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Unformatted text preview: B/MapleTA." MATH 135 1119 Algebra for Honours Mathematics : Grades —
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Score: Duration: 1 hrs 42 min
Started: 9/13/11 12:40:07 PM
11,12'0 ﬁnished: 9/13/11 2:22:55 PM
V Bill Zimou Peng
Login: bzpeng Assignments
Email: [email protected] Completed: 9 Active: 0
Student ID: 20430954 To Be Reviewed: 0 Passed: 7
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_‘ Question 1 Question Grade:
Weighted Grade: F 2 Question Grade:
Weighted Gade: 1.0
(1/1.0) 1.0
(1/1.0) Your response Consider the following statement P. I“ If X is a positive real number, then X2 > X Which ofthe following is true of P ? P is a false statement. (100%) Comment: P is a false statement. To see this, consider X = E. 2 l 1 1
Wh — :— —.
eave(2] 4} 2 Thus,we have a real number X which does notsatisfy X2 > X, and so the statement P is false. Your response
Consider the statement: I? Let ﬂX} = “i 3 X . Then ﬂX} = X forsomereal number X with U S X S l . correct Which choice corresponds to the conclusion ofthe above statement?
There exists a real number X , U S X S 1 such that ﬂX} = X . (100%) Comment: When working with mathematical statements it is important to be able to distinguish which parts form the hypothesis
and which form the conclusion. In a statement“lf P,then Q“, P is the hypothesis and Q is the conclusion.Awayofthinking about which parts
belong where is to ask 'What is being given to me“ (for the hypothesis) and “What is the statementtrying to show" (for
the conclusion). \ﬁewed in this light itshould be clear thatthe statements claim was thatthere is some number X in the interval [0, 1]
such that ﬂX} = X. Hence the correct choice of conclusion is:
"There exists a real number X, 0 S X g 1 such that ﬂX} = X." 3 Question Grade:
Weighted Grade: I 4 Question Grade:
Weighted Grade: 1.0
(1/10) 0.0
(0/1 .0) Note,the hypothesis was simplythat ﬂx} = ,f 3 X . Your response Consider the statement: J“ If AD bisects ASAC of aSACJhen — :
AC AB SD Correct
DC ' Which choice corresponds to the hypothesis of the above statement? SAC is a triangle and
AD bisects A SAC. (100%) Comment: When working with mathematical statements it is important to be able to distinguish which parts form the hypothesis
and which form the conclusion. In a statement "If, P then Q ", P is the hypothesis and Q is the conclusion.Awayofthinking aboutwhich parts
belong where is to ask 'What is being given to me" (for the hypothesis) and "What is the statementtrying to show" (for the conclusion). Hence, in the statement If AD bisects LSAC of aBAC,then — 2
AC AS SD
DC I We have that the hypothesis is “ AD bisects A SAC of A SAC,“.
This is equivalent to the correct option: SAC is atriangle and AD bisects A SAC. Your response
Considerthe statement: Let fand g be realvalued
functions with ﬁx} } QM} for
all X E [R . fixolhtxo) > 5iixoihlxol f°’ anyrealvalued function h and
any X0 E IR. Which choice corresponds to the hypothesis of
the above statement?
f and g are realvalued functions with ﬂx}> 9pc} for all x E IR
._(0%) Correct response Consider the statement: ‘xl‘
Let fend g be realvalued Incorrect
functions with ﬂx} > ﬁx} for all
x E [R . flxolhlxo) > thoihlxo) ”my realvalued function h and any XOER. Which choice corresponds to the hypothesis ofthe
above statement? f, 9 and h are realvalued functions
with ﬂx}> g{x} for all x E IR. XOEIR. Total grade:0.0><1l1 = 0% Comment: When working with mathematical statements it is important to be able to distinguish which parts form the hypothesis
and which form the conclusion. In a statement"lf P,then Q", P is the hypothesis and Q is the conclusion.Awayofﬂ1inking about which parts
belong where is to ask 'What is being given to me" (for the hypothesis) and "What is the statementtrying to show" (for the conclusion). \ﬁewed in this light it is clearthat the hypothesis was composed ofthese bits ofgiven information: 1. f, g and h are realwlued functions ﬂx}> g{x}foraii x 6 IR XOEIR Hence the correct choice offor the hypothesis is: " f, g and h are realvalued functions with ﬂx}> QM} for all X E IR. x0 E IR " 5 Question Grade:
Weighted Gade: 6 Question Grade:
Weighted Grade: 7 Question Grade: Weighted Grade: 1.0
(1/1.0) 1.0
(1/1.0) 1.0 (1/1.0) The conclusion was simply " ftxo)h(x0) } g(x0)h(x0)" which happens to be false in this example. Your response Which choice mostcloselycorresponds to the following statement? Everyquadratic polynomial ﬁx} = x2 + bx + C can be factored into a productof “ two linearfactors. If ﬂx} = x2 + bx + C, then fix) = (x— r1) (x — r2) for some numbers r1 and r2
.(100%) Comment: This statementbegins with the hypothesis that fis a quadratic bystating ﬂx} = x2 + bx + C, and then concludes
that fis the product oftwo linear factors bywriting flx} = (x — Fl) (x — r2). Thus, the oorrectanswer is: "If ﬂx} = x2 + bx + C,then fix) = (x — r1) (x — r2) forsome numbers F1 and r2 ." Your response In orderto prove the implication "If A , then B", we assume that A is true, and use that “ assumption to prove that B is true. In orderto use the implication "If A, then B" we mustﬁrst Correct establish that A is true. If A is true, we can conclude that B is true. An element C E IR is called a root ofthe real polynomial ﬁx) 2 {2an +    + alx + :20 if
ﬂc} = 0. Considerthe following proposition: If C is a root ofthe polynomial ﬂx}, then {x — C} is a factor of ﬁx}. If fl—l} = 0,then which ofthefollowingisafactorof fix} = x3 — 12 x2 + 23 x + 36 ?
[x + 1} (100%) Comment: We are given that ﬂ—l} = 0 , and so —1 is a root ofthe polynomial. [Verifythis yourself. You can ﬁnd a root by plugging values of x into ﬁx}. Start close to zero and work your way
outward (e.g., x = :l: l, :l: 2, ). Usingthisapproachweﬁnd: ﬂ—l} = [—1}3 — 12 (—1}2 + 23 (—1} + 36 = 0.] Thus, bythe proposiiton we have that { x — {1)} = [x + 1} is afactorof ﬁx}. Your response In orderto prove the implication "If A , then B", we assume that A is true, and use that assumption v to prove that B is true. In orderto use the implication “If A,then B“ we mustﬁrstestablish that A Correct is true. If A is true, then we can conclude that B is true.
Consider the following statement about planar geometry: "If C is acircle and Iis a Iine,then C and I intersectin atmosttwo points." Using onlythis proposition, underwhat circumstances could you correctlyconclude that there are at
mosttwo points ofintersection? All of the above (100%) Comment: ...
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