OCR C2 (Core Mathematics 2) 2006 June

1 Find the binomial expansion of \(( 3 x - 2 ) ^ { 4 }\).

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

1. Write down the values of \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\).
2. Find \(\sum _ { n = 1 } ^ { 100 } u _ { n }\).

3 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 x ^ { - \frac { 1 } { 2 } }\), and the curve passes through the point (4,5). Find the equation of the curve.

4
\includegraphics[max width=\textwidth, alt={}, center]{367db494-294e-4b53-b9e8-fd2a69fb6069-2_634_670_1123_740} The diagram shows the curve \(y = 4 - x ^ { 2 }\) and the line \(y = x + 2\).

1. Find the \(x\)-coordinates of the points of intersection of the curve and the line.
2. Use integration to find the area of the shaded region bounded by the line and the curve.

5 Solve each of the following equations, for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).

1. \(2 \sin ^ { 2 } x = 1 + \cos x\).
2. \(\sin 2 x = - \cos 2 x\).

6

1. John aims to pay a certain amount of money each month into a pension fund. He plans to pay \(\pounds 100\) in the first month, and then to increase the amount paid by \(\pounds 5\) each month, i.e. paying \(\pounds 105\) in the second month, \(\pounds 110\) in the third month, etc. If John continues making payments according to this plan for 240 months, calculate
   (a) how much he will pay in the final month,
   (b) how much he will pay altogether over the whole period.
2. Rachel also plans to pay money monthly into a pension fund over a period of 240 months, starting with \(\pounds 100\) in the first month. Her monthly payments will form a geometric progression, and she will pay \(\pounds 1500\) in the final month. Calculate how much Rachel will pay altogether over the whole period.

7
\includegraphics[max width=\textwidth, alt={}, center]{367db494-294e-4b53-b9e8-fd2a69fb6069-3_476_1018_1000_566} The diagram shows a triangle \(A B C\), and a sector \(A C D\) of a circle with centre \(A\). It is given that \(A B = 11 \mathrm {~cm} , B C = 8 \mathrm {~cm}\), angle \(A B C = 0.8\) radians and angle \(D A C = 1.7\) radians. The shaded segment is bounded by the line \(D C\) and the arc \(D C\).

1. Show that the length of \(A C\) is 7.90 cm , correct to 3 significant figures.
2. Find the area of the shaded segment.
3. Find the perimeter of the shaded segment.

8 The cubic polynomial \(2 x ^ { 3 } + a x ^ { 2 } + b x - 10\) is denoted by \(\mathrm { f } ( x )\). It is given that, when \(\mathrm { f } ( x )\) is divided by \(( x - 2 )\), the remainder is 12 . It is also given that ( \(x + 1\) ) is a factor of \(\mathrm { f } ( x )\).

1. Find the values of \(a\) and \(b\).
2. Divide \(\mathrm { f } ( x )\) by ( \(x + 2\) ) to find the quotient and the remainder.

9

1. Sketch the curve \(y = \left( \frac { 1 } { 2 } \right) ^ { x }\), and state the coordinates of any point where the curve crosses an axis.
2. Use the trapezium rule, with 4 strips of width 0.5 , to estimate the area of the region bounded by the curve \(y = \left( \frac { 1 } { 2 } \right) ^ { x }\), the axes, and the line \(x = 2\).
3. The point \(P\) on the curve \(y = \left( \frac { 1 } { 2 } \right) ^ { x }\) has \(y\)-coordinate equal to \(\frac { 1 } { 6 }\). Prove that the \(x\)-coordinate of \(P\) may be written as $$1 + \frac { \log _ { 10 } 3 } { \log _ { 10 } 2 }$$