Kerala University 2005 B.Sc Mathematics Part3 - Question Paper

Please see the attachment below

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Name........................................................

FIRST YEAR B.Sc. DEGREE EXAMINATION, APRIL/MAY 2005

Part IIIMathematics Subsidiary Paper IMATHEMATICS (Common for all subsidiary subjects)

Time : Three Hours    Maximum : 70 Marks

A maximum of 14 marks will be awarded from each unit All questions carry 5 marks each.

Unit I

1.    Find the sum of infinity of the series 1 - +     5 . 7_ ₊

4 4.8 4 . 8 . 12

( lY* ( 1 11

2.    If n is large show that 1 + = ell - - + -=- approximately.

I n) V 2/i 24 n )

3.    If a, P, y are the roots of the equation x³ + px + <j = 0, find the equation whose roots are (a - p)¹, (P-y)²,(y-a)²-

4.    Solve the biquadratic equation x⁴-4x³ + 7x²-6x + 2 = 0.

Unit II

5.    If tan (x + iy) = u + iv, prove that =

v sinh2y

13.    Find the area lying above the x-axis and included between the circle x² + y² 2 x = 0 and the parabola y² = x.

14.    Find the area of the surface formed by the rotation of the curve y² = 8 x about the x-axis from * = 2 to x =s 7.

15.    Evaluate JJ(*² y²) dx dy over the region for which x 0, y 'z 0 and x + y < 1.

16. Evaluate [ Vtan 0 dQ .

o

UnitV

12 2'

17. Show that 2 1 - 2 is orthogonal. -2 2 - 1

18. Test for consistency and solve :

x+2y-z=3 3x-y+2z=l

2 x-2y + 32 = 2

3-0 0

19. Find the eigen vectors of the matrix 5 4 0

3 6 1

, find A⁴ using Cayley-Hamilton theorem.

2 4

1

   3

c    c

6.    Sum to infinity the series c sin a + cos 2 a + cos 3 a + . . .

2!    3!

7.    Transform the equation 6x^s + 24xy-y² = 0 into another in which the xy term is absent.

8.    The asymptotes of a hyperbola are parallel to the lines 2 x + 3 y = 0 and 3 x - 2 y = 0 and its centre is at (1, 2). Find its equation if it passes through the point (5, 3). Find also the equation to the conjugate hyperbola.

Unit m

9.    Expand sin x as an infinite series.

\tan2x

10.    Evaluate liw (tan x)

11.    Find the maxima and minima of the function x³ - 18 x² + 96 x + 4.

12.    Find the radius of curvature of the curve x¹ + s 1 at the point    .

Attachment: kerala-university-2005-b-sc-mathematics-39212-1.pdf

Earning:   Approval pending.