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Pre-U Pre-U 9795/1 2015 June Q13
10 marks Challenging +1.2
  1. By sketching a suitable triangle, show that \(\tan^{-1} a + \tan^{-1} \left(\frac{1}{a}\right) = \frac{1}{4}\pi\), for \(a > 0\). [1]
  2. Given that \(a\) and \(b\) are positive and less than 1, express \(\tan(\tan^{-1} a \pm \tan^{-1} b)\) in terms of \(a\) and \(b\). [2]
  3. By letting \(a = \frac{1}{n-1}\) and \(b = \frac{1}{n+1}\), use the method of differences to prove that $$\sum_{n=1}^{\infty} \tan^{-1} \left(\frac{2}{n^2}\right) = \frac{3}{4}\pi.$$ [7]
Pre-U Pre-U 9794/2 2016 June Q1
3 marks Easy -1.3
  1. Find the remainder when \(x^3 + 2x\) is divided by \(x + 2\). [2]
  2. Write down the value of \(k\) for which \(x + 2\) is a factor of \(x^3 + 2x + k\). [1]
Pre-U Pre-U 9794/2 2016 June Q2
4 marks Easy -1.2
Solve the equation \(4 \times 3^x = 5\), giving the solution in an exact form. [4]
Pre-U Pre-U 9794/2 2016 June Q3
4 marks Moderate -0.8
The graph of \(\log_{10} y\) against \(x\) is a straight line with gradient 2 and the intercept on the vertical axis at 4. Write down an equation for this straight line and show that \(y = 10000 \times 100^x\). [4]
Pre-U Pre-U 9794/2 2016 June Q4
6 marks Moderate -0.8
The complex numbers \(z_1\) and \(z_2\) are given by \(z_1 = 2 + i\) and \(z_2 = 3 + 4i\).
  1. Verify that \(|z_1| + |z_2| > |z_1 + z_2|\). [4]
  2. Sketch on an Argand diagram the locus \(|z - z_1| = 2\). [2]
Pre-U Pre-U 9794/2 2016 June Q5
7 marks Moderate -0.3
  1. Show that \(\frac{3}{x+2} + \frac{1}{x+1} \equiv \frac{4x+5}{x^2+3x+2}\). [2]
  2. Differentiate \(\frac{4x+5}{x^2+3x+2}\) with respect to \(x\). [3]
  3. Hence show that the function given by $$f(x) = \frac{4x+5}{x^2+3x+2}, \quad x \neq -1, x \neq -2,$$ is a decreasing function. [2]
Pre-U Pre-U 9794/2 2016 June Q6
7 marks Moderate -0.8
The points \(A\) and \(B\) are at \((2, 3, 5)\) and \((8, 2, 4)\) with respect to the origin \(O\).
  1. Find the size of angle \(AOB\). [4]
  2. Show that triangle \(AOB\) is isosceles. [3]
Pre-U Pre-U 9794/2 2016 June Q7
11 marks Moderate -0.3
  1. Use a change of sign to verify that the equation \(\cos x - x = 0\) has a root \(\alpha\) between \(x = 0.7\) and \(x = 0.8\). [2]
  2. Sketch, on a single diagram, the curve \(y = \cos x\) and the line \(y = x\) for \(0 \leqslant x \leqslant \frac{1}{2}\pi\), giving the coordinates of all points of intersection with the coordinate axes. [2]
An iteration of the form \(x_{n+1} = \cos(x_n)\) is to be used to find \(\alpha\).
