This is the entrance exam paper for ISI MSQE Entrance Exam for the year 2006. Much more information on the ISI MSQE Entrance Exam and ISI MSQE Entrance preparation help available on http://crackdse.com
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This is the entrance exam paper for ISI MSQE Entrance Exam for the year 2004. Much more information on the ISI MSQE Entrance Exam and ISI MSQE Entrance preparation help available on http://crackdse.com
This is the entrance exam paper for ISI MSQE Entrance Exam for the year 2009. Much more information on the ISI MSQE Entrance Exam and ISI MSQE Entrance preparation help available on http://crackdse.com
This is the entrance exam paper for ISI MSQE Entrance Exam for the year 2010. Much more information on the ISI MSQE Entrance Exam and ISI MSQE Entrance preparation help available on http://crackdse.com
This is the entrance exam paper for ISI MSQE Entrance Exam for the year 2008. Much more information on the ISI MSQE Entrance Exam and ISI MSQE Entrance preparation help available on http://crackdse.com
This is the entrance exam paper for ISI MSQE Entrance Exam for the year 2011. Much more information on the ISI MSQE Entrance Exam and ISI MSQE Entrance preparation help available on http://crackdse.com
This is the entrance exam paper for ISI MSQE Entrance Exam for the year 2005. Much more information on the ISI MSQE Entrance Exam and ISI MSQE Entrance preparation help available on http://crackdse.com
This is the entrance exam paper for ISI MSQE Entrance Exam for the year 2013. Much more information on the ISI MSQE Entrance Exam and ISI MSQE Entrance preparation help available on http://crackdse.com
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JNU MA Economics Entrance Test Paper (2013)CrackDSE
This is the entrance exam paper for JNU MA Economics Entrance Exam for the year 2013. Much more information on the JNU MA Economics Entrance Exam and JNU MA Economics Entrance preparation help available on http://crackdse.com
II PUC (MATHEMATICS) ANNUAL MODEL QUESTION PAPER FOR ALL SCIENCE STUDENTS WHO...Bagalkot
My dear Students,
Wishing you all happy SHIVRATRI. & ALL THE BEST IN YOUR ANNUAL EXAMS-2014
Here I have uploaded II- P.U.C MATHEMATICS MODEL QUESTION PAPER FOR the year 2014 Which i have designed according to New syllabus of CBSE. I hope this model paper will be helpful to all the students who are writing annual exams on 18-March-2014.
wish you all the best
Regards,
A. NAGARAJ
Director-Faculty
Shree Susheela Tutorials
BAGALKOT-587101
mob: 9845222682
Motivated by presenting mathematics visually and interestingly to common people based on calculus and its extension, parametric curves are explored here to have two and three dimensional objects such that these objects can be used for demonstrating mathematics.
Epicycloid, hypocycloid are particular curves that are implemented in MATLAB programs and the motifs are presented here. The obtained curves are considered to be domains for complex mappings to have new variation of Figures and objects. Additionally Voronoi mapping is also implemented to some parametric curves and some resulting complex mappings.
Some obtained 3 dimensional objects are considered as flowers and animals inspiring to be mathematical ornaments of hypocycloid dance which is also illustrated here.
1. understand the terms function, domain, range, one-one function,inverse function and composition of functions
2. identify the range of a given function in simple cases, and find the
composition of two given functions
3. determine whether or not a given function is one-one, and find the inverse of a one-one function in simple cases
4. illustrate in graphical terms the relation between a one-one function and its inverse.
This learner's module discusses or talks about the topic of Quadratic Functions. It also discusses what is Quadratic Functions. It also shows how to transform or rewrite the equation f(x)=ax2 + bx + c to f(x)= a(x-h)2 + k. It will also show the different characteristics of Quadratic Functions.
The pattern of question paper in the subject Mathematics has been changed in CBSE,India.I am uploading the paper with marking scheme so that students will be benefitted-Pratima Nayak,KVS
I am Frank P. I am a Statistics Coursework Expert at statisticsassignmenthelp.com. I hold a master's in Statistics from Malacca, Malaysia. I have been helping students with their assignments for the past 10 years. I solve assignments related to Statistics. Visit statisticsassignmenthelp.com or email info@statisticsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Statistics Assignments.
Chapter wise important questions in Mathematics for Karnataka 2 year PU Science students. This is taken from the PU board website and compiled together.
MATH 107 FINAL EXAMINATIONMULTIPLE CHOICE1. Deter.docxTatianaMajor22
MATH 107 FINAL EXAMINATION
MULTIPLE CHOICE
1. Determine the domain and range of the piecewise function.
A. Domain [–2, 2];
B. Domain [–1, 1];
C. Domain [–1, 3];
D. Domain [–3/2, –1/2];
2. Solve:
A. 3
B. 3,7
C. 9
D. No solution
3. Determine the interval(s) on which the function is increasing.
A. (−1.3, 1.3)
B. (1, 3)
C. (−∞,−1)and (3,∞)
D. (−2.5, 1)and (4.5,∞)
4. Determine whether the graph of y = 2|x| + 1 is symmetric with respect to the origin,
the x-axis, or the y-axis.
A. symmetric with respect to the origin only
B. symmetric with respect to the x-axis only
C. symmetric with respect to the y-axis only
D. not symmetric with respect to the origin, not symmetric with respect to the x-axis, and
not symmetric with respect to the y-axis
5. Solve, and express the answer in interval notation: | 9 – 7x | ≤ 12.
A. (–∞, –3/7]
B. (–∞, −3/7] ∪ [3, ∞) C. [–3, 3/7]
D. [–3/7, 3]
6. Which of the following represents the graph of 7x + 2y = 14 ?
A. B.
C. D.
7. Write a slope-intercept equation for a line parallel to the line x – 2y = 6 which passes through the point (10, – 4).
A.
B.
C.
D.
8. Which of the following best describes the graph?
A. It is the graph of a function and it is one-to-one.
B. It is the graph of a function and it is not one-to-one.
C. It is not the graph of a function and it is one-to-one.
D. It is not the graph of a function and it is not one-to-one.
9. Express as a single logarithm: log x + log 1 – 6 log (y + 4)
A.
B.
C.
D.
10. Which of the functions corresponds to the graph?
A.
B.
C.
D.
11. Suppose that a function f has exactly one x-intercept.
Which of the following statements MUST be true?
A. f is a linear function.
B. f (x) ≥ 0 for all x in the domain of f.
C. The equation f(x) = 0 has exactly one real-number solution.
D. f is an invertible function.
12. The graph of y = f(x) is shown at the left and the graph of y = g(x) is shown at the right. (No formulas are given.) What is the relationship between g(x) and f(x)?
y = f (x) y = g(x)
A. g(x) = f (x – 3) + 1
B. g(x) = f (x – 1) + 3
C. g(x) = f (x + 3) – 1
D. g(x) = f (x + 1) .
Pre-Calculus Midterm Exam
Score: ______ / ______
Name: ____________________________
Student Number: ___________________
Short Answer: Type your answer below each question. Show your work.
1
Verify the identity. Show your work.
cot θ ∙ sec θ = csc θ
cot= cos/ sin
sec= 1/cos
cos/sin*1/cos
1/sin x
2
A gas company has the following rate schedule for natural gas usage in single-family residences:
Monthly service charge $8.80
Per therm service charge
1st 25 therms $0.6686/therm
Over 25 therms $0.85870/therm
What is the charge for using 25 therms in one month? Show your work.
