AQA C2 (Core Mathematics 2) 2013 June

1 A geometric series has first term 80 and common ratio \(\frac { 1 } { 2 }\).

1. Find the third term of the series.
2. Find the sum to infinity of the series.
3. Find the sum of the first 12 terms of the series, giving your answer to two decimal places.

2 The diagram shows a sector \(O A B\) of a circle with centre \(O\).
\includegraphics[max width=\textwidth, alt={}, center]{f4f090a1-7e36-4993-a49e-b6e7e8589057-2_341_371_968_815} The radius of the circle is 20 cm and the angle \(A O B = 0.8\) radians.

1. Find the length of the arc \(A B\).
2. Find the area of the sector \(O A B\).
3. A line from \(B\) meets the radius \(O A\) at the point \(D\), as shown in the diagram below.
   \includegraphics[max width=\textwidth, alt={}, center]{f4f090a1-7e36-4993-a49e-b6e7e8589057-2_344_371_1747_815} The length of \(B D\) is 15 cm . Find the size of the obtuse angle \(O D B\), in radians, giving your answer to three significant figures.

3

1. 1. Using the binomial expansion, or otherwise, express \(( 2 + y ) ^ { 3 }\) in the form \(a + b y + c y ^ { 2 } + y ^ { 3 }\), where \(a , b\) and \(c\) are integers.
   2. Hence show that \(\left( 2 + x ^ { - 2 } \right) ^ { 3 } + \left( 2 - x ^ { - 2 } \right) ^ { 3 }\) can be expressed in the form \(p + q x ^ { - 4 }\), where \(p\) and \(q\) are integers.
2. 1. Hence find \(\int \left[ \left( 2 + x ^ { - 2 } \right) ^ { 3 } + \left( 2 - x ^ { - 2 } \right) ^ { 3 } \right] \mathrm { d } x\).
   2. Hence find the value of \(\int _ { 1 } ^ { 2 } \left[ \left( 2 + x ^ { - 2 } \right) ^ { 3 } + \left( 2 - x ^ { - 2 } \right) ^ { 3 } \right] \mathrm { d } x\).

4

1. Sketch the graph of \(y = 9 ^ { x }\), indicating the value of the intercept on the \(y\)-axis.
   (2 marks)
2. Use logarithms to solve the equation \(9 ^ { x } = 15\), giving your value of \(x\) to three significant figures.
3. The curve \(y = 9 ^ { x }\) is reflected in the \(y\)-axis to give the curve with equation \(y = \mathrm { f } ( x )\). Write down an expression for \(\mathrm { f } ( x )\).
   (l mark)

5

1. Use the trapezium rule with five ordinates (four strips) to find an approximate value for \(\int _ { 0 } ^ { 2 } \sqrt { 8 x ^ { 3 } + 1 } \mathrm {~d} x\), giving your answer to three significant figures.
2. Describe the single transformation that maps the graph of \(y = \sqrt { 8 x ^ { 3 } + 1 }\) onto the graph of \(y = \sqrt { x ^ { 3 } + 1 }\).
3. The curve with equation \(y = \sqrt { x ^ { 3 } + 1 }\) is translated by \(\left[ \begin{array} { c } 2
   - 0.7 \end{array} \right]\) to give the curve with equation \(y = \mathrm { g } ( x )\). Find the value of \(\mathrm { g } ( 4 )\).
   (3 marks)

6 A curve has the equation $$y = \frac { 12 + x ^ { 2 } \sqrt { x } } { x } , \quad x > 0$$

1. Express \(\frac { 12 + x ^ { 2 } \sqrt { x } } { x }\) in the form \(12 x ^ { p } + x ^ { q }\).
2. 1. Hence find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
   2. Find an equation of the normal to the curve at the point on the curve where \(x = 4\).
   3. The curve has a stationary point \(P\). Show that the \(x\)-coordinate of \(P\) can be written in the form \(2 ^ { k }\), where \(k\) is a rational number.

7 The \(n\)th term of a sequence is \(u _ { n }\). The sequence is defined by $$u _ { n + 1 } = p u _ { n } + q$$ where \(p\) and \(q\) are constants.
The first two terms of the sequence are given by \(u _ { 1 } = 96\) and \(u _ { 2 } = 72\).
The limit of \(u _ { n }\) as \(n\) tends to infinity is 24 .

1. Show that \(p = \frac { 2 } { 3 }\).
2. Find the value of \(u _ { 3 }\).

8

1. Given that \(\log _ { a } b = c\), express \(b\) in terms of \(a\) and \(c\).
2. By forming a quadratic equation, show that there is only one value of \(x\) which satisfies the equation \(2 \log _ { 2 } ( x + 7 ) - \log _ { 2 } ( x + 5 ) = 3\).

9

1. 1. On the axes given below, sketch the graph of \(y = \tan x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
   2. Solve the equation \(\tan x = - 1\), giving all values of \(x\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
2. 1. Given that \(6 \tan \theta \sin \theta = 5\), show that \(6 \cos ^ { 2 } \theta + 5 \cos \theta - 6 = 0\).
   2. Hence solve the equation \(6 \tan 3 x \sin 3 x = 5\), giving all values of \(x\) to the nearest degree in the interval \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
      \includegraphics[max width=\textwidth, alt={}, center]{f4f090a1-7e36-4993-a49e-b6e7e8589057-5_720_1367_806_390}