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Math 308 Section 101 Final Exam - December 2010, University of British Columbia, Exams of Mathematics

Acharya Nagarjuna UniversityMathematics

This is a final exam for math 308, section 101, instructor tai-peng tsai, at the university of british columbia, held in december 2010. The exam consists of 9 problems covering various topics in mathematics, including ellipses, parabolas, conics, affine transformations, projective transformations, collinearity, cross ratio, inversion, and stereographic projection.

Typology: Exams

2012/2013

Uploaded on 02/21/2013

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Be sure this exam has 12 pages including the cover
The University of British Columbia
MATH 308, Section 101, Instructor Tai-Peng Tsai
Final Exam – December 2010
Family Name Given Name
Student Number Signature
No notes nor calculators.
Rules Governing Formal Examinations:
1. Each candidate must be prepared to produce, upon request, a
UBCcard for identification;
2. Candidates are not permitted to ask questions of the invigila-
tors, except in cases of supposed errors or ambiguities in exami-
nation questions;
3. No candidate shall be permitted to enter the examination room
after the expiration of one-half hour from the scheduled starting
time, or to leave during the first half hour of the examination;
4. Candidates suspected of any of the following, or similar, dishon-
est practices shall be immediately dismissed from the examination
and shall be liable to disciplinary action;
(a) Having at the place of writing any books, papers or
memoranda, calculators, computers, sound or image play-
ers/recorders/transmitters (including telephones), or other mem-
ory aid devices, other than those authorized by the examiners;
(b) Speaking or communicating with other candidates;
(c) Purposely exposing written papers to the view of other can-
didates or imaging devices. The plea of accident or forgetfulness
shall not be received;
5. Candidates must not destroy or mutilate any examination ma-
terial; must hand in all examination papers, and must not take
any examination material from the examination room without
permission of the invigilator; and
6. Candidates must follow any additional examination rules or
directions communicated by the instructor or invigilator.
problem max score
1. 10
2. 10
3. 10
4. 10
5. 10
6. 15
7. 15
8. 10
9. 10
total 100
pf3
pf4
pf5
pf8
pf9
pfa

Partial preview of the text

Download Math 308 Section 101 Final Exam - December 2010, University of British Columbia and more Exams Mathematics in PDF only on Docsity!

Be sure this exam has 12 pages including the cover The University of British Columbia MATH 308, Section 101, Instructor Tai-Peng Tsai Final Exam – December 2010

Family Name Given Name Student Number Signature

No notes nor calculators.

Rules Governing Formal Examinations:

  1. Each candidate must be prepared to produce, upon request, aUBCcard for identification;
  2. Candidates are not permitted to ask questions of the invigila-tors, except in cases of supposed errors or ambiguities in exami- nation questions;
  3. No candidate shall be permitted to enter the examination roomafter the expiration of one-half hour from the scheduled starting time, or to leave during the first half hour of the examination;
  4. Candidates suspected of any of the following, or similar, dishon-est practices shall be immediately dismissed from the examination and shall be liable to disciplinary action;(a) Having at the place of writing any books, papers or memoranda,ers/recorders/transmitters (including telephones), or other mem- calculators, computers, sound or image play- ory aid devices, other than those authorized by the examiners;(b) Speaking or communicating with other candidates;

didates or imaging devices. The plea of accident or forgetfulness(c) Purposely exposing written papers to the view of other can- shall not be received;

  1. Candidates must not destroy or mutilate any examination ma-terial; must hand in all examination papers, and must not take any examination material from the examination room withoutpermission of the invigilator; and 6.directions communicated by the instructor or invigilator. Candidates must follow any additional examination rules or

problem max score

  1. 10
  2. 10
  3. 10
  4. 10
  5. 10
  6. 15
  7. 15
  8. 10
  9. 10 total 100

(10 points) 1. Find the two pairs of focuses and directrices of the ellipse x^2 + 4y^2 + 2x = 0.

  1. Classify the conics in R^2 with the following equations. You do not need to find their axes. (5 points) (a) 3 x^2 − 8 xy + 2y^2 − 2 x + 4y − 16 = 0

(5 points) (b)x^2 + 8xy + 16y^2 − x + 8y − 12 = 0

(10 points) 4. On a trianglepoints on side ABC CA so that, D and ∠ DCBE′^ are points on side = ∠E′BA, and BC F andso that F ′ (^) are points on sides ∠BAD = ∠D′AC ,AB E and so that E′^ are ∠ACF = ∠F ′CB. Suppose that AD, BE, CF are concurrent. Show that AD′, BE′, CF ′^ are also concurrent. Hint: You may use BDDC = ABAC^ sinsin^ ∠∠BADDAC.

(5 points) 6. (a)Find the affine transformation which maps the points

and

to the points

and

, respectively.

(5 points) (b)Find the projective transformation which maps the Points [1[1, 1 , 1] to the Points [2, 1 , 0], [1, 0 , −1], [0, 3 , −1] and [3, − 1 , 2]., 0 , 0], [0, 1 , 0], [0, 0 , 1] and

(5 points) (c)Find the Point of intersection of the Lines L 1 : x + 2y + 5z = 0 and L 2 : 3x − y + z = 0.

  1. Let A = [2, 1 , 1], B = [− 1 , 1 , −1], C = [1, 2 , 0] and D = [− 1 , 4 , −2] be four points in RP^2. (5 points) (a)Verify that A, B, C, D are collinear and find the equation of the Line.

(10 points) 8. Find the imagies of the circles |z| = 1 and |z| = 2 under the M¨obius transformation t(z) = z z^ −+^2 i.

(10 points) 9. Letthe inversion of C be a circle with radius P in C. Then the line r and centered at p through O P. For any ordinary point ′ (^) and perpendicular to OP PP 6 = O′ (^) is called the, let P ′^ be polar of P for the circle C. Show that, if the polar of P passes through a point Q, then the polar of Q passes through P.