Edexcel C2 (Core Mathematics 2)

f(x) = px³ + 6x² + 12x + q. Given that the remainder when f(x) is divided by (x - 1) is equal to the remainder when f(x) is divided by (2x + 1),

1. find the value of p. [4]

Given also that q = 3, and p has the value found in part (a),

1. find the value of the remainder. [1]

Figure 1 \includegraphics{figure_1} The circle C, with centre (a, b) and radius 5, touches the x-axis at (4, 0), as shown in Fig. 1.

1. Write down the value of a and the value of b. [1]
2. Find a cartesian equation of C. [2]

A tangent to the circle, drawn from the point P(8, 17), touches the circle at T.

1. Find, to 3 significant figures, the length of PT. [3]

1. Expand (2√x + 3)². [2]
2. Hence evaluate $$\int_1^{2^2} (2\sqrt{x} + 3)^2 \, dx$$, giving your answer in the form a + b√2, where a and b are integers. [5]

The first three terms in the expansion, in ascending powers of x, of (1 + px)ⁿ, are 1 - 18x + 36p²x². Given that n is a positive integer, find the value of n and the value of p. [7]

Find all values of θ in the interval 0 ≤ θ < 360 for which

1. cos (θ + 75)° = 0. [3]
2. sin 2θ° = 0.7, giving your answers to one decimal place. [5]

Given that log₂ x = a, find, in terms of a, the simplest form of

1. log₂ (16x), [2]
2. log₂ \(\left(\frac{x⁴}{2}\right)\). [3]

1. Hence, or otherwise, solve $$\log_2 (16x) - \log_2 \left(\frac{x^4}{2}\right) = \frac{1}{2},$$ giving your answer in its simplest surd form. [4]

The curve C has equation \(y = \cos \left(x + \frac{\pi}{4}\right)\), \(0 \leq x \leq 2\pi\).

1. Sketch C. [2]
2. Write down the exact coordinates of the points at which C meets the coordinate axes. [3]

1. Solve, for x in the interval \(0 \leq x \leq 2\pi\), $$\cos \left(x + \frac{\pi}{4}\right) = 0.5,$$ giving your answers in terms of π. [4]

Figure 2 \includegraphics{figure_2} Figure 2 shows part of the curve with equation $$y = x³ - 6x² + 9x.$$ The curve touches the x-axis at A and has a maximum turning point at B.

1. Show that the equation of the curve may be written as $$y = x(x - 3)²,$$ and hence write down the coordinates of A. [2]
2. Find the coordinates of B. [5]

The shaded region R is bounded by the curve and the x-axis.

1. Find the area of R. [5]

\includegraphics{figure_3} Fig. 3 Triangle ABC has AB = 9 cm, BC = 10 cm and CA = 5 cm. A circle, centre A and radius 3 cm, intersects AB and AC at P and Q respectively, as shown in Fig. 3.

1. Show that, to 3 decimal places, ∠BAC = 1.504 radians. [3]

Calculate,

1. the area, in cm², of the sector APQ, [2]
2. the area, in cm², of the shaded region BPQC, [3]
3. the perimeter, in cm, of the shaded region BPQC. [4]

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