ANDHRA
LOYOLA INSTITUTE OF ENGINEERING AND TECHNOLOGY:: VIJAYAWADA
First
B.Tech First Semester (R19 Regulation)
Course: MATHEMATICS-1
Subject Code: 19BS1101
2020-21 (R19)
|
INDEX |
PAGE NO. |
|
|
1 |
Assignment Schedule |
3 |
|
2 |
19BS1101-Mathematics-1 |
|
|
2.1 |
Course information sheet |
|
|
2.2 |
Course Plan |
|
|
2.3 |
Question Bank |
|
|
2.4 |
Assignment and Day Test / Tutorial
Questions |
|
1.
ASSIGNMENT
SCHEDULE
|
Subject |
Week / Date |
|
19BS1101-Mathematics-1
|
Week-4 |
|
Week-12 |
· Assignment-1is
on Units-1,2,3.5 for the weightage of 5 marks
· Assignment-2
is on Units-3,4,5 for the weightage of 5 marks
2.1 COURSE INFORMATION SHEET
|
PROGRAMME:
MECHANICAL ENGINEERING |
DEGREE: B.TECH |
|
COURSE-
Mathematics-1 |
SEMESTER-1 CREDITS-3 |
|
COURSE
CODE- R19BS1101 Year of
introduction - 2019 |
COURSE TYPE -
CORE |
|
COURSE AREA/DOMAIN :: MATHEMATICS |
CONTACT HOURS:
3-1-0 |
|
UNIT |
DETAILS |
HOURS
(Lecture) |
HOURS
(Tutorial) |
|
I |
Sequences, Series and Mean value theorems:Sequences and Series: Convergences and
divergence – Ratio test – Comparison tests – Integral test – Cauchy’s root
test – Alternate series – Leibnitz’s rule. Mean Value Theorems (without
proofs): Rolle’s Theorem – Lagrange’s mean value theorem – Cauchy’s mean
value theorem – Taylor’s and Maclaurin’s theorems with remainders. |
10 |
3 |
|
II |
Differential equations of first order and
first degree: Linear differential equations – Bernoulli’s
equations – Exact equations and equations reducible to exact form. Applications:
Newton’s Law of cooling – Law of natural growth and decay – Orthogonal
trajectories – Electrical circuits. |
10 |
2 |
|
III |
Linear differential equations of higher
order:
Non-homogeneous equations of higher order with constant coefficients – with
non-homogeneous term of the type eax, sin ax, cos ax, polynomials in xn ,eax
V(x) and xnV(x) – Method of Variation of parameters. Applications: LCR
circuit, Simple Harmonic motion. |
10 |
3 |
|
IV |
Partial differentiation:
Introduction – Homogeneous function – Euler’s theorem – Total
derivative – Chain rule – Jacobian – Functional dependence – Taylor’s and Mc
Laurent’s series expansion of functions of two variables. Applications:
Maxima and Minima of functions of two variables without constraints and
Lagrange’s method (with constraints). |
10 |
2 |
|
V |
Multiple integrals: Double and Triple integrals – Change of
order of integration – Change of variables. Applications: Finding Areas and
Volumes. |
8 |
2 |
|
TOTAL |
48 |
12 |
|
TEXT/REFERENCE/ADDITIONAL
BOOKS:
|
T/R |
BOOK TITLE/AUTHORS/PUBLICATION |
|
T |
B. S.
Grewal, Higher Engineering Mathematics, 43rd Edition, Khanna Publishers. |
|
T |
B. V. Ramana, Higher Engineering Mathematics, 2007 Edition,
Tata Mc. Graw Hill Education. |
|
R |
Erwin Kreyszig, Advanced Engineering Mathematics, 10th
Edition, Wiley-India. |
|
R |
Joel Hass, Christopher Heil and
Maurice D. Weir, Thomas calculus, 14th Edition, Pearson. |
|
R |
Lawrence Turyn, Advanced Engineering Mathematics, CRC
Press, 2013. |
|
R |
Srimantha Pal, S C Bhunia, Engineering Mathematics, Oxford
University Press. |
WEB
SOURCE REFERENCES: (Detailed Topic link)
|
W1 |
|
|
W2 |
|
|
W3 |
Course
Objectives
In this course the students are introduced to
some basic tools in Mathematics which are useful in modelling and analysing
physical phenomena involving continuous changes of variables or parameters. The
differential and integral calculus of functions of one or more variables and functions
of several variables taught in this course have applications across all
branches of engineering. This course will also provide basic training in
plotting and visualizing graphs of functions and intuitively understanding
their properties.
