OCR C2 (Core Mathematics 2) 2007 June

1 A geometric progression \(\mathrm { u } _ { 1 } , \mathrm { u } _ { 2 } , \mathrm { u } _ { 3 } , \ldots\) is defined by $$u _ { 1 } = 15 \quad \text { and } \quad u _ { n + 1 } = 0.8 u _ { n } \text { for } n \geqslant 1 .$$

1. Write down the values of \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\).
2. Find \(\sum _ { n = 1 } ^ { 20 } u _ { n }\).

2 Expand \(\left( x + \frac { 2 } { x } \right) ^ { 4 }\) completely, simplifying the terms.

3 U se logarithms to solve the equation \(3 ^ { 2 x + 1 } = 5 ^ { 200 }\), giving the value of \(x\) correct to 3 significant figures.

4
\includegraphics[max width=\textwidth, alt={}, center]{e429080f-8634-46bc-b451-7b13b871e518-2_543_857_1155_644} The diagram shows the curve \(\mathrm { y } = \sqrt { 4 \mathrm { X } + 1 }\).

1. Use the trapezium rule, with strips of width 0.5 , to find an approximate value for the area of the region bounded by the curve \(y = \sqrt { 4 x + 1 }\), the \(x\)-axis, and the lines \(x = 1\) and \(x = 3\). Give your answer correct to 3 significant figures.
2. State with a reason whether this approximation is an under-estimate or an over-estimate.

5

1. Show that the equation $$3 \cos ^ { 2 } \theta = \sin \theta + 1$$ can be expressed in the form $$3 \sin ^ { 2 } \theta + \sin \theta - 2 = 0$$
2. Hence solve the equation $$3 \cos ^ { 2 } \theta = \sin \theta + 1 ,$$ giving all values of \(\theta\) between \(0 ^ { \circ }\) and \(360 ^ { \circ }\).

6

1. 1. Find \(\int x \left( x ^ { 2 } - 4 \right) d x\)
   2. Hence evaluate \(\int _ { 1 } ^ { 6 } x \left( x ^ { 2 } - 4 \right) d x\).
2. Find \(\int \frac { 6 } { x ^ { 3 } } d x\)

7

1. In an arithmetic progression, the first term is 12 and the sum of the first 70 terms is 12915 . Find the common difference.
2. In a geometric progression, the second term is - 4 and the sum to infinity is 9 . Find the common ratio.

8
\includegraphics[max width=\textwidth, alt={}, center]{e429080f-8634-46bc-b451-7b13b871e518-3_300_744_1046_703} The diagram shows a triangle \(A B C\), where angle \(B A C\) is 0.9 radians. \(B A D\) is a sector of the circle with centre A and radius AB .

1. The area of the sector \(B A D\) is \(16.2 \mathrm {~cm} ^ { 2 }\). Show that the length of \(A B\) is 6 cm .
2. The area of triangle \(A B C\) is twice the area of sector \(B A D\). Find the length of \(A C\).
3. Find the perimeter of the region \(B C D\).

9 The polynomial \(f ( x )\) is given by $$f ( x ) = x ^ { 3 } + 6 x ^ { 2 } + x - 4 .$$

1. (a) Show that ( \(\mathrm { x } + 1\) ) is a factor of \(\mathrm { f } ( \mathrm { x } )\).
   (b) Hence find the exact roots of the equation \(f ( x ) = 0\).
2. (a) Show that the equation $$2 \log _ { 2 } ( x + 3 ) + \log _ { 2 } x - \log _ { 2 } ( 4 x + 2 ) = 1$$ can be written in the form \(f ( x ) = 0\).
   (b) Explain why the equation $$2 \log _ { 2 } ( x + 3 ) + \log _ { 2 } x - \log _ { 2 } ( 4 x + 2 ) = 1$$ has only one real root and state the exact value of this root.