2020. Complex Numbers. Rangitoto PDF

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Summary

complex numbers example for calculus level 3...

Description

NAME:

TEACHER:

Level 3 Topic Test Calculus 2020 91577 (3.5) Apply the algebra of complex numbers in solving problems Credits: Five (practice)

Time: 55 minutes Make sure you have a copy of the formulae and tables booklet. You should attempt ALL questions in this booklet. Show ALL working You must hand this booklet to your teacher at the end of the test. In class, closed book test.

Total:

Calculators are permitted.

Achievement Criteria Achievement Apply the algebra of complex numbers in solving problems.

For Assessor’s use only Achievement with Merit Apply the algebra of complex numbers, with relational thinking, in solving problems.

Achievement with Excellence Apply the algebra of complex numbers, with extended and abstract thinking, in solving problems.

Overall Level of Performance

Assessor’s use only

QUESTION ONE a) Write

√5−√7 √5+√7

in the form a + b√c where a, b and c are rational numbers.

________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ b)

Let 𝒖 = −𝟓 − 𝟒𝒊 and = 𝟑 − 𝟖𝒊 . Express 𝒖 − 𝒗 in rectangular form and plot it on the argand diagram above. ________________________________________________________________ ________________________________________________________________ ________________________________________________________________

Assessor’s use only

c) Solve 𝒌√𝒙 + 𝟓 − 𝟐√𝒙 = 𝟎 for x in terms of k. ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________

d) 𝒎 is the complex number 𝟑 − 𝟖𝒊 𝒏 is the complex number −𝟒 + 𝟕𝒊 Find the real numbers 𝒂 and 𝒃 such that 𝒂𝒎 − 𝒃𝒏 = −𝟏𝟒 + 𝟏𝟗𝒊 . ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________

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e) Find the values of 𝒌 for which the equation 𝒙 − 𝟕 − 𝟐√𝒌𝒙 + 𝟑 = 𝟎 has real roots. ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________

Assessor’s use only

QUESTION TWO a) If 𝒂 = 𝟐 + 𝒊 and 𝒃 = −𝟑 + 𝟒𝒊 find the modulus of 𝒂 × 𝒃. ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ 𝟑𝝅

b) If 𝒖 = 𝟐𝒄𝒊𝒔 (

𝟒

𝝅

𝒖

) and 𝒗 = 𝟔𝒄𝒊𝒔 ( 𝟐 ), write in polar form. 𝒗

________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ c) Solve the equation 𝒛𝟒 = −𝟓 + 𝟏𝟐𝒊 . Write your solutions in polar form. ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________

Assessor’s use only

d) For |𝒛 − 𝟐| = |𝒛 + 𝟓 − 𝒊| , find and describe the Cartesian equation for the locus of 𝒛 . ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ e) Find all possible values of k that make 𝒖 =

𝒌−𝟏𝟐ii 𝟑−𝒌ii

a purely real number.

________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________

Assessor’s use only

QUESTION THREE a) Solve the equation 𝒙𝟐 − 𝟖𝒙 − 𝟒 = 𝟎 . Write your answer in the form 𝒂 ± 𝒃√𝟓 where a, b and c are integers and 𝒃 ≠ 𝟏 . ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________

b) Find the remainder when 𝟐𝒙𝟑 + 𝟑𝒙𝟐 + 𝟑𝒙 + 𝟏𝟎 is divided by (𝒙 + 𝟑) . ________________________________________________________________ ________________________________________________________________ ________________________________________________________________

c) One root of the equation 𝒛𝟑 − 𝟐𝒛 + 𝒑 = 𝟎 is 𝒛 = 𝟑 − 𝟓𝒊. If 𝒑 is a real number find the value of 𝒑 and the other roots of the equation. ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________

Assessor’s use only

________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________

d) Given 𝟐𝒙𝟒 + 𝟕𝒙𝟑 − 𝟒𝒙𝟐 − 𝟐𝟕𝒙 − 𝟐𝟎 = 𝟐𝒙𝟑 + 𝑨𝒙𝟐 + 𝑩𝒙 + 𝟗 −

𝑪

𝒙−𝟐

,

find the values of A, B and C. _______________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________

Assessor’s use only

𝝅

e) If 𝒖 = 𝟖 + 𝒌𝒊 and 𝒗 = 𝟏𝟐 + 𝒌𝒊, find 𝒌 if 𝑨𝒓𝒈(𝒖𝒗) = 𝟒 . ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________

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Extra working space ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________ ________________________________________________________________...