AQA C2 (Core Mathematics 2) 2014 June

1 The diagram shows a triangle \(A B C\). The size of angle \(B A C\) is \(47 ^ { \circ }\) and the lengths of \(A B\) and \(A C\) are 5 cm and 12 cm respectively.

1. Calculate the area of the triangle \(A B C\), giving your answer to the nearest \(\mathrm { cm } ^ { 2 }\).
2. Calculate the length of \(B C\), giving your answer, in cm , to one decimal place.
   [0pt] [3 marks]

2

1. Find \(\int \left( 1 + 3 x ^ { \frac { 1 } { 2 } } + x ^ { \frac { 3 } { 2 } } \right) \mathrm { d } x\).
2. 1. The expression \(( 1 + y ) ^ { 3 }\) can be written in the form \(1 + 3 y + n y ^ { 2 } + y ^ { 3 }\). Write down the value of the constant \(n\).
   2. Hence, or otherwise, expand \(( 1 + \sqrt { x } ) ^ { 3 }\).
3. Hence find the exact value of \(\int _ { 0 } ^ { 1 } ( 1 + \sqrt { x } ) ^ { 3 } \mathrm {~d} x\).

3 The first term of a geometric series is 54 and the common ratio of the series is \(\frac { 8 } { 9 }\).

1. Find the sum to infinity of the series.
2. Find the second term of the series.
3. Show that the 12th term of the series can be written in the form \(\frac { 2 ^ { p } } { 3 ^ { q } }\), where \(p\) and \(q\) are integers.
   [0pt] [3 marks]

4 A curve has equation \(y = \frac { 1 } { x ^ { 2 } } + 4 x\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. The point \(P ( - 1 , - 3 )\) lies on the curve. Find an equation of the normal to the curve at the point \(P\).
3. Find an equation of the tangent to the curve that is parallel to the line \(y = - 12 x\).
   [0pt] [5 marks]

5 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]{30ccdbe9-0c91-4011-a3f9-3ce01862215d-10_346_360_360_824} The angle \(A O B\) is \(\theta\) radians.
The area of the sector is \(12 \mathrm {~cm} ^ { 2 }\).
The perimeter of the sector is four times the length of the \(\operatorname { arc } A B\).
Find the value of \(r\).
[0pt] [6 marks]

6

1. Sketch, on the axes given below, the graph of \(y = \sin x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
2. Describe the geometrical transformation that maps the graph of \(y = \sin x\) onto the graph of \(y = \sin 5 x\).
3. Describe the single geometrical transformation that maps the graph of \(y = \sin 5 x\) onto the graph of \(y = \sin \left( 5 x + 10 ^ { \circ } \right)\).
   [0pt] [2 marks]
4. 
   \includegraphics[max width=\textwidth, alt={}, center]{30ccdbe9-0c91-4011-a3f9-3ce01862215d-12_675_1417_906_370}

7

1. Given that \(\frac { \cos ^ { 2 } x + 4 \sin ^ { 2 } x } { 1 - \sin ^ { 2 } x } = 7\), show that \(\tan ^ { 2 } x = \frac { 3 } { 2 }\).
2. Hence solve the equation \(\frac { \cos ^ { 2 } 2 \theta + 4 \sin ^ { 2 } 2 \theta } { 1 - \sin ^ { 2 } 2 \theta } = 7\) in the interval \(0 ^ { \circ } < \theta < 180 ^ { \circ }\), giving your values of \(\theta\) to the nearest degree.
   [0pt] [4 marks]

8 An arithmetic series has first term \(a\) and common difference \(d\). The sum of the first 5 terms of the series is 575 .

1. Show that \(a + 2 d = 115\).
2. Given also that the 10th term of the series is 87, find the value of \(d\).
3. The \(n\)th term of the series is \(u _ { n }\). Given that \(u _ { k } > 0\) and \(u _ { k + 1 } < 0\), find the value of \(\sum _ { n = 1 } ^ { k } u _ { n }\).
   [0pt] [5 marks]

9 A curve has equation \(y = 3 \times 12 ^ { x }\).

1. The point ( \(k , 6\) ) lies on the curve \(y = 3 \times 12 ^ { x }\). Use logarithms to find the value of \(k\), giving your answer to three significant figures.
2. Use the trapezium rule with four ordinates (three strips) to find an approximate value for \(\int _ { 0 } ^ { 1.5 } 3 \times 12 ^ { x } \mathrm {~d} x\), giving your answer to two significant figures.
3. The curve \(y = 3 \times 12 ^ { x }\) is translated by the vector \(\left[ \begin{array} { l } 1
   p \end{array} \right]\) to give the curve \(y = \mathrm { f } ( x )\). Given that the curve \(y = \mathrm { f } ( x )\) passes through the origin ( 0,0 ), find the value of the constant \(p\).
4. The curve with equation \(y = 2 ^ { 2 - x }\) intersects the curve \(y = 3 \times 12 ^ { x }\) at the point \(T\). Show that the \(x\)-coordinate of \(T\) can be written in the form \(\frac { 2 - \log _ { 2 } 3 } { q + \log _ { 2 } 3 }\), where \(q\) is an integer. State the value of \(q\).
   [0pt] [5 marks]
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