OCR C2 (Core Mathematics 2) 2009 June

1 The lengths of the three sides of a triangle are \(6.4 \mathrm {~cm} , 7.0 \mathrm {~cm}\) and 11.3 cm .

1. Find the largest angle in the triangle.
2. Find the area of the triangle.

2 The tenth term of an arithmetic progression is equal to twice the fourth term. The twentieth term of the progression is 44 .

1. Find the first term and the common difference.
2. Find the sum of the first 50 terms.

3 Use logarithms to solve the equation \(7 ^ { x } = 2 ^ { x + 1 }\), giving the value of \(x\) correct to 3 significant figures.

4

1. Find the binomial expansion of \(\left( x ^ { 2 } - 5 \right) ^ { 3 }\), simplifying the terms.
2. Hence find \(\int \left( x ^ { 2 } - 5 \right) ^ { 3 } \mathrm {~d} x\).

5 Solve each of the following equations for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).

1. \(\sin 2 x = 0.5\)
2. \(2 \sin ^ { 2 } x = 2 - \sqrt { 3 } \cos x\)

6 The gradient of a curve is given by \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { 2 } + a\), where \(a\) is a constant. The curve passes through the points \(( - 1,2 )\) and \(( 2,17 )\). Find the equation of the curve.

7 The polynomial \(\mathrm { f } ( x )\) is given by \(\mathrm { f } ( x ) = 2 x ^ { 3 } + 9 x ^ { 2 } + 11 x - 8\).

1. Find the remainder when \(\mathrm { f } ( x )\) is divided by ( \(x + 2\) ).
2. Use the factor theorem to show that ( \(2 x - 1\) ) is a factor of \(\mathrm { f } ( x )\).
3. Express \(\mathrm { f } ( x )\) as a product of a linear factor and a quadratic factor.
4. State the number of real roots of the equation \(\mathrm { f } ( x ) = 0\), giving a reason for your answer.

8 \begin{figure}[h] \end{figure} Fig. 1 shows a sector \(A O B\) of a circle, centre \(O\) and radius \(O A\). The angle \(A O B\) is 1.2 radians and the area of the sector is \(60 \mathrm {~cm} ^ { 2 }\).

1. Find the perimeter of the sector. A pattern on a T-shirt, the start of which is shown in Fig. 2, consists of a sequence of similar sectors. The first sector in the pattern is sector \(A O B\) from Fig. 1, and the area of each successive sector is \(\frac { 3 } { 5 }\) of the area of the previous one. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{3836b0e7-95e6-4634-bb1e-c99b7ae3c8ba-3_362_1011_1263_568} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure}
2. (a) Find the area of the fifth sector in the pattern.
   (b) Find the total area of the first ten sectors in the pattern.
   (c) Explain why the total area will never exceed a certain limit, no matter how many sectors are used, and state the value of this limit.

9

1. Sketch the graph of \(y = 4 k ^ { x }\), where \(k\) is a constant such that \(k > 1\). State the coordinates of any points of intersection with the axes.
2. The point \(P\) on the curve \(y = 4 k ^ { x }\) has its \(y\)-coordinate equal to \(20 k ^ { 2 }\). Show that the \(x\)-coordinate of \(P\) may be written as \(2 + \log _ { k } 5\).
3. (a) Use the trapezium rule, with two strips each of width \(\frac { 1 } { 2 }\), to find an expression for the approximate value of $$\int _ { 0 } ^ { 1 } 4 k ^ { x } \mathrm {~d} x$$ (b) Given that this approximate value is equal to 16 , find the value of \(k\).