Total No. of Questions :9] SEAT No.
PA-4296 [Total No. of Pages :4
[5924|;5
FE.
ENGINEERING MATHEMATICS-I
(2019 1 (Semester - I)(End- Sem)
Time :2% Hours] [Max. Marks :70
Instructions to the canaidates
1) Attempt ,ompulšory,0.2 or Q.3, Q.4 or Q.5, Q.6 or Q.7, 0.8 or 0.9.
2) Use of elecronic pocket caleulator is allowed.
3) Assne suitabte data, ifnecessary.
4) Figurés to the right indicate full marks.
Q1)Write theCorrect option for the following multiplechoice questions. [10]
09:4:03
u
a) If u=ty then is eq [21
ôx ôy
3
20503/2023
i) 2 0
CEGNO18311
b) Ifx =-,y=2uvánd =4(u'+v') then
O(u,v)
is equal to
iu,v) O(x,y)
21
114.79125.146
i) 4 (+y) i) 4( +v) 09:21:03'3
iv) 1
4(u' +v')
C) For cxt cx, = 0 where, x,, x, are non-zeto vectors and e, C, are
(2]
constants then x, x, are linearly independentif
14620/03/2023
C, #0, c, 0 0,=0
T?5.
i) c, =0, C, #0 4CHGNO18911
0,,=0
11479.
PIO
Galaxy F15 5G
[1 2 31
to the matrix M=| 2 -4 6 is [2]
form corresponding
d) The quadratic
3 6 5]
) Q)=xg +4xx, ++6xx, +12x,X,
Q)
im) Q;;+5x; +2xx, + 3x +6x,x,
iv)
e) uyt
If 2x,
ây
is equal to [11
i)2r + 2 i) 2y
ii) 2x + 2y +2 i) 2
f) If for a square matrix M of order 2 súm of diagonal elements =4 and
M-3 then. Characteristic equaton) of Ads
629,09:21:03 [1]
i) 22-32 + 4 =0 A2- 42+3 =0
i) 22+ 32 +4 =0 22+ 42 +3=0
2043
146
02) a) CEGNO1Š
Ifu= 2x +3y ,v=k'2rnd value of
U4.79.25.
b) If u= cosec-1
+ then that
09:21:0å
1%,
ou
+ 2xy
o'u
ôxôy
,2 O'u tan 4
144 Ltan' u +13
89 (5]
20/03/2023
C) 1x=utvtW, y=uv +vw +uw, z=andPas function is of x, y,z
then prove that u
ôu
+y + CEGN
WFZ5. +3
OR
14.75.
(5924]-5
2
Galaxy F15 5G
the cquations x +ytz= 1, 2r+y+4z=k,
07) a) Determine values ofK for which [5]
4x+y 10z+ =are inconsistent.
b) Examine whether the vectors
=
1,-4), x, (2, 2,-3), x,
x,3, (0, -4,1) =
If dependent, find the relation
are linearly independent or dependent. [5]
between them.
(5]
tbefollowing network.
c) Determine the currents in
14 -10
GEGNDÍS9
Findthe eigenvalues and eigen vectors of the matrix
A= .[5]
08) a) 5 -1
i514020t
By using cayley Hamilton theorem, find the Dverse of the matrix
b)
21 1]
01 0,if it exists.
09:21 [5]
11 2 8911
3 1
c) Reduce the matrix 0 tosts diagonal form by finding modal
Sa03/2023
matrix P. [5]
CHGNOI
.15.146[1 0 -1
09) a) Find the cigen valuès of A= 2 1 1 Also find eigenpector
2;03
2 2 3 09:24;
(5]
corresponding to the largest eigen value ofA.
|1 2] I51
b) Verify cayley Hamilton thcorem for -Ntence indA".
c) Find the transformation which redues thequadratic
29/42023 form
7+2xj +3x; +2x, x, -2x, X, +2x,x,tho cangical form by using
to
form. (51
congruent transformations. Also write,tha,canonial
114.79,17
[5924]-5 4
Galaxy F15 5G
8'z
93) a) If z = tan (y+ax)-(y-ax) then find value of [5]
(
ôy?
au a'u
b) Ifu=log +y-ry) then fìnd value of x +2xy +y².
.
@xôy
(5]
ôu Ôu Ou
c) Ifu=f then findvalue of x. +y: -+Z [5]
Ox ôy
20/03202
04) a) If x
5.ty6
ÇGNIs911
w²+u', z =#+ find
O(,y,z)
[5]
b
In calculating the volume ofa right circular cylinder, using the fornmula :
V= mh,
errots. of 2% and 1% are made in measuring the height and radius of
base respectively. Find the error in the calculatedvolume. [5]
Find stationary points of :
f,y)=3*-y+* and find
fhereifexists. b [5]
05) a) Ifx =u+ v, y=+w, z + wthen find
Ou
using jacobian. [5]
b) Examine forfunctional
ehndetsé :
u= -,V= tan
1-xy [5]
c) Find stationary value ofu=x+y² +z? underthe condition 1
using Lagrange's method.
[5]
092[\(3
06) a) Solve the following
+ t +7w=
6x 3y 4z
system of linear equations. 3w= 0,
5,2r
+y+w=-1. (51
b) Examine whether the
vectors x, (2,2, 1), =
incarly independent or dependent. If dependet,
them.
NNO18911=
3),x, (1,2,2)are
fiphe relation between
(51
c)
CE® -8 4 al
Find thevalucs
of a,b, cifAis =1 4 b.(5]
orthogonal/ere A
114.2.175.19/03/2023
47 c
[59241-5 OR
Galaxy F15 5G 3