8.8+25*0.6686
16.715+8.8
25.52
What is the charge for using 45 therms in one month? Show your work.
8.8+25*0.6686+20*0.85870
25.52+17.174
42.694
Construct a function that gives the monthly charge C for x therms of gas.
C(x)=8.8+0.6686x
If 0<=x<=25
25.515+0.85870(x-25)
If x>25
25.515= 8.8 + 25*0.6686
3
The wind chill factor represents the equivalent air temperature at a standard wind speed that would produce the same heat loss as the given temperature and wind speed. One formula for computing the equivalent temperature is
W(t) =
where v represents the wind speed (in meters per second) and t represents the air temperature . Compute the wind chill for an air temperature of 15°C and a wind speed of 12 meters per second. (Round the answer to one decimal place.) Show your work.
6.0c
4
Complete the following:
(a) Use the Leading Coefficient Test to determine the graph's end behavior.
The leading coefficient is positive x->infinity
(b) Find the x-intercepts. State whether the graph crosses the x-axis or touches the x-axis and turns around at each intercept.
0 with multiplicity 2 and -2 with multiplicity 1 this it touches at 0 and goes through at 2
(c) Find the y-intercept. Y intercept= the values you get when x=0, therefore the y-intercept is 0
f(x) = x2(x + 2)
5
For the data set shown by the table,
a. Create a scatter plot for the data. (You do not need to submit the scatter plot)
b. Use the scatter plot to determine whether an exponential function or a logarithmic function is the best choice for modeling the data.
Logarithmic function
Number of Homes Built in a Town by Year
6
Verify the identity. Show your work.
(1 + tan2u)(1 - sin2u) = 1
Sec^2u*cos^2u=1/cos^2u*cos^2u
7
Verify the identity. Show your work.
cot2x + csc2x = 2csc2x – 1
Csc x is cosecx
Cot^2x+1=cosec^2x
Lhs=cot^2x+cosec^2x+cosec^2x
Cot^2x+cosec^x=1+2cosec^2x
8
Verify the identity. Show your work.
1 + sec2xsin2x = sec2x
Sec^2x=1/cos^2x
Sec^2xsin^x=sin^2x/cos^2x=tan^2x
1+tan^2x=sex^2x
9
Verify the identity. Show your work.
cos(α - β) - cos(α + β) = 2 sin α sin β
cos(a-b)=cos(a)*cos(b)+sin(a)*sin(b)
cos(a+b)=cos(.
JNU MA Economics Entrance Test Paper (2013)CrackDSE
This is the entrance exam paper for JNU MA Economics Entrance Exam for the year 2013. Much more information on the JNU MA Economics Entrance Exam and JNU MA Economics Entrance preparation help available on http://crackdse.com
II PUC (MATHEMATICS) ANNUAL MODEL QUESTION PAPER FOR ALL SCIENCE STUDENTS WHO...Bagalkot
My dear Students,
Wishing you all happy SHIVRATRI. & ALL THE BEST IN YOUR ANNUAL EXAMS-2014
Here I have uploaded II- P.U.C MATHEMATICS MODEL QUESTION PAPER FOR the year 2014 Which i have designed according to New syllabus of CBSE. I hope this model paper will be helpful to all the students who are writing annual exams on 18-March-2014.
wish you all the best
Regards,
A. NAGARAJ
Director-Faculty
Shree Susheela Tutorials
BAGALKOT-587101
mob: 9845222682
Motivated by presenting mathematics visually and interestingly to common people based on calculus and its extension, parametric curves are explored here to have two and three dimensional objects such that these objects can be used for demonstrating mathematics.
Epicycloid, hypocycloid are particular curves that are implemented in MATLAB programs and the motifs are presented here. The obtained curves are considered to be domains for complex mappings to have new variation of Figures and objects. Additionally Voronoi mapping is also implemented to some parametric curves and some resulting complex mappings.
Some obtained 3 dimensional objects are considered as flowers and animals inspiring to be mathematical ornaments of hypocycloid dance which is also illustrated here.
1. understand the terms function, domain, range, one-one function,inverse function and composition of functions
2. identify the range of a given function in simple cases, and find the
composition of two given functions
3. determine whether or not a given function is one-one, and find the inverse of a one-one function in simple cases
4. illustrate in graphical terms the relation between a one-one function and its inverse.
This learner's module discusses or talks about the topic of Quadratic Functions. It also discusses what is Quadratic Functions. It also shows how to transform or rewrite the equation f(x)=ax2 + bx + c to f(x)= a(x-h)2 + k. It will also show the different characteristics of Quadratic Functions.
The pattern of question paper in the subject Mathematics has been changed in CBSE,India.I am uploading the paper with marking scheme so that students will be benefitted-Pratima Nayak,KVS
I am Frank P. I am a Statistics Coursework Expert at statisticsassignmenthelp.com. I hold a master's in Statistics from Malacca, Malaysia. I have been helping students with their assignments for the past 10 years. I solve assignments related to Statistics. Visit statisticsassignmenthelp.com or email info@statisticsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Statistics Assignments.
Chapter wise important questions in Mathematics for Karnataka 2 year PU Science students. This is taken from the PU board website and compiled together.
MATH 107 FINAL EXAMINATIONMULTIPLE CHOICE1. Deter.docxTatianaMajor22
MATH 107 FINAL EXAMINATION
MULTIPLE CHOICE
1. Determine the domain and range of the piecewise function.
A. Domain [–2, 2];
B. Domain [–1, 1];
C. Domain [–1, 3];
D. Domain [–3/2, –1/2];
2. Solve:
A. 3
B. 3,7
C. 9
D. No solution
3. Determine the interval(s) on which the function is increasing.
A. (−1.3, 1.3)
B. (1, 3)
C. (−∞,−1)and (3,∞)
D. (−2.5, 1)and (4.5,∞)
4. Determine whether the graph of y = 2|x| + 1 is symmetric with respect to the origin,
the x-axis, or the y-axis.
A. symmetric with respect to the origin only
B. symmetric with respect to the x-axis only
C. symmetric with respect to the y-axis only
D. not symmetric with respect to the origin, not symmetric with respect to the x-axis, and
not symmetric with respect to the y-axis
5. Solve, and express the answer in interval notation: | 9 – 7x | ≤ 12.
A. (–∞, –3/7]
B. (–∞, −3/7] ∪ [3, ∞) C. [–3, 3/7]
D. [–3/7, 3]
6. Which of the following represents the graph of 7x + 2y = 14 ?
A. B.
C. D.
7. Write a slope-intercept equation for a line parallel to the line x – 2y = 6 which passes through the point (10, – 4).
A.
B.
C.
D.
8. Which of the following best describes the graph?
A. It is the graph of a function and it is one-to-one.
B. It is the graph of a function and it is not one-to-one.
C. It is not the graph of a function and it is one-to-one.
D. It is not the graph of a function and it is not one-to-one.
9. Express as a single logarithm: log x + log 1 – 6 log (y + 4)
A.
B.
C.
D.
10. Which of the functions corresponds to the graph?
A.
B.
C.
D.
11. Suppose that a function f has exactly one x-intercept.
Which of the following statements MUST be true?
A. f is a linear function.
B. f (x) ≥ 0 for all x in the domain of f.
C. The equation f(x) = 0 has exactly one real-number solution.
D. f is an invertible function.
12. The graph of y = f(x) is shown at the left and the graph of y = g(x) is shown at the right. (No formulas are given.) What is the relationship between g(x) and f(x)?
y = f (x) y = g(x)
A. g(x) = f (x – 3) + 1
B. g(x) = f (x – 1) + 3
C. g(x) = f (x + 3) – 1
D. g(x) = f (x + 1) .
Pre-Calculus Midterm Exam
Score: ______ / ______
Name: ____________________________
Student Number: ___________________
Short Answer: Type your answer below each question. Show your work.
