Questions — OCR MEI (4455 questions)

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OCR MEI C4 2011 June Q7
18 marks Standard +0.3
A piece of cloth ABDC is attached to the tops of vertical poles AE, BF, DG and CH, where E, F, G and H are at ground level (see Fig. 7). Coordinates are as shown, with lengths in metres. The length of pole DG is \(k\) metres. \includegraphics{figure_7}
  1. Write down the vectors \(\overrightarrow{AB}\) and \(\overrightarrow{AC}\). Hence calculate the angle BAC. [6]
  2. Verify that the equation of the plane ABC is \(x + y - 2z + d = 0\), where \(d\) is a constant to be determined. Calculate the acute angle the plane makes with the horizontal plane. [7]
  3. Given that A, B, D and C are coplanar, show that \(k = 3\). Hence show that ABDC is a trapezium, and find the ratio of CD to AB. [5]
OCR MEI C4 2011 June Q8
18 marks Standard +0.8
Water is leaking from a container. After \(t\) seconds, the depth of water in the container is \(x\) cm, and the volume of water is \(V\) cm\(^3\), where \(V = \frac{1}{3}x^3\). The rate at which water is lost is proportional to \(x\), so that \(\frac{dV}{dt} = -kx\), where \(k\) is a constant.
  1. Show that \(x \frac{dx}{dt} = -k\). [3]
Initially, the depth of water in the container is 10 cm.
  1. Show by integration that \(x = \sqrt{100 - 2kt}\). [4]
  2. Given that the container empties after 50 seconds, find \(k\). [2]
Once the container is empty, water is poured into it at a constant rate of 1 cm\(^3\) per second. The container continues to lose water as before.
  1. Show that, \(t\) seconds after starting to pour the water in, \(\frac{dx}{dt} = \frac{1-x}{x^2}\). [2]
  2. Show that \(\frac{1}{1-x} - x - 1 = \frac{x^2}{1-x}\). Hence solve the differential equation in part (iv) to show that $$t = \ln\left(\frac{1}{1-x}\right) - \frac{1}{2}x^2 - x.$$ [6]
  3. Show that the depth cannot reach 1 cm. [1]
OCR MEI C4 2011 June Q1
1 marks Easy -2.5
In lines 59 and 60, the text says "In that case the proportion suffering such an attack would be 6.4%." Explain how this figure was obtained. [1]
OCR MEI C4 2011 June Q2
5 marks Easy -1.2
  1. In lines 8 to 10, the article says "Some banks do not allow numbers that begin with zero, numbers in which the digits are all the same (such as 5555) or numbers in which the digits are consecutive (such as 2345 or 8765)." How many different 4-digit PINs can be made when all these rules are applied? [3]
  2. At the time of writing, the world population is \(6.7 \times 10^9\) people. Assuming that, on average, each person has one card with a 4-digit PIN (subject to the rules in part (i) of this question), estimate the average number of people holding cards with any given PIN. Give your answer to an appropriate degree of accuracy. [2]
OCR MEI C4 2011 June Q3
2 marks Easy -2.0
In lines 46 and 47, the text says "Of the 11 people with unauthorised transactions, 3 could explain them as breaches of card security (typically losing the card) but 9 could not ... ." Place numbers in the three regions of the diagram consistent with the information in this sentence. [2] \includegraphics{figure_3}
OCR MEI C4 2011 June Q4
2 marks Easy -1.8
In lines 101 and 102, the text says "The total number of transactions for those who responded has been estimated as 100 000 for the \(3\frac{1}{2}\) years covered by the survey." Estimate the number of transactions per person per day that would give this figure. [2]
OCR MEI C4 2011 June Q5
1 marks Easy -2.5
The survey described in the article was based on a small sample. State one conclusion which is unlikely to be influenced by the size of the sample. [1]
OCR MEI C4 2011 June Q6
7 marks Moderate -0.3
A bank has detection software that can be set at two different levels, 'Mild' and 'Severe'. • When it is set at Mild, 0.1% of all transactions are queried. • When it is set at Severe 0.5% of all transactions are queried.
