AQA C2 (Core Mathematics 2) 2005 June

1 The diagram shows a triangle \(A B C\).
\includegraphics[max width=\textwidth, alt={}, center]{37627fc4-a90b-4f3b-9b10-0a9e200f8485-2_423_707_612_657} The lengths of \(A C\) and \(B C\) are 5 cm and 4.8 cm respectively.
The size of the angle \(B C A\) is \(30 ^ { \circ }\).

1. Calculate the area of the triangle \(A B C\).
2. Calculate the length of \(A B\), giving your answer to three significant figures.

2 The diagram shows a sector \(O A B\) of a circle with centre \(O\) and radius \(r \mathrm {~cm}\).
\includegraphics[max width=\textwidth, alt={}, center]{37627fc4-a90b-4f3b-9b10-0a9e200f8485-2_486_381_1686_739} The angle \(A O B\) is 1.5 radians. The perimeter of the sector is 56 cm .

1. Show that \(r = 16\).
2. Find the area of the sector.

3 The \(n\)th term of an arithmetic sequence is \(u _ { n }\), where $$u _ { n } = 90 - 3 n$$

1. Find the value of \(u _ { 1 }\) and the value of \(u _ { 2 }\).
2. Write down the common difference of the arithmetic sequence.
3. Given that \(\sum _ { n = 1 } ^ { k } u _ { n } = 0\), find the value of \(k\).

4 The diagram shows a curve \(C\) with equation \(y = \sqrt { x }\). The point \(O\) is the origin \(( 0,0 )\).
\includegraphics[max width=\textwidth, alt={}, center]{37627fc4-a90b-4f3b-9b10-0a9e200f8485-3_488_1136_1009_443} The region bounded by the curve \(C\), the \(x\)-axis and the vertical lines \(x = 1\) and \(x = 4\) is shown shaded in the diagram.

1. 1. Write \(\sqrt { x }\) in the form \(x ^ { p }\), where \(p\) is a constant.
   2. Find \(\int \sqrt { x } \mathrm {~d} x\).
   3. Hence find the area of the shaded region.
2. The point on \(C\) for which \(x = 4\) is \(P\). The tangent to \(C\) at the point \(P\) intersects the \(x\)-axis and the \(y\)-axis at the points \(A\) and \(B\) respectively.
   1. Find an equation for the tangent to the curve \(C\) at the point \(P\).
   2. Find the area of the triangle \(A O B\).
3. Describe the single geometrical transformation by which the curve with equation \(y = \sqrt { x - 1 }\) can be obtained from the curve \(C\).
4. Use the trapezium rule with four ordinates (three strips) to find an approximation for \(\int _ { 1 } ^ { 4 } \sqrt { x - 1 } \mathrm {~d} x\), giving your answer to three significant figures.

5 The sum to infinity of a geometric series is four times the first term of the series.

1. Show that the common ratio, \(r\), of the geometric series is \(\frac { 3 } { 4 }\).
2. The first term of the geometric series is 48 . Find the sum of the first 10 terms of the series, giving your answer to four decimal places.
3. The \(n\)th term of the geometric series is \(u _ { n }\) and the ( \(2 n\) )th term of the series is \(u _ { 2 n }\).
   1. Write \(u _ { n }\) and \(u _ { 2 n }\) in terms of \(n\).
   2. Hence show that \(\log _ { 10 } \left( u _ { n } \right) - \log _ { 10 } \left( u _ { 2 n } \right) = n \log _ { 10 } \left( \frac { 4 } { 3 } \right)\).
   3. Hence show that the value of $$\log _ { 10 } \left( \frac { u _ { 100 } } { u _ { 200 } } \right)$$ is 12.5 correct to three significant figures.

6

1. Using the binomial expansion, or otherwise, express \(( 1 + x ) ^ { 4 }\) in ascending powers of \(x\).
2. 1. Hence show that \(( 1 + \sqrt { 5 } ) ^ { 4 } = 56 + 24 \sqrt { 5 }\).
   2. Hence show that \(\log _ { 2 } ( 1 + \sqrt { 5 } ) ^ { 4 } = k + \log _ { 2 } ( 7 + 3 \sqrt { 5 } )\), where \(k\) is an integer.

7 A curve is defined, for \(x > 0\), by the equation \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = \frac { x ^ { 8 } - 1 } { x ^ { 3 } }$$

1. Express \(\frac { x ^ { 8 } - 1 } { x ^ { 3 } }\) in the form \(x ^ { p } - x ^ { q }\), where \(p\) and \(q\) are integers.
2. 1. Hence differentiate \(\mathrm { f } ( x )\) to find \(\mathrm { f } ^ { \prime } ( x )\).
   2. Hence show that f is an increasing function.
3. Find the gradient of the normal to the curve at the point \(( 1,0 )\).

8

1. 1. Show that the equation $$4 \tan \theta \sin \theta = 15$$ can be written as $$4 \sin ^ { 2 } \theta = 15 \cos \theta$$ (1 mark)
   2. Use an appropriate identity to show that the equation $$4 \sin ^ { 2 } \theta = 15 \cos \theta$$ can be written as $$4 \cos ^ { 2 } \theta + 15 \cos \theta - 4 = 0$$
2. 1. Solve the equation \(4 c ^ { 2 } + 15 c - 4 = 0\).
   2. Hence explain why the only value of \(\cos \theta\) which satisfies the equation $$4 \cos ^ { 2 } \theta + 15 \cos \theta - 4 = 0$$ is \(\cos \theta = \frac { 1 } { 4 }\).
   3. Hence solve the equation \(4 \tan \theta \sin \theta = 15\) giving all solutions to the nearest \(0.1 ^ { \circ }\) in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
3. Write down all the values of \(x\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 90 ^ { \circ }\) for which $$4 \tan 4 x \sin 4 x = 15$$ giving your answers to the nearest degree.