Name of the University : Alagappa University
Degree : B.Sc
Department : Mathematics
Subject Code/Name : Classical Algebra
Document Type : Model Question Papers
Website : alagappauniversity.ac.in
Download Model/Sample Question Paper : May 2008 :www.pdfquestion.in/uploads/alagappauniversity.ac.in/4071-B.Sc.(Maths).docx
Alagappa Classical Algebra Question Paper
Time : Three hours
Maximum : 100 marks
1.Prove that Every convergent sequence is a Cauchy sequence.
Related : Alagappa University Algebra-I M.Sc Model Question Papers : www.pdfquestion.in/3996.html
2.State and prove Cauchy’s General principles of convergence theorem.
3.Sum to infinity the series
4.Find the sum to Infinity the series
5.Show that the equation
6.Show that the sum of the power of the roots equation where
7.Prove that (ii)If are three distinct positive real numbers then prove that
8.If and Verify that where is the transpose of A.
CALCULUS
1.If , prove that
2.Prove that the maximum rectangle inscribed in a circle is a square.
3.Evaluate .
4.Prove that and evaluate .
5.Solve : . .
7.Evaluate by using Laplace transform.
8.Eliminate f and – from .
ANALYTICAL GEOMETRY AND VECTOR CALCULUS
1.Show that the equation of the pair of straight lines
2.Find the equation to the two circles that cut orthogonally the circles
3.Show that the straight lines whose D.C’s are given by the equations
4.Find the equations of the image of the line in the plane .
5.Find the shortest distance and the equation
6.Find the equation of the sphere having the circle .
7.(i) If find if .
8.Evaluate where and the curve C is the rectangle in xy plane bounded
MECHANICS
1.Derive the analytical expression for the resultant of two forces.
2.Find the magnitude and direction of the resultant of any number
3.Find the resultant of two like parallel forces acting on a rigid body.
4.Explain the following :(i)Statical, dynamical and limiting friction
5.Show that the greatest height which a particlend angle of projection.
6.A short of mass m penetrates a thickness moving along a straight line with velocities and .
7.The equation of a simple harmonic motion is . s is along an ellipse.
8.Obtain the equation to the central orbit in the form .
ANALYSIS
1.Define a metric space.
2.Let be a metric space. Let then prove that A is open iff .
3.State and prove intermediate value theorem.
4.State and prove Cantor’s intersection theorem.
5.State and prove Heine Borel theorem.
6.Prove that a closed subspace of a compact metric space is compact.
7.Define uniform convergence defined
8.State and prove contraction mapping theorem. (10)
DISTANCE EDUCATION :
1.Define conditional probability.
2.Find the mean and standard deviation for the following
3.Fit a curve of the form to the following data.
4.Derive the formula for rank correlation coefficient.
5.Derive recurrence relation for central moments of binomial distribution.
6.Prove that the moment generating function
7.A machine puts out 16 imperfect articles in a sample of 500 articles.
Discrete Mathematics :
Time : Three hours
Maximum : 100 marks
Answer any FIVE questions.
1. (a) Draw the parsing tree
(b) Construct the truth table
2. (a) Show that is a tautology.
(b) Obtain a disjunctive normal for
3. (a) Derive , using the rule of conditional proof if necessary,
4. (a) Write each of the following in symbolic form
(i) All men are giants
(ii) No men are giants
(iii) Some men are giants
(iv) Some men are not giants.
(b) Verify the validity of the following arguments :
Lions are dangerous animals. There are lions.
Therefore there are dangerous animals.
5. (a) (i) Define a ‘Simple graph’.
(ii) Let G be a graph, then , where .
(b) A graph G is disconnected if and only if its vertex set V can be partitioned into two non-empty subsets and such that there exists no edge in G whose one end vertex is in and other in prove.
6. (a) Let r be a positive integers. Let A be the adjacency matrix of a simple graph G. Then the entry in is the number of different walks of length r between the vertices and – Prove.
(b) Obtain the adjacency matrix A of digraph given below. Find the elementary paths of length 1 and 2 from to .
7. (a) (i) Define a ‘cut-set’.
(ii) Every cut-set in a connected graph G must contain atleast one edge of every spanning tree of G – Prove.
(b) (i) Define ‘Fundamental cut set’.
(ii) Let T be a spanning tree of a connected graph G. Let C be a chord of T and be the fundamental circuit determined by C. Then C occurs in every fundamental cut set associated with the edges in and in no other ¬ – Prove.
8. (a) State and prove the ‘Max-flow min-cut’ theorem.
(b) Prove that, if a connected plane graph G has n vertices, e edges and f faces, then .