Two Mathematics-I papers (AS 101) from BUTEX affiliated colleges' 2019 Level-1 Term-I finals, worth 105 marks. Questions include matrix inversion, Cauchy's inequalities, De Moivre's theorem, maxima and curvature problems, and mean value theorem proofs.
Q1
Level-1 Term-I, Final Exám-2019
Subject: Mathematics-I (Code: AS 101)
Time: 3.0 Hrs. Full Marks: 105
(Use separate answer script for Part: A and Part: B)
(All parts of a question must be answered consecutively)
Part: A
(Answer any three questions)
1. (a) Define: Diagonal matrix, identify matrix, symmetric matrix, rank of matrix and skew symmetric matrix.
(b) Define Inverse matrix and find the inverse matrix of this matrix A = .
(c) If A and B are non-singular of same order, then AB is also a non-singular matrix and .
[7.5+6+4=17.5]
2. (a) State and proof Cauchy's Inequalities.
(b) Prove that .
(c) Test the convergency .
[6+6+5.5=17.5]
3. (a) If then prove that .
(b) State and prove De-Moiver’s theorem rational values of n.
(c) Find the sum to n terms 1.3.5+2.4.6+3.5.7+... ... ... .
[6+6+5.5=17.5]
Q2
Level-1 Term-I, Final Exám-2019
Subject: Mathematics-I (Code: MS 101)
Time: 3.0 Hrs. Full Marks: 72
(Use separate answer script for Part: A and Part: B)
(All parts of a question must be answered consecutively)
Part: A
(Answer any three questions)
1. (a) State and proof Euler’s theorem on homogeneous function.
(b) If u(x,y) be a homogeneous function of degree ‘n’ then prove that .
(c) Define function. Final the domain and range of the following function f(x)=.
[4+4+4=12]
2. (a) State and prove Roll’s theorem.
(b) Verify mean value theorem of over (0,1).
(c) Find the maximum and minimum value of .
[4+4+4=12]
3. (a) Prove that .
(b) Integrate of the following .
(c) Integrate of the following .
[4+4+4=12]