OCR AS Pure (AS Pure Mathematics) 2017 Specimen

1 Given that \(\mathrm { f } ( x ) = 6 x ^ { 3 } - 5 x\), find

1. \(\mathrm { f } ^ { \prime } ( x )\),
2. \(f ^ { \prime \prime } ( 2 )\).

2 Points \(A\) and \(B\) have coordinates \(( 3,0 )\) and \(( 9,8 )\) respectively. The line \(A B\) is a diameter of a circle.

1. Find the coordinates of the centre of the circle.
2. Find the equation of the tangent to the circle at the point \(B\).

3 The points \(P , Q\) and \(R\) have coordinates \(( - 1,6 ) , ( 2,10 )\) and \(( 11,1 )\) respectively. Find the angle \(P R Q\).

4 The curve \(y = 2 x ^ { 3 } + 3 x ^ { 2 } - k x + 4\) has a stationary point where \(x = 2\).

1. Determine the value of the constant \(k\).
2. Determine whether this stationary point is a maximum or a minimum point.

5

1. Find \(\int \left( x ^ { 3 } - 6 x \right) \mathrm { d } x\).
2. 1. Find \(\int \left( \frac { 4 } { x ^ { 2 } } - 1 \right) \mathrm { d } x\).
   2. The diagram shows part of the curve \(y = \frac { 4 } { x ^ { 2 } } - 1\).
      \includegraphics[max width=\textwidth, alt={}, center]{35d8bb6d-ff0f-4590-b13d-46e4869e2587-04_707_1283_708_415} The curve crosses the \(x\)-axis at \(( 2,0 )\).
      The shaded region is bounded by the curve, the \(x\)-axis, and the lines \(x = 1\) and \(x = 5\). Calculate the area of the shaded region.

6 In this question you must show detailed reasoning. The cubic polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 4 x ^ { 3 } + 4 x ^ { 2 } + 7 x - 5\).

1. Show that \(( 2 x - 1 )\) is a factor of \(\mathrm { f } ( x )\).
2. Hence solve the equation \(4 \sin ^ { 3 } \theta + 4 \sin ^ { 2 } \theta + 7 \sin \theta - 5 = 0\) for \(0 ^ { \circ } \leq \theta \leq 360 ^ { \circ }\).

7

1. Sketch the curve \(y = 2 x ^ { 2 } - x - 3\).
2. Hence, or otherwise, solve \(2 x ^ { 2 } - x - 3 < 0\).
3. Given that the equation \(2 x ^ { 2 } - x - 3 = k\) has no real roots, find the set of possible values of k .

8 A club secretary wishes to survey a sample of members of his club. He uses all members present at a particular meeting as his sample.

1. Explain why this sample is likely to be biased. Later the secretary decides to choose a random sample of members.
   The club has 253 members and the secretary numbers the members from 1 to 253 . He then generates random 3-digit numbers on his calculator. The first six random numbers generated are 156, 965, 248, 156, 073 and 181. The secretary uses each number, where possible, as the number of a member in the sample.
2. Find possible numbers for the first four members in the sample.

9 The probability distribution of a random variable \(X\) is given in the table. Two values of \(X\) are chosen at random. Find the probability that the second value is greater than the first.

10

1. Write down and simplify the first four terms in the expansion of \(( x + y ) ^ { 7 }\).
   Give your answer in ascending powers of \(x\).
2. Given that the terms in \(x ^ { 2 } y ^ { 5 }\) and \(x ^ { 3 } y ^ { 4 }\) in this expansion are equal, find the value of \(\frac { x } { y }\).
3. A hospital consultant has seven appointments every day. The number of these appointments which start late on a randomly chosen day is denoted by \(L\).
   The variable \(L\) is modelled by the distribution \(\mathrm { B } \left( 7 , \frac { 3 } { 8 } \right)\). Show that, in this model, the hospital consultant is equally likely to have two appointments start late or three appointments start late.

11 The scatter diagram below shows data taken from the 2011 UK census for each of the Local Authorities in the North East and North West regions.
The scatter diagram shows the total population of the Local Authority and the proportion of its workforce that travel to work by bus, minibus or coach.
\includegraphics[max width=\textwidth, alt={}, center]{35d8bb6d-ff0f-4590-b13d-46e4869e2587-07_938_1136_664_260}

1. Samuel suggests that, with a few exceptions, the data points in the diagram show that Local Authorities with larger populations generally have higher proportions of workers travelling by bus, minibus or coach. On the diagram in the Printed Answer Booklet draw a ring around each of the data points that Samuel might regard as an exception.
2. Jasper suggests that it is possible to separate these Local Authorities into more than one group with different relationships between population and proportion travelling to work by bus, minibus or coach. Discuss Jasper's suggestion, referring to the data and to how differences between the Local Authorities could explain the patterns seen in the diagram.
   [0pt] [3]

12 It is known that under the standard treatment for a certain disease, \(9.7 \%\) of patients with the disease experience side effects within one year. In a trial of a new treatment, 450 patients with this disease were selected and the number, \(X\), that experienced side effects within one year was noted. It was found that 51 of the 450 patients experienced side effects within one year.

1. Test, at the \(10 \%\) significance level, whether the proportion of patients experiencing side effects within one year is greater under the new treatment than under the standard treatment.
2. It was later discovered that all 450 patients selected for the trial were treated in the same hospital. Comment on the validity of the model used in part (a).