IB Mathematics AA HL - Questionbank
Complex Numbers
Different Forms, Roots, De Moivre’s Theorem, Argand Diagram, Geometric Applications…
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Question 1
[Maximum mark: 4]
On the Argand diagram below, the point A represents the complex number and the point B represents the complex number . The shape ABCD is a square.
Determine the complex number represented by:
-
the point C; [2]
-
the point D. [2]
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Question 2
[Maximum mark: 6]
Solve the equation , giving your answers in Cartesian form.
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Question 3
[Maximum mark: 6]
A circle of radius 3 and centre (0,3) is drawn on an Argand diagram. The tangent to the circle from the point B meets the circle at the point A as shown. Let .
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Show that . [2]
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Find . [2]
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Hence write in the form where . [2]
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Question 4
[Maximum mark: 6]
In this question give all angles in radians.
Let and .
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Find . [1]
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Find:
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;
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. [3]
-
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Find , the angle shown on the diagram below. [2]
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Question 5
[Maximum mark: 6]
Consider the complex number where and .
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Express and in modulus-argument form and write down
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the modulus of ;
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the argument of . [4]
-
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Find the smallest positive integer value of such that is a real number. [2]
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Question 6
[Maximum mark: 8]
Let and .
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Find . [2]
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Illustrate , and on the same Argand diagram. [3]
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Let be the angle between and . Find , giving your answer in radians.[3]
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Question 7
[Maximum mark: 6]
Let where .
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For ,
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express and in the form where ;
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draw and on the following Argand diagram. [4]
-
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Given that the integer powers of lie on a unit circle centred
at the origin, find the value of . [2]
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Question 8
[Maximum mark: 6]
Consider the equation , where
and , .
Find the value of and the value of .
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Question 9
[Maximum mark: 6]
Let where .
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For ,
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express and in the form where ;
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draw and on the following Argand diagram. [4]
-
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Given that the integer powers of lie on a unit circle centred at the origin, find the value of . [2]
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Question 10
[Maximum mark: 6]
The complex numbers and correspond to the points A and B as shown on the diagram below.
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Find the exact value of . [2]
-
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Find the exact perimeter of triangle AOB.
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Find the exact area of triangle AOB. [4]
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Question 11
[Maximum mark: 7]
Points A and B represent the complex numbers and as shown on the Argand diagram below.
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Find the angle AOB. [3]
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Find the argument of . [1]
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Given that the real powers of , for , all lie on a unit circle centred at the origin, find the exact value of . [3]
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Question 12
[Maximum mark: 7]
The complex numbers and satisfy the equations
Find and in the form where .
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Question 13
[Maximum mark: 6]
Let where . Find the modulus and argument of , expressing your answers in terms of .
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Question 14
[Maximum mark: 8]
Let .
-
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Write , and in the form where .
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Draw , and on an Argand diagram. [6]
-
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Find the smallest integer such that is a real number. [2]
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Question 15
[Maximum mark: 7]
Consider the equation where and .
Three of the roots of the equation are , and , where .
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Find the value of .[4]
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Hence, or otherwise, find the value of .[3]
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Question 16
[Maximum mark: 9]
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Find three distinct roots of the equation , , giving your answers in modulus-argument form. [6]
The roots are represented by the vertices of a triangle in an Argand diagram.
- Show that the area of the triangle is .
[3]
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Question 17
[Maximum mark: 7]
Consider the complex numbers and .
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Given that , express in the form where . [4]
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Find and express it in the form . [3]
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Question 18
[Maximum mark: 7]
Consider the equation , where and . Two of the roots of the equation are and and the sum of all the roots is .
Show that .
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Question 19
[Maximum mark: 18]
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Express in the form , where and . [5]
Let the roots of the equation be , and .
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Find , and expressing your answers in the form , where and . [5]
On an Argand diagram, , and are represented by the points A, B and C, respectively.
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Find the area of the triangle ABC. [4]
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By considering the sum of the roots , and , show that
[4]
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Question 20
[Maximum mark: 12]
Consider the complex numbers and .
