A01
What does it mean for two matrices A and B to be similar? Show that the matrix
 0 2
M 

2 0
is similar to a diagonal matrix.
A02
Use general recurrence relation to find y1 and the derivative method for y2
(2x2)y''+ (2x2+x)y' -y = 0
A03
Find two different isomorphisms from Z20 to Z4 + Z5.
Please explain how you got your answer.
A04
For what rightarrow u is T(rightarrow u) =9 1? Let T be a linear transformation
from R3into R2, T(rightarrow el)=1 1.T (rightarrow e 2),1 2,. T(rightarrow e 3)= 1
3.Here 'rightarrow e1, rightarrow e2, rightarrow e3 is the standard basis of R3
(a) Find T(rightarrow U) for rightarrow U x y Z (b) For what rightarrow U is
T(rightarrow U)= rightarrow O? (c) For what rightarrow U is T(rightarrow U)=3 4?
Let D be a linear transformation from F into F D(f)
A05
Find
A06
 z
1
dz and  z dz for the circular path   t   3eit ,0  t  2

(1pt) Take the Laplace transform of the following initial value problem and solve for
Y (s) = L{y(t)}:
y " 4 y ' 18 y  T t 
Where T(t) =
 t,
 1t ,

0 t
1
2
1
 t 1
2
y  0  0, y '  0  0
, T  t  1  T  t  .
Y(s) =
Graph of T (t) (a triangular wave function):
A07
Evaluate the integral
 4z dz
3
c
where C is the arc in the complex plane for 0  t  1 described by
(a) [10 points]  (t) = 3ej2t
(b) [10 points]  (t) = ej4t.
A08
Find the general solution of the equation y"-3xy'+(x2 +1)y=0 in terms of power
series centered about the point x0 = 0. Give atleast 3 non-zero terms in each of
the linearly independent solutions.
A09
Find Fourier cosine series for

2
f  x  
0

A10
0 x

2

2
 x
Note that v1 is perpendicular to v2.
Find the orthogonal projection of a vector (1, 1, 1, 1)T onto the subspace
spanned by the vectors v1 = (1, 3, 1, 1)T and v1 = (2, –1, 1, 0)T (note that v1v2).
A11
Hello
I am having trouble with this question.
dx  2 4 

