AQA C2 (Core Mathematics 2) 2015 June

1 The diagram shows a sector \(O A B\) of a circle with centre \(O\) and radius 5 cm .
\includegraphics[max width=\textwidth, alt={}, center]{24641e66-b73b-4323-98c8-349727151aba-02_378_451_648_790} The angle \(A O B\) is \(\theta\) radians and the area of the sector is \(15 \mathrm {~cm} ^ { 2 }\).
Find the perimeter of the sector.
[0pt] [4 marks]

2 The diagram shows a triangle \(A B C\). The size of angle \(B A C\) is \(72 ^ { \circ }\) and the size of angle \(A B C\) is \(48 ^ { \circ }\). The length of \(B C\) is 20 cm .

1. Show that the length of \(A C\) is 15.6 cm , correct to three significant figures.
2. The midpoint of \(B C\) is \(M\). Calculate the length of \(A M\), giving your answer, in cm , to three significant figures.
   [0pt] [4 marks]

3 The first term of an infinite geometric series is 48 . The common ratio of the series is 0.6 .

1. Find the third term of the series.
2. Find the sum to infinity of the series.
3. The \(n\)th term of the series is \(u _ { n }\). Find the value of \(\sum _ { n = 4 } ^ { \infty } u _ { n }\).

4 A curve is defined for \(x > 0\). The gradient of the curve at the point \(( x , y )\) is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 } { x ^ { 2 } } - \frac { x } { 4 }$$

1. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
2. The curve has a stationary point \(M\) whose \(y\)-coordinate is \(\frac { 5 } { 2 }\).
   1. Find the \(x\)-coordinate of \(M\).
   2. Use your answers to parts (a) and (b)(i) to show that \(M\) is a maximum point.
   3. Find the equation of the curve.

5 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 second term of the sequence is 160 . The third term of the sequence is 132 .
The limit of \(u _ { n }\) as \(n\) tends to infinity is 20 .

1. Find the value of \(p\) and the value of \(q\).
2. Hence find the value of the first term of the sequence.

6

1. Solve the equation \(\sin ( x + 0.7 ) = 0.6\) in the interval \(- \pi < x < \pi\), giving your answers in radians to two significant figures.
2. It is given that \(5 \cos ^ { 2 } \theta - \cos \theta = \sin ^ { 2 } \theta\).
   1. By forming and solving a suitable quadratic equation, find the possible values of \(\cos \theta\).
   2. Hence show that a possible value of \(\tan \theta\) is \(2 \sqrt { 2 }\).

7 The diagram shows a sketch of two curves.
\includegraphics[max width=\textwidth, alt={}, center]{24641e66-b73b-4323-98c8-349727151aba-14_448_527_370_762} The equations of the two curves are \(y = 1 + \sqrt { x }\) and \(y = 4 ^ { \frac { x } { 9 } }\).
The curves meet at the points \(P ( 0,1 )\) and \(Q ( 9,4 )\).

1. 1. Describe the geometrical transformation that maps the graph of \(y = \sqrt { x }\) onto the graph of \(y = 1 + \sqrt { x }\).
   2. Describe the geometrical transformation that maps the graph of \(y = 4 ^ { x }\) onto the graph of \(y = 4 ^ { \frac { x } { 9 } }\).
2. 1. Given that \(\int _ { 0 } ^ { 9 } \sqrt { x } \mathrm {~d} x = 18\), find the value of \(\int _ { 0 } ^ { 9 } ( 1 + \sqrt { x } ) \mathrm { d } x\).
   2. Use the trapezium rule with five ordinates (four strips) to find an approximate value for \(\int _ { 0 } ^ { 9 } 4 ^ { \frac { x } { 9 } } \mathrm {~d} x\). Give your answer to one decimal place.
   3. Hence find an approximate value for the area of the shaded region bounded by the two curves and state, with an explanation, whether your approximation will be an overestimate or an underestimate of the true value for the area of the shaded region.
      [0pt] [3 marks]

8 The point \(A\) lies on the curve with equation \(y = x ^ { \frac { 1 } { 2 } }\). The tangent to this curve at \(A\) is parallel to the line \(3 y - 2 x = 1\). Find an equation of this tangent at \(A\).
[0pt] [5 marks]

9

1. Use logarithms to solve the equation \(2 ^ { 3 x } = 5\), giving your value of \(x\) to three significant figures.
2. Given that \(\log _ { a } k - \log _ { a } 2 = \frac { 2 } { 3 }\), express \(a\) in terms of \(k\).
3. 1. By using the binomial expansion, or otherwise, express \(( 1 + 2 x ) ^ { 3 }\) in ascending powers of \(x\).
   2. It is given that $$\log _ { 2 } \left[ ( 1 + 2 n ) ^ { 3 } - 8 n \right] = \log _ { 2 } ( 1 + 2 n ) + \log _ { 2 } \left[ 4 \left( 1 + n ^ { 2 } \right) \right]$$ By forming and solving a suitable quadratic equation, find the possible values of \(n\). [5 marks] \includegraphics[max width=\textwidth, alt={}, center]{24641e66-b73b-4323-98c8-349727151aba-20_1581_1714_1126_153}
      \includegraphics[max width=\textwidth, alt={}, center]{24641e66-b73b-4323-98c8-349727151aba-24_2488_1728_219_141}