OCR MEI C2 (Core Mathematics 2)

1 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 - 5 x\).
Find the equation of the curve given that it passes through the point \(( 0,1 )\).

2

1. Write \(\log _ { 2 } 5 + \log _ { 2 } 1.6\) as an integer.
2. Solve the equation \(2 ^ { x } = 3\), giving your answer correct to 4 decimal places.

3 On his \(1 ^ { \text {st } }\) birthday, John was given \(\pounds 5\) by his Uncle Fred. On each succeeding birthday, Uncle Fred gave a sum of money that was \(\pounds 3\) more than the amount he gave on the last birthday.

1. How much did Uncle Fred give John on his \(8 { } ^ { \text {th } }\) birthday?
2. On what birthday did the gift from Uncle Fred result in the total sum given on all birthdays exceeding £200?

4 Find the equation of the tangent to the curve \(y = x ^ { 3 } + 2 x - 7\) at the point where it cuts the \(y\) axis.

5

1. Express \(2 \sin ^ { 2 } \theta + 3 \cos \theta\) as a quadratic function of \(\cos \theta\).
2. Hence solve the equation \(2 \sin ^ { 2 } \theta + 3 \cos \theta = 3\), giving all values of \(\theta\) correct to the nearest degree in the range \(0 ^ { \circ } \leq \theta \leq 360 ^ { \circ }\).

6 The angle of a sector of a circle is 2 radians and the length of the arc of the sector is 45 cm .
Find

1. the radius of the circle,
2. the area of the sector.

7 The first two terms of a geometric series are 5 and 4.
Find

1. the sum of the first 10 terms,
2. the sum to infinity.

8 In the triangle \(\mathrm { ABC } , \mathrm { AB } = 5 \mathrm {~cm} , \mathrm { AC } = 6 \mathrm {~cm}\) and angle \(\mathrm { BAC } = 110 ^ { \circ }\).
Find the length of the side BC .

9 The equation of a curve is given by \(y = ( x - 1 ) ^ { 2 } ( x + 2 )\).

1. Write \(( x - 1 ) ^ { 2 } ( x + 2 )\) in the form \(x ^ { 3 } + p x ^ { 2 } + q x + r\) where \(p , q\) and \(r\) are to be determined.
2. Show that the curve \(y = ( x - 1 ) ^ { 2 } ( x + 2 )\) has a maximum point when \(x = - 1\) and find the coordinates of the minimum point.
3. Sketch the curve \(y = ( x - 1 ) ^ { 2 } ( x + 2 )\).
4. For what values of \(k\) does \(( x - 1 ) ^ { 2 } ( x + 2 ) = k\) have exactly one root.

10 A function \(y = \mathrm { f } ( x )\) may be modelled by the equation \(y = a x ^ { b }\).

1. Show why, if this is so, then plotting \(\log y\) against \(\log x\) will produce a straight line graph. Explain how \(a\) and \(b\) may be determined experimentally from the graph.
2. Values of \(x\) and \(y\) are given below. By plotting a graph of logy against log \(x\), show that the model above is appropriate for this set of data and find values of \(a\) and \(b\) given that \(a\) is an integer and \(b\) can be written as a fraction with a denominator less than 10 .

   \(x\)	2	3	4	5	6
   \(y\)	4.6	5.0	5.3	5.5	5.7

3. Use your formula from part (ii) to estimate the value of \(y\) when \(x = 2.8\).

11 The cross-section of a brick wall built on horizontal ground is given, for \(0 \leq x \leq 6\), by the following function $$\begin{array} { l l } 0 \leq x \leq 2 & y = 1 \\ 2 \leq x \leq 4 & y = - \frac { 1 } { 2 } x ^ { 2 } + 3 x - 3 \\ 4 \leq x \leq 6 & y = 1 \end{array}$$ Units are metres.

1. Show that the highest point on the wall is 1.5 metres above the ground.
2. Find the area of the cross-section of the wall.