MATH INVESTIGATIONS 4
Problem Set: 2
Teacher (circle): Condie Dowling Loo
Due Thursday, September 8
Fall 2016
ID Number____________
Mods: __________
Recall that (NC) means that calculators and other technology MAY NOT be used on that problem (except for basic
arithmetic.) On all other problems, you may use your calculator to graph or find values of built-in functions (like sine or
logarithm or square root) but not to use CAS features (like the solve function) to do the work for you.
1)
In how many different ways can 7 boys and 5 girls stand in a line under the following condition? Show
thinking!
2)
a.
Anyone can stand next to anyone else (i.e., there are no special conditions).
b.
No one is standing next to someone of the same gender.
c.
The seven boys must stand together and the five girls must stand together
d.
The five girls must stand together.
Find the values of constants A and B so that
6
 3n  4  3n  1

A
B

. Note: this is called partial
3n  4 3n  1
fraction decomposition and is used throughout mathematics and is often used to help solve problems.
3)
Solve the equations: (Find exact answers. Show your work- graphing is not acceptable work.)
3ln 2 ( x) 14ln( x)  8  0
a.
4)
A and B working together can do a job in 24 hours. If A works alone for 8 hours and B then finishes the job in 30
hours, how many hours would it take each working alone to do the job?
(NC) 5)
Find exact values for each of the following:

3
 5 
cos  sin –1    tan –1   
5
 12  

a)
6)
7)
b. tan4(3x) – 4 tan2(3x) + 3 = 0, 0 ≤ x ≤ π
Solve:
2
2
log x (9) log 9 ( x)
b)

1
 5 
sin  cos –1    sin –1    
 3
 8 

 3 . (show thinking)
Consider the ellipse given by 4x2 + 9y2 – 32x + 18y + 37 = 0.
a)
(x  h)2 (y  k)2

1
Put this equation in standard form:
2
2
a
b
b)
Find the center of the graph and the four vertices of the ellipse. (These are the four points above, below, left,
and right of the center.)
c)
Sketch the graph.
MATH INVESTIGATIONS 4
Problem Set: 2
Teacher (circle): Condie Dowling Loo
Due Thursday, September 8
Fall 2016
ID Number____________
Mods: __________
n
r
The notation   , read "n choose r", is the number of combinations of n objects taken r at a time. That is,
it is the number of ways to select a group of r objects from a set of n distinct objects. It is equal to
n!
n(n -1) (n - r +1)
=
r !(n  r )!
r(r -1) 2×1
where n and r are non-negative integers and n  r. It is also called a binomial coefficient.
8)
n
 18   18 
 if 
=

6
 n  n  2
a) Find: 
b)
 n – 1
n – 3  n – 5
  
 
  80
 3 
 3   3 
Solve for n: 
9) Recall that i  1
a.

Write out the terms of (1  i) n

10
n 1
in a+bi form.
b. Carefully describe at least two patterns you see in the sequence above.