LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

XZ 54

SECOND SEMESTER – APRIL 2008

MT 2500 – ALGEBRA, ANAL.GEO & CALCULUS – II

 

 

 

Date : 23/04/2008                Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

 

PART – A

Answer ALL questions.:                                                                               (10 x 2 = 20)

1. Evaluate
2. Write the value of
3. Is exact?
4. Solve
5. State Raabe’s test.
6. Define uniform convergence of a sequence.
7. Find the Coefficient of in the expansion of
8. Write down the last term in the expansion of
9. Write the intercept and normal forms of the equation of a plane.
10. Find the Centre and radius of the sphere

 

 

PART – B

Answer any FIVE  questions.                                                                      (5 x 8 = 40)

 

1. Evaluate
2. Solve
3. Test the Convergence of
4. Find the sum to infinity of the series
5. Sum the series
6. If a, b, c denote three Consecutive integers, show that

1. The foot of the perpendicular drawn form the origin to the plane is (12,-4,-3); find the equation of the plane.
2. Find the equation to the sphere through the four points (0,0,0), (a,0,0), (0,b,0), (0,0,c) and determine its radius.

 

 

 

 

PART – C

Answer any TWO   questions.                                                                      (2 x 20 = 40)

1. a) Evaluate

1. b) Find the area of the cardioid (12+8)

1. a) Prove that the series

is convergent if and

1. b) Sum the series

1. Find the image of the point (1,3,4) in the plane . Hence prove that the image of the line is .
2. Through the circle of intersection of the sphere and the plane two spheres and  are drawn to touch the place . Find the equations of the spheres.

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

SECOND SEMESTER – APRIL 2011

MT 2501/MT 2500 – ALGEBRA, ANAL.GEO & CALCULUS – II

 

 

 

Date : 08-04-2011              Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

PART – A

Answer ALL questions:                                                                                                     (10 x 2 = 20)

1. Evaluate .
2. Evaluate .
3. Solve: .
4. Solve: .
5. Prove that the series is convergent.
6. Test for convergency the series .
7. Find the general term in the expansion of .
8. Prove that the coefficient of in the expansion of is .
9. Find the equation of the sphere which has its centre at the point and touches the

plane  .

10. Find the distance between the parallel planes and

PART – B

Answer any FIVE questions:                                                                                            (5 x 8 = 40)

11. Prove that .
12. If ( n being a positive integer), prove that .

Also evaluate  and .

13. Solve: .

 

14. Solve
15. Test for convergency and divergency the series
16. Show that the sum of the series .
17. Show that if

 

18. Find the equation of the plane passing through the points

.

 

PART – C

Answer any TWO questions:                                                                                           (2 x 20 = 40)

19. a) Evaluate (10 marks)
20. b) Find the area and the perimeter of the cardiod .                    (10 marks)
21. a) Solve: . (10 marks)
22. b) Discuss the convergence of the series for

positive values of .                                                                                       (10 marks)

 

21.a) Show that the error in taking  as  an approximation to  is

approximately equal to  when  is small.                                                   (10 marks)

1. b) show that (10 marks)

 

22. a) A sphere of constant radius passes through the origin and meets the axes in A, B, C.

Prove that the centroid of the triangle ABC lies on the sphere

(10 marks)

1. b) Find the shortest distance between the lines

.

Also find the equation of the line of shortest distance.                              (10 marks)

 

 

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – MATHEMATICS

FIRST SEMESTER – NOVEMBER 2012

MT 1500 – ALGEBRA, ANALY. GEO., CALCULUS & TRIGONOMETRY

 

 

 

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

Time : 1:00 – 4:00

 

PART – A

 

Answer ALL the questions:                                                                                     (10 x 2 = 20 marks)

 

1. Write the n^th derivative of
2. If y = a show that
3. Define the evolute of a curve.
4. Find the p-r equation of the curve r = a sin q.
5. Determine the quadratic equation having 3 – 2 i as a root.
6. Diminish the roots of by 2.
7. Show that
8. Express in locus of logarithmic function.
9. Define a rectangular hyporbola.
10. Write down the angle between the asymptotes of the hyperbola

PART – B

Answer any FIVE questions:                                                                                   (5 x 8 = 40 marks)

 

11. Show that in the parabola the subtangent at any point is double the abscissa and the subnormal is a constant.
12. Find the radius of curvature at the point ‘O’ on
13. Show that if the roots of
14. Find the p-r equation of the curve with respect to the focus as the pole.
15. Separate into real and uniaguinary parts.
16. Find the sum of the series
17. Find the locus of poles of ale Laugets to with respect to
18. Derive the polar equation of a comic.

 

PART – C

Answer any TWO questions:                                                                             ( 2 x 20 = 40 marks)

 

19. a) If prove that
20. b) Show that r = a sec² and r = b cosec² intersect at right angles.
21. a) Find the minimum value of
22.        b) Find the radius of  curvature of .
23. a) Solve: given that the roots are in geometric progression.

 

1. b) Solve: .

 

22. a) Express cos8q in locus of power of sinq.

 

1. b) If e₁ and e₂ are the eccentricities of a hyperbola and its conjugate show that .

 

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