Argand Diagrams (Cambridge (CIE) A Level Maths: Pure 3): Exam Questions

The solutions to the quadratic equation  are  and .

Work out the values of  and , giving your answers in the form  where  and  are integers.

On an Argand diagram, show the complex numbers and  in position vector form.

Show that 

Find, showing your working:

(i)

(ii) , giving your answer in radians to 2 decimal places.

Show  and  on an Argand diagram.

The complex numbers  and are such that  and

Show that .

Given that ,  find the two possible values of .

Given additionally that ,  find the precise value of .

Express the complex number  in the form  ,  where  is a positive real number and  is given in radians correct to 2 decimal places.  The value for  should be given in the interval  .

A complex number has modulus 8 and argument  .  Express the number in the form  , where  and  are real numbers.

(i) Find  , giving your answer in the form    where  and  are real numbers.

(ii) Calculate the modulus and argument of each of the complex numbers and , and show that these satisfy the standard results

(i) Using your answers to part (a) and the standard results

       calculate the values of  and .

(ii) Hence express   in the form  ,  where  is a positive real number and  is given in radians correct to 2 decimal places.

Work out

(i)

(ii)

giving your answers in modulus-argument form.

Given that  :

(i) On an Argand diagram, sketch the locus (i.e., set of points) for which the equation is true.

(ii) Shade the region of your diagram that satisfies the inequality .

Given that :

(i) On an Argand diagram, sketch the locus (i.e., set of points) for which the equation is true.

(ii) Shade the region of your diagram that satisfies the inequality  .

Show that  and  are roots of the cubic equation , and write down the third root of the equation.

Verify that there is a constant  such that all three roots of the cubic equation  satisfy

.

(i) Draw an Argand diagram showing the locus of points representing all complex numbers for which .

Mark the points corresponding to the three roots of the cubic equation .

(ii) Shade the region of your diagram that satisfies the inequality .

The solutions to the cubic equation  are  and , where  .

Given that , work out the other two solutions to the equation.

(i) On an Argand diagram, show the complex numbers and  as position vectors.

(ii) Describe the geometric transformation that maps  to .

Find, showing your working:

(i)

(ii)

(iii) , giving your answer in radians to 2 decimal places.

The complex numbers  and are such that  and 

Find  in the form  , giving the real numbers  and  in terms of .

Given that ,  find the possible values of .

Given additionally that   in radians to 2 decimal places, determine the exact value of  .

Express the complex number   in the form  ,  where  is a positive real number and  is given in radians correct to 2 decimal places.  The value for  should be given in the interval  .

A complex number has modulus  and argument  . Express the number in the form  , where  and  are real numbers.

For the general complex numbers

given in modulus-argument form, use algebra and the appropriate trigonometric compound angle formulae to prove the results

Work out 

(i)

(ii)

giving your answers in modulus-argument form with  in the interval  .

Given that :

(i) On an Argand diagram, sketch the locus (i.e., set of points) for which the equation is true.

(ii) Shade the region of your diagram that satisfies the inequality                                      .

On an Argand diagram, sketch the loci (i.e., sets of points) for which each of the following equations is true:

(i)

(ii)

Shade the region of your diagram that satisfies both of the following inequalities:

You are given that  satisfies the cubic equation

Find the other two roots of the cubic equation.

Verify that there is a constant  such that all three roots of the cubic equation satisfy

.

Draw an Argand diagram showing the locus of points representing all complex numbers  for which  .

Mark the points corresponding to the three roots of the cubic equation.

Given that  is a root of the equation  ,  use algebra to solve  completely.

Show all three solutions on an Argand diagram.

For a general complex number ,  where  and  :

(i) Show that

(ii) Show that

(iii) Given that , write down in radians the principal argument of  .

The complex numbers  and are such that  and 

Given that , find the possible values of .  Be sure to show clear algebraic working.

Given additionally that   in radians to 2 decimal places, find the exact value of .

Express  in the form  , where  is a positive real number and  is given as an exact value in radians.  The value for  should be given in the interval  .

A complex number is such that  and .

Express  in the form  , where  and  are real numbers.

For the general complex numbers

given in modulus-argument form, where  ,  use algebra and the appropriate trigonometric compound angle formulae to prove the results

Re-express  and  in correct modulus-argument form with in the interval  .

Work out

(i)

(ii)

giving your answers in modulus-argument form with  in the interval  .

(i) On an Argand diagram, show the complex numbers  and  as position vectors.

(ii) Use your diagram to explain briefly why we speak of a ‘parallelogram rule’ for complex number addition.

On an Argand diagram, shade the region which satisfies both of the following inequalities:

On an Argand diagram, shade the region which satisfies all three of the following inequalities:

Given that

find all three roots of the cubic equation . Be sure to show clear algebraic working.

Verify that there is a constant  such that all three roots of the cubic equation satisfy

Draw an Argand diagram showing the locus of points representing all complex numbers  for which .

Mark the points corresponding to the three roots of the cubic equation  .