OCR C2 (Core Mathematics 2) 2013 January

1 The diagram shows triangle \(A B C\), with \(A C = 14 \mathrm {~cm} , B C = 10 \mathrm {~cm}\) and angle \(A B C = 63 ^ { \circ }\).

1. Find angle \(C A B\).
2. Find the length of \(A B\).

2 A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { 1 } = 7 \text { and } u _ { n + 1 } = u _ { n } + 4 \text { for } n \geqslant 1 .$$

1. Show that \(u _ { 17 } = 71\).
2. Show that \(\sum _ { n = 1 } ^ { 35 } u _ { n } = \sum _ { n = 36 } ^ { 50 } u _ { n }\).

3 A curve has an equation which satisfies \(\frac { \mathrm { d } y } { \mathrm {~d} x } = k x ( 2 x - 1 )\) for all values of \(x\). The point \(P ( 2,7 )\) lies on the curve and the gradient of the curve at \(P\) is 9 .

1. Find the value of the constant \(k\).
2. Find the equation of the curve.

4

1. Find the binomial expansion of \(( 2 + x ) ^ { 5 }\), simplifying the terms.
2. Hence find the coefficient of \(y ^ { 3 }\) in the expansion of \(\left( 2 + 3 y + y ^ { 2 } \right) ^ { 5 }\).

5

1. Show that the equation \(2 \sin x = \frac { 4 \cos x - 1 } { \tan x }\) can be expressed in the form $$6 \cos ^ { 2 } x - \cos x - 2 = 0 .$$
2. Hence solve the equation \(2 \sin x = \frac { 4 \cos x - 1 } { \tan x }\), giving all values of \(x\) between \(0 ^ { \circ }\) and \(360 ^ { \circ }\).

6

1. The first three terms of an arithmetic progression are \(2 x , x + 4\) and \(2 x - 7\) respectively. Find the value of \(x\).
2. The first three terms of another sequence are also \(2 x , x + 4\) and \(2 x - 7\) respectively.
   (a) Verify that when \(x = 8\) the terms form a geometric progression and find the sum to infinity in this case.
   (b) Find the other possible value of \(x\) that also gives a geometric progression.

7
\includegraphics[max width=\textwidth, alt={}, center]{87012792-fa63-4003-875d-b8e7739037f1-3_412_707_751_680} The diagram shows two circles of radius 7 cm with centres \(A\) and \(B\). The distance \(A B\) is 12 cm and the point \(C\) lies on both circles. The region common to both circles is shaded.

1. Show that angle \(C A B\) is 0.5411 radians, correct to 4 significant figures.
2. Find the perimeter of the shaded region.
3. Find the area of the shaded region.

8
\includegraphics[max width=\textwidth, alt={}, center]{87012792-fa63-4003-875d-b8e7739037f1-4_489_697_274_667} The diagram shows the curves \(y = \log _ { 2 } x\) and \(y = \log _ { 2 } ( x - 3 )\).

1. Describe the geometrical transformation that transforms the curve \(y = \log _ { 2 } x\) to the curve \(y = \log _ { 2 } ( x - 3 )\).
2. The curve \(y = \log _ { 2 } x\) passes through the point ( \(a , 3\) ). State the value of \(a\).
3. The curve \(y = \log _ { 2 } ( x - 3 )\) passes through the point ( \(b , 1.8\) ). Find the value of \(b\), giving your answer correct to 3 significant figures.
4. The point \(P\) lies on \(y = \log _ { 2 } x\) and has an \(x\)-coordinate of \(c\). The point \(Q\) lies on \(y = \log _ { 2 } ( x - 3 )\) and also has an \(x\)-coordinate of \(c\). Given that the distance \(P Q\) is 4 units find the exact value of \(c\).

9 The positive constant \(a\) is such that \(\int _ { a } ^ { 2 a } \frac { 2 x ^ { 3 } - 5 x ^ { 2 } + 4 } { x ^ { 2 } } \mathrm {~d} x = 0\).

1. Show that \(3 a ^ { 3 } - 5 a ^ { 2 } + 2 = 0\).
2. Show that \(a = 1\) is a root of \(3 a ^ { 3 } - 5 a ^ { 2 } + 2 = 0\), and hence find the other possible value of \(a\), giving your answer in simplified surd form.