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OCR FS1 AS 2017 Specimen Q6
13 marks Moderate -0.3
Sabrina counts the number of cars passing her house during randomly chosen one minute intervals. Two assumptions are needed for the number of cars passing her house in a fixed time interval to be well modelled by a Poisson distribution.
  1. State these two assumptions. [2]
  2. For each assumption in part (i) give a reason why it might not be a reasonable assumption for this context. [2]
Assume now that the number of cars that pass Sabrina's house in one minute can be well modelled by the distribution \(\text{Po}(0.8)\).
    1. Write down an expression for the probability that, in a given one minute period, exactly \(r\) cars pass Sabrina's house. [1]
    2. Hence find the probability that, in a given one minute period, exactly 2 cars pass Sabrina's house. [1]
  2. Find the probability that, in a given 30 minute period, at least 28 cars pass Sabrina's house. [3]
  3. The number of bicycles that pass Sabrina's house in a 5 minute period is a random variable with the distribution \(\text{Po}(1.5)\). Find the probability that, in a given 10 minute period, the total number of cars and bicycles that pass Sabrina's house is between 12 and 15 inclusive. State a necessary condition. [4]
OCR FS1 AS 2017 Specimen Q7
4 marks Standard +0.3
The discrete random variable \(X\) is equally likely to take values 0, 1 and 2. \(3N\) observations of \(X\) are obtained, and the observed frequencies corresponding to \(X = 0\), \(X = 1\) and \(X = 2\) are given in the following table.
\(x\)012
Observed frequency\(N - 1\)\(N - 1\)\(N + 2\)
The test statistic for a chi-squared goodness of fit test for the data is 0.3. Find the value of \(N\). [4]
OCR FS1 AS 2017 Specimen Q8
10 marks Standard +0.3
The following table gives the mean per capita consumption of mozzarella cheese per annum, \(x\) pounds, and the number of civil engineering doctorates awarded, \(y\), in the United States in each of 10 years.
\(x\)9.39.79.79.79.910.210.511.010.610.6
\(y\)480501540552547622655701712708
source: www.tylervigen.com
  1. Find the equation of the regression line of \(y\) on \(x\). [2]
You are given that the product moment correlation coefficient is 0.959.
  1. Explain whether this value would be different if \(x\) is measured in kilograms instead of pounds. [1]
It is desired to carry out a hypothesis test to investigate whether there is correlation between these two variables.
  1. Assume that the data is a random sample of all years.
    1. Carry out the test at the 10\% significance level. [6]
    2. Explain whether your conclusion suggests that manufacturers of mozzarella cheese could increase consumption by sponsoring doctoral candidates in civil engineering. [1]
Pre-U Pre-U 9794/1 2010 June Q1
3 marks Easy -1.2
Solve the equation \(2^x = 4^{2x+1}\). [3]
Pre-U Pre-U 9794/1 2010 June Q2
3 marks Standard +0.3
The equation \(x^3 - 5x + 3 = 0\) has a root between \(x = 0\) and \(x = 1\).
  1. The equation can be rearranged into the form \(x = g(x)\) where \(g(x) = px^3 + q\). State the values of \(p\) and \(q\). [1]
  2. By considering \(|g'(x)|\), show that the iterative form \(x_{n+1} = g(x_n)\) with a suitable starting value converges to the root between \(x = 0\) and \(x = 1\). [You are not required to find this root.] [2]
Pre-U Pre-U 9794/1 2010 June Q3
6 marks Moderate -0.3
Let \(f(x) = x^2(x - 2)\) and \(g(x) = 2x - 1\) for all real \(x\).
  1. Sketch the graph of \(y = f(x)\) and explain briefly why the function f has no inverse. [2]
  2. Write down \(g^{-1}(x)\). [1]
  3. On the same diagram, sketch the graphs of \(y = f(x - 1) - 3\) and \(y = g^{-1}(x)\) and state the number of real roots of the equation \(f(x - 1) - 3 = g^{-1}(x)\). [3]
Pre-U Pre-U 9794/1 2010 June Q4
5 marks Moderate -0.3
Using the substitution \(u = 1 + \sqrt{x}\), or otherwise, find \(\int \frac{1}{1 + \sqrt{x}} dx\) giving your answer in terms of \(x\). [5]
Pre-U Pre-U 9794/1 2010 June Q5
7 marks Standard +0.3
The parametric equations of a curve are \(x = \frac{1}{1 + t^2}\) and \(y = \frac{t}{1 + t^2}\), \(t \in \mathbb{R}\).
  1. Find \(\frac{dy}{dx}\) in terms of \(t\). [5]
  2. Hence find the coordinates of the stationary points of the curve. [2]
Pre-U Pre-U 9794/1 2010 June Q6
7 marks Standard +0.3
A geometric progression with common ratio \(r\) consists of positive terms. The sum of the first four terms is five times the sum of the first two terms.
  1. Find an equation in \(r\) and deduce that \(r = 2\). [3]
  2. Given that the fifth term is 192, find the value of the first term. [1]
  3. Find the smallest value of \(n\) such that the sum of the first \(n\) terms of the progression exceeds \(10^{64}\). [3]
Pre-U Pre-U 9794/1 2010 June Q7
9 marks Standard +0.3
Let \(f(x) = \frac{1 + x^2}{\sqrt{4 - 3x}}\)
  1. Obtain in ascending powers of \(x\) the first three terms in the expansion of \(\frac{1}{\sqrt{4 - 3x}}\) and state the values of \(x\) for which this expansion is valid. [5]
  2. Hence obtain an approximation to \(f(x)\) in the form \(a + bx + cx^2\) where \(a\), \(b\) and \(c\) are constants. [2]
  3. Use your approximation to estimate \(\int_0^{0.1} f(x) dx\). [2]
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]