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
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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 ya.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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