
	Trending ▼ ICSE CBSE 10th ISC CBSE 12th CTET GATE UGC NET Vestibulares ResFinder

Pune University - FY BSc MATHEMATICS - I, April 2010

4 pages, 32 questions, 2 questions with responses, 2 total responses,

	pune_sci	+Fave Message
	Profile Timeline Uploads Q & A

Home > pune_sci >

Instantly get Model Answers to questions on this ResPaper. Try now!
NEW ResPaper Exclusive!

Formatting page ...

Total No. of Questions : 5] [Total No. of Pages : 4 [3717] - 1 P179 F.Y. B.Sc. MATHEMATICS Algebra and Geometry (Paper - I) (New Course) Time : 3 Hours] Instructions to the candidates : 1) All questions are compulsory. 2) Figures to the right indicate full marks. [Max. Marks : 80 Q1) Attempt all the subquestions : a) Define power set of a set and equivalence relation on a set. b) [16] Let f be the function defined by f(x) = 3x 4. 1 Find f and f(a + h). 3 c) Show that (3) (10) (4) is a perfect square, where is Euler s function. d) Using synthetic division find the quotient and remainder when 7x4 + 3x3 + 8x + 1 is divided by x + 2. e) Discuss the nature of the conic x4 4xy 2y2 + 10x + 4y = 0. f) Obtain the equation of a line joining the points ( 2, 1, 3) and (3, 1, 2). g) Show that the line x 4 y+7 z+3 l ies wholly in the plane = = 3 6 4 2x y + 3z 6 = 0. h) Find x so that rank of the following matrix A is not equal to 3 2 3 4 A = 3 1 2 . x 2 2 P.T.O. Q2) Attempt any four of the following : [16] a) Let g : R R be a function defined by g(x) = 5x 2. Show that the function g is bijective. Also find a formula for g 1. b) Define ~ on R2, the set of all points in XY - plane as : for (x, y), (x1, y1) R2, (x, y) ~ (x1, y1) if x + y = x1 + y1. Show that ~ is an equivalence relation. Interprete the equivalence classes geometrically. c) If a, b, c are integers such that a bc and (a, b) = 1 then show that a c . d) If z 1, z 2 a re two complex numbers then prove that z1 z1 a nd = z2 z2 z arg 1 = arg z1 arg z2, where z2 0 . z2 e) In z12, calculate i) ii) f) (2 9 + 1) 5 ( 4 + 5 ). 1 If u + iv, v 0 is a root of the real polynomial equation f(x) = 0 then prove that u iv is also a root of f(x) = 0. Q3) Attempt any two of the following : a) b) c) [16] If a and b are any two integers with b 0 , then prove that there exist unique integers q and r such that a = bq + r, where 0 r < b . a b = 1 if and only if either a = 1 or b = 1, where a and b 1 ab are complex numbers. Prove that Show that 4999 and 1109 are relatively prime. ii) Find a polynomial equation of least degree with rational coefficients having roots 1, 2 + 3 , 2 3 i. i) Let ~ be an equivalence relation on a set X. for x, y X, prove that x y if and only if x = y . ii) d) i) By using De Moivre s theorem prove that sin 5 = [3717] -1 1 [sin 5 5sin 3 + 10 sin ]. 16 2 Q4) Attempt any four of the following : [16] a) Shift the origin to a suitable point so that the equation x2 6x 4y 1 = 0 will be in the form x2 = 4by. State the value of b. b) If , , are the angles made by the line with positive direction of co-ordinate axes then prove that cos 2 + cos 2 + cos 2 = 1. c) Prove that the straight line d) Derive equation of the plane in the normal form. e) Examine for consistency and solve if consistent x +1 y 2 z 2 = = touches the sphere 4 1 1 x2 + y2 + z2 = 9. Find the point of contact. x y + z = 1 x 3y + 4z = 6 4x + 3y 2z = 3 7x 4y + 7z = 16. f) Find the centre and the radius of the circle x2 + y2 + z2 2y 4z = 11, x + 2y + 2z = 15. Q5) Attempt any two of the following : [16] a) Reduce the equation 5x2 + 6xy + 5y2 10x 6y 3 = 0 to the standard form and name the conic. b) i) ii) c) x x1 y y1 z z1 = = and the plane l m n ax + by + cz + d = 0, where l, m, n are direction ratios of a line. Find angle between the line Find angle between two lines whose direction cosines are connected by the relations 2l m + 2n = 0, mn + nl + lm = 0. Show that for every real number the equation S + U = 0 represents a sphere containing the circle of intersection of the sphere S x2 + y2 + z2 + 2ux + 2vy + 2wz + d = 0 and U ax + by + cz + d = 0. Hence find equation of the sphere through the circle x2 + y2 + z2 + 6x 4y 6z 14 = 0, x + y z = 0 and passing through the point (1, 1, 1). [3717] -1 3 d) i) Find non-trivial solution of the following system x 4y + 5 z = 0 2x y + 3z = 0 3x + 2y + z = 0 ii) [3717] -1 Find equation of the plane which passes through (2, 0, 3) and makes intercepts on the axes which are in the ratio 3 : 1 : 2. 4

Formatting page ...

Top Contributors
to this ResPaper
(answers/comments)

Abhi Kumbhar
(1)

chandra
(1)

Formatting page ...

Additional Info : F.Y. B.Sc. MATHEMATICS : Algebra and Geometry (Paper - I) (New Course), Pune University
Tags : pune university exam papers, university of pune question papers, pune university science, pune university courses, bsc pune university, msc pune university, pune university solved question papers, pune university model question paper, pune university paper pattern, pune university syllabus, old question papers pune university

pune_sci chat

Horizontal lines at:
Guest Horizontal lines at:
AutoRM Data:
Box geometries: