OCR C2 (Core Mathematics 2)

\includegraphics{figure_2} The diagram shows the curve with equation \(y = 2^x\). Use the trapezium rule with four intervals, each of width 1, to estimate the area of the shaded region bounded by the curve, the \(x\)-axis and the lines \(x = -2\) and \(x = 2\). [4]

1. Given that $$5 \cos \theta - 2 \sin \theta = 0,$$ show that \(\tan \theta = 2.5\) [2]
2. Solve, for \(0 \leq x \leq 180\), the equation $$5 \cos 2x° - 2 \sin 2x° = 0,$$ giving your answers to 1 decimal place. [4]

1. Given that \(y = \log_2 x\), find expressions in terms of \(y\) for
   1. \(\log_2 \left(\frac{x}{2}\right)\), [2]
   2. \(\log_2 (\sqrt{x})\). [2]
2. Hence, or otherwise, solve the equation $$2 \log_2 \left(\frac{x}{2}\right) + \log_2 (\sqrt{x}) = 8.$$ [3]

\includegraphics{figure_5} The diagram shows the sector \(OAB\) of a circle, centre \(O\), in which \(\angle AOB = 2.5\) radians. Given that the perimeter of the sector is 36 cm,

1. find the length \(OA\), [2]
2. find the perimeter and the area of the shaded segment. [6]

\includegraphics{figure_6} The diagram shows the curve with equation \(y = 4x^{\frac{1}{3}} - x\), \(x \geq 0\). The curve meets the \(x\)-axis at the origin and at the point \(A\) with coordinates \((a, 0)\).

1. Show that \(a = 8\). [3]
2. Find the area of the finite region bounded by the curve and the positive \(x\)-axis. [5]

1. Evaluate $$\sum_{r=10}^{30} (7 + 2r).$$ [4]
2. 1. Write down the formula for the sum of the first \(n\) positive integers. [1]
   2. Using this formula, find the sum of the integers from 100 to 200 inclusive. [3]
   3. Hence, find the sum of the integers between 300 and 600 inclusive which are divisible by 3. [2]

The first three terms of a geometric series are \((x - 2)\), \((x + 6)\) and \(x^2\) respectively.

1. Show that \(x\) must be a solution of the equation $$x^3 - 3x^2 - 12x - 36 = 0. \quad \text{(I)}$$ [3]
2. Verify that \(x = 6\) is a solution of equation (I) and show that there are no other real solutions. [6]

Using \(x = 6\),

1. find the common ratio of the series, [1]
2. find the sum of the first eight terms of the series. [2]

1. Evaluate $$\int_1^3 (3 - \sqrt{x})^2 \, dx,$$ giving your answer in the form \(a + b\sqrt{3}\), where \(a\) and \(b\) are integers. [6]
2. The gradient of a curve is given by $$\frac{dy}{dx} = 3x^2 + 4x + k,$$ where \(k\) is a constant. Given that the curve passes through the points \((0, -2)\) and \((2, 18)\), show that \(k = 2\) and find an equation for the curve. [7]