Questions — OCR MEI (4455 questions)

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OCR MEI C4 Q2
19 marks Standard +0.3
Archimedes, about 2200 years ago, used regular polygons inside and outside circles to obtain approximations for \(\pi\).
  1. Fig. 8.1 shows a regular 12-sided polygon inscribed in a circle of radius 1 unit, centre O. AB is one of the sides of the polygon. C is the midpoint of AB. Archimedes used the fact that the circumference of the circle is greater than the perimeter of this polygon. \includegraphics{figure_1}
    1. Show that \(\text{AB} = 2 \sin 15°\). [2]
    2. Use a double angle formula to express \(\cos 30°\) in terms of \(\sin 15°\). Using the exact value of \(\cos 30°\), show that \(\sin 15° = \frac{1}{2}\sqrt{2 - \sqrt{3}}\). [4]
    3. Use this result to find an exact expression for the perimeter of the polygon. Hence show that \(\pi > 6\sqrt{2 - \sqrt{3}}\). [2]
  2. In Fig. 8.2, a regular 12-sided polygon lies outside the circle of radius 1 unit, which touches each side of the polygon. F is the midpoint of DE. Archimedes used the fact that the circumference of the circle is less than the perimeter of this polygon. \includegraphics{figure_2}
    1. Show that \(\text{DE} = 2 \tan 15°\). [2]
    2. Let \(t = \tan 15°\). Use a double angle formula to express \(\tan 30°\) in terms of \(t\). Hence show that \(t^2 + 2\sqrt{3}t - 1 = 0\). [3]
    3. Solve this equation, and hence show that \(\pi < 12(2 - \sqrt{3})\). [4]
  3. Use the results in parts (i)(C) and (ii)(C) to establish upper and lower bounds for the value of \(\pi\), giving your answers in decimal form. [2]
OCR MEI C4 Q3
7 marks Moderate -0.3
Express \(\sin \theta - 3 \cos \theta\) in the form \(R \sin (\theta - \alpha)\), where \(R\) and \(\alpha\) are constants to be determined, and \(0° < \alpha < 90°\). Hence solve the equation \(\sin \theta - 3 \cos \theta = 1\) for \(0° \leqslant \theta \leqslant 360°\). [7]
OCR MEI C4 Q4
16 marks Standard +0.3
\includegraphics{figure_3} In a theme park ride, a capsule C moves in a vertical plane (see Fig. 8). With respect to the axes shown, the path of C is modelled by the parametric equations $$x = 10 \cos \theta + 5 \cos 2\theta, \quad y = 10 \sin \theta + 5 \sin 2\theta, \quad (0 \leqslant \theta < 2\pi),$$ where \(x\) and \(y\) are in metres.
  1. Show that \(\frac{\text{d}y}{\text{d}x} = -\frac{\cos \theta + \cos 2\theta}{\sin \theta + \sin 2\theta}\). Verify that \(\frac{\text{d}y}{\text{d}x} = 0\) when \(\theta = \frac{1}{3}\pi\). Hence find the exact coordinates of the highest point A on the path of C. [6]
  2. Express \(x^2 + y^2\) in terms of \(\theta\). Hence show that $$x^2 + y^2 = 125 + 100 \cos \theta.$$ [4]
  3. Using this result, or otherwise, find the greatest and least distances of C from O. [2]
You are given that, at the point B on the path vertically above O, $$2 \cos^2 \theta + 2 \cos \theta - 1 = 0.$$
  1. Using this result, and the result in part (ii), find the distance OB. Give your answer to 3 significant figures. [4]
OCR MEI C4 Q5
7 marks Standard +0.3
Show that \(\cot 2\theta = \frac{1 - \tan^2 \theta}{2 \tan \theta}\). Hence solve the equation $$\cot 2\theta = 1 + \tan \theta \quad \text{for } 0° < \theta < 360°.$$ [7]
OCR MEI C4 Q1
18 marks Standard +0.3
The upper and lower surfaces of a coal seam are modelled as planes ABC and DEF, as shown in Fig. 8. All dimensions are metres. \includegraphics{figure_1} Relative to axes \(Ox\) (due east), \(Oy\) (due north) and \(Oz\) (vertically upwards), the coordinates of the points are as follows. A: \((0, 0, -15)\) \quad B: \((100, 0, -30)\) \quad C: \((0, 100, -25)\) D: \((0, 0, -40)\) \quad E: \((100, 0, -50)\) \quad F: \((0, 100, -35)\)
  1. Verify that the cartesian equation of the plane ABC is \(3x + 2y + 20z + 300 = 0\). [3]
  2. Find the vectors \(\overrightarrow{DE}\) and \(\overrightarrow{DF}\). Show that the vector \(2\mathbf{i} - \mathbf{j} + 20\mathbf{k}\) is perpendicular to each of these vectors. Hence find the cartesian equation of the plane DEF. [6]
  3. By calculating the angle between their normal vectors, find the angle between the planes ABC and DEF. [4]
It is decided to drill down to the seam from a point R \((15, 34, 0)\) in a line perpendicular to the upper surface of the seam. This line meets the plane ABC at the point S.