x
dt  4 2 
4
with x  0    
2
Give your solution in real form.
x1 =
x2 =
A12
y'''-2y''-y'+2y=e4t solve by using the method of variation of parameters
thanks!
A13
Suppose that a population p(t) whose dynamic are given by the differential
equation
dp
 p3  4 p 2  4 p
dt
(a) Find and classify all equilibrium solutions according to it stability.
(b) Draw phase line
(c) Draw a sketch of the family of solutions on the t-p plane
A14
Solve the non-homogeneous linear recurrence relation: an = 5an-1 - 6an-2 + 3n +
2 with a0 = 4 and a1 = -3
A15
Help!!!
1 if x is rational
Let f ( x)  
1 if x is irrational
Show that f does not have a limit at any point. Note that |f| has a limit
everywhere.
Hint: for c in R, let (Xn)n in N be a sequence in Q\(c) and (Yn)n in N be a
sequence in (R\Q)\(c) with Xn --> c and Yn -->c. then f(Xn) --> 1 and f(Yn) --> -1
A16
Investigate the phenomenon of resonance in a mechanical system that is
described by the differential equation
d2x
dt
2
2
dx
 26 x  20cos t 
dt
and then calculate the amplitude of the VELOCITY corresponding to that value
of the external force frequency o at which the phenomenon of resonance occurs
in this mechanical system. A student solved the problem and figured out that the
answer rounded-off to four significant decimal digits was as follows
A17
Sketch the phase line of the given differential equation ....
Sketch the phase line of the differential equation y’ = f(y), where the graph of f is
given in Figure 1.15. Classify the equilibria into sinks, sources, and nodes.
A18
Are the vectors v1 = (1, 1, 2, 4), v2 = (2, -1, -5, 2) v3 = ( 1, -1, -4, 0), v4 = ( 2, 1, 1,
6) Linearly independent in the 4-space R4?
A19
If the differential equation:
 2  x  y ''  x  1 y ' y  0
2
is solved by a power series about x0 = 1, then the radius of convergence is at
least
(a) 1
A20
Find
two
(b)
2
linearly
(c)
3;
independent
(d) 2;
(e) 5
solutions
of 2 x2 y '' xy '  6 x  1 y  0, x  0
of the form
y1  x r1 1  a1 x  a2 x 2  a3 x3  ...
y2  x r2 1  b1 x  b2 x 2  b3 x 3  ...
where r1 > r2.
Enter
r1 
a1 
a2 
a3 
r2 
b1 
b2 
b3 
A21
Let f : B1  B2 be an isomorphism of Boolean Algebras. Prove a is an atom of
B1 if and only if f(a) is an atom of B2. Suggestion: Use Theorem 3.10.1 from
Notes 3.3 and the definition of an isomorphism of Boolean Algebras.
Theorem:
Theorem 3.10.1. A nonzero element a in a Boolean algebra B is an atom if and
only if for every x, y  B with a = x + y we must have a = x or a= y. In otherwords,
a is indecomposable.
A22
Let X = {a ,b, c} Find two relations, E and O on X such that E is an
equivalence relation with two equivalence classes and O is a total order.
A23
Need a Step Response RLC circuit to show cases of overdamped, undamped,
and critical, and to display i(t), v(t), vr(t), and vc(t), and display the plots for all
three voltages and current. Ask user for input values
For the following circuit;
Solve for i(t) when v(t)=0. Choose appropriate values of R, C and L for your
solution to represent:
(i)
(ii)
(iii)
Over-damped case
Under-damped case
Critical case
Write expressions for VR(t), VC(t), VL(t) and i(t). Choose your own initial
conditions. Then write a program (Language of your preference) to display the
plots of VR(t), VC(t), VL(t) and i(t).
A25
Evaluate each of the following integrals using two-point and four-point Gaussian
quadrature. Compare each result with the corresponding analytical solution and
determine the error for each case.
1
 sin
1
1
x
1
A26
2
 x  dx
2x
2
dx
1
<S,*> where S={ (a)^(1/3) in R | a is in Z} and (a)^(1/3)*(b)^(1/3)=(a+b)^(1/3)
Is this group cyclic and if so, what are the generators?
A27
10) AWM Corporation expects its common stock dividend to grow at a constant
5 percent for the foreseeable future. The company recently paid a $4.00
dividend. If the required rate of return on the company's stock is 11 percent,
what is the value (price) of the stock today?
A28
Find the solution of the linear system of equations
 x '  2 x  y

y '  x  4y
and sketch the phase portrait showing this solution.
Extra Credit:
Find the value(s) of m so that the linear system
 1 m

 m 2 
 '
has more than one equilibrium positions. How many equilibrium positions there
are for such value(s)?
A29
let W be a subset of Rn be a subspace, and let orthogonal complement W :={y 
Rn | x * y = 0 for all x  W} Then orthogonal complement W is a subspace of Rn,
True or False and prove.
A30
A) Show that the positive integers less than 11, except 1 and 10, can be split into
pairs of into pairs of integers such that each pair consists of integers that are
inverses of each other modulo 11.
b) Use part (a) to show that 10! is congruent to -1(mod11)
A31
Inference Rules: DN, CONJ, MT, ADD
Argument: (¬P ? Q), (¬R ? Q), ¬Q ? ((R ? P) ? (R ? Q))
How do you write this using the rules of derivations?
A32
Need help solving:
Binary relation P on set S= {a,b,c,d,e}
P = {(a,c),(a,e),(b,a),(e,d)}
What is the matrix representation of reflexive closure of P? Is this possible?
Is P symmetric relation?
Do I close the gaps by showing reflexive closure
{(c,a),(a,c),(b,a),(a,b),(e,d),(d,e)}?
Is P symmetric relation?
A33
Find the critical path and the project completion time for the following list of
activities. Calculate the probabilities of completing the following project in:
a) 11 days or less
b) 12 days or less
c) 13 days or more
d) between 12 and 14 days.
Immediate
Activity Predecessor
A34
Optimistic
Duration
Most Probable
Duration
Pessimistic
Duration
A
-
2
3
9
B
-
2
4
4
C
A, B
2
4
6
D
A, B
2
5
8
E
C, D
1
3
10
I appreciate your time, please show every step I have a midterm coming up.
Let Gm,n be the grid graph. For example, G3,5 is
Prove that Gm,n has a Hamilton cycle if and only mn is even.

