Edexcel C2 (Core Mathematics 2)

1. Find the coefficient of \(x ^ { 2 }\) in the expansion of

$$( 1 + x ) ( 1 - x ) ^ { 6 }$$

1. A geometric series has common ratio \(\frac { 1 } { 3 }\).

Given that the sum of the first four terms of the series is 200,

1. find the first term of the series,
2. find the sum to infinity of the series.

3. \begin{figure}[h] \end{figure} Figure 1 shows the curve \(y = \mathrm { f } ( x )\) where $$f ( x ) = 4 + 5 x + k x ^ { 2 } - 2 x ^ { 3 }$$ and \(k\) is a constant. The curve crosses the \(x\)-axis at the points \(A , B\) and \(C\).
Given that \(A\) has coordinates ( \(- 4,0\) ),

1. show that \(k = - 7\),
2. find the coordinates of \(B\) and \(C\).

4. (a) (i) Sketch the curve \(y = \sin ( x - 30 ) ^ { \circ }\) for \(x\) in the interval \(- 180 \leq x \leq 180\).
(ii) Write down the coordinates of the turning points of the curve in this interval.
(b) Find all values of \(x\) in the interval \(- 180 \leq x \leq 180\) for which $$\sin ( x - 30 ) ^ { \circ } = 0.35$$ giving your answers to 1 decimal place.

5. (a) Evaluate $$\log _ { 3 } 27 - \log _ { 8 } 4$$ (b) Solve the equation $$4 ^ { x } - 3 \left( 2 ^ { x + 1 } \right) = 0 .$$

1. \(\quad \mathrm { f } ( x ) = 2 - x + 3 x ^ { \frac { 2 } { 3 } } , \quad x > 0\).
   1. Find \(f ^ { \prime } ( x )\) and \(f ^ { \prime \prime } ( x )\).
   2. Find the coordinates of the turning point of the curve \(y = \mathrm { f } ( x )\).
   3. Determine whether the turning point is a maximum or minimum point.
   4. The points \(P , Q\) and \(R\) have coordinates \(( - 5,2 ) , ( - 3,8 )\) and \(( 9,4 )\) respectively.
   5. Show that \(\angle P Q R = 90 ^ { \circ }\).
   Given that \(P , Q\) and \(R\) all lie on circle \(C\),
2. find the coordinates of the centre of \(C\),
3. show that the equation of \(C\) can be written in the form $$x ^ { 2 } + y ^ { 2 } - 4 x - 6 y = k$$ where \(k\) is an integer to be found.

8. \begin{figure}[h] \end{figure} Figure 2 shows a circle of radius 12 cm which passes through the points \(P\) and \(Q\). The chord \(P Q\) subtends an angle of \(120 ^ { \circ }\) at the centre of the circle.

1. Find the exact length of the major arc \(P Q\).
2. Show that the perimeter of the shaded minor segment is given by \(k ( 2 \pi + 3 \sqrt { 3 } ) \mathrm { cm }\), where \(k\) is an integer to be found.
3. Find, to 1 decimal place, the area of the shaded minor segment as a percentage of the area of the circle.

9. The finite region \(R\) is bounded by the curve \(y = 1 + 3 \sqrt { x }\), the \(x\)-axis and the lines \(x = 2\) and \(x = 8\).

1. Use the trapezium rule with three intervals of equal width to estimate to 3 significant figures the area of \(R\).
2. Use integration to find the exact area of \(R\) in the form \(a + b \sqrt { 2 }\).
3. Find the percentage error in the estimate made in part (a).