1
Verify the identity. Show your work.
cot θ ∙ sec θ = csc θ
cot= cos/ sin
sec= 1/cos
cos/sin*1/cos
1/sin x
2
A gas company has the following rate schedule for natural gas usage in single-family residences:
Monthly service charge $8.80
Per therm service charge
1st 25 therms $0.6686/therm
Over 25 therms $0.85870/therm
What is the charge for using 25 therms in one month? Show your work.
8.8+25*0.6686
16.715+8.8
25.52
What is the charge for using 45 therms in one month? Show your work.
8.8+25*0.6686+20*0.85870
25.52+17.174
42.694
Construct a function that gives the monthly charge C for x therms of gas.
C(x)=8.8+0.6686x
If 0<=x<=25
25.515+0.85870(x-25)
If x>25
25.515= 8.8 + 25*0.6686
3
The wind chill factor represents the equivalent air temperature at a standard wind speed that would produce the same heat loss as the given temperature and wind speed. One formula for computing the equivalent temperature is
W(t) =
where v represents the wind speed (in meters per second) and t represents the air temperature . Compute the wind chill for an air temperature of 15°C and a wind speed of 12 meters per second. (Round the answer to one decimal place.) Show your work.
6.0c
4
Complete the following:
(a) Use the Leading Coefficient Test to determine the graph's end behavior.
The leading coefficient is positive x->infinity
(b) Find the x-intercepts. State whether the graph crosses the x-axis or touches the x-axis and turns around at each intercept.
0 with multiplicity 2 and -2 with multiplicity 1 this it touches at 0 and goes through at 2
(c) Find the y-intercept. Y intercept= the values you get when x=0, therefore the y-intercept is 0
f(x) = x2(x + 2)
5
For the data set shown by the table,
a. Create a scatter plot for the data. (You do not need to submit the scatter plot)
b. Use the scatter plot to determine whether an exponential function or a logarithmic function is the best choice for modeling the data.
Logarithmic function
Number of Homes Built in a Town by Year
6
Verify the identity. Show your work.
(1 + tan2u)(1 - sin2u) = 1
Sec^2u*cos^2u=1/cos^2u*cos^2u
7
Verify the identity. Show your work.
cot2x + csc2x = 2csc2x – 1
Csc x is cosecx
Cot^2x+1=cosec^2x
Lhs=cot^2x+cosec^2x+cosec^2x
Cot^2x+cosec^x=1+2cosec^2x
8
Verify the identity. Show your work.
1 + sec2xsin2x = sec2x
Sec^2x=1/cos^2x
Sec^2xsin^x=sin^2x/cos^2x=tan^2x
1+tan^2x=sex^2x
9
Verify the identity. Show your work.
cos(α - β) - cos(α + β) = 2 sin α sin β
cos(a-b)=cos(a)*cos(b)+sin(a)*sin(b)
cos(a+b)=cos(.
Name ____________________________Student Number ________________.docxTanaMaeskm
Name: ____________________________
Student Number: ___________________
Short Answer:
Type your answer below each question. Show your work.
1
Verify the identity.
Show your work.
cot θ ∙ sec θ = csc θ
2
A gas company has the following rate schedule for natural gas usage in single-family residences:
Monthly service charge $8.80
Per therm service charge
1st 25 therms $0.6686/therm
Over 25 therms $0.85870/therm
What is the charge for using 25 therms in one month? Show your work.
What is the charge for using 45 therms in one month? Show your work.
Construct a function that gives the monthly charge C for x therms of gas.
3
The wind chill factor represents the equivalent air temperature at a standard wind speed that would produce the same heat loss as the given temperature and wind speed. One formula for computing the equivalent temperature is
W(t) =
where v represents the wind speed (in meters per second) and t represents the air temperature . Compute the wind chill for an air temperature of 15°C and a wind speed of 12 meters per second. (Round the answer to one decimal place.) Show your work.
4
Complete the following:
(a) Use the Leading Coefficient Test to determine the graph's end behavior.
(b) Find the x-intercepts. State whether the graph crosses the x-axis or touches the x-axis and turns around at each intercept.
Show your work.
(c) Find the y-intercept.
Show your work.
f(x) = x
2
(x + 2)
(a).
(b).
(c).
5
For the data set shown by the table,
a. Create a scatter plot for the data. (You do not need to submit the scatter plot)
b. Use the scatter plot to determine whether an exponential function or a logarithmic function is the best choice for modeling the data.
Number of Homes Built in a Town by Year
6
Verify the identity. Show your work.
(1 + tan
2
u)(1 - sin
2
u) = 1
7
Verify the identity
. Show your work.
cot
2
x + csc
2
x = 2csc
2
x - 1
8
Verify the identity. Show your work.
1 + sec
2
xsin
2
x = sec
2
x
9
Verify the identity.
Show your work.
cos(α - β) - cos(α + β) = 2 sin α sin β
10
The following data represents the normal monthly precipitation for a certain city.
Draw a scatter diagram of the data for one period. (You do not need to submit the scatter diagram). Find the sinusoidal function of the form that fits the data. Show your work.
.
11.
The graph below shows the percentage of students enrolled in the College of Engineering at State University. Use the graph to answer the question.
Does the graph represent a function? Explain
12.
Find the vertical asymptotes, if any, of the graph of the rational function. Show your work.
f(x) =
13.
The formula A = 118e
0.024t
models the popula.
I am Bella A. I am a Statistical Method In Economics Assignment Expert at economicshomeworkhelper.com/. I hold a Ph.D. in Economics. I have been helping students with their homework for the past 9 years. I solve assignments related to Economics Assignment.
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1 College Algebra Final Examination---FHSU Math & C.S.docxjoyjonna282
1
College Algebra
Final Examination---FHSU Math & C.S. Department
Form G Name ____________________________
Multiple Choice:
Mark through (horizontally) the space on the answer card corresponding to the best answer for each
problem.
1. Solve the equation by factoring: 2 9 18 0x x
[A] 6, 3 [B] 6, 3 [C] 6, 3 [D] 6, 3
2. Write the product in the standard form a + bi: (2 6 )(2 9 )i i
[A] 58 6i [B] 50 30i [C] 58 6i [D]
2
54 6 4i i
3. Find the real solutions of the equation 14 21 2 .x x
[A] 5 [B] 3 [C] 4 [D] 5
4. Solve the inequality 3 7 2 4x x . Express your answer using interval notation.
[A] 11, [B] [3, ) [C] 3, [D] ( , 3]
5. Solve the equation 3 3 18.x
[A] 3, 9 [B] 3 [C] 9, 3 [D] no solution
6. Solve the inequality 5 1 5x . Express your answer using interval notation.
[A] 645 5, or , [B]
64
5 5
,
[C] 645 5, [D]
64
5 5
, or ,
7. A bank loaned $68,000, part of it at a rate of 15% per year and the rest at a rate of 5% per year. If the
interest received was $6600, how much was loaned at 15%?
