IB Math AA HL - Questionbank
Complex Numbers
Different Forms, Roots, De Moivre’s Theorem, Argand Diagram, Geometric Applications…
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Question 1
no calculator
easy
[Maximum mark: 6]
Solve the equation , giving your answers in Cartesian form.
easy
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Question 2
no calculator
easy
[Maximum mark: 6]
Consider the complex number where and .
-
Express and in modulus-argument form and write down
-
the modulus of ;
-
the argument of . [4]
-
-
Find the smallest positive integer value of such that is a real number. [2]
easy
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Question 3
no calculator
medium
[Maximum mark: 7]
The complex numbers and satisfy the equations
Find and in the form where .
medium
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Question 4
no calculator
medium
[Maximum mark: 6]
-
Verify that and are the second roots of . [2]
-
Hence, find two distinct roots of the equation , ,
giving your answers in the form where . [4]
medium
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Question 5
no calculator
medium
[Maximum mark: 6]
Consider the equation , where
and , .
Find the value of and the value of .
medium
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Question 6
no calculator
medium
[Maximum mark: 7]
Consider the complex numbers and .
-
Given that , express in the form where . [4]
-
Find and express it in the form . [3]
medium
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Question 7
no calculator
medium
[Maximum mark: 9]
-
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]
medium
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Question 8
calculator
medium
[Maximum mark: 5]
Consider where , .
Show that .
medium
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Question 9
no calculator
medium
[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 .
medium
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Question 10
calculator
medium
[Maximum mark: 7]
Consider the complex numbers
where .
- Express and in modulus-argument form and write down
-
the modulus of .
-
the argument of in terms of .[3]
-
Suppose that .
-
-
Find the minimum value of .
-
For the value of found in part (i), find the value of .[4]
-
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Question 11
no calculator
medium
[Maximum mark: 12]
Consider the complex numbers and .
-
Calculate giving your answer both in modulus-argument form and
Cartesian form. [7]
-
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 12
no calculator
medium
[Maximum mark: 18]
-
Express in the form , where and . [5]
Let the roots of the equation be , and .
-
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.
-
Find the area of the triangle ABC. [4]
-
By considering the sum of the roots , and , show that
[4]
medium
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Question 13
no calculator
medium
[Maximum mark: 8]
Find the six distinct roots of the equation , , giving your answers in the form where .
medium
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Question 14
no calculator
medium
[Maximum mark: 7]
Consider the polynomial equation where .
Two of the roots of this equation are and where .
-
Write down the other two roots in terms of and . [1]
-
Find the possible values of . [6]
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Question 15
calculator
medium
[Maximum mark: 7]
Two distinct roots for the polynomial equation are and where and .
-
Write down the other two roots in terms of and . [1]
-
Find the value of . [2]
-
Hence, or otherwise, find the four roots of the polynomial. [4]
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Question 16
no calculator
hard
[Maximum mark: 15]
Consider where and .
-
If ,
-
write in the form ;
-
find the value of . [5]
-
-
Show that in general,
[4]
-
Find condition under which . [2]
-
State condition under which is:
-
real;
-
purely imaginary. [2]
-
-
Find the modulus of given that . [2]
hard
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Question 17
no calculator
hard
[Maximum mark: 19]
-
-
Expand by using the binomial theorem.
-
Hence use de Moivre's theorem to prove that
-
State a similar expression for in terms of and . [6]
-
Let
,
where is measured in degrees, be the solution
of which has the
smallest positive argument.
-
Find the modulus and argument of . [4]
-
Use (a) (ii) and your answer from (b) to show that . [4]
-
Hence express in the form where . [5]
hard
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Question 18
no calculator
hard
[Maximum mark: 19]
Let , for .
-
-
Find using the binomial theorem.
-
Use de Moivre's theorem to show that and . [8]
-
-
Hence show that . [6]
-
Given that , find the exact value of . [5]
hard
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Question 19
no calculator
hard
[Maximum mark: 22]
-
Solve , for . [5]
-
Show that . [3]
-
Let , for , .
-
Find the modulus and argument of in terms of .
-
Hence find the fourth roots of in modulus-argument form. [14]
-
hard
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Question 20
no calculator
hard
[Maximum mark: 17]
-
Solve the equation , , giving your answer in the form
and in the form where . [6]
-
Consider the complex numbers .
-
Write in the form .
-
Calculate and write in the form where .
-
Hence find the value of in the form where .
-
Find the smallest such that is a positive real number. [11]
-
hard
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Question 21
no calculator
hard
[Maximum mark: 16]
-
Find the roots of which satisfy the condition ,
expressing your answer in the form , where . [5]
-
Let be the sum of the roots found in part (a).
-
Show that .
-
By writing as , find the value of in the form ,
where and are integers to be determined.
-
Hence, or otherwise, show that . [11]
-
hard
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Question 22
no calculator
hard
[Maximum mark: 18]
-
-
Verify that and are the second roots of .
-
Find two distinct roots of the equation , , giving your answers in the form where . [5]
-
-
Find six distinct roots of the equation , , giving your answers in the form where . [8]
On an Argand diagram, , and are represented by the points A, B and C, respectively.
-
-
Verify that , and are the roots of the equation , .
-
Show that the area of the triangle ABC is . [5]
-
hard
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Question 23
no calculator
hard
[Maximum mark: 20]
-
Solve the equation , . [5]
-
Show that . [4]
-
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]
-
hard
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Question 24
no calculator
hard
[Maximum mark: 21]
-
Use de Moivre's theorem to find the value of . [2]
-
Use mathematical induction to prove that
[6]
Let .
-
Find an expression in terms of for , , where is the complex conjugate of . [2]
-
-
Show that .
-
Write down and simplify the binomial expansion of in terms of and .
-
Hence show that . [5]
-
-
Hence solve for . [6]
hard
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Frequently Asked Questions
What is the IB Math AA HL Questionbank?
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.
Where should I start in the AA HL Questionbank?
The AA HL Questionbank is designed to help IB students practice AA HL exam style questions in a specific topic or concept. Therefore, a good place to start is by identifying a concept that you would like to practice and improve in and go to that area of the AA HL Question bank. For example, if you want to practice AA HL exam style questions that involve Complex Numbers, you can go to AA SL Topic 1 (Number & Algebra) and go to the Complex Numbers area of the question bank. On this page there is a carefully designed set of IB Math AA HL exam style questions, progressing in order of difficulty from easiest to hardest. If you’re just getting started with your revision, you could start at the top of the page with Question 1, or if you already have some confidence, you could start at the medium difficulty questions and progress down.
How should I use the AA HL Questionbank?
The AA HL Questionbank is perfect for revising a particular topic or concept, in-depth. For example, if you wanted to improve your knowledge of Counting Principles (Combinations & Permutations), there is a set of full length IB Math AA HL exam style questions focused specifically on this concept. Alternatively, Revision Village also has an extensive library of AA HL Practice Exams, where students can simulate the length and difficulty of an IB exam with the Mock Exam sets, as well as AA HL Key Concepts, where students can learn and revise the underlying theory, if missed or misunderstood in class.
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With an extensive and growing library of full length IB Math Analysis and Approaches (AA) HL exam style questions in the AA HL Question bank, finishing all of the questions would be a fantastic effort, and you will be in a great position for your final exams. If you were able to complete all the questions in the AA HL Question bank, then a popular option would be to go to the AA HL Practice Exams section on Revision Village and test yourself with the Mock Exam Papers, to simulate the length and difficulty of an actual IB Mathematics AA HL exam.