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OCR C1 Q1
3 marks Easy -1.8
  1. Solve the inequality
$$4 ( x - 2 ) < 2 x + 5$$
OCR C1 Q2
4 marks Moderate -0.8
2. $$f ( x ) = 2 - x - x ^ { 3 } .$$ Show that \(\mathrm { f } ( x )\) is decreasing for all values of \(x\).
OCR C1 Q3
5 marks Moderate -0.3
3.
  1. Solve the equation $$y ^ { 2 } + 8 = 9 y .$$
  2. Hence solve the equation $$x ^ { 3 } + 8 = 9 x ^ { \frac { 3 } { 2 } } .$$
OCR C1 Q4
6 marks Moderate -0.8
  1. Given that
$$y = \frac { x ^ { 4 } - 3 } { 2 x ^ { 2 } } ,$$
  1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
  2. show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \frac { x ^ { 4 } - 9 } { x ^ { 4 } }\).
OCR C1 Q5
7 marks Moderate -0.5
5. Find the pairs of values \(( x , y )\) which satisfy the simultaneous equations $$\begin{aligned} & 3 x ^ { 2 } + y ^ { 2 } = 21 \\ & 5 x + y = 7 \end{aligned}$$
OCR C1 Q6
7 marks Moderate -0.8
  1. Evaluate \(\left( 5 \frac { 4 } { 9 } \right) ^ { - \frac { 1 } { 2 } }\).
  2. Find the value of \(x\) such that $$\frac { 1 + x } { x } = \sqrt { 3 } ,$$ giving your answer in the form \(a + b \sqrt { 3 }\) where \(a\) and \(b\) are rational.
OCR C1 Q7
8 marks Moderate -0.8
7. The straight line \(l\) passes through the point \(P ( - 3,6 )\) and the point \(Q ( 1 , - 4 )\).
  1. Find an equation for \(l\) in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers. The straight line \(m\) has the equation \(2 x + k y + 7 = 0\), where \(k\) is a constant.
    Given that \(l\) and \(m\) are perpendicular,
  2. find the value of \(k\).
OCR C1 Q8
9 marks Standard +0.3
8.
  1. Describe fully a single transformation that maps the graph of \(y = \frac { 1 } { x }\) onto the graph of \(y = \frac { 3 } { x }\).
  2. Sketch the graph of \(y = \frac { 3 } { x }\) and write down the equations of any asymptotes.
  3. Find the values of the constant \(c\) for which the straight line \(y = c - 3 x\) is a tangent to the curve \(y = \frac { 3 } { x }\).
OCR C1 Q9
10 marks Moderate -0.3
9. The circle \(C\) has the equation $$x ^ { 2 } + y ^ { 2 } - 12 x + 8 y + 16 = 0$$
  1. Find the coordinates of the centre of \(C\).
  2. Find the radius of \(C\).
  3. Sketch \(C\). Given that \(C\) crosses the \(x\)-axis at the points \(A\) and \(B\),
  4. find the length \(A B\), giving your answer in the form \(k \sqrt { 5 }\).
OCR C1 Q10
13 marks Moderate -0.3
10. \includegraphics[max width=\textwidth, alt={}, center]{76efaf91-a6f3-4493-88d4-3654b023441d-3_646_773_986_477} The diagram shows the curve \(y = x ^ { 2 } - 3 x + 5\) and the straight line \(y = 2 x + 1\). The curve and line intersect at the points \(P\) and \(Q\).
  1. Using algebra, show that \(P\) has coordinates \(( 1,3 )\) and find the coordinates of \(Q\).
  2. Find an equation for the tangent to the curve at \(P\).
  3. Show that the tangent to the curve at \(Q\) has the equation \(y = 5 x - 11\).
  4. Find the coordinates of the point where the tangent to the curve at \(P\) intersects the tangent to the curve at \(Q\).
OCR S2 2014 June Q1
5 marks Moderate -0.8
1 The random variable \(F\) has the distribution \(B ( 50,0.7 )\). Use a suitable approximation to find \(\mathbf { P } \boldsymbol { ( } \mathbf { F > } \mathbf { 4 0 } \boldsymbol { ) }\). [5]
OCR S2 2014 June Q2
7 marks Standard +0.3
2 The events organiser of a school sends out invitations to \(\mathbf { 1 5 0 }\) people to attend its prize day. From past experience the organiser knows that the number of those who will come to the prize day can be modelled by the distribution \(\mathbf { B } ( \mathbf { 1 5 0 } , \mathbf { 0 . 9 8 } )\).