Course
Outcomes
|
1 |
Ability tounderstand and explain
concepts of convergence and mean value theorems |
|
2 |
Student’s
gain knowledge on solving
differential equations and its applications to various Engineering fields. |
|
3.A |
Ability to apply the basic knowledge of differential equations |
|
3.B |
Ability to apply the basic knowledge of Linear differential equations in
electrical systems |
|
4 |
Ability to apply
Euler’s theorem for multivariable function and to find extreme values. |
|
5 |
Students are introduced finding areas and volumes using
integrals. |
COURSE OUTCOMES VS POs
MAPPING (DETAILED; HIGH:3; MEDIUM:2; LOW:1):
|
SNO |
PO1 |
PO2 |
PO3 |
PO4 |
PO5 |
PO6 |
PO7 |
PO8 |
PO9 |
PO10 |
PO11 |
PO12 |
PSO1 |
PSO2 |
PSO3 |
|
C1101.1 |
3 |
3 |
-- |
-- |
-- |
-- |
-- |
-- |
-- |
-- |
-- |
-- |
-- |
-- |
-- |
|
C1101.2 |
3 |
3 |
-- |
-- |
-- |
-- |
-- |
-- |
-- |
-- |
-- |
3 |
-- |
-- |
-- |
|
C1101.3.A |
2 |
3 |
-- |
-- |
-- |
3 |
-- |
-- |
-- |
-- |
-- |
-- |
-- |
-- |
-- |
|
C1101.3.B |
2 |
3 |
-- |
-- |
-- |
3 |
-- |
-- |
-- |
-- |
-- |
-- |
-- |
-- |
-- |
|
C1101.4 |
3 |
3 |
-- |
-- |
-- |
-- |
-- |
-- |
-- |
-- |
-- |
-- |
-- |
-- |
-- |
|
C1101.5 |
3 |
3 |
-- |
-- |
-- |
-- |
-- |
-- |
-- |
-- |
-- |
-- |
-- |
-- |
-- |
* Average of the correlation values of each CO mapped to the
particular PO/PSO, corrected to the nearest whole number
Justification for the correlation
level assigned in each cell of the table above
|
C1101.1-PO1 |
fundamental knowledge in series functions will help in analyzing engineering problems
very easily |
|
C1101.1-PO2 |
Fundamental knowledge in mean value
theorems can be used to formulate engineering problems. |
|
C1101.2-PO1 |
Basic knowledge in periodic functions
is necessary for the development of mathematical modeling |
|
C1101.2-PO2 |
Fundamental knowledge in differential
equation can be used to formulate engineering principles. |
|
C1101.2-PO12 |
DE is a mathematical field which needs
lot of research |
|
C1101.3.A-PO1 |
Working principles in typical
electrical systems are based on fundamental laws of DE |
|
C1101.3.A-PO2 |
Basic knowledge in differential
equation can be used to formulate engineering principles. |
|
C1101.3.A-PO6 |
DE can address various problems of
society in fields like health safety etc. |
|
C1101.3.B-PO1 |
Basic knowledge in linear differential equation can be used to formulate engineering principles |
|
C1101.3.B-PO2 |
Working principles in typical electrical systems are based on fundamental laws of DE |
|
C1101.3.B-PO6 |
DE can address various problems of
society in fields like health safety etc. |
|
C1101.4-PO1 |
Basic knowledge in continuous change
in variable s will help to model various engineering problems |
|
C1101.4-PO2 |
Fundamental knowledge in Partial
differentiation can be used to formulate engineering problems. |
|
C1101.5-PO1 |
basic knowledge in finding areas and
volume s is used for solving complex engineering problems |
|
C1101.5-PO2 |
Fundamental knowledge in integral
calculus can be used to formulate engineering problems. |
POs & PSO REFERENCE:
|
PO1 |
Engineering
Knowledge |
PO6 |
Engineer
& Society |
PO11 |
Project
Mgt. & Finance |
|
PO2 |
Problem
Analysis |
PO7 |
Environment & Sustainability |
PO12 |
Life
Long Learning |
|
PO3 |
Design
& Development |
PO8 |
Ethics |
PSO1 |
………………………… |
|
PO4 |
Investigations |
PO9 |
Individual
& Team Work |
PSO2 |
…………………………. |
|
PO5 |
Modern
Tools |
PO10 |
Communication
Skills |
PSO3 |
………………………….. |
GAPS IN THE SYLLABUS -
TO MEET COs, POs & PSOs:
|
SNO |
GAP |
PROPOSEDACTIONS |
PROPOSED RESOURCE |
CO |
PO / PSO |
|
1 |
-- |
-- |
-- |
-- |
-- |
TOPICS BEYOND SYLLABUS:
Additional
course material / learning material / Lab Experiments / Projects
|
S.No |
Description |
PROPOSEDACTIONS |
PROPOSED RESOURCE |
CO |
PO / PSO |
|
1 |
Integral calculus |
Chalk
& Talk |
-- |
CO 5 |
1,2 |
Web Link of the Course
Material: _______________________________
INSTRUCTIONAL
METHODOLOGIES:
|
x |
CHALK
& TALK |
x |
STUD.
