OCR C2 (Core Mathematics 2) 2015 June

1 A geometric progression has first term 3 and second term - 6 .

1. State the value of the common ratio.
2. Find the value of the eleventh term.
3. Find the sum of the first twenty terms.

2

1. Use the trapezium rule, with 4 strips each of width 1.5, to estimate the value of $$\int _ { 4 } ^ { 10 } \sqrt { 2 x - 1 } \mathrm {~d} x ,$$ giving your answer correct to 3 significant figures.
2. Explain how the trapezium rule could be used to obtain a more accurate estimate.

3
\includegraphics[max width=\textwidth, alt={}, center]{6dd10d03-5fe2-4a70-b5a2-03347dff0360-2_576_599_1062_733} The diagram shows a sector \(A O B\) of a circle with centre \(O\) and radius 8 cm . The angle \(A O B\) is 1.2 radians. The points \(C\) and \(D\) lie on \(O A\) and \(O B\) respectively such that \(O C = 5.2 \mathrm {~cm}\) and \(O D = 2.6 \mathrm {~cm} . C D\) is a straight line.

1. Find the area of the shaded region \(A C D B\).
2. Find the perimeter of the shaded region \(A C D B\).

4

1. Find and simplify the first three terms in the binomial expansion of \(( 2 + a x ) ^ { 6 }\) in ascending powers of \(x\).
2. In the expansion of \(( 3 - 5 x ) ( 2 + a x ) ^ { 6 }\), the coefficient of \(x\) is 64 . Find the value of \(a\).

5 A curve has an equation which satisfies \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 3 x ^ { - \frac { 1 } { 2 } }\) for all positive values of \(x\). The point \(P ( 4,1 )\) lies on the curve, and the gradient of the curve at \(P\) is 5 . Find the equation of the curve.

6 The cubic polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = x ^ { 3 } - 19 x + 30\).

1. Given that \(x = 2\) is a root of the equation \(\mathrm { f } ( x ) = 0\), express \(\mathrm { f } ( x )\) as the product of 3 linear factors.
2. Use integration to find the exact value of \(\int _ { - 5 } ^ { 3 } \mathrm { f } ( x ) \mathrm { d } x\).
3. Explain with the aid of a sketch why the answer to part (ii) does not give the area enclosed by the curve \(y = \mathrm { f } ( x )\) and the \(x\)-axis for \(- 5 \leqslant x \leqslant 3\).

7 In an arithmetic progression the first term is 5 and the common difference is 3 . The \(n\)th term of the progression is denoted by \(u _ { n }\).

1. Find the value of \(u _ { 20 }\).
2. Show that \(\sum _ { n = 10 } ^ { 20 } u _ { n } = 517\).
3. Find the value of \(N\) such that \(\sum _ { n = N } ^ { 2 N } u _ { n } = 2750\).

8

1. Use logarithms to solve the equation $$2 ^ { n - 3 } = 18000$$ giving your answer correct to 3 significant figures.
2. Solve the simultaneous equations $$\log _ { 2 } x + \log _ { 2 } y = 8 , \quad \log _ { 2 } \left( \frac { x ^ { 2 } } { y } \right) = 7$$

9
\includegraphics[max width=\textwidth, alt={}, center]{6dd10d03-5fe2-4a70-b5a2-03347dff0360-4_406_625_248_721} The diagram shows part of the curve \(y = 2 \cos \frac { 1 } { 3 } x\), where \(x\) is in radians, and the line \(y = k\).

1. The smallest positive solution of the equation \(2 \cos \frac { 1 } { 3 } x = k\) is denoted by \(\alpha\). State, in terms of \(\alpha\),
   (a) the next smallest positive solution of the equation \(2 \cos \frac { 1 } { 3 } x = k\),
   (b) the smallest positive solution of the equation \(2 \cos \frac { 1 } { 3 } x = - k\).
2. The curve \(y = 2 \cos \frac { 1 } { 3 } x\) is shown in the Printed Answer Book. On the diagram, and for the same values of \(x\), sketch the curve of \(y = \sin \frac { 1 } { 3 } x\).
3. Calculate the \(x\)-coordinates of the points of intersection of the curves in part (ii). Give your answers in radians correct to 3 significant figures. \section*{END OF QUESTION PAPER}