Edexcel A Level Maths C3 · Core Mathematics 3 June 2009 question paper 6665

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

is used.

Taking x₀ = 2.5, find the values of x₁, x₂, x₃ and x₄. Give your answers to 3 decimal places where appropriate.

Show that α = 2.359 correct to 3 decimal places.

Use the identity cos²θ + sin²θ = 1 to prove that tan²θ = sec² θ – 1.

Solve, for 0 ≤ θ < 360°, the equation

Rabbits were introduced onto an island. The number of rabbits, P, t years after they were introduced is modelled by the equation

Write down the number of rabbits that were introduced to the island.

Find the number of years it would take for the number of rabbits to first exceed 1000.

Differentiate with respect to x

A curve C has the equation

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.

The curve meets the coordinate axes at the points A(0,1–k) and B (–¹₂ k ln,0), where k is a constant and k 1, as shown in Figure 2. On separate diagrams, sketch the curve with equation

y = f ¹(x).

Show on each sketch the coordinates, in terms of k, of each point at which the curve meets or cuts the axes.

Given that f (x) = e^2x − k,

state the range of f,

write down the domain of f ¹.

Use the identity cos(A + B) = cos Acos B sin Asin B, to show that

The curves C₁ and C₂ have equations

Show that the x-coordinates of the points where C₁ and C₂ intersect satisfy the equation

Express 4cos2x + 3sin 2x in the form R cos (2x – α), where R > 0 and 0 < α < 90°, giving the value of α to 2 decimal places.

Hence find, for 0 x 180°, all the solutions of

giving your answers to 1 decimal place.

The function f is defined by

The function g is defined by

Find the exact values of x for which g ( x) = 1

Write down sin 2 x in terms of sin x and cos x.

Find, for 0 < x < π, all the solutions of the equation

giving your answers to 2 decimal places.