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MATH1002: Mathematics

创建日期：2024-06-17 14:23:38

所属分类：数学mathematics

标签： MATH1002

Mathematics

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MATH1002: Mathematics

This individual assignment is due by 11:59pm Sunday 12 May 2024, via Canvas.
It should include your SID. Please make sure you review
your submission carefully. What you see is exactly how the marker will see your
assignment. Submissions can be overwritten until the due date. To ensure compliance
with our anonymous marking obligations, please do not under any circumstances
include your name in any area of your assignment; only your SID should be present.
The School of Mathematics and Statistics encourages some collaboration between
students when working on problems, but students must write up and submit their
own version of the solutions. If you have technical difficulties with your submission,
see the University of Sydney Canvas Guide, available from the Help section of Canvas.
This assignment is worth 10% of your final assessment for this course. Your answers should be
well written, neat, thoughtful, mathematically concise, and a pleasure to read. Please cite any
resources used and show all working. Present your arguments clearly using words of explanation
and diagrams where relevant. After all, mathematics is about communicating your ideas. This
is a worthwhile skill which takes time and effort to master. The marker will give you feedback
and allocate an overall mark to your assignment using the criteria available through Canvas.
Copyright © 2024 The University of Sydney 1
Please justify your answers. Correct answers without adequate justification will not receive full
marks. A plot on its own is not considered adequate justification.
1. Let ℓ1 and ℓ2 be two lines in space defined by the parametric equations:
ℓ1 : x = 3− s, y = 4 + 5s, z = 3 + s (s ∈ R)
ℓ2 : x = 2 + t, y = −3 + t, z = −2 + 2t (t ∈ R)
Let P be the plane that contains ℓ1 and ℓ2.
(a) Find the point of intersection of ℓ1 and ℓ2.
(b) Find a general equation for P .
2. Let λ ∈ R, and consider the system of linear equations in the variables x, y, z given by
x − 5z = −4
x − λy − 2z = 2
x + 2y + λz = 2
(a) Row reduce the corresponding augmented matrix to row echelon form.
(b) Find the values of the constant λ for which the system has
(i) no solutions
(ii) exactly one solution
(iii) infinitely many solutions
3. There are 2800 MATH1061 students, and they all do one of three things on a given
night: they study linear algebra, they study calculus, or they watch netflix. We say a
MATH1061 student
is in State 1 if they study linear algebra;
is in State 2 if they study calculus; and
is in State 3 if they watch netflix.
MATH1061 students change their habits from night to night according to the following
rules:
If a student studies linear algebra one night, they have an 80% chance of studying
linear algebra the next night; a 10% chance of studying calculus the next night; and
a 10% chance of watching netflix the next night.
If a student studies calculus one night, they have a 20% chance of studying linear
algebra the next night; a 60% chance of studying calculus the next night; and a 20%
chance of watching netflix the next night.
If a student watches netflix one night, they have a 40% chance of studying linear
algebra the next night; a 40% chance of studying calculus the next night; and a 20%
chance of watching netflix the next night.
We encode the collection of probabilities of moving from one state to another in the
matrix P = (pij)3×3, where
pij is the probability of moving from State j one night to State i the next night.
2
This means P is the matrix
P =
p11 0.2 p13p21 0.6 p23
p31 0.2 p33
 ,
where the middle column has been filled in for you.
(a) Finish writing down the matrix P .
(b) For night n we define the vector xn =
xy
z
, where
x is the number of students in State 1;
y is the number of students in State 2; and
z is the number of students in State 3.
This means that for night n+ 1 we have
xn+1 = Pxn.
Suppose initially we have 1000 students in State 1, 1000 students in State 2, and
800 students in State 3; or in other words,
x0 =
10001000
800
 .
Find the number of students in each state on night two, i.e. find x2.
(c) By considering the system
(P − I3)x = 0,
find all the vectors x that satisfy Px = x.
(d) Suppose that instead of the initial conditions in part (b), we initially have 1600
students studying linear algebra, 800 studying calculus, and 400 watching net-
flix. Find the number of students studying linear algebra, the number of students
studying calculus, and the number of students watching netflix on night n = 100.

