Edexcel A Level Maths C4 · Core Mathematics 4 June 2011 question paper 6666

Find the values of the constants A, B and C.

Find the first three non-zero terms of the binomial expansion of f (x) in ascending powers of x. Give each coefficient as a simplified fraction.

A hollow hemispherical bowl is shown in Figure 1. Water is flowing into the bowl. When the depth of the water is h m, the volume V m³ is given by

Water flows into the bowl at a rate of m s.

Find the rate of change of h, in m s, when h = 0.1

Figure 2 shows a sketch of the curve with equation y = x³ ln (x² + 2), x0. The finite region R, shown shaded in Figure 2, is bounded by the curve, the x-axis and the line x = ˲ 2. The table below shows corresponding values of x and y for y = x³ ln (x² + 2).

Complete the table above giving the missing values of y to 4 decimal places.

Use the trapezium rule, with all the values of y in the completed table, to obtain an estimate for the area of R, giving your answer to 2 decimal places.

Use the substitution u = x²+ 2 to show that the area of R is

Hence, or otherwise, find the exact area of R.

Find the gradient of the curve with equation ln y = 2 x ln x, x 0, y 0 at the point on the curve where x = 2. Give your answer as an exact value.

With respect to a fixed origin O, the lines l and l are given by the equations

where and — are scalar parameters.

Show that l₁ and l₂ meet and find the position vector of their point of intersection A.

Find, to the nearest 0.1, the acute angle between l₁ and l₂.

Show that B lies on l₁.

Find the shortest distance from B to the line l₂, giving your answer to 3 significant figures.

Figure 3 shows part of the curve C with parametric equations

The point P lies on C and has coordinates ˲ 3, 3.

Find the value of at the point P.

The line l is a normal to C at P. The normal cuts the x-axis at the point Q.

Show that Q has coordinates k ˲ 3, 0, giving the value of the constant k.

The finite shaded region S shown in Figure 3 is bounded by the curve C, the line x ˲3 and the x-axis. This shaded region is rotated through 2 radians about the x-axis to form a solid of revolution.

Find the volume of the solid of revolution, giving your answer in the form p ¥ 3 + q ², where p and q are constants.

Given that y 1.5 at x = – 2, solve the differential equation

giving your answer in the form y = f (x).