antiflu. [3]
  2. Sketch, on the same diagram, the graphs of concentration against time for each drug. [2]
  3. 20 hours after the first injection, Diane is given a second injection. Determine the concentration of Coldcure 10 hours later. [2]
Pre-U Pre-U 9794/2 2011 June Q6
8 marks Standard +0.3
  1. Using the substitution \(u = x^2\), or otherwise, find the numerical value of $$\int_0^{\sqrt{\ln 4}} xe^{-\frac{1}{2}x^2} \, dx.$$ [4]
  2. Determine the exact coordinates of the stationary points of the curve \(y = xe^{-\frac{1}{2}x^2}\). [4]
Pre-U Pre-U 9794/2 2011 June Q7
9 marks Moderate -0.3
Functions f, g and h are defined for \(x \in \mathbb{R}\) by $$f : x \mapsto x^2 - 2x,$$ $$g : x \mapsto x^2,$$ $$h : x \mapsto \sin x.$$
    1. State whether or not f has an inverse, giving a reason. [2]
    2. Determine the range of the function f. [2]
    1. Show that gh(x) can be expressed as \(\frac{1}{2}(1 - \cos 2x)\). [2]
    2. Sketch the curve C defined by \(y = \text{gh}(x)\) for \(0 \leqslant x \leqslant 2\pi\). [3]
Pre-U Pre-U 9794/2 2011 June Q8
15 marks Challenging +1.3
  1. A curve \(C_1\) is defined by the parametric equations $$x = \theta - \sin \theta, \quad y = 1 - \cos \theta,$$ where the parameter \(\theta\) is measured in radians.
    1. Show that \(\frac{dy}{dx} = \cot \frac{1}{2}\theta\), except for certain values of \(\theta\), which should be identified. [5]
    2. Show that the points of intersection of the curve \(C_1\) and the line \(y = x\) are determined by an equation of the form \(\theta = 1 + A \sin(\theta - \alpha)\), where \(A\) and \(\alpha\) are constants to be found, such that \(A > 0\) and \(0 < \alpha < \frac{1}{2}\pi\). [4]
    3. Show that the equation found in part (b) has a root between \(\frac{1}{4}\pi\) and \(\pi\). [2]
  2. A curve \(C_2\) is defined by the parametric equations $$x = \theta - \frac{1}{2} \sin \theta, \quad y = 1 - \frac{1}{2} \cos \theta,$$ where the parameter \(\theta\) is measured in radians. Find the y-coordinates of all points on \(C_2\) for which \(\frac{d^2y}{dx^2} = 0\). [4]
Pre-U Pre-U 9794/2 2011 June Q9
15 marks Challenging +1.2
The curve \(y = x^3\) intersects the line \(y = kx\), \(k > 0\), at the origin and the point \(P\). The region bounded by the curve and the line, between the origin and \(P\), is denoted by \(R\).
  1. Show that the area of the region \(R\) is \(\frac{1}{6}k^3\). [3]
The line \(x = a\) cuts the region \(R\) into two parts of equal area.
  1. Show that \(k^3 - 6a^2k + 4a^3 = 0\). [3]
The gradient of the line \(y = kx\) increases at a constant rate with respect to time \(t\). Given that \(\frac{dk}{dt} = 2\),
  1. determine the value of \(\frac{da}{dt}\) when \(a = 1\) and \(k = 2\), [4]
  2. determine the value of \(\frac{da}{dt}\) when \(a = 1\) and \(k = 2\), expressing your answer in the form \(p + q\sqrt{3}\), where \(p\) and \(q\) are integers. [5]
Pre-U Pre-U 9794/2 2011 June Q10
8 marks Standard +0.3
The points \(A\), \(B\) and \(C\) lie in a vertical plane and have position vectors \(4\mathbf{i}\), \(3\mathbf{j}\) and \(7\mathbf{i} + 4\mathbf{j}\), respectively. The unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal and vertically upwards, respectively. The units of the components are metres.
  1. Show that angle \(BAC\) is a right angle. [2]
\includegraphics{figure_10} Strings \(AB\) and \(AC\) are attached to \(B\) and \(C\), and joined at \(A\). A particle of weight 20 N is attached at \(A\) (see diagram). The particle is in equilibrium.
  1. By resolving in the directions \(AB\) and \(AC\), determine the magnitude of the tension in each string. [3]
  2. Express the tension in the string \(AB\) as a vector, in terms of \(\mathbf{i}\) and \(\mathbf{j}\). [3]
Pre-U Pre-U 9794/2 2011 June Q11
10 marks Standard +0.3
\includegraphics{figure_11} A projectile is fired from a point \(O\) in a horizontal plane, with initial speed \(V\), at an angle \(\theta\) to the horizontal (see diagram).
  1. Show that the range of the projectile on the horizontal plane is $$\frac{2V^2 \sin \theta \cos \theta}{g}.$$ [4]
There are two vertical walls, each of height \(h\), at distances 30 m and 70 m, respectively, from \(O\) with bases on the horizontal plane. The value of \(\theta\) is \(45°\).
  1. If the projectile just clears both walls, state the range of the projectile. [1]
  2. Hence find the value of \(V\) and of \(h\). [5]
Pre-U Pre-U 9794/2 2011 June Q12
11 marks Standard +0.3
\includegraphics{figure_12} A particle \(P\) of mass 2 kg can move along a line of greatest slope on a smooth plane, inclined at \(30°\) to the horizontal. \(P\) is initially at rest at a point on the plane, and a force of constant magnitude 20 N is applied to \(P\) parallel to and up the slope (see diagram).
