Edexcel C2 (Core Mathematics 2) 2006 January

1. \(\mathrm { f } ( x ) = 2 x ^ { 3 } + x ^ { 2 } - 5 x + c\), where \(c\) is a constant.

Given that \(\mathrm { f } ( 1 ) = 0\),

1. find the value of \(c\),
2. factorise \(\mathrm { f } ( x )\) completely,
3. find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(2 x - 3\) ).

2. (a) Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of $$( 1 + p x ) ^ { 9 }$$ where \(p\) is a constant. These first 3 terms are \(1,36 x\) and \(q x ^ { 2 }\), where \(q\) is a constant.
(b) Find the value of \(p\) and the value of \(q\).

3. \begin{figure}[h] \end{figure} In Figure \(1 , A ( 4,0 )\) and \(B ( 3,5 )\) are the end points of a diameter of the circle \(C\). Find

1. the exact length of \(A B\),
2. the coordinates of the midpoint \(P\) of \(A B\),
3. an equation for the circle \(C\).

1. The first term of a geometric series is 120 . The sum to infinity of the series is 480 .
   1. Show that the common ratio, \(r\), is \(\frac { 3 } { 4 }\).
   2. Find, to 2 decimal places, the difference between the 5th and 6th term.
   3. Calculate the sum of the first 7 terms.
   The sum of the first \(n\) terms of the series is greater than 300 .
2. Calculate the smallest possible value of \(n\).

5. \begin{figure}[h] \end{figure} In Figure \(2 O A B\) is a sector of a circle radius 5 m . The chord \(A B\) is 6 m long.

1. Show that \(\cos A \hat { O } B = \frac { 7 } { 25 }\).
2. Hence find the angle \(A \hat { O } B\) in radians, giving your answer to 3 decimal places.
3. Calculate the area of the sector \(O A B\).
4. Hence calculate the shaded area.

1. The speed, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of a train at time \(t\) seconds is given by

$$v = \sqrt { } \left( 1.2 ^ { t } - 1 \right) , \quad 0 \leqslant t \leqslant 30$$ The following table shows the speed of the train at 5 second intervals.

1. Complete the table, giving the values of \(v\) to 2 decimal places. The distance, \(s\) metres, travelled by the train in 30 seconds is given by $$s = \int _ { 0 } ^ { 30 } \sqrt { } \left( 1.2 ^ { t } - 1 \right) \mathrm { d } t$$
2. Use the trapezium rule, with all the values from your table, to estimate the value of \(s\).
   (3)

7. The curve \(C\) has equation $$y = 2 x ^ { 3 } - 5 x ^ { 2 } - 4 x + 2$$

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Using the result from part (a), find the coordinates of the turning points of \(C\).
3. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
4. Hence, or otherwise, determine the nature of the turning points of \(C\).

1. (a) Find all the values of \(\theta\), to 1 decimal place, in the interval \(0 ^ { \circ } \leqslant \theta < 360 ^ { \circ }\) for which

$$5 \sin \left( \theta + 30 ^ { \circ } \right) = 3$$ (b) Find all the values of \(\theta\), to 1 decimal place, in the interval \(0 ^ { \circ } \leqslant \theta < 360 ^ { \circ }\) for which $$\tan ^ { 2 } \theta = 4$$

9. \begin{figure}[h] \end{figure} Figure 3 shows the shaded region \(R\) which is bounded by the curve \(y = - 2 x ^ { 2 } + 4 x\) and the line \(y = \frac { 3 } { 2 }\). The points \(A\) and \(B\) are the points of intersection of the line and the curve. Find

1. the \(x\)-coordinates of the points \(A\) and \(B\),
2. the exact area of \(R\).