Questions — OCR MEI (4301 questions)

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OCR MEI C1 Q3
3 Solve the inequality \(2 x ^ { 2 } - 7 x \geq 4\).
OCR MEI C1 Q4
4 Simplify the following.
  1. \(x ^ { \frac { 5 } { 2 } } \times \sqrt { x }\)
  2. \(12 x ^ { - 5 } \div 3 x ^ { - 2 }\)
OCR MEI C1 Q5
5 The vertices of a triangle have coordinates ( 1,5 ), ( \(- 3,7\) ) and ( \(- 2 , - 1\) ).
Show that the triangle is right-angled.
OCR MEI C1 Q6
6 Find the coefficient of \(x ^ { 2 }\) in the expansion of \(( 3 - 2 x ) ^ { 5 }\).
OCR MEI C1 Q7
7 Find the coordinates of the points where the line \(y = 3 x - 2\) cuts the curve \(y = x ^ { 2 } + 4 x - 8\).
OCR MEI C1 Q8
8 The lines \(y = 5 x - a\) and \(y = 2 x + 18\) meet at the point ( \(7 , b\) ).
Find the values of \(a\) and \(b\).
OCR MEI C1 Q9
9 The graph shows the function \(y = x ^ { 2 } + b x + c\) where \(b\) and \(c\) are constants.
The point \(\mathrm { M } ( - 3 , - 16 )\) on the graph is the minimum point of the graph.
\includegraphics[max width=\textwidth, alt={}, center]{3b6291ef-bef9-49de-a20f-591e621bed65-2_478_948_1871_588}
  1. Write down the function \(y = \mathrm { f } ( x )\) in completed square form.
  2. Hence find the coordinates of the points where the curve cuts the axes.
OCR MEI C1 Q11
11 In this question \(\mathrm { f } ( x ) = x ^ { 3 } - 2 x ^ { 2 } - 4 x + k\).
  1. You are asked to find the values of \(k\) which satisfy the following conditions.
    (A) The graph of \(y = \mathrm { f } ( x )\) goes through the origin.
    (B) The graph of \(y = \mathrm { f } ( x )\) intersects with the \(y\) axis at ( \(0 , - 2\) ).
    (C) ( \(x - 2\) ) is a factor of \(\mathrm { f } ( x )\).
    (D) The remainder when \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\) is 5 .
    (E) The graph of \(y = \mathrm { f } ( x )\) is as shown in the diagram below.
    \includegraphics[max width=\textwidth, alt={}, center]{3b6291ef-bef9-49de-a20f-591e621bed65-3_373_788_2131_584}
  2. Find the solution of the equation \(\mathrm { f } ( x ) = 0\) when \(k = 8\). Sketch a graph of \(y = \mathrm { f } ( x )\) in this case.
OCR MEI C1 Q12
12 ABCD is a parallelogram. The coordinates of \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D are (-2, 3), (2, 4), (8, -3) and ( \(4 , - 4\) ) respectively.
\includegraphics[max width=\textwidth, alt={}, center]{3b6291ef-bef9-49de-a20f-591e621bed65-4_592_725_387_492}
  1. Prove that AB and BD are perpendicular.
  2. Find the lengths of AB and BD and hence find the area of the parallelogram ABCD
  3. Find the equation of the line CD and show that it meets the \(y\)-axis at \(\mathrm { X } ( 0 , - 5 )\).
  4. Show that the lines BX and AD bisect each other.
  5. Explain why the area of the parallelogram ABCD is equal to the area of the triangle BXC.
    Find the length of BX and hence calculate exactly the perpendicular distance of C from BX .
OCR MEI C1 Q1
1
  1. Statement P is \(a + b = 4\).
    Statement Q is \(\quad a = 1\) and \(b = 3\).
    Which one of the following is correct? $$\mathrm { P } \Rightarrow \mathrm { Q } , \quad \mathrm { P } \Leftrightarrow \mathrm { Q } , \quad \mathrm { P } \Leftarrow \mathrm { Q }$$
  2. Statement R is \(\quad x = 2\). Statement S is \(\quad x ^ { 2 } = 4\). Which one of the following is correct? $$R \Rightarrow S , \quad R \Leftrightarrow S , \quad R \Leftarrow S$$
OCR MEI C1 Q2
2 Find the equation of the straight line which is parallel to the line \(y = 3 x + 5\) and which goes through the point \(( 2,12 )\).
