Questions Pre-U 9794/1 (194 questions)

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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 9794/1 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]
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]