Edexcel A Level Maths C3 · Core Mathematics 3 June 2013 question paper 6665R

Express

as a single fraction in its simplest form. (4)

asymptote to the curve. On separate diagrams, sketch the curve with equation

(3) Indicate clearly on each sketch the coordinates of the point at which the curve crosses or meets the x-axis.

find the exact value of R and the value of a to one decimal place. (3)

Hence solve the equation

State the values of & for which the equation

The functions f and g are defined by

Find g", the inverse function of g. (2)

Solve the equation

Differentiate

Show that (sec? 3x) can be written in the form

where yu is a constant. (3)

Use an appropriate double angle formula to show that

and state the value of the constant /. (3)

You must show all your working. Give your answers in terms of z. (6)

The curve cuts the x-axis at points A and B as shown in Figure 2.

Calculate the x coordinate of A and the x coordinate of B, giving your answers to 3 decimal places. (2)

Find f/(x). (3) The curve has a minimum turning point at the point P as shown in Figure 2.

Show that the x coordinate of P is the solution of

Use the iteration formula

to calculate the values of x), x. and x3, giving your answers to 3 decimal places. (3)

The x coordinate of P is a.

The population of a town is being studied. The population P, at time ¢ years from the start of the study, is assumed to be

wherek is a positive constant. The graph of P against is shown in Figure 3. Use the given equation to

find the population at the start of the study, (2)

find a value for the expected upper limit of the population. (1)

Given also that the population reaches 2500 at 3 years from the start of the study,

calculate the value of k to 3 decimal places. (5) Using this value for k,

find the population at 10 years from the start of the study, giving your to 3 significant figures. (2)

Find, using the rate at which the population is growing at 10 years from the start of the study. (3)