For the “if” statement, a generic picture that shows how to construct the cycle is

fine.
The “only if” statement is harder to figure out, but there should be an “a-ha”
moment and the proof should be 2 or 3 sentences.
Hints*
A35
A36
9b. Explain why Hamilton cycle is length mn, and then explain why the cycle must have
even length.
let f(z)= (3+4i)z -2+i find the image of the disk |z-1|<1
In a given population of men and women, 90 of the men are married and 25% of
the women are married. What is the percentage of the adult population is
married. Assume that, in this particular population, the number of married men is
the same as the number of married women. Answer is 35.29
Is there an easy way to work this problem?????
A37
 16
 25

A=  12

 25
12 
25 

9

25 
(2x2 matrix). (Projection onto the line through the origin with direction vector:
 4
 5
d=   (2x1 matrix)
3
 
5
A38
A39
Find the eigenvalues and eigenvectors of A geometrically
A 50g mass stretches a spring 10cm. As the system moves through the air, a
resistive force is supplied that is proportional to, but opposite the velocity, with
magnitude .01v. The system is hooked to a machine that applies a driving
force to the mass that is equal to F(t) = 5 cos(4:4t). If the system is started
from equilibrium (no displacement, no velocity) SET UP BUT DO NO SOLVE
an ODE and two ICs to describe the motion. (Remember 1000g=1kg and
100cm=1m).
Extra Credit: Find the position of the mass as a function of time, i.e.
solve the ODE
A projectile of constant mass m is fired vertically from Earth. According to
Newton's law of gravitation, the motion of the projectile is governed by the
equation dv/dt = -gR2/r2 where g is a constant, R is the radius of Earth, and r is
the distance between the projectile and the center of Earth. Show that if the initial
velocity v0 is greater than root2gR then the projectile will not fall back to Earth
due to the gravity. (Hint: Use the chain rule dv/dt=dv/dr dr/dt.)
A40
Each front tire on a particular type of vehicle is supposed to be filled to a
pressure of 26 psi. Suppose the actual air pressure in each tire is a random
variable—X for the right tire and Y for the left tire, with joint pdf given below.
f(x, y) =
K(x2 + y2)
20 ? x ? 30, 20 ? y ? 30
0
otherwise
(a) Compute the covariance between X and Y. (Round your answer to four
decimal places.)
Cov(X, Y) =
(b) Compute the correlation coefficient p for this X and Y. (Round your answer to
four decimal places.)
?=
A41
For Insurance Company 1 the probability that no claims will be made next year is
0.55. If any claims are made, based on recent history the total claim amount will
have a normal distribution with mean $11,000 and standard deviation $2,200.
For Insurance Company 2, the probability of no claims is 0.75. If any claims are
made, the distribution of the total claim amount will have a normal distribution
with mean $9,500 and standard deviation $2,100. If we assume that the total
claims of Company 1 and Company 2 are independent, what is the probability
that Company 1’s total claim amount will be larger than that of Company 2?
[Hint: Company 1’s total can only be larger than Company 2’s total if claims are
made for Company 1. So, independence gives us part of the answer as
P(T1>0)P(T2=0). The other part may involve P(T1>T2) when P(T2>0) which in
turn may involve an integral for which you may need your calculator or Maple.]
A42
In a sample of 56 addicts, 28 were given a new drug to help them overcome their
