IB Mathematics AI HL - Questionbank
Complex Numbers
Cartesian/ Polar/ Euler Form, Operations (+-x/) & Powers, Argand Diagrams, Applications with Sin/ Cos Waves...
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Question 1
[Maximum mark: 5]
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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. [2]
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Find , the angle shown on the diagram below. [2]
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Question 2
[Maximum mark: 7]
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.[2]
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Question 3
[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 4
[Maximum mark: 6]
Let , , and be represented by the points
A, B and C on an Argand diagram as shown below.
The shape OABC is a rectangle.
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Show that . [1]
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Find . [1]
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Express in modulus-argument form. [4]
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Question 5
[Maximum mark: 6]
A circle is drawn on an Argand diagram as shown below. 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 6
[Maximum mark: 6]
On an Argand diagram, the complex numbers , and are represented by the vertices of a triangle.
Find the area of the triangle.
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Question 7
[Maximum mark: 8]
Consider two power sources with voltages and respectively, where is the frequency of the power source in Hertz (Hz) and is time in seconds.
It is given that
When the two power sources are combined, the resultant voltage is given by
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Find . Give your answer in the form , where .[6]
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Hence, determine the difference in phase shift between and .[2]
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Question 8
[Maximum mark: 6]
Points A and B represent the complex numbers and as shown on an Argand diagram below.
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Find the angle AOB. [2]
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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 9
[Maximum mark: 8]
Let and , where .
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Find the value of . Give your answer in the form , where , . [2]
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Find the value of for . Give your answer in the form , where , . [3]
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Find the smallest integer such that . [3]
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Question 10
[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 11
[Maximum mark: 6]
Let where .
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For ,
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express and in the form where ;
-
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 12
[Maximum mark: 5]
Two voltage sources are connected to a circuit. At time milliseconds (ms), the voltage from the first source is and the voltage from the second source is , where both and are measured in volts.
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Write, in the form , an expression for the total voltage in the circuit at time ms. [4]
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Hence write down the highest voltage in the circuit. [1]
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Question 13
[Maximum mark: 6]
Ali is swimming in a public pool with some of his friends. At time seconds, he some waves with height from big Bobby jumping into the pool, and waves of height from small Suzie jumping into the pool. Both and are measured in metres.
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Write, in the form , an expression for the total height of the waves Ali encounters at time seconds. [3]
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Find the times in the first seconds when Ali isn't affected by any waves. [2]
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Find the first time when the waves reaching Ali has maximum height. [1]
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Question 14
[Maximum mark: 6]
The revenues of a four seasons hotel can be modelled by the function
,
where is the number of days after midnight on December.
In a similar way, the operating costs of the hotel can be modelled by the function
.
Both and are measured in thousand dollars.
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Show that the profits of the hotel can be modelled by the function . [3]
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According to the model, find:
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the highest profit the hotel will make;
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the date on which the highest profit will occur. [3]
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Question 15
[Maximum mark: 6]
In an unbalanced three-phase electrical circuit, the current at time ms is given by
,
where is measured in milliamperes (mA).
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Write in the form . [4]
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Hence find the highest current flowing through the circuit, and the time it first occurs. [2]
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