Mathematics T (Number and Sets)

[AnniceAng]

Can someone help me to solve these question??? I dunno how to do...:(:( Thanks in advance!

1. z is a complex number such that z/(z+2)=2-i, find the modulus and argument of z.

2. Determine the value of x if (surd 5 + xi)/(1 + surd 5i) is a real number and find this value.

[kudis]

this is all algebra:

1.
z/(z+2)=2-i
=> z=(2-i)(z+2)
=> z=(2-i)z+4-i2
=> 0=z-iz+4-i2
since z is complex, z=x+iy(where x and y are real variables). plug that into the last eq:
x+iy-ix+y+4-i2=0
group the real and imaginary terms:
(x+y+4)+i(y-x-2)=0
since each term must satisfy the null condition independently, you'll get 2 eqs and 2 unknowns:
x+y=-4
-x+y=2
once you solved x, y, finding the modulus and arg is straightforward.

2.
say (surd 5 + xi)/(1 + surd 5i)=y
for ease of typing, I'll define surd 5=a
hence,
(a+ix)/(1+ia)=y
the strategy here is to get rid of the complex term in the denominator by multiplying the top and bottom by it's complex conjugate:
[(a+ix)/(1+ia)][(1-ia)/(1-ia)]=y
=> (a-ia^2+ix+ax)/(1+a^2)=y
rearrange the numerator to real and imaginary parts:
[(a+ax)+i(x-a^2)]/(1+a^2)=y
since y is real, it follows that x=a^2=5
so y=6a/6=a=surd 5

[AnniceAng]

hey guys....I got another question...

show a^2 + b^2 more than or equal to 2ab.
If x+y+z=c, show that x^2+y^2+z^2 more than or equal to 1/3c^2.

[markwongsk]

hey guys....I got another question...

show a^2 + b^2 more than or equal to 2ab.
If x+y+z=c, show that x^2+y^2+z^2 more than or equal to 1/3c^2.

any square number is positive or equal to zero. Therefore (a-b)^2 >= 0
a^2 + b^2 >= 2ab

square (x+y+z) = c and use the result you have (a^2 + b^2 >= 2ab) to simplify

[crystalapple]

hi, i need help for Math T too...
1) Given a and b are real numbers where a>1 and b>1, prove that 1/logb a + 1/loga b >= 2.
2) Given A=(-1 2) and B=(1 0). P takes the general form of (m -n).
( 2 -1) (0 -3) (m n)
Find the matrix P if P(transposition)AP=B and P(transposition)P=I.

[Allmaths]

hi, i need help for Math T too...
1) Given a and b are real numbers where a>1 and b>1, prove that 1/logb a + 1/loga b >= 2.
2) Given A=(-1 2) and B=(1 0). P takes the general form of (m -n).
( 2 -1) (0 -3) (m n)
Find the matrix P if P(transposition)AP=B and P(transposition)P=I.

For question 1,

Let z=log_a (b) where z>0 *(log_a means log base a)

(z-1)

[Sillyboy]

[QUOTE=Allmaths;342549]For question 1,

Let z=log_a (b) where z>0 *(log_a means log base a)

(z-1)