ASSIGNMENT |
X |
WEB
RESOURCES |
|
LCD/SMART
BOARDS |
|
|
STUD.
SEMINARS |
|
ADD-ON
COURSES |
|
ANY
OTHER (SPECIFY) |
|
|
INNOVATIVE /
PEDAGOGICAL INITIATIVES:
|
X |
REAL
WORLD EXAMPLES |
|
COLLABORATIVE
LEARNING |
|
QUALITY
LAB EXPERIMENTS |
|
OBSERVATIONS
RECORDED |
|
|
INDUSTRY
INTERNSHIP |
|
SUMMER
TRAINING |
|
EXPERT
GUEST LECTURES |
|
PROJECTS |
|
|
USE
OF ICT |
|
ANY
OTHER (SPECIFY) |
|
|
|
|
ASSESSMENT
METHODOLOGIES-DIRECT
|
X |
EXAM
QUESTIONS |
x |
TUTORIAL
QUESTIONS |
X |
ASSIGNMENTS |
|
LABORATORY
TESTS |
|
|
PROJECT
EVALUATION |
|
STUDENT
ARTIFACTS |
|
ORAL
EXAMS |
|
PROJECT
PRESENTATIONS |
|
|
INTERNALLY
DEVELOPED EXAMS |
|
ANY
OTHER (SPECIFY) |
|
|
|
|
2.2 Course Plan
ANDHRA LOYOLA INSTITUTE OF ENGINEERING AND TECHNOLOGY:: VIJAYAWADAAcademic Year 2019-20Lesson PlanName
of the Faculty: B. Ravi Shankar
Course: MATHEMATICS-1 Subject Code: 19BS1101
Programme: B.Tech Class:
I MECHANICAL Semester:
ODD
|
Session No. |
Topics to be covered |
Date of planning |
Teaching Method |
|
|
UNIT I: Sequences, Series and Mean value theorems: |
||||
|
1. |
Introduction to sequence and series |
26/08/2019 |
Chalk & Talk |
|
|
2. |
Limit of a sequence |
27/08/2019 |
Chalk & Talk |
|
|
3. |
Types of sequences – definitions |
30/08/2019 |
Chalk & Talk |
|
|
4. |
Geometric series test – problems |
31/08/2019 |
Chalk & Talk |
|
|
5. |
P-series Test – problems |
03/09/2019 |
Chalk & Talk |
|
|
6. |
Day Test / Tutorial -1 |
04/09/2019 |
|
|
|
7. |
Comparison Test – problems |
05/09/2019 |
Chalk & Talk |
|
|
8. |
Cauchy’s integral test – problems |
06/09/2019 |
Chalk & Talk |
|
|
9. |
D-Alembert’s ratio test – problems |
07/09/2019 |
Chalk & Talk |
|
|
10. |
Day Test / Tutorial -2 |
11/09/2019 |
|
|
|
11. |
Leibnitz test – problems |
12/09/2019 |
Chalk & Talk |
|
|
12 |
Absolute convergence – problems |
13/09/2019 |
Chalk & Talk |
|
|
13 |
Conditional convergence – problems |
14/09/2019 |
Chalk & Talk |
|
|
14 |
Rolle’s theorem - problems |
16/09/2019 |
Chalk & Talk |
|
|
15 |
Lagranges theorem – problems |
17/09/2019 |
Chalk & Talk |
|
|
16 |
Day Test / Tutorial -3 |
18/09/2019 |
|
|
|
17 |
Cauchy’s mean value theorem – problems |
19/09/2019 |
Chalk & Talk |
|
|
18 |
Taylor’s series with Remainder terms – problems |
20/09/2019 |
Chalk & Talk |
|
|
19 |
McLaren’s series with Remainder terms – problems |
21/09/2019 |
Chalk & Talk |
|
|
UNIT II: Differential equations of first order and
first degree: |
||||
|
20 |
Introduction on Ordinary differential equations |
23/09/2019 |
Chalk & Talk |
|
|
21 |
Linear differential equations – problems |
24/09/2019 |
Chalk & Talk |
|
|
22 |
Day Test / Tutorial -4 |
25/09/2019 |
|
|
|
23 |
Bernoulli’s differential equations – problems |
26/09/2019 |
Chalk & Talk |
|
|
24 |
