Edexcel C2 (Core Mathematics 2)

Given that \(p = \log_q 16\), express in terms of \(p\),

1. \(\log_q 2\). [2]
2. \(\log_q (8q)\). [4]

The expansion of \((2 - px)^6\) in ascending powers of \(x\), as far as the term in \(x^2\), is $$64 + Ax + 135x^2.$$ Given that \(p > 0\), find the value of \(p\) and the value of \(A\). [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. Sketch, for \(0 \leq x \leq 360°\), the graph of \(y = \sin (x + 30°)\). [2]
2. Write down the coordinates of the points at which the graph meets the axes. [3]
3. Solve, for \(0 \leq x < 360°\), the equation $$\sin (x + 30°) = -\frac{1}{2}.$$ [3]

\includegraphics{figure_1} The shape of a badge is a sector \(ABC\) of a circle with centre \(A\) and radius \(AB\), as shown in Fig 1. The triangle \(ABC\) is equilateral and has a perpendicular height 3 cm.

1. Find, in surd form, the length \(AB\). [2]
2. Find, in terms of \(\pi\), the area of the badge. [2]
3. Prove that the perimeter of the badge is \(\frac{2\sqrt{3}}{3}(\pi + 6)\) cm. [2]

\(f(x) = 6x^3 + px^2 + qx + 8\), where \(p\) and \(q\) are constants. Given that \(f(x)\) is exactly divisible by \((2x - 1)\), and also that when \(f(x)\) is divided by \((x - 1)\) the remainder is \(-7\),

1. find the value of \(p\) and the value of \(q\). [6]
2. Hence factorise \(f(x)\) completely. [3]

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]

A circle \(C\) has centre \((3, 4)\) and radius \(3\sqrt{2}\). A straight line \(l\) has equation \(y = x + 3\).

1. Write down an equation of the circle \(C\). [2]
2. Calculate the exact coordinates of the two points where the line \(l\) intersects \(C\), giving your answers in surds. [5]
3. Find the distance between these two points. [2]

\includegraphics{figure_2} The curve \(C\), shown in Fig. 2, 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]