LOYOLA COLLEGE (Autonomous), chennai – 600 034

B.Sc.  degree examination – physics

third semester -april 2003

 Mt  3100/ MAT 100 mathematics for physics

28.04.2003                                                                                     Max.: 100 Marks

9.00 – 12.00

 

PART – A                                       (10 ´ 2 = 20 Marks)

1. Define Laplace transform of f(t) and prove that L(e^–at) = .
2. Find .
3. Prove that the mean of the Poisson distribution P_r =, r = 0, 1, 2, 3 ….. is equal to m.
4. Mention any two significance of the normal distribution.
5. Find the .
6. Find L (1+ t)² .
7. Find L^-1.
8. Write down the real part of sin .
9. Prove that in the R.H. xy = c², the subnormal varies as the cube of the ordinate.
10. If y = log (ax +b), find y

 

PART – B                                          (5 ´ 8 = 40 Marks)

Answer any FIVE questions.  Each question carries EIGHT marks

11. (a) Find L^-1

           

• Find L .

12. Find L  .
13. Define orthoganal matrix and prove that the matrix is orthoganal.
14. Verify cayley-Hamilton theorem and hence find the inverse of
15. (i)  Prove that

• Find the sum to infinity of series

 

 

 

 

 

16. (i) Find q approximately to the nearest minute if cos q =

(ii)   Determine a, b, c such that     .

 

17. If cos (x + iy) = cos q + i sinq,  Show that cos 2x + cosh 2y =2.

 

18. What is the rank of .

 

PART – C                                      (2 ´ 20 = 40 Marks)

Answer any TWO questions. Each question carries twenty marks.

 

19. (a) If y = .

 

• Find the angle of intersection of the cardioids r = a(1+cosq) and r = b(1-cosq).

20. (a) Certain mass -produced articles of which 0.5 percent are defective, are packed

in  cartons each containing 130 article.  What Proportion of cartons are free

from defective articles, and what proportion contain 2 or more defectives

(given e^-2.2 = 0.6065)

 

• Of a large group of men 5 percent are under 60 inches in height and 40 percent are between 60 and 65 inches. Assuming a normal distribution find the mean height and standard deviation.

 

21. (a) Find the sum to infinity of the series

 

• From a solid sphere, matter is scooped out so as to form a conical cup, with vertex of the cup on the surface of the sphere, Find when the volume of the cup is maximum.

 

22. a) Prove that sin⁵q =
23. b) Prove that sin⁴q cos²q =

 

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – PHYSICS

 

AB 03

THIRD SEMESTER – November 2008

MT 3102/MT 3100 – MATHEMATICS FOR PHYSICS

 

 

 

Date : 11-11-08                       Dept. No.                                          Max. : 100 Marks

Time : 9:00 – 12:00

Section A

Answer ALL questions:                                                                                                  (10 x 2 = 20)

1. If   then show that .
2. Prove that the subtangent to the curve  is of constant length.

 

1. Show that

 

4. Find the rank of the matrix .
5. Find .

1. Find .
2. Write the expansion of tan nq in terms of tanq.
3. Prove that cosh²x – sinh²x = 1.
4. A bag contains 3 red, 6 white, 7 blue balls. What is probability that two balls drawn are white and blue balls?
5. A Poisson variate X is such that 2 P(X = 1) = 2P(X =2). Find the mean.

 

Section B

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

11. Find the derivative of .
12. Find the maxima and minima of .
13. Prove that .
14. Find .
15. Ifprove that .
16. Expand in terms of cosq.
17. Find the mean and standard deviation for the following data:

 

Years under	10	20	30	40	50	60
No. of people	15	32	51	78	97	109

18. X is normally distributed with mean 12 and standard deviation 4. Find the probability of the following:

(i)            X ³ 20             (ii)  X £ 20             (iii) 0 £ X £ 12.

given that z_2.0 = 0. 4772, z_3.0 = 0. 4987, z_4.0 = 0. 4999.

 

Section C

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

 

19. (a) Find the sum of the series to infinity:

(b) If   then prove that  and hence prove .                                          (10 +10) 

 

20. (a) Find the characteristic roots and characteristic vectors of the matrix

.

(b)  Verify Cayley Hamilton Theorem for matrix .

