CPhill

f(1)  =   - (1)^2  =  -1

f (f(1) )  =  f (-1)  = -1 + 8  =  7

f (f (f (1) ) )  =  f ( 7)  = - (7)^2  = -49

 f ( f ( f( f(1) ) ) )   =  f (-49)  = -49 + 8  = -41

f ( f ( f ( f ( f ( 1) ) ) ) )  =  f (-41)  =  - (-41)^2  = -1681

From A to C we have all the possible arrangements of this set

( North, North, East)  =  C(3,1)   = 3

From C to B we have all the possible arrangements of this set

( North , North, East, East, East, East )   =  C(6,2)  = 15

So...the total number of paths from  A to B  through C are  3 * 15  = 45

We can answer this with a Venn Diagram :

There are   [ 4 + 10 + 11 + 7 ] = 32 total students

10 play football and cricket....so...the probability that a student plays football and cricket  is

10 / 32   =    5 / 16  =  .3125  =  31.25 %   

s(t)  =  t^3  - 3t

The velocity is given by the derivative of this function  =

s ' (t)   =  3t^2  - 3

Setting the velocity to 0 and solving for t, we have

3t^2  -  3  = 0       factor

3 (t^2 - 1)   = 0        divide through  by 3

t^2  - 1   =0

(t + 1) ( t - 1)  = 0

Setting both factors to 0 and solving for t, we have that

t  = - 1 seconds    or  t  =  1 second

Assuming  that the time is  ≥ 0....its velocity is  0 at 1 second  and its position  is

(1)^3  -  3(1)    =    -2   ⇒  (2 meters to the left of its starting position)

Here are the possible outcomes :

1,1   1,2   1,3   1,4    1,5    1,6

2,1  2,2    2,3   2,4    2,5    2,6

3,1  3,2    3,3   3,4    3,5    3,6

4,1  4,2    4,3   4,4    4,5    4,6

5,1  5,2   5,3    5,4    5,5    5,6

We have 30 possible outcomes

Those which are multiples of 4 and  factors of 15   (in any order) are in red  = 6 outcomes

So....the probability is   6 / 30   =  1 / 5

Note that....on a 12-hour clock....the number opposite  any integer N < 7 is given by :

N  +  (12/2)

And the number opposite any integer N  > 6  is given by :

N - (12/2)

So...this implies that on a 142-hour clock, since 27 is  less than 1/2 of 142, the integer directly opposite 27  is given by :

27  +  (142/2)  =

27  + 71  =

98

In 20 years, it will have

37000 ( 1.05)^20  ≈

98,172  inhabitants

We can choose    1 of any 26 shirts, 1 of any 26 pairs of pants  and 1 of any 26 pairs of shoes...so

C(26,1)  * C(26,1)  * C (26,1)   =

[ C (26, 1) ] ^3  =

26^3   =

17,576 different outfits

2 sin A  = 3 cos A     square both sides

4 sin^2 A  = 9 cos^2 A

4 sin^2 A = 9 [ 1 - sin^2 A]

4 sin ^2 A  = 9 - 9 sin^2 A     rearrange as

13 sin^2 A    = 9

sin^2 A  = 9 /13       sinc A is acute and  < 90°   ...take the positive root

sin A  = 3 / √13   =  ( 3 √13 ) / 13

2/15  =

1/8  + 1 / 120

1/ 9  +   1 / 45

1/10  +  1  /30

1/ 12  + 1 / 20

I can generate the first through something called "Egyptian Fractions"....I don't know how the other three are obtained

You can read here : http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fractions/egyptian.html