Download DBATU B-Tech 1st Year 2017 Dec Engineering Mathematics I 3 Question Paper

Download DBATU (Dr. Babasaheb Ambedkar Technological University) B.Tech First Year 2017 Dec Engineering Mathematics I 3 Question Paper

DR. BABASAHEB AMBEDKAR TECHNOLOGICAL
UNIVERSITY, LONERE ? RAIGAD - 402 103
Semester Examination: December ? 2017
Branch: All Courses Semester: I
Subject with Subject Code: Engineering Mathematics?I Marks: 60
(MATH101)
Date: 11/12/2017 Time: 3 Hrs.
Instructions to the Students:-
1. Each question carries 12 marks.
2. Attempt any five questions of the following.
3. Illustrate your answers with neat sketches, diagram etc., wherever necessary.
4. If some part or parameter is noticed to be missing, you may appropriately
assume it and should mention it clearly.
(Marks)
Q.1. (a) For What value of A the following system of linear equations is (06)
consistent and solve it completely in each case:
x+y+z=1,x+2y+4z=}\,x+4y+102=}\2 .
(b) Find the eigen values and the eorresponding eigen vectors for the matrix
1 0 ?4
A: 0 5 4 (06)
- 4 4 3
1
Q2. (:1) If y=Sin PX+ COS PX.. ,_then prove that y": pn[1+(_1)nsm[2px)]2 . (04)
(b) If y=e"??5_l" , then prove that (04)
(1? X2)yn+2?(2n+1)~xyn+1_("2+a2]yn:0 -
(C) Expand y = 10g1COSX] about the point X 2% up to third degree by using
Taylor?s series. (04)

Q.3. Attempt Any Three: (12)
2
(a) If Xxyyzz=c ,thenprove that atpoint Xzy? ? (a:azy_~{xmexw '
3 3 2 2 2
(b) If u=tan_1(::::) ,provethat X2-37121+y2:?;2+2xya::y=sin4u?sin2u _
(c)If X2=au+bv,y2=au-bv and Z=f{u,VJ ,thenprovethat
Oz 62_ ? 2
x6x+y$_2(uau+vav) '
(d)If "zsm(%) where x=e',y=t2 :then?nd %'
Q.4. Attempt Any Three: (12) "
(a) If ux:yz,vy=zx,WZ:Xy , then prove that J J* = 1 Where J=SE%:I??g and
J* : 6(X,y,z) .
6(u,v,w)
(b) If the sides and angles of a plane triangle vary in such a that its circum?radius
. th th da + db + dc _ O
remalns constant, en prove at _cos A cos B C05 C ? .
(c) A rectangular box open at the top is to have volume of 32cubic units. Find the
dimensions of the box requiring the least material for its construction by
Lagrange?s method of undetermined multipliers.
((1) Expand f(x,yJ=x? as for as second degree in the powers of [x? 1)
and [y ~ 1) using Taylor's theorem.
Q.5. Attempt Any Three: (12)
(a) Change the order of integration and evaluate I = j} cosy dxdy .
0 x y
2
:1 2 1
(b) Use elliptical polar form to evaluate I = ffxy(%+zg)zdxdy , where R is
R
D?
the region of ellipse in positive quadrant.
d ?\
1!). y'of[x2+yz+zz)2 '

/\
(?5
(d) Find the centroid of the positive loop of the curve r2=a2c0529 .
?? 2
(2.6. (a) Test the convergence of the series ; (%+%) . (04)
(b) Test the convergence of the series 21: In;+)1,x . (04)
(C) Test the absolute convergence of the series 22: rdtTlriiz . (04)