Edexcel C2 (Core Mathematics 2)

1. Using the factor theorem, show that \((x + 3)\) is a factor of \(x^3 - 3x^2 - 10x + 24\). [2]
2. Factorise \(x^3 - 3x^2 - 10x + 24\) completely. [4]

1. Expand \((2\sqrt{x} + 3)^2\). [2]
2. Hence evaluate \(\int_1^2 (2\sqrt{x} + 3)^2 \, dx\), giving your answer in the form \(a + b\sqrt{2}\), where \(a\) and \(b\) are integers. [5]

The first three terms in the expansion, in ascending powers of \(x\), of \((1 + px)^n\), are \(1 - 18x + 36p^2x^2\). Given that \(n\) is a positive integer, find the value of \(n\) and the value of \(p\). [7]

A circle \(C\) has equation \(x^2 + y^2 - 6x + 8y - 75 = 0\).

1. Write down the coordinates of the centre of \(C\), and calculate the radius of \(C\). [3]

A second circle has centre at the point \((15, 12)\) and radius \(10\).

1. Sketch both circles on a single diagram and find the coordinates of the point where they touch. [4]

1. Differentiate \(2x^2 + \sqrt{x} + \frac{x^2 + 2x}{x^2}\) with respect to \(x\) [5]
2. Evaluate \(\int_1^4 \left(\frac{x}{2} + \frac{1}{x^2}\right) dx\). [5]

A geometric series has first term \(1200\). Its sum to infinity is \(960\).

1. Show that the common ratio of the series is \(-\frac{1}{4}\). [3]
2. Find, to 3 decimal places, the difference between the ninth and tenth terms of the series. [3]
3. Write down an expression for the sum of the first \(n\) terms of the series. [2]

Given that \(n\) is odd,

1. prove that the sum of the first \(n\) terms of the series is \(960(1 + 0.25^n)\). [2]

On a journey, the average speed of a car is \(v\) m s\(^{-1}\). For \(v \geq 5\), the cost per kilometre, \(C\) pence, of the journey is modelled by \(C = \frac{160}{v} + \frac{v^2}{100}\). Using this model,

1. show, by calculus, that there is a value of \(v\) for which \(C\) has a stationary value, and find this value of \(v\). [5]
2. Justify that this value of \(v\) gives a minimum value of \(C\). [2]
3. Find the minimum value of \(C\) and hence find the minimum cost of a 250 km car journey. [3]

\includegraphics{figure_1} The curve \(C\), shown in Fig. 1, represents the graph of \(y = \frac{x^2}{25}\), \(x \geq 0\). The points \(A\) and \(B\) on the curve \(C\) have \(x\)-coordinates \(5\) and \(10\) respectively.

1. Write down the \(y\)-coordinates of \(A\) and \(B\). [1]
2. Find an equation of the tangent to \(C\) at \(A\). [4]

The finite region \(R\) is enclosed by \(C\), the \(y\)-axis and the lines through \(A\) and \(B\) parallel to the \(x\)-axis.

1. For points \((x, y)\) on \(C\), express \(x\) in terms of \(y\). [2]
2. Use integration to find the area of \(R\). [5]

1. Solve, for \(0° < x < 180°\), the equation \(\sin (2x + 50°) = 0.6\), giving your answers to 1 d. p. [7]
2. In the triangle \(ABC\), \(AC = 18\) cm, \(\angle ABC = 60°\) and \(\sin A = \frac{1}{3}\).
   1. Use the sine rule to show that \(BC = 4\sqrt{3}\). [4]
   2. Find the exact value of \(\cos A\). [2]