M.E.S ASMABI COLLEGE, P.VEMBALLUR

FIRST INTERNAL EXAMINATION JANUARY 2019

SIXTH SEMESTER B.Sc MATHEMATICS

MAT 6B 12 –NUMBER THEORY AND LINEAR ALGEBRA

Time: 2 hrs Maximum: 50 Marks

Section-A

Answer all thethreequestions.

Each question carries 1 mark

1. Find gcd (143,227).

2. Find the canonical form of 360.

3. Express 109 in binary system.

Section-B

Answer any Fourquestions from among the questions 6 to 12.

Each question carries 5 marks

4. Use the Euclidean Algorithm to obtain integers x and y satisfying gcd (56, 72)= 56x+72y.

5. Prove that is irrational.

6. Find lcm(306, 657).

7. Find the remainder when is divided by 7.

8. Find remainder when 1!+2!+…+99!+100! Is divided by 12.

9. If p be a prime number then find Ф(p^k).

Section-C

Answer any two questions from among the questions 13 to 16.

Each question carries 6 marks

10.Given integers a and b, not both of which are zero, prove that there exists integers x and y such that gcd (a, b)=ax+by.

11. State and prove Euclid Theorem.

12. Using sieve of Eratosthenes find the number of prime not exceeding 50.

Section-D

Answer any one question from among the questions 17 to 19.

Question carries 15 marks

13. State and prove Division Algorithm. Illustrate with an example.

14. State and prove Chinese Remainder Theorem.

15. (a) State and prove Wilsons theorem.

(b) if n is an odd integer then prove that Ф(2n)=Ф(n).

M.E.S ASMABI COLLEGE, P.VEMBALLUR

SECOND INTERNAL EXAMINATION JANUARY 2019

SIXTH SEMESTER B.Sc MATHEMATICS

MAT 6B 13 (E 02) –LINEAR PROGRAMMING

Time: 2 hrs Maximum: 50 Marks

Section-A (Answer all the questions. Each question carries 1 mark) ( 8 x 1 = 8 )

1.Define convex hull of a set.

2. Define slack and surplus variables.

3. Reduce the following LPP to its standard form:

Maximize Z = x₁ – 3x₂

Subject to -x₁ + 2x₂ ≤ 15

x₁ + 3x₂ = 10

x₁, x₂ ≥ 0

4.True or False : The union of two convex sets is convex.

5. What is the optimality condition for the BFS of a maximization LPP to be optimal ?

6. In a mathematical programming problem, the function to be maximized or minimized is called an _____.

7. What is the standard form of an LPP ?

8. Define a convex set in Rⁿ.

Section-B (Answer all the six questions. Each question carries 2 marks) ( 6 x 2 = 12 )

6.Prove that a hyperplane in Rⁿ is convex.

7.Show that the set K = {(x,y) : x+ 7y = 12} is convex.

8.Define a half space in Rⁿ.

9. When is Charne’s method used to solve a linear programming problem? Define penalty.

11. Find all the basic solutions of the system : x₁ + 2x₂ + x₃ = 4, 2x₁ + x₂ + 5x₃ = 5.

12. Solve graphically, Maximize Z = 2x₁ + 3x₂ subject to the constraints x₁ + x₂ ≤ 1, 3x₁+ x₂ ≤ 4,

x₁ , x₂ ≥ 0 .

Section-C (Answer any four questions. Each question carries 5 marks) ( 4 x 5 = 20 )

13.Use simplex method to solve Maximize Z = 2x₁ + 3x₂

Subject to x₁ + x₂ ≤ 1

3x₁ + x₂ ≤ 4

x₁, x₂ ≥ 0

14. Use graphical method to solve Maximize Z = 5x₁ + 7x₂

Subject to x₁ + x₂ ≤ 4

3x₁ + 8x₂ ≤ 24

10x₁ + 7x₂ ≤ 35

x₁, x₂ ≥ 0

15. Prove that the set of all convex combinations of a finite number of vectors x₁, x₂, ...,x_k in Rⁿ is a convex set.

16. Explain Charne’s Big-M method.

17. Let A⊆Rⁿ be any set. Prove that <A> , the convex hull of A, is the set of all finite convex combinations of vectors in A.

