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05 geometry - 8073 THE UNIVERSITY or SYDNEY Faculties of...

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Unformatted text preview: 8073 THE UNIVERSITY or SYDNEY Faculties of Arts, Economics, Education, Engineering 85 Science MATH 3006 GEOMETRY November 2005 Time allowed: Two hours plus ten minutes reading time Lecturer: Jenny Henderson All questions may be attempted. The total number of marks for this exam is 76. No notes or books are allowed in this exam. University calculators are supplied; no other calculators may be used. 8073 1. [13 marks] (2') Give the definitions of the following terms: (a) a transformation of S, (b) an affine transformation, (0) an isometry, (d) an odd isometry. (ii) Consider the afiine transformation a : 8 —> 5 given by 06(.73,y) = (—31,—56 + 2)- (a) Using the derivative 01* or otherwise, show that a is an isometry. (b) By considering the determinant of 01* and the fixed point set of a, say exactly which type of isometry a is. (iii) If fig is an odd isometry with more than one fixed point, what type of isometry is ,8, and why? [13 marks] (1') Find on (u) when a is a halfturn and u is any vector. (ii) Show that if n 1 G={L,p, ,p 7 ,0,pa, Hon—10} is a symmetry group of type D”, then O'pk = p‘ka for all integers k. Use this result to compute (pa)(p20) in the group D3. (iii) What conditions on E, m, n guarantee that woman is a glide reflection? (iv) Find the line b and the vector u such that ammo” = 719,11 when E is the line ac = O, m is the line 3; = a: and n is the line y = —m + 2. 8073 3 3. [13 marks] (2') Find the formula for the unique affine transformation which maps (1,0) to (3, —4), (0,1) to (5, —7) and (1,2) to (9, —6). (ii) Express the point 5 = (6, 3) as a weighted sum of points P = (1,1), Q = (3,0), R = (0,0). (iii) In the following diagram (which is not to scale), m 2 gm, B—I: = §§6 and the lines AL, BM, C'N are concurrent. Use Ceva’s Theorem to find N as a weighted sum of A and B. 4. [13 marks] (1') Find all symmetries of the set of points {(2, 0), (0,1), (—2, 0), (0,—1)}. (2'1) Find all isometries which map the line :10 = 0 to the line y = 0. (iii) Classify the frieze group .7: of the following frieze pattern, and describe all elements of f. /\/\/\/\/\/\ (iv) Prove that if a frieze group .7: contains one halfturn, then it contains infinitely many halfturns with centres spaced half a period apart. /4 8073 5. [13 marks] (2') Find the point of intersection in ”P of the lines a: + 2y — 52 = 0 and :1: + 2y — 42 = 0. (ii) Find the equation in 73 of the line joining (1 : 2 : 1) and (3 : 2 : 0). (iii) By considering the symmetric matrix associated with the conic :02 — 2y2 + 322 — 2361/ + 41/2 — 22m = 0, show that the conic is non—singular. (iv) Find all points at infinity on the conic in the previous part, and hence classify the conic as an ellipse, hyperbola or parabola. (1)) Find the image of the line at +3; — 32 = 0 under the collineation (15 M, where 1%: GNP—l HI—‘N i—Ioi—I [11 marks] (2') Find a collineation which maps the hyperbola xy = 22 to the ellipse g: + 2 _ 22 4 y — ' 1 2 1 (ii) Let ¢A be the collineation with matrix A = 0 —1 ——1 . Find all 0 0 1 the fixed points of qSA and Show that qfiA has an axis and a centre. THIS IS THE END OF THE EXAMINATION PAPER ...
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