Edexcel A Level Maths C4 · Core Mathematics 4 June 2013 question paper 6666

Use the binomial expansion to show that

Give your answer in the form where a and b are integers.

Figure 1 shows the finite region R bounded by the x-axis, the y-axis, the line x and the curve with equation

Complete the table above giving the missing value of y to 6 decimal places.

Using the trapezium rule, with all of the values of y from the completed table, find an approximation for the area of R, giving your answer to 4 decimal places.

Region R is rotated through 2 radians about the x-axis.

Use calculus to find the exact volume of the solid formed.

A curve C has parametric equations

Find at the point where t =

Find a cartesian equation for C in the form

stating the value of the constant k.

Write down the range of f(x).

Use the substitution x = u², u > 0, to show that

Hence show that

where a and b are integers to be determined.

Water is being heated in a kettle. At time t seconds, the temperature of the water is °C. The rate of increase of the temperature of the water at any time t is modelled by the differential equation

where is a positive constant. Given that θ = 20 when t = 0,

solve this differential equation to show that

When the temperature of the water reaches 100 °C, the kettle switches off.

Given that λ = 0.01, find the time, to the nearest second, when the kettle switches off.

A curve is described by the equation

A point Q lies on the curve. The tangent to the curve at Q is parallel to the y-axis. Given that the x coordinate of Q is negative,

use your answer to part (a) to find the coordinates of Q.

With respect to a fixed origin O, the line l has equation

The point A lies on l and has coordinates (3, – 2, 6). The point P has position vector (– p i + 2 p k) relative to O, where p is a constant. Given that vector PA is perpendicular to l,

find the value of p.

Given also that B is a point on l such that BPA = 45°,

find the coordinates of the two possible positions of B.