18. Show that S = (x₁, x_2, x₃) : 2x₁ - x₂ + x₃ ≤ 4 in R³ is convex.

Section-D (Answer any one question. Each question carries 10 marks) ( 1 x 10 = 10 )

19. Use Two-Phase method to Maximize Z = 3x₁ + 2x₂

Subject to 2x₁ + x₂ ≤ 2

3x₁ + 4x₂ ≥ 12

x₁, x₂ ≥ 0

20.Use Charne’s method to solve Minimize Z = 4x₁ + 2x₂

Subject to 3x₁ + x₂ ≥27

x₁ + x₂ ≥ 21

x₁ +2 x₂ ≥30

x₁, x₂ ≥ 0

M.E.S . ASMABI COLLEGE, P. VEMBALLUR

First internal examination, August 2018

Third semester Mathematics (Complementary course)

STS 3C O3-STATISTICAL INFERENCE

Time: 2hrs Total Marks: 50

Part A (Answer all questions. Each carries 1 mark) (1x6=6)

1. What is the variance of a chi-square distribution with 2 df ?

2. If t₁ and t₂ are two estimators such that v(t₁)=v(t₂), then ............

3. Factorization theorem for sufficiency is known as .........

4. Mgf of chi-square distribution is.............

5. Give the pdf of t distribution with 2 df.

6. Define unbiasedness of an estimator.

Part B (Answer all questions. Each carries 2 marks) (4x2=8)

7. How we compare the efficiencies of two estimators?

8. Let be a random sample from N( ). Give a point estimator of .

9. What are the properties of MLE.

10. Give one example of a statistic following F distribution.

Part C (Answer any two questions. Each carries 4 marks) (2x4=8)

11. Define chi-square and obtain its mean and variance.

12. Describe the desirable properties of good estimator.

13. Explain the method of moment estimation.

14. Define sampling distribution.

Part D (Answer any three questions. Each carries 6 marks) (3x6=18)

15. Obtain the distribution of the sample mean from normal population.

16. Examine whether the sample variance is an unbiased estimator of the population variance for a normal population.

17. State the interrelation among Normal, Chi-square, t and F distribution.

18. Define t-distribution. State its properties.

19. Let , ......, is a random sample from normal distribution N(µ,1). Show that t = is an unbiased estimator of +1.

Part E (Answer any one questions. Each carries 10 marks) (1x10=10)

20. Let , ......, be a random sample from normal distribution N( ). Find the MLE of µ and .

21. Examine whether is an unbiased, consistent and sufficient estimator of µ in case of a N( ), is known.

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M.E.S ASMABI COLLEGE, P.VEMBALLUR

I ST INTERNAL EXAMINATION AUGUST- 2018

III SEMESTER B.Sc. MATHEMATICS (CORE COURSE)

MM3B03 - ANALYTIC GEOMETRY AND CALCULUS

Time: 2 hrs Max. Mark: 50

PART-A (Answer all the questions) ( 8 x 1= 8)

1. Write the nth term of the sequence 0, -1/2, 2/3, -3/4, -------.

2. Define the convergence of a sequence.

3. State True / False : The sequence { (-1)ⁿ } is divergent.

4. Write the least upper bound for the sequence 1, -1, 1, -1,….,(-1)ⁿ⁺¹,….

5. The directrix of x ² = - 8y is ---

6. The eccentricity of 9 x ² + 10 y ² = 90 is ---

7. Directrices of 8 y ² - 2 x ² = 16 are ---

8. Identify the graph of x ² - 3 xy + 3 y ² + 6y - 7 = 0

PART-B (Answer all the questions) (6 x2 =12)

9. Show that the sequence (0, 2, 0, 2,….., 0, 2,….) does not converge to 0.

10.Show that = 2.

11. State Sandwich theorem and show that {} converges to 0.

12. Find the equation of the tangent to the curve x = 4 sin t , y = 2 cos t at t =

13. Replace the polar equation r = by equivalent Cartesian equation..

14. Find the polar equation of the hyperbola with eccentricity 3/2 and x = 2 .

PART-C (Answer any 4 questions) (4 x 5 =20)

15. Does the sequence whose nth term is u_n = converge? If so, find u_n .

16. Show that the geometric series a + ar + ar² +….+ ar^n-1+…. Converges if | r | < 1 and diverges if | r | 1.

17. Show that the series diverges.

18. The coordinate axes are to be rotated through an angle to produce an equation for the curve 2 x ²

+ xy + y ² – 10 = 0 and has no cross product term. Find and the new equation.

