Questions — Pre-U (885 questions)

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Pre-U Pre-U 9794/1 2010 June Q8
9 marks Standard +0.3
The points \(A\) and \(B\) have position vectors \(\mathbf{i} - \mathbf{j} + \mathbf{k}\) and \(2\mathbf{i} + \mathbf{j} + 3\mathbf{k}\) respectively, relative to the origin \(O\). The point \(C\) is on the line \(OA\) extended so that \(\overrightarrow{AC} = 2\overrightarrow{OA}\) and the point \(D\) is on the line \(OB\) extended so that \(\overrightarrow{BD} = 3\overrightarrow{OB}\). The point \(X\) is such that \(OCXD\) is a parallelogram.
  1. Show that a vector equation of the line \(AX\) is \(\mathbf{r} = \mathbf{i} - \mathbf{j} + \mathbf{k} + \lambda(5\mathbf{i} + 7\mathbf{k})\) and find an equation of the line \(CD\) in a similar form. [5]
  2. Prove that the lines \(AX\) and \(CD\) intersect and find the position vector of their point of intersection. [4]
Pre-U Pre-U 9794/1 2010 June Q9
9 marks Standard +0.3
A curve has equation \(x^2 - xy + y^2 = 1\).
  1. Find \(\frac{dy}{dx}\) in terms of \(x\) and \(y\). [4]
  2. Find the coordinates of the points on the curve in the second and fourth quadrants where the tangent is parallel to \(y = x\). [5]
Pre-U Pre-U 9794/1 2010 June Q10
10 marks Standard +0.3
  1. Solve the equation \((2 + i)z = (4 + in)\). Give your answer in the form \(a + ib\), expressing \(a\) and \(b\) in terms of the real constant \(n\). [4]
  2. The roots of the equation \(z^2 + 8z + 25 = 0\) are denoted by \(z_1\) and \(z_2\).
    1. Find \(z_1\) and \(z_2\) and show these roots on an Argand diagram. [3]
    2. Find the modulus and argument in radians of each of \((z_1 + 1)\) and \((z_2 + 1)\). [3]
Pre-U Pre-U 9794/1 2010 June Q11
11 marks Challenging +1.2
  1. Write down an identity for \(\tan 2\theta\) in terms of \(\tan \theta\) and use this result to show that $$\tan 3\theta = \frac{3 \tan \theta - \tan^3 \theta}{1 - 3 \tan^2 \theta}.$$ [4]
  2. Given that \(0 < \theta < \frac{1}{2}\pi\) and \(\theta = \sin^{-1}\left(\frac{1}{\sqrt{10}}\right)\), show that \(\tan 3\theta = \frac{13}{3}\). [3]
  3. Show that the solutions of the equation $$\tan(3 \sin^{-1} x) = \frac{13}{3}$$ for \(0 < x < 2\pi\) are $$x = \frac{\sqrt{10}}{10} \quad \text{and} \quad x = \frac{\sqrt{10(1 + 3\sqrt{3})}}{20}.$$ [4]
Pre-U Pre-U 9794/1 2010 June Q12
7 marks Moderate -0.3
  1. Events \(A\) and \(B\) are such that \(\mathrm{P}(A' \cap B') = \frac{1}{6}\).
    1. Find \(\mathrm{P}(A \cup B)\). [2]
    2. Given that \(\mathrm{P}(A | B) = \frac{1}{4}\) and \(\mathrm{P}(B) = \frac{1}{3}\), find \(\mathrm{P}(A \cap B)\) and \(\mathrm{P}(A)\). [3]
  2. In playing the UK Lottery, a set of 6 different integers is chosen irrespective of order from the integers 1 to 49 inclusive. How many different sets of 6 integers can be chosen? [2]
Pre-U Pre-U 9794/1 2010 June Q13
10 marks Moderate -0.3
A survey was conducted into the annual salary offered for 19 different jobs in 2008. The results were as follows, in thousands of pounds.
15161819213636384141
4347515556606264110
It was decided to undertake a further study to see if self-esteem was correlated with level of annual salary. A random sample of 11 employees was taken and self-esteem was rated on a scale of 1 to 10 with the highest self-esteem being 10. The results were as follows.
Salary in £10 000's1234567891011
Self-esteem435177851079
Pre-U Pre-U 9794/1 2010 June Q14
12 marks Standard +0.3
\begin{enumerate}[label=(\alph*)] \item In a game show contestants are asked up to five questions in succession to qualify for the next round. An incorrect answer eliminates a contestant from the game show. Let \(X\) denote the number of questions correctly answered by a contestant. The probability distribution of \(X\) is given below.