Exact differential equations – problems |
27/09/2019 |
Chalk & Talk |
|
|
25 |
Non-Exact Differential equations Type-I,II |
28/09/2019 |
Chalk & Talk |
|
|
26 |
Non-Exact Differential equations Type-III,IV,V |
30/09/2019 |
Chalk & Talk |
|
|
27 |
Newton’s law of cooling - problems |
01/10/2019 |
Chalk & Talk |
|
|
28 |
Natural law of growth – problems |
03/10/2019 |
Chalk & Talk |
|
|
29 |
Natural law of decay – problems |
04/10/2019 |
Chalk & Talk |
|
|
30 |
RL & RC circuit - problems |
05/10/2019 |
Chalk & Talk |
|
|
31 |
Day Test / Tutorial -5 |
09/10/2019 |
|
|
|
32 |
RL
& RC circuit - problems |
10/10/2019 |
Chalk & Talk |
|
|
33 |
Orthogonal Trajectories – Cartesian form – problems |
11/10/2019 |
Chalk & Talk |
|
|
34 |
Polar form, self orthogonal – problems |
12/10/2019 |
Chalk & Talk |
|
|
UNIT III: Linear differential equations of higher
order: |
||||
|
35 |
Introduction on Higher order linear differential equations |
14/10/2019 |
Chalk & Talk |
|
|
36 |
Solution of homogeneous liner differential equation |
15/10/2019 |
Chalk & Talk |
|
|
37 |
Day Test / Tutorial -6 |
16/10/2019 |
|
|
|
38 |
When
Q(x) = |
17/10/2019 |
Chalk & Talk |
|
|
39 |
When
Q(x) = |
18/10/2019 |
Chalk & Talk |
|
|
40 |
When
Q(x) = |
19/10/2019 |
Chalk & Talk |
|
|
41 |
When
Q(x) = |
04/11/2019 |
Chalk & Talk |
|
|
42 |
When
Q(x) = |
05/11/2019 |
Chalk & Talk |
|
|
43 |
When
Q(x) = |
05/11/2019 |
Chalk & Talk |
|
|
44 |
Method of variation of parameters – problems |
05/11/2019 |
Chalk & Talk |
|
|
45 |
Day Test / Tutorial -7 |
06/11/2019 |
|
|
|
46 |
LCR-Circuits – problems |
07/11/2019 |
Chalk & Talk |
|
|
47 |
Simple Harmonic motion - problems |
07/11/2019 |
Chalk & Talk |
|
|
UNIT IV: Partial differentiation: |
||||
|
48 |
Introduction to partial differentiation |
08/11/2019 |
Chalk & Talk |
|
|
49 |
Taylors theorem – problems |
08/11/2019 |
Chalk & Talk |
|
|
50 |
Mc Laurent’s theorem - problems |
08/11/2019 |
Chalk & Talk |
|
|
51 |
Jacobian, functional dependent and independent |
11/11/2019 |
Chalk & Talk |
|
|
52 |
Functional relation – problems |
11/11/2019 |
Chalk & Talk |
|
|
53 |
Chain – rule problems |
12/11/2019 |
Chalk & Talk |
|
|
54 |
Maxima and Minima (without constraint) - problems |
12/11/2019 |
Chalk & Talk |
|
|
55 |
Day Test / Tutorial -8 |
13/11/2019 |
|
|
|
56 |
Maxima and Minima (with constraint) – problems |
14/11/2019 |
Chalk & Talk |
|
|
57 |
Lagrange method of Undetermined multipliers-problems |
15/11/2019 |
Chalk & Talk |
|
|
58 |
Problems on the above |
16/11/2019 |
Chalk & Talk |
|
|
59 |
Euler’s Theorem - problems |
18/11/2019 |
Chalk & Talk |
|
|
UNIT V: Multiple integrals: |
||||
|
60 |
Introduction to line,surface,volume integrals |
18/11/2019 |
Chalk & Talk |
|
|
61 |
Evaluation of Double integrals – problems |