(12+8)

21. (a) Find the Laplace transform of

 

(b) Using Laplace transform, solve the equation y¢¢ + 2y¢ – 3y = sin t, given that y = y¢ = 0 when t = 0.

(8+ 12)

22. (a) Expand sin³qcos⁵q in a series of sines of multiplies of q.

 

(b) In the long run 3 vessels out of every 100 are sunk. If 10 vessels are out, what is

the probability that (i) exactly 6 will arrive safely.  (ii) at least 6 will arrive safely.

(10 +10)

 

 

 

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      LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – PHYSICS

ZA 06

THIRD SEMESTER – April 2009

MT 3102 / 3100 – MATHEMATICS FOR PHYSICS

 

 

 

Date & Time: 17/04/2009 / 1:00 – 4:00      Dept. No.                                                       Max. : 100 Marks

 

 

SECTION A

Answer ALL questions:                                                                   (10 x 2 = 20)

1. Find the n th derivative of
2. Find the polar subtangent and subnormal for the curve .
3. Prove that
4. Find the rank of the matrix
5. Find .
6. Find .
7. Expand .
8. Prove that
9. A letter of English alphabet is chosen at random. Find the probability that the letter so chosen follows m and is a vowel.
10. With the usual notation state Poisson distribution.

 

SECTION B

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

11. Find the nth derivative of .
12. Find the angle of intersection of the curves and .
13. Find the sum to infinity the series
14. Find the inverse Laplace transform of

1. If prove that .

16. Expand in terms of cosq.

1. 25 books are placed at random in a shelf. Find the probability that a pair of books shall be always together.
2. If two dice are thrown, what is the probability that the sum is greater than 8.

 

SECTION C

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

 

19. (a)If prove that

(b) Find the maxima and minima of .                   (10+10)

20. (a) Find the characteristic roots and the associated characteristic vectors of the matrix

 

(b) Verify Cayley Hamilton Theorem for matrix  .

(12+8)

• (a) Express in a series of cosines of multiples of θ.

(b) Find .

(c) Separate into real and imaginary parts of .                                  (5+5+10)

22  (a) Solve , given  and .

(b) A coin is tossed six times. What is the probability of obtaining four or more heads?

 

(12+8)

 

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – PHYSICS

THIRD SEMESTER – APRIL 2012

MT 3102/3100 – MATHEMATICS FOR PHYSICS

 

 

Date : 28-04-2012              Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

Section A

Answer ALL questions:                                                                                 (10 ´ 2 = 20)

1. Find the n^th derivative of e^4x.
2. Show that in the curve rq = a, the polar sub tangent is constant.
3. Expand in ascending powers of x, ‘a’ being positive.
4. Define a symmetric matrix and give an example.

1. Find the Laplace transform of t² + 2t + 3.

6. Find .
7. Prove that .
8. Write down the expansion of and  in a series of ascending powers of .
9. Two dice are thrown. What is the probability that the sum of the numbers is greater than 8?
10. Write a short note on binomial distribution.

 

Section B

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

11. Find the n^th differential coefficient of sinx sin2x sin3x.

 

12. Find the angle of intersection of curves rⁿ = aⁿcosnq and rⁿ = aⁿ sinnq.

 

13. Show that .
14. Show that the matrix is orthogonal.

 

15. Find the Laplace transform of

 

16. Separate into real and imaginary parts of .
17. Prove that .

 

18. Calculate the mean and standard deviation for the following frequency distribution:

Class Interval	0 – 8	8 – 16	16 – 24	24 – 32	32 – 40	40 – 48
Frequency	8	7	16	24	15	7

 

Section C

 

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

 

19. a) If y = acos(logx) + bsin(logx), prove that x²y_{n + 2} + (2n + 1)xy_{n + 1} + (n² + 1)y_n = 0.
20. b) Find the sum to infinity of the series .

(12 + 8)

20.a) Find the characteristic roots of the matrix .

1. b) Verify Cayley Hamilton Theorem for matrix and also find .                                                                                                              (6 + 14)
2. a) Find .
3. b) Solve the equation given that when t = 0.

(5 + 15)

22. a) Ifprove that .
23. b) Express in a series of sines of multiples of θ.
24. c) A car hire firm has two cars, which it hires out day by day. The number of demands for a car on each day is distributed as a Poisson distribution with mean 1.5. Calculate the proportion of days on which (i) neither car is used, and (ii) the proportion of days on which some demand is refused.                                                                                                                                            (8+5+7)

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – PHYSICS

FIRST SEMESTER – NOVEMBER 2012

MT 1100 – MATHEMATICS FOR PHYSICS

 

 

Date : 03/11/2012             Dept. No.                                        Max. : 100 Marks

Time : 1:00 – 4:00

 

 

SECTION A

ANSWER ALL THE QUESTIONS:                                                                                         (10×2 =20)

 

• Find the n^th derivative of.
• Write down the formula for subtangent and subnormal.
• Prove that .
• Find the rank of the matrix.
• Show that .
• State the formula for Laplace transformation of a periodic function.
• Write down the expansion for.
• If Show that
• What is the chance that a leap year selected at random will contain 53 Sundays?
• Define Binomial distribution.

 

SECTION B

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

 

• Find the angle of intersection of cardioidsand.
• Find the minimum and maximum value of the function.
• Find the sum to infinity series.
• Show that the system of equations

 

are consistent and solve them.

• Find the L(f(t)) if
• Find a)   b) .
• Prove that cos8θ = 1- 32sin² θ + 160sin⁴ θ-256sin⁶ θ+128 sin⁸ θ.
• Find the moment generating function for the Poisson distribution and hence find its

mean and variance.

 

SECTION C

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

• a) If then Prove that. b) Find the n^th derivative of. (10+10)
• If then

1. a) Find the characteristic value and characteristic vector of the matrix.
2. b) Verify Cayley Hamilton Theorem and find A^-1. (10+10)

• a) Express cos⁵θ sin³θ in terms of sines of multiples of θ.

1. b) Separate into real and imaginary parts of tan^-1(α+iβ). (10+10)

• a) Solve with using Laplace transform.

1. b) An urn contains 6 white, 4 red and 9 black balls. If 3 balls are drawn at random, find the probability that: (i) two of the ball drawn is white; (ii) one is of each colour,

(iii) none is red.

(14+6)

 

 

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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

B.Sc. DEGREE EXAMINATION – PHYSICS

THIRD SEMESTER – NOVEMBER 2012

MT 3102/3100 – MATHEMATICS FOR PHYSICS

 

 

Date : 07/11/2012             Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

SECTION A

ANSWER ALL QUESTIONS.                                                                               (10 x 2 = 20)

1. Find the n^th derivative of .
2. Find the slope of the curve at .
3. Write the expansion for .
4. Find the rank of the matrix .
5. Find the Laplace transform of .
6. Find .
7. Write down the expansion of in powers of .
8. Show that .
9. Two dice are thrown. What is the probability that the sum of the numbers is greater than 8?
10. Define Normal distribution.

 

SECTION B

ANSWER ANY FIVE QUESTIONS.                                                                 (5 x 8 = 40)

11. Find the n^th differential coefficient of .
12. Find the maximum value of for positive values of x.
13. Find the characteristic roots of the matrix .
14. Find the Laplace transform of .
15. Express in a series of sines of multiple of .
16. Four cards are drawn at random from a pack of 52 cards. Find the probability that

(i) they are a king, a queen, a jack and an ace.

(ii) two are kings and two are queens.

(iii) two are black and two are red.

17. A car hire firm has two cars, which it hires out day by day. The number of demands for a car on each day is distributed as a Poisson distribution with mean 1.5. Calculate the proportion of days on which (i) neither car is used, (ii) the proportion of days on which some demand is refused.
18. X is a normal variable with mean 30 and standard deviation 5. Find the probabilities that

(i) 26  X  40, (ii) X  45.

 

SECTION C

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

19. (a) If , then prove that .

(b) Find the length of subtangent and subnormal at any point t on the curve  and .                                                        (12 + 8)

20. (a) Verify Cayley-Hamilton theorem for the matrix and also find .

(b) Find the sum to infinity of the series . (12 + 8)

21. (a) Express in terms of .

(b) Find the mean and standard deviation for the following data:

x	10	20	30	40	50	60
f	15	32	51	78	97	109

(10 + 10)

22. (a) Solve the equation given that when .

(b) Ten coins are thrown simultaneously. Find the probability of getting at least seven

heads?                                                                                                                   (12 + 8)

 

 

 

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