19. Find the length of the asteroid x = cos ³ t , y = sin ³ t , 0t /2

20. Graph the cardioid r = 1 – cos

PART-D (Answer any one question) (Weightage: 1 x 10 = 10)

21. Using the Integral test, show that the p-series = + + +….+ +….(p a real constant) converges if p1 diverges if p1.

22. Find the area of the surface of the solid formed by the revolution of the curve x = (2/3) ,

y = 2 , 0 t about y axis.

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M ES ASMABI COLLEGE, P.VEMBALLUR

First Internal Examination, August 2018

3^rd semester Physics (Complimentary Course)

MM3C03-Mathematics

Time: 2 hrs Marks: 50

Part A (Answer all questions. Each carries 1 mark) (8 x 1 = 8)

1. The degree of the differential equation is ----------.

2. Write the order of the differential equation y’’’ + 2(y’’) ² – y’ + y = 0.

3. Check whether y= is solution of the differential equation y ’ = 2y?

4. Solve the differential equation 10yy ’ + 3x = 0.

5. When are two vectors a and b said to be orthogonal?

6. What is the divergence of a =[ 3xz , 2xy , -yz ² ]

7. Prove that div r = 3.

8. When curl of a vector point function f is a zero vector , then f is called .....

Part B (Answer all questions 2 marks each) (6 x 2 = 12)

9. Solve the differential equation + 2xy = 0.

10. Define Orthogonal Trajectories.

11. .Solve the initial value problem y’ + 5y = 20, y(0) = 2.

12. Prove that Curl ( grad f ) = 0.

13. If v = 6x² zi + 2x² yj - yz² k, find div v.

14. Find the work done by p = [2,6,6] if it displaces a body from A = (3,4,0) to B = (5,8,0).

Part C (Answer any 4questions. 5 mark each) (4 x 5 = 20)

15. Solve: xy’ + y = xy³.

16. Find the Orthogonal Trajectories of the family of curves x² + y² = c² .

17. Show that the equation (1 + 4xy + 2y²) dx + (1 + 4xy + 2x²) dy = 0 is exact &solve it.

18. Find the unit normal to the surface x ² y + 2xz = 4 at the point ( 2, 2 , 3 ).

19. Prove that div(fv)=f div v+ v. grad f, where f is a scalar function.

20. Find the equation for the plane determine by the points (2,-1, 1), (3, 2, -1) and (-1, 3, 2).

Part D (Answer any 1 question) (1 x 10 = 10)

21. Solve .

22. Find the angle between the surfaces x² + y² + z² = 9 and z = x ² + y² - 3 at the point (2,-1,2).

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M.E.S ASMABI COLLEGE, P.VEMBALLUR

FIRST INTERNAL EXAMINATION AUGUST- 2018

FIFTH SEMESTER B.Sc.MATHEMATICS (CORE COURSE)

MAT5B 05 – VECTOR CALCULUS

Time: 2 hrs Max. Mark: 50

PART-A (Answer all the questions) (8 x 1 = 8)

1. Define the gradient of f(x,y).

2. Define the Linearization of f(x,y) at the point ( , ).

3. State Euler’s mixed derivative theorem.

4. Define the continuity of a function at a point.

5. Define the Level Surface of a function.

6. State first derivative test for local extreme values.

7. State Orthogonal gradient theorem.

8. Define critical point of a function.

PART-B (Answer all the questions) (6 x 2 = 12)

9. Find dy/dx if x + sin xy + y – ln 2 = 0.

10. Find , and if f (x,y,z) = ln ( x + 2y + 3z ).

11. Show that f(x,y) = x / have no limit as (x,y) .

12. Evaluate .

13. Find the domain and range of f(x,y,z) = xy ln z.

14. Find the local extreme values of the function f (x,y) = x² + y².

PART-C (Answer any four questions) (4 x 5 = 20)

15. Show that the function f(x,y) = , (x,y) and equal to 0 when (x,y) = (0,0) , is continuous at every point except at origin.

16. Find the tangent plane and normal line of the surface f(x,y,z) = at the point (1,1,1).

17. Find the derivative of f(x,y) = x + cos (xy) at the point (2,0) in the direction of the vector i + j .

18. If w = and z = , evaluate the following: (i) (ii) (iii) (iv) .

19. Find points closest to the origin on the hyperbolic cylinder x² – z² – 1 = 0.

20. Find positive numbers x, y, z such that x + y + z = 18 and xyz is a maximum.

PART-D (Answer any one question) (1 x 10 = 10)

21. Find the Linearization of f(x,y,z) = xz – 3yz + 2 at the point (1,1,2). Find an upper bound for the error on the region R: | x – 1 | , | y - 1| , | z - 2 | .

