Edexcel C2 (Core Mathematics 2) 2005 June

Find the coordinates of the stationary point on the curve with equation \(y = 2 x ^ { 2 } - 12 x\).

Solve

1. \(5 ^ { x } = 8\), giving your answer to 3 significant figures,
2. \(\log _ { 2 } ( x + 1 ) - \log _ { 2 } x = \log _ { 2 } 7\).

1. Use the factor theorem to show that \(( x + 4 )\) is a factor of \(2 x ^ { 3 } + x ^ { 2 } - 25 x + 12\).
2. Factorise \(2 x ^ { 3 } + x ^ { 2 } - 25 x + 12\) completely.

1. Write down the first three terms, in ascending powers of \(x\), of the binomial expansion of \(( 1 + p x ) ^ { 12 }\), where \(p\) is a non-zero constant. Given that, in the expansion of \(( 1 + p x ) ^ { 12 }\), the coefficient of \(x\) is \(( - q )\) and the coefficient of \(x ^ { 2 }\) is \(11 q\),
2. find the value of \(p\) and the value of \(q\).

5. Solve, for \(0 \leqslant x \leqslant 180 ^ { \circ }\), the equation

1. \(\quad \sin \left( x + 10 ^ { \circ } \right) = \frac { \sqrt { } 3 } { 2 }\),
2. \(\cos 2 x = - 0.9\), giving your answers to 1 decimal place.

6. A river, running between parallel banks, is 20 m wide. The depth, \(y\) metres, of the river measured at a point \(x\) metres from one bank is given by the formula $$y = \frac { 1 } { 10 } x \sqrt { } ( 20 - x ) , \quad 0 \leqslant x \leqslant 20$$

1. Complete the table below, giving values of \(y\) to 3 decimal places.

   \(x\)	0	4	8	12	16	20
   \(y\)	0		2.771			0

2. Use the trapezium rule with all the values in the table to estimate the cross-sectional area of the river. Given that the cross-sectional area is constant and that the river is flowing uniformly at \(2 \mathrm {~ms} ^ { - 1 }\),
3. estimate, in \(\mathrm { m } ^ { 3 }\), the volume of water flowing per minute, giving your answer to 3 significant figures.

7. In the triangle \(A B C , A B = 8 \mathrm {~cm} , A C = 7 \mathrm {~cm} , \angle A B C = 0.5\) radians and \(\angle A C B = x\) radians.

1. Use the sine rule to find the value of \(\sin x\), giving your answer to 3 decimal places. Given that there are two possible values of \(x\),
2. find these values of \(x\), giving your answers to 2 decimal places.

8. The circle \(C\), with centre at the point \(A\), has equation \(x ^ { 2 } + y ^ { 2 } - 10 x + 9 = 0\). Find

1. the coordinates of \(A\),
2. the radius of \(C\),
3. the coordinates of the points at which \(C\) crosses the \(x\)-axis. Given that the line \(l\) with gradient \(\frac { 7 } { 2 }\) is a tangent to \(C\), and that \(l\) touches \(C\) at the point \(T\),
4. find an equation of the line which passes through \(A\) and \(T\).

9. (a) A geometric series has first term \(a\) and common ratio \(r\). Prove that the sum of the first \(n\) terms of the series is $$\frac { a \left( 1 - r ^ { n } \right) } { 1 - r } .$$ Mr. King will be paid a salary of \(\pounds 35000\) in the year 2005 . Mr. King's contract promises a \(4 \%\) increase in salary every year, the first increase being given in 2006, so that his annual salaries form a geometric sequence.
(b) Find, to the nearest \(\pounds 100\), Mr. King's salary in the year 2008. Mr. King will receive a salary each year from 2005 until he retires at the end of 2024.
(c) Find, to the nearest \(\pounds 1000\), the total amount of salary he will receive in the period from 2005 until he retires at the end of 2024.

10. \begin{figure}[h] \end{figure} Figure 1 shows part of the curve \(C\) with equation \(y = 2 x + \frac { 8 } { x ^ { 2 } } - 5 , x > 0\).
The points \(P\) and \(Q\) lie on \(C\) and have \(x\)-coordinates 1 and 4 respectively. The region \(R\), shaded in Figure 1, is bounded by \(C\) and the straight line joining \(P\) and \(Q\).

1. Find the exact area of \(R\).
2. Use calculus to show that \(y\) is increasing for \(x > 2\).