  1. Copy and complete the diagram, showing all forces acting on \(P\). [1]
  2. Find the velocity of \(P\) in terms of time \(t\) seconds, whilst the force of 20 N is applied. [4]
After 3 seconds the force is removed at the instant that \(P\) collides with a particle of mass 1 kg moving down the slope with speed 5 m s\(^{-1}\). The coefficient of restitution between the particles is 0.2.
  1. Express the velocity of \(P\) as a function of time after the collision. [6]
Pre-U Pre-U 9794/2 2011 June Q13
12 marks Challenging +1.2
\includegraphics{figure_13} Particles \(A\) and \(B\) of masses \(2m\) and \(m\), respectively, are attached to the ends of a light inextensible string. The string passes over a smooth fixed pulley \(P\). The particle \(A\) rests in equilibrium on a rough plane inclined at an angle \(\alpha\) to the horizontal, where \(\alpha \leqslant 45°\) and \(B\) is above the plane. The vertical plane defined by \(APB\) contains a line of greatest slope of the plane, and \(PA\) is inclined at angle \(2\alpha\) to the horizontal (see diagram).
  1. Show that the normal reaction \(R\) between \(A\) and the plane is \(mg(2 \cos \alpha - \sin \alpha)\). [3]
  2. Show that \(R \geqslant \frac{1}{2}mg\sqrt{2}\). [3]
The coefficient of friction between \(A\) and the plane is \(\mu\). The particle \(A\) is about to slip down the plane.
  1. Show that \(0.5 < \tan \alpha \leqslant 1\). [3]
  2. Express \(\mu\) as a function of \(\tan \alpha\) and deduce its maximum value as \(\alpha\) varies. [3]
Pre-U Pre-U 9794/2 2012 June Q1
5 marks Easy -1.3
  1. Solve the equation \(x^2 - 8x + 4 = 0\), giving your answer in the form \(p \pm q\sqrt{3}\), where \(p\) and \(q\) are integers. [2]
  2. Expand and simplify \((6 + 2\sqrt{3})(2 - \sqrt{3})\). [3]
Pre-U Pre-U 9794/2 2012 June Q2
9 marks Moderate -0.8
\includegraphics{figure_2} The diagram shows a triangle \(ABC\). The vertices have coordinates \(A(3, -7)\), \(B(9, 1)\) and \(C(-1, -5)\).
    1. Find the length of the side \(AB\). [2]
    2. Find the coordinates of the mid-point of \(AB\). [1]
    3. A circle has diameter \(AB\). Find the equation of the circle in the form \((x - a)^2 + (y - b)^2 = r^2\), where \(a\), \(b\) and \(r\) are constants to be found. [3]
  2. Find the equation of the line \(l\) passing through \(B\) parallel to \(AC\). [3]
Pre-U Pre-U 9794/2 2012 June Q3
4 marks Easy -1.2
Find the exact value of \(\int_0^1 (e^x - x) dx\). [4]
Pre-U Pre-U 9794/2 2012 June Q4
4 marks Easy -1.2
Use logarithms to solve the equation \(2^{2x-1} = 5\). [4]
Pre-U Pre-U 9794/2 2012 June Q5
3 marks Easy -2.0
Sketch, on separate diagrams, the graphs of the following functions for \(0 \leqslant x \leqslant 2\pi\) giving the coordinates of all points of intersection with the axes.
  1. \(y = \sin x\). [1]
  2. \(y = \sin\left(x + \frac{1}{6}\pi\right)\). [2]
Pre-U Pre-U 9794/2 2012 June Q6
8 marks Moderate -0.8
  1. An arithmetic sequence has first term 5 and fifth term 37.
    1. Find an expression for \(u_n\), the \(n\)th term of the sequence, in terms of \(n\). [4]
    2. Find an expression for \(S_n\), the sum of the first \(n\) terms of this sequence, in terms of \(n\). [2]
  2. Hence, or otherwise, calculate \(\sum_{n=5}^{25} (8n - 3)\). [2]
Pre-U Pre-U 9794/2 2012 June Q7
5 marks Moderate -0.8
Let \(y = (2x - 3)e^{-2x}\).
  1. Find \(\frac{dy}{dx}\), giving your answer in the form \(e^{-2x}(ax + b)\), where \(a\) and \(b\) are integers. [3]
  2. Determine the set of values of \(x\) for which \(y\) is increasing. [2]
Pre-U Pre-U 9794/2 2012 June Q8
6 marks Moderate -0.3
Solve the differential equation \(\frac{dy}{dx} = -y^2 x^3\), where \(y = 2\) when \(x = 1\), expressing your solution in the form \(y = f(x)\). [6]
Pre-U Pre-U 9794/2 2012 June Q9
9 marks Moderate -0.3
\includegraphics{figure_9} The diagram shows a sector of a circle, \(OMN\). The angle \(MON\) is \(2x\) 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 perimeter, \(P\), of the sector. [2]
  2. Given that \(P = 20\), show that \(A = \frac{100x}{(1 + x)^2}\). [2]
  3. Find \(\frac{dA}{dx}\), and hence find the value of \(x\) for which the area of the sector is a maximum. [5]