22. Using Taylor’s formula find a cubic approximation of the function f(x,y) = sin x sin y near the origin. Estimate the error in the approximation if | x | 0.1 and | y | 0.1 .

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M.E.S ASMABI COLLEGE, P.VEMBALLUR

FIRST INTERNAL EXAMINATION AUGUST- 2018

FIFTH SEMESTER B.Sc.MATHEMATICS (CORE COURSE)

MAT5B 06 – ABSTRACT ALGEBRA

Time: 2 hrs Max. Mark: 50

PART-A (Answer all the questions) (5 x 1.5= 7.5)

1. Find the sum of 23 & 31 modulo 45

2. Define isomorphism

3. Write down the nontrivial proper subgroup of Z₄

4. Define cyclic group

5. Define permutation group

PART- B (Answer all the questions) (5 x 2.5 = 12.5)

6. Find all generators of Z₁₂

7. Prove that every cyclic group is abelian

8. State and prove the reversal law

9. Find all residue classes modulo 5

10. show that (Z₄,+₄) is isomorphic to (U₄ , .)

PART-C (Answer any three questions) (3 x 6 = 18)

11. show that the group Z₄ is cyclic while the Klein 4 group is not.

12. Prove that a subgroup of a cyclic group is cyclic.

13. Find all subgroups of Z₁₈ and draw the lattice diagram. Also find all generators of each subgroup.

14. If = & = find

15. Prove that HK is a subgroup of G if and only if H K or K⊆ H

PART-D (Answer any one question) (1 x 12= 12)

16. Let G be a group and a G. then prove that H = {aⁿ : n Z} is a subgroup of G and is the smallest subgroup of G that contains a .

17. (a). Let (G,*) and (G,*’) be two isomorphic groups and be a group isomorphism. Show that if G is cyclic, then G’ is also cyclic.

18. (b). Show that the infinite cyclic group has exactly two generators.

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M.E.S ASMABI COLLEGE, P.VEMBALLUR

FIRST INTERNAL EXAMINATION AUGUST- 2018

FIFTH SEMESTER B.Sc.MATHEMATICS (CORE COURSE)

MAT5B 07 BASIC MATHEMATICAL ANALYSIS

Time: 2 hrs Max. Mark: 50

PART-A (Answer all the questions) (8 x 1= 8)

1. State well ordering property of N

2. If f( x ) = x ³ and g ( x ) = sin x find gof

3. State True or False “ Let f : X Y and let A and B be any two subsets of X , then f ( A B )=

f(A) f ( B )”.

4. Define an neighbourhood of a.

5. Find the supremum and infimum of the set { 1 – 1/n: n N}

6. =...

7. Give an example of a monotone sequence.

8. =...

PART-B (Answer all the questions) (4 x 2= 12)

9. Show that is not a rational number.

10. State and prove Bernoulli’s inequality

11. State Archemedian property.

12. Prove that lim(bⁿ) = 0 if 0 b 1

13. Show that lim = 0

14.If a R and a 0, then show that a ² > 0.

PART-C (Answer any three questions) (3 x 5= 15)

15.Prove that =.

16.Prove that convergent sequences are bounded.

17. State and prove squeeze theorem

18. State and prove monotone convergence theorem

19. State and prove Cauchy’s inequality.

PART-D (Answer any one question) ( 1 x 10 = 10)

20. State and prove Bolzano Weistrass theorem

21. State and prove Nested Interval property.

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M.E.S ASMABI COLLEGE, P.VEMBALLUR

FIRST INTERNAL EXAMINATION AUGUST- 2018

FIFTH SEMESTER B.Sc.MATHEMATICS (CORE COURSE)

MAT5B08 - DIFFERENTIAL EQUATIONS

Time: 2 hrs Max. Mark: 50

PART-A (Answer all the questions) ( 8 x 1= 8)

1. What is the order and degree of y’’’ +2(y’’)² = e^t.

2. Write the general form of a first order linear differential equation .

3. Check whether y= is solution of the differential equation y ’ = 2y?

4. Solve the differential equation 10yy ’ + 3x = 0.

5. Find L^-1[]

6. Define Laplace transform of a function .

7. What is the fundamental period of sin 7t?

8. The function x | x | is … function. a) even b) odd c) both d) neither

PART-B (Answer all the questions) ( 6 x 2= 12)

9. Find the integrating factor & solve y’+2y = 6e^2t

10. Solve y’ = (1+ x)(1+ y²)

11. Show that the differential equation of all parabolas having x-axis as the axis of symmetry is

y (d²y/dx²) + (dy/dx)² = 0.