[A] $36, 000 [B] $33, 000 [C] $32, 000 [D] $35, 000
8. Find the distance between the points 1 2(2, 2) and ( 10, 3).PP
[A] 14 [B] 26 [C] 169 [D] 13
9. List the intercepts for the graph of the equation
2
16 0.x y
[A] ( 4, 0), (0,16), (4, 0) [B] (4, 0), (0,16), (0, 16)
[C] (0, 4), (16, 0), (0, 4) [D] ( 4, 0), (0, 16), (4, 0)
2
10. Find the slope-intercept form of the equation of the line that is parallel with the given properties:
( 2, 3); and 9 20y x
[A] 9 15y x [B] 9 20y x [C]
1
20
9
y x [D]
1
20
9
y x
11. Write the standard from of the equation of the circle with radius 5 and center (1, 4) .
[A]
2 2
4 1 25x y [B]
2 2
1 4 5x y
[C]
2 2
4 1 25x y [D]
2 2
1 4 5x y
12. Write a general formula to describe the variation.
The illumination I produced on a surface by a source of light varies directly as the candlepower c of
the source and inversely as the square of the distance d between the source and the surface.
[A]
2
I kcd [B]
2
2
kc
I
d
[C]
2
kc
I
d
[D]
2
kd
I
c
13. For the given functions ( ) 6 3 and ( ) 7 9f x x g x x , determine ( )( ).f g x
[A]
2
( )( ) 4 27f g x x [B]
2
( )( ) 13 33 6f g x x x
[C]
2
( )( ) 42 33 27f g x x x [D]
2
( )( ) 42 12 27f g x x x
14. The graph of a function f is given. For what values of x is
( ) 0f x ?
[A] 15,17.5
[B] 15,17.5, 25
[C] 15
[D] 25, 15,17.5, 25
15. Determine whether the function
3 2
( ) 4 9f x x is even, odd, neither, or ...
Pre-Calculus Midterm Exam
Score: ______ / ______
Name: ____________________________
Student Number: ___________________
Short Answer: Type your answer below each question. Show your work.
1
Verify the identity. Show your work.
cot θ ∙ sec θ = csc θ
2
A gas company has the following rate schedule for natural gas usage in single-family residences:
Monthly service charge $8.80
Per therm service charge
1st 25 therms $0.6686/therm
Over 25 therms $0.85870/therm
What is the charge for using 25 therms in one month? Show your work.
What is the charge for using 45 therms in one month? Show your work.
Construct a function that gives the monthly charge C for x therms of gas.
3
The wind chill factor represents the equivalent air temperature at a standard wind speed that would produce the same heat loss as the given temperature and wind speed. One formula for computing the equivalent temperature is
W(t) =
where v represents the wind speed (in meters per second) and t represents the air temperature . Compute the wind chill for an air temperature of 15°C and a wind speed of 12 meters per second. (Round the answer to one decimal place.) Show your work.
4
Complete the following:
(a) Use the Leading Coefficient Test to determine the graph's end behavior.
(b) Find the x-intercepts. State whether the graph crosses the x-axis or touches the x-axis and turns around at each intercept. Show your work.
(c) Find the y-intercept. Show your work.
f(x) = x2(x + 2)
(a).
(b).
(c).
5
For the data set shown by the table,
a. Create a scatter plot for the data. (You do not need to submit the scatter plot)
b. Use the scatter plot to determine whether an exponential function or a logarithmic function is the best choice for modeling the data.
Number of Homes Built in a Town by Year
6
Verify the identity. Show your work.
(1 + tan2u)(1 - sin2u) = 1
7
Verify the identity. Show your work.
cot2x + csc2x = 2csc2x - 1
8
Verify the identity. Show your work.
1 + sec2xsin2x = sec2x
9
Verify the identity. Show your work.
cos(α - β) - cos(α + β) = 2 sin α sin β
10
The following data represents the normal monthly precipitation for a certain city.
Draw a scatter diagram of the data for one period. (You do not need to submit the scatter diagram). Find the sinusoidal function of the form that fits the data. Show your work.
.
11.
The graph below shows the percentage of students enrolled in the College of Engineering at State University. Use the graph to answer the question.
Does the graph represent a function? Explain
12.
Find the vertical asymptotes, if any, of the graph of the rational function. Show your work.
f(x) =
13.
T.
1.Evaluate the function at the indicated value of x. Round your.docxpaynetawnya
1.
Evaluate the function at the indicated value of x. Round your result to three decimal places.
Function: f(x) = 0.5^x Value: x = 1.7
-0.308
1.7
0.308
0.5
2.
Solve for x. 3x = 81
7
3
4
-3
3.
Logarithms are the inverse of exponentials.
True
False
4.
The Logarithm Quotient Rule states:
logb(x / y) = logb(x) + logb(y)
logb(x / y) = logb(x) - logb(y)
logb(x y) = y ∙ logb(x)
logb(c) = 1 / logc(b)
5.
Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. Assume all variables are positive.
log3 9x
log3 9 * log3 x
log3 9 + log3 x
log3 9 - log3
none of these
6.
Select the graph of the function. Indicate which graph is correct: 1st, 2nd, 3rd, or 4th
f(x) = 5x-1
7.
Evaluate the function at the indicated value of x. Round your result to three decimal places.
Value: x=2
8.
The exponential equation y=bx is equivalent to the logarithmic equation x=logby
True
False
9.
Use the One-to-One property to solve the equation for x.
e(3x+5) = e6
x = -1/3
x2 = 6
x = 1/3
x = 3
10.
Write the logarithmic equation in exponential form.
log8 64 = 2
82 = 16
82 = 88
82 = 64
864 = 2
11.
Write the exponential equation in logarithmic form.
43 = 64
log64 4 = 3
log4 64 = 3
log4 64 = -3
log4 3 = 64
12.
The given x-value is a solution (or an approximate solution) of the equation.
42x-7 = 16
x = 5
True
False
13.
Find the magnitude R of each earthquake of intensity I (let I0=1). (Hint: R=log (I/I0)
I = 19000
3.28
5.28
4.28
2.38
14.
pH is a measure of the hydrogen ion concentration of a solution. It is defined as the negative logarithm of the hydrogen ion concentration. The equation is:
pH = - log [H+]
If an acid has an H+ concentration of 10-4, what's the pH?
15.
A general formula for exponential Growth can be given by:
A = P ekt
In your textbook, or using another reliable source, research what values, P, A, k and t represent and write your answer. (Hint: What do each of the variables stand for?)
16.
Continuously compounded interest means your principal is earning interest and you keep earning interest on the interest earned. Research the formula for Continuously Compounded Interest and write it below.
17.
$2500 is invested in an account at interest rate r, compounded continuously. Find the time required for the amount to double. (Approximate the result to two decimal places.)
r = 0.0570
13.16 years
10.16 years
11.16 years
12.16 years
18.
Write Eulers Number (e) to three decimal places.
19.
Exponential functions often involve the rate of increase or decrease of something such as a population, for example. If there is a population increase, it is a _______ function and when there is a decrease, it is a ________ function.
20.
Do some research on important numbers in mathematics. Choose one that is interesting to you (it can be on e or pi or a different number)
In your post, share with us what you ...
1. Write an equation in standard form of the parabola that has th.docxKiyokoSlagleis
1.
Write an equation in standard form of the parabola that has the same shape as the graph of f(x) = 2x
2
, but with the given point as the vertex (5, 3).
