Pre-U Pre-U 9794/2 (Pre-U Mathematics Paper 2) 2013 June

1 Vectors \(\mathbf { u }\) and \(\mathbf { v }\) are given by \(\mathbf { u } = \binom { 4 } { 6 }\) and \(\mathbf { v } = \binom { - 3 } { 2 }\).

1. Find \(\mathbf { u } + \mathbf { v }\) and \(\mathbf { u } - \mathbf { v }\).
2. Show that \(| \mathbf { u } + \mathbf { v } | = | \mathbf { u } - \mathbf { v } |\).

2

1. An arithmetic sequence has first term 3 and common difference 2. Find the twenty-first term of this sequence.
2. Find the sum to infinity of a geometric progression with first term 162 and second term 54.
3. A sequence is given by the recurrence relation \(u _ { 1 } = 3 , u _ { n + 1 } = 2 - u _ { n } , n = 1,2,3 , \ldots\). Find \(u _ { 2 } , u _ { 3 }\), \(u _ { 4 } , u _ { 5 }\) and describe the behaviour of this sequence.

3

1. Express \(x ^ { 2 } + 2 x - 3\) in the form \(( x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are integers to be found.
2. Sketch the graph of \(y = x ^ { 2 } + 2 x - 3\) giving the coordinates of the vertex and of any intersections with the coordinate axes.

4

1. Verify that \(z = - 1\) is a root of the equation \(z ^ { 3 } + 5 z ^ { 2 } + 9 z + 5 = 0\).
2. Find the two complex roots of the equation \(z ^ { 3 } + 5 z ^ { 2 } + 9 z + 5 = 0\).
3. Show all three roots on an Argand diagram.

5 The curve \(C\) has equation \(x ^ { 2 } + x y + y ^ { 2 } = 19\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { - 2 x - y } { x + 2 y }\).
2. Hence find the equation of the normal to \(C\) at the point \(( 2,3 )\) in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

6 The table below gives the population of breeding pairs of red kites in Yorkshire from 2001 to 2008. Source: \href{http://www.gigrin.co.uk}{www.gigrin.co.uk}
The following model for the population has been proposed: $$N = a \times b ^ { t } ,$$ where \(N\) is the number of breeding pairs \(t\) years after the year 2000, and \(a\) and \(b\) are constants.

1. Show that the model can be transformed to a linear relationship between \(\log _ { 10 } N\) and \(t\).
2. On graph paper, plot \(\log _ { 10 } N\) against \(t\) and draw by eye a line of best fit. Use your line to estimate the values of \(a\) and \(b\) in the equation for \(N\) in terms of \(t\).
3. What values of \(N\) does the model give for the years 2008 and 2020?
4. In which year will the number of breeding pairs first exceed 500 according to the model?
5. Comment on the suitability of the model to predict the population of breeding pairs of red kites in Yorkshire.

7 It is given that \(y = x ^ { 2 } \mathrm { e } ^ { - x }\).

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = x \mathrm { e } ^ { - x } ( 2 - x )\).
2. Hence find the exact coordinates of the stationary points on the curve \(y = x ^ { 2 } \mathrm { e } ^ { - x }\).

8 Evaluate the following, giving your answers in exact form.

1. \(\sum _ { n = 1 } ^ { 30 } \frac { 1 } { n } - \sum _ { n = 2 } ^ { 29 } \frac { 1 } { n }\).
2. \(\sum _ { n = 1 } ^ { 100 } n \times ( - 1 ) ^ { n }\).

9

1. Prove that \(\operatorname { cosec } 2 x - \cot 2 x \equiv \tan x\) and hence find an exact value for \(\tan \left( \frac { 3 } { 8 } \pi \right)\).
2. Find the exact value of \(\int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 3 } { 8 } \pi } ( \operatorname { cosec } 2 x - \cot 2 x ) ^ { 2 } \mathrm {~d} x\).

10 A tank with vertical sides and rectangular cross-section is initially full of water. The water is leaking out of a hole in the base of the tank at a rate which is proportional to the square root of the depth of the water. \(V \mathrm {~m} ^ { 3 }\) is the volume of water in the tank at time \(t\) hours.

1. Show that \(\frac { \mathrm { d } V } { \mathrm {~d} t } = a \sqrt { V }\), where \(a\) is a constant.
2. Given that the tank is half full after one hour, show that \(V = V _ { 0 } \left( \left( \frac { 1 } { \sqrt { 2 } } - 1 \right) t + 1 \right) ^ { 2 }\), where \(V _ { 0 } \mathrm {~m} ^ { 3 }\) is the initial volume of water in the tank.
3. Hence show that the tank will be empty after approximately 3 hours and 25 minutes.