Bengal Engineering and Science University 2007 B.E Computer Science and Engineering Mathematics-IIIC - Question Paper
The ques. paper is with the attachment.
Ex/BESUS/ MA-303/07 B.E. (CST) Part-II 3rd Semester Examination, 2007
Time : 3 hours Full Marks : 70
Use separate answerscript for each half.
FIRST HALF
(Answer any THREE Questions.
Two marks are reserved for general profjciencu.)
1. a) State and prove the Cauchy Integral Formula. (A formal proof is needed),
b) Suppose f(z) is defined by the equations f(z) = ( 1 when y < 0 ( 4y when y > 0
and C is the arc from z=-l- i to z=l+i along the curve y=x3. Evaluate Jf(z)dz. [6+5]
2. a) State Dirichlets conditions.
b) Obtain the Fourier series generated by the function f(x) = x2, -n < x < n. Does this function satisfy Dirichlets conditions? If so, why? Discuss the convergence of the Fourier series you have obtained.
1 1 IT2
c) Prove that 1 + + ji +.........= [3+6+2]
iz2
3. a) Integrate f(z) = e around a suitable contour to evaluate the integrals
00
J cos(x2)dx and J sin(x2)dx.
b) Use residue calculus to show that 271 A
a + bsmG Va2- b
4. a) State and prove Jordons lemma.
b) Integrate e around the rectangle whose vertices are-R, R, R + ia,-R + ia, where a is real and positive. Hence show that
CO 2 2
J e-x cos (2ax) dx = Vn e-a [5+6]
1I
5. A string of length X is tied to two fixed points. The string is of uniform tension along its length and has uniform mass per unit length. It is given an initial displacement y = a sin3 where y is the displacement at a distance x from one end of the string and released from rest. Find the motion of the string. [11]
SECOND HALF (Answer O.No.6 and TWO from the rest.)
6. i) Let a, b, c be integers such that g.c.d. (a, c) = g.c.d. (b, c) = 1. Prove that
g.c.d. (ab, c) = 1.
ii) Let R be a relation on the set A ={ 1,2, 3,4, 5,6, 7} defined by R = {(a, b) e A x A : 4 divides a-b} then find domain and range of R and R_l.
iii) Determine which of the mappings f: R -> R are one-one and which are onto R.
(a)f(x) = x+4, (b)f(x) = x2 VxeR.
iv) Write the proof if the following statements are true, otherwise give a counter example.
a) Every group of four elements is commutative.
b) Every finite ring with unit element 1 is an integral domain. [15]
7. a) Let R = {(a, b) ] a, b are rationals and a-b is an integer}
Prove that R is an equivalence relation on set of rationals.
b) Let H be a subgroup of G. If x2 e H for all x e G then prove that H is a normal subgroup of G and G/H is abelian.
c) Let G = (a) be a finite cyclic group of order n. Show that ak is a generator of G iff g.c.d. (k, n) = 1 where k is a positive integer. [10]
8. a) Let G be a finite cyclic group of order m. Then prove that for every positive
divisor d of m, there exists a unique subgroup of G of order d.
b) In the ring z8 and z6 find the following elements.
i) the invertible elements
ii) the nilpotent elements
iii) the zero divisors.
c) Prove that a ring R is commutative iff (a + b)2 = a2 + 2ab + b2 for any a, b 6 R.
[10]
9. a) Examine whether the set of vectors {(3, 0, 2), (7, 0, 9), (4, 1, 2)} are linearly
independent or not?
b) Show that the set V of all ordered pairs of positive real numbers with operations defined by
(x,, x2) + (yl5 y2) = (x, yb x2y2) c(xl5 x2) = (x,c, x2) is a vector space.
c) Show that the set W = {(a, 0, b, 0) \ where a, b are reals} is a subspace of R4.
[10]
10. a) Prove that intersection of two subspaces of a vectorspace V(F) is always
subspace.
b) Show that the number of vertices of odddegree in a graph is always even.
c) What is a simple graph? Give an example.
d) Verify whether the set of all even integers form a field or not. [10]
9. a) Examine whether the set of vectors {(3, 0, 2), (7, 0, 9), (4, 1, 2)} are linearly
independent or not?
b) Show that the set V of all ordered pairs of positive real numbers with operations defined by
(xi, x2) + (yb y2) = (xjyi, x2y2) c(xj, x2) = (x,c, x2) is a vector space.
c) Show that the set W = {(a, 0, b, 0) j where a, b are reals} is a subspace of R4.
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10. a) Prove that intersection of two subspaces of a vectorspace V(F) is always
subspace.
b) Show that the number of vertices of odddegree in a graph is always even.
c) What is a simple graph? Give an example.
d) Verify whether the set of all even integers form a field or not. [10]
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