AQA C2 (Core Mathematics 2) 2010 January

1 The diagram shows a sector \(O A B\) of a circle with centre \(O\).
\includegraphics[max width=\textwidth, alt={}, center]{961ff4d6-b62a-4fab-8204-8a33a969d343-2_444_373_541_804} The radius of the circle is 15 cm and angle \(A O B = 1.2\) radians.

1. 1. Show that the area of the sector is \(135 \mathrm {~cm} ^ { 2 }\).
   2. Calculate the length of the arc \(A B\).
2. The point \(P\) lies on the radius \(O B\) such that \(O P = 10 \mathrm {~cm}\), as shown in the diagram below.
   \includegraphics[max width=\textwidth, alt={}, center]{961ff4d6-b62a-4fab-8204-8a33a969d343-2_449_378_1436_799} Calculate the perimeter of the shaded region bounded by \(A P , P B\) and the arc \(A B\), giving your answer to three significant figures.
   (5 marks)

2 At the point \(( x , y )\) on a curve, where \(x > 0\), the gradient is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 7 \sqrt { x ^ { 5 } } - 4$$

1. Write \(\sqrt { x ^ { 5 } }\) in the form \(x ^ { k }\), where \(k\) is a fraction.
2. Find \(\int \left( 7 \sqrt { x ^ { 5 } } - 4 \right) \mathrm { d } x\).
3. Hence find the equation of the curve, given that the curve passes through the point \(( 1,3 )\).

3

1. Find the value of \(x\) in each of the following:
   1. \(\quad \log _ { 9 } x = 0\);
   2. \(\quad \log _ { 9 } x = \frac { 1 } { 2 }\).
2. Given that $$2 \log _ { a } n = \log _ { a } 18 + \log _ { a } ( n - 4 )$$ find the possible values of \(n\).

4 An arithmetic series has first term \(a\) and common difference \(d\).
The sum of the first 31 terms of the series is 310 .

1. Show that \(a + 15 d = 10\).
2. Given also that the 21st term is twice the 16th term, find the value of \(d\).
3. The \(n\)th term of the series is \(u _ { n }\). Given that \(\sum _ { n = 1 } ^ { k } u _ { n } = 0\), find the value of \(k\).

5 A curve has equation \(y = \frac { 1 } { x ^ { 3 } } + 48 x\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Hence find the equation of each of the two tangents to the curve that are parallel to the \(x\)-axis.
3. Find an equation of the normal to the curve at the point \(( 1,49 )\).

6

1. Sketch the curve with equation \(y = 2 ^ { x }\), indicating the coordinates of any point where the curve intersects the coordinate axes.
2. 1. Use the trapezium rule with five ordinates (four strips) to find an approximate value for \(\int _ { 0 } ^ { 2 } 2 ^ { x } \mathrm {~d} x\), giving your answer to three significant figures.
   2. State how you could obtain a better approximation to the value of the integral using the trapezium rule.
3. Describe a geometrical transformation that maps the graph of \(y = 2 ^ { x }\) onto the graph of \(y = 2 ^ { x + 7 } + 3\).
4. The curve \(y = 2 ^ { x + k } + 3\) intersects the \(y\)-axis at the point \(A ( 0,8 )\). Show that \(k = \log _ { m } n\), where \(m\) and \(n\) are integers.

7

1. The first four terms of the binomial expansion of \(( 1 + 2 x ) ^ { 7 }\) in ascending powers of \(x\) are \(1 + a x + b x ^ { 2 } + c x ^ { 3 }\). Find the values of the integers \(a , b\) and \(c\).
2. Hence find the coefficient of \(x ^ { 3 }\) in the expansion of \(\left( 1 - \frac { 1 } { 2 } x \right) ^ { 2 } ( 1 + 2 x ) ^ { 7 }\).

8

1. Solve the equation \(\tan \left( x + 52 ^ { \circ } \right) = \tan 22 ^ { \circ }\), giving the values of \(x\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
2. 1. Show that the equation $$3 \tan \theta = \frac { 8 } { \sin \theta }$$ can be written as $$3 \cos ^ { 2 } \theta + 8 \cos \theta - 3 = 0$$
   2. Find the value of \(\cos \theta\) that satisfies the equation $$3 \cos ^ { 2 } \theta + 8 \cos \theta - 3 = 0$$
   3. Hence solve the equation $$3 \tan 2 x = \frac { 8 } { \sin 2 x }$$ giving all values of \(x\) to the nearest degree in the interval \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).