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# Janardan Rai Nagar Rajasthan Vidyapeeth 2006 B.Tech MATHEMATICS-II - Question Paper

Tuesday, 14 May 2013 02:55Web

JRN Rajasthan Vidyapeeth University

B.Tech

SEMESTER 2

MATHEMATICS-II

DR

Model Question Paper

#### Time : 3 hrs Maximum marks : 75

Instruction:

1. Question paper is divided into Group A, Group B and Group C

2. Each Group is 25 Marks

3. Figure to the right in bracket indicates marks.

4. Good Handwriting is expected

5. Assume suitable data if necessary.

Group A (25 Marks)

Answer any three questions (Question 1 is compulsory)

 (5)

 (10)

Q.1 Prove that the inverse of a matrix is unique .

Q.2 (a) show that (10)

(b) (10)

Q.3 (a) Prove that E = 1 + (b) Prove that E = (1- )-1 ?

 (10)

Q.4 Derive cauchys integral formula ?

Q.5 Solve the difference equation

Un+2 16 Un = cos n P/2

Group B (25 Marks)

 (5)

Answer any three questions (Questions 6 is compulsory)

Q.6 Solve with the help of matrices the simultaneous equation x + y + z =3 ,

x +2y +3z = 4, x+4y+9z = 6

 (10)

Q.7 Find the inverse of matrix by row transformation method of

 (10)

A = Q.8 Evaluate the following i) dz where c is the circle =2 ii) where C is the circle (10)

Q.9 Determine the analytic function whose real part is

 (10)

i) log ii) cos x cos h y

Q.10. If f (z) = u + iv is an analytic function them show that u and v

are both harmonic functions.

Group C (25 Marks)

All Questions are Compulsory.

Q.11 Fill in the blanks ( each question carry 2 marks)

1.The order of the difference equation Yn + 1 + 2 Yn-1 = 1 is ____________

2.The backward difference operator symbol and inverse shifting operator E-1 is related
by __________________

3.If y is a polynomial of nth degree, then nth differences are ________________

4.On putting x = ez, the transformed differential equation of x2 + x + y = x is __________

5.If A is a non-zero column vector (nx1), then the rank of matrix AAT is _____________

Q.11 Multiple choice question. (Each question carry 2 marks)

1. If u = x2- y2 , then analytic function is

a) t2 b) t2 +c c) t/2 +c d) None of these

2. w = log t is an analytic function every where except at t =

a) 3 b) 0 c) 2 d) 1

3. If A is a skew-symmetric matrix of odd order then the determinant of A is ..

a) -1 b) 0 c) 1 d) a real number

4. The differential equation (ay2+x+x8)dx + (y8-y+bxy)dy=0 is exact if

a)b=a b) b=2a c) a=1, b=3 d) b 2a

5. If A and B are square matrices of the same order explain in general.

a) (A+B)2 A2 + 2AB + B2 , b) (A-B)2 A2 - 2AB + B2

c) (A+B) (A-B) A2 B2

Q.11 True or false (each question carry 1 marks).

1. The A-1 exist only for singular matrix.

2. The determinant of a matrix is zero, then matrix is non-singular.

3. 4. E = 1- 5. (A1)-1 = (A-1)1

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