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CS (Main) Exam:20l5
G-AVZ-O-NBUA
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MATHEMATICS
Paper—I
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Time Allowed: Three Hours
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Maximum Marks : 250
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QUESTION PAPER SPECIFIC INSTRUCTIONS
Please read each of the following instructions carefully before attempting questions :
There are EIGHT questions divided in Two Sections and printed both in HINDI and in ENGLISH.
Candidate has to attempt FIVE questions in all
Question Nos. I and 5 are compulsory and out of the remaining, THREE are to be attempted choosing at
least ONE from each Section.
The number of marks carried by a question/part is indicated against it.
Answers must be written in the medium authorized in the Admission Certificate which must be stated clearly
on the cover of this Question-cum-Answer (QCA) Booklet in the space provided. No marks will be given
for answers written in a medium other than the authorized one.
Assume suitable data, if considered necessary, and indicate the same clearly.
Unless and otherwise indicated, symbols and notations carry their usual standard meaning.
Attempts of questions shall be counted in sequential order. Unless struck off attempt of a question shall
be counted even if attempted partly. Any page or portion of the page left blank in the Question-cum-Answer
Booklet must be clearly struck off.
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1
+
-<
$ & l& t
to
V2 = (-1, 2, 1)
i
3
+
V3 = (3, -1, 1)
Let V = R3 and T € A(V), for all a} e A(V), be defined by
T(ap a^ a3) = (2a{ + 5 ^ + ^
3 s l + % - a3, - a j + 2a2 + 3a3)
-
]
What is the matrix T relative to the basis
V, - (1, 0, 1)
Q. 3(b)
V2 = (-1, 2, 1)
x2 + y2 + z2 — 1
v 3 - (3, -1, 1) ?
f^^IcT
12
(2, 1 , 3 ) ^ 3rf£jdH
1? I
Which point of the sphere x2 + y2 + z2 = 1 is at the maximum distance from the point
(2, 1, 3) ?
Q. 3(c)
(i)
13
Wrier
fr'+lfcTi'
(2, 3, 1) ^
(4, -5 , 3) %
t *T
$ w ^ rr 1 1
x -m
Obtain the equation of the plane passing through the points (2, 3 ,1 ) and (4, -5, 3)
parallel to x-axis.
6
(ii) f i n i t e
:
x~a+d_ y -a _ z-a-d
a -8
1 1 ^
a
x-b+c= y~b= z-b-c
a +5
ft, ^
P~Y
P
P +Y
^mcfcT
ftw 1 1
Verify if the lines :
x -a +d _ y -a
a -5
z-a-d
a
a +5
x-b+c_ y -b _ z-b-c
anC*
P~Y
P
P +Y
are coplanar. If yes, then find the equation of the plane in which they lie.
Q. 3(d) f ^ T SHPficH
^
7
;
JJ ( x - y ) 2cos2(x + y)dxdy
R
t,
ittf
W5WT (71, 0) (271, 7t) (71, 271) (0, 7l) t !
Evaluate the integral
JJ (x - y)2cos2(x + y) dx dy
R
where R is the rhombus with successive vertices as (n, 0) (2?t, 71) (71, 27t)(0, 7t).
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4
+
12
#
Q. 4(a)
PlH frftekl
JTFT Rf^rfcfq :
/ / V l y - x 7] d x d y
WT R = [-1, 1 ; 0, 2].
Evaluate l i V l y - ^ 2 1 dxdy
R
where R = [-1, 1 ; 0, 2].
Q. 4(b)
R4 ^ t
13
f^ n - m
{(1,
*rg^RT
0, 0, 0), (0, 1, 0, 0), (1, 2, 0, 1), (0, 0, 0,
STTT f ^ n f e r 11 ?T cW ^
1)}
m iK
Find thedimension o f the subspace of R4, spanned by the set
{(1,
0, 0, 0), (0, 1, 0, 0), (1, 2, 0, 1), (0, 0, 0,
1)}
Hence find its basis.
