OCR C2 (Core Mathematics 2) 2006 January

1 The 20th term of an arithmetic progression is 10 and the 50th term is 70 .

1. Find the first term and the common difference.
2. Show that the sum of the first 29 terms is zero.

2 Triangle \(A B C\) has \(A B = 10 \mathrm {~cm} , B C = 7 \mathrm {~cm}\) and angle \(B = 80 ^ { \circ }\). Calculate

1. the area of the triangle,
2. the length of \(C A\),
3. the size of angle \(C\).

3

1. Find the first three terms of the expansion, in ascending powers of \(x\), of \(( 1 - 2 x ) ^ { 12 }\).
2. Hence find the coefficient of \(x ^ { 2 }\) in the expansion of $$( 1 + 3 x ) ( 1 - 2 x ) ^ { 12 } .$$

4
\includegraphics[max width=\textwidth, alt={}, center]{58680cd3-8744-42ee-83d4-35056592b2d0-2_647_797_1323_680} The diagram shows a sector \(O A B\) of a circle with centre \(O\). The angle \(A O B\) is 1.8 radians. The points \(C\) and \(D\) lie on \(O A\) and \(O B\) respectively. It is given that \(O A = O B = 20 \mathrm {~cm}\) and \(O C = O D = 15 \mathrm {~cm}\). The shaded region is bounded by the arcs \(A B\) and \(C D\) and by the lines \(C A\) and \(D B\).

1. Find the perimeter of the shaded region.
2. Find the area of the shaded region.

5 In a geometric progression, the first term is 5 and the second term is 4.8 .

1. Show that the sum to infinity is 125 .
2. The sum of the first \(n\) terms is greater than 124 . Show that $$0.96 ^ { n } < 0.008$$ and use logarithms to calculate the smallest possible value of \(n\).

6

1. Find \(\int \left( x ^ { \frac { 1 } { 2 } } + 4 \right) \mathrm { d } x\).
2. 1. Find the value, in terms of \(a\), of \(\int _ { 1 } ^ { a } 4 x ^ { - 2 } \mathrm {~d} x\), where \(a\) is a constant greater than 1 .
   2. Deduce the value of \(\int _ { 1 } ^ { \infty } 4 x ^ { - 2 } \mathrm {~d} x\).

7

1. Express each of the following in terms of \(\log _ { 10 } x\) and \(\log _ { 10 } y\).
   (a) \(\log _ { 10 } \left( \frac { x } { y } \right)\)
   (b) \(\log _ { 10 } \left( 10 x ^ { 2 } y \right)\)
2. Given that $$2 \log _ { 10 } \left( \frac { x } { y } \right) = 1 + \log _ { 10 } \left( 10 x ^ { 2 } y \right)$$ find the value of \(y\) correct to 3 decimal places.

8 The cubic polynomial \(2 x ^ { 3 } + k x ^ { 2 } - x + 6\) is denoted by \(\mathrm { f } ( x )\). It is given that \(( x + 1 )\) is a factor of \(\mathrm { f } ( x )\).

1. Show that \(k = - 5\), and factorise \(\mathrm { f } ( x )\) completely.
2. Find \(\int _ { - 1 } ^ { 2 } f ( x ) \mathrm { d } x\).
3. Explain with the aid of a sketch why the answer to part (ii) does not give the area of the region between the curve \(y = \mathrm { f } ( x )\) and the \(x\)-axis for \(- 1 \leqslant x \leqslant 2\). \section*{[Question 9 is printed overleaf.]}