# Edexcel Core Mathematics C3 June 2009

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### C3 Mathematics Edexcel June 2009 Question 1

Figure 1 shows part of the curve with equation y = -x3 + 2x2 + 2, which intersects the x-axis at the point A where x = α.
To find an approximation to α, the iterative formula
xn+1 = 2/(xn)2 + 2
is used.
(a) Taking x0= 2.5, find the values of x1, x2, x3 and x4
(b) Show that α = 2.359 correct to 3 decimal places.

### C3 Mathematics Edexcel June 2009 Question 2

2. (a) Use the identity cos2 θ + sin2 θ = 1 to prove that tan2 θ = sec2 θ – 1.
(b) Solve, for 0 ≤ θ < 360°, the equation
2 tan2 θ + 4 sec θ + sec2 θ = 2

### C3 Mathematics Edexcel June 2009 Question 3

3. Rabbits were introduced onto an island. The number of rabbits, P, t years after they were introduced is modelled by the equation
P = 80e1/3t, t ∈ ℝ, t ≥ 0
(a) Write down the number of rabbits that were introduced to the island.
(b) Find the number of years it would take for the number of rabbits to first exceed 1000.
(c) Find dP/dt
(d) Find P when dP/dt 50.

### C3 Mathematics Edexcel June 2009 Question 4

4. (i) Differentiate with respect to x
(a) x2 cos 3x
(b) ln(x2 + 1)/(x2 + 1)

(ii) A curve C has the equation
y = √(4x + 1), x > -¼ , y > 0
The point P on the curve has x-coordinate 2. Find an equation of the tangent to C at P in the form ax + by + c = 0, where a, b and c are integers.

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