OCR MEI C1 Q3
3 Find the term which has the highest coefficient in the expansion of \(( 1 + x ) ^ { 8 }\).
OCR MEI C1 Q4
4 The surface area of the surface of a cylinder is given by the formula $$A = 2 \pi r ( r + h )$$ Rearrange this formula so that \(h\) is the subject.
OCR MEI C1 Q5
5 Solve the following equations.
  1. \(\quad 2 ^ { x } = \frac { 1 } { 8 }\).
  2. \(\quad x ^ { - \frac { 1 } { 2 } } = \frac { 1 } { 4 }\)
OCR MEI C1 Q6
6 Find the positive integer values of \(x\) for which $$\frac { 1 } { 2 } ( 26 - 2 x ) \geq 2 ( 3 + x )$$
OCR MEI C1 Q7
7 The remainder when \(x ^ { 3 } - 2 x + 4\) is divided by ( \(x - 2\) ) is twice the remainder when \(x ^ { 2 } + x + k\) is divided by ( \(x + 1\) ).
Find the value of \(k\).
OCR MEI C1 Q8
8 Find the values of \(a\) and \(b\) for which \(\frac { 4 } { ( 2 \sqrt { 3 } - 1 ) } = a + b \sqrt { 3 }\).
OCR MEI C1 Q9
9 Find the coordinates of the points where the curve \(y = x ^ { 2 } - 2 x - 8\) meets the line \(y = x + 2\).
OCR MEI C1 Q10
10 The diagram shows the graph of \(y = \mathrm { f } ( x )\).
\includegraphics[max width=\textwidth, alt={}, center]{4c556b8e-1a19-4480-bf2a-0ef9e67f98b4-3_507_1085_933_383} A is the minimum point of the curve at \(( 3 , - 4 )\) and B is the point \(( 5,0 )\).
On separate diagrams on graph paper, draw the graphs of the following. In each case give the coordinates of the images of the points A and B .
  1. \(\quad y = \mathrm { f } ( x ) + 2\),
  2. \(y = \mathrm { f } ( x + 2 )\).
OCR MEI C1 Q11
11 Fig. 11 shows the graph of \(y = a x ^ { 2 } + b x + c\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4c556b8e-1a19-4480-bf2a-0ef9e67f98b4-4_572_1509_465_285} \captionsetup{labelformat=empty} \caption{Fig. 11}
\end{figure}
  1. Explain why a must be negative.
  2. State two factors of \(y = a x ^ { 2 } + b x + c\).
  3. Hence, or otherwise, find the values of \(a , b\) and \(c\). Another function is given by \(y = x ^ { 2 } - 4 x + 10\).
  4. Write this in completed square form.
  5. Explain why the graphs of these two functions never meet.
OCR MEI C1 Q12
12 The function \(\mathrm { f } ( x )\) is given by \(\mathrm { f } ( x ) = x ^ { 3 } + 6 x ^ { 2 } + 5 x - 12\).
  1. Show that \(( x + 3 )\) is a factor of \(\mathrm { f } ( x )\).
  2. Find the other factors of \(\mathrm { f } ( x )\).
  3. State the coordinates where the graph of \(y = \mathrm { f } ( x )\) cuts the \(x\) axis. Hence sketch the graph of \(y = \mathrm { f } ( x )\).
  4. On the same graph sketch also \(y = x ( x - 1 ) ( x - 2 )\) Label the two points of intersection of the two curves A and B .
  5. By equating the two curves, show that the \(x\) coordinates of A and B satisfy the equation \(3 x ^ { 2 } + x - 4 = 0\).
    Solve this equation to find the \(x\)-coordinates of A and B .
OCR MEI C1 Q13
13 In Fig.13, XP and XQ are the perpendicular bisectors of AC and BC respectively. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4c556b8e-1a19-4480-bf2a-0ef9e67f98b4-5_409_768_383_604} \captionsetup{labelformat=empty} \caption{Fig. 13}
\end{figure}
  1. Find the coordinates of X .
  2. Hence show that \(\mathrm { AX } = \mathrm { BX } = \mathrm { CX }\).
  3. The circumcircle of a triangle is the circle which passes through the vertices of the triangle.
    Write down the equation of the circumcircle of the triangle ABC .