A. f(x) = (2x - 4) + 4
B. f(x) = 2(2x + 8) + 3
C. f(x) = 2(x - 5)
2
+ 3
D. f(x) = 2(x + 3)
2
+ 3
2 of 20
5.0 Points
Find the coordinates of the vertex for the parabola defined by the given quadratic function.
f(x) = 2(x - 3)
2
+ 1
A. (3, 1)
B. (7, 2)
C. (6, 5)
D. (2, 1)
3 of 20
5.0 Points
Find the vertical asymptotes, if any, and the values of x corresponding to holes, if any, of the graph of the following rational function.
g(x) = x + 3/x(x + 4)
A. Vertical asymptotes: x = 4, x = 0; holes at 3x
B. Vertical asymptotes: x = -8, x = 0; holes at x + 4
C. Vertical asymptotes: x = -4, x = 0; no holes
D. Vertical asymptotes: x = 5, x = 0; holes at x - 3
4 of 20
5.0 Points
"Y varies directly as the n
th
power of x" can be modeled by the equation:
A. y = kx
n
.
B. y = kx/n.
C. y = kx
*n
.
D. y = kn
x
.
5 of 20
5.0 Points
40 times a number added to the negative square of that number can be expressed as:
A.
A(x) = x
2
+ 20x.
B. A(x) = -x + 30x.
C.
A(x) = -x
2
- 60x.
D.
A(x) = -x
2
+ 40x.
6 of 20
5.0 Points
The graph of f(x) = -x
3
__________ to the left and __________ to the right.
A. rises; falls
B. falls; falls
C. falls; rises
D. falls; falls
Solve the following formula for the specified variable:
V = 1/3 lwh for h
7 of 20
Write an equation that expresses each relationship. Then solve the equation for y.
x varies jointly as y and z
A. x = kz; y = x/k
B. x = kyz; y = x/kz
C. x = kzy; y = x/z
D. x = ky/z; y = x/zk
8 of 20
8 times a number subtracted from the squared of that number can be expressed as:
A. P(x) = x + 7x.
B.P(x) = x
2
- 8x.
C. P(x) = x - x.
P(x) = x
2
+ 10x.
9of 20
Find the x-intercepts. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each intercept.
f(x) = x
4
- 9x
2
A. x = 0, x = 3, x = -3; f(x) crosses the x-axis at -3 and 3; f(x) touches the x-axis at 0.
B. x = 1, x = 2, x = 3; f(x) crosses the x-axis at 2 and 3; f(x) crosses the x-axis at 0.
C. x = 0, x = -3, x = 5; f(x) touches the x-axis at -3 and 5; f(x) touches the x-axis at 0.
D. x = 1, x = 2, x = -4; f(x) crosses the x-axis at 2 and -4; f(x) touches the x-axis at 0.
10 of 20
Find the domain of the following rational function.
f(x) = x + 7/x
2
+ 49
A. All real numbers < 69
B. All real numbers > 210
C. All real numbers ≤ 77
D. All real numbers
11 of 20
Write an equation in standard form of the parabola that has the same shape as the graph of f(x) = 3x
2
or g(x) = -3x
2
, but with the given maximum or minimum.
Minimum = 0 at x = 11
A. f(x) = 6(x - 9)
B. f(x) = 3(x - 11)
2
C. f(x) = 4(x + 10)
D. f(x) = 3(x
2
- 15)
2
12 of 20
Solve the following polynomial inequality.
3x
2
+ 10x - 8 ≤ 0
A. [6, 1/3]
B. [-4, 2/3]
C. [-9, 4/5]
D. [8, 2/7]
13 of 20
Find the coordinate.
Short Answer Type your answer below each question. Show your work.docxbjohn46
Short Answer:
Type your answer below each question. Show your work.
1
Verify the identity.
cot θ ∙ sec θ = csc θ
2
A gas company has the following rate schedule for natural gas usage in single-family residences:
Monthly service charge $8.80
Per therm service charge
1st 25 therms $0.6686/therm
Over 25 therms $0.85870/therm
What is the charge for using 25 therms in one month?
What is the charge for using 45 therms in one month?
Construct a function that gives the monthly charge C for x therms of gas.
3
The wind chill factor represents the equivalent air temperature at a standard wind speed that would produce the same heat loss as the given temperature and wind speed. One formula for computing the equivalent temperature is
W(t) =
where v represents the wind speed (in meters per second) and t represents the air temperature . Compute the wind chill for an air temperature of 15°C and a wind speed of 12 meters per second. (Round the answer to one decimal place.)
4
Complete the following:
(a) Use the Leading Coefficient Test to determine the graph's end behavior.
(b) Find the x-intercepts. State whether the graph crosses the x-axis or touches the x-axis and turns around at each intercept.
(c) Find the y-intercept.
f(x) = x
2
(x + 2)
(a).
(b).
(c).
5
For the data set shown by the table,
a. Create a scatter plot for the data. (You do not need to submit the scatter plot)
b. Use the scatter plot to determine whether an exponential function or a logarithmic function is the best choice for modeling the data.
Number of Homes Built in a Town by Year
6
Verify the identity.
(1 + tan
2
u)(1 - sin
2
u) = 1
7
Verify the identity
.
cot
2
x + csc
2
x = 2csc
2
x - 1
8
Verify the identity.
1 + sec
2
xsin
2
x = sec
2
x
9
Verify the identity.
cos(α - β) - cos(α + β) = 2 sin α sin β
10
The following data represents the normal monthly precipitation for a certain city.
Draw a scatter diagram of the data for one period. (You do not need to submit the scatter diagram). Find the sinusoidal function of the form
that fits the data.
Multiple Choice:
Type your answer choice in the blank next to each question.
_____11.
The graph below shows the percentage of students enrolled in the College of Engineering at State University. Use the graph to answer the question.
Does the graph represent a function?
A. Yes
B. No
_____12.
Find the vertical asymptotes, if any, of the graph of the rational function.
f(x) =
A. x = 0 and x = 4
B. x = 0
C. x = 4
D. no vertical asymptote
_____13.
The formula A = 118e
0.024t
models the population of a particular city, in thousands, t years after 1998. When will the population of the city reach 140 thousand?
A. 2008
B. 2005
C. 2006
.
Question 1 Aggregate Demand and Aggregate Supply (This question i.docxIRESH3
Question 1: Aggregate Demand and Aggregate Supply (This question is worth 20 points if correctly answered.)
Assume the U.S. economy is in long-run equilibrium. Analyze each of the following events independently and include answers to the following in your analysis: (1) Explain whether AD or SAS changes and why the change occurred. (2) Explain what happens to the equilibrium price level and equilibrium output in the U.S. in the short run. (3) Describe the type of gap facing the economy. (4) Draw a graph to illustrate your answer.
a. The bubble in the housing market bursts, and prices of houses quickly begin to fall.
b. With plenty of slack in the labor market, firms lower wages.
c. Anticipating the possibility of war, the government increases its purchases of military equipment.
d. Productivity in the U.S. continues to increase.
Question 2: More Aggregate Demand/Aggregate Supply (This question is worth 10 points if correctly answered.)
a. Suppose the United States’ economy is in short run equilibrium producing RGDP equal to $150 billion. Potential GDP equals $250 billion. The marginal propensity to consume in the U.S. is 0.5. Draw a graph illustrating the U.S. economy. Is the economy characterized by a recessionary gap or an inflationary gap? What problems does the gap present for United States?
b. You are an economic advisor to the President. He asks you to design a fiscal policy to close the gap. What fiscal policy do you propose? Why did you choose this particular policy? Explain how your policy works. Draw a graph illustrating your answer.
c. Describe any costs the United States may bear in the long run due to the implementation of the policy you designed in part (b).
d. If your policy is not acceptable to Congress, describe the self-correction mechanism by which the economy could return to long-run equilibrium. Draw a graph illustrating the self-correction process. Describe any costs the United States may pay with self-correction.