addiction, while the remaining 28 were given the current drug. The following is
the output from a randomization test for the difference in the proportion of addicts
suffering relapses.
a) What is the difference in proportions in the original data?
b) What is the p-value and what does it mean in terms of this specific problem?
A43
A clinical trial is run to assess the effects of different forms of regular exercise on
HDL levels in persons between the ages of 18 and 29. Participants in the study are
randomly assigned to one of three exercise groups - Weight training, Aerobic exercise
or Stretching/Yoga – and instructed to follow the program for 8 weeks. Their HDL
levels are measured after 8 weeks and are summarized below.
Exercise Group
N
Mean
Std Dev
Weight Training
20
49.7
10.2
Aerobic Exercise
20
43.1
11.1
Stretching/Yoga
20
57.0
12.5
Is there a significant difference in mean HDL levels among the exercise groups?
Run the test at a 5% level of significance. HINT: SSwithin + 21,860
A44
A45
The lifetime of a certain type of electric bulb has a mean of 500 hours and a
standard deviation of 60 hours.
(a) Determine the sampling distribution of the sample mean for a sample size of
64 bulbs.
(b) Determine the percentage of all samples of 64 bulbs that have mean lifetime
within 10 hours of the population mean lifetime of 500 hours. Interpret your
answer in terms of sampling error.
In an experiment to learn if substance M can help restore memory, the brains of
20 rats were treated to damage their memories. The rats were trained to run a
maze. After a day, 10 rats were given M and 7 of them succeeded in the maze;
only 2 of the 10 control rats were successful. The plus four 90% confidence for
the difference between the proportion of rats that succeed when given M and the
proportion that succeed without it is
Answer
a.
0.417 +/- 0.185
b.
0.417 +/- 0.304
c.
0.455 +/- 0.312
d.
0.455 +/- 0.312
A46
Let f(x) = x + 1 for -1 < x < 0, f(x) = 1 - x for 0 ≤ x < 1, and f(x) = 0 elsewhere, be
the p.d.f. of X. Find pi0.65.
Not sure how to do this homework prob. Please help and explain!
A47
The PIRON Software Company currently develops marketing software for
primarily service-based organizations. They obtained a sample of potential bids
from the Phoenix market (shown in thousands below). The company's average
or mean contract is $55,000.
30 42 37 41 26 33 65 50 35 70
Using the scenario above, identify the sample error and determine the impact of
the error on potential decision making by PIRON Software Company leaders.
What should the company do based on the information calculated?
A48
A49
A vending machine dispenses coffee into a twelve-ounce cup. The amount of
coffee dispensed into the cup is normally distributed with a standard deviation of
0.08 ounces. You can allow the cup to overfill 5percent of the time. What amount
should you set as the mean amount of coffee to be dispensed? How many
ounces?
3
  k 1  k
Calculate  k 5 
 k and show all work.
 k  5 2
A50
You are a new hire at Laurel Woods Real Estate which specializes in selling foreclosed
homes via public auction. Your boss has asked you to use the following data (mortage
balance, monthly payments, payments made before default, and final auction price) on
a random sample of recent sales in order to estimate what the actual auction price will
be. Add a new variable that describes the potential interaction between the loan
amount and the number of payments made.
Loan
Monthly
Payments Auction