\(x\)012345
\(\mathrm{P}(X = x)\)0.300.250.200.160.060.03
  1. Find the expected number of correctly answered questions and the variance of the distribution. [3]
  2. Find the probability that a randomly selected contestant will correctly answer 3 or more questions. [1]
  3. Each show had two contestants. Find the probability that both the contestants will correctly answer at least one question. [2]
\item In a promotion, a newspaper included a token in every copy of the newspaper. A proportion, 0.002, are winning tokens and occur randomly. A reader keeps buying copies of the newspaper until he buys one with a winning token and then stops. Let \(Y\) denote the number of copies bought.
  1. Explain briefly why this situation may be modelled by a geometric distribution and write down a formula for \(\mathrm{P}(Y = y)\). [2]
  2. Find the probability that the reader gets a winning token with the twentieth copy bought. [2]
  3. Find the probability that the reader will not have to buy more than three copies in order to get a winning token. [2] \end{enumerate]
Pre-U Pre-U 9794/1 2010 June Q15
12 marks Standard +0.3
A manufacturer produces components designed with length \(L\) mm such that \(12 < L < 15\). The Quality Control department finds that 15% of the components sampled are longer than 15 mm while 8% are shorter than 12 mm. Assume that \(L\) is normally distributed with mean \(\mu\) and standard deviation \(\sigma\).
  1. Calculate \(\mu\) and \(\sigma\). [6]
  2. The shortest 5% of components are rejected. Find the minimum length which a component may have before it is rejected. [3]
  3. It was found in a random sample that 10% of components were longer than 16 mm. Determine whether this finding is consistent with the assumption that \(L\) is normally distributed with the \(\mu\) and \(\sigma\) found in part (i). [3]
Pre-U Pre-U 9794/2 2010 June Q1
3 marks Easy -1.8
Find the exact value of $$\int_1^4 \left(10x^2 - 3x^2\right) dx.$$ [3]
Pre-U Pre-U 9794/2 2010 June Q2
5 marks Standard +0.8
Solve the inequality $$\log_3(2x^2 - x) - \log_3(2x^2 - 3x + 1) > 1.$$ [5]
Pre-U Pre-U 9794/2 2010 June Q3
6 marks Standard +0.3
An arithmetic progression has 13th term equal to 60 and 31st term equal to 141.
  1. Find the first term and common difference of the progression. [3]
A second arithmetic progression has first term 1.5 and common difference 3.
    1. Write down the first four terms of each progression. [1]
    2. Prove that the two progressions have an infinite number of terms in common. [2]
Pre-U Pre-U 9794/2 2010 June Q4
6 marks Standard +0.3
  1. Show that $$\cos^4 x - \sin^4 x = 2\cos^2 x - 1.$$ [2]
  2. Hence find the solutions of $$\cos^4 x - \sin^4 x = \cos x,$$ where \(0° \leqslant x \leqslant 360°\). [4]
Pre-U Pre-U 9794/2 2010 June Q5
9 marks Standard +0.8
It is given that $$y = \frac{1}{x+1} + \frac{1}{x-1},$$ where \(x\) and \(y\) are real and positive, and \(i^2 = -1\).
  1. Show that $$x = \frac{1 \pm \sqrt{1-y^2}}{y} \quad \text{and} \quad y \leqslant 1.$$ [4]
  2. Deduce that $$xy < 2.$$ [2]
  3. Indicate the region in the \(x\)-\(y\) plane defined by $$y \leqslant 1 \quad \text{and} \quad xy < 2.$$ [3]
Pre-U Pre-U 9794/2 2010 June Q6
10 marks Standard +0.3
  1. Express \(\frac{x-1}{x^2+2x+1}\) in the form \(\frac{A}{x+1} + \frac{B}{(x+1)^2}\), where \(A\) and \(B\) are integers. [2]
  2. Find the quotient and remainder when \(2y^2 + 1\) is divided by \(y + 1\). [2]
  3. A curve in the \(x\)-\(y\) plane passes through the point \((0, 2)\) and satisfies the differential equation $$(2y^2 + 1)(x^2 + 2x + 1)\frac{dy}{dx} = (x - 1)(y + 1).$$ By solving the differential equation find the equation of the curve in implicit form. [6]
Pre-U Pre-U 9794/2 2010 June Q7
12 marks Standard +0.3
Let \(y = (x - 1)\left(\frac{2}{x^2} + t\right)\) define \(y\) as a function of \(x\) (\(x > 0\)), for each value of the real parameter \(t\).
