OCR PURE

1

1. Find \(\frac { \mathrm { d } } { \mathrm { d } x } \left( x ^ { 3 } - 3 x + \frac { 5 } { x ^ { 2 } } \right)\).
2. Find \(\int \left( 6 x ^ { 2 } - \frac { 2 } { x ^ { 3 } } \right) \mathrm { d } x\).

2 Points \(A\) and \(B\) have position vectors \(\binom { - 3 } { 4 }\) and \(\binom { 1 } { 2 }\) respectively.
Point \(C\) has position vector \(\binom { p } { 1 }\) and \(A B C\) is a straight line.

1. Find \(p\). Point \(D\) has position vector \(\binom { q } { 1 }\) and angle \(A B D = 90 ^ { \circ }\).
2. Determine the value of \(q\).

3 In this question you must show detailed reasoning.

1. Solve the equation \(4 \sin ^ { 2 } \theta = \tan ^ { 2 } \theta\) for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
2. Prove that \(\frac { \sin ^ { 2 } \theta - 1 + \cos \theta } { 1 - \cos \theta } \equiv \cos \theta \quad ( \cos \theta \neq 1 )\).

4

1. Expand \(( 1 + x ) ^ { 4 }\).
2. Use your expansion to determine the exact value of \(1002 ^ { 4 }\).

5 The function f is defined by \(\mathrm { f } ( x ) = ( x + a ) ( x + 3 a ) ( x - b )\) where \(a\) and \(b\) are positive integers.

1. On the axes in the Printed Answer Booklet, sketch the curve \(y = \mathrm { f } ( x )\).
2. On your sketch show, in terms of \(a\) and \(b\), the coordinates of the points where the curve meets the axes. It is now given that \(a = 1\) and \(b = 4\).
3. Find the total area enclosed between the curve \(y = \mathrm { f } ( x )\) and the \(x\)-axis.

6 In this question you must show detailed reasoning.

1. Solve the inequality \(x ^ { 2 } + x - 6 > 0\), giving your answer in set notation.
2. Solve the equation \(x ^ { 3 } - 7 x ^ { \frac { 3 } { 2 } } - 8 = 0\).
3. Find the exact solution of the equation \(\left( 3 ^ { x } \right) ^ { 2 } = 3 \times 2 ^ { x }\).

7 Determine the points of intersection of the curve \(3 x y + x ^ { 2 } + 14 = 0\) and the line \(x + 2 y = 4\).

8 The histogram shows information about the lengths, \(l\) centimetres, of a sample of worms of a certain species.
\includegraphics[max width=\textwidth, alt={}, center]{4c6b7c92-2fc9-4d4f-a199-8e70f34e5eed-5_904_1284_488_242} The number of worms in the sample with lengths in the class \(3 \leqslant l < 4\) is 30 .

1. Find the number of worms in the sample with lengths in the class \(0 \leqslant l < 2\).
2. Find an estimate of the number of worms in the sample with lengths in the range \(4.5 \leqslant l < 5.5\).

9 A researcher is studying changes in behaviour in travelling to work by people who live outside London, between 2001 and 2011. He chooses the 15 Local Authorities (LAs) outside London with the largest decreases in the percentage of people driving to work, and arranges these in descending order. The table shows the changes in percentages from 2001 to 2011 in various travel categories, for these Local Authorities.

1. Explain why these LAs are not necessarily the 15 LAs with the largest decreases in the percentage of people driving to work.
2. The researcher wants to talk to those LAs outside London which have been most successful in encouraging people to change to cycling or walking to work.
   Suggest four LAs that he should talk to and why.
3. The researcher claims that Waverley is the LA outside London which has had the largest increase in the number of people working mainly at or from home.
   Does the data support his claim? Explain your answer.
4. Which two categories have replaced driving to work for the highest percentages of workers in these LAs? Support your answer with evidence from the table.
5. The researcher suggested that there would be strong correlation between the decrease in the percentage driving to work and the increase in percentage working mainly at or from home. Without calculation, use data from the table to comment briefly on this suggestion.

10 Some packets of a certain kind of biscuit contain a free gift. The manufacturer claims that the proportion of packets containing a free gift is 1 in 4 . Marisa suspects that this claim is not true, and that the true proportion is less than 1 in 4 . She chooses 20 packets at random and finds that exactly 1 contains the free gift.

1. Use a binomial model to test the manufacturer's claim, at the \(2.5 \%\) significance level. The packets are packed in boxes, with each box containing 40 packets. Marisa chooses three boxes at random and finds that one box contains 19 packets with the free gift and the other two boxes contain no packets with the free gift.
2. Give a reason why this suggests that the binomial model used in part (a) may not be appropriate.

11 In this question you must show detailed reasoning. A biased four-sided spinner has edges numbered \(1,2,3,4\). When the spinner is spun, the probability that it will land on the edge numbered \(X\) is given by
\(P ( X = x ) = \begin{cases} \frac { 1 } { 2 } - \frac { 1 } { 10 } x & x = 1,2,3,4 ,
0 & \text { otherwise } . \end{cases}\)

1. Draw a table showing the probability distribution of \(X\). The spinner is spun three times and the value of \(X\) is noted each time.
2. Find the probability that the third value of \(X\) is greater than the sum of the first two values of \(X\).