CPhill

Excellent, tanmai.....!!!!

We have   C(7,2)  = 21 possible sets formed by choosing any two of the seven numbers

There are 6 sets  where the positive difference is < 2

(1,2) (2,3) (3,4) (4,5) (5,6) (6, 7)

So....the probability that two numbers are drawn whose difference is ≥ 2 is

[ 21 - 6 ] / 21    =   15/ 21   =   5 / 7

g ( f(x) ) =   2 [ ax + b ] - 5   = 2ax + 2b - 5

So....we have

2ax + 2b -  5  = 3x + 4

This implies that

2ax = 3x  ⇒   2a = 3 ⇒  a = 3/2

And

2b - 5 = 4    add 5 to both sides

2b = 9       divide both sides by 2

b = 9/2

So

a + b   =     3/2 + 9/2    =  12 / 2    =   6

Using the Law of Sines

sin B            sin C

____    =    _____

 AC             AB

sin 145           sin C

______   =    _______

2035             1017.15

sin C   =  (1017.5) * sin (145) / 2035

arcsin   [ (1017.5) * sin (145) / 2035 ] =  C = 16.67° ≈ [1.67 x 10^1 ] ° 

The regression formula is

f(x) = 1.476 (2.017)^x       when x is 7, we have

f(x) = 1.476 (2.017)^7   = 200.64

Thanks, tertre.....!!!!!

7x^2 - 3y^2 + 4z^2 = 8            (1)

16x^2 -7y^2 + 9z^2 = -3          (2)

Multiply the first equation through  by 7   and the second through by -3

49x^2 - 21y^2 + 28z^2 = 56

-48x^2 + 21y^2 - 27z^2 = 9        add these

x^2   +  z^2 = 65

Possible integer values for    (x , z) are ( 4, 7)   or  ( 7, 4)

Since x^2, z^2  are arbitrary....we can sub (4, 7)  into (1) for (x, z)

And we have

7^3 - 3y^2 + 4^3 = 8

-3y^3 + 407 = - 3

-3y^2 =  - 410

This does not produce an integer for y

Sub (7,4) = (x, y)  into (1)

7(4)^2 - 3y^2 + 4(7)^2 = 8

112 - 3y^2 + 196 = 8

-3y^2 =  -300

y^2 = 100

y = 10

So

x^2 + y^2 + z^2   =

65 + 100   =

165

We have that 

(a - 1) a ( a + 1) (a + 2)      for    a ≥ 2  where a is an integer

(a - 1) ( a + 1) a (a + 2)

(a^2 - 1) ( a^2 + 2a)

If this were a perfect square....both factors would have to  be equal

But...the first factor will always be <  the second factor for any   a ≥ 2

So....this cannot be a perfect square

We found this regression equation for the data to be

f(x) = .003415(2.0399)^x

When x = 14

.003415 (2.0399)^14  ≈  73.5   

Excellent, tertre...your LaTex presentations are good...