  1. When \(t = 0\),
    1. determine the set of values of \(x\) for which \(y\) is positive and an increasing function, [3]
    2. locate the stationary point of \(y\), and determine its nature. [2]
  2. It is given that \(t = 2\) and \(y = -2\).
    1. Show that \(x\) satisfies \(f(x) = 0\), where \(f(x) = x^3 + x - 1\). [1]
    2. Prove that \(f\) has no stationary points. [2]
    3. Use the Newton-Raphson method, with \(x_0 = 1\), to find \(x\) correct to 4 significant figures. [4]
Pre-U Pre-U 9794/2 2010 June Q8
14 marks Standard +0.8
The point \(F\) has coordinates \((0, a)\) and the straight line \(D\) has equation \(y = b\), where \(a\) and \(b\) are constants with \(a > b\). The curve \(C\) consists of points equidistant from \(F\) and \(D\).
  1. Show that the cartesian equation of \(C\) can be expressed in the form $$y = \frac{1}{2(a-b)}x^2 + \frac{1}{2}(a+b).$$ [3]
  2. State the \(y\)-coordinate of the lowest point of the curve and prove that \(F\) and \(D\) are on opposite sides of \(C\). [2]
    1. The point \(P\) on the curve has \(x\)-coordinate \(\sqrt{a^2 - b^2}\), where \(|a| > |b|\). Show that the tangent at \(P\) passes through the origin. [4]
    2. The tangent at \(P\) intersects the line \(D\) at the point \(Q\). In the case that \(a = 12\) and \(b = -8\), find the coordinates of \(P\) and \(Q\). Show that the length of \(PQ\) can be expressed as \(p\sqrt{q}\), where \(p = 2q\). [5]
Pre-U Pre-U 9794/2 2010 June Q9
15 marks Challenging +1.2
  1. Show that $$\int x^n \ln x \, dx = \frac{x^{n+1}}{(n+1)^2}\left((n+1)\ln a - 1\right) + \frac{1}{(n+1)^2},$$ where \(n \neq -1\) and \(a > 1\). [6]
    1. Determine the \(x\)-coordinate of the point of intersection of the curves \(y = x^3 \ln x\) and \(y = x \ln 2^x\), where \(x > 0\). [2]
    2. Find the exact value of the area of the region enclosed between these two curves, the line \(x = 1\) and their point of intersection. Express your answer in the form \(b + c \ln 2\), where \(b\) and \(c\) are rational. [4]
  3. The curve \(y = (x^3 \ln x)^{0.5}\), for \(1 < x < e\), is rotated through \(2\pi\) radians about the \(x\)-axis. Determine the value of the resulting volume of revolution, giving your answer correct to 4 significant figures. [3]
Pre-U Pre-U 9794/2 2010 June Q10
9 marks Standard +0.8
A particle is projected from a point \(P\) on an inclined plane, up the line of greatest slope through \(P\), with initial speed \(V\). The angle of the plane to the horizontal is \(\theta\).
  1. If the plane is smooth, and the particle travels for a time \(\frac{2V}{g}\cos\theta\) before coming instantaneously to rest, show that \(\theta = \frac{1}{4}\pi\). [4]
  2. If the same plane is given a roughened surface, with a coefficient of friction 0.5, find the distance travelled before the particle comes instantaneously to rest. [5]
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 9794/1 2011 June Q1
3 marks Easy -1.8
Find the equation of the line passing through the points \((-2, 5)\) and \((4, -7)\). Give your answer in the form \(y = mx + c\). [3]
Pre-U Pre-U 9794/1 2011 June Q2
4 marks Moderate -0.8
\includegraphics{figure_2} The diagram shows a sector \(OAB\) of a circle with centre \(O\) and radius \(r\) cm in which angle \(AOB\) is \(\theta\) radians. The sector has a perimeter of 18 cm.
  1. Show that \(\theta = \frac{18 - 2r}{r}\). [2]
  2. Find the area of the sector in terms of \(r\), simplifying your answer. [2]
Pre-U Pre-U 9794/1 2011 June Q3
3 marks Moderate -0.5
Solve the equation \(3 + 2x = |7 - 4x|\). [3]
Pre-U Pre-U 9794/1 2011 June Q4
6 marks Moderate -0.8
  1. Show that \(4 \ln x - \ln(3x - 2) - \ln x^2 = \ln\left(\frac{x^2}{3x - 2}\right)\), where \(x > \frac{2}{3}\). [3]
  2. Hence solve the equation \(4 \ln x - \ln(3x - 2) - \ln x^2 = 0\). [3]