Edexcel A Level Maths C4 · Core Mathematics 4 June 2013 question paper 6666R

Express in partial fractions

The curve C has equation Show that at the point (1, 3) on the curve C can be written in the form where A and yw are integers to be found. (7)

value of

Find the binomial expansion of

in ascending powers ofx, up to and including the term in x°. Give each coefficient as

Use your expansion to estimate an approximate value for 7100, giving your to 4 decimal places. State the value ofx, which you use in your expansion, and show all your working. (3)

3 decimal places.

Use the trapezium rule with all the values of x in the completed table to obtain an estimate for the area of the region R, giving your to 2 decimal places. (3)

Use calculus to find the exact value for the area of R. (6)

Find the difference between the values obtained in part (b) and part (c), giving your to 2 decimal places. (1)

The line / has vector equation

where a, b and c are constants and 4 is a parameter. Given that the point A lies on the line /,

find the value of a. (3) Given also that the vector 4B is perpendicular to /

find the distance AB. (2) The image of the point B after reflection in the line / is the point B’.

Find the position vector of the point B’. (2)

Figure 2 showsa sketch of the curve C with parametric equations

Show that the cartesian equation of C may be written in the form

stating the values of a and b. (3)

shown shaded in Figure 3. This region is rotated through 27 radians about the x-axis to form a solid of revolution.

Use calculus to find the exact value of the volume of the solid of revolution. (5)

In an experiment testing solid rocket fuel, some fuel is burned and the waste products are collected. Throughout the experiment the sum of the masses of the unburned fuel and waste products remains constant. Letx be the mass ofwaste products, in kg, at time ¢ minutes after the start ofthe experiment. It is known that at time ¢ minutes, the rate of increase of the mass of waste products, inconstant.kg per minute, is k times the mass of unburned fuel remaining, where k is a positive

The differential equation connecting x and ¢ may be written in the form

Explain, in the context of the problem, what and M represent. (2)

Given that initially the mass of waste products is zero,

solve the differential equation, expressing x in terms of k, M and ¢. (6)