LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIRST SEMESTER – NOVEMBER 2012

MT 1502 – ALGEBRA AND CALCULUS – I

 

 

 

Date : 08/11/2012             Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

 

PART – A

ANSWER ALL QUESTIONS:                                                                (10×2=20)

 

1. Find the n^th derivative of
2. Show that, in the curve, the polar sub tangent varies as the square of the

radius vector and the polar subnormal is a constant.

3. Write the conditions for maxima and minima of two variables.
4. What is the radius of curvature of the curve at the point (1, 1)?
5. Find the co-ordinates of the centre of curvature of the curve at the point

(2, 1).

6. Form a rational cubic equation which shall have the roots 1,
7. If are the roots of the biquadratic equation

find

8. State Newton’s theorem on the sum of the powers of the roots.
9. State Descartes’ rule of signs for negative roots.
10. Determine if Cardon’s method can be applied to solve the equation

 

PART – B                               

 

ANSWER ANY FIVE QUESTIONS:                                                      (5×8=40)

11. a) Find the angle at which the radius vector cuts the curve
12. b) Find the slope of the tangent with the initial line for the cardioid

at                                                                                            (4 + 4)

12. Discuss the maxima and minima of the function

 

13. Prove that the (p-r) equation of the cardioid is and hence

prove that its radius of curvature is

14. Show that the evolute of the cycloid ; is another

cycloid.

15. Solve the equation
16. Show that the sum of the eleventh powers of the roots of is zero.
17. a) If are the roots of the equation find the value of

 

1. b) Determine completely the nature of the roots of the equation

(5 + 3)

18. If be a real root of the cubic equation of which the coefficients

are real, show that the other two roots of the equation are real, if

 

PART – C                               

 

ANSWER ANY TWO QUESTIONS:                                                      (2 x 20 = 40)

 

19. a) Find the n^th differential coefficient of .
20. b) If, prove that               (10 +10)

 

20. A tent having the form of a cylinder surmounted by a cone is to contain a given

volume. If the canvass required is minimum, show that the altitude of the cone is

twice that of the cylinder.

 

21. a) Find the asymptotes of

 

1. b) Show that the roots of the equation are in Arithmetical

progression if  Show that the above condition is satisfied by the

equation and hence solve it.                                              (10 + 10)

 

22. Determine the root of the equation which lies between 1 and 2

correct to three places of decimals by Horner’s method.

 

 

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