Karunya University 2009 B.Sc Mathematics Supple , ./t,, – I : Calculus, Analytic Geometry, Algebra and Trigonometry - Question Paper

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Reg. No. : .....................................
Name : ..........................................
B.Sc. Supple. Degree Examination, Aug./Sept. 2009
Part III : Group – I : MATHEMATICS
Paper – I : Calculus, Analytic Geometry, Algebra and Trigonometry
(2006 Admission Onwards)
Time : three Hours Max. Marks : 65

Instructions : A maximum of 13 marks can be earned from every Unit. every ques. carries five marks.


UNIT – I

1. If y = Sin m (Sin–1 x), prove that

i) (1 – x2)y2 – xy, + m2 y = 0 and

ii) (1 – x2) yn+2 – (2n + 1) xyn+1 + (m2 – n2) yn = 0

2. Verify that ? 2u ? ? 2u –1 y
, for u = Sin .
?x ?y ?y ?x x
3. Prove that the radius of curvature at any point of the cycloid x = a ( ? + Sin ? ) and

?
y = a (1 – Cos ? ) is 4a Cos 2 .
4. Find the asymptotes of the curve x3 + y3 = 3axy.




P.T.O.


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UNIT – II

5. Find the number and sum of all the divisors of 360.

6. State Euclidean algorithm. obtain the g.c.d. of 12378 and 3054.

7. If 'n' is any number, and 'a' is prime to n, then prove that a (n) = one (mod n).
8. State and prove Wilson's theorem.

UNIT – III

9. Show that in an formula with rational coefficients, irrational roots occur in pairs.

10. If a, ß, ? are the roots of the equation
x3 + px2 + qx + r = 0, obtain the formula whose roots are a ? ß, ß ? ? and ? + .
a
11. Remove the fractional coefficients from the formula x 3 - 1 x 2 ? 1 3 x – one = 0.
4


12. State Desearte's rule of signs.
Determine completely the nature of the roots of the formula x7 – 3x4 + 2x3 – one = 0.

UNIT – IV

13. Find the formula of the tangent at the point (x1, y1) on the parabola y2 = 4ax.

14. Find the formula of the polar of a provided point (x1, y1) with respect to the ellipse

x 2 y 2

a two ? ? 1.
b2
15. Determine the translation of axes that will transform the formula

3x2 – 4y2 + 6x + 24 y = 135, into 1 in which the coefficients of the 1st degree term is zero.

16. obtain the polar formula of a straight line.


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UNIT – V

17. Expand tan n ? in terms of powers of tan ? .

18. Prove that

1
Cos7 ? = 64 [Cos seven ? + seven Cos five ? + 21 Cos three ? + 35 Cos ? ]

19. Separate tan (x + iy) into real and imaginary parts.

20. If sin (A + iB) = x + iy , show that

i) x2 ? y2 = 1

Cos h2 B Sin h2 B
x2 y2
ii) - = 1.

Sin2 A Cos2 A




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Earning:   Approval pending.