OCR MEI C3 (Core Mathematics 3)

1 John asserts that the expression \(n ^ { 2 } + n + 11\) is prime for all positive integer values of \(n\). Show that John is wrong in his assertion.

2

1. Show that \(\mathrm { f } ( x ) = \left| x ^ { 3 } \right|\) is an even function.
2. It is suggested that the function \(\mathrm { g } ( x ) = ( x - 1 ) ^ { 3 }\) is odd. Prove that this is false.

1. \(\quad y = 2 \mathrm { f } ( x )\),
2. \(y = \mathrm { f } ( 2 x )\).

4 The volume of a sphere, \(V \mathrm {~cm} ^ { 3 }\) is given by the formula \(V = \frac { 4 } { 3 } \pi r ^ { 3 }\) where \(r \mathrm {~cm}\) is the radius.
The radius of a sphere increases at a constant rate of 2 cm per second.
Find the rate of increase of \(V\) when \(r = 10 \mathrm {~cm}\).

5 The equation of a circle is \(x ^ { 2 } + y ^ { 2 } = 25\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { x } { y }\).
2. Hence find the equation of the normal to the circle at the point ( 3,4 ).

6

1. Find \(\int x \cos 2 x d x\).
2. Using the substitution \(u = x ^ { 2 } + 1\), or otherwise, find the exact value of \(\int _ { 2 } ^ { 3 } \frac { x } { x ^ { 2 } + 1 } \mathrm {~d} x\).

7 Fig. 7 shows the graphs of the curves \(y = \mathrm { e } ^ { - x }\) and \(y = \mathrm { e } ^ { - x } \sin x\) for \(0 \leq x \leq \pi\). \begin{figure}[h] \end{figure} The maximum point on \(y = \mathrm { e } ^ { - x } \sin x\) is at A , and the curves touch at B .
\(\mathrm { A } ^ { \prime }\) and \(\mathrm { B } ^ { \prime }\) are the points on the \(x\)-axis such that \(\mathrm { A } ^ { \prime } \mathrm { A }\) and \(\mathrm { B } ^ { \prime } \mathrm { B }\) are parallel to the \(y\)-axis.
Show that \(\mathrm { OA } ^ { \prime } = \mathrm { A } ^ { \prime } \mathrm { B } ^ { \prime }\).

8 Fig. 8 shows part of the graph of the function \(y = 5 x ( 2 x - 1 ) ^ { 3 }\). \begin{figure}[h] \end{figure}

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence find the \(x\)-coordinate of S , the turning point of the curve.
2. Find the area of the shaded region enclosed between the curve and the \(x\)-axis.
3. Given that \(\mathrm { f } ( x ) = 5 x ( 2 x - 1 ) ^ { 3 }\), show that \(\mathrm { f } ( x + 0.5 ) = 40 x ^ { 3 } ( x + 0.5 )\).
4. Find \(\int _ { - \frac { 1 } { 2 } } ^ { 0 } 40 x ^ { 3 } ( x + 0.5 ) \mathrm { d } x\).
5. Explain, with the aid of a sketch, the connection between your answer to parts (ii) and (iv).

9 Answer parts (ii) and (iii) of this question on the Insert provided. The bat population of a colony is being investigated and data are collected of the estimated number of bats in the colony at the beginning of each year. It is thought that the population may be modelled by the formula $$P = P _ { 0 } \mathrm { e } ^ { k t }$$ where \(P _ { 0 }\) and \(k\) are constants, \(P\) is the number of bats and \(t\) is the number of years after the start of the collection of data.

1. Explain why a graph of \(\ln P\) against \(t\) should give a straight line. State the gradient and intercept of this line.
2. The data collected are as follows.

   Time \(( t\) years \()\)	0	1	2	3	4
   Number of bats, \(P\)	100	170	300	340	360

   Using the first three pairs of data in the table, plot \(\ln P\) against \(t\) on the axes given on the Insert, and hence estimate values for \(P _ { 0 }\) and \(k\).
   (Work to three significant figures.) This model assumes exponential growth, and assumes that once born a bat does not die, continuing to reproduce. This is unrealistic and so a second model is proposed with formula $$P = 150 \arctan ( t - 1 ) + 170$$ (You are reminded that arctan values should be given in radians.)
3. Plot on a single graph on the Insert the curves \(P = P _ { 0 } \mathrm { e } ^ { k t }\) for your values of \(P _ { 0 }\) and \(k\) and \(P = 150 \arctan ( t - 1 ) + 170\). The data pairs in the table above have been plotted for you.
4. Using the second model calculate an estimate of the number of years it is before the bat population exceeds 375. \section*{Insert for question 3.}
5. Sketch the graph of \(y = 2 \mathrm { f } ( x )\)
   \includegraphics[max width=\textwidth, alt={}, center]{3853d1e7-ae1f-4eca-93c7-96f03b6d31c3-6_641_1431_541_354}
6. Sketch the graph of \(y = \mathrm { f } ( 2 x )\).
   \includegraphics[max width=\textwidth, alt={}, center]{3853d1e7-ae1f-4eca-93c7-96f03b6d31c3-6_691_1539_1468_374} \section*{Insert for question 9.}
7. Plot \(\ln P\) against \(t\).
   \includegraphics[max width=\textwidth, alt={}, center]{3853d1e7-ae1f-4eca-93c7-96f03b6d31c3-7_704_1442_443_338}
8. Plot the curves \(P = P _ { 0 } \mathrm { e } ^ { k t }\) and \(P = 150 \arctan ( t - 1 ) + 170\) for your values of \(P _ { 0 }\) and \(k\). The data pairs are plotted on the graph.
   \includegraphics[max width=\textwidth, alt={}, center]{3853d1e7-ae1f-4eca-93c7-96f03b6d31c3-7_780_1399_1546_333}