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

Model Question Paper

BF7-MATHEMATICS-II

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

U_n+2 16 U_n = 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 Y_{n + 1} + ² Y_n-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 n^th degree, then n^th differences are ________________

4.On putting x = e^z, 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 AA^T 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) b2a

 

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

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