12. Show that the sum of two even functions is even.

13. Find Laplace transformation of sinh at .

14. Find

PART-C (Answer any three questions) ( 4 x 5= 20)

15. Solve: Sin 2t = y + tan t .

16. By the method of variation of parameters solve y’- y = t .

17. Solve .

18. Find the inverse transform of .

19. If f(t) = t sin at , find L[f(t)].

20. Solve y’ +3y = 10 sin t , y ( 0 ) = 0, by Laplace transforms

PART-D (Answer any one question) ( 1 x 10 = 10)

21. Solve: + t sin 2y = t³ cos²y.

22. Using Laplace transforms solve

y’’ – 3y’ + 2y = 4t , y(0) = 1 , y’(0) = -1.

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M.E.S ASMABI COLLEGE, P.VEMBALLUR

FIRST INTERNAL EXAMINATION AUGUST- 2018

FIFTH SEMESTER MATHEMATICS (OPEN COURSE)

MAT5D19 - MATHEMATICS FOR SOCIAL SCIENCES

Time: 1 hrs Max. Mark: 25

PART-A (Answer any 5 the questions) ( 5 x 1= 5)

1. Find the x- intercept of y = 9x – 72.

2. Solve = 2.

3. If f(x ) = x ³ + 5x – 6 , then f( 3 ) is …

4. Evaluate .

5. If y = , find .

6. Solve 4 x ² - 1 = 0.

PART-B (Answer any 3 questions) ( 3 x 2= 6)

7. Find the slope of the line joining ( 1,7 ) and ( 5, 15 ).

8. Solve 11 x ² + x – 12 = 0.

9. Find the equation of the line passing through ( -2 ,5 ) and parallel to y = 3x + 7 .

10. Differentiate x ³ + 5x – 6

11..Find the equation of the line passing through the origin and slope 4 .

PART-C (Answer any 3 questions) ( 3 x 3= 9)

12. Show that the set of points (-1 ,6 ) , ( -10 ,12 ) , ( -16 , 16 ) are collinear

13. Graph the function f (x ) = 5 – x ²

14. Examine the continuity of f (x ) = 14x + 6 at x = 3 .

15. Given y = x ² e ^x , find .

16. Write down the equation of the line passing through (1,-2) and perpendicular to 3x – 7y =5.

PART-D (Answer any one question) ( 1 x 10 = 10)

17. a) If y = Find .

b) .

18. Find the length of the medians of a triangle whose vertices are ( 1,2) , ( 2,-1) , ( 3,4) .

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M.E.S ASMABI COLLEGE, P.VEMBALLUR

II ND INTERNAL EXAMINATION OCTOBER - 2018

V SEMESTER B.Sc.MATHEMATICS (CORE COURSE)

MM5B07 – BASIC MATHEMATICAL ANAYSIS

Time:2 hrs Max.Marks: 50

PART-A (Answer all the questions) ( 8 x 1 = 8)

1. ____ is an example of a bounded sequence which is not convergent .

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M.E.S ASMABI COLLEGE, P.VEMBALLUR

SECOND INTERNAL EXAMINATION OCTOBER- 2018

FIFTH SEMESTER B.Sc.MATHEMATICS (CORE COURSE)

MAT5B05-VECTOR CALCULUS

Time: 2 hrs Max. Mark: 50

PART-A (Answer all the questions) (8 x 1= 8)

1. State Fubini’s theorem (first form).

2. Define a line integral.

3. Write the formula for radius of gyration about a line.

4. Define a vector field.

5. .

