Practice Paper 3 (Pure 3) (CIE A Level Maths: Pure 1)

Given that :

Shade the region of your diagram that satisfies the inequality                                      .

Find an expression for    in terms of t for the parametric equations

Verify that the graph of x against y passes through the point (0 , 1) and find the gradient at that point.

Solve the equation

giving your answer correct to 3 significant figures.

Solve the equation

giving your answers in the form .

The function, is defined by

Show that the equation can be written in the form

On the same diagram sketch the graphs of and .

The equation has a root, α, close to .
The iterative formula  x_n+1=e^-x_n+1 with x₀=2 is to be used to find correct to three significant figures.

Show, using a diagram and your answer to part (b), that this formula and initial x value will converge to the root  .

Use the identity

sincos sin sin cos 

to show that

sin  cos

can be written as

sin

Hence, or otherwise, solve the equation sin cos for  .
Give your answers to three significant figures.

Show that   is a root of the cubic equation

Find the other two roots of the cubic equation in part (a), being sure to show clear algebraic working.

Find the particular solution to the following differential equation, using the given boundary conditions.

Express  in partial fractions.

Express as the first three terms of a binomial expansion in ascending powers of .

Write    in the form  where , and are all linear functions of .

Hence show that

The coordinates of three points are (—3, 4, 0), (2, —2, —3) and (—l, —1, —4).

Calculate the angle between  and . Give your answer in degrees, accurate to 1 decimal place.

The coordinates of three points are (—2, 3, —5), (2, —5, —9) and (—10, —1, —1).

The point  is the midpoint of , and the point  lies on .

Given that  = , find the equation of the line through points  and in vector form.

Show that

where  and  are constants.