  1. By considering the gradient of \(y = \cos x\), show that this iteration will converge. [3]
  2. On a copy of your sketch from part (ii), illustrate how this iteration converges to \(\alpha\). [2]
  3. Use a change of sign to verify that \(\alpha = 0.7391\) to 4 decimal places. [2]
Pre-U Pre-U 9794/2 2016 June Q8
5 marks Standard +0.3
\(P\) and \(Q\) are points on the circumference of a circle with centre \(O\) and radius \(r\). The angle \(POQ\) is \(\theta\) radians. Given that the chord \(PQ\) has length 4, find an expression for the length of the arc \(PQ\) in terms of \(\theta\) of only. [5]
Pre-U Pre-U 9794/2 2016 June Q9
11 marks Challenging +1.2
  1. Show that \(\frac{\sin x}{1 + \sin x} \equiv \sec x \tan x - \sec^2 x + 1\). [5]
  2. Hence show that \(\int_0^{\frac{\pi}{4}} \frac{\sin x}{1 + \sin x} \, dx = \frac{1}{4}\pi + \sqrt{2} - 2\). [6]
Pre-U Pre-U 9794/2 2016 June Q10
10 marks Challenging +1.2
  1. Using the substitution \(u = \frac{1}{x}\), or otherwise, find \(\int \frac{\sin\left(\frac{1}{x}\right)}{x^2} \, dx\). [4]
  2. Evaluate \(\int_{\frac{1}{2\pi}}^{\frac{1}{\pi}} \frac{\sin\left(\frac{1}{x}\right)}{x^2} \, dx\) and \(\int_{\frac{1}{2\pi}}^{\frac{1}{\pi}} \frac{\sin\left(\frac{1}{x}\right)}{x^2} \, dx\). [3]
  3. Show that, when \(n\) is a positive integer, the integral \(\int_{\frac{1}{(n+1)\pi}}^{\frac{1}{n\pi}} \frac{\sin\left(\frac{1}{x}\right)}{x^2} \, dx\) takes one of the two values found in part (ii), distinguishing between the two cases. [3]
Pre-U Pre-U 9794/2 2016 June Q11
12 marks Standard +0.3
The function f is defined by \(f(x) = \sqrt{x}, x > 0\).
  1. Use differentiation from first principles to find an expression for \(f'(x)\). [5]
The lines \(l_1\) and \(l_2\) are the tangents to the curve \(y = f(x)\) at the points \(A\) and \(B\) where \(x = a\) and \(x = b\) respectively, \(a \neq b\).
    1. Show that the tangents intersect at the point \(\left(\sqrt{ab}, \frac{1}{2}(\sqrt{a} + \sqrt{b})\right)\). [5]
    2. Given that \(l_1\) and \(l_2\) intersect at a point with integer coordinates, write down a possible pair of values for \(a\) and \(b\). [2]
Pre-U Pre-U 9794/3 2016 June Q1
4 marks Moderate -0.8
The following data refer to the annual rate of inflation and the annual percentage pay increase measured on 10 randomly chosen occasions.
Inflation rate (\%)0.91.21.61.51.73.04.13.72.84.2
Pay increase (\%)4.84.73.84.45.65.52.40.40.61.7
Show that, for these data, the product moment correlation coefficient between the rate of inflation and the annual pay increase is \(-0.679\), correct to 3 significant figures. [4]
Pre-U Pre-U 9794/3 2016 June Q2
8 marks Moderate -0.8
The weights of pineapples on sale at a wholesaler are normally distributed with mean \(1.349\) kg and standard deviation \(0.236\) kg. Before going on sale the pineapples are classified as 'Small', 'Medium', 'Large' and 'Extra Large'.
  1. A pineapple is classified as 'Small' if it weighs less than \(1.100\) kg. Find the probability that a randomly chosen pineapple will be classified as 'Small'. [5]
  2. \(10\%\) of pineapples are classified as 'Extra Large'. Find the minimum weight required for a pineapple to be classified as 'Extra Large'. [3]
Pre-U Pre-U 9794/3 2016 June Q3
11 marks Moderate -0.3
Chris plays for his local hockey club. In his first 20 games for the club, the mean number of goals per game he has scored is \(0.7\), with a standard deviation of \(0.9\). In the next 5 games he scores \(0, 1, 0, 2, 1\) goals.
  1. Find the mean and standard deviation for the number of goals per game Chris has scored in all 25 games. [7]
  2. A sponsor pays Chris £65 each time he plays for the club and a further £25 for each goal he scores. Find the mean and standard deviation of the amount per game he earns from the sponsor for all 25 games. [4]
Pre-U Pre-U 9794/3 2016 June Q4
8 marks Moderate -0.3
A certain type of sweet is made in a variety of colours. \(20\%\) of the sweets made are blue. Sweets of the various colours are thoroughly mixed before being put into packets.
