Edexcel C2 (Core Mathematics 2)

f(x) = ax³ + bx² - 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 marks] Given also that (x + 2) is a factor of f(x),
2. find the values of a and b. [4 marks]

1. Differentiate with respect to x $$2x^3 + \sqrt{x} + \frac{x^2 + 2x}{x^2}.$$ [5 marks]
2. Evaluate $$\int_1^4 \left(\frac{x}{2} + \frac{1}{x^2}\right) dx.$$ [5 marks]

1. An arithmetic series has first term a and common difference d. Prove that the sum of the first n terms of the series is $$\frac{1}{2}n[2a + (n - 1)d].$$ [4 marks] A company made a profit of £54000 in the year 2001. A model for future performance assumes that yearly profits will increase in an arithmetic sequence with common difference £d. This model predicts total profits of £619200 for the 9 years 2001 to 2009 inclusive.
2. Find the value of d. [4 marks] Using your value of d,
3. find the predicted profit for the year 2011. [2 marks] An alternative model assumes that the company's yearly profits will increase in a geometric sequence with common ratio 1.06. Using this alternative model and again taking the profit in 2001 to be £54000,
4. find the predicted profit for the year 2011. [3 marks]

1. Write down formulae for sin (A + B) and sin (A - B). Using X = A + B and Y = A - B, prove that $$\sin X + \sin Y = 2 \sin \frac{X + Y}{2} \cos \frac{X - Y}{2}.$$ [4 marks]
2. Hence, or otherwise, solve, for 0 ≤ θ < 360, $$\sin 40° + \sin 20° = 0.$$ [5 marks]

\includegraphics{figure_1} Figure 1 shows a gardener's design for the shape of a flower bed with perimeter ABCD. AD is an arc of a circle with centre O and radius 5 m. BC is an arc of a circle with centre O and radius 7 m. OAB and ODC are straight lines and the size of ∠AOD is θ radians.

1. Find, in terms of θ, an expression for the area of the flower bed. [3 marks] Given that the area of the flower bed is 15 m²,
2. show that θ = 1.25. [2 marks]
3. calculate, in m, the perimeter of the flower bed. [3 marks] The gardener now decides to replace arc AD with the straight line AD.
4. Find, to the nearest cm, the reduction in the perimeter of the flower bed. [2 marks]

1. Given that $$(2 + x)^5 + (2 - x)^5 ≡ A + Bx^2 + Cx^4,$$ Find the values of the constants A, B and C. [6 marks]
2. Using the substitution y = x² and your answers to part (a), solve, $$(2 + x)^5 + (2 - x)^5 = 349.$$ [5 marks]

\includegraphics{figure_2} Figure 2 shows part of the curve C with equation y = f(x), where $$f(x) = x^3 - 6x^2 + 5x.$$ The curve crosses the x-axis at the origin O and at the points A and B.

1. Factorise f(x) completely [3 marks]
2. Write down the x-coordinates of the points A and B. [1 marks]
3. Find the gradient of C at A. [3 marks] The region R is bounded by C and the line OA, and the region S is bounded by C and the line AB.
4. Use integration to find the area of the combined regions R and S, shown shaded in Fig. 2. [7 marks]