Question 3: The Federal Reserve System (the Fed). (This question is worth 10 points if correctly answered.)
a. Describe the structure of the Federal Reserve System.
b. The government of Turtleville uses measures of monetary aggregates similar to the United States, and the central bank of Turtleville imposes a required reserve ratio of 10%. Given the following information, answer the questions below.
Bank deposits at the central bank = $200 million
Currency held by the public = $150 million
Checkable bank deposits = $500 million
Currency in bank vaults = $100 million
Traveler’s checks = $10 million
1. M1 = _____________
2. The Monetary Base = _____________
3. Excess Reserves = _______________
4. The amount by which commercial banks in Turtleville can increase checkable deposits: _____________
1
Solving Differential Equations:
1. Solve the following diffe ...
1) Use properties of logarithms to expand the following logarithm.docxdorishigh
1) Use properties of logarithms to expand the following logarithmic expression as much as possible.
Logb (√xy3 / z3)
A. 1/2 logb x - 6 logb y + 3 logb z
B. 1/2 logb x - 9 logb y - 3 logb z
C. 1/2 logb x + 3 logb y + 6 logb z
D. 1/2 logb x + 3 logb y - 3 logb z
2) Solve the following logarithmic equation. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, to two decimal places, for the solution.
2 log x = log 25
A. {12}
B. {5}
C. {-3}
D. {25}
3) Write the following equation in its equivalent logarithmic form.
2-4 = 1/16
A. Log4 1/16 = 64
B. Log2 1/24 = -4
C. Log2 1/16 = -4
D. Log4 1/16 = 54
4) Use properties of logarithms to condense the following logarithmic expression. Write the expression as a single logarithm whose coefficient is 1.
log2 96 – log2 3
A. 5
B. 7
C. 12
D. 4
5) Use the exponential growth model, A = A0ekt, to show that the time it takes a population to double (to grow from A0 to 2A0 ) is given by t = ln 2/k.
A. A0 = A0ekt; ln = ekt; ln 2 = ln ekt; ln 2 = kt; ln 2/k = t
B. 2A0 = A0e; 2= ekt; ln = ln ekt; ln 2 = kt; ln 2/k = t
C. 2A0 = A0ekt; 2= ekt; ln 2 = ln ekt; ln 2 = kt; ln 2/k = t
D. 2A0 = A0ekt; 2 = ekt; ln 1 = ln ekt; ln 2 = kt; ln 2/k = toe
6) Find the domain of following logarithmic function.
f(x) = log (2 - x)
A. (∞, 4)
B. (∞, -12)
C. (-∞, 2)
D. (-∞, -3)
7) An artifact originally had 16 grams of carbon-14 present. The decay model A = 16e -0.000121t describes the amount of carbon-14 present after t years. How many grams of carbon-14 will be present in 5715 years?
A. Approximately 7 grams
B. Approximately 8 grams
C. Approximately 23 grams
D. Approximately 4 grams
8) Use properties of logarithms to expand the following logarithmic expression as much as possible.
logb (x2 y) / z2
A. 2 logb x + logb y - 2 logb z
B. 4 logb x - logb y - 2 logb z
C. 2 logb x + 2 logb y + 2 logb z
D. logb x - logb y + 2 logb z
9) The exponential function f with base b is defined by f(x) = __________, b > 0 and b ≠ 1. Using interval notation, the domain of this function is __________ and the range is __________.
A. bx; (∞, -∞); (1, ∞)
B. bx; (-∞, -∞); (2, ∞)
C. bx; (-∞, ∞); (0, ∞)
D. bx; (-∞, -∞); (-1, ∞)
10) Approximate the following using a calculator; round your answer to three decimal places.
3√5
A. .765
B. 14297
C. 11.494
D. 11.665
11) Write the following equation in its equivalent exponential form.
4 = log2 16
A. 2 log4 = 16
B. 22 = 4
C. 44 = 256
D. 24 = 16
12) Solve the following exponential equation by expressing each side as a power of the same base and then equating exponents.
31-x = 1/27
A. {2}
B. {-7}
C. {4}
D. {3}
13) Use properties of logarithms to expand the following logarithmic expression as much as possible.
logb (x2y)
A. 2 logy x + logx y
B. 2 logb x + logb y
C. logx - logb y
D. logb x – ...
Due Week 10 and worth 250 pointsIn preparation for this assignme.docxjacksnathalie
Due Week 10 and worth 250 points
In preparation for this assignment, please view the Jurisville scenarios and resulting simulations from Weeks 8 through 10 in the Corrections unit.
In the scenarios and resulting simulations, Robert Donovan, a Jurisville probation officer, discusses the intricacies of probation. Kris, the defendant, is offered an intensive supervised probation plan to follow. Brennan Brooke, a senior criminologist, discusses the tailoring of the inmate to the appropriate facility. Finally, Orlando Boyce, a sergeant at the fictional Deephall correctional facility, discusses measures that could conceivably make prison life effective and thus decrease the likelihood of recidivism.
Write a three to four (3-4) page paper in which you:
1. Outline your findings from your review of the file of Kris, for whom Robert is considering probation. State whether or not your results from the file review match Robert’s. Explain two (2) instances in which your views and those of Robert are both similar and different.
2. Develop a profile of the so-called perfect candidate to participate in an intensive supervised probation program. The profile should contain at least three (3) attributes that you believe make this defendant the perfect candidate for this type of probation.
3. Defend or critique the strategy of matching the inmate to the correctional facility as a response to the legal concept of cruel and unusual punishment. Provide a rationale for your position with concrete examples.
4. Defend or critique whether programs and amenities geared to making prison life effective—which run the gamut from hiring extra officers, to counseling and therapy, to building a garden—are time and taxpayer money well spent.
5. Use at least three (3) quality resources in this assignment. Note: Wikipedia and similar Websites do not qualify as quality resources.
Your assignment must follow these formatting requirements:
· Be typed, double spaced, using Times New Roman font (size 12), with one-inch margins on all sides; citations and references must follow APA or school-specific format. Check with your professor for any additional instructions.
· Include a cover page containing the title of the assignment, the student’s name, the professor’s name, the course title, and the date. The cover page and the reference page are not included in the required assignment page length.
The specific course learning outcomes associated with this assignment are:
· Outline the major characteristics and purposes of prisons, including prisoners’ rights and prison society.
· Use technology and information resources to research issues in criminal justice.
· Write clearly and concisely about criminal justice using proper writing mechanics and APA style conventions.
Grading for this assignment will be based on answer quality, logic / organization of the paper, and language and writing skills, using the following rubric.
MULTIPLE CHOICE. Choose the one alternative that best complete ...
Multiple Choice Type your answer choice in the blank next to each.docxadelaidefarmer322
Multiple Choice:
Type your answer choice in the blank next to each question number.
_____1.
Find the indicated sum.
A. 2
B. 54
C. 46
D. -54
_____2.
Graph the ellipse and locate the foci.
A.
foci at (0,
6) and (0, -6)
C.
foci at (
, 0) and (-
, 0)
B.
foci at ( 5, 0) and (-5, 0)
D.
foci at (0,
5) and (0, -5)
_____3.
Solve the system by the substitution method.
2y - x = 5
x2 + y2 - 25 = 0
A.
B.