6. Define gradient field.

7. Define flow integral and circulation.

8. Define flux across a plane curve.

PART-B (Answer any four questions) (4 x 3 = 12)

9. Find the area enclosed by the lemniscate = 4 cos 2 .

10. Find the volume of the ice cream cone D cut from the solid sphere by the cone .

11. Find the line integral of f (x, y, z) = x + y + z over the straightline segment from (1,2,3) to (0,-1,1).

12. Evaluate dz dy dx .

13. Find the work done by the force F = ( y - along the curve C : r(t) = t i + t² j + t³ k from (0,0,0) to (1,1,1).

14. Find the flux of F = ( x - y across the circle x² + y² = 1 in the xy – plane .

PART-C (Answer any four questions) (4 x 5= 20)

15. Evaluate .

16. Change the order of integration and evaluate .

17. Using triple integral find the volume of the prism whose base is the triangle in the xy – plane bounded by

y = 0, y = x, x = 1 and x + y + z = 3.

18. Find the area which lies inside the cardioid r = 1 + cos and outside the circle r = 1.

19. A slender metal arch denser at the bottom than top lies along the semi circle in the yz – plane. Find the centre of the arch’s mass if the density at a point (x, y, z) on the arch is .

20. Find the average value of f(x,y,z) = x cos(xy) over the rectangle 0 .

PART-D (Answer any one question) ( 1 x 10 = 10)

21. Evaluate by applying transformation u = and v = .

22. (a) Find gradient field of f, if f (x, y, z) = .

(b) Find the circulation of F = 2x i + 2z j + 2y k around the closed path consisting of the following 3 curves transversed in the direction of increasing t : : r(t) = cos t i + sin t j + t k , 0 t

: r(t) = j + (1-t) k , 0 t and : r(t) = t i + (1-t) j , 0 t

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M.E.S ASMABI COLLEGE, P.VEMBALLUR

II ND INTERNAL EXAMINATION OCTOBER - 2018

V SEMESTER B.Sc.MATHEMATICS (CORE COURSE)

MM5B07 – BASIC MATHEMATICAL ANAYSIS

Time:2 hrs Max.Marks: 50

PART-A (Answer all the questions) ( 8 x 1 = 8)

1. ____ is an example of a bounded sequence which is not convergent .

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M.E.S ASMABI COLLEGE, P.VEMBALLUR

SECOND INTERNAL EXAMINATION OCTOBER- 2018

FIFTH SEMESTER B.Sc.MATHEMATICS (CORE COURSE)

MAT5B05-VECTOR CALCULUS

Time: 2 hrs Max. Mark: 50

PART-A (Answer all the questions) (8 x 1= 8)

1. State Fubini’s theorem (first form).

2. Define a line integral.

3. Write the formula for radius of gyration about a line.

4. Define a vector field.

5. .

6. Define gradient field.

7. Define flow integral and circulation.

8. Define flux across a plane curve.

PART-B (Answer any four questions) (4 x 3 = 12)

9. Find the area enclosed by the lemniscate = 4 cos 2 .

10. Find the volume of the ice cream cone D cut from the solid sphere by the cone .

11. Find the line integral of f (x, y, z) = x + y + z over the straightline segment from (1,2,3) to (0,-1,1).

12. Evaluate dz dy dx .

13. Find the work done by the force F = ( y - along the curve C : r(t) = t i + t² j + t³ k from (0,0,0) to (1,1,1).

14. Find the flux of F = ( x - y across the circle x² + y² = 1 in the xy – plane .

PART-C (Answer any four questions) (4 x 5= 20)

15. Evaluate .

16. Change the order of integration and evaluate .

17. Using triple integral find the volume of the prism whose base is the triangle in the xy – plane bounded by

y = 0, y = x, x = 1 and x + y + z = 3.

18. Find the area which lies inside the cardioid r = 1 + cos and outside the circle r = 1.

19. A slender metal arch denser at the bottom than top lies along the semi circle in the yz – plane. Find the centre of the arch’s mass if the density at a point (x, y, z) on the arch is .

20. Find the average value of f(x,y,z) = x cos(xy) over the rectangle 0 .

PART-D (Answer any one question) ( 1 x 10 = 10)

21. Evaluate by applying transformation u = and v = .

22. (a) Find gradient field of f, if f (x, y, z) = .

