7/24/2019 Module 1 Assignment 2

1/3

Module 1Assignment Due:

Tuesday 10thNovember 2015

1. (a) ind the modulus and argument o!

w=9+3i

12i

.

"#$

(b) %olve the e&uation 2z22 iz5=0.

"'$

() iven that z is a real number* +nd the ,ossible values o! -here

=+4 i

1+i , R .

"'$

2. The ubi e&uation 2z3z

2+4z+p=0,pR , is satis+ed by z=1+2 i .

(a) ind the other t-o roots o! the e&uation.

"'$

(b) Determine the value o! ,.

"$

. (a) iven that z1=10i231i32

, e/,ressz

1 in the !orm p+iq , -here

pandq are real. "'$

(b) iven thatz

1=r(cos+i sin ) , -here r>0and

7/24/2019 Module 1 Assignment 2

2/3

'. The om,le/ number 1+i2 is denoted by u . The ,olynomial

x4+x

2+2x+6 is denoted by p (x ) .

(a) %ho-ing your -oring* veri!y that u is a root o! the e&uation p (x )=0.

"$

(b) rite do-n a seond om,le/ root o! the e&uation p (x )=0.

"1$

() ind the other t-o roots o! the e&uation p (x )=0.

"$

5. (a) %olve the e&uation i w2=(22i )2.

"'$

(b) ind the least value o! |z1| i!

(i)arg (z+1 )=

4

"$

(ii) |z+3i|=2 "$

#. (a) ind the values o! z i! z3=443 i , giving your ans-ers in ,olar !orm

r(cos+i sin ) , r>0,

7/24/2019 Module 1 Assignment 2

3/3

The ,ro+t* P , in dollars* on sales o! the ne- games and systems* is given by

P=20 g100s0.01g s ,

-here g is the number o! games sold and s is the number o! ne- game

systems sold.

(a) 6! the om,any sells no ne- systems* ho- muh ,ro+t do they mae ,er ne-

game sold7 "2$(b) 8o- many ne- systems -ould the om,any have to sell be!ore they start

losing money regardless o! ho- many ne- games they sell7

"$

Marginal proft means the instantaneous rate o change o the proft. The

marginal proft, per new game sold, is the instantaneous rate o change o the

proft, with respect to the number o new games sold, holding constant the

number o new systems sold.

() 6! the om,any has +/ed sales o! 9250* 000 ne- games system* -hat is the

marginal ,ro+t* ,er ne- game sold* -hen the number o! ne- games sold is 1*

000* 0007 "$(d) 6! the om,any immediately sells 10* 000 ne- games* and never sells another

one* -hat is the marginal ,ro+t ,er ne- game system sold* -hen the number

o! ne- games sold is 10* 0007"'$

Total 3 12 mars