Edexcel C2 (Core Mathematics 2)

1. A circle \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 10 x + 6 y - 15 = 0\).
   1. Find the coordinates of the centre of \(C\).
   2. Find the radius of \(C\).
      [0pt] [P3 June 2001 Question 1]
   3. \(\mathrm { f } ( x ) \equiv a x ^ { 3 } + b x ^ { 2 } - 7 x + 14\), where \(a\) and \(b\) are constants.
   Given that when \(\mathrm { f } ( x )\) is divided by \(( x - 1 )\) the remainder is 9 ,
2. write down an equation connecting \(a\) and \(b\). Given also that \(( x + 2 )\) is a factor of \(\mathrm { f } ( x )\),
3. find the values of \(a\) and \(b\).
   [0pt] [P3 June 2001 Question 2]

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

1. \(\cos ( \theta + 75 ) ^ { \circ } = 0\).
2. \(\sin 2 \theta ^ { \circ } = 0.7\), giving your answers to one decima1 place.

4.
\includegraphics[max width=\textwidth, alt={}, center]{ffa0b566-6448-491b-96d7-d3806bcfe063-2_639_1408_1315_212} Fig. 1 shows the curve with equation \(y = 5 + 2 x - x ^ { 2 }\) 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 \(B\). The shaded region \(R\) is bounded by the curve and the line.
2. Find the area of \(R\).

5. 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. the first term of the series,
3. the sum to infinity of the series.
4. Calculate the difference between the sum to infinity of the series and the sum of the first 25 terms of the series.

6.
\includegraphics[max width=\textwidth, alt={}, center]{ffa0b566-6448-491b-96d7-d3806bcfe063-3_684_1237_685_239} Triangle \(A B C\) has \(A B = 9 \mathrm {~cm} , B C 10 \mathrm {~cm}\) and \(C A = 5 \mathrm {~cm}\). A circle, centre \(A\) and radius 3 cm , intersects \(A B\) and \(A C\) at \(P\) and \(Q\) respectively, as shown in Fig. 2.

1. Show that, to 3 decimal places, \(\angle B A C = 1.504\) radians. Calculate,
2. the area, in \(\mathrm { cm } ^ { 2 }\), of the sector \(A P Q\),
3. the area, in \(\mathrm { cm } ^ { 2 }\), of the shaded region \(B P Q C\),
4. the perimeter, in cm , of the shaded region \(B P Q C\).

7. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} Fig. 3 shows the cross-sections of two drawer handles. Shape \(X\) is a rectangle \(A B C D\) joined to a semicircle with \(B C\) as diameter. The length \(A B = d \mathrm {~cm}\) and \(B C = 2 d \mathrm {~cm}\). Shape \(Y\) is a sector \(O P Q\) of a circle with centre \(O\) and radius \(2 d \mathrm {~cm}\). Angle \(P O Q\) is \(\theta\) radians. Given that the areas of the shapes \(X\) and \(Y\) are equal,

1. prove that \(\theta = 1 + \frac { 1 } { 4 } \pi\). Using this value of \(\theta\), and given that \(d = 3\), find in terms of \(\pi\),
2. the perimeter of shape \(X\),
3. the perimeter of shape \(Y\).
4. Hence find the difference, in mm , between the perimeters of shapes \(X\) and \(Y\).

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

1. express \(h\) in terms of \(x\),
2. show that the surface area, \(A \mathrm {~cm} ^ { 2 }\), of a carton is given by \(A = 4 x ^ { 2 } + \frac { 3090 } { x }\). The manufacturer needs to minimise the surface area of a carton.
3. Use calculus to find the value of \(x\) for which \(A\) is a minimum.
4. Calculate the minimum value of \(A\).
5. Prove that this value of \(A\) is a minimum.