AQA C2 (Core Mathematics 2) 2007 June

1

1. Simplify:
   1. \(x ^ { \frac { 3 } { 2 } } \times x ^ { \frac { 1 } { 2 } }\);
   2. \(x ^ { \frac { 3 } { 2 } } \div x\);
   3. \(\left( x ^ { \frac { 3 } { 2 } } \right) ^ { 2 }\).
2. 1. Find \(\int 3 x ^ { \frac { 1 } { 2 } } \mathrm {~d} x\).
   2. Hence find the value of \(\int _ { 1 } ^ { 9 } 3 x ^ { \frac { 1 } { 2 } } \mathrm {~d} x\).

2 The \(n\)th term of a geometric sequence is \(u _ { n }\), where $$u _ { n } = 3 \times 4 ^ { n }$$

1. Find the value of \(u _ { 1 }\) and show that \(u _ { 2 } = 48\).
2. Write down the common ratio of the geometric sequence.
3. 1. Show that the sum of the first 12 terms of the geometric sequence is \(4 ^ { k } - 4\), where \(k\) is an integer.
   2. Hence find the value of \(\sum _ { n = 2 } ^ { 12 } u _ { n }\).

3 The diagram shows a sector \(O A B\) of a circle with centre \(O\) and radius 20 cm . The angle between the radii \(O A\) and \(O B\) is \(\theta\) radians.
\includegraphics[max width=\textwidth, alt={}, center]{ad574bde-3bf1-45be-a454-9c723088b357-3_453_499_429_804} The length of the \(\operatorname { arc } A B\) is 28 cm .

1. Show that \(\theta = 1.4\).
2. Find the area of the sector \(O A B\).
3. The point \(D\) lies on \(O A\). The region bounded by the line \(B D\), the line \(D A\) and the arc \(A B\) is shaded.
   \includegraphics[max width=\textwidth, alt={}, center]{ad574bde-3bf1-45be-a454-9c723088b357-3_440_380_1372_806} The length of \(O D\) is 15 cm .
   1. Find the area of the shaded region, giving your answer to three significant figures.
      (3 marks)
   2. Use the cosine rule to calculate the length of \(B D\), giving your answer to three significant figures.
      (3 marks)

4 An arithmetic series has first term \(a\) and common difference \(d\).
The sum of the first 29 terms is 1102.

1. Show that \(a + 14 d = 38\).
2. The sum of the second term and the seventh term is 13 . Find the value of \(a\) and the value of \(d\).

5 A curve is defined for \(x > 0\) by the equation $$y = \left( 1 + \frac { 2 } { x } \right) ^ { 2 }$$ The point \(P\) lies on the curve where \(x = 2\).

1. Find the \(y\)-coordinate of \(P\).
2. Expand \(\left( 1 + \frac { 2 } { x } \right) ^ { 2 }\).
3. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
4. Hence show that the gradient of the curve at \(P\) is - 2 .
5. Find the equation of the normal to the curve at \(P\), giving your answer in the form \(x + b y + c = 0\), where \(b\) and \(c\) are integers.

6 The diagram shows a sketch of the curve with equation \(y = 3 \left( 2 ^ { x } + 1 \right)\).
\includegraphics[max width=\textwidth, alt={}, center]{ad574bde-3bf1-45be-a454-9c723088b357-5_465_851_390_607} The curve \(y = 3 \left( 2 ^ { x } + 1 \right)\) intersects the \(y\)-axis at the point \(A\).

1. Find the \(y\)-coordinate of the point \(A\).
2. Use the trapezium rule with four ordinates (three strips) to find an approximate value for \(\int _ { 0 } ^ { 6 } 3 \left( 2 ^ { x } + 1 \right) d x\).
3. The line \(y = 21\) intersects the curve \(y = 3 \left( 2 ^ { x } + 1 \right)\) at the point \(P\).
   1. Show that the \(x\)-coordinate of \(P\) satisfies the equation $$2 ^ { x } = 6$$
   2. Use logarithms to find the \(x\)-coordinate of \(P\), giving your answer to three significant figures.

7

1. Sketch the graph of \(y = \tan x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
2. Write down the two solutions of the equation \(\tan x = \tan 61 ^ { \circ }\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
3. 1. Given that \(\sin \theta + \cos \theta = 0\), show that \(\tan \theta = - 1\).
   2. Hence solve the equation \(\sin \left( x - 20 ^ { \circ } \right) + \cos \left( x - 20 ^ { \circ } \right) = 0\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
4. Describe the single geometrical transformation that maps the graph of \(y = \tan x\) onto the graph of \(y = \tan \left( x - 20 ^ { \circ } \right)\).
5. The curve \(y = \tan x\) is stretched in the \(x\)-direction with scale factor \(\frac { 1 } { 4 }\) to give the curve with equation \(y = \mathrm { f } ( x )\). Write down an expression for \(\mathrm { f } ( x )\).

8

1. It is given that \(n\) satisfies the equation $$\log _ { a } n = \log _ { a } 3 + \log _ { a } ( 2 n - 1 )$$ Find the value of \(n\).
2. Given that \(\log _ { a } x = 3\) and \(\log _ { a } y - 3 \log _ { a } 2 = 4\) :
   1. express \(x\) in terms of \(a\);
   2. express \(x y\) in terms of \(a\).