  1. Write down a vector equation of the line RS. Find the coordinates of S. [5]
OCR MEI C4 Q2
4 marks Easy -1.2
Write down normal vectors to the planes \(2x + 3y + 4z = 10\) and \(x - 2y + z = 5\). Hence show that these planes are perpendicular to each other. [4]
OCR MEI C4 Q3
7 marks Moderate -0.3
Verify that the point \((-1, 6, 5)\) lies on both the lines $$\mathbf{r} = \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} \quad \text{and} \quad \mathbf{r} = \begin{pmatrix} 0 \\ 6 \\ 3 \end{pmatrix} + \mu \begin{pmatrix} 1 \\ 0 \\ 2 \end{pmatrix}.$$ Find the acute angle between the lines. [7]
OCR MEI C4 Q4
18 marks Standard +0.3
A computer-controlled machine can be programmed to make cuts by entering the equation of the plane of the cut, and to drill holes by entering the equation of the line of the hole. A \(20\text{ cm} \times 30\text{ cm} \times 30\text{ cm}\) cuboid is to be cut and drilled. The cuboid is positioned relative to \(x\)-, \(y\)- and \(z\)-axes as shown in Fig. 8.1. \includegraphics{figure_2} First, a plane cut is made to remove the corner at E. The cut goes through the points P, Q and R, which are the midpoints of the sides ED, EA and EF respectively.
  1. Write down the coordinates of P, Q and R. Hence show that \(\overrightarrow{PQ} = \begin{pmatrix} 0 \\ 0 \\ -15 \end{pmatrix}\) and \(\overrightarrow{PR} = \begin{pmatrix} -15 \\ 0 \\ 1 \end{pmatrix}\). [4]
  2. Show that \(\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}\) is perpendicular to the plane through P, Q and R. Hence find the cartesian equation of this plane. [5]
A hole is then drilled perpendicular to triangle PQR, as shown in Fig. 8.2. The hole passes through the triangle at the point T which divides the line PS in the ratio \(2:1\), where S is the midpoint of QR.
  1. Write down the coordinates of S, and show that the point T has coordinates \((-5, 16, 25)\). [4]
  2. Write down a vector equation of the line of the drill hole. Hence determine whether or not this line passes through C. [5]
OCR MEI C4 Q5
17 marks Standard +0.3
A tent has vertices ABCDEF with coordinates as shown in Fig. 7. Lengths are in metres. The \(Oxy\) plane is horizontal. \includegraphics{figure_3}
  1. Find the length of the ridge of the tent DE, and the angle this makes with the horizontal. [4]
  2. Show that the vector \(\mathbf{i} - 4\mathbf{j} + 5\mathbf{k}\) is normal to the plane through A, D and E. Hence find the equation of this plane. Given that B lies in this plane, find \(a\). [7]
  3. Verify that the equation of the plane BCD is \(x + z = 8\). Hence find the acute angle between the planes ABDE and BCD. [6]
OCR MEI S1 2010 January Q1
8 marks Easy -1.3
A camera records the speeds in miles per hour of 15 vehicles on a motorway. The speeds are given below. $$73 \quad 67 \quad 75 \quad 64 \quad 52 \quad 63 \quad 75 \quad 81 \quad 77 \quad 72 \quad 68 \quad 74 \quad 79 \quad 72 \quad 71$$
  1. Construct a sorted stem and leaf diagram to represent these data, taking stem values of 50, 60, ... . [4]
  2. Write down the median and midrange of the data. [2]
  3. Which of the median and midrange would you recommend to measure the central tendency of the data? Briefly explain your answer. [2]
OCR MEI S1 2010 January Q2
8 marks Moderate -0.8
In her purse, Katharine has two £5 notes, two £10 notes and one £20 note. She decides to select two of these notes at random to donate to a charity. The total value of these two notes is denoted by the random variable \(£X\).