  1. One day the bank has 500 000 transactions. The software is set on 'Mild'. There are 480 false positives. Only \(\frac{1}{4}\) of the unauthorised transactions are queried. Complete the table. [3]
  2. What is the ratio of false positives to false negatives? [1]
  3. If the software had been set on 'Severe' for the same set of 500 000 transactions, with the total numbers of authorised and unauthorised transactions the same as in part (i) of this question, the number of false negatives would have been 5. What would the ratio of false positives to false negatives have been with this setting? [3]
OCR MEI C4 2012 June Q1
5 marks Moderate -0.3
Solve the equation \(\frac{4x}{x+1} - \frac{3}{2x+1} = 1\). [5]
OCR MEI C4 2012 June Q2
5 marks Moderate -0.8
Find the first four terms in the binomial expansion of \(\sqrt{1+2x}\). State the set of values of \(x\) for which the expansion is valid. [5]
OCR MEI C4 2012 June Q3
8 marks Standard +0.3
The total value of the sales made by a new company in the first \(t\) years of its existence is denoted by \(£V\). A model is proposed in which the rate of increase of \(V\) is proportional to the square root of \(V\). The constant of proportionality is \(k\).
  1. Express the model as a differential equation. Verify by differentiation that \(V = (\frac{1}{2}kt + c)^2\), where \(c\) is an arbitrary constant, satisfies this differential equation. [4]
  2. The value of the company's sales in its first year is £10000, and the total value of the sales in the first two years is £40000. Find \(V\) in terms of \(t\). [4]
OCR MEI C4 2012 June Q4
4 marks Moderate -0.5
Prove that \(\sec^2\theta + \cosec^2\theta = \sec^2\theta \cosec^2\theta\). [4]
OCR MEI C4 2012 June Q5
6 marks Moderate -0.3
Given the equation \(\sin(x + 45°) = 2\cos x\), show that \(\sin x + \cos x = 2\sqrt{2}\cos x\). Hence solve, correct to 2 decimal places, the equation for \(0° < x < 360°\). [6]
OCR MEI C4 2012 June Q6
8 marks Standard +0.3
Solve the differential equation \(\frac{dy}{dx} = \frac{y}{x(x+1)}\), given that when \(x = 1\), \(y = 1\). Your answer should express \(y\) explicitly in terms of \(x\). [8]
OCR MEI C4 2012 June Q7
19 marks Standard +0.3
Fig. 7a shows the curve with the parametric equations $$x = 2\cos\theta, \quad y = \sin 2\theta, \quad -\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}.$$ The curve meets the \(x\)-axis at O and P. Q and R are turning points on the curve. The scales on the axes are the same. \includegraphics{figure_7a}
  1. State, with their coordinates, the points on the curve for which \(\theta = -\frac{\pi}{2}\), \(\theta = 0\) and \(\theta = \frac{\pi}{2}\). [3]
  2. Find \(\frac{dy}{dx}\) in terms of \(\theta\). Hence find the gradient of the curve when \(\theta = \frac{\pi}{2}\), and verify that the two tangents to the curve at the origin meet at right angles. [5]
  3. Find the exact coordinates of the turning point Q. [3]
When the curve is rotated about the \(x\)-axis, it forms a paperweight shape, as shown in Fig. 7b. \includegraphics{figure_7b}
  1. Express \(\sin^2\theta\) in terms of \(x\). Hence show that the cartesian equation of the curve is \(y^2 = x^2(1 - \frac{1}{4}x^2)\). [4]
  2. Find the volume of the paperweight shape. [4]
OCR MEI C4 2012 June Q8
17 marks Standard +0.3
With respect to cartesian coordinates \(Oxyz\), a laser beam ABC is fired from the point A(1, 2, 4), and is reflected at point B off the plane with equation \(x + 2y - 3z = 0\), as shown in Fig. 8. A' is the point (2, 4, 1), and M is the midpoint of AA'. \includegraphics{figure_8}
  1. Show that AA' is perpendicular to the plane \(x + 2y - 3z = 0\), and that M lies in the plane. [4]
The vector equation of the line AB is \(\mathbf{r} = \begin{pmatrix} 1 \\ 2 \\ 4 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 1 \\ 2 \end{pmatrix}\).
