IMSA
1)
MI-3
Prob. Set #5
a.
When is the projection of a segment one half the length of the segment?
b.
When is the length of the projection equal to 1/ 2 the length of the segment?
c.
Find the projection of the GRE if G(3,–4), R(–3,6) and E(8,9)
onto the x-axis ____  x  ____
9
Compute (no calculator): a.
2
ii. onto the y-axis  y  ____
   1
13
n 1
b.
n4
3)
Spring '12
The projection of a point onto a line is the foot of the
perpendicular from the point to the line.
The projection of a segment onto a line is the set of
projections of all of its points onto the line.
i.
2)
DUE: Tuesday, March 6
n
 n2
n 5

What is the maximum number of different lines that can be determined by:
a.
4 points
b.
9 points
c.
n points
d.
What property must these points have to realize this maximum number?
g (3  h)  g (3  h)
2h
4)
Let g ( x)  x 2  4 x  9 . Simplify completely:
5)
Given that y  f ( x) is a function whose domain is [–3, 6] and range is [–2, 9], what is the domain
and range of each of the following functions?
6)
a.
g ( x)  2 f ( x)
c.
k ( x)   f (2 x)
b.
h( x)  f ( x  2)
d.
j ( x)  f ( x)  2 5)
An arrow is to be shot from a hill 60 feet above level ground. The arrow's distance, s, in
feet above sea level at any time, t in seconds, is found by the equation:
s  16t 2  80t  50
a.
After how many seconds will the arrow hit the ground? [Give time to 4 decimal places]
b.
Find the maximum height reached and the time it reaches that maximum.
[Algebraic justifications required.]
PS 5.1
Rev. S12
IMSA
7)
8)
9)
MI-3
Prob. Set #5
DUE: Tuesday, March 6
Spring '12
Given that the roots of the polynomial P( x)  x3  2 x 2  3x  5 are r1, r2 and r3, find (no calculator):
a.
r1 + r2 + r3
b.
r1r2 + r1r3 + r2r3
c.
r1r2r3
d.
1
1
1


r1r2 r1r3 r2 r3
x y
  1 , k > 0,
5 k
Let A be the x-intercept, B be the y-intercept, and O be the origin. Let  be the angle of inclination.
Given the linear equation:
a.
For what values of k will the area of ∆ABO be between 4 and 18?
b.
For what values of k is 30° ≤  ≤ 60°? [Give your answer to 3 decimal places.]
How many different phone numbers of 9 digits can be formed if:
a.
b.
c.
there are no restrictions on the digits?
the number cannot start with 0 or 1?
the number cannot start with 0 or 1 or 911 or 411?
10)
Two six sided dice are tossed. What is the probability that the sum of the sides facing upward is:
a. 8 or 5
b.
not 2, 5, 10, or 12
11)
The following equation may be considered a quadratic equation in terms of x or in terms of y:
3x2  2 xy  y 2  20  0
a.
Using the quadratic formula, "solve" the equation for y. Simplify your answer.
b.
Graph your solutions. Label all intercepts.
12) Solve for x: (no calculator)
a. log3 (log8 ( x))  1
b.
log 9 (log 4 ( x)) 
1
2
13)
Determine the arithmetic mean, geometric mean and harmonic mean of:
1, x 2 y and xy 2
a.
4, 20, and 40
b.
14)
Solve the equations (no calculator):
a.
x2 13 x  30  0
b.
PS 5.2
x 4  10 x 2  16  0
Rev. S12
IMSA
MI-3
Prob. Set #5
DUE: Tuesday, March 6
Spring '12
The following section (questions 15-19) is designed to introduce you to the Binomial Theorem. You will
be working problems involving this theorem in remaining problem sets.
15) a. Complete the table on the answer sheet by expanding each of the binomials. Note: The TI-89 or
Wolframalpha.com might come in handy here. Feel free to use either.
b. Look at the coefficients in your table. In this arrangement, they form a famous triangle, called ??
16) Determine the numbers in the next row of the triangle, corresponding to the expansion of
without doing the actual expansion. Explain your method.
 x  y 8 ,
When expanding  x  y    x  y  x  y  , each factor of  x  y  "donates" either an x or a y to the
2
resulting product. That is, when we obtain x 2  xy  yx  y 2 for the expansion, each term in this
expression is comprised of either an x or a y from the first factor and either an x or a y from the second
factor.
17) Consider the expansion of  x  y    x  y  x  y  x  y  x  y  x  y  x  y  .
6
a.
In how many ways can you choose
4 factors to "donate" x's to the expansion
from the 6 factors? Explain.
b.
In how many ways can you choose
2 factors to "donate" x's to the expansion
from the 6 factors? Explain.
b.
In how many ways can you choose
5 factors to "donate" y's to the expansion
from the 8 factors? Explain.
18) Now consider the expansion of  x  y  .
8
a.
In how many ways can you choose
5 factors to "donate" x's to the expansion
from the 8 factors? Explain.
\
Binomial Theorem
 a  b n  C  n,0 anb0  C  n,1 an1b1  C  n, 2 a n2b2  ...  C  n, n 1 a1bn1  C  n, n  a0bn
where n is a non-negative integer.
19) Explain/show how to use the Binomial Theorem to determine the coefficient of the term with a6b4 in
the expansion of  a  b  . (Do not simply expand the binomial.)
10
PS 5.3
Rev. S12