    1. Show that P(X = 10) = 0.1. [1]
    2. Show that P(X = 30) = 0.2. [2]
    The table shows the probability distribution of X.
    \(r\)1015202530
    P(X = r)0.10.40.10.20.2
  2. Find E(X) and Var(X). [5]
OCR MEI S1 2010 January Q3
8 marks Easy -1.2
In a survey, a large number of young people are asked about their exercise habits. One of these people is selected at random. • \(G\) is the event that this person goes to the gym. • \(R\) is the event that this person goes running. You are given that P(G) = 0.24, P(R) = 0.13 and P(G ∩ R) = 0.06.
  1. Draw a Venn diagram, showing the events \(G\) and \(R\), and fill in the probability corresponding to each of the four regions of your diagram. [3]
  2. Determine whether the events \(G\) and \(R\) are independent. [2]
  3. Find P(R | G). [3]
OCR MEI S1 2010 January Q4
5 marks Moderate -0.8
In a multiple-choice test there are 30 questions. For each question, there is a 60% chance that a randomly selected student answers correctly, independently of all other questions.
  1. Find the probability that a randomly selected student gets a total of exactly 20 questions correct. [3]
  2. If 100 randomly selected students take the test, find the expected number of students who get exactly 20 questions correct. [2]
OCR MEI S1 2010 January Q5
3 marks Easy -1.2
My credit card has a 4-digit code called a PIN. You should assume that any 4-digit number from 0000 to 9999 can be a PIN.
  1. If I cannot remember any digits and guess my number, find the probability that I guess it correctly. [1]
In fact my PIN consists of four different digits. I can remember all four digits, but cannot remember the correct order.
  1. If I now guess my number, find the probability that I guess it correctly. [2]
OCR MEI S1 2010 January Q6
4 marks Easy -1.2
Three prizes, one for English, one for French and one for Spanish, are to be awarded in a class of 20 students. Find the number of different ways in which the three prizes can be awarded if
  1. no student may win more than 1 prize, [2]
  2. no student may win all 3 prizes. [2]
OCR MEI S1 2010 January Q7
19 marks Moderate -0.8
A pear grower collects a random sample of 120 pears from his orchard. The histogram below shows the lengths, in mm, of these pears. \includegraphics{figure_7}
  1. Calculate the number of pears which are between 90 and 100 mm long. [2]
  2. Calculate an estimate of the mean length of the pears. Explain why your answer is only an estimate. [4]
  3. Calculate an estimate of the standard deviation. [3]
  4. Use your answers to parts (ii) and (iii) to investigate whether there are any outliers. [4]
  5. Name the type of skewness of the distribution. [1]
  6. Illustrate the data using a cumulative frequency diagram. [5]
OCR MEI S1 2010 January Q8
17 marks Standard +0.3
An environmental health officer monitors the air pollution level in a city street. Each day the level of pollution is classified as low, medium or high. The probabilities of each level of pollution on a randomly chosen day are as given in the table.
Pollution levelLowMediumHigh
Probability0.50.350.15
  1. Three days are chosen at random. Find the probability that the pollution level is
    1. low on all 3 days, [2]
    2. low on at least one day, [2]
    3. low on one day, medium on another day, and high on the other day. [3]
  2. Ten days are chosen at random. Find the probability that
    1. there are no days when the pollution level is high, [2]
    2. there is exactly one day when the pollution level is high. [3]
The environmental health officer believes that pollution levels will be low more frequently in a different street. On 20 randomly selected days she monitors the pollution level in this street and finds that it is low on 15 occasions.
  1. Carry out a test at the 5% level to determine if there is evidence to suggest that she is correct. Use hypotheses \(H_0: p = 0.5\), \(H_1: p > 0.5\), where \(p\) represents the probability that the pollution level in this street is low. Explain why \(H_1\) has this form. [5]
OCR MEI S1 2011 January Q1
3 marks Easy -1.8
The stem and leaf diagram shows the weights, rounded to the nearest 10 grams, of 25 female iguanas. \begin{align} 8 &| 3 \quad 9
9 &| 3 \quad 5 \quad 6 \quad 6 \quad 6 \quad 8 \quad 9 \quad 9
10 &| 0 \quad 2 \quad 2 \quad 3 \quad 4 \quad 6 \quad 9
11 &| 2 \quad 4 \quad 7 \quad 8
12 &| 3 \quad 4 \quad 5
13 &| 2 \end{align} Key: \(11|2\) represents 1120 grams
  1. Find the mode and the median of the data. [2]
  2. Identify the type of skewness of the distribution. [1]
OCR MEI S1 2011 January Q2
4 marks Moderate -0.8
The table shows all the possible products of the scores on two fair four-sided dice.
