Edexcel A Level Maths C4 · Core Mathematics 4 June 2016 question paper 6666

Use the binomial series to find the expansion of

in ascending powers of x, up to and including the term in x³. Give each coefficient as a fraction in its simplest form.

Figure 1 shows a sketch of part of the curve with equation y = x²ln x, x 1 The finite region R, shown shaded in Figure 1, 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 = x²ln x

Complete the table above, giving the missing value 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 3 decimal places.

Use integration to find the exact value for the area of R.

The curve C has equation 2x²y + 2x + 4y – cos () = 17

Use implicit differentiation to find in terms of x and y. The point P with coordinates 3, lies on C. The normal to C at P meets the x-axis at the point A.

Find the x coordinate of A, giving your answer in the form , where a, b, c and d are integers to be determined.

The rate of decay of the mass of a particular substance is modelled by the differential equation

where x is the mass of the substance measured in grams and t is the time measured in days. Given that x = 60 when t = 0,

solve the differential equation, giving x in terms of t. You should show all steps in your working and give your answer in its simplest form.

Find the time taken for the mass of the substance to decay from 60 grams to 20 grams. Give your answer to the nearest minute.

Figure 2 shows a sketch of the curve C with parametric equations

The point P lies on C and has coordinates 4 .

Find the exact value of at the point P. Give your answer as a simplified surd.

The point Q lies on the curve C, where 0

Find the exact coordinates of the point Q.

Given that y 0, find

Use the substitution x = 4sin² to show that

Hence use integration to find

giving your answer in the form a + b, where a and b are exact constants.

Find

giving your answer in its simplest form.

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

The curve C cuts the line y = 8 at the point P with coordinates (k, 8), where k is a constant.

Find the value of k.

The finite region S, shown shaded in Figure 3, is bounded by the curve C, the x-axis, the y-axis and the line y = 8. This region is rotated through 2 radians about the x-axis to form a solid of revolution.

Find the exact value of the volume of the solid generated.

With respect to a fixed origin O, the line l₁ is given by the equation

where is a scalar parameter. The point A lies on l₁ where = 1

Find the coordinates of A.

1

The line l₂ passes through the point P and is parallel to the line l₁

Write down a vector equation for the line l₂

Find the exact value of the dista AP.

The acute angle between AP and l₂ is .

Find the value of cos

A point E lies on the line l₂ Given that AP = PE,

find the area of triangle APE,

find the coordinates of the two possible positions of E.