AQA C2 (Core Mathematics 2)

4 The triangle \(A B C\), shown in the diagram, is such that \(A C = 8 \mathrm {~cm} , C B = 12 \mathrm {~cm}\) and angle \(A C B = \theta\) radians. The area of triangle \(A B C = 20 \mathrm {~cm} ^ { 2 }\).

1. Show that \(\theta = 0.430\) correct to three significant figures.
2. Use the cosine rule to calculate the length of \(A B\), giving your answer to two significant figures.
3. The point \(D\) lies on \(C B\) such that \(A D\) is an arc of a circle centre \(C\) and radius 8 cm . The region bounded by the arc \(A D\) and the straight lines \(D B\) and \(A B\) is shaded in the diagram.
   \includegraphics[max width=\textwidth, alt={}, center]{48c5470e-6489-4b25-98a6-1b4e101ab01c-004_417_883_1436_557} Calculate, to two significant figures:
   1. the length of the \(\operatorname { arc } A D\);
   2. the area of the shaded region.

5 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 three terms of the sequence are given by $$u _ { 1 } = 200 \quad u _ { 2 } = 150 \quad u _ { 3 } = 120$$

1. Show that \(p = 0.6\) and find the value of \(q\).
2. Find the value of \(u _ { 4 }\).
3. The limit of \(u _ { n }\) as \(n\) tends to infinity is \(L\). Write down an equation for \(L\) and hence find the value of \(L\).

6

1. Describe the geometrical transformation that maps the curve with equation \(y = \sin x\) onto the curve with equation:
   1. \(y = 2 \sin x\);
   2. \(y = - \sin x\);
   3. \(\quad y = \sin \left( x - 30 ^ { \circ } \right)\).
2. Solve the equation \(\sin \left( \theta - 30 ^ { \circ } \right) = 0.7\), giving your answers to the nearest \(0.1 ^ { \circ }\) in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
3. Prove that \(( \cos x + \sin x ) ^ { 2 } + ( \cos x - \sin x ) ^ { 2 } = 2\).

8 A curve, drawn from the origin \(O\), crosses the \(x\)-axis at the point \(A ( 9,0 )\). Tangents to the curve at \(O\) and \(A\) meet at the point \(P\), as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{48c5470e-6489-4b25-98a6-1b4e101ab01c-006_763_879_466_577} The curve, defined for \(x \geqslant 0\), has equation $$y = x ^ { \frac { 3 } { 2 } } - 3 x$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. 1. Find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at the point \(O\) and hence write down an equation of the tangent at \(O\).
   2. Show that the equation of the tangent at \(A ( 9,0 )\) is \(2 y = 3 x - 27\).
   3. Hence find the coordinates of the point \(P\) where the two tangents meet.
3. Find \(\int \left( x ^ { \frac { 3 } { 2 } } - 3 x \right) \mathrm { d } x\).
4. Calculate the area of the shaded region bounded by the curve and the tangents \(O P\) and \(A P\).