Edexcel C2 (Core Mathematics 2) 2015 June

1. Find the first 3 terms, in ascending powers of \(x\), of the binomial expansion of

$$\left( 2 - \frac { x } { 4 } \right) ^ { 10 }$$ giving each term in its simplest form.

2. A circle \(C\) with centre at the point \(( 2 , - 1 )\) passes through the point \(A\) at \(( 4 , - 5 )\).

1. Find an equation for the circle \(C\).
2. Find an equation of the tangent to the circle \(C\) at the point \(A\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

3. \(\mathrm { f } ( x ) = 6 x ^ { 3 } + 3 x ^ { 2 } + A x + B\), where \(A\) and \(B\) are constants. Given that when \(\mathrm { f } ( x )\) is divided by \(( x + 1 )\) the remainder is 45 ,

1. show that \(B - A = 48\) Given also that ( \(2 x + 1\) ) is a factor of \(\mathrm { f } ( x )\),
2. find the value of \(A\) and the value of \(B\).
3. Factorise f(x) fully.

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of a design for a scraper blade. The blade \(A O B C D A\) consists of an isosceles triangle \(C O D\) joined along its equal sides to sectors \(O B C\) and \(O D A\) of a circle with centre \(O\) and radius 8 cm . Angles \(A O D\) and \(B O C\) are equal. \(A O B\) is a straight line and is parallel to the line \(D C . D C\) has length 7 cm .

1. Show that the angle \(C O D\) is 0.906 radians, correct to 3 significant figures.
2. Find the perimeter of \(A O B C D A\), giving your answer to 3 significant figures.
3. Find the area of \(A O B C D A\), giving your answer to 3 significant figures.

1. 1. All the terms of a geometric series are positive. The sum of the first two terms is 34 and the sum to infinity is 162

Find

1. the common ratio,
2. the first term.
   (ii) A different geometric series has a first term of 42 and a common ratio of \(\frac { 6 } { 7 }\). Find the smallest value of \(n\) for which the sum of the first \(n\) terms of the series exceeds 290

6. (a) Find $$\int 10 x \left( x ^ { \frac { 1 } { 2 } } - 2 \right) \mathrm { d } x$$ giving each term in its simplest form. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve \(C\) with equation $$y = 10 x \left( x ^ { \frac { 1 } { 2 } } - 2 \right) , \quad x \geqslant 0$$ The curve \(C\) starts at the origin and crosses the \(x\)-axis at the point \(( 4,0 )\). The area, shown shaded in Figure 2, consists of two finite regions and is bounded by the curve \(C\), the \(x\)-axis and the line \(x = 9\)
(b) Use your answer from part (a) to find the total area of the shaded regions.

7. (i) Use logarithms to solve the equation \(8 ^ { 2 x + 1 } = 24\), giving your answer to 3 decimal places.
(ii) Find the values of \(y\) such that $$\log _ { 2 } ( 11 y - 3 ) - \log _ { 2 } 3 - 2 \log _ { 2 } y = 1 , \quad y > \frac { 3 } { 11 }$$

8. (i) Solve, for \(0 \leqslant \theta < \pi\), the equation $$\sin 3 \theta - \sqrt { 3 } \cos 3 \theta = 0$$ giving your answers in terms of \(\pi\).
(ii) Given that $$4 \sin ^ { 2 } x + \cos x = 4 - k , \quad 0 \leqslant k \leqslant 3$$

1. find \(\cos x\) in terms of \(k\).
2. When \(k = 3\), find the values of \(x\) in the range \(0 \leqslant x < 360 ^ { \circ }\)

9. A solid glass cylinder, which is used in an expensive laser amplifier, has a volume of \(75 \pi \mathrm {~cm} ^ { 3 }\).
The cost of polishing the surface area of this glass cylinder is \(\pounds 2\) per \(\mathrm { cm } ^ { 2 }\) for the curved surface area and \(\pounds 3\) per \(\mathrm { cm } ^ { 2 }\) for the circular top and base areas. Given that the radius of the cylinder is \(r \mathrm {~cm}\),

1. show that the cost of the polishing, \(\pounds C\), is given by $$C = 6 \pi r ^ { 2 } + \frac { 300 \pi } { r }$$
2. Use calculus to find the minimum cost of the polishing, giving your answer to the nearest pound.
3. Justify that the answer that you have obtained in part (b) is a minimum.