Maths MC(need detailed steps)

1) P(at most 4 draws) = 1 - P(need 5 draws)
= 1 - P(not green at the first 4 draws)
= 1 - 4/6 * 3/5 * 2/4 * 1/3
= 1 - 1/15
= 14/15 (C)

2) x^2 - 3x + k = 0
a^2 - 3a + k = 0...(1)
b^2 - 3b + k = 0...(2)
(1)-(2) :
(a^2+3b) - (b^2+3a) = 0
a^2 + 3b = b^2 + 3a
(1)+(2) :
a^2 - 3b + b^2 - 3a + 2k = 0
a^2 + 3b + b^2 + 3a + 2k = 6(a+b)
2(a^2 + 3b) + 2k = 6([ - (-3)/1 ] = 18 (Tips : Sum of two roots)
(a^2 + 3b) + k = 9
a^2 + 3b = 9 - k (C)

3) He lost 10% * $ 999 999 / ( 1 - 10%) = lost $ 111 111
He gained 10% * $ 999 999 / (1 + 10%) = gained $ 90909
So he lost $ 111 111 - 90909 = $ 20 202 (D)

1. The answer is C.

P(Peter needs at most 4 times)
= P(green at the 1st time) + P(green at the 2nd time) +
__P(green at the 3rd time) + P(green at the 4th time)
= 2/6 + (4/6)(2/5) + (4/6)(3/5)(2/4) + (4/6)(3/5)(2/4)(2/3)
= 1/3 + 4/15 + 1/5 + 2/15
= 14/15


2. The answer is C.

Since a and b are the roots of the equation x^2 - 3x + k,
we have a^2 - 3a + k = 0 ---------- (1)
____& b^2 - 3b + k = 0 ---------- (2)

Sub. (2) into (1) : a^2 - 3a + k = b^2 - 3b + k
_________________a^2 + 3b = b^2 + 3a ---------- (3)

From (1) : a = [3 + sqrt(9 - 4k)]/2 or [3 - sqrt(9 - 4k)]/2
From (2) : b = [3 - sqrt(9 - 4k)]/2 or [3 + sqrt(9 - 4k)]/2

Then, take a = [3 + sqrt(9 - 4k)]/2 & b = [3 - sqr b t(9 - 4k)]/2

So, from (3) : a^2 + 3b = b^2 + 3a
__________________= {[3 - sqr b t(9 - 4k)]/2}^2 + 3{[3 + sqrt(9 - 4k)]/2}
__________________= [9 - 6sqrt(9 - 4k) + 9 - 4k]/4 + [9 + 3sqrt(9 - 4k)]/2
__________________= [18 - 6sqrt(9 - 4k) - 4k + 18 + 6sqrt(9 - 4k)]/4
__________________= (36 - 4k)/4
__________________= 9 - k


3. The answer is D.

Peter bought the flat that he lost 10% at :
$999,999 / (1 - 10%) = $1,111,110

So, Peter's lost = $1,111,110 - $999,999
___________= $111,111

Peter bought the flat that he gain 10% at :
$999,999 / (1 + 10%) = $909,090

So, Peter's gain = $999,999 - $909,090
___________= $90,909

Then, after the two transactions, Peter lost :
$111,111 - $90,909 = $20,202