IMSA
1)
MI-3
Prob. Set #4
DUE: Tues., Feb. 28
Spring '12
A Locus is a set of points with a common characteristic. For example, the locus of points 5
units from a given point would be a circle with a radius of 5. The locus of points equidistant
from two given points is the perpendicular bisector of the segment determined by the two points.
a.
What is the locus of points equidistant from two parallel lines?
b.
What is the locus of points equidistant from the sides of an angle?
c.
What is the locus of points equidistant from three non-collinear points?
d.
Derive the equation of the locus of all points that are equidistant from points A and B if
A = (–2,–5) and B = (2,7)
2)
Consider the quadratic equation f ( x)  ax 2  8x  9 .
a.
Determine the coordinates of the vertex if a = 1 and if a = 4
b. Write an equation to describe the locus of points determined by the vertices of
f ( x)  ax 2  8x  9 as a varies.
3)
a.
b.
4)
Find all values of x for which H ( x)  2 x3  7 x2  3x 18 intersects the x-axis.
Use the results of (a) to factor H ( x ) .
Go to math2.org/math/algebra/conics.htm or your favorite (trustworthy) mathematics internet
page to learn a little about the conic sections, which arise from quadratic equations. The
quadratic equation, in general form, is written without any parentheses and set equal to zero:
Ax 2  Bxy  Cy 2  Dx  Ey  F  0 .
If B  0 and AC  0 , the equation can be rewritten in standard form:
( x  h) 2 ( y  k ) 2

 1.
a2
b2
Complete the squares to rewrite each of the following in standard form, and classify the conic section
(parabola, circle, ellipse, or hyperbola) :
a.
5)
4 x2  y 2  16 x  6 y  11  0
b.
5 x 2  12 y 2  50 x  24 y  53  0
What is the area of the union of two circles
(circle O and circle C) of radius 1 whose centers are 1 unit
apart?
PS 4.1
Rev. S12
IMSA
6)
7)
MI-3
Prob. Set #4
DUE: Tues., Feb. 28
Spring '12
Ten students are working at the table in the math office. How many ways can they be arranged
differently around the table? Note: Two arrangements are NOT different if one arrangement is just
the other arrangement with each person rotated one (or more) chair(s) clockwise or counterclockwise
around the table.
Probability is simply a fraction that relates
number of "successes"
.
total number of possibilities
a.
A number is randomly chosen from the set 1, 2,3, 4,5,...,50
What is the probability that the number is a PRIME ?
b.
If a point is randomly chosen in the interval [1,8] ,
what is the probability that it is
between 2 and 6 ?
8)
9)
10)
a.
If the average of 3 angles of a quadrilateral is 95°, what is the size of the remaining angle?
b.
If the average of 4 angles of a pentagon is 95°, what is the size of the remaining angle?
c.
If the average of 5 angles of a hexagon is 95°, what is the size of the remaining angle?
d.
If the average of (n – 1) angles of an n-gon is 95°, what is the size of the remaining angle?
For what values of n will your formula hold true?
A charm bracelet has 10 charms. In how many different ways may the charms be arranged around the
bracelet? Note: This problem is similar to, but not exactly the same as problem #6. Here, an
additional restriction you need to remember is that, if one arrangement is the same as another
arrangement you get when you flip the charm bracelet over, then these are not two different
arrangements of the charms.
A tangent to a circle is perpendicular to the radius drawn to the point of tangency.
A circle has center (4,3).
a. Find the equation of the tangent to the circle at (6,-1).
b. Find the radius of the circle.
11) Solve for x:
1
1
1
1
+
+
+
= log17 (289)
log 3 ( x)
log 5 ( x)
log 2 ( x )
log 4 ( x )
12) Find the values of x, and z, if the diagonal
of the square is 6 units.
[Review 45-45-90 and 30-60-90 triangles]
13)
The sides of a rectangle are log 3 ( x) and log5 (125) .
Determine all x so that the perimeter of the rectangle is less than 8.
PS 4.2
Rev. S12
IMSA
NC 14)
MI-3
Prob. Set #4
DUE: Tues., Feb. 28
Spring '12
g ( x)  2 x 2  x  4
a.
b.
g ( x  h)  g ( x )
h
Explain what happens to the expression when h gets "really-really" close to 0.
Substitute and simplify the expression:
15) Given f ( x)  log 2  x 5  1 , where x > –1.
a. Find an expression for f 1 ( x) .
b. Find the domain and range of both f ( x) and f 1 ( x) .
16)
Given 2 (non-opposite) rays with a common endpoint,
1 angle is determined.
Given 4 rays (no two opposite) with a common
endpoint, 6 angles are determined.
How many angles are formed by:
a. 5 such rays?
b. 17 such rays?
17)
Means:
c. n such rays?
Given a set of n-numbers,
their arithmetic mean is their sum divided by n,
their geometric mean is the nth root of their product, and
their harmonic mean is the reciprocal of the average of their reciprocals.
PS 4.3
a.
Determine the coordinates of point M.
b.
The value, M, is called the _______?_____ of
a and b.
c.
Compute the harmonic mean of x and y.
Rev. S12
IMSA
MI-3
Prob. Set #4 DUE: Tues., Feb. 28
28) Find the "projection" of the GRE if G(3,–4), R(–3,6) and E(8,9).
onto the x-axis ____  x  ____
a.
10.
 y  ____
onto the y-axis
Given that the roots of the polynomial P(x) = x3 + 2x2 – 3x + 5 are r1,r2 and r3, find:
a.
r1 + r2 + r3
b.
r1r2 + r1r3 + r2r3
c.
16)
b.
Spring '12
r1r2r3
d.
1
r1r2
1
+ rr
1 3
1
+ rr
2 3
If S = 1 + 2 + 3 + 4 + . . . . . . + 100 then S = 100 + 99 + 98 + 97+. . . . . . + 1 so
a.
S =
1 + 2 + 3 + 4 + . . . . . . + 100
+ S
= 100 + 99 + 98 + 97 + . . . . . . + 1
150
b. Use the result to find  (n)
n=1
2S = (?)
Find S.
30
c.
50
Find  (2n)
d. Find  (2n – 1)
n=1
20)
n=1
Let r1,r2, and r3 be the roots of: x3 + 6x2 – 4x + 2 = 0
a.
1
1
1
Find r + r + r
1
2
3
b.
PS 4.4
r1
Find r
2
r1
+ r
3
r2
+ r
1
r2
+ r
3
r3
r3
+ r + r
1
2
Rev. S12