[0pt]
  1. Explain why this distribution cannot be well approximated by either a normal or a Poisson distribution. [3]
    [0pt]
  2. By considering the number of those who do not attend, use a suitable approximation to find the probability that fewer than 146 people attend. [4]
OCR S2 2014 June Q3
7 marks Standard +0.3
3 The random variable \(G\) has the distribution \(\mathbf { N } \left( \mu , \boldsymbol { \sigma } ^ { 2 } \right)\). One hundred observations of \(G\) are taken. The results are summarised in the following table.
Interval\(G < 40.0\)\(40.0 \leqslant G < 60.0\)\(G \geqslant 60.0\)
Frequency175825
  1. By considering \(\mathrm { P } ( G < 40.0 )\), write down an equation involving \(\mu\) and \(\sigma\). [2]
  2. Find a second equation involving \(\mu\) and \(\sigma\). Hence calculate values for \(\mu\) and \(\sigma\). [4]
    [0pt]
  3. Explain why your answers are only estimates. [1]
OCR S2 2014 June Q4
7 marks Easy -1.2
4 A zoologist investigates the number of snakes found in a given region of land. The zoologist intends to use a Poisson distribution to model the number of snakes.
[0pt]
  1. One condition for a Poisson distribution to be valid is that snakes must occur at constant average rate. State another condition needed for a Poisson distribution to be valid. [1] Assume now that the number of snakes found in 1 acre of a region can be modelled by the distribution Po(4).
    [0pt]
  2. Find the probability that, in 1 acre of the region, at least 6 snakes are found. [2]
    [0pt]
  3. Find the probability that, in 0.77 acres of the region, the number of snakes found is either 2 or 3. [4]
OCR S2 2014 June Q5
13 marks Moderate -0.3
5 A continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} \frac { 1 } { 2 } \pi \sin ( \pi x ) & 0 \leqslant x \leqslant 1 \\ 0 & \text { otherwise } \end{cases}$$
  1. Show that this is a valid probability density function. [4]
  2. Sketch the curve \(\boldsymbol { y } = \mathbf { f } ( \boldsymbol { x } )\) and write down the value of \(\mathbf { E } \boldsymbol { ( } \boldsymbol { X } \boldsymbol { ) }\). [3]
  3. Find the value \(q\) such that \(\mathrm { P } ( X > q ) = 0.75\). [3]
  4. Write down an expression, including an integral, for \(\operatorname { Var } ( X )\). (Do not attempt to evaluate the integral.) [2]
  5. A student states that " \(X\) is more likely to occur when \(x\) is close to \(\mathrm { E } ( X )\)." Give an improved version of this statement. [1]
OCR S2 2014 June Q6
12 marks Standard +0.3
6 In a city the proportion of inhabitants from ethnic group \(\mathbf { Z }\) is known to be \(\mathbf { 0 . 4 }\). A sample of \(\mathbf { 1 2 }\) employees of a large company in this city is obtained and it is found that 2 of them are from ethnic group \(Z\). A test is carried out, at the \(5 \%\) significance level, of whether the proportion of employees in this company from ethnic group \(Z\) is less than in the city as a whole.
[0pt]
  1. State an assumption that must be made about the sample for a significance test to be valid. [1]
    [0pt]
  2. Describe briefly an appropriate way of obtaining the sample. [2]
    [0pt]
  3. Carry out the test. [7]
  4. A manager believes that the company discriminates against ethnic group \(Z\). Explain whether carrying out the test at the 10\% significance level would be more supportive or less supportive of the manager's belief. [2]
OCR S2 2014 June Q7
15 marks Standard +0.3
7 An examination board is developing a new syllabus and wants to know if the question papers are the right length. A random sample of 50 candidates was given a pre-test on a dummy paper. The times, \(t\) minutes, taken by these candidates to complete the paper can be summarised by \(n = 50\), \(\sum \boldsymbol { t } = \mathbf { 4 0 5 0 }\), \(\sum \boldsymbol { t } ^ { \mathbf { 2 } } \boldsymbol { = } \mathbf { 3 2 9 8 0 0 }\).
Assume that times are normally distributed.
