Edexcel C2 (Core Mathematics 2)

A circle \(C\) has equation \(x^2 + y^2 - 10x + 6y - 15 = 0\).

1. Find the coordinates of the centre of \(C\). [2]
2. Find the radius of \(C\). [2]

\(f(x) = ax^3 + bx^2 - 7x + 14\), where \(a\) and \(b\) are constants. Given that when \(f(x)\) is divided by \((x - 1)\) the remainder is 9,

1. write down an equation connecting \(a\) and \(b\). [2]

Given also that \((x + 2)\) is a factor of \(f(x)\),

1. find the values of \(a\) and \(b\). [4]

Find all values of \(\theta\) in the interval \(0 \leq \theta < 360\) for which

1. \(\cos(\theta + 75)^\circ = 0\). [3]
2. \(\sin 2\theta^\circ = 0.7\), giving your answers to one decimal place. [5]

\includegraphics{figure_1} Fig. 1 shows the curve with equation \(y = 5 + 2x - x^2\) and the line with equation \(y = 2\). The curve and the line intersect at the points \(A\) and \(B\).

1. Find the \(x\)-coordinates of \(A\) and \(B\). [3]

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

1. Find the area of \(R\). [6]

The third and fourth terms of a geometric series are 6.4 and 5.12 respectively. Find

1. the common ratio of the series, [2]
2. the first term of the series, [2]
3. the sum to infinity of the series. [2]
4. Calculate the difference between the sum to infinity of the series and the sum of the first 25 terms of the series. [4]

\includegraphics{figure_2} Triangle \(ABC\) has \(AB = 9\) cm, \(BC = 10\) cm and \(CA = 5\) cm. A circle, centre \(A\) and radius 3 cm, intersects \(AB\) and \(AC\) at \(P\) and \(Q\) respectively, as shown in Fig. 2.

1. Show that, to 3 decimal places, \(\angle BAC = 1.504\) radians. [3]

Calculate,

1. the area, in cm\(^2\), of the sector \(APQ\), [2]
2. the area, in cm\(^2\), of the shaded region \(BPQC\), [3]
3. the perimeter, in cm, of the shaded region \(BPQC\). [4]

\includegraphics{figure_3} Fig. 3 shows the cross-sections of two drawer handles. Shape \(X\) is a rectangle \(ABCD\) joined to a semicircle with \(BC\) as diameter. The length \(AB = d\) cm and \(BC = 2d\) cm. Shape \(Y\) is a sector \(OPQ\) of a circle with centre \(O\) and radius \(2d\) cm. Angle \(POQ\) is \(\theta\) radians. Given that the areas of the shapes \(X\) and \(Y\) are equal,

1. prove that \(\theta = 1 + \frac{1}{4}\pi\). [5]

Using this value of \(\theta\), and given that \(d = 3\), find in terms of \(\pi\),

1. the perimeter of shape \(X\), [2]
2. the perimeter of shape \(Y\). [3]
3. Hence find the difference, in mm, between the perimeters of shapes \(X\) and \(Y\). [2]

\includegraphics{figure_3} A manufacturer produces cartons for fruit juice. Each carton is in the shape of a closed cuboid with base dimensions \(2x\) cm by \(x\) cm and height \(h\) cm, as shown in Fig. 3. Given that the capacity of a carton has to be 1030 cm\(^3\),

1. express \(h\) in terms of \(x\), [2]
2. show that the surface area, \(A\) cm\(^2\), of a carton is given by \(A = 4x^2 + \frac{3090}{x}\). [3]

The manufacturer needs to minimise the surface area of a carton.

1. Use calculus to find the value of \(x\) for which \(A\) is a minimum. [5]
2. Calculate the minimum value of \(A\). [2]
3. Prove that this value of \(A\) is a minimum. [2]