Edexcel C2 (Core Mathematics 2)

1. (a) Write down the first four terms of the binomial expansion, in ascending powers of \(x\), of \(( 1 + 3 x ) ^ { n }\), where \(n > 2\).

Given that the coefficient of \(x ^ { 3 }\) in this expansion is ten times the coefficient of \(x ^ { 2 }\),
(b) find the value of \(n\),
(c) find the coefficient of \(x ^ { 4 }\) in the expansion.

2. \(\mathrm { f } ( x ) = x ^ { 3 } + a x ^ { 2 } + b x - 10\), where \(a\) and \(b\) are constants. When \(\mathrm { f } ( x )\) is divided by \(( x - 3 )\), the remainder is 14 . When \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\), the remainder is - 18 .

1. Find the value of \(a\) and the value of \(b\).
2. Show that \(( x - 2 )\) is a factor of \(\mathrm { f } ( x )\).
   [0pt] [P3 June 2002 Question 1]

3. Given that \(\mathrm { f } ( x ) = 15 - 7 x - 2 x ^ { 2 }\),

1. find the coordinates of all points at which the graph of \(y = \mathrm { f } ( x )\) crosses the coordinate axes.
2. Sketch the graph of \(y = \mathrm { f } ( x )\).
3. Calculate the coordinates of the stationary point of \(\mathrm { f } ( x )\).
   [0pt] [P1 June 2002 Question 3]

4. $$\mathrm { f } ( x ) = 5 \sin 3 x ^ { \circ } , \quad 0 \leq x \leq 180$$

1. Sketch the graph of \(\mathrm { f } ( x )\), indicating the value of \(x\) at each point where the graph intersects the \(x\) axis.
2. Write down the coordinates of all the maximum and minimum points of \(\mathrm { f } ( x )\).
3. Calculate the values of \(x\) for which \(\mathrm { f } ( x ) = 2.5\)
   [0pt] [P1 June 2002 Question 5]

5. (a) Given that \(3 + 2 \log _ { 2 } x = \log _ { 2 } y\), show that \(y = 8 x ^ { 2 }\).
(b) Hence, or otherwise, find the roots \(\alpha\) and \(\beta\), where \(\alpha < \beta\), of the equation $$3 + 2 \log _ { 2 } x = \log _ { 2 } ( 14 x - 3 )$$ (c) Show that \(\log _ { 2 } \alpha = - 2\).
(d) Calculate \(\log _ { 2 } \beta\), giving your answer to 3 significant figures.

6. Given that \(\mathrm { f } ( x ) = \left( 2 x ^ { \frac { 3 } { 2 } } - 3 x ^ { - \frac { 3 } { 2 } } \right) ^ { 2 } + 5 , x > 0\),

1. find, to 3 significant figures, the value of \(x\) for which \(\mathrm { f } ( x ) = 5\).
2. Show that \(\mathrm { f } ( x )\) may be written in the form \(A x ^ { 3 } + \frac { B } { x ^ { 3 } } + C\), where \(A , B\) and \(C\) are constants to be found.
3. Hence evaluate \(\int _ { 1 } ^ { 2 } f ( x ) d x\).

7. \begin{figure}[h] \end{figure} Fig. 1 shows the cross-section \(A B C D\) of a chocolate bar, where \(A B , C D\) and \(A D\) are straight lines and \(M\) is the mid-point of \(A D\). The length \(A D\) is 28 mm , and \(B C\) is an arc of a circle with centre \(M\). Taking \(A\) as the origin, \(B , C\) and \(D\) have coordinates (7,24), (21,24) and (28,0) respectively.

1. Show that the length of \(B M\) is 25 mm .
2. Show that, to 3 significant figures, \(\angle B M C = 0.568\) radians.
3. Hence calculate, in \(\mathrm { mm } ^ { 2 }\), the area of the cross-section of the chocolate bar. Given that this chocolate bar has length 85 mm ,
4. calculate, to the nearest \(\mathrm { cm } ^ { 3 }\), the volume of the bar.

8. (a) An arithmetic series has first term \(a\) and common difference \(d\). Prove that the sum of the first \(n\) terms of the series is \(\frac { 1 } { 2 } n [ 2 a + ( n - 1 ) d ]\). A company made a profit of \(\pounds 54000\) in the year 2001. A model for future performance assumes that yearly profits will increase in an arithmetic sequence with common difference \(\pounds d\). This model predicts total profits of \(\pounds 619200\) for the 9 years 2001 to 2009 inclusive.
(b) Find the value of \(d\). Using your value of \(d\),
(c) find the predicted profit for the year 2011. An alternative model assumes that the company's yearly profits will increase in a geometric sequence with common ratio 1.06 . Using this alternative model and again taking the profit in 2001 to be \(\pounds 54000\),
(d) find the predicted profit for the year 2011.