Edexcel A Level Maths C4 · Core Mathematics 4 June 2015 question paper 6666

Find the binomial expansion of

in ascending powers of x, up to and including the term in x². Give each coefficient in its simplest form.

Find the exact value of (4 5x when x =

Substitute x = into your binomial expansion from part (a) and hence find an

Give your answer in the form where p and q are integers.

The curve C has equation

Find the coordinates of the points on C where = 0 (Solutions based entirely on graphical or numerical methods are not acceptable.)

The curve meets the x-axis at the origin O and cuts the x-axis at the point A.

Find, in terms of ln2, the x coordinate of the point A.

Find

The finite region R, shown shaded in Figure 1, is bounded by the x-axis and the curve with equation

Find, by integration, the exact value for the area of R. Give your answer in terms of ln2

With respect to a fixed origin O, the lines l₁ and l₂ are given by the equations 5 0 8 3

where λ and are scalar parameters and p is a constant. The lines l₁ and l₂ intersect at the point A.

Find the coordinates of A.

Find the value of the constant p.

Find the acute angle between l₁ and l₂, giving your answer in degrees to 2 decimal places.

The point B lies on l₂ where = 1

Find the shortest distance from the point B to the line l₁, giving your answer to 3 significant figures.

A curve C has parametric equations

Find the value of at the point on C where t = 2, giving your answer as a fraction in its simplest form.

Show that the cartesian equation of the curve C can be written in the form

where a and b are integers to be determined.

The finite region R, shown shaded in Figure 2, is bounded by the curve, the x-axis, and the y-axis.

Use the substitution x = 1 + 2sinθ to show that

where k is a constant to be determined.

Hence find, by integration, the exact area of R.

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A team of biologists is studying a population of a particular species of animal. The population is modelled by the differential equation

where P is the population in thousands, and t is the time measured in years since the start of the study. Given that P = 3 when t = 0,

solve this differential equation to show that

find the time taken for the population to reach 4000 for the first time. Give your answer in years to 3 significant figures.

Figure 3 shows a sketch of part of the curve C with equation

The point P lies on C and has coordinates (2, 9). The line l is a tangent to C at P. The line l cuts the x-axis at the point Q.

Find the exact value of the x coordinate of Q.

The finite region R, shown shaded in Figure 3, is bounded by the curve C, the x-axis, the y-axis and the line l. This region R is rotated through 360° about the x-axis.

Use integration to find the exact value of the volume of the solid generated. Give your answer in the form where p and q are exact constants. [You may assume the formula V = r₂h for the volume of a cone.]