OCR PURE 2066

1 It is given that \(\mathrm { f } ( x ) = 3 x - \frac { 5 } { x ^ { 3 } }\).
Find

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

2 The circle \(x ^ { 2 } + y ^ { 2 } - 4 x + k y + 12 = 0\) has radius 1.
Find the two possible values of the constant \(k\).

3 In this question you must show detailed reasoning.

1. The polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 2 x ^ { 3 } + 3 x ^ { 2 } - 8 x + 3\).
   1. Show that \(f ( 1 ) = 0\).
   2. Solve the equation \(\mathrm { f } ( x ) = 0\).
2. Hence solve the equation \(2 \sin ^ { 3 } \theta + 3 \sin ^ { 2 } \theta - 8 \sin \theta + 3 = 0\) for \(0 ^ { \circ } \leqslant \theta < 360 ^ { \circ }\).

4

1. Find the coordinates of the stationary points on the curve \(y = x ^ { 3 } - 6 x ^ { 2 } + 9 x\).
2. The equation \(x ^ { 3 } - 6 x ^ { 2 } + 9 x + k = 0\) has exactly one real root. Using your answers from part (a) or otherwise, find the range of possible values of \(k\).

5

1. Prove that the following statement is not true.
   \(m\) is an odd number greater than \(1 \Rightarrow m ^ { 2 } + 4\) is prime.
2. By considering separately the case when \(n\) is odd and the case when \(n\) is even, prove that the following statement is true.
   \(n\) is a positive integer \(\Rightarrow n ^ { 2 } + 1\) is not a multiple of 4 .

6
\includegraphics[max width=\textwidth, alt={}, center]{68f1107f-f188-4698-934e-8fd593b25418-4_442_661_840_260} The diagram shows triangle \(A B C\), with \(A B = x \mathrm {~cm} , A C = y \mathrm {~cm}\) and angle \(B A C = 60 ^ { \circ }\). It is given that the area of the triangle is \(( x + y ) \sqrt { 3 } \mathrm {~cm} ^ { 2 }\).

1. Show that \(4 x + 4 y = x y\). When the vertices of the triangle are placed on the circumference of a circle, \(A C\) is a diameter of the circle.
2. Determine the value of \(x\) and the value of \(y\).

7

1. Write down an expression for the gradient of the curve \(y = \mathrm { e } ^ { k x }\).
2. The line L is a tangent to the curve \(y = \mathrm { e } ^ { \frac { 1 } { 2 } x }\) at the point where \(x = 2\). Show that L passes through the point \(( 0,0 )\).
3. Find the coordinates of the point of intersection of the curves \(y = 3 \mathrm { e } ^ { x }\) and \(y = 1 - 2 \mathrm { e } ^ { \frac { 1 } { 2 } x }\).

8

1. Joseph drew a histogram to show information about one Local Authority. He used data from the "Age structure by LA 2011" tab in the large data set. The table shows an extract from the data that he used.

   Age group	0 to 4
   Frequency	2143

   Joseph used a scale of \(1 \mathrm {~cm} = 1000\) units on the frequency density axis. Calculate the height of the histogram block for the 0 to 4 class.
2. Magdalene wishes to draw a statistical diagram to illustrate some of the data from the "Method of travel by LA 2011" tab in the large data set. State why she cannot draw a histogram.

9 The table shows information about the number of days absent last year by students in class 2A at a certain school.

1. Calculate an estimate of the mean for these data.
2. Find the median of these data. The headteacher is writing a report on the numbers of absences at her school. She wishes to include a figure for the average number of absences in class 2A. A governor suggests that she should quote the mean. The class teacher suggests that she should quote the median, because it is lower than the mean.
3. Give another reason for using the median rather than the mean for the average number of absences in class 2A.

10 The table shows extracts from the "Method of travel by LA" tabs for 2001 and 2011 in the large data set.

1. In one of these four LAs a new tram system was opened in 2004. Suggest, with a reason taken from the data, which LA this could have been.
2. Julian suggests that the figures for "Bus, minibus or coach" for LA1 show that some new bus routes were probably introduced in this LA between 2001 and 2011. Use data from the table to comment on this suggestion.
3. In one of these four LAs a congestion charge on vehicles was introduced in 2003. Suggest, with a reason taken from the data, which LA this could have been.

11 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, a random sample of 450 patients with this disease was selected and the number \(X\) who experienced side effects within one year was noted.

1. State one assumption needed in order to use a binomial model for \(X\). It was found that 51 of the 450 patients experienced side effects within one year.
2. 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.

12 The Venn diagram shows the numbers of students studying various subjects, in a year group of 100 students.
\includegraphics[max width=\textwidth, alt={}, center]{68f1107f-f188-4698-934e-8fd593b25418-7_554_910_347_244} A student is chosen at random from the 100 students. Then another student is chosen from the remaining students. Find the probability that the first student studies History and the second student studies Geography but not Psychology. \section*{END OF QUESTION PAPER} \section*{OCR} Oxford Cambridge and RSA