  1. In a packet that contains 10 sweets, find the probability that the packet contains
    1. at most 3 blue sweets, [1]
    2. exactly 3 blue sweets, [2]
    3. at least 1 blue sweet. [2]
  2. What is the smallest number of sweets that a packet should contain in order to be at least \(95\%\) certain of having at least 1 blue sweet? [3]
Pre-U Pre-U 9794/3 2016 June Q5
4 marks Moderate -0.3
The letters of the word 'SEPARATE' are to be rearranged. Find the probability that, in a randomly chosen rearrangement, the two letters 'A' are not next to each other. [4]
Pre-U Pre-U 9794/3 2016 June Q6
5 marks Moderate -0.8
\(A\) and \(B\) are independent events. \(P(A) = \frac{3}{4}\) and \(P(A \cap B) = \frac{1}{4}\). Find \(P(A' \cap B)\) and \(P(A' \cap B')\). [5]
Pre-U Pre-U 9794/3 2016 June Q7
5 marks Moderate -0.8
A stone that weighs 15 kg is propelled across the ice in an ice rink with an initial speed of \(4 \text{ m s}^{-1}\). The coefficient of friction between the stone and the ice is \(0.017\). How far does the stone slide before it comes to rest? [5]
Pre-U Pre-U 9794/3 2016 June Q8
8 marks Moderate -0.3
A particle is projected with speed \(U \text{ m s}^{-1}\) at an angle \(\theta\) above the horizontal, where \(\sin \theta = \frac{12}{13}\), and reaches its maximum height after \(2.4\) seconds.
  1. Find \(U\) and the maximum height reached by the particle. [4]
  2. Find the horizontal range of the particle. [4]
Pre-U Pre-U 9794/3 2016 June Q9
8 marks Standard +0.8
A particle of mass \(0.01\) kg is projected vertically upwards from a point \(G\) at ground level with speed \(165 \text{ m s}^{-1}\) and reaches a maximum height of \(1237.5\) m. Throughout its motion it experiences a constant resistance.
  1. Find the acceleration of the particle as it ascends and hence the magnitude of the resistance. [4]
  2. During its descent back to \(G\) the particle experiences the same constant resistance. Find the time taken for the descent. [4]
Pre-U Pre-U 9794/3 2016 June Q10
7 marks Standard +0.3
  1. A particle \(A\) of mass \(m\) travelling with speed \(u\) on a smooth horizontal surface collides directly with a particle \(B\) of mass \(3m\) travelling with speed \(\frac{2u}{5}\) in the opposite direction. After the collision, \(A\) travels at speed \(\frac{2u}{5}\) and \(B\) travels at speed \(\frac{4u}{15}\), both in the same direction as \(B\) before the collision. Find \(A\) and the coefficient of restitution between the two particles. [4]
  2. A particle of mass 3 kg moving with velocity \((2\mathbf{i} + 3\mathbf{j} - 2\mathbf{k}) \text{ m s}^{-1}\) receives an impulse of \((6\mathbf{i} - 6\mathbf{j} - 9\mathbf{k})\) N s. Find the velocity of the particle after the impulse. [3]
Pre-U Pre-U 9794/3 2016 June Q11
12 marks Standard +0.3
\includegraphics{figure_11} The diagram shows a particle, \(A\), of mass \(m_1\) at rest on a rough slope at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac{3}{5}\). Particle \(A\) is connected by a light inextensible string to another particle, \(B\), of mass \(m_2\). The string passes over a smooth peg at the top of the slope and particle \(B\) is hanging freely.
  1. In the case when \(m_2 = \frac{1}{4}m_1\), particle \(A\) is on the point of sliding down the slope.
    1. Draw a fully labelled diagram to show all the forces acting on the particles. [2]
    2. Find the coefficient of friction between \(A\) and the slope. [6]
  2. In the case when \(m_2 = m_1\), find the acceleration of the particles. [4]
Pre-U Pre-U 9795/1 2018 June Q1
5 marks Moderate -0.3
  1. Express \(\frac{3}{(3r-1)(3r+2)}\) in partial fractions. [2]
  2. Using the method of differences, prove that \(\sum_{r=1}^{n} \frac{3}{(3r-1)(3r+2)} = \frac{1}{2} - \frac{1}{3n+2}\). [2]
  3. Deduce the value of \(\sum_{r=1}^{\infty} \frac{1}{(3r-1)(3r+2)}\). [1]
Pre-U Pre-U 9795/1 2018 June Q2
10 marks Standard +0.3
  1. Determine the asymptotes and turning points of the curve with equation \(y = \frac{x^2+3}{x+1}\). [7]
  2. Sketch the curve. [3]