Q. 4(c)
12
x2 + y2 = 2z ^
X= 0
^
^TRcTcT
'Mi?! !? I OT
fwRT ^TT
^PR?TcT
} y) * (o, o)
(x, y) = (0, 0)
3T^epfteT
For the function
x2 - x ,/ y
2
» (x, y )* (0 , 0)
x +y
f(x, y) =
0
(x, y) = (0, 0)
Examine the continuity and differentiability.
5
+
12
+
SECTION—B
Q. 5(a)
PlHlrlftRT 3T^ kT MWTTW ^
FeT
:
x co sx — + y(xsinx + cosx) = l.
dx
Solve the differential equation :
dy
x c o s x ~ + y(xsm x + cosx) = l.
dx 7
Q. 5(b) PlHfclfad 3T^cT
10
FcT
:
(2xy4ey + 2xy3 + y)dx + (xV ^y - x2y2 - 3x)dy = 0.
Solve the differential equation :
(2xy4ey + 2xy3 + y)dx + (x2) ^ - x2y2 - 3x)dy = 0.
Q. 5(c) ^
^
^cT 3TT^rf ^
(WFT/qJT.) ^
2
-a
7TTST f^«TfcT $
T ^cfT «rrq, ^
»
TFT t ,
^
10
3MFT ‘a’ ^ 3||ckf+ld ‘J* f |
%1T fcT^m
j^TT xSTFT, ^
3TR?jwr
TO 3MHT TTTcJTT ^tf^T I
A body moving under SHM has an amplitude ‘a’ and time period *T\ If the velocity is
*2 ’
trebled, when the distance from mean position is —a >the period being unaltered, find
3
the new amplitude.
Q. 5(d)
8 kg
*fTT
w zm
^ f s r r rTcT ^
^
M
I
*rrc w
10
w
j
M
^ \ % ^RT ^
11 T ttft ^
HK
Wt ^
% b
"'TT W ^TPT 11
TR, 3;sf f^TT
W
^
^ ^
$
i
A rod o f 8 kg is movable in a vertical plane about a hinge at one end, another end
is fastened a weight equal to half of the rod, this end is fastened by a string of
length / to a point at a height b above the hinge vertically. Obtain the tension in the string.
-
Q. 5(e)
t <+>f'on
m
^
4 kg t ,
^
$ w r ^ i< m
f^RT^T ^sftrr yJTT^ ^ f |
I I f^PTT
$
$ f^ScT W#
^rnr i
Two equal ladders o f weight 4 kg each are placed so as to lean at A against each other
with their ends resting on a rough floor, given the coefficient o f friction is |i. The ladders
at A make an angle 60° with each other. Find what weight on the top would cause them
to slip.
13
Q. 6(c) qft ^ ^ 5 X x 2 - nyz = (X + 2)x
eTt X^ n
4 x ^ + z3 = 4(1,-1 , 2)TTT
PHlfcIHI
Find the value o f X and ji so that the surfaces X x 2 - p y z =(X + 2)x and 4x2y + z3 = 4
may intersect orthogonally at (1, -1 , 2).
Q. 6(d)
^
t
13r^fa, ^
^ r r 11
12
3n^f% ^ ? tt t ; ‘a’
^
arrrm
w
htcjr-
^ tf^ r i
A mass starts from rest at a distance ‘a’ from the centre of forcewhich attractsinversely
as the distance. Find the time of arriving at the centre.
Q. 7(a)
(i)
ft'Hfctf&d
dMIfl f ^ T
T *?)
(ii) ciiKiiti ^HlcK
13
W M X STTRT ^tf^T :
s
■e""5
s2 +25
"STzihT
Pl*-ifeiRaa
y" + y = t, y(0) = 1, y'(0) = - 2
?cr f^rfcrm
(i)
Obtain Laplace Inverse transform of
(ii) Using Laplace transform, solve
y" + y = t, y(0) = 1, y'(0) - - 2.
Tj-znKZ-o-rtSr&t
7
+
6+6=12
Q. 7(b) XP’ ^ ul ^
^ 3TTE1R
H*5f fTTT^R ¥F§
$ 3TTOTC *TT
?t,