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1300A ENGG 1300 OLET1139 STAT 3503 STAT 206 FEM11149 ELEN 4720 COMP20003 SOLA1070 CS574 C11CS LINB29H3 MAST90083 ECON7200 STAT 610 LINB29 DSM-5 COMP3161 EEEN40010 MATH 4431 K353 STA 3386 IERG4300 COMS W4167 ENG5298 AMME 2302 CS3910 COMPSCI 320 STATS 369 GR5242 COSC 328 CSCI435 MGTS1301 INFS2200 COMP3900 CIS 375 Module AE03 COMP3121 CS260 HAM503 MGMT5050 INT3075 COSC 285 GEOL0029 GEGE2X01 STAT1201 MATH1061 MMF1941H STAT 5060 CS 520 COMP 4320 MSIN0104 000T FINS3648 ECMT2160 ECON4003 Phys 129L CSE 142 CS 3130 MGEB01 EECS 340 Finance (20443/20444) ECMM149 COMP5310 EEEN60180 CSC 111 AT2 CSCI 4430 STAB57 MA/CSC 427 MATH322 D100 CSCI544 CITS3004 CI6227 CSE 565 APCO 1P01 852L1 ELEC S411F COMP3207 STATS330 ECON7530 FIT5149 CSI4104 CMPT 371 COMP201 SEHS4696 FI4003 MATH6182 CG1 WS MATH401 MGT 205 QTM 110 COMP 0137 MACE12201 FIT5210 ECE 544 BISM7206 HUDM 6055 ECON3203 STATS 210 MSCI 570 MATH6173 COMPSCI4039 Accounting COM3503 AOE2024 ECON213 PSYC10004 Control Data 100/200A PHIL2617 PHYS3116 MIE250 BCPM0008 ECN 620 QBUS6830 ECSE-415 ECON5102 MGMT20005 MAT 653 ECO 480 COMP 5/600 CSSE 5600 ME 571 7CCMMS61 CS 550 COMP2865 MATH3075 STATS 506 QTM 220 ISYS2120 FIT5106 STAT 153 MKTG3112 ECON2102 ELE2018 COMP3031 MSIN0010 SCC461 MATH237 CS246 CSC3065 CSE 101 ICTWEB411 COMP9313 ECOS 2025 FN620 MATH11111 MATH 3018 KE1068 FINA 6112 C++ MISM 6202 MSIN0105 CSC8631 MECO6935 COMP4121 ACCT2011 DPBS1150 EEL 3701C F19PL OPM 661 CSE 310 CS918 BIOL30001 ACCT3563 BANK3600 MET CS622 MATH367 ACCT5907 Continuing COMP5048 COMP61021 CS3334 GSOE9210 CSCI-570 COMP 5322 HW 2 ALY-6020 IEOR 4706 DATA 200 EBUS504 CGE12412 COMP4108 7SSMM800 ECMM171P MATE50002 CS225 RS 6003 BH2274 ELEC3111 MAT6669 ECON-UA 227 PHIL 1012 7SSMM603 TB028A P6 ECS708 CSC 427 CS6476 CSE 250A A1A2 ITS61504 EECS 376 model HW6 UV0010 ECON 6074 MG4F7 CMP-7030Y INFO3008 BFIP013 ECON 5068 MATH6002 CS3483 PSTAT 131 C31FM CSOR W4246 AAEC 5307 DATA ELEC362 COMSM0047 ECN6006 MA20224 BST351 MATH 323 MATH323 AA2021 COMP3811 COMP2013 IB9X60 INVP001 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ATMS 201 COMP3425 MATH7861 SOLA2060 MA 325 7CCEMCTH ECON 432 INFT2051 SOLA 2060 ECOS2002 Stats141XP BIOA02H3 S2022 MA4601 K5 COMP5329 EG 284 IB3A70 WECO1020 COMP3027 Energy ACS61011 MAST20026 STAT 533 MATA31 ENVS257 BEM2031 EC 486 COMP0172 CS3214 EECE5644 IB2B20 COSC2536 COSC1147 COMP6214 5CCM242A COMP0130 AMME2000 MTMG50: EG238 ACF6006 QBUS2810 COSC2391 ECON7560 MATH377 STA305 CS 486 COMP1117B EL648 ELEC3909 ECON 241 COMP 4901W BUSI4428 ENEN20002 ENGN4524 ELEC3115 CSC 413 PSYC 100B ECON1002 MATH256 ANSC20005 MTHS2008 MATH40082 COSC 2406 FIT3174 STA 238 CMT202 Differential Equations MTHM017 GARCH 101 MATH2069 CSC336S ACSC12 ETF2121 CEME 2001 DB B474F ARHT1001 EECS 3311 3220 STATS 786 CS 484 COMP20008 MATH4091 ECE 455 CSCI 204 BUS B200F BA 875 BU5562 COSC 1147 X11 GEOG 101 IMM250F CSE 584A MATH38172 LAB 4 ELEC5882M FINS5523 EE 430 COMP 465 6CCS3PRE 7CCSMNSE FINANCE ACMB02H3 ENGG2112 ELEC5616 SCM B371 CHEM3120 CS22 ECMT2130 MAST30011 ADSPBF706 CENG0013 ACCOUNTING BUSS1040 CHEM 181 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