AQA C2 (Core Mathematics 2) 2009 January

1 The diagram shows a sector \(O A B\) of a circle with centre \(O\) and radius 10 cm .
\includegraphics[max width=\textwidth, alt={}, center]{0e19665b-5ee5-49e4-8de2-6c8dd17f61eb-2_371_378_555_824} The angle \(A O B\) is 0.8 radians.

1. Find the area of the sector.
2. 1. Find the perimeter of the sector \(O A B\).
   2. The perimeter of the sector \(O A B\) is equal to the perimeter of a square. Find the area of the square.

2

1. Use the trapezium rule with four ordinates (three strips) to find an approximate value for $$\int _ { 1.5 } ^ { 6 } x ^ { 2 } \sqrt { x ^ { 2 } - 1 } \mathrm {~d} x$$ giving your answer to three significant figures.
2. State how you could obtain a better approximation to the value of the integral using the trapezium rule.
   (1 mark)

3 The diagram shows a triangle \(A B C\). The size of angle \(A\) is \(63 ^ { \circ }\), and the lengths of \(A B\) and \(A C\) are 7.4 m and 5.26 m respectively.

1. Calculate the area of triangle \(A B C\), giving your answer in \(\mathrm { m } ^ { 2 }\) to three significant figures.
2. Show that the length of \(B C\) is 6.86 m , correct to three significant figures.
3. Find the value of \(\sin \boldsymbol { B }\) to two significant figures.

4 The diagram shows a sketch of the curves with equations \(y = 2 x ^ { \frac { 3 } { 2 } }\) and \(y = 8 x ^ { \frac { 1 } { 2 } }\).
\includegraphics[max width=\textwidth, alt={}, center]{0e19665b-5ee5-49e4-8de2-6c8dd17f61eb-3_433_720_1452_644} The curves intersect at the origin and at the point \(A\), where \(x = 4\).

1. 1. For the curve \(y = 2 x ^ { \frac { 3 } { 2 } }\), find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(x = 4\).
      (2 marks)
   2. Find an equation of the normal to the curve \(y = 2 x ^ { \frac { 3 } { 2 } }\) at the point \(A\).
2. 1. Find \(\int 8 x ^ { \frac { 1 } { 2 } } \mathrm {~d} x\).
   2. Find the area of the shaded region bounded by the two curves.
3. Describe a single geometrical transformation that maps the graph of \(y = 2 x ^ { \frac { 3 } { 2 } }\) onto the graph of \(y = 2 ( x + 3 ) ^ { \frac { 3 } { 2 } }\).
   (2 marks)

5

1. By using the binomial expansion, or otherwise, express \(( 1 + 2 x ) ^ { 4 }\) in the form $$1 + a x + b x ^ { 2 } + c x ^ { 3 } + 16 x ^ { 4 }$$ where \(a\), \(b\) and \(c\) are integers.
2. Hence show that \(( 1 + 2 x ) ^ { 4 } + ( 1 - 2 x ) ^ { 4 } = 2 + 48 x ^ { 2 } + 32 x ^ { 4 }\).
3. Hence show that the curve with equation $$y = ( 1 + 2 x ) ^ { 4 } + ( 1 - 2 x ) ^ { 4 }$$ has just one stationary point and state its coordinates.

6

1. Write each of the following in the form \(\log _ { a } k\), where \(k\) is an integer:
   1. \(\log _ { a } 4 + \log _ { a } 10\);
   2. \(\log _ { a } 16 - \log _ { a } 2\);
   3. \(3 \log _ { a } 5\).
2. Use logarithms to solve the equation \(( 1.5 ) ^ { 3 x } = 7.5\), giving your value of \(x\) to three decimal places.
3. Given that \(\log _ { 2 } p = m\) and \(\log _ { 8 } q = n\), express \(p q\) in the form \(2 ^ { y }\), where \(y\) is an expression in \(m\) and \(n\).

7

1. Solve the equation \(\sin x = 0.8\) in the interval \(0 \leqslant x \leqslant 2 \pi\), giving your answers in radians to three significant figures.
2. The diagram shows the graph of the curve \(y = \sin x , 0 \leqslant x \leqslant 2 \pi\) and the lines \(y = k\) and \(y = - k\).
   \includegraphics[max width=\textwidth, alt={}, center]{0e19665b-5ee5-49e4-8de2-6c8dd17f61eb-5_497_780_552_689} The line \(y = k\) intersects the curve at the points \(P\) and \(Q\), and the line \(y = - k\) intersects the curve at the points \(R\) and \(S\). The point \(M\) is the minimum point of the curve.
   1. Write down the coordinates of the point \(M\).
   2. The \(x\)-coordinate of \(P\) is \(\alpha\). Write down the \(x\)-coordinate of the point \(Q\) in terms of \(\pi\) and \(\alpha\).
   3. Find the length of \(R S\) in terms of \(\pi\) and \(\alpha\), giving your answer in its simplest form.
3. Sketch the graph of \(y = \sin 2 x\) for \(0 \leqslant x \leqslant 2 \pi\), indicating the coordinates of points where the graph intersects the \(x\)-axis and the coordinates of any maximum points.

8 The 25th term of an arithmetic series is 38 .
The sum of the first 40 terms of the series is 1250 .

1. Show that the common difference of this series is 1.5 .
2. Find the number of terms in the series which are less than 100 .