C. {( 5, 0), ( -5, 0), ( 3, 4)}
D. {( -5, 0), ( 3, 4)}
_____4.
Graph the function. Then use your graph to find the indicated limit.
f(x) = 5x - 3,
f(x)
A. 5
B. 25
C. 2
D. 22
_____5.
Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists.
4x - y + 3z = 12
x + 4y + 6z = -32
5x + 3y + 9z = 20
A. {(8, -7, -2)}
B. {(-8, -7, 9)}
C.
∅
D. {(2, -7, -1)}
_____6.
Solve the system of equations using matrices. Use Gaussian elimination with back-substitution.
x + y + z
= -5
x - y + 3z
= -1
4x + y + z = -2
A. {( 1, -4, -2)}
B. {( -2, 1, -4)}
C. {( 1, -2, -4)}
D. {( -2, -4, 1)}
_____7.
A woman works out by running and swimming. When she runs, she burns 7 calories per minute. When she swims, she burns 8 calories per minute. She wants to burn at least 336 calories in her workout. Graph an inequality that describes the situation. Let x represent the number of minutes running and y the number of minutes swimming. Because x and y must be positive, limit the graph to quadrant I only.
A.
C.
B.
D.
Short Answer Questions:
Type your answer below each question. Show your work.
8
A statement S
n
about the positive integers is given. Write statements S
1
, S
2
, and S
3
, and show that each of these statements is true.
S
n
: 1
2
+ 4
2
+ 7
2
+ . . . + (3n - 2)
2
=
9
A statement
S
n
about the positive integers is given. Write statements
S
k
and
S
k+1
, simplifying
S
k+1
completely.
S
n
: 1 ∙ 2 + 2 ∙ 3 + 3 ∙ 4 + . . . +
n
(
n
+ 1) = [
n
(
n
+ 1)(
n
+ 2)]/3
10
Joely's Tea Shop, a store that specializes in tea blends, has available 45 pounds of A grade tea and 70 pounds of B grade tea. These will be blended into 1 pound packages as follows: A breakfast blend that contains one third of a pound of A grade tea and two thirds of a pound of B grade tea and an afternoon tea that contains one half pound of A grade tea and one half pound of B grade tea. If Joely makes a profit of $1.50 on each pound of the breakfast blend and $2.00 profit on each pound of the afternoon blend, how many pounds of each blend should she make to maximize profits? What is the maximum profit?
11
Your computer supply store sells two types of laser printers. The first type, A, has a cost of $86 and you make a $45 profit on each one. The second type, B, has a cost of $130 and you make a $35 profit on each one. You expect to sell at least 100 laser printers this month a.
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1. Test code: ME I/ME II, 2006
Syllabus for ME I, 2006
Matrix Algebra: Matrices and Vectors, Matrix Operations.
Permutation and Combination.
Calculus: Functions, Limits, Continuity, Differentiation of functions of
one or more variables, Unconstrained optimization, Definite and Indefinite
Integrals: integration by parts and integration by substitution, Constrained
optimization of functions of not more than two variables.
Linear Programming: Formulations, statements of Primal and Dual
problems, Graphical solutions.
Theory of Polynomial Equations (up to third degree).
Elementary Statistics: Measures of central tendency; dispersion,
correlation, Elementary probability theory, Probability mass function,
Probability density function and Distribution function.
Sample Questions for ME I (Mathematics), 2006
For each of the following questions four alternative answers are provided.
Choose the answer that you consider to be the most appropriate for a
question.
f x equals
( ) log 1 , 0 < x < 1, then
f (x) ; (C) ( f (x))2 ; (D) none of these.
1
f x x
−
1. If
+
=
x
1
1+ 2
2
x
(A) 2 f (x) ; (B)
2
2. If u =φ (x − y, y − z, z − x) , then
u
z
u
y
u
x
∂
+
∂
∂
+
∂
∂
∂
equals
2. (A) 0; (B) 1; (C) u; (D) none of these.
3. Let A and B be disjoint sets containing m and n elements, respectively,
and let C = AU B . The number of subsets S of C that contain k
elements and that also have the property that S I A contains i
elements is
k k
2
i
(A)
m
i
; (B)
n
m
; (C) − i
n
k i
n
−
m
; (D)
k i
i
.
4. The number of disjoint intervals over which the function
f (x) = 0.5x2 − x is decreasing is
(A) one; (B) two; (C) three; (D) none of these.
5. For a set of real numbers n x , x ,........, x 1 2 , the root mean square (RMS)
defined as RMS =
2
1
1 n
2
1
Σ=
i
i x
N
is a measure of central tendency. If
AM denotes the arithmetic mean of the set of numbers, then which of
the following statements is correct?
(A) RMS < AM always; (B) RMS > AM always;
(C) RMS < AM when the numbers are not all equal;
(D) RMS > AM when numbers are not all equal.
6. Let f(x) be a function of real variable and let Δf be the function
Δf (x) = f (x +1) − f (x) . For k > 1, put Δk f = Δ(Δk−1 f ) . Then
Δk f (x) equals
k k
(A) ( 1) ( )
0
f x j
j
j
j +
− Σ=
; (B) ( 1) ( )
0
1 f x j
j
j
j +
− Σ=
+ ;
k k
(C) ( 1) ( )
0
f x k j
j
j
j + −
− Σ=
k k
; (D) ( 1) ( )
0
1 f x k j
j
j
j + −
− Σ=
+ .
3. I xne xdx
n , where n is some positive integer. Then In equals
3
∞
= −
7. Let ∫
0
(A) n! – nIn-1; (B) n! + nIn-1; (C) nIn-1; (D) none of these.
8. If x3 = 1, then
Δ =
a b c
b c a
c a b
equals
(A) (cx2 + bx + a)
b c
1
x c a
2
x a b
; (B) (cx2 + bx + a)
x b c
c a
1 ;
2
x a b
(C) (cx2 + bx + a)
2
x b c
x c a
a b
1
; (D) (cx2 + bx + a)
b c
1
2
x c a
x a b
.
9. Consider any integer I = m2 + n2 , where m and n are any two odd
integers. Then
(A) I is never divisible by 2;
(B) I is never divisible by 4;
(C) I is never divisible by 6;
(D) none of these.
10. A box has 10 red balls and 5 black balls. A ball is selected from the
box. If the ball is red, it is returned to the box. If the ball is black, it
and 2 additional black balls are added to the box. The probability that a
second ball selected from the box will be red is
(A)
47 ; (B)
72
25 ; (C)
72
55 ; (D)
153
98 .
153
4.
log 1+ log 1
− −
1 − 1 ; (B) p + q ; (C)
4
11. Let f (x) =
x
x
q
x
p
, x ≠ 0. If f is continuous at
x = 0, then the value of f(0) is
(A)
p q
1 + 1 ; (D) none of these.
p q
12. Consider four positive numbers 1 x , 2 x , 1 y , 2 y such that 1 y 2 y > 1 x 2 x .
Consider the number 1 2 2 1 1 2 S = (x y + x y ) − 2x x . The number S is
(A) always a negative integer;
(B) can be a negative fraction;
(C) always a positive number;
(D) none of these.
13. Given x ≥ y ≥ z, and x + y + z = 12, the maximum value of
x + 3y + 5z is
(A) 36; (B) 42; (C) 38; (D) 32.
14. The number of positive pairs of integral values of (x, y) that solves
2xy − 4x2 +12x − 5y = 11 is
(A) 4; (B) 1; (C) 2; (D) none of these.
15. Consider any continuous function f : [0, 1] → [0, 1]. Which one of
the following statements is incorrect?
