IMSA
1)
Prob. Set #9
DUE: Friday, April 13
Spring, ‘12
Given the equations:
 x  y  3

 x  3 y  8
a.
b.
2)
MI-3
Graph the region bounded by the equations. Label the vertices.
Find the area of the region.
5
If x   ln(2k  3) , what would be the value of e x ?
2
3)
4)
5)
If a point is chosen at random in a 6 cm by 6 cm square,
what is the probability that the point is within 2 cm of a
corner?
2
2
2
2
a.
The points A(1, 2), B(-4, 1), and C(2, -5) determine a triangle ABC. What is
the equation of the line containing the median from A to side BC ?
b.
Write the equation of the line that contains all of the points in the plane that
are equidistant from the points A(-6, 3) and B(5, -8).
The graph at the right is
1
f(x) = x . Approximate
the area under the graph
from x = 1 to x = 2.2 by
taking circumscribed
rectangles of width 0.2.
1
P.S. 9.1
2.2
6)
If the sum of two angles of a triangle is given to be  3 x  30  , what are the
restrictions on x? (Give your answer in interval notation.)
7)
Solve for (x,y):
8)
 2 x   5 3 
 3 1   x 1 =



A4
6 y4 
11 y 2  4 


Each triangle CAnAn + 1 is a right
A3
triangle with An An + 1= n . If CA1 =
1, find the length of:
A2
Find (A, B) such that
CA2, CA3, CA4
b.
CAn + 1
A1
C
9)
a.
3x  1
A
B
=
+
x  2 x  15
x 5
x3
2
[Hint: Write the right hand side as a single fraction. Use the fact that if
Mx + N = Ux + V, then M = u and N = v.]
10)
Given the function f ( x)  x 2  3x  4 .
f ( x  h)  f ( x )
If m( x, h) is defined to be m( x, h) 
,
h
a.
find a simplified expression for m( x, h)
b.
find m(4, h)
m(4,1)
m(4, 0.1)
c.
Explain why m(4, h) is undefined for h = 0. What value should be assigned
to m(4, 0) so that the graph of y  m(4, h) will contain no holes?
P.S. 9.2
11)
B
E
Given: Right triangles ABC and CDE, AB
= CD, find the perimeter and area of
Quadrilateral ABED. (The figure is not
C
drawn to scale.)
20
12
D
A
12) (a) A secretary types four letters to four people and addresses the four envelopes. If the
secretary inserts the letters at random, each in a different envelope, what is the probability that
exactly three letters will go into right envelope? Explain your answer.
(b) A pencil with five pentagonal cross-section has a maker’s logo imprinted on one of its faces.
If the pencil is rolled on the table, what is the probability that it stops with logo facing up?
Explain your anseer.
13)
14)
15)
16)
17)

a.
x 2  7 x  10  0
b.
x 2/5  7 x1/5  10  0
c.
34 x  7  32 x   10  0
d.
 log3 x 
2
 7 log 3 x  10  0
Let z1  p  qi and z2  r  si represent two complex numbers.
a.
Find z1 and z 2 , the conjugates.
b.
Find z1  z2 , the product of the conjugates.
c. Find z1  z2 and z1  z2 , the product and the conjugate of the product.
d. What is the relationship between the product of the conjugates and the
conjugate of the product?
List all solutions(over the complex numbers), if x 4  2 x3  5 x 2  6 x  2  0 and if
one of the roots is x  2  3 .
Write the equation of a polynomial of minimal degree with rational coefficients
and the following conditions: (Give your answer in general form.)
Known zeroes are: 3, -2, and 3 + i. The graph passes through (–1,–51).
If the length of each edge in a cube is increased by the same percent and the
result was a cube that had 60% more volume than it originally had, by what
percentage were the edges increased? (Give the answer rounded to two decimal
places).

P.S. 9.3
16)
Let
y  log 2 t and x  log8 t where t < 100.
a. Eliminate the variable t and rewrite this relationship so that y is a function of
x. Then give the domain and range.
b. Graph the function. Be sure to include important details.
20)
Let f ( x)  x3  2 x 2  3x  5 .
a. Find the slope of the secant line between the points ( 1, f(1) ) and
( 1+h , f(1+h) ) [Your answer will be in terms of h.]
b. Evaluate the expression in part a for h = .01.
[This is an approximation of the slope of the line tangent to the curve at ( 1, f (1) ).]
c. What number does the slope approach as h gets very, very, close to 0?
3)
Combinations
n!
Remember: nCr = r!(n – r)!
[The number of combinations of n things taken r at a time.]
n!
n
Another notation for nCr is  r  = r!(n – r)!
 
Simplify the following:
11!

a.
6!5!
b.
13 C5 
c.
101


 99 
P.S. 9.4
12)
Derive the equation of the locus of
all points whose distance to the
point F(–5,4) is half the distance to
the line y = –4. Leave your answer
in general form, (i.e. multiplied and
simplified and equal to zero.)
P.S. 9.5