AQA C2 (Core Mathematics 2) 2013 January

1 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]{bfe96138-9587-4efb-95c5-84c4d5eadfbe-2_382_351_379_826} The angle \(A O B\) is 1.25 radians. The perimeter of the sector is 39 cm .

1. Show that \(r = 12\).
2. Calculate the area of the sector \(O A B\).

2

1. Use the trapezium rule with five ordinates (four strips) to find an approximate value for $$\int _ { 1 } ^ { 5 } \frac { 1 } { x ^ { 2 } + 1 } \mathrm {~d} x$$ giving your answer to three significant figures.
2. 1. Find \(\int \left( x ^ { - \frac { 3 } { 2 } } + 6 x ^ { \frac { 1 } { 2 } } \right) \mathrm { d } x\), giving the coefficient of each term in its simplest form.
   2. Hence find the value of \(\int _ { 1 } ^ { 4 } \left( x ^ { - \frac { 3 } { 2 } } + 6 x ^ { \frac { 1 } { 2 } } \right) \mathrm { d } x\).

3 The diagram shows a triangle \(A B C\).
\includegraphics[max width=\textwidth, alt={}, center]{bfe96138-9587-4efb-95c5-84c4d5eadfbe-3_273_622_356_708} The lengths of \(A C\) and \(B C\) are 5 cm and 6 cm respectively.
The area of triangle \(A B C\) is \(12.5 \mathrm {~cm} ^ { 2 }\), and angle \(A C B\) is obtuse.

1. Find the size of angle \(A C B\), giving your answer to the nearest \(0.1 ^ { \circ }\).
2. Find the length of \(A B\), giving your answer to two significant figures.

4 Given that $$\log _ { a } N - \log _ { a } x = \frac { 3 } { 2 }$$ express \(x\) in terms of \(a\) and \(N\), giving your answer in a form not involving logarithms.
(3 marks)

5 The point \(P ( 2,8 )\) lies on a curve, and the point \(M\) is the only stationary point of the curve. The curve has equation \(y = 6 + 2 x - \frac { 8 } { x ^ { 2 } }\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Show that the normal to the curve at the point \(P ( 2,8 )\) has equation \(x + 4 y = 34\).
3. 1. Show that the stationary point \(M\) lies on the \(x\)-axis.
   2. Hence write down the equation of the tangent to the curve at \(M\).
4. The tangent to the curve at \(M\) and the normal to the curve at \(P\) intersect at the point \(T\). Find the coordinates of \(T\).

6

1. A geometric series begins \(420 + 294 + 205.8 + \ldots\).
   1. Show that the common ratio of the series is 0.7 .
   2. Find the sum to infinity of the series.
   3. Write the \(n\)th term of the series in the form \(p \times q ^ { n }\), where \(p\) and \(q\) are constants.
2. The first term of an arithmetic series is 240 and the common difference of the series is - 8 . The \(n\)th term of the series is \(u _ { n }\).
   1. Write down an expression for \(u _ { n }\).
   2. Given that \(u _ { k } = 0\), find the value of \(\sum _ { n = 1 } ^ { k } u _ { n }\).

7

1. Describe a geometrical transformation that maps the graph of \(y = 4 ^ { x }\) onto the graph of \(y = 3 \times 4 ^ { x }\).
2. Sketch the curve with equation \(y = 3 \times 4 ^ { x }\), indicating the value of the intercept on the \(y\)-axis.
3. The curve with equation \(y = 4 ^ { - x }\) intersects the curve \(y = 3 \times 4 ^ { x }\) at the point \(P\). Use logarithms to find the \(x\)-coordinate of \(P\), giving your answer to three significant figures.

8

1. Expand \(\left( 1 + \frac { 4 } { x } \right) ^ { 2 }\).
2. The first four terms of the binomial expansion of \(\left( 1 + \frac { x } { 4 } \right) ^ { 8 }\) in ascending powers of \(x\) are \(1 + a x + b x ^ { 2 } + c x ^ { 3 }\). Find the values of the constants \(a , b\) and \(c\).
3. Hence find the coefficient of \(x\) in the expansion of \(\left( 1 + \frac { 4 } { x } \right) ^ { 2 } \left( 1 + \frac { x } { 4 } \right) ^ { 8 }\).

9

1. Write down the two solutions of the equation \(\tan \left( x + 30 ^ { \circ } \right) = \tan 79 ^ { \circ }\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
   (2 marks)
2. Describe a single geometrical transformation that maps the graph of \(y = \tan x\) onto the graph of \(y = \tan \left( x + 30 ^ { \circ } \right)\).
3. 1. Given that \(5 + \sin ^ { 2 } \theta = ( 5 + 3 \cos \theta ) \cos \theta\), show that \(\cos \theta = \frac { 3 } { 4 }\).
   2. Hence solve the equation \(5 + \sin ^ { 2 } 2 x = ( 5 + 3 \cos 2 x ) \cos 2 x\) in the interval \(0 < x < 2 \pi\), giving your values of \(x\) in radians to three significant figures.