Math 431 Exam 1
10/4/02
1. Show that the projective transformation f :
2. Show that the lines
3. Compute
= e01 + e02 and
@e01 + e12 , e01 - e03 D when
2
2
i1 0 0 y
j
z
j
z
j
z
with matrix j 2 3 4 z maps ideal points to ideal points.
j
z
j
z
j
z
k5 6 7 cfw
Math 431, Practice Final
1. If A ips n + 1 and B ips n fair coins, nd the probability that A gets more heads
than B .
Solution. Denote by Ai the random variable which counts the number of heads
that A has obtained after the i th ip; dene Bi similarly. Let
Math 431 Exam 1
2/28/03
1.How many distinct points of
2
are represented by the following list of homogeneous coordinates?
H1, - 2, 1L, H0, 0, 47L, H1, 1, 1L, H1, 1, - 1L, H0, 0, - 39L, H- .5, 1, - .5L
2. Let l be the point in 3 passing through the points
Math 431 Exam 2
4/4/03
1. Let l be the line in 2 with bivector = - 2 e01 + 3 e02 + 6 e12 .
a) Find the affine coordinates of the intersection of l with the x -and y -axes.
b) Let f : 2 2 is the translation that maps an affine point with coordinates Hx, yL
Math 431 Exam 3
5/2/03
1. Recall that an altitude of a triangle is a line through a vertex perpendicular to the opposite side. Find a bivector
!
for the altitude of the triangle with vertices A : H- 1, 0L , B : H0, - 1L , C : I0, 3 M in 2 .
2. Find a posi
Math 431 Exam 1
10/4/02
1. Show that the projective transformation f :
2. Show that the lines
3. Compute
= e01 + e02 and
@e01 + e12 , e01 - e03 D when
2
2
i1 0 0 y
j
z
j
z
j
z
with matrix j 2 3 4 z maps ideal points to ideal points.
j
z
j
z
j
z
k5 6 7 cfw
Math 431 Exam 2
11/1/02
1. Find a bivector for the line through the point with affine coordinates H0, 0, 1L in
bivector = 2 e01 - 3 e02 + 4 e03 - 5 e12 + 6 e13 - 7 e23 .
2. Show that the envelope of the family of lines
2.
3
and parallel to the line with
H
Math 431 Exam 3
12/6/02
1. Let l be the line in 2 passing through the points with affine coordinates H0, 0L and H2, 1L . Find a bivector for
the line through H0, 3L normal to l .
2. Find a bivector for the tangent line to the ellipse sHtL = e0 + cos t e1
Math 431 Final Exam
12/19/02
Do the first problem (50%) and any 5 of the remaining problems. Only 6 problems will be graded. Write
the numbers of the problems to be graded on the front of the exam booklet. The default is problems 1
through 6.
1. In 3
a. F
Math 431 Final Exam
5/20/02
Do any 8 problemsonly 8 solutions will be graded. Write the numbers of the problems to be graded on the
front of the exam booklet. The default is problems 1 through 8.
1. Find a trivector for the plane through the point with af
Math 431, Practice Final
1. If A ips n + 1 and B ips n fair coins, nd the probability that A gets more heads
thab B .
2. Show that for any two events E and F ,
P (E |E F ) P (E |F ).
3. A total of 2n cards, of which 2 are aces, are to be randomly divided
Math 431
Practice Problems for Test 1
There are more questions here than on the actual test, which will have ve
(possibly multi-part) questions.
1. Experience shows that 20% of the people reserving tables at a particular restaurant
never show up . If the
Math 431 Exam 1
2/28/03
1.How many distinct points of
2
are represented by the following list of homogeneous coordinates?
H1, - 2, 1L, H0, 0, 47L, H1, 1, 1L, H1, 1, - 1L, H0, 0, - 39L, H- .5, 1, - .5L
2. Let l be the point in 3 passing through the points
Math 431 Exam 2
11/1/02
1. Find a bivector for the line through the point with affine coordinates H0, 0, 1L in
bivector = 2 e01 - 3 e02 + 4 e03 - 5 e12 + 6 e13 - 7 e23 .
2. Show that the envelope of the family of lines
2.
3
and parallel to the line with
H
Math 431 Exam 2
4/4/03
1. Let l be the line in 2 with bivector = - 2 e01 + 3 e02 + 6 e12 .
a) Find the affine coordinates of the intersection of l with the x -and y -axes.
b) Let f : 2 2 is the translation that maps an affine point with coordinates Hx, yL
Math 431 Exam 3
12/6/02
1. Let l be the line in 2 passing through the points with affine coordinates H0, 0L and H2, 1L . Find a bivector for
the line through H0, 3L normal to l .
2. Find a bivector for the tangent line to the ellipse sHtL = e0 + cos t e1
Math 431 Exam 3
5/2/03
1. Recall that an altitude of a triangle is a line through a vertex perpendicular to the opposite side. Find a bivector
!
for the altitude of the triangle with vertices A : H- 1, 0L , B : H0, - 1L , C : I0, 3 M in 2 .
2. Find a posi
Math 431 Final Exam
12/19/02
Do the first problem (50%) and any 5 of the remaining problems. Only 6 problems will be graded. Write
the numbers of the problems to be graded on the front of the exam booklet. The default is problems 1
through 6.
1. In 3
a. F
Exam 1: Math 431, Fall 2010
Name
Kowalski
Version A
There are 5 questions on this exam, totalling 100 points. Answer each question using the space provided, being sure
to answer all parts of the question and to show all work! Use the free space on this te
Final Examination: Math 431, Fall 2010
Name
Kowalski
Part
There are 6 questions on this exam, each worth 25 points. Answer each question using the space provided, being
sure to answer all parts of the question and to show all work! Use the free space on