MATH 54 1/31/2013 QUIZ SOLUTIONS
You have 15 minutes to complete the following quiz. Notice that there are two problems, one on
each side. No notes are allowed. You may use a 4-function calculator if
Chaptel 1.2, Problem GEE
Mar. -X.)+2y(y. -y3)=X. -x, +y. -y]A
Com ment
Step 6 of 15 A
Thus, the intersectinnoi L. and L, is the solution (1?) to the lollcmhg linear system:
2:"(xJ x,)+2y(y, y,)= x,
55 Chapter 1.2, Problem GEE I] Bonkmark Show all steps: a:
Problem
A perpendicular bisecmr at a line segment is a line through the midpoint of the segment,
perpendicular to the segment (Figure 149), P
55 Chapter 1.2, Problem GEE D Bookmark Showallsteps: m :
Slepll Df 15 A
Similarlywe have
y=i[(.'l|2 71:(1, 7x)*(yxl 711. II)+(J"JI 7x12)"l 7%)]-
Slep 13 of15 A
Now ifmcnmpute the coordinata x' and y'
Chaptel 1.2, Problem GEE
19-232 +xzz+y-2m+yz-x-2u,+xa'+y-2m+y. -
Step 4 of 15 A
Subtracting the right side from the left yields
2m, -X:)+2y(y. -y:)+1: -x3 +ya m =0-
whence
24: _x1)+2)'(yl y1)=x): 33:]
Chapter 1.2, Problem GEE
D Bookmark Showallsteps: LIE- :
perpendicular bisectors Ll of EC. [1 cl CA . and L] of AB are given bythe fallawmg
equations:
11=cfw_[w)ER=(r-r=+(y-yzl1=(1-13)2+(yya]::
L, =[i
EE Chapter 1.2, Pmbbem GEE D Bookmark Showallsteps: m a:
= (xwz am my. '13.": Hay. in)
If. y. l
4: y: l
x, y. l
=44.
Com ment
Step 9 of 15 A
Note that detd \s nonzero since the vemces A, B, and C ar
Chapter 1.2, Problem GEE
Step 1 0f 15 A
We wish to show that the perpendicular bisectprs of the three sides of a triangle are
concurrent,
Com ment
Step 2 of 15 A
[t A(z.y.). Man). and C(lel be the ver
55 Chapter 1.2, Problem GEE l1 Buokmark Showasteps: LEE- :5
Slepl of 15 A
Now ifmcnmpute the coordinates x' and y' of intersection point (1'0) 01 L, and 1.
we obtain the same formulas m fur x andy wit
MATH 54 FINAL EXAM PRACTICE QUESTIONS
1. Let M be an m n matrix. In terms of the pivots of M , how do we tell if the matrix equation
M x = b always has at least one solution? At most one solution?
2.
MATH 54 QUIZ 2/28
1. Let P2 be the set of polynomials of degree 2. Find the change-of-basis matrix P from the
CB
basis B = cfw_1, t, t2 to the basis C = cfw_1 2t + t2 , 3 5t + 4t2 , 2t + 3t2 .
Soluti
MATH 54 2/7/2013 QUIZ
You have 15 minutes to complete the following quiz. Notice that there are problems on each
side. No notes are allowed. You may use a 4-function calculator if you wish, but nothin
MATH 54 MIDTERM 1 REVIEW
1. Make sure you review the denitions of the following terms.
(1) augmented matrix vs. coecient matrix
(2) echelon form and reduced echelon form
(3) pivots, pivot rows, pivot
MATH 54 3/14 QUIZ
1. Which of the following are orthogonal matrices? Circle all that apply.
0 1/2
a.
0 1/ 2
b.
0 1
1 0
c.
1 2
2 1
d.
1 0
0 1
Solution: Only b and d are orthogonal matrices. A matrix ha
MATH 54 3/7 QUIZ
You have 20 minutes to complete this quiz. No notes or books are allowed.
1.
Let P2 be (as usual) the vector space of polynomials of degree 2. Let D : P2 P2 be the linear
transformati
MATH 54 4/11 QUIZ
1. Find a general solution to the following dierential equation:
y y = e2t + tet
1
2
MATH 54 4/11 QUIZ
2. Suppose that yp and yq are two solutions to the dierential equation
y + 2y +
MATH 54 4/4 QUIZ
There is only one problem on this quiz.
1. Find all dierential functions y satisfying the following conditions:
y 4y + 4y = 0
y (0) = 1
y (1) = 0
Solution: The general solution to the
55 Chapter 1.2, Prohbem GEE D Bookmark Showallsteps: m :
Ax=b.
where
_ 2(cfw_1'32 2(yy'yz) _ _ x _ x12xzz*112J12
4' ' b 3 3 z a
20m) 2(n-J5) y x. -x; +y. -y;
Comment
Stepof 15 A
Nowwe have
detA:
2