Payments
Made
Price
$85,607
$1,088.69
1
$22,675
110,548
942.65
35
19,150
108,591
743.28
12
45,225
A51
115,962
896.31
8
16,600
97,600
904.71
16
40,700
104,400
983.27
23
63,100
113,800
1,075.54
21
72,600
116,400
1.087.16
35
72,300
100,000
900.01
33
58,100
92,800
683.11
36
37,100
105,200
915.24
34
52,600
105,900
905.67
38
51,900
94,700
810.70
25
43,200
105,600
891.33
20
52,600
104,100
864.38
7
42,700
85,700
1,074.73
30
22,200
113,600
871.61
24
77,000
119,400
1,021.23
58
69,000
90,600
836.46
3
35,600
104,500
1,056.37
22
63,000
Can you help me break down the steps in solving this questions.
Q: The assessment data were gathered from 20 homes in Milwaukee, Wisconsin,
neighborhood. Fit the regression model.
Total Dwelling
size (100ft^2)
Assessed Value
(Thousands)
Selling Price
(thousands)
15.31
57.3
74.8
15.20
63.8
74
16.25
65.4
72.9
14.33
57
70
14.57
63.8
74.9
17.33
63.2
76
14.48
60.2
72
14.91
57.7
73.5
15.25
56.4
74.5
13.89
55.6
73.5
15.18
62.6
71.5
14.44
63.4
71
14.87
60.2
78.9
18.63
67.2
86.5
15.20
57.1
68
25.76
89.6
102
19.05
68.6
84
15.37
60.1
69
18.06
66.3
88
16.35
65.8
76
a) Assess the goodness of fit. Determine if the model is significant and determine
which variables are significant.
b)Define and discuss correlation
A52
A53
To evaluate the success of a 1-year experimental program designed to increase
the mathematical achievement of underprivileged high school seniors, a random
sample of participants in the program will be selected and their mathematics
scores will be compared with the previous year’s statewide average of 525 for
underprivileged seniors. The researchers want to determine whether the
experimental program has increased the mean achievement level over the
previous year’s statewide average. If alpha = .05, what sample size is needed to
have a probability of Type II error of at most .025 if the actual mean is increased
to 550? From previous results, s = 80. Suppose a random sample of 100
students is selected yielding y(bar) = 542 and s = 76. Is there sufficient evidence
to conclude that the mean mathematics achievement level has been increased?
Explain.
Below are the records on the weights of adult grey wolves from two regions in
the Southwest:
Chihuahua Region Grey Wolf Weights
86
75
91
70
79
80
68
71
74
64
Durango Region Grey Wolf Weights
68
72
79
68
77
89
62
55
68
68
59
63
66
58
54
71
59
67
a. Find an 85% confidence interval for the mean weights of the two Southwest
Regions
b. What does this mean when comparing the mean weight of grey wolves from
the two regions
A54
Consider the problem facing security personnel at a military facility in the
Southwest. Their job is to detect infiltrators (spies trying to break in). The facility
has an alarm system to assist the security officers. However, sometimes the
alarm doesn’t work properly, and sometimes the officers don’t notice a real
alarm. In general, the security personnel must decide between these two
alternatives at any given time: H0: Everything is fine; no one is attempting an
illegal entry. H1: There are problems; someone is trying to break into the facility.
Based on this information, fill in the blanks in these statements: a. A “missed
alarm” is a Type ___ error, and its probability of occurrence is denoted as ___. b.
A “false alarm” is a Type ___ error.
A55
Least squares as Maximum Likelihood Estimator (with no y intercept) proof
Given
n
n
1 n
L  ( ) ln(2 )  ( ) ln( 2 )(  2 ) ( yi  1 xi ) 2
2
2
2 i 1
Show
L  1  n
     yi  1 xi xi
1   2  i 1
and
n
n
L
n
1
2

