OCR C2 (Core Mathematics 2) 2012 January

1
\includegraphics[max width=\textwidth, alt={}, center]{ad3083ae-caa6-42d8-a1f2-e984150cb104-2_319_454_246_810} The diagram shows a sector \(A O B\) of a circle with centre \(O\) and radius 12 cm . The reflex angle \(A O B\) is 4.2 radians.

1. Find the perimeter of the sector.
2. Find the area of the sector.

2
\includegraphics[max width=\textwidth, alt={}, center]{ad3083ae-caa6-42d8-a1f2-e984150cb104-2_536_917_1016_577} The diagram shows the curve \(y = \log _ { 10 } ( 2 x + 1 )\).

1. Use the trapezium rule with 4 strips each of width 1.5 to find an approximation to the area of the region bounded by the curve, the \(x\)-axis and the lines \(x = 4\) and \(x = 10\). Give your answer correct to 3 significant figures.
2. Explain why this approximation is an under-estimate.

3 One of the terms in the binomial expansion of \(( 4 + a x ) ^ { 6 }\) is \(160 x ^ { 3 }\).

1. Find the value of \(a\).
2. Using this value of \(a\), find the first two terms in the expansion of \(( 4 + a x ) ^ { 6 }\) in ascending powers of \(x\).

4
\includegraphics[max width=\textwidth, alt={}, center]{ad3083ae-caa6-42d8-a1f2-e984150cb104-3_622_513_244_776} The diagram shows two points \(A\) and \(B\) on a straight coastline, with \(A\) being 2.4 km due north of \(B\). A stationary ship is at point \(C\), on a bearing of \(040 ^ { \circ }\) and at a distance of 2 km from \(B\).

1. Find the distance \(A C\), giving your answer correct to 3 significant figures.
2. Find the bearing of \(C\) from \(A\).
3. Find the shortest distance from the ship to the coastline.

5 The cubic polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 2 x ^ { 3 } + 3 x ^ { 2 } - 17 x + 6\).

1. Find the remainder when \(\mathrm { f } ( x )\) is divided by \(( x - 3 )\).
2. Given that \(\mathrm { f } ( 2 ) = 0\), express \(\mathrm { f } ( x )\) as the product of a linear factor and a quadratic factor.
3. Determine the number of real roots of the equation \(\mathrm { f } ( x ) = 0\), giving a reason for your answer.

6 A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by \(u _ { n } = 85 - 5 n\) for \(n \geqslant 1\).

1. Write down the values of \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\).
2. Find \(\sum _ { n = 1 } ^ { 20 } u _ { n }\).
3. Given that \(u _ { 1 } , u _ { 5 }\) and \(u _ { p }\) are, respectively, the first, second and third terms of a geometric progression, find the value of \(p\).
4. Find the sum to infinity of the geometric progression in part (iii).

7

1. Find \(\int \left( x ^ { 2 } + 4 \right) ( x - 6 ) \mathrm { d } x\).
2. 
   \includegraphics[max width=\textwidth, alt={}, center]{ad3083ae-caa6-42d8-a1f2-e984150cb104-4_449_551_349_758} The diagram shows the curve \(y = 6 x ^ { \frac { 3 } { 2 } }\) and part of the curve \(y = \frac { 8 } { x ^ { 2 } } - 2\), which intersect at the point \(( 1,6 )\). Use integration to find the area of the shaded region enclosed by the two curves and the \(x\)-axis.

8

1. Use logarithms to solve the equation \(7 ^ { w - 3 } - 4 = 180\), giving your answer correct to 3 significant figures.
2. Solve the simultaneous equations $$\log _ { 10 } x + \log _ { 10 } y = \log _ { 10 } 3 , \quad \log _ { 10 } ( 3 x + y ) = 1$$

9

1. Sketch the graph of \(y = \tan \left( \frac { 1 } { 2 } x \right)\) for \(- 2 \pi \leqslant x \leqslant 2 \pi\) on the axes provided.
   On the same axes, sketch the graph of \(y = 3 \cos \left( \frac { 1 } { 2 } x \right)\) for \(- 2 \pi \leqslant x \leqslant 2 \pi\), indicating the point of intersection with the \(y\)-axis.
2. Show that the equation \(\tan \left( \frac { 1 } { 2 } x \right) = 3 \cos \left( \frac { 1 } { 2 } x \right)\) can be expressed in the form $$3 \sin ^ { 2 } \left( \frac { 1 } { 2 } x \right) + \sin \left( \frac { 1 } { 2 } x \right) - 3 = 0$$ Hence solve the equation \(\tan \left( \frac { 1 } { 2 } x \right) = 3 \cos \left( \frac { 1 } { 2 } x \right)\) for \(- 2 \pi \leqslant x \leqslant 2 \pi\).