  4. Find the coordinates of the points where the circle cuts the \(x\) axis.
OCR MEI S2 2006 January Q1
5 marks
1 A roller-coaster ride has a safety system to detect faults on the track.
  1. State conditions for a Poisson distribution to be a suitable model for the number of faults occurring on a randomly selected day. Faults are detected at an average rate of 0.15 per day. You may assume that a Poisson distribution is a suitable model.
  2. Find the probability that on a randomly chosen day there are
    (A) no faults,
    (B) at least 2 faults.
  3. Find the probability that, in a randomly chosen period of 30 days, there are at most 3 faults. There is also a separate safety system to detect faults on the roller-coaster train itself. Faults are detected by this system at an average rate of 0.05 per day, independently of the faults detected on the track. You may assume that a Poisson distribution is also suitable for modelling the number of faults detected on the train.
  4. State the distribution of the total number of faults detected by the two systems in a period of 10 days. Find the probability that a total of 5 faults is detected in a period of 10 days.
    [0pt]
  5. The roller-coaster is operational for 200 days each year. Use a suitable approximating distribution to find the probability that a total of at least 50 faults is detected in 200 days. [5]
OCR MEI S2 2006 January Q2
2 The drug EPO (erythropoetin) is taken by some athletes to improve their performance. This drug is in fact banned and blood samples taken from athletes are tested to measure their 'hematocrit level'. If the level is over 50 it is considered that the athlete is likely to have taken EPO and the result is described as 'positive'. The measured hematocrit level of each athlete varies over time, even if EPO has not been taken.
  1. For each athlete in a large population of innocent athletes, the variation in measured hematocrit level is described by the Normal distribution with mean 42.0 and standard deviation 3.0.
    (A) Show that the probability that such an athlete tests positive for EPO in a randomly chosen test is 0.0038 .
    (B) Find the probability that such an athlete tests positive on at least 1 of the 7 occasions during the year when hematocrit level is measured. (These occasions are spread at random through the year and all test results are assumed to be independent.)
    (C) It is standard policy to apply a penalty after testing positive. Comment briefly on this policy in the light of your answer to part (i)(B).
  2. Suppose that 1000 tests are carried out on innocent athletes whose variation in measured hematocrit level is as described in part (i). It may be assumed that the probability of a positive result in each test is 0.0038 , independently of all other test results.
    (A) State the exact distribution of the number of positive tests.
    (B) Use a suitable approximating distribution to find the probability that at least 10 tests are positive.
  3. Because of genetic factors, a particular innocent athlete has an abnormally high natural hematocrit level. This athlete's measured level is Normally distributed with mean 48.0 and standard deviation 2.0. The usual limit of 50 for a positive test is to be altered for this athlete to a higher value \(h\). Find the value of \(h\) for which this athlete would test positive on average just once in 200 occasions.
OCR MEI S2 2006 January Q3
3 A researcher is investigating the relationship between temperature and levels of the air pollutant nitrous oxide at a particular site. The researcher believes that there will be a positive correlation between the daily maximum temperature, \(x\), and nitrous oxide level, \(y\). Data are collected for 10 randomly selected days. The data, measured in suitable units, are given in the table and illustrated on the scatter diagram.
\(x\)13.317.216.918.718.419.323.115.020.614.4
\(y\)911142643255215107
\includegraphics[max width=\textwidth, alt={}, center]{794b337f-6306-4d2e-bb5e-af8cedc9742e-4_823_1234_774_370}
  1. Calculate the value of Spearman's rank correlation coefficient for these data.
  2. Perform a hypothesis test at the \(5 \%\) level to check the researcher's belief, stating your hypotheses clearly.
  3. It is suggested that it would be preferable to carry out a test based on the product moment correlation coefficient. State the distributional assumption required for such a test to be valid. Explain how a scatter diagram can be used to check whether the distributional assumption is likely to be valid and comment on the validity in this case.
  4. A statistician investigates data over a much longer period and finds that the assumptions for the use of the product moment correlation coefficient are in fact valid. Give the critical region for the test at the \(1 \%\) level, based on a sample of 60 days.
  5. In a different research project, into the correlation between daily temperature and ozone pollution levels, a positive correlation is found. It is argued that this shows that high temperatures cause increased ozone levels. Comment on this claim.