Edexcel C2 (Core Mathematics 2)

f(x) = x³ + ax² + bx - 10, where a and b are constants. When f(x) is divided by (x - 3), the remainder is 14. When f(x) is divided by (x + 1), the remainder is -18.

1. Find the value of a and the value of b. [5]
2. Show that (x - 2) is a factor of f(x). [2]

1. Write down the first four terms of the binomial expansion, in ascending powers of x, of \((1 + ax)^n\), where \(n > 2\). [2]

Given that, in this expansion, the coefficient of x is 8 and the coefficient of x² is 30,

1. find the value of n and the value of a, [4]
2. find the coefficient of x³. [2]

A population of deer is introduced into a park. The population P at t years after the deer have been introduced is modelled by $$P = \frac{2000a^t}{4 + a^t},$$ where a is a constant. Given that there are 800 deer in the park after 6 years,

1. calculate, to 4 decimal places, the value of a, [4]
2. use the model to predict the number of years needed for the population of deer to increase from 800 to 1800. [4]
3. With reference to this model, give a reason why the population of deer cannot exceed 2000. [1]

Given that \(\text{f}(x) = (2x^{\frac{1}{3}} - 3x^{-\frac{1}{2}})^2 + 5\), \(x > 0\),

1. find, to 3 significant figures, the value of x for which f(x) = 5. [3]
2. Show that f(x) may be written in the form \(Ax^{\frac{2}{3}} + \frac{B}{x} + C\), where A, B and C are constants to be found. [3]
3. Hence evaluate \(\int_1^2 \text{f}(x) \, \text{dx}\). [5]

\includegraphics{figure_1} Figure 1 shows the cross-section ABCD of a chocolate bar, where AB, CD and AD are straight lines and M is the mid-point of AD. The length AD is 28 mm, and BC is an arc of a circle with centre M. Taking A as the origin, B, C and D have coordinates (7, 24), (21, 24) and (28, 0) respectively.

1. Show that the length of BM is 25 mm. [1]
2. Show that, to 3 significant figures, \(\angle BMC = 0.568\) radians. [3]
3. Hence calculate, in mm², the area of the cross-section of the chocolate bar. [5]

Given that this chocolate bar has length 85 mm,

1. calculate, to the nearest cm³, the volume of the bar. [2]

\includegraphics{figure_2} Figure 1 shows the curve with equation \(y = 5 + 2x - x^2\) 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. [3]

The shaded region R is bounded by the curve and the line.

1. Find the area of R. [6]

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

1. \(\cos(\theta - 10°) = \cos 15°\), [3]
2. \(\tan 2\theta = 0.4\), [5]
3. \(2 \sin \theta \tan \theta = 3\). [6]