(A) f always has at least one maximum in the interval [0, 1];
(B) f always has at least one minimum in the interval [0, 1];
(C) ∃ x ∈ [0, 1] such that f(x) = x;
(D) the function f must always have the property that f(0) ∈ {0, 1},
f(1) ∈ {0, 1} and f(0) + f(1) =1.
5. Syllabus for ME II (Economics), 2006
Microeconomics: Theory of consumer behaviour, Theory of production,
Market forms (Perfect competition, Monopoly, Price Discrimination,
Duopoly – Cournot and Bertrand (elementary problems)) and Welfare
economics.
Macroeconomics: National income accounting, Simple model of income
determination and Multiplier, IS – LM model (with comparative statics),
Harrod – Domar and Solow models, Money, Banking and Inflation.
Sample questions for ME II (Economics), 2006
1.(a) There are two sectors producing the same commodity. Labour is
perfectly mobile between these two sectors. Labour market is competitive
and the representative firm in each of the two sectors maximizes profit. If
there are 100 units of labour and the production function for sector i is:
( ) i F L = 15 i L , i = 1,2, find the allocation of labour between the two
sectors.
(b) Suppose that prices of all variable factors and output double. What
will be its effect on the short-run equilibrium output of a competitive firm?
Examine whether the short-run profit of the firm will double.
(c) Suppose in year 1 economic activities in a country constitute only
production of wheat worth Rs. 750. Of this, wheat worth Rs. 150 is
exported and the rest remains unsold. Suppose further that in year 2 no
production takes place, but the unsold wheat of year 1 is sold domestically
and residents of the country import shirts worth Rs. 250. Fill in, with
adequate explanation, the following chart :
Year GDP = Consumption + Investment + Export - Import
1 ____ ____ ____ ____ ____
2 ____ ____ ____ ____ ____
2. A price-taking farmer produces a crop with labour L as the only input.
His production function is: F(L) = 10 L − 2L . He has 4 units of labour
5
6. in his family and he cannot hire labour from the wage labour market. He
does not face any cost of employing family labour.
(a) Find out his equilibrium level of output.
(b) Suppose that the government imposes an income tax at the rate of
10 per cent. How does this affect his equilibrium output?
(c) Suppose an alternative production technology given by:
F(L) = 11 L − L −15 is available. Will the farmer adopt this
alternative technology? Briefly justify your answer.
3. Suppose a monopolist faces two types of consumers. In type I there is
only one person whose demand for the product is given by : Q P I = 100 − ,
where P represents price of the good. In type II there are n persons, each
of whom has a demand for one unit of the good and each of them wants to
pay a maximum of Rs. 5 for one unit. Monopolist cannot price
discriminate between the two types. Assume that the cost of production for
the good is zero. Does the equilibrium price depend on n ? Give reasons
for your answer.
4. The utility function of a consumer is: U(x, y) = xy . Suppose income
of the consumer (M) is 100 and the initial prices are Px = 5, Py = 10. Now
suppose that Px goes up to 10, Py and M remaining unchanged. Assuming
Slutsky compensation scheme, estimate price effect, income effect and
substitution effect.
5. Consider an IS-LM model for a closed economy. Private consumption
depends on disposable income. Income taxes (T) are lump-sum. Both
private investment and speculative demand for money vary inversely with
interest rate (r). However, transaction demand for money depends not on
income (y) but on disposable income (yd). Argue how the equilibrium
values of private investment, private saving, government saving,
disposable income and income will change, if the government raises T.
6. An individual enjoys bus ride. However, buses emit smoke which he
dislikes. The individual’s utility function is: U =U(x, s) , where x is the
distance (in km) traveled by bus and s is the amount of smoke consumed
from bus travel.
6
7. (a) What could be the plausible alternative shapes of indifference curve
u = c 2 h , where c is the household’s
1 1
2 C , 0<θ < 1,ρ > 0 ,
− −
1
7
between x and s?
(b) Suppose, smoke consumed from bus travel is proportional to the
distance traveled: s = α x ( α is a positive parameter). Suppose further
that the bus fare per km is p and that the individual has money
income M to spend on bus travel. Show the budget set of the consumer
in an (s, x) diagram.
(c) What can you say about an optimal choice of the individual? Will he
necessarily exhaust his entire income on bus travel?
7. (a) Suppose the labour supply (l) of a household is governed by
1
maximization of its utility (u): 3
3
consumption and h is leisure enjoyed by the household (with h + l =
24). Real wage rate (w) is given and the household consumes the
entire labour income (wl). What is the household’s labour supply?
Does it depend on w?
(b) Consider now a typical Keynesian (closed) economy producing a
single good and having a single household. There are two types of
final expenditure – viz., investment autonomously given at 36 units
and household consumption (c) equalling the household’s labour
income (wl). It is given that w = 4 . Firms produce aggregate output
(y) according to the production function: y = 24 l . Find the
equilibrium level of output and employment. Is there any involuntary
unemployment? If so, how much?
8. Suppose an economic agent’s life is divided into two periods, the first
period constitutes her youth and the second her old age. There is a single
consumption good, C , available in both periods and the agent’s utility
function is given by
( , ) 1 2 u C C =
1 1
1 C +
θ
−
− −
1
θ
1
1+ ρ
θ
−
θ
where the first term represents utility from consumption during youth.
The second term represents discounted utility from consumption in old
age, 1/(1+ρ ) being the discount factor. During the period, the agent has a
8. unit of labour which she supplies inelastically for a wage rate w . Any
savings (i.e., income minus consumption during the first period) earns a
rate of interest r , the proceeds from which are available in old age in units
of the only consumption good available in the economy. Denote savings
by s . The agent maximizes utility subjects to her budget constraint.
i) Show that θ represents the elasticity of marginal utility with
respect to consumption in each period.
ii) Write down the agent’s optimization problem, i.e., her problem
of maximizing utility subject to the budget constraint.
iii) Find an expression for s as a function of w and r .
iv) How does s change in response to a change in r ? In particular,
show that this change depends on whether θ exceeds or falls short
of unity.
v) Give an intuitive explanation of your finding in (iv)
9. A consumer consumes only two commodities 1 x and 2 x . Suppose that
her utility function is given by U( , ) 1 2 x x = min (2 , ) 1 2 x x .
(i) Draw a representative indifference curve of the consumer.
(ii) Suppose the prices of the commodities are Rs.5 and Rs.10
respectively while the consumer’s income is Rs. 100. What commodity
bundle will the consumer purchase?
(iii) Suppose the price of commodity 1 now increases to Rs. 8.
Decompose the change in the amount of commodity 1 purchased into
income and substitution effects.
10. A price taking firm makes machine tools Y using labour and capital
according to the production function Y = K 0.25L0.25 . Labour can be hired
at the beginning of every week while capital can be hired only at the
beginning of every month. Let one month be considered as long run period
and one week as short run period. Further assume that one month equals
four weeks. The wage rate per week and the rental rate of capital per
month are both 10.
8
9. (i) Given the above information, find the short run and the long run
9
cost functions of the firm.
(ii) At the beginning of the month of January, the firm is making long
run decisions given that the price of machine tools is 400. What is
the long run profit maximizing number of machine tools? How
many units of labour and capital should the firm hire at the
beginning of January?
11. Consider a neo-classical one-sector growth model with the production
function Y = KL . If 30% of income is invested and capital stock
depreciates at the rate of 7% and labour force grows at the rate of 3%, find
out the level of per capita income in the steady-state equilibrium.