Edexcel A Level Maths C4 · Core Mathematics 4 January 2008 question paper 6666

Use the trapezium rule, with all the values in the completed table, to obtain an estimate for the area of the region R. Give your answer to 4 decimal places.

Use the binomial theorem to expand

simplified fraction.

curve, the x-axis and the lines x = a and x = b is shown shaded in Figure 2. This region is rotated through 360° about the x- axis to generate a solid of revolution. Find the volume of the solid generated. Express your answer as a single simplified fraction, in terms of a and b.

Find the exact value of sin²x dx.

A curve is described by the equation

Find the coordinates of the two points on the curve where x = –8.

Find the gradient of the curve at each of these points.

The points A and B have position vectors 2 i + 6 j – k and 3 i + 4 j + k respectively.

Find the position vector of the point C.

The curve C has parametric equations

The finite region R between the curve C and the x-axis, bounded by the lines with equations x = ln 2 and x = ln 4, is shown shaded in Figure 3.

Show that the area of R is given by the integral

Hence find an exact value for this area.

Find a cartesian equation of the curve C, in the form y = f(x).

State the domain of values for x for this curve.

Liquid is pouring into a large vertical circular cylinder at a constant rate of 1600 cm³ s^–1 and is leaking out of a hole in the base, at a rate proportional to the square root of the height of the liquid already in the cylinder. The area of the circular cross section of the cylinder is 4000 cm².

Show that at time t seconds, the height h cm of liquid in the cylinder satisfies the differential equation

When h = 25, water is leaking out of the hole at 400 cm³ s^–1.

Show that k = 0.02

Separate the variables of the differential equation

to show that the time taken to fill the cylinder from empty to a height of 100 cm is given by

Hence find the time taken to fill the cylinder from empty to a height of 100 cm, giving your answer in minutes and seconds to the nearest second.