Edexcel C2 (Core Mathematics 2)

Find the coefficient of \(x^2\) in the expansion of $$(1 + x)(1 - x)^6.$$ [4]

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, [3]
2. find the sum to infinity of the series. [2]

\includegraphics{figure_1} Figure 1 shows the curve \(y = f(x)\) where $$f(x) = 4 + 5x + kx^2 - 2x^3,$$ 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]
2. find the coordinates of \(B\) and \(C\). [5]

1. 1. Sketch the curve \(y = \sin (x - 30)°\) for \(x\) in the interval \(-180 \leq x \leq 180\).
   2. Write down the coordinates of the turning points of the curve in this interval. [4]
2. Find all values of \(x\) in the interval \(-180 \leq x \leq 180\) for which $$\sin (x - 30)° = 0.35,$$ giving your answers to 1 decimal place. [4]

1. Evaluate $$\log_3 27 - \log_3 4.$$ [4]
2. Solve the equation $$4^x - 3(2^{x+1}) = 0.$$ [5]

$$f(x) = 2 - x + 3x^{\frac{1}{2}}, \quad x > 0.$$

1. Find \(f'(x)\) and \(f''(x)\). [3]
2. Find the coordinates of the turning point of the curve \(y = f(x)\). [4]
3. Determine whether the turning point is a maximum or minimum point. [2]

The points \(P\), \(Q\) and \(R\) have coordinates \((-5, 2)\), \((-3, 8)\) and \((9, 4)\) respectively.

1. Show that \(\angle PQR = 90°\). [4]

Given that \(P\), \(Q\) and \(R\) all lie on circle \(C\),

1. find the coordinates of the centre of \(C\), [3]
2. show that the equation of \(C\) can be written in the form $$x^2 + y^2 - 4x - 6y = k,$$ where \(k\) is an integer to be found. [3]

\includegraphics{figure_2} Figure 2 shows a circle of radius 12 cm which passes through the points \(P\) and \(Q\). The chord \(PQ\) subtends an angle of \(120°\) at the centre of the circle.

1. Find the exact length of the major arc \(PQ\). [2]
2. Show that the perimeter of the shaded minor segment is given by \(k(2\pi + 3\sqrt{3})\) cm, where \(k\) is an integer to be found. [4]
3. Find, to 1 decimal place, the area of the shaded minor segment as a percentage of the area of the circle. [4]

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 \(R\). [6]
2. Use integration to find the exact area of \(R\) in the form \(a + b\sqrt{2}\). [5]
3. Find the percentage error in the estimate made in part (a). [2]