Edexcel C2 (Core Mathematics 2)

The point \(A\) has coordinates \((2, 5)\) and the point \(B\) has coordinates \((-2, 8)\). Find, in cartesian form, an equation of the circle with diameter \(AB\). [4]

\includegraphics{figure_1} The circle \(C\), with centre \((a, b)\) and radius \(5\), touches the \(x\)-axis at \((4, 0)\), as shown in Fig. 1.

1. Write down the value of \(a\) and the value of \(b\). [1]
2. Find a cartesian equation of \(C\). [2]

A tangent to the circle, drawn from the point \(P(8, 17)\), touches the circle at \(T\).

1. Find, to 3 significant figures, the length of \(PT\). [3]

\(\text{f}(n) = n^3 + pn^2 + 11n + 9\), where \(p\) is a constant.

1. Given that f\((n)\) has a remainder of \(3\) when it is divided by \((n + 2)\), prove that \(p = 6\). [2]
2. Show that f\((n)\) can be written in the form \((n + 2)(n + q)(n + r) + 3\), where \(q\) and \(r\) are integers to be found. [3]
3. Hence show that f\((n)\) is divisible by \(3\) for all positive integer values of \(n\). [2]

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]

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]

Find, in degrees, the value of \(\theta\) in the interval \(0 \leq \theta < 360°\) for which $$2\cos^2\theta - \cos\theta - 1 = \sin^2\theta.$$ Give your answers to \(1\) decimal place where appropriate. [8]

\includegraphics{figure_2} Fig. 2 shows the sector \(OAB\) of a circle of radius \(r\) cm. The area of the sector is \(15\) cm\(^2\) and \(\angle AOB = 1.5\) radians.

1. Prove that \(r = 2\sqrt{5}\). [3]
2. Find, in cm, the perimeter of the sector \(OAB\). [2]

The segment \(R\), shaded in Fig 1, is enclosed by the arc \(AB\) and the straight line \(AB\).

1. Calculate, to 3 decimal places, the area of \(R\). [3]

\includegraphics{figure_3} Fig. 3 shows the line with equation \(y = 9 - x\) and the curve with equation \(y = x^2 - 2x + 3\). The line and the curve intersect at the points \(A\) and \(B\), and \(O\) is the origin.

1. Calculate the coordinates of \(A\) and the coordinates of \(B\). [5]

The shaded region \(R\) is bounded by the line and the curve.

1. Calculate the area of \(R\). [7]

For the curve \(C\) with equation \(y = x^4 - 8x^2 + 3\),

1. find \(\frac{dy}{dx}\), [2]
2. find the coordinates of each of the stationary points, [5]
3. determine the nature of each stationary point. [3]

The point \(A\), on the curve \(C\), has \(x\)-coordinate \(1\).

1. Find an equation for the normal to \(C\) at \(A\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [5]