OCR C2 (Core Mathematics 2) 2007 January

In an arithmetic progression the first term is 15 and the twentieth term is 72. Find the sum of the first 100 terms. [4]

\includegraphics{figure_2} The diagram shows a sector \(OAB\) of a circle, centre \(O\) and radius 8 cm. The angle \(AOB\) is \(46°\).

1. Express \(46°\) in radians, correct to 3 significant figures. [2]
2. Find the length of the arc \(AB\). [1]
3. Find the area of the sector \(OAB\). [2]

1. Find \(\int (4x - 5) dx\). [2]
2. The gradient of a curve is given by \(\frac{dy}{dx} = 4x - 5\). The curve passes through the point \((3, 7)\). Find the equation of the curve. [3]

In a triangle \(ABC\), \(AB = 5\sqrt{2}\) cm, \(BC = 8\) cm and angle \(B = 60°\).

1. Find the exact area of the triangle, giving your answer as simply as possible. [3]
2. Find the length of \(AC\), correct to 3 significant figures. [3]

1. 1. Express \(\log_3(4x + 7) - \log_3 x\) as a single logarithm. [1]
   2. Hence solve the equation \(\log_3(4x + 7) - \log_3 x = 2\). [3]
2. Use the trapezium rule, with two strips of width 3, to find an approximate value for $$\int_3^9 \log_{10} x \, dx,$$ giving your answer correct to 3 significant figures. [4]

1. Find and simplify the first four terms in the expansion of \((1 + 4x)^7\) in ascending powers of \(x\). [4]
2. In the expansion of $$(3 + ax)(1 + 4x)^7,$$ the coefficient of \(x^2\) is 1001. Find the value of \(a\). [3]

1. 1. Sketch the graph of \(y = 2 \cos x\) for values of \(x\) such that \(0° \leq x \leq 360°\), indicating the coordinates of any points where the curve meets the axes. [2]
   2. Solve the equation \(2 \cos x = 0.8\), giving all values of \(x\) between \(0°\) and \(360°\). [3]
2. Solve the equation \(2 \cos x = \sin x\), giving all values of \(x\) between \(-180°\) and \(180°\). [3]

The polynomial f(x) is defined by \(f(x) = x^3 - 9x^2 + 7x + 33\).

1. Find the remainder when f(x) is divided by \((x + 2)\). [2]
2. Show that \((x - 3)\) is a factor of f(x). [1]
3. Solve the equation f(x) = 0, giving each root in an exact form as simply as possible. [6]

On its first trip between Maltby and Grenlish, a steam train uses 1.5 tonnes of coal. As the train does more trips, it becomes less efficient so that each subsequent trip uses 2% more coal than the previous trip.

1. Show that the amount of coal used on the fifth trip is 1.624 tonnes, correct to 4 significant figures. [2]
2. There are 39 tonnes of coal available. An engineer wishes to calculate \(N\), the total number of trips possible. Show that \(N\) satisfies the inequality $$1.02^N < 1.52.$$ [4]
3. Hence, by using logarithms, find the greatest number of trips possible. [4]

\includegraphics{figure_10} The diagram shows the graph of \(y = 1 - 3x^{-\frac{1}{2}}\).

1. Verify that the curve intersects the \(x\)-axis at \((9, 0)\). [1]
2. The shaded region is enclosed by the curve, the \(x\)-axis and the line \(x = a\) (where \(a > 9\)). Given that the area of the shaded region is 4 square units, find the value of \(a\). [9]