OCR H240/02 (Pure Mathematics and Statistics) 2018 December

\includegraphics{figure_1} The diagram shows the curve \(y = \sqrt{x - 3}\). The shaded region is bounded by the curve and the two axes. Find the exact area of the shaded region. [4]

\(\text{f}(x)\) is a cubic polynomial in which the coefficient of \(x^3\) is 1. The equation \(\text{f}(x) = 0\) has exactly two roots.

1. Sketch a possible graph of \(y = \text{f}(x)\). [2]

It is now given that the two roots are \(x = 2\) and \(x = 3\).

1. Find, in expanded form, the two possible polynomials \(\text{f}(x)\). [3]

\includegraphics{figure_3} The diagram shows the graph of \(y = \text{g}(x)\). In the printed answer booklet, using the same scale as in this diagram, sketch the curves

1. \(y = \frac{3}{2}\text{g}(x)\), [2]
2. \(y = \text{g}\left(\frac{1}{2}x\right)\). [2]

In this question you must show detailed reasoning.

1. Show that \(\cos A + \sin A \tan A = \sec A\). [3]
2. Solve the equation \(\tan 2\theta = 3 \tan \theta\) for \(0° \leqslant \theta \leqslant 180°\). [7]

Points \(A\) and \(B\) have position vectors \(\mathbf{a}\) and \(\mathbf{b}\). Point \(C\) lies on \(AB\) such that \(AC : CB = p : 1\).

1. Show that the position vector of \(C\) is \(\frac{1}{p+1}(\mathbf{a} + p\mathbf{b})\). [3]

It is now given that \(\mathbf{a} = 2\mathbf{i} + 3\mathbf{j} - 4\mathbf{k}\) and \(\mathbf{b} = -6\mathbf{i} + 4\mathbf{j} + 12\mathbf{k}\), and that \(C\) lies on the \(y\)-axis.

1. Find the value of \(p\). [4]
2. Write down the position vector of \(C\). [1]

The table shows information about three geometric series. The three geometric series have different common ratios.

1. Find \(n_1\). [2]
2. Given that \(r_2\) is an integer less than 10, find the value of \(r_2\) and the value of \(n_2\). [2]
3. Given that \(r_3\) is not an integer, find a possible value for the sum of all the terms in Series 3. [4]

1. Show that, if \(n\) is a positive integer, then \((x^n - 1)\) is divisible by \((x - 1)\). [1]
2. Hence show that, if \(k\) is a positive integer, then \(2^{8k} - 1\) is divisible by 17. [4]

Use a suitable trigonometric substitution to find \(\int \frac{x^2}{\sqrt{1-x^2}} \text{d}x\). [7]

Research has shown that drug A is effective in 32% of patients with a certain disease. In a trial, drug B is given to a random sample of 1000 patients with the disease, and it is found that the drug is effective in 290 of these patients. Test at the 2.5% significance level whether there is evidence that drug B is effective in a lower proportion of patients than drug A. [7]

Using the 2001 UK census results and some software, Javid intended to calculate the mean number of people who travelled to work by underground, metro, light rail or tram (UMLT) for all 348 Local Authorities. However, Javid noticed that for one LA the entry in the UMLT column is a dash, rather than a 0. See the extract below. Javid felt that it was not clear how this LA was to be treated so he decided to omit it from his calculation.

1. Explain how the omission of this LA affects Javid's calculation of the mean. [1]

The value of the mean that Javid obtained was 2046.3.

1. Calculate the value of the mean when this LA is not removed. [2]

Javid finds that the corresponding mean for all Local Authorities for 2011 is 2860.8. In order to compare the means for the two years, Javid also finds the total number of employees in each of these years. His results are given below.

1. Show that a higher proportion of employees used the metro to travel to work in 2011 than in 2001. [2]
2. Suggest a reason for this increase. [1]

Laxmi wishes to test whether there is linear correlation between the mass and the height of adult males.

1. State, with a reason, whether Laxmi should use a 1-tail or a 2-tail test. [1]

Laxmi chooses a random sample of 40 adult males and calculates Pearson's product-moment correlation coefficient, \(r\). She finds that \(r = 0.2705\).

1. Use the table below to carry out the test at the 5% significance level. [5]

Critical values of Pearson's product-moment correlation coefficient.

Paul drew a cumulative frequency graph showing information about the numbers of people in various age-groups in a certain region X. He forgot to include the scale on the cumulative frequency axis, as shown below. \includegraphics{figure_12}

1. Find an estimate of the median age of the population of region X. [1]
2. Find an estimate of the proportion of people aged over 60 in region X. [2]

Sonika drew similar cumulative graphs for another two regions, Y and Z, but she included the scales on the cumulative frequency axes, as shown below. \includegraphics{figure_12b}

1. Find an age group, of width 20 years, in which region Z has approximately 3 times as many people as region Y. [1]
2. State one advantage and one disadvantage of using Sonika's two diagrams to compare the populations in Regions Y and Z. [2]
3. Without calculation state, with a reason, which of regions Y or Z has the greater proportion of people aged under 40. [1]

The marks of 24 students in a test had mean \(m\) and standard deviation \(\sqrt{6}\). Two new students took the same test. Their marks were \(m - 4\) and \(m + 4\). Show that the standard deviation of the marks of all 26 students is 2.60, correct to 3 significant figures. [3]

Mr Jones has 3 tins of beans and 2 tins of pears. His daughter has removed the labels for a school project, and the tins are identical in appearance. Mr Jones opens tins in turn until he has opened at least 1 tin of beans and at least 1 tin of pears. He does not open any remaining tins.

1. Draw a tree diagram to illustrate this situation, labelling each branch with its associated probability. [3]
2. Find the probability that Mr Jones opens exactly 3 tins. [3]
3. It is given that the last tin Mr Jones opens is a tin of pears. Find the probability that he opens exactly 3 tins. [5]

A fair dice is thrown 1000 times and the number, \(X\), of throws on which the score is 6 is noted.

1. 1. State the distribution of \(X\). [1]
   2. Explain why a normal distribution would be an appropriate approximation to the distribution of \(X\). [1]
2. Use a normal distribution to find two positive integer values, \(a\) and \(b\), such that \(\text{P}(a < X < b) \approx 0.4\). [5]
3. For your two values of \(a\) and \(b\), use the distribution of part (a)(i) to find the value of \(\text{P}(a < X < b)\), correct to 3 significant figures. [2]