Edexcel A Level Maths C4 · Core Mathematics 4 January 2009 question paper 6666

A curve C has the equation y² – 3y = x³ + 8.

Hence find the gradient of C at the point where y = 3.

Figure 1 shows part of the curve y The region R is bounded by the curve, the x-axis, and the lines x = 0 and x = 2, as shown shaded in Figure 1.

Use integration to find the area of R.

The region R is rotated 360° about the x-axis.

Use integration to find the exact value of the volume of the solid formed.

₂

Given that f (x) can be expressed in the form

find the values of B and C and show that A = 0.

Hence, or otherwise, find the series expansion of f (x), in ascending powers of x, up to and including the term in x². Simplify each term.

Find the percentage error made in using the series expansion in part (b) to estimate the value of f (0.2). Give your answer to 2 significant figures.

With respect to a fixed origin O the lines l₁ and l₂ are given by the equations

where λ and μ are parameters and p and q are constants. Given that l₁ and l₂ are perpendicular,

show that q = –3.

Given further that l₁ and l₂ intersect, find

the value of p,

the coordinates of the point of intersection.

Given that a circle, with centre C, cuts the line l₁ at the points A and B,

find the position vector of B.

A container is made in the shape of a hollow inverted right circular cone. The height of the container is 24 cm and the radius is 16 cm, as shown in Figure 2. Water is flowing into the container. When the height of water is h cm, the surface of the water has radius r cm and the volume of water is V cm³.

[ The volume V of a right circular cone with vertical height h and base radius r is given

Water flows into the container at a rate of 8 cm³ s^–1.

Find, in terms of π, the rate of change of h when h = 12.

Use the substitution u^x to show that = 1 + e

where k is a constant.

The curve C shown in Figure 3 has parametric equations

where t is a parameter. Given that the point A has parameter t = –1,

find the coordinates of A.

The line l is the tangent to C at A.

Show that an equation for l is 2 x – 5 y – 9 = 0.

The line l also intersects the curve at the point B.

Find the coordinates of B.