  1. Find the coordinates of B, and a vector equation of the line A'B. [6]
  2. Given that A'BC is a straight line, find the angle \(\theta\). [4]
  3. Find the coordinates of the point where BC crosses the \(Oxz\) plane (the plane containing the \(x\)- and \(z\)-axes). [3]
OCR MEI C4 2013 June Q1
8 marks Moderate -0.3
  1. Express \(\frac{x}{(1 + x)(1 - 2x)}\) in partial fractions. [3]
  2. Hence use binomial expansions to show that \(\frac{x}{(1 + x)(1 - 2x)} = ax + bx^2 + ...\), where \(a\) and \(b\) are constants to be determined. State the set of values of \(x\) for which the expansion is valid. [5]
OCR MEI C4 2013 June Q2
7 marks Standard +0.3
Show that the equation \(\cos ec x + 5 \cot x = 3 \sin x\) may be rearranged as $$3 \cos^2 x + 5 \cos x - 2 = 0.$$ Hence solve the equation for \(0° \leq x \leq 360°\), giving your answers to 1 decimal place. [7]
OCR MEI C4 2013 June Q3
7 marks Moderate -0.8
Using appropriate right-angled triangles, show that \(\tan 45° = 1\) and \(\tan 30° = \frac{1}{\sqrt{3}}\). Hence show that \(\tan 75° = 2 + \sqrt{3}\). [7]
OCR MEI C4 2013 June Q4
8 marks Moderate -0.3
  1. Find a vector equation of the line \(l\) joining the points \((0, 1, 3)\) and \((-2, 2, 5)\). [2]
  2. Find the point of intersection of the line \(l\) with the plane \(x + 3y + 2z = 4\). [3]
  3. Find the acute angle between the line \(l\) and the normal to the plane. [3]
OCR MEI C4 2013 June Q5
6 marks Standard +0.3
The points A, B and C have coordinates \(A(3, 2, -1)\), \(B(-1, 1, 2)\) and \(C(10, 5, -5)\), relative to the origin O. Show that \(\overrightarrow{OC}\) can be written in the form \(\lambda\overrightarrow{OA} + \mu\overrightarrow{OB}\), where \(\lambda\) and \(\mu\) are to be determined. What can you deduce about the points O, A, B and C from the fact that \(\overrightarrow{OC}\) can be expressed as a combination of \(\overrightarrow{OA}\) and \(\overrightarrow{OB}\)? [6]
OCR MEI C4 2013 June Q6
18 marks Standard +0.3
The motion of a particle is modelled by the differential equation $$v \frac{dv}{dt} + 4x = 0,$$ where \(x\) is its displacement from a fixed point, and \(v\) is its velocity. Initially \(x = 1\) and \(v = 4\).
  1. Solve the differential equation to show that \(v^2 = 20 - 4x^2\). [4]
Now consider motion for which \(x = \cos 2t + 2 \sin 2t\), where \(x\) is the displacement from a fixed point at time \(t\).
  1. Verify that, when \(t = 0\), \(x = 1\). Use the fact that \(v = \frac{dx}{dt}\) to verify that when \(t = 0\), \(v = 4\). [4]
  2. Express \(x\) in the form \(R \cos(2t - \alpha)\), where \(R\) and \(\alpha\) are constants to be determined, and obtain the corresponding expression for \(v\). Hence or otherwise verify that, for this motion too, \(v^2 = 20 - 4x^2\). [7]
  3. Use your answers to part (iii) to find the maximum value of \(x\), and the earliest time at which \(x\) reaches this maximum value. [3]
OCR MEI C4 2013 June Q7
18 marks Standard +0.3
Fig. 7 shows the curve BC defined by the parametric equations $$x = 5 \ln u, \quad y = u + \frac{1}{u}, \quad 1 \leq u \leq 10.$$ The point A lies on the \(x\)-axis and AC is parallel to the \(y\)-axis. The tangent to the curve at C makes an angle \(\theta\) with AC, as shown. \includegraphics{figure_7}
  1. Find the lengths OA, OB and AC. [5]
  2. Find \(\frac{dy}{dx}\) in terms of \(u\). Hence find the angle \(\theta\). [6]
  3. Show that the cartesian equation of the curve is \(y = e^{x/5} + e^{-x/5}\). [2]
An object is formed by rotating the region OACB through \(360°\) about Ox.
  1. Find the volume of the object. [5]
OCR MEI C4 2013 June Q1
2 marks Easy -2.0
The diagram is a copy of Fig. 4. R is a place with latitude \(45°\) north and longitude \(60°\) west. Show the position of R on the diagram. M is the sub-solar point. It is on the Greenwich meridian and the declination of the sun is \(+20°\). Show the position of M on the diagram. [2] \includegraphics{figure_4}
OCR MEI C4 2013 June Q2
3 marks Easy -1.2
Use Fig. 8 to estimate the difference in the length of daylight between places with latitudes of \(30°\) south and \(60°\) south on the day for which the graph applies. [3]