Edexcel C2 (Core Mathematics 2)

2. (a) Expand \(( 2 \sqrt { } x + 3 ) ^ { 2 }\).
(b) Hence evaluate \(\int _ { 1 } ^ { 2 } ( 2 \sqrt { } x + 3 ) ^ { 2 } \mathrm {~d} x\), giving your answer in the form \(a + b \sqrt { } 2\), where \(a\) and \(b\) are integers.

3. Every \(\pounds 1\) of money invested in a savings scheme continuously gains interest at a rate of \(4 \%\) per year. Hence, after \(x\) years, the total value of an initial \(\pounds 1\) investment is \(\pounds y\), where $$y = 1.04 ^ { x }$$

1. Sketch the graph of \(y = 1.04 ^ { x } , x \geq 0\).
2. Calculate, to the nearest \(\pounds\), the total value of an initial \(\pounds 800\) investment after 10 years.
3. Use logarithms to find the number of years it takes to double the total value of any initial investment.

4. \begin{figure}[h] \end{figure} Fig. 1 shows the sector \(A O B\) of a circle, with centre \(O\) and radius 6.5 cm , and \(\angle A O B = 0.8\) radians.

1. Calculate, in \(\mathrm { cm } ^ { 2 }\), the area of the sector \(A O B\).
2. Show that the length of the chord \(A B\) is 5.06 cm , to 3 significant figures. The segment \(R\), shaded in Fig. 1, is enclosed by the arc \(A B\) and the straight line \(A B\).
3. Calculate, in cm , the perimeter of \(R\).

5. (a) Write down the first 4 terms of the binomial expansion, in ascending powers of \(x\), of $$( 1 + a x ) ^ { n } , n > 2 .$$ Given that, in this expansion, the coefficient of \(x\) is 8 and the coefficient of \(x ^ { 2 }\) is 30 ,
(b) calculate the value of \(n\) and the value of \(a\),
(c) find the coefficient of \(x ^ { 3 }\).
[0pt] [P2 November 2003 Question 3]

6. A container made from thin metal is in the shape of a right circular cylinder with height \(h \mathrm {~cm}\) and base radius \(r \mathrm {~cm}\). The container has no lid. When full of water, the container holds \(500 \mathrm {~cm} ^ { 3 }\) of water.

1. Show that the exterior surface area, \(A \mathrm {~cm} ^ { 2 }\), of the container is given by \(A = \pi r ^ { 2 } + \frac { 1000 } { r }\).
2. Find the value of \(r\) for which \(A\) is a minimum.
3. Prove that this value of \(r\) gives a minimum value of \(A\).
4. Calculate the minimum value of \(A\), giving your answer to the nearest integer.

7. \begin{figure}[h] \end{figure} Fig. 2 shows part of the curve \(C\) with equation \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = x ^ { 3 } - 6 x ^ { 2 } + 5 x\).
The curve crosses the \(x\)-axis at the origin \(O\) and at the points \(A\) and \(B\).

1. Factorise \(\mathrm { f } ( x )\) completely.
2. Write down the \(x\)-coordinates of the points \(A\) and \(B\).
3. Find the gradient of \(C\) at \(A\). The region \(R\) is bounded by \(C\) and the line \(O A\), and the region \(S\) is bounded by \(C\) and the line \(A B\).
4. Use integration to find the area of the combined regions \(R\) and \(S\), shown shaded in Fig.2. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{e4d38009-aa70-4c55-9765-c54044ffaa31-4_736_727_338_402}
   \end{figure} Fig. 3 shows a sketch of part of the curve \(C\) with equation \(y = x ^ { 3 } - 7 x ^ { 2 } + 15 x + 3 , x \geq 0\). The point \(P\), on \(C\), has \(x\)-coordinate 1 and the point \(Q\) is the minimum turning point of \(C\).
5. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
6. Find the coordinates of \(Q\).
7. Show that \(P Q\) is parallel to the \(x\)-axis.
8. Calculate the area, shown shaded in Fig. 3, bounded by \(C\) and the line \(P Q\).