CPhill

These are dependent events.....the second card that is drawn is dependent upon the card that is drawn first  [Hint....any time we have "without replacement"....the event will be dependent ]

Second one

We have 13 Hearts.....the probability that one of them is a King  is just  1/13

Last one

We have a total of  70 + 50  = 120  students who arrived by car

The probability that  one of these is a girl  is   

[ Number of girls that arrived by  car  ]  /  [ Total Arrivals by car ]  =

[ 50  / 120 ]   =  5/12  ≈    .42  =   42%

Taking two marbles from a box without replacement

P(Q)  = .41      P(R)  = .44

P(Q  ∩ R )   =   .41 * .44   =  0.1804

Mutually exclusive means that it impossible for both things to happen at the same time

Obviously...it is impossible to  roll an even number and an odd number at the same time

Here's the answer to  the second one, LF :  https://web2.0calc.com/questions/please-help-asap_22

First one

P(A)  = 5/8       P(B)  =  3/4    P (A  ∩ B)  =  15/32 

D is the true statement because

P(A  ∩ B)  = P(A) P(B)

15/32   =  (5/8) (3/4)

15/32  = 15/32

Last  one

Prob of a boy chosen at random does not wear glasses  =  33/45  = 11/15  = 73%

y  = x - tan x

y' = 1 - sec^2x

At  pi/3   the slope is  1 - sec^2 (pi/3)   =  1  - (2)^2  =   1 - 4  = -3

And...at pi/3, y  =   pi/3  - tan (pi/3)    =  pi/3  - √3

So....the equation  of the tangent line at    (pi/3, pi/3 - √3 )  is

y  = -3 (x - pi/3) + pi/3 - √3      simplify

y = -3x + pi + pi/3  - √3

y = -3x + (4/3)pi - √3

Here's the graph, Julius :   https://www.desmos.com/calculator/aheivaxqtb

You are correct....!!!!

 35°  is a first quadrant angle....and all the trig functions are positive in the first quadrant

But  145°  is a second quadrant angle....and the cosine is negative in this quadrant

              Red      White

Jar 1       72        108

Jar 2     117          63

Explanation....

Jar 1 contains 40%  red  marbles of 180   = .40 * 180  =  72

So...Jar 1  also contains 180 - 72  = 108 white marbles

Jar 2 contains  35% white  = .35 * 180  =  63

And it must also contain 180 - 63  = 117 red

What are the events  ???

Since the center of dilation is the origin, this one is easy.....

The scale factor  is   9  because

(-3, 4) * scale factor  = (-3, 4) * 9  =  (-27, 36)

So

W'  = W * scale factor  =  (3,5) * 9   =   (27, 45 )

\( |z|^2-2\bar z+iz=2i. \)  

z*z(bar) - 2(a - bi) + i (a + bi)  =  2i

(a + bI) ( a - bi) - 2a + 2bi + ai + bi^2  =  2i

a^2+ b^2 - 2a + 2bi + ai + bi^2  = 2i

a^2 + b^2 - 2a + 2bi + ai - b  = 2i      

(a^2 + b^2 - 2a - b) + (2b + a)i  =   0 +  2i     equate coefficients

Thus

a^2 + b^2 - 2a - b  =  0   (1)

2b + a  = 2  ⇒  a  = 2 - 2b  ⇒  a = 2 (1 - b)   (2)

Sub (2)  into (1)

[4 (1 - b)^2] + b^2 - 2 [ 2 (1 - b) ] - b  = 0

4 (b^2 - 2b + 1) + b^2 - 4 + 4b  - b  = 0

4b^2 - 8b + 4 + b^2 - 4 + 4b - b  = 0

5b^2 - 5b = 0

b^2 - b  = 0

b(b - 1)  = 0      set each factor to 0 and solve for b

b  =  0     or  b  = 1

So when b = 0, a  =  2 (1 - 0)  = 2   

And when b  = 1,   a  = 2 ( 1 - 1)  = 0

So  the solutions are

z = 2 + 0i ,   z = 0 +1i

We would have the graph of the lines

x  = 3y

x = -3y

y = 3x

y = -3x

Look at the following graph : https://www.desmos.com/calculator/gkpfp6lqhy

These split  the  plane into 8 regions