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

Find the value of at the point where x = 2 on the curve with equation

Express f (x) as a single fraction in its simplest form.

Figure 1 shows the graph of y = f (x), 1 < x < 9. The points T (3, 5) and S (7, 2) are turning points on the graph. Sketch, on separate diagrams, the graphs of

y = 2f (x) – 4,

Indicate on each diagram the coordinates of any turning points on your sketch.

Find the equation of the tangent to the curve x cos(2y ) at 0 . Give your answer in the form y = ax + b, where a and b are constants to be found.

The functions f and g are defined by

Write down the range of g.

Show that the composite function fg is defined by

Write down the range of fg.

By writing 3 = (2 + ), show that

Hence, or otherwise, for 0 solve

Give your answers in terms of π.

Using sin( ) sin cos cos sin , or otherwise, show that

f (x) 3xe^x 1 The curve with equation y = f (x) has a turning point P.

Find the exact coordinates of P.

The equation f (x) = 0 has a root between x = 0.25 and x = 0.3

Use the iterative formula

with x₀ = 0.25 to find, to 4 decimal places, the values of x₁, x₂ and x₃.

By choosing a suitable interval, show that a root of f (x) = 0 is x = 0.2576 correct to 4 decimal places.

Express 3 cos θ + 4 sin θ in the form R cos(θ – α), where R and α are constants, R > 0 and 0 < α < 90°.

Hence find the maximum value of 3 cos θ + 4 sin θ and the smallest positive value of θ for which this maximum occurs.

The temperature, f (t), of a warehouse is modelled using the equation f (t) = 10 + 3 cos(15 t)° + 4 sin(15 t)°, where t is the time in hours from midday and 0 t < 24.

Calculate the minimum temperature of the warehouse as given by this model.

Find the value of t when this minimum temperature occurs.