AQA C2 (Core Mathematics 2) 2011 June

1 The triangle \(A B C\), shown in the diagram, is such that \(A C = 9 \mathrm {~cm} , B C = 10 \mathrm {~cm}\), angle \(A B C = 54 ^ { \circ }\) and the acute angle \(B A C = \theta\).

1. Show that \(\theta = 64 ^ { \circ }\), correct to the nearest degree.
2. Calculate the area of triangle \(A B C\), giving your answer to the nearest square centimetre.

2 The diagram shows a sector \(O A B\) of a circle with centre \(O\).
\includegraphics[max width=\textwidth, alt={}, center]{258f0400-6e3b-406c-9f86-acc9fff4e094-2_440_392_1500_826} The radius of the circle is 6 cm and the angle \(A O B = 0.5\) radians.

1. Find the area of the sector \(O A B\).
2. 1. Find the length of the arc \(A B\).
   2. Hence show that the perimeter of the sector \(O A B = k \times\) the length of the \(\operatorname { arc } A B\) where \(k\) is an integer.

3

1. The expression \(\left( 2 + x ^ { 2 } \right) ^ { 3 }\) can be written in the form $$8 + p x ^ { 2 } + q x ^ { 4 } + x ^ { 6 }$$ Show that \(p = 12\) and find the value of the integer \(q\).
2. 1. Hence find \(\int \frac { \left( 2 + x ^ { 2 } \right) ^ { 3 } } { x ^ { 4 } } \mathrm {~d} x\).
      (5 marks)
   2. Hence find the exact value of \(\int _ { 1 } ^ { 2 } \frac { \left( 2 + x ^ { 2 } \right) ^ { 3 } } { x ^ { 4 } } \mathrm {~d} x\).
      (2 marks)

4

1. Sketch the curve with equation \(y = 4 ^ { x }\), indicating the coordinates of any point where the curve intersects the coordinate axes.
   (2 marks)
2. Describe the geometrical transformation that maps the graph of \(y = 4 ^ { x }\) onto the graph of \(y = 4 ^ { x } - 5\).
3. 1. Use the substitution \(Y = 2 ^ { x }\) to show that the equation \(4 ^ { x } - 2 ^ { x + 2 } - 5 = 0\) can be written as \(Y ^ { 2 } - 4 Y - 5 = 0\).
   2. Hence show that the equation \(4 ^ { x } - 2 ^ { x + 2 } - 5 = 0\) has only one real solution. Use logarithms to find this solution, giving your answer to three decimal places.
      (4 marks)

5 The diagram shows part of a curve with a maximum point \(M\).
\includegraphics[max width=\textwidth, alt={}, center]{258f0400-6e3b-406c-9f86-acc9fff4e094-4_480_645_354_694} The curve is defined for \(x \geqslant 0\) by the equation $$y = 6 x - 2 x ^ { \frac { 3 } { 2 } }$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
   (3 marks)
2. 1. Hence find the coordinates of the maximum point \(M\).
   2. Write down the equation of the normal to the curve at \(M\).
3. The point \(P \left( \frac { 9 } { 4 } , \frac { 27 } { 4 } \right)\) lies on the curve.
   1. Find an equation of the normal to the curve at the point \(P\), giving your answer in the form \(a x + b y = c\), where \(a , b\) and \(c\) are positive integers.
   2. The normals to the curve at the points \(M\) and \(P\) intersect at the point \(R\). Find the coordinates of \(R\).
      \(6 \quad\) A curve \(C\), defined for \(0 \leqslant x \leqslant 2 \pi\) by the equation \(y = \sin x\), where \(x\) is in radians, is sketched below. The region bounded by the curve \(C\), the \(x\)-axis from 0 to 2 and the line \(x = 2\) is shaded.
      \includegraphics[max width=\textwidth, alt={}, center]{258f0400-6e3b-406c-9f86-acc9fff4e094-5_441_789_466_612}

1. The area of the shaded region is given by \(\int _ { 0 } ^ { 2 } \sin x \mathrm {~d} x\), where \(x\) is in radians. Use the trapezium rule with five ordinates (four strips) to find an approximate value for the area of the shaded region, giving your answer to three significant figures.
2. Describe the geometrical transformation that maps the graph of \(y = \sin x\) onto the graph of \(y = 2 \sin x\).
3. Use a trigonometrical identity to solve the equation $$2 \sin x = \cos x$$ in the interval \(0 \leqslant x \leqslant 2 \pi\), giving your solutions in radians to three significant figures.

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 \(u _ { 1 } = 60\) and \(u _ { 2 } = 48\).
The limit of \(u _ { n }\) as \(n\) tends to infinity is 12 .

1. Show that \(p = \frac { 3 } { 4 }\) and find the value of \(q\).
2. Find the value of \(u _ { 3 }\).

8 Prove that, for all values of \(x\), the value of the expression $$( 3 \sin x + \cos x ) ^ { 2 } + ( \sin x - 3 \cos x ) ^ { 2 }$$ is an integer and state its value.

9 The first term of a geometric series is 12 and the common ratio of the series is \(\frac { 3 } { 8 }\).

1. Find the sum to infinity of the series.
2. Show that the sixth term of the series can be written in the form \(\frac { 3 ^ { 6 } } { 2 ^ { 13 } }\).
3. The \(n\)th term of the series is \(u _ { n }\).
   1. Write down an expression for \(u _ { n }\) in terms of \(n\).
   2. Hence show that $$\log _ { a } u _ { n } = n \log _ { a } 3 - ( 3 n - 5 ) \log _ { a } 2$$ (4 marks)