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Calculate giving your answer both in modulus-argument form and
Cartesian form. [7]
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Use your results from part (a) to find the exact value of ,
giving your answer in the form where . [5]
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Question 21
[Maximum mark: 15]
Consider where and .
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If ,
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write in the form ;
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find the value of . [5]
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Show that in general,
[4]
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Find condition under which . [2]
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State condition under which is:
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real;
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purely imaginary. [2]
-
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Find the modulus of given that . [2]
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Question 22
[Maximum mark: 6]
On an Argand diagram, the complex numbers , and are represented by the vertices of a triangle.
The exact area of the triangle can be expressed in the form . Find the value of and of .
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Question 23
[Maximum mark: 19]
Let , for .
-
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Find using the binomial theorem.
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Use de Moivre's theorem to show that and . [8]
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Hence show that . [6]
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Given that , find the exact value of . [5]
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Question 24
[Maximum mark: 19]
-
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Expand by using the binomial theorem.
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Hence use de Moivre's theorem to prove that
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State a similar expression for in terms of and . [6]
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Let
,
where is measured in degrees, be the solution
of which has the
smallest positive argument.
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Find the modulus and argument of . [4]
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Use (a) (ii) and your answer from (b) to show that . [4]
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Hence express in the form where . [5]
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Question 25
[Maximum mark: 22]
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Solve , for . [5]
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Show that . [3]
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Let , for , .
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Find the modulus and argument of in terms of .
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Hence find the fourth roots of in modulus-argument form. [14]
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Question 26
[Maximum mark: 16]
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Find the roots of which satisfy the condition ,
expressing your answer in the form , where . [5]
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Let be the sum of the roots found in part (a).
-
Show that .
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By writing as , find the value of in the form ,
where and are integers to be determined.
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Hence, or otherwise, show that . [11]
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Question 27
[Maximum mark: 17]
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Solve the equation , , giving your answer in the form
and in the form where . [6]
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Consider the complex numbers .
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Write in the form .
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Calculate and write in the form where .
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Hence find the value of in the form where .
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Find the smallest such that is a positive real number. [11]
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Question 28
[Maximum mark: 21]
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Use de Moivre's theorem to find the value of . [2]
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Use mathematical induction to prove that
[6]
Let .
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Find an expression in terms of for , , where is the complex conjugate of . [2]
-
-
Show that .
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Write down and simplify the binomial expansion of in terms of and .
-
Hence show that . [5]
-
-
Hence solve for . [6]
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Question 29
[Maximum mark: 20]
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Solve the equation , . [5]
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Show that . [4]
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Let , for , .
-
Find the modulus and argument of . Express each answer
in its simplest form. -
Hence find the fourth roots of in modulus-argument form. [11]
-
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Question 30
[Maximum mark: 28]
This questions will investigate a method to obtain Cardano's formula using elementary properties of complex numbers and algebraic expressions.
- The roots, of magnitude , described above can be written polar form such that . Where, the argument , is .
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Show that and find the remaining roots, and , in polar form.
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Show that .
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Show that .
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Hence, show that in Cartesian form. [7]
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Consider the cubic equation for , .
Assume we can write it in a factorised form such that .
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- Identify and sum the roots of the factorised form above and then use some of the relations relations found in part (a) to show that
Another formula for the roots , and of the cubic equation states that .
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Using the roots identified from (i), the formula above and some of the relations found in part (a) show that .
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Using parts (i) and (ii), write and in terms of and .
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Hence, show that is a root of the cubic equation . [12]
Now, consider a more general cubic equation , where , .
-
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By substituting and prove Cardano's formula, which states that
is a root of the cubic equation .
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Use Cardano's formula to find the exact solution to .
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Using the substitution and Cardano's formula, find an exact solution to .[9]
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The IB Math Analysis and Approaches (AA) HL Questionbank is a comprehensive set of IB Mathematics exam style questions, categorised into syllabus topic and concept, and sorted by difficulty of question. The bank of exam style questions are accompanied by high quality step-by-step markschemes and video tutorials, taught by experienced IB Mathematics teachers. The IB Mathematics AA HL Question bank is the perfect exam revision resource for IB students looking to practice IB Math exam style questions in a particular topic or concept in their AA Higher Level course.
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