DR. BABASAHEB AMBEDI'..AR TECHNOLOGICAL  
SEMESTER EXAMINATION:  
UNIVERSITY, LONERE  
.f!tJt:.{ ;'~ (d7-  
Mccha nica I/Electrka IIExTC/Chem ical/P et rochem ica IIC0 mpute rI ITIC iviI  
enggsolution.com  
Subject  
Time  
: Engincering Mathematics-I (Ncw)  
: 03 Hrs  
SClllcster  
: I  
Max. Marks  
: 60  
2 MAY 2017  
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INSTRUCTION: ATTEMPT ANY FIV!!: QUESTIONS.  
[
Q1. (a) Find the rank ofthc matrix A = ,~ 3  
1
2
3
5
by reducing it to normal form.  
]
1
4iVlarks]  
1
.
1
3
5
(
b) Fcr what valucs of k is the following systcm of equations consistent. and hence solvc l'or 14Marks]  
x + Y+ z = 1;x + 2y + 4z = k; x + 4y + 10z = k2.  
1
(c) Find the eigen  
values anc! elgen vectors of the matrix  
2
461.  
[4 Marks]  
o 5  
A = [~  
Q2. (a) Find the nth derivative of tar:'" (2lX-x,)- in terms of ,. and e.  
[4 Marks]  
[4 Marks]  
(
b) If Y = (x2 - 1)n. prove that (x2 - 1)Yn+2 + 2x}'u+' - n(n + 1)yu = O.  
c) Expand [(x + h) = tan-'(x + h)  
(
m powers of h and hence lind thc value of [4 Marks]  
ta]]-'(1.003)  
upto five places of decimal.  
x'  
y '  
b2+u  
z'  
(au)2 + (a-ouy)2 + (a-ouz)2 =2 [au ilu  
x-+y-+z-.  
lJx Dy  
aozu] [4 Marks]  
Q3. (a) If --+--+--=1,  
pro\'cthat  
C2+11  
-
aX  
aZ+u  
(b) If z is  
a
homogeneous  
ftlnction of degree n in x, y  
then prove that [4 Marksj  
a2z a2z  
a2z  
x2 ax' + 2xy axay + y2 ay2  
=
n(n - 1)z.  
(
c) If F = F(x,y,z) where x  
=
U,C L'  
+
W, y = uv + vw + Wll, z = xyz . then show that [4 Marks]  
aF aF aF aF  
aF  
of  
u a-u + L'-;;--  
+
W
J - = x a-x + 2. va-y + 3 zo-z .  
avow  
--.  
P.T.O.  
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Q4. (a) Expand [(x, y) = cos x sin y •a.sfar as the terms of third degree.  
[4 Marks]  
4 Marks]  
••  
b) Iflhe sides and angles ora plane t~i&ngle/vary ~ such a way that its circum-radius remains  
• •  
[
(
da  
db  
de  
constant, prove that -- + -- +-- = 0, where da, db, dc are smaller increments  
cosA  
r.osD  
cosC  
in' he sides a, b, c respectively.  
c) f'i"d thc maximum and minimum distances from the origin to thc curve [4 Marks]  
lx + 4xy + 6y2 = 140.  
(
:
2
[4 Marks]  
[4 Marks]  
[4 Marks]  
Q5.  
()  
a
CIlange to poIar co-orcI'mates and <:\aIuatc  
I -- f.0oof.0"' e-(x'+y') dx dy.  
I = f.01 I i y '  
r;;  
=
"x "y'-x  
(b) EVlluate  
dxcly  
by changing the order of integration.  
[4 Marks]  
Q6. (a) f'it,d the interval of convergence of the series :~n:n-~!.xn.  
(
b) Teitthe convergence of the scrie:,  
c) Teitthe convergence of the serie:i  
[4 Marks]  
[4 Marks]  
~
O O  
n!  
(
L m = l (nn)2  
.
*******HU*************************  
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