OCR MEI C4 (Core Mathematics 4) 2011 June

Express \(\frac{1}{(2x + 1)(x^2 + 1)}\) in partial fractions. [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]

Find the first three terms in the binomial expansion of \(\sqrt{1 + 3x}\) in ascending powers of \(x\). State the set of values of \(x\) for which the expansion is valid. [5]

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]

Express \(2 \sin \theta - 3 \cos \theta\) in the form \(R \sin(\theta - \alpha)\), where \(R\) and \(\alpha\) are constants to be determined, and \(0 < \alpha < \frac{1}{2}\pi\). Hence write down the greatest and least possible values of \(1 + 2 \sin \theta - 3 \cos \theta\). [6]

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}

A curve has parametric equations $$x = 2 \sin \theta, \quad y = \cos 2\theta.$$

1. Find the exact coordinates and the gradient of the curve at the point with parameter \(\theta = \frac{1}{4}\pi\). [5]
2. Find \(y\) in terms of \(x\). [2]

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]

Solve the equation \(\cosec^2 \theta = 1 + 2 \cot \theta\), for \(-180° \leqslant \theta \leqslant 180°\). [6]

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]

Fig. 6 shows the region enclosed by part of the curve \(y = 2x^2\), the straight line \(x + y = 3\), and the \(y\)-axis. The curve and the straight line meet at P (1, 2). \includegraphics{figure_6} The shaded region is rotated through \(360°\) about the \(y\)-axis. Find, in terms of \(\pi\), the volume of the solid of revolution formed. [7] [You may use the formula \(V = \frac{1}{3}\pi r^2 h\) for the volume of a cone.]

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]

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]

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]