OCR C2 (Core Mathematics 2)

\includegraphics{figure_1} The diagram shows the sector \(OAB\) of a circle of radius 9.2 cm and centre \(O\). Given that the area of the sector is 37.4 cm\(^2\), find to 3 significant figures

1. the size of \(\angle AOB\) in radians, [2]
2. the perimeter of the sector. [2]

$$f(x) = x^3 + kx - 20.$$ Given that f(x) is exactly divisible by \((x + 1)\),

1. find the value of the constant \(k\), [2]
2. solve the equation \(f(x) = 0\). [4]

Given that $$\frac{dy}{dx} = 3\sqrt{x} - x^2,$$ and that \(y = \frac{4}{3}\) when \(x = 1\), find the value of \(y\) when \(x = 4\). [7]

A geometric progression has third term 36 and fourth term 27. Find

1. the common ratio, [2]
2. the fifth term, [2]
3. the sum to infinity. [4]

1. Solve the equation $$\log_2 (6 - x) = 3 - \log_2 x.$$ [4]
2. Find the smallest integer \(n\) such that $$3^{n-2} > 8^{250}.$$ [4]

$$f(x) = \cos 2x, \quad 0 \leq x \leq \pi.$$

1. Sketch the curve \(y = f(x)\). [2]
2. Write down the coordinates of any points where the curve \(y = f(x)\) meets the coordinate axes. [3]
3. Solve the equation \(f(x) = 0.5\), giving your answers in terms of \(\pi\). [3]

1. Find $$\int \left( x + 5 + \frac{3}{\sqrt{x}} \right) dx.$$ [4]
2. Evaluate $$\int_{-2}^{0} (3x - 1)^2 dx.$$ [5]

1. An arithmetic series has a common difference of 7. Given that the sum of the first 20 terms of the series is 530, find
   1. the first term of the series, [3]
   2. the smallest positive term of the series. [2]
2. The terms of a sequence are given by $$u_n = (n + k)^2, \quad n \geq 1,$$ where \(k\) is a positive constant. Given that \(u_2 = 2u_1\),
   1. find the value of \(k\), [4]
   2. show that \(u_3 = 11 + 6\sqrt{2}\). [2]

\includegraphics{figure_9} The diagram shows the curve \(y = 2x^2 + 6x + 7\) and the straight line \(y = 2x + 13\).

1. Find the coordinates of the points where the curve and line intersect. [4]
2. Show that the area of the shaded region bounded by the curve and line is given by $$\int_{-3}^{1} (6 - 4x - 2x^2) dx.$$ [2]
3. Hence find the area of the shaded region. [5]