Score on second die
1234
\multirow{4}{*}{\rotatebox{90}{Score on first die}} 11234
\cline{2-5} 22468
\cline{2-5} 336912
\cline{2-5} 4481216
  1. Find the probability that the product of the two scores is less than 10. [1]
  2. Show that the events 'the score on the first die is even' and 'the product of the scores on the two dice is less than 10' are not independent. [3]
OCR MEI S1 2011 January Q3
6 marks Moderate -0.8
There are 13 men and 10 women in a running club. A committee of 3 men and 3 women is to be selected.
  1. In how many different ways can the three men be selected? [2]
  2. In how many different ways can the whole committee be selected? [2]
  3. A random sample of 6 people is selected from the running club. Find the probability that this sample consists of 3 men and 3 women. [2]
OCR MEI S1 2011 January Q4
7 marks Standard +0.3
The probability distribution of the random variable \(X\) is given by the formula $$\text{P}(X = r) = kr(r + 1) \quad \text{for } r = 1, 2, 3, 4, 5.$$
  1. Show that \(k = \frac{1}{70}\). [2]
  2. Find E\((X)\) and Var\((X)\). [5]
OCR MEI S1 2011 January Q5
8 marks Moderate -0.8
Andy can walk to work, travel by bike or travel by bus. The tree diagram shows the probabilities of any day being dry or wet and the corresponding probabilities for each of Andy's methods of travel. \includegraphics{figure_5} A day is selected at random. Find the probability that
  1. the weather is wet and Andy travels by bus, [2]
  2. Andy walks or travels by bike, [3]
  3. the weather is dry given that Andy walks or travels by bike. [3]
OCR MEI S1 2011 January Q6
8 marks Moderate -0.8
A survey is being carried out into the carbon footprint of individual citizens. As part of the survey, 100 citizens are asked whether they have attempted to reduce their carbon footprint by any of the following methods.
  • Reducing car use
  • Insulating their homes
  • Avoiding air travel
The numbers of citizens who have used each of these methods are shown in the Venn diagram. \includegraphics{figure_6} One of the citizens is selected at random.
  1. Find the probability that this citizen
    1. has avoided air travel, [1]
    2. has used at least two of the three methods. [2]
  2. Given that the citizen has avoided air travel, find the probability that this citizen has reduced car use. [2]
Three of the citizens are selected at random.
  1. Find the probability that none of them have avoided air travel. [3]
OCR MEI S1 2011 January Q7
19 marks Moderate -0.3
The incomes of a sample of 918 households on an island are given in the table below.
Income (x thousand pounds)\(0 \leqslant x \leqslant 20\)\(20 < x \leqslant 40\)\(40 < x \leqslant 60\)\(60 < x \leqslant 100\)\(100 < x \leqslant 200\)
Frequency23836514212845
  1. Draw a histogram to illustrate the data. [5]
  2. Calculate an estimate of the mean income. [3]
  3. Calculate an estimate of the standard deviation of the incomes. [4]
  4. Use your answers to parts (ii) and (iii) to show there are almost certainly some outliers in the sample. Explain whether or not it would be appropriate to exclude the outliers from the calculation of the mean and the standard deviation. [4]
  5. The incomes were converted into another currency using the formula \(y = 1.15x\). Calculate estimates of the mean and variance of the incomes in the new currency. [3]
OCR MEI S1 2011 January Q8
17 marks Standard +0.3
Mark is playing solitaire on his computer. The probability that he wins a game is 0.2, independently of all other games that he plays.
  1. Find the expected number of wins in 12 games. [2]
  2. Find the probability that
    1. he wins exactly 2 out of the next 12 games that he plays, [3]
    2. he wins at least 2 out of the next 12 games that he plays. [3]
  3. Mark's friend Ali also plays solitaire. Ali claims that he is better at winning games than Mark. In a random sample of 20 games played by Ali, he wins 7 of them. Write down suitable hypotheses for a test at the 5\% level to investigate whether Ali is correct. Give a reason for your choice of alternative hypothesis. Carry out the test. [9]