(

)

(
)
(
y


x
)
ei 2

i
1 i
2
2
2 2 

2
2( ) i 1
i 1
where e = y - y-hat (the residual at point i)
G02
Verify one of DeMorgan's Laws for sets:
{ s \ u: u U} = s \
G03
G07
{u: u U}
Consider the usual budget line: p1q1 + p2q2 + … + pnqn = y, where the pi are prices, the qi
are quantities, and y is income. Interpreting p and q as vectors and y as a scalar, we can
1
6
state this as the dot product: p · q = y. Suppose p    and q    . Graph the budget
 2
3
line and show that p is orthogonal it. Add the vector p to your graph. (Note: Plot an
accurate graph, not the usual freehand sketch.)
1. Estimate the area under the graph of f(x) = x2 + 5 between x = -2 and x = 1using 3
rectangles and
1. Right endpoints
2. Left endpoints
3. Midpoints
show all work please
G09
The contour plots for four different surfaces are given in the figure. Select the contour
diagram for the function f  x, y  that satisfies the following conditions at the point P.
df
dx
df
2.
dx
df
3.
dx
df
4.
dx
1.
G11
df
is negative.
dy
df
is negative and
is negative.
dy
df
is negative and
is positive.
dy
df
is positive and
is positive.
dy
is positive and
The contour map given below for a function f shows also a path r (t) traversed counter-
clockwise as indicated.
Which of the following properties does the derivative
G12
I.
II.
III.
negative at Q,
positive at R,
positive at P.
1.
2.
3.
4.
5.
6.
7.
8.
I only
all of them
none of them
I and II only
II only
II and III only
I and III only
III only.
Verify that f satisfies
d
f  r  t   have?
dt
 f  ndA  0 over the tetrahedron with vertices
s
(0, 0, 0), (0, 0, 1), (0, 1, 0) and (1, 0, 0).
f  2 x2  3 y 2  5z 2 .
G13
Evaluate triple integral x2eydV where E is bounded by parabolic cylinder y=x2 and
planes z=0,x=1,x=-1
G14
The rectangular enclosure for a loudspeaker system is to have an internal
volume of 2.4 ft3. For aesthetic reasons the height of the enclosure is to be 1.5
times its width. If the top, bottom, and sides of the enclosure are to be
constructed of veneer costing 80 cents per square foot and the front and rear are
to be constructed of particle board costing 40 cents per square foot, find the
dimensions of the enclosure that can be constructed at a minimum cost. (Round
your answers to two decimal places.)
width
height
depth
G15
ft
ft
ft
Sketch the region of integration and convert each polar integral or sum of
integrals to a cartesian integral or sum of integrals. Do not evaluate the integral.

 
2

6
csc
1
r 2 cos  dr d
G16
Find the volume of the solid region in the first octant that is bounded by
2 x  2 y  z  1  0, y  x and x  2.
G17
(a) Use Newton’s Method and the function f  x   xn  a to obtain a general rule for
approximating x  n a .
xi + 1 =
(b) Use the general rule found in part (a) to approximate
places.
4
5
3
19 
G18
4
5 and 3 19 to three decimal
let f  x    sin  5t 2  dt
x
0
Find the MacLaurin polynomial of degree 7 for f(x).
Use this polynomial to estimate the value of

