Questions — Pre-U (885 questions)

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Pre-U Pre-U 9794/1 2011 June Q5
5 marks Standard +0.3
A circle has equation \(x^2 + y^2 = 16\). Find the volume generated when the region in the first quadrant which is bounded by the circle and the lines \(x = 1\) and \(x = 2\) is rotated through \(2\pi\) radians about the \(x\)-axis. [5]
Pre-U Pre-U 9794/1 2011 June Q6
7 marks Standard +0.3
  1. Sketch, on a single diagram, the graphs of \(y = e^{3x}\) and \(y = x\) and state the number of roots of the equation \(e^{3x} = x\). [3]
  2. Use the Newton-Raphson method with \(x_0 = 0\) to determine the value of a root of the equation \(e^{3x} = x\) correct to 3 decimal places. [4]
Pre-U Pre-U 9794/1 2011 June Q7
7 marks Standard +0.3
  1. Given that the point \((-1, -2, 4)\) lies on both the lines $$\mathbf{r} = \begin{pmatrix} 2 \\ -3 \\ a \end{pmatrix} + \lambda \begin{pmatrix} -3 \\ 2 \\ 1 \end{pmatrix} \quad \text{and} \quad \mathbf{r} = \begin{pmatrix} 2 \\ 4 \\ b \end{pmatrix} + \mu \begin{pmatrix} -1 \\ -2 \\ 1 \end{pmatrix},$$ find \(a\) and \(b\). [3]
  2. Find the acute angle between the lines. [4]
Pre-U Pre-U 9794/1 2011 June Q8
8 marks Standard +0.3
  1. Find and simplify the first three terms in the expansion of \((1 - 4a)^{\frac{1}{2}}\) in ascending powers of \(a\), where \(|a| < \frac{1}{4}\). [4]
  2. Hence show that the roots of the quadratic equation \(x^2 - x + a = 0\) are approximately \(1 - a - a^2\) and \(a + a^2\), where \(a\) is small. [4]
Pre-U Pre-U 9794/1 2011 June Q9
9 marks Standard +0.8
  1. Prove that \(\sin 3\theta = 3 \sin \theta - 4 \sin^3 \theta\) and deduce that $$\sin \theta + \sin 3\theta = 4 \sin \theta \cos^2 \theta.$$ [5]
  2. Hence find the values of \(\theta\) such that \(0° < \theta < 180°\) that satisfy the equation $$\cot^2 \theta = \sin \theta + \sin 3\theta.$$ [4]
Pre-U Pre-U 9794/1 2011 June Q10
9 marks Moderate -0.3
  1. The complex number \(z\) is such that \(|z| = 2\) and \(\arg z = -\frac{3}{4}\pi\). Find the exact value of the real part of \(z\) and of the imaginary part of \(z\). [2]
  2. The complex numbers \(u\) and \(v\) are such that $$u = 1 + ia \quad \text{and} \quad v = b - i,$$ where \(a\) and \(b\) are real and \(a < b\). Given that \(uv = 7 + 9i\), find the values of \(a\) and \(b\). [7]
Pre-U Pre-U 9794/1 2011 June Q11
9 marks Standard +0.3
An arithmetic progression has first term \(a\) and common difference \(d\). The first, ninth and fourteenth terms are, respectively, the first three terms of a geometric progression with common ratio \(r\), where \(r \neq 1\).
  1. Find \(d\) in terms of \(a\) and show that \(r = \frac{5}{3}\). [7]
  2. Find the sum to infinity of the geometric progression in terms of \(a\). [2]
Pre-U Pre-U 9794/1 2011 June Q12
10 marks Moderate -0.8
Find the general solution of the differential equation $$\frac{dy}{dx} = \frac{x}{x(1 + x^2)}$$ giving your answer in the form \(y = f(x)\). [10]
Pre-U Pre-U 9794/1 2011 June Q13
7 marks Moderate -0.3
  1. A random sample of young people in a certain town comprised 312 boys and 253 girls. Denoting a boy's age by \(x\) years and a girl's age by \(y\) years, the following data were obtained: $$\sum x = 4618, \quad \sum x^2 = 68812, \quad \sum y = 3719, \quad \sum y^2 = 55998.$$
    1. Calculate the mean and standard deviation of the ages of the boys in the sample and also of the girls in the sample. [3]
    2. Use these results to comment on the distribution of the ages of the boys and girls in the sample. [1]
  2. How many arrangements of the letters of the word DEFEATED are there in which the Es are separated from each other? [3]
Pre-U Pre-U 9794/1 2011 June Q14
9 marks Standard +0.3
  1. The table below relates the values of two variables \(x\) and \(y\).
    \(x\)1\(A\)\(A + 3\)10
    \(y\)2\(A - 1\)\(A\)5
    \(A\) is a positive integer and \(\sum xy = 92\).
    1. Calculate the value of \(A\). [3]
    2. Explain how you can tell that the product-moment correlation coefficient is 1. [1]
  2. A music society has 300 members. 240 like Puccini, 100 like Wagner and 50 like neither.
    1. Calculate the probability that a member chosen at random likes Puccini but not Wagner. [3]
    2. Calculate the probability that a member chosen at random likes Puccini given that this member likes Wagner. [2]
Pre-U Pre-U 9794/1 2011 June Q15
12 marks Standard +0.8
A firm produces chocolate bars whose weights are normally distributed with mean 120 g and standard deviation 6 g.
  1. Bars which weigh more than 114 g are sold at a profit of 15p per bar. The remaining bars are sold at no profit. Show that the expected profit per 100 bars is £12.62. [5]
  2. It is subsequently decided that bars which weigh more than \(x\) g should be sold at a profit of 20p per bar. Those which weigh \(x\) g or less are sold to employees at a profit of 3p per bar. The expected profit per 100 bars is £19.17. Find the value of \(x\). [7]
Pre-U Pre-U 9794/1 2011 June Q16
12 marks Standard +0.8
In a factory, computer chips are produced in large batches. A quality control procedure is used for each batch which requires a random sample of 8 chips to be tested. If no faulty chip is found, the batch is accepted. If two or more are faulty, the batch is rejected. If one is faulty, a further sample of 4 is selected and the batch is accepted if none of these is faulty. The probability of any chip being faulty is \(q\).
  1. Show that the probability of accepting a batch is \(p^8(1 + 8p^3 - 8p^4)\), where \(p = 1 - q\). [6]
  2. Find the expected number of chips sampled per batch, giving your answer in terms of \(p\). Hence show that when \(p = 0.75\), the expected number of chips sampled per batch is approximately 9. [6]
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]