19/11/2019 |
Chalk & Talk |
|
|
62 |
Day Test / Tutorial -9 |
20/11/2019 |
|
|
|
63 |
Change the order of integration – problems |
21/11/2019 |
Chalk & Talk |
|
|
64 |
Changing in to polar co-ordinates –
problems |
26/11/2019 |
Chalk & Talk |
|
|
65 |
Changing of variables – problems |
27/11/2019 |
Chalk & Talk |
|
|
66 |
Volume integrals – Evaluation |
29/11/2019 |
Chalk & Talk |
|
|
67 |
Mass of triple integrals – problems |
30/11/2019 |
Chalk & Talk |
|
|
68 |
Volume of cylindrical co-ordinates – problems |
02/12/2019 |
Chalk & Talk |
|
|
69 |
Problems on the above |
03/12/2019 |
Chalk & Talk |
|
|
70 |
Day Test / Tutorial -10 |
04/12/2019 |
Chalk & Talk |
|
|
71 |
Problems on volume integrals |
05/12/2019 |
Chalk & Talk |
|
|
72 |
Tutorial |
06/12/2019 |
|
|
|
73 |
Model paper solving(Revision) |
07/12/2019 |
Chalk & Talk |
|
|
74 |
Model paper solving(Revision) |
09/12/2019 |
Chalk & Talk |
|
|
75 |
Model paper solving(Revision) |
10/12/2019 |
Chalk & Talk |
|
2.3
Question Bank
Cognitive levels
L1– Remember, L2-Understanding, L3- Applying
/Analyzing
Question – Bank
UNIT -1
|
Q.No |
Question |
Marks |
Cognitive level |
|
1 |
a)Apply comparison Test to find the nature of b) Test for convergence of |
5M
5M |
L2
L1 |
|
2 |
a) Discuss the nature of b)
Verify Rolle′s theorem for f(x) = |
5M 5M
|
L3
L3 |
|
3 |
a) Test for convergence of b) Verify whether Rolle′s
theorem can be applied to the
following 1)f(x) = tanx in |
5M
5M |
L2
L3 |
|
4 |
a) State comparison test. b) Test for convergence of |
3M
7M |
L1
L2 |
|
5 |
a) Define D’ALEMBERT’S
ratio test. b) Examine the convergence
of1+ |
3M 7M |
L1 L3 |
|
6 |
a) Define the convergence
of geometric series. b) Test for convergence of |
3M 7M |
L1 L2 |
|
7 |
a) Examine the convergence
of b) Find ‘c’ of the
Lagrange′s theorem of f(x) = (x-1) (x-2) (x-3) on
|
6M
4M |
L3
L2 |
|
8 |
a) Test for convergence of b) Discuss the nature of
the series |
5M
5M |
L2
L3 |
|
9 |
a) Discuss the applicability of Cauchy′s mean
value theorem for f(x)= b) Obtain the Maclaurin′s
series expansion of |
4M
6M |
L3
L2 |
|
10 |
a) Explain the convergence
of P-series. b) Obtain the Taylor’s
series expansion of f(x) = |
2M 8M |
L1 L2 |
UNIT -2
|
Q.No |
Question |
Marks |
Cognitive level |
|
1 |
a) Solve ( b) A bacterial culture,
growing exponentially increases from 200 to 500 grams in the period from 6am
to 9am. How many grams will be present at noon? |
5M
5M |
L1
L3 |
|
2 |
a) Find the equation of the system of
orthogonal trajectories of the family of curves 𝑟𝑛𝑠𝑖𝑛𝑛 b) Solve |
5M
5M
|
L3
L3 |
|
3 |
a) Solve b) Solve
|
5M
5M |
L2
L2 |
|
4 |
a) Solve b) Examine whether the
system of rectangular hyperbolas
|
5M
5M |
L2
L3 |
|
5 |
a) Solve b) Prove that the system of parabolas 𝑦2=4(𝑥+𝑎) is self-orthogonal.
