AQA C2 (Core Mathematics 2) 2010 June

1 The diagram shows a sector \(O A B\) of a circle with centre \(O\).
\includegraphics[max width=\textwidth, alt={}, center]{f9a7a4dd-f7fd-4135-8872-2c1270d46a14-2_383_472_566_778} The radius of the circle is 8 m and the angle \(A O B\) is 1.4 radians.

1. Find the area of the sector \(O A B\).
2. 1. Find the perimeter of the sector \(O A B\).
   2. The perimeter of the sector \(O A B\) is equal to the circumference of a circle of radius \(x \mathrm {~m}\). Calculate the value of \(x\) to three significant figures.

2 The \(n\)th term of a sequence is \(u _ { n }\).
The sequence is defined by $$u _ { n + 1 } = 6 + \frac { 2 } { 5 } u _ { n }$$ The first term of the sequence is given by \(u _ { 1 } = 2\).

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

3 The triangle \(A B C\), shown in the diagram, is such that \(A B = 6 \mathrm {~cm} , B C = 15 \mathrm {~cm}\), angle \(B A C = 150 ^ { \circ }\) and angle \(A C B = \theta\).
\includegraphics[max width=\textwidth, alt={}, center]{f9a7a4dd-f7fd-4135-8872-2c1270d46a14-4_376_867_406_584}

1. Show that \(\theta = 11.5 ^ { \circ }\), 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.

4

1. The expression \(\left( 1 - \frac { 1 } { x ^ { 2 } } \right) ^ { 3 }\) can be written in the form $$1 + \frac { p } { x ^ { 2 } } + \frac { q } { x ^ { 4 } } - \frac { 1 } { x ^ { 6 } }$$ Find the values of the integers \(p\) and \(q\).
2. 1. Hence find \(\int \left( 1 - \frac { 1 } { x ^ { 2 } } \right) ^ { 3 } \mathrm {~d} x\).
   2. Hence find the value of \(\int _ { \frac { 1 } { 2 } } ^ { 1 } \left( 1 - \frac { 1 } { x ^ { 2 } } \right) ^ { 3 } \mathrm {~d} x\).

      REFEREN	REFERENCE
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5

1. An infinite geometric series has common ratio \(r\).
   The first term of the series is 10 and its sum to infinity is 50 .
   1. Show that \(r = \frac { 4 } { 5 }\).
   2. Find the second term of the series.
2. The first and second terms of the geometric series in part (a) have the same values as the 4th and 8th terms respectively of an arithmetic series.
   1. Find the common difference of the arithmetic series.
   2. The \(n\)th term of the arithmetic series is \(u _ { n }\). Find the value of \(\sum _ { n = 1 } ^ { 40 } u _ { n }\).

6 A curve \(C\) has the equation $$y = \frac { x ^ { 3 } + \sqrt { x } } { x } , \quad x > 0$$

1. Express \(\frac { x ^ { 3 } + \sqrt { x } } { x }\) in the form \(x ^ { p } + x ^ { q }\).
2. 1. Hence find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
   2. Find an equation of the normal to the curve \(C\) at the point on the curve where \(x = 1\).
3. 1. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
   2. Hence deduce that the curve \(C\) has no maximum points.
      \includegraphics[max width=\textwidth, alt={}, center]{f9a7a4dd-f7fd-4135-8872-2c1270d46a14-7_1463_1707_1244_153}

7

1. Sketch the graph of \(y = \cos x\) in the interval \(0 \leqslant x \leqslant 2 \pi\). State the values of the intercepts with the coordinate axes.
2. 1. Given that $$\sin ^ { 2 } \theta = \cos \theta ( 2 - \cos \theta )$$ prove that \(\cos \theta = \frac { 1 } { 2 }\).
   2. Hence solve the equation $$\sin ^ { 2 } 2 x = \cos 2 x ( 2 - \cos 2 x )$$ in the interval \(0 \leqslant x \leqslant \pi\), giving your answers in radians to three significant figures.

8 The diagram shows a sketch of the curve \(y = 2 ^ { 4 x }\).
\includegraphics[max width=\textwidth, alt={}, center]{f9a7a4dd-f7fd-4135-8872-2c1270d46a14-9_435_814_374_623} The curve intersects the \(y\)-axis at the point \(A\).

1. Find the value of the \(y\)-coordinate of \(A\).
2. Use the trapezium rule with six ordinates (five strips) to find an approximate value for \(\int _ { 0 } ^ { 1 } 2 ^ { 4 x } \mathrm {~d} x\), giving your answer to two decimal places.
3. Describe the geometrical transformation that maps the graph of \(y = 2 ^ { 4 x }\) onto the graph of \(y = 2 ^ { 4 x - 3 }\).
4. The curve \(y = 2 ^ { 4 x }\) is translated by the vector \(\left[ \begin{array} { c } 1
   - \frac { 1 } { 2 } \end{array} \right]\) to give the curve \(y = \mathrm { g } ( x )\). The curve \(y = \mathrm { g } ( x )\) crosses the \(x\)-axis at the point \(Q\). Find the \(x\)-coordinate of \(Q\).
5. 1. Given that $$\log _ { a } k = 3 \log _ { a } 2 + \log _ { a } 5 - \log _ { a } 4$$ show that \(k = 10\).
   2. The line \(y = \frac { 5 } { 4 }\) crosses the curve \(y = 2 ^ { 4 x - 3 }\) at the point \(P\). Show that the \(x\)-coordinate of \(P\) is \(\frac { 1 } { 4 \log _ { 10 } 2 }\).