University of Delhi 2009 B.A Mathematics , SOL, - Question Paper

4

MATHEMATICS

Paper 1 *: ' .    (Algebra and Calculus) (New Course : Admissions of 2004 and onwards)

Time : 3 Hours    Maximum Marks : 75

(yVrite your Roll No. onthe top immediately on receipt of this question paper.)

Note : The maximum marks printed oh the question paper are applicable for the students of the regular colleges (Cat. A). These marks will, however, be Scaled up proportionately in respect of the students of NCWEB at the time of posting of awards for compilation of result.

All sections are compulsory and have equal marks.

[P.TO.

Section I

(a) Find non-singular matrices P and Q such that PAQ is in the normal form where

5 -2

1-4,11 -19 j

(b) Show that the set S = \x, y, 2x - 3y) x, y are real numbers} is a subspace of 1R³.

Or , . .

(a) Using Cayley-Hamilton theorem determine the inverse of the matrix

3 4 T5

0 -6 -7

(b)

x - 3y + 2z = 0 Ix - 21y+ I4z = 0 -3x + 9 v - 6z = 0

2.    (a) Express sin⁵0 cos²0 in a series of sines of

multiples of 0.

(b) If a, (3, y, 8 are the roots of the equations x⁴ + px³ + qx² + rx + s = 0,s0 ; find the values of    f

(i) Z oc² (ii) X ot²p²    

Or

(a)    Solve the equation x⁴ + 4x* + 6x² + Ax + 5 = 0, whose one root is i.

(b)    Prove that

1 + cos 9 0 = (1 + cos 0) [16 cos⁴ 0-8 cos³ 0 -12 cos²0 + 4 cos 0 + l]².    ,

Section III

3.    (a) Examine the function

fix)

for derivability at the origin. Also determine m when/' 0;) is continuous at origin.

(b) IfV = sin

yfx +

av .av i

+:y~-~- - tan V = 0. ox oy 2

Or

(a) Prove that the function defined as

r-v v n ] fix) = X .v'11 ,x?= U 0 if x = 0

is not derivable at x = 0.

Further sho w that / is continuous at x

(b) (i) If z = x tan¹

Show that

dz dz x--h y z

yvdx, w:/

2-^:- ' 2

. *    X + V    ;

(ii) If u - log --, prove that

du du ,

x~ + y~ = l dy

4.    (a) Prove that the equation of the normal to the

asteroid

x² + j;2/3 #2/3

may be written in the form x sin (() - v cos <|) + a cos 2(j) = 0.    \

(b) Determine the position and nature of the double points on the curve . x³ - y² - lx² + 4y + 15* -13 = 0

(a)    Trace the curve :

x (x² + y²) = a (x² -y¹)

(b)    Find the asymptotes of the curve jc³ + 3x²y - 4>³ ~x + y + 3 = 0

/ Section V

5.    (a) State and prove Cauchys Mean Value theorem.

(b) Find the extreme points of the function

where xe [0, n].

Or

Section VI

/2

6. (a) If = f cot 0 dO then show that

' '.71/4': '

> ¹¹ (J₊i+J_;i-i) = 1. Hence deduce

(b) Show that the length of a loop of the curve r² = a² cos 2 0 is 2a f

Or

(a) Evaluate

{2x + 3)dx

(i rj

-4.x²

(ii)J

(b)

the loop of the curve 9ay¹ = x (3a - x)² about A'-axis. .    ; _:

933    7    1500

Attachment: university-of-delhi-2009-b-a--35044-1.pdf

Earning:   Approval pending.