AQA C2 (Core Mathematics 2) 2009 June

1 The triangle \(A B C\), shown in the diagram, is such that \(A B = 7 \mathrm {~cm} , A C = 5 \mathrm {~cm}\), \(B C = 8 \mathrm {~cm}\) and angle \(A B C = \theta\).

1. Show that \(\theta = 38.2 ^ { \circ }\), correct to the nearest \(0.1 ^ { \circ }\).
2. Calculate the area of triangle \(A B C\), giving your answer, in \(\mathrm { cm } ^ { 2 }\), to three significant figures.

2

1. Write down the value of \(n\) given that \(\frac { 1 } { x ^ { 4 } } = x ^ { n }\).
2. Expand \(\left( 1 + \frac { 3 } { x ^ { 2 } } \right) ^ { 2 }\).
3. Hence find \(\int \left( 1 + \frac { 3 } { x ^ { 2 } } \right) ^ { 2 } \mathrm {~d} x\).
4. Hence find the exact value of \(\int _ { 1 } ^ { 3 } \left( 1 + \frac { 3 } { x ^ { 2 } } \right) ^ { 2 } \mathrm {~d} x\).

3 The \(n\)th term of a sequence is \(u _ { n }\).
The sequence is defined by $$u _ { n + 1 } = k u _ { n } + 12$$ where \(k\) is a constant.
The first two terms of the sequence are given by $$u _ { 1 } = 16 \quad u _ { 2 } = 24$$

1. Show that \(k = 0.75\).
2. Find the value of \(u _ { 3 }\) and the value of \(u _ { 4 }\).
3. The limit of \(u _ { n }\) as \(n\) tends to infinity is \(L\).
   1. Write down an equation for \(L\).
   2. Hence find the value of \(L\).

5 The diagram shows part of a curve with a maximum point \(M\).
\includegraphics[max width=\textwidth, alt={}, center]{22f2da99-0878-48a6-a2b7-1ba339d3c7e4-06_472_791_358_630} The equation of the curve is $$y = 15 x ^ { \frac { 3 } { 2 } } - x ^ { \frac { 5 } { 2 } }$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Hence find the coordinates of the maximum point \(M\).
3. The point \(P ( 1,14 )\) lies on the curve. Show that the equation of the tangent to the curve at \(P\) is \(y = 20 x - 6\).
4. The tangents to the curve at the points \(P\) and \(M\) intersect at the point \(R\). Find the length of \(R M\).

6 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]{22f2da99-0878-48a6-a2b7-1ba339d3c7e4-07_447_424_360_790} The angle \(A O B\) is 1.2 radians. The area of the sector is \(33.75 \mathrm {~cm} ^ { 2 }\).
Find the perimeter of the sector.

7 A geometric series has second term 375 and fifth term 81.

1. 1. Show that the common ratio of the series is 0.6 .
   2. Find the first term of the series.
2. Find the sum to infinity of the series.
3. The \(n\)th term of the series is \(u _ { n }\). Find the value of \(\sum _ { n = 6 } ^ { \infty } u _ { n }\).
   □ .......... \(\_\_\_\_\)

8

1. Given that \(\frac { \sin \theta - \cos \theta } { \cos \theta } = 4\), prove that \(\tan \theta = 5\).
2. 1. Use an appropriate identity to show that the equation $$2 \cos ^ { 2 } x - \sin x = 1$$ can be written as $$2 \sin ^ { 2 } x + \sin x - 1 = 0$$
   2. Hence solve the equation $$2 \cos ^ { 2 } x - \sin x = 1$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

      PART	REFERENCE
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      	\includegraphics[max width=\textwidth, alt={}]{22f2da99-0878-48a6-a2b7-1ba339d3c7e4-09_33_1698_2682_155}

9

1. 1. Find the value of \(p\) for which \(\sqrt { 125 } = 5 ^ { p }\).
   2. Hence solve the equation \(5 ^ { 2 x } = \sqrt { 125 }\).
2. Use logarithms to solve the equation \(3 ^ { 2 x - 1 } = 0.05\), giving your value of \(x\) to four decimal places.
3. It is given that $$\log _ { a } x = 2 \left( \log _ { a } 3 + \log _ { a } 2 \right) - 1$$ Express \(x\) in terms of \(a\), giving your answer in a form not involving logarithms.
   (4 marks)