[0pt]
  1. Estimate the proportion of candidates that could not complete the paper within 90 minutes. [6]
  2. Test, at the \(10 \%\) significance level, whether the mean time for all candidates to complete this paper is \(\mathbf { 8 0 }\) minutes. Use a two-tail test. [7]
  3. Explain whether the assumption that times are normally distributed is necessary in answering
    1. part (i),
      [0pt]
    2. part (ii). [2]
OCR S2 2014 June Q8
6 marks Challenging +1.2
8 The random variable \(W\) has the distribution \(\operatorname { Po } ( \lambda )\). A significance test is carried out of the null hypothesis \(\mathrm { H } _ { 0 } : \lambda = 3.60\), against the alternative hypothesis \(\mathrm { H } _ { 1 } : \lambda < 3.60\). The test is based on a single observation of \(W\). The critical region is \(W = 0\).
[0pt]
  1. Find the significance level of the test. [2]
  2. It is known that, when \(\boldsymbol { \lambda } = \boldsymbol { \lambda } _ { \mathbf { 0 } }\), the probability that the test results in a Type II error is \(\mathbf { 0 . 8 }\). Find the value of \(\lambda _ { 0 }\). [4] \section*{END OF QUESTION PAPER}
OCR S2 Specimen Q1
5 marks Moderate -0.8
1 The standard deviation of a random variable \(F\) is 12.0. The mean of \(n\) independent observations of \(F\) is denoted by \(\bar { F }\).
  1. Given that the standard deviation of \(\bar { F }\) is 1.50 , find the value of \(n\).
  2. For this value of \(n\), state, with justification, what can be said about the distribution of \(\bar { F }\).
OCR S2 Specimen Q2
5 marks Easy -1.2
2 A certain neighbourhood contains many small houses (with small gardens) and a few large houses (with large gardens). A sample survey of all houses is to be carried out in this neighbourhood. A student suggests that the sample could be selected by sticking a pin into a map of the neighbourhood the requisite number of times, while blindfolded.
  1. Give two reasons why this method does not produce a random sample.
  2. Describe a better method.
OCR S2 Specimen Q3
6 marks Moderate -0.3
3 Sixty people each make two throws with a fair six-sided die.
  1. State the probability of one particular person obtaining two sixes.
  2. Using a suitable approximation, calculate the probability that at least four of the sixty obtain two sixes.
OCR S2 Specimen Q4
9 marks Standard +0.3
4 The random variable \(G\) has mean 20.0 and standard deviation \(\sigma\). It is given that \(\mathrm { P } ( G > 15.0 ) = 0.6\). Assume that \(G\) is normally distributed.
  1. (a) Find the value of \(\sigma\).
    (b) Given that \(\mathrm { P } ( G > g ) = 0.4\), find the value of \(\mathrm { P } ( G > 2 g )\).
  2. It is known that no values of \(G\) are ever negative. State with a reason what this tells you about the assumption that \(G\) is normally distributed.
OCR S2 Specimen Q5
10 marks Standard +0.3
5 The mean solubility rating of widgets inserted into beer cans is thought to be 84.0, in appropriate units. A random sample of 50 widgets is taken. The solubility ratings, \(x\), are summarised by $$n = 50 , \quad \Sigma x = 4070 , \quad \Sigma x ^ { 2 } = 336100$$ Test, at the \(5 \%\) significance level, whether the mean solubility rating is less than 84.0 .
OCR S2 Specimen Q6
11 marks Standard +0.8
6 On average a motorway police force records one car that has run out of petrol every two days.
  1. (a) Using a Poisson distribution, calculate the probability that, in one randomly chosen day, the police force records exactly two cars that have run out of petrol.
    (b) Using a Poisson distribution and a suitable approximation to the binomial distribution, calculate the probability that, in one year of 365 days, there are fewer than 205 days on which the police force records no cars that have run out of petrol.
  2. State an assumption needed for the Poisson distribution to be appropriate in part (i), and explain why this assumption is unlikely to be valid.
OCR S2 Specimen Q7
12 marks Standard +0.3
7 The time, in minutes, for which a customer is prepared to wait on a telephone complaints line is modelled by the random variable \(X\). The probability density function of \(X\) is given by $$\mathrm { f } ( x ) = \begin{cases} k x \left( 9 - x ^ { 2 } \right) & 0 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = \frac { 4 } { 81 }\).
  2. Find \(\mathrm { E } ( X )\).
  3. (a) Show that the value \(y\) which satisfies \(\mathrm { P } ( X < y ) = \frac { 3 } { 5 }\) satisfies $$5 y ^ { 4 } - 90 y ^ { 2 } + 243 = 0 .$$ (b) Using the substitution \(w = y ^ { 2 }\), or otherwise, solve the equation in part (a) to find the value of \(y\).