CPhill

Very nice, BB!!!!

1 - k(-3) = -3(-3) + 4 - 7(-3)

1 + 3k  = 9 + 4 + 21

1 + 3k =  34

3k =  33

k  = 11

If B is in the second quadrant, the tangent of this angle must be  negative

Assuming that everyone takes at least one of the subjects :

Number in School =   N(Chinese) + N(Spanish) - N (Both)

374  =  x  + 2x  - 100

474  = 3x

x = 474 /  3  =      158  = total taking Chinese

2x  = 2 (158)  = 316  = total taking Spanish

Proof  (you can set up in a Venn diagram)

58 take only Chinese   +   100 take both  +  216 take only Spanish

p - 2q  = 0    →   p = 2q             (1)

q - 2r  =  8  →    r  = (q - 8) / 2     (2)

p + r = 5     (3)

Sub (1) , (2) into (3)

2q  + (q - 8) / 2 =  5

4q + q - 8   =  10

5q  - 8  = 10

5q = 18

q = 18/5           p =  2(18/5) =  36/5         r  =  (18/5 - 8 ) / 2  =  -11/5

4x  -6y   = - 2    →   mult through by   9   →     36x - 54y  =  -18      

kx - 9y    =  3   →    mult through  by  6   →      6kx - 54y =    18

This will have no solution when

6k =  36

k  = 6

DUH!!!

Nice work figuring out the arithmetic series  !!!

b₁₉₉₉ =  2(1999) + 1  =  3999

b ₂₀₀₀ =  2(2000)   +1  =  4001

b_n  =      1  / [ (a_n )^2  - 1] 

Note that we can write this as partal fractions as follows :

b_n  =   1 / [ ( a_n - 1) (a_n + 1)]     =   

1/ [ 2 (a_n -1) ]  - 1/ [2(a_n + 1) ]  =

 (1/2) [ 1/(a_n - 1) - 1/(a_n + 1)]

So  we can write

(1/2)  [  ( 1/2 - 1/4 ) +  (1/4  - 1/6) + ( 1/6 - 1/8) + ( 1/8 - 1/10)  +  ....... + ( 1/3998 - 1/4000) + (1/ 4000 - 1/4002 )

Notice that most of the terms  "cancel"   and we are left with

(1/2) ( 1/2  - 1/4002)  =

(1/2)  [ 4002 - 2 ]  / ( 8004)    =

(1/2) (4000)  / 8004  =

2000 / 8004

500/ 2001    =  p/q

p - q    =   500 - 2001   =  -1501

a= 32     b = 81   c   = 64    d  = 25

You should be able to finish this now....

3x^2 + 4x + 12  =  2x^2  - 7x  + 1        rearrange as

x^2 + 11x + 11  =  0

r + s  =  -11     square both sides     

r^2 + 2rs + s^2    =121

And  rs =  11    so .....  2rs  =  22

So

r^2 + 22 + s^2  =  121

r^2 + s^2  + 22  =  121

r^2 + s^2   = 99