(b) Find the circulation of F = 2x i + 2z j + 2y k around the closed path consisting of the following 3 curves transversed in the direction of increasing t : : r(t) = cos t i + sin t j + t k , 0 t

: r(t) = j + (1-t) k , 0 t and : r(t) = t i + (1-t) j , 0 t

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M.E.S ASMABI COLLEGE, P.VEMBALLUR

SECOND INTERNAL EXAMINATION OCTOBER 2018

FIFTH SEMESTER B.Sc.MATHEMATICS (CORE COURSE)

MAT5B 06 – ABSTRACT ALGEBRA

Time: 2 hrs Max. Mark: 50

PART-A (Answer all the questions) (8 x 1 = 8)

1. What is the order of the cycle (1,4,5,7) in S₈?

2. Find all cosets of the subgroup 4Z of 2Z.

3. The index of < 3 > in Z₂₄ is ...........

4. Define kernel of a homomorphism.

5. Write all units in the ring Z₄.

6. Define orbits of a permutation.

7. Define a permutation of a set.

8. Define Index of a subgroup?

PART- B (Answer all the questions) (6 x 2 = 12)

9. Give an example to show that multiplication of permutations is not commutative..

10. Let be a homomorphism of a group G into a group G^’ ; if e is the identity element in G, then prove that (e) is the identity element e^’ in G^’.

11. Let be a homomorphism of a group G into a group G^’. if a G then prove that (a^{- 1}) =[]^{- 1}

12. Prove that every group of prime order is cyclic.

13. Let be a homomorphism of a group G into a group G^’. If G is abelian, then prove that G’ is also abelian.

14. If = & = find

PART-C (Answer any FOUR questions) (4 x 5 = 20)

15. State and prove Lagrange’s theorem.

16. If n 2, then prove that the collection of all even permutations of {1,2,3...........n} forms a subgroup of order n/2 of the symmetric group S_n.

17. Prove that (Z₅,+_5, x₅) is a ring.

18. Every permutation is a product of disjoint cycles.

19. Let be a homomorphism of a group G into a group G^’_.if K’ is a normal subgroup of [G], then prove that ^{- 1}[K’] is a normal subgroup of G.

20. Find all cosets of the subgroup (2) in Z₁₂.

PART-D (Answer any one question) (1 x 10 = 10)

21. Let be a homomorphism of a group G into a group G^’ and let H = ker. Let a G. then prove that [{}] = {x G / } is the left coset aH of H and is the right coset Ha of H.

22. State and prove Cayley’s Theorem.

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M.E.S ASMABI COLLEGE, P.VEMBALLUR

FIRST INTERNAL EXAMINATION OCTOBER- 2018

FIRST SEMESTER B.Sc.MATHEMATICS (CORE COURSE)

MAT1B01-FOUNDATIONS OF MATHEMATICS

Time: 2 hrs Max. Mark: 50

PART-A (Answer all the questions) (8 x 1= 8)

1. A = ?

2. State De Morgan’s law.

3. Find x, y if (x+2, 4) = (5, 2x+y)

4. Give an example of an equivalence relation.

5. If f (x) = x² + 2x . Then (f o f) (2) is .....

6. Find the domain of the real valued function f (x) = 1/(x-2) .

7. Find .

8. The graph of y = x² is shifted to 2 units to the left and 2 units up, write the equation of the new graph.

PART-B (Answer all the questions) (6 x 2= 12)

9. What are different types of relations. If A={1,2,3} find a relation which is both symmetric and anti symmetric .

10. Define partial order relation on a set. Show that the relation “divides” is not a partial order relation on the set of integers.

11. f(x) = x²+3x+1 , g(x) = 2x-3 . Find gof and fog .

12. Find the centre and radius of the circle x² + y² + 4x – 6y – 3 = 0.

13. Evaluate .

14. If and find

PART-C (Answer any four questions) (4 x 5= 20)

15. Prove that (A B)\ (A B) = (A \ B)(B \ A)

16. Given the relation R on set of positive integers as x+ 3y = 12. Write R as a set of ordered pairs. Draw directed graph of R and find RoR.

17. Let A = {1,2,3,4} , B = {3,4,5,6,7} , C = {6,7,8,9}. Find A B and B C.

18. State Sandwich theorem and show that if then .

19. Show that if f(x) = x² , x 2 and f(x) = 1, x = 2.

20. Graph the parabola y = (- x²/2 )– x + 4 . Label the vertex, axis and intercepts .

PART-D (Answer any one question) ( 1 x 10 = 10)

21. a) Show that x ≡ y(mod m) is an equivalence relation on Z

b) Let A = {1,2,3,......,14,15} and R is an equivalence relation on A given by

congruence modulo 4. Find equivalence classes determined by R.

22. If f(x) = x , x < 1 and f(x) = x+1 , x > 1 then show that (a)

(b) (c) .