Consider , where and
Express in the form .
Write the complex numbers and in the form .
Express in the form
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Consider , where and
Express in the form .
Write the complex numbers and in the form .
Express in the form
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Consider the equation , where .
Find the four distinct roots of the equation, giving your answers in the form , where
Represent the roots found in part (a) on the Argand diagram below.
Find the area of the polygon whose vertices are represented by the four roots on the Argand diagram.
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Let and
Giving your answers in the form find
Write and in the form
Find giving your answer in the form
It is given that and are the complex conjugates of and respectively.
Find giving your answer in the form
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Let and .
Express
Find giving your answer in the form .
Find giving your answer in the form
Sketch and on a single Argand diagram.
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It is given that that and
Find the value of for
Find the least value of such that
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Consider the complex number where and
Express in the form
Sketch and on the Argand diagram below.
Find the smallest positive integer value of such that is a real number.
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Consider the complex numbers and .
Write and in the form , where and .
Find the modulus and argument of .
Write down the value of .
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Write in the form where .
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Consider the equation where .
Find the value of for which one of the two distinct roots is .
Find the range of values of for which the equation has two distinct, real roots.
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Let , where .
Find when
On an Argand diagram the point can be transformed to the point by two transformations. Describe the two transformations and the order in which they are applied.
Hence, or otherwise, find the value of when .
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Consider where .
Show that .
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It is given that and
Giving your answers in the form , use technology to find the values of
Find the least value of such that .
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Let
Express in the form , where , giving the exact values of and .
Let and .
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Consider the complex numbers and .
Express
Find the exact value of .
Find , giving your answer in the form , where and .
Without drawing an Argand diagram, describe the geometrical relationship between and .
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Use technology to find all the powers
Find the area of the shape made by the powers when plotted on an Argand diagram.
Give your answer as an exact value.
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Let .
Write down the value of .
Let and
Prove the results
Using technology, or otherwise, show that
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The current, , in an AC circuit can be modelled by the equation where is the frequency and is the phase shift.
Two AC voltage sources of the same frequency generate currents and .
Write down the maximum value and phase shift of the two currents and when they are each connected to the circuit alone.
The two AC voltage sources are connected to the circuit at the same time and the total current can be expressed as .
Write down the maximum value and phase shift of .
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The height of a wave in metres, relative to a particular boat, can be modelled by the function , where is the time in seconds. Observers on the boat are tracking a jumping dolphin. The height of the dolphin’s jumps can be modelled by the function .
Find an expression for the height the dolphin can reach, at time seconds, when the height of the dolphin’s jump is affected by the height of the waves. Give your answer in the form
Use technology to find the time when the dolphin first reaches its maximum height and write down the maximum height the dolphin reaches.
Find the time interval in the first two seconds when the height of dolphin will be above the wave
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Let .
Find the exact value of such that successive integer powers of the complex number lie on a unit circle.
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Consider the complex numbers and , where
Use geometrical reasoning to find the two possibilities for w, giving your answers in exponential form.
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Consider the complex numbers and .
Write and in the form where , giving exact values of and .
A scale model of a triangular patio is planned by representing the vertices of the triangle on an Argand diagram as the points and . Find the area of the triangle in the model.
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Consider the complex number .
Use technology to find the value of . Give your answer in the form , where , giving the exact value of and the exact value of .
Find the smallest positive integer such that is
Explain why there is no possible integer value of such that is purely imaginary.
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Given the points 1 and on an Argand diagram, where is a complex number, explain how to find each of the following points by geometrical construction. In each case provide a sketch to illustrate your answer.
.
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Consider the equations and , where . Find giving your answer in the form , where and .
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By first expressing and in the form where and , show that .
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Two power sources are connected to a single electrical circuit. At time seconds, the voltage, , provided by the first power source is modelled by , and the voltage, , from the second power source can be modelled by the function .
The total voltage in the circuit, , is given by .
Find an expression for in the form , where
Write down the maximum voltage and the phase shift provided by the second power source, giving your answers correct to 2 decimal places.
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A popular fast-food chain is considering opening a new restaurant in an up and coming part of New York. They have looked at competition in the area and predict that the costs , in thousands of USD, to run the new restaurant for the first 100 days after opening could be modelled by the function
where is the number of days since opening.
The CEO of the company will only give permission for the new restaurant to open if the model predicts that the revenue will be greater than the costs by the 80th day after opening.
The revenue , in thousands of USD, for the first 100 days is predicted to follow the model
According to the model found in part (a) (ii), find
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The primary square root of a complex number is defined as , where and . If then the value for is chosen such that . Note that the other square root of will then be given by .
Show that
Given that , derive a formula for in terms of and , and explain why in this case will always have the same sign (positive, negative, or zero) as .
Hence show that in general
with the choice of the positive or negative value being dependent on the properties of .
Explain what must be true of for each of the following to be true:
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