Edexcel C2 (Core Mathematics 2) 2013 January

1. Find the first 3 terms, in ascending powers of \(x\), in the binomial expansion of

$$( 2 - 5 x ) ^ { 6 }$$ Give each term in its simplest form.

2. \(\mathrm { f } ( x ) = a x ^ { 3 } + b x ^ { 2 } - 4 x - 3\), where \(a\) and \(b\) are constants. Given that \(( x - 1 )\) is a factor of \(\mathrm { f } ( x )\),

1. show that $$a + b = 7$$ Given also that, when \(\mathrm { f } ( x )\) is divided by \(( x + 2 )\), the remainder is 9 ,
2. find the value of \(a\) and the value of \(b\), showing each step in your working.

3. A company predicts a yearly profit of \(\pounds 120000\) in the year 2013 . The company predicts that the yearly profit will rise each year by \(5 \%\). The predicted yearly profit forms a geometric sequence with common ratio 1.05

1. Show that the predicted profit in the year 2016 is \(\pounds 138915\)
2. Find the first year in which the yearly predicted profit exceeds \(\pounds 200000\)
3. Find the total predicted profit for the years 2013 to 2023 inclusive, giving your answer to the nearest pound.

4. Solve, for \(0 \leqslant x < 180 ^ { \circ }\), $$\cos \left( 3 x - 10 ^ { \circ } \right) = - 0.4$$ giving your answers to 1 decimal place. You should show each step in your working.

5. The circle \(C\) has equation $$x ^ { 2 } + y ^ { 2 } - 20 x - 24 y + 195 = 0$$ The centre of \(C\) is at the point \(M\).

1. Find
   1. the coordinates of the point \(M\),
   2. the radius of the circle \(C\). \(N\) is the point with coordinates \(( 25,32 )\).
2. Find the length of the line \(M N\). The tangent to \(C\) at a point \(P\) on the circle passes through point \(N\).
3. Find the length of the line \(N P\).

6. Given that $$2 \log _ { 2 } ( x + 15 ) - \log _ { 2 } x = 6$$

1. Show that $$x ^ { 2 } - 34 x + 225 = 0$$
2. Hence, or otherwise, solve the equation $$2 \log _ { 2 } ( x + 15 ) - \log _ { 2 } x = 6$$

7. \begin{figure}[h] \end{figure} The triangle \(X Y Z\) in Figure 1 has \(X Y = 6 \mathrm {~cm} , Y Z = 9 \mathrm {~cm} , Z X = 4 \mathrm {~cm}\) and angle \(Z X Y = \alpha\). The point \(W\) lies on the line \(X Y\). The circular arc \(Z W\), in Figure 1 is a major arc of the circle with centre \(X\) and radius 4 cm .

1. Show that, to 3 significant figures, \(\alpha = 2.22\) radians.
2. Find the area, in \(\mathrm { cm } ^ { 2 }\), of the major sector \(X Z W X\). The region enclosed by the major arc \(Z W\) of the circle and the lines \(W Y\) and \(Y Z\) is shown shaded in Figure 1. Calculate
3. the area of this shaded region,
4. the perimeter \(Z W Y Z\) of this shaded region.

8. The curve \(C\) has equation \(y = 6 - 3 x - \frac { 4 } { x ^ { 3 } } , x \neq 0\)

1. Use calculus to show that the curve has a turning point \(P\) when \(x = \sqrt { } 2\)
2. Find the \(x\)-coordinate of the other turning point \(Q\) on the curve.
3. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
4. Hence or otherwise, state with justification, the nature of each of these turning points \(P\) and \(Q\).

9. \(y\) \begin{figure}[h] \end{figure} The finite region \(R\), as shown in Figure 2, is bounded by the \(x\)-axis and the curve with equation $$y = 27 - 2 x - 9 \sqrt { } x - \frac { 16 } { x ^ { 2 } } , \quad x > 0$$ The curve crosses the \(x\)-axis at the points \(( 1,0 )\) and \(( 4,0 )\).

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

   \(x\)	1	1.5	2	2.5	3	3.5	4
   \(y\)	0	5.866		5.210		1.856	0

2. Use the trapezium rule with all the values in the completed table to find an approximate value for the area of \(R\), giving your answer to 2 decimal places.
3. Use integration to find the exact value for the area of \(R\).