AQA C2 (Core Mathematics 2) 2011 January

1 The diagram shows a sector \(O A B\) of a circle with centre \(O\) and radius 5 cm .
\includegraphics[max width=\textwidth, alt={}, center]{1c06ba04-575c-4eb8-b4aa-0a7510838cd2-02_415_525_550_794} The angle between the radii \(O A\) and \(O B\) is \(\theta\) radians.
The length of the \(\operatorname { arc } A B\) is 4 cm .

1. Find the value of \(\theta\).
2. Find the area of the sector \(O A B\).

2

1. Write down the values of \(p , q\) and \(r\) given that:
   1. \(8 = 2 ^ { p }\);
   2. \(\frac { 1 } { 8 } = 2 ^ { q }\);
   3. \(\sqrt { 2 } = 2 ^ { r }\).
2. Find the value of \(x\) for which \(\sqrt { 2 } \times 2 ^ { x } = \frac { 1 } { 8 }\).

3 The triangle \(A B C\), shown in the diagram, is such that \(A B = 5 \mathrm {~cm} , A C = 8 \mathrm {~cm}\), \(B C = 10 \mathrm {~cm}\) and angle \(B A C = \theta\).

1. Show that \(\theta = 97.9 ^ { \circ }\), correct to the nearest \(0.1 ^ { \circ }\).
2. 1. Calculate the area of triangle \(A B C\), giving your answer, in \(\mathrm { cm } ^ { 2 }\), to three significant figures.
   2. The line through \(A\), perpendicular to \(B C\), meets \(B C\) at the point \(D\). Calculate the length of \(A D\), giving your answer, in cm , to three significant figures.

4

1. Use the trapezium rule with four ordinates (three strips) to find an approximate value for \(\int _ { 0 } ^ { 1.5 } \sqrt { 27 x ^ { 3 } + 4 } \mathrm {~d} x\), giving your answer to three significant figures.
2. The curve with equation \(y = \sqrt { 27 x ^ { 3 } + 4 }\) is stretched parallel to the \(x\)-axis with scale factor 3 to give the curve with equation \(y = \mathrm { g } ( x )\). Write down an expression for \(\mathrm { g } ( x )\).
   (2 marks)
   \includegraphics[max width=\textwidth, alt={}]{1c06ba04-575c-4eb8-b4aa-0a7510838cd2-05_1988_1717_719_150}

5

1. Using the binomial expansion, or otherwise, express \(( 1 - x ) ^ { 3 }\) in ascending powers of \(x\).
2. Show that the expansion of $$( 1 + y ) ^ { 4 } - ( 1 - y ) ^ { 3 }$$ is $$7 y + p y ^ { 2 } + q y ^ { 3 } + y ^ { 4 }$$ where \(p\) and \(q\) are constants to be found.
3. Hence find \(\int \left[ ( 1 + \sqrt { x } ) ^ { 4 } - ( 1 - \sqrt { x } ) ^ { 3 } \right] \mathrm { d } x\), expressing each coefficient in its simplest form.

6 A geometric series has third term 36 and sixth term 972.

1. 1. Show that the common ratio of the series is 3 .
   2. Find the first term of the series.
2. The \(n\)th term of the series is \(u _ { n }\).
   1. Show that \(\sum _ { n = 1 } ^ { 20 } u _ { n } = 2 \left( 3 ^ { 20 } - 1 \right)\).
   2. Find the least value of \(n\) such that \(u _ { n } > 4 \times 10 ^ { 15 }\).
      \(7 \quad\) A curve \(C\) is defined for \(x > 0\) by the equation \(y = x + 3 + \frac { 8 } { x ^ { 4 } }\) and is sketched below.
      \includegraphics[max width=\textwidth, alt={}, center]{1c06ba04-575c-4eb8-b4aa-0a7510838cd2-08_602_799_447_632}

1. Given that \(y = x + 3 + \frac { 8 } { x ^ { 4 } }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Find an equation of the tangent at the point on the curve \(C\) where \(x = 1\).
3. The curve \(C\) has a minimum point \(M\). Find the coordinates of \(M\).
4. 1. Find \(\int \left( x + 3 + \frac { 8 } { x ^ { 4 } } \right) \mathrm { d } x\).
   2. Hence find the area of the region bounded by the curve \(C\), the \(x\)-axis and the lines \(x = 1\) and \(x = 2\).
5. The curve \(C\) is translated by \(\left[ \begin{array} { l } 0
   k \end{array} \right]\) to give the curve \(y = \mathrm { f } ( x )\). Given that the \(x\)-axis is a tangent to the curve \(y = \mathrm { f } ( x )\), state the value of the constant \(k\).
   (1 mark)

8

1. Given that \(2 \log _ { k } x - \log _ { k } 5 = 1\), express \(k\) in terms of \(x\). Give your answer in a form not involving logarithms.
2. Given that \(\log _ { a } y = \frac { 3 } { 2 }\) and that \(\log _ { 4 } a = b + 2\), show that \(y = 2 ^ { p }\), where \(p\) is an expression in terms of \(b\).
   \includegraphics[max width=\textwidth, alt={}]{1c06ba04-575c-4eb8-b4aa-0a7510838cd2-09_2102_1717_605_150}

9

1. Solve the equation \(\tan x = - 3\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\), giving your answers to the nearest degree.
2. 1. Given that $$7 \sin ^ { 2 } \theta + \sin \theta \cos \theta = 6$$ show that $$\tan ^ { 2 } \theta + \tan \theta - 6 = 0$$
   2. Hence solve the equation \(7 \sin ^ { 2 } \theta + \sin \theta \cos \theta = 6\) in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\), giving your answers to the nearest degree.
      (4 marks)