0.67
0
sin  5x2  dx .
(Write as many decimal digits as possible)
G19
a) find the volume enclosed by the coordinate planes and the following surface
z=5+(x-4)2 + 2y , x=3 , y= 4
b) find the volume enclosed by the following surfaces z=7-6x2 -6y2 , z=1
G20
A trust fund is established by depositing $500,000 into an account that earns
continuously compounded interest at an annual rate of 4%. Money is then
withdrawn from the account in a continuous stream at a constant rate of $80,000
per year. Let B(t) denote the balance of the account after t years.
A) give a differential equation and all initial conditions. Use it to determine the
amount of money in the account after two years.
B) Suppose you desired to keep the trust fund constant at $500,000. What
continuous compounding rate is needed to accomplish this?
G21
G22
find the volume bounded Y=
1
, y  0, x  1 rotated about the x-axis
x 1
Evaluate the triple integral 
Zdv
4
, where t is the solid bounded by the surface
Z  2 x  y and the plane Z = 4.
2
2
Evaluate the triple integral 
Zdv
, where T is the solid that lies between the
Zdv
spheres x2+y2+z2=1 and x2+y2+z2=9 and above the cone 
, Z  x2  y 2 .
G23
You are designing a 1000 right circular cylindrical can whose manufacturer will take
waste into account. There is no waste in cutting the aluminum for the side, but the top
and bottom of radius r will be cut from squares that measure 2r units on a side. The total
amount of aluminum used up by the can will therefore be
A=8r2 + 2rh
Find the radius of the most economical can. Just find the critical point, this will be the
radius of the most economical can. Don’t worry about the endpoints of the interval.
G24
(From 13.2) Shown is a contour plot for a function z  f  x, y  .
(Though not labeled on the graph, the horizontal axis is the x-axis, and the vertical axis is
the y-axis.)
In each part, explain your answers.
a) The contour plot doesn’t give perfect information about the function, but as best
we can tell from the plot, is fy (28, 28) positive or negative?
b) Is fx (28, 28) positive or negative?
c) Give a rough estimate of each of fy (28, 28) and fx (28, 28).
G25
Sketch and use double integration in polar coordinates to find the area of one “petal” of
a rose-shaped region bounded by the curve r = cos 3θ.
G27
Evaluate
 sin y dA where D is the triangle with vertices (0, 0), (–2, 2) and (2, 2).
2
D
G28
Use cylindrical coordinates to find the volume of the portion of the unit ball
x2 + y2 + z2 ≤ 1 above the plane z =
 2
2
.
G29
Calculate
0 x
region:
the

3
double
,0  y 

integral

R
x cos  2 x  y  dA
where R is
4.
G30
Use the Divergence Theorem to calculate the surface integral

S
F · dS;
that is, calculate the flux of F across S.
F(x, y, z) = x4i ? x3z2j + 4xy2zk,
S is the surface of the solid bounded by the cylinder
x2 + y2 = 4
and the planes
z = x + 6 and z = 0.
G31
Find the volume of the wedge-shaped region (Figure 1) contained in the
2
2
cylinder x  y  4 and bounded above by the plane z = x and below by the
xy - plane.
V=
the
G32
Suppose there are n lines in the plane such that every three of them can be
intersected with a unit circle.
Prove that all of them can be intersected with a unit circle.
G33
G34
Prove that the marked vertices method solves the point- in polygon problem.
Create a nest of 5 sets whose largest set in the set of all polygons. For each set
but the largest set in the nest, write a definition that precisely describes the
objects in that set in terms of the objects in the next larger level.
G35
Prove that if triangle ABC is an equilateral triangle, then the centre of the ninepoint circle of triangle ABC is the circumcentre.
(You should solve the problem starting directly from the definition of the 9-point
circle.)
G36
G37
Prove the identity V = summation of Vi[xi] where x1,x2,x3 are the natural
coordinate functions.
Calculate the second moment of area of a 2-inch-diameter shaft about the x-x
and y-y axes, as shown.
Use the image below:
G46
G48
Test r=2sin4 for symmetry with respect to  = pi/2, the polar axis, and the pole.
Give the vertex, focus/foci, directrix, and asymptotes if applicable to the following
equations and graph.
25(x-1)2 -9(x+3)2 - 225 =0
6(y-3) = -(x+1)2
G49
1) The graph of y = cos(x) is vertically stretched by a factor of 11, shifted a
distance of 2 units to the left, and translated 1 units downward.
Find a formula for the function whose graph is the resulting graph
f(x)=
2) On the graph of f (x) = 6 sin (3x), points P and Q are at consecutive lowest
and highest points with P occurring before Q. Find the slope of the line which
passes through P and Q.
Slope =
G50
A sporting goods store sells 105 pool tables per year. It costs $20 to store one pool
table for a year. To reorder, there is a fixed cost of $42 per shipment plus $19 for each
pool table. How many times per year should the store order pool tables, and in what lot
size, in order to minimize invenory costs?
The store should order
G51
pool tables
times per year to minimize inventory costs.
coefficient h3 (1-x)n+2 / coefficient h3 (x-1)n = 21/10
Find the value of n??