|
5M
5M |
L2
L3 |
|
6 |
a.) Solve b) A body is originally at
80
|
5M
5M |
L2
L3 |
|
7 |
a) solve b) State Newton’s law of
cooling and derive its equation
|
6M
4M
|
L2
L2 |
|
8 |
a) solve b) If 30% of a radioactive
substance disappears in 10days, how long will it take for 90% of it to
disappear?
|
5M
5M |
L2
L3 |
|
9 |
a) solve b)when a switch is closed in a circuit
containing a battery E, a resistance R and an inductance L, the current ‘i’ builds up at a rate given by L
|
5M
5M |
L2
L3 |
|
10 |
a) solve b) Solve
|
5M
5M |
L2
L2 |
UNIT – 3
|
Q.No |
Question |
Marks |
Cognitive level |
|
1 |
a) Solve b) Solve ( |
5M
5M
|
L2
L2 |
|
2 |
a)solve ( b) Solve
|
7M
3M
|
L2
L1 |
|
3 |
a) Solve ( b) Solve
|
7M
3M |
L2
L2 |
|
4 |
a) Solve ( b) Solve
|
6M 4M |
L2 L1 |
|
5 |
a) Solve b) Solve
|
6M 4M |
L3 L1 |
Unit - 3
|
Q.No. |
Questions |
Marks |
Cognitive Level |
|
1. |
(a) Solve by method of variation of
parameters (b) Solve
|
5M
5M |
L2
L1 |
|
2. |
(a) The charge q(t) on the capacitor is
given by differential equation Find the charge on the capacitor for t >0. (b) Solve |
5M 5M |
L3
L3 |
|
3. |
(a) Write the equation of SHM and find its solution (b) Solve |
5M
5M |
L2
L3 |
|
4. |
(a) Solve (b) Solve
|
5M
5M |
L1
L2 |
|
5. |
(a) Find the solution of (b) Solve |
5M 5M |
L1 L3 |
|
6. |
(a) Solve (b) Solve |
5M 5M |
L1 L2 |
|
7. |
(a) Solve (b) Solve by method of variation of parameters |
5M 5M |
L3 L3 |
Unit -4
|
Q.No |
Question |
Marks |
Cognitive level |
|
1 |
(a)
Find the stationary points of (b)
If |
5M
5M |
L1
L3 |
|
2 |
(a) Find the dimensions of a rectangular box open at the top of max capacity whose surface area is 108 sq inches. (b)
Expand |
5M
5M |
L3
L3 |
|
3 |
(a)
Find (b)
Expand |
5M
5M |
L2
L2 |
|
4 |
(a) Find the point in the plane 2x+3y-z = 5 which is nearest to the origin. (b)
If |
5M
5M |
L2 L3 |
|
5 |
(a)
If (b)
Find the maximum and minimum values |
5M
5M |
L2
L3 |
|
6 |
(a)
Prove that (b)
Examine the function for extreme values |
5M
5M |
L2 L3 |
|
7 |
(a)
If (b)
Find |
5M
5M |
L2
L2 |
Unit 5
|
Q.No |
Question |
Marks |
Cognitive level |
|
1 |
(a)
Evaluate (b)
Calculate |
5M
5M |
L2 L2 |
|
2 |
(a)
Evaluate (b)
Evaluate |
7M
3M |
L2
L1 |
|
3 |
(a)
Evaluate , by changing its order of
integration (b)
Evaluate by change of order of integration |
7M
3M |
L2
L2 |
|
4 |
Evaluate |
10M |
L3 |
|
5 |
(a)
Evaluate (b)
Find the area enclosed by the curves
|
6M 4M |
L3 L1 |
|
6 |
Find the volume of the tetrahedron bounded by the plane |
10M |
L2 |
2.4 Day Test / Tutorial Questions :
·
individual
faculty conducting day test/Tutorials and assignment questions add here
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