ias main 2012 math paper ii
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8/19/2019 Ias Main 2012 Math Paper II
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8/19/2019 Ias Main 2012 Math Paper II
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Section A
1. (a) How many elements of order 2 are there in the
group of order 16 generated by a and
b
such
that the order of a is 8, the order of
b is 2 and
-i
bab = a
1
.
2
(b)
Let
0 if
<
n +1
1
sin—, if —
x
1
n +1
1
0,
f x > n
f,,(x).
Show that
f„ x )
converges to a continuous
function but not uniformly.
2
(c) Show that the function defined by
{x
3
y
5
(x + iy)
f (z) =
x
6 + y1 0 Z °
0, z—0
is not analytic at the origin though it satisfies
Cauchy-Riemann equations at the origin. 12
F-DTN-M-NUIB
Contd.)
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1.
CO
a
aft b
WRTwftie
a1r
1 6
k fig
A ' ct re
2%ffm1
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ft
aft4P.
2*311Tba/;
1
=iiierl
2
1
0,
ift x <
n +1
1
sin, ffr
. —
LE X tS .
x
n 1
1
0, a. x >—,
fn
(x) =
T 4 6
-
47 k f
n x)
t ioo T ER ff
1fir iTiCa rdT t,
t f t9 t;chtt i l ici
I 2
(Ti) Tfri-
4 7
x y 5 (x
+ i y
Z
#0
f(z) =
6
4'
y
10
0 z=
k
r i rg r rEf f
E F F 9
fa A ch 9
4 14
W f .1ttet
1:R40
4 1 4 1 1 1 4 - 1 4 1 c h t u i 1 1
wli7
T h-
K
-r
t
2
F-DTN-M-NUIB
Contd.)
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(d)
For each hour per day that Ashok studies
mathematics, it yields him 10 marks and for
each hour that he studies physics, it yields him
5 marks. He can study at most 14 hours a day
and he must get at least 40 marks in each.
Determine graphically how many hours a day
he should stud y ma thema t ics and physics each,
in order to maximize his marks ?
2
(e)
Show that the series i —
g n
n6
is conver-
„=1(g
+1
gent .
2
2. (a) How many conjugacy classes does the per-
mutation group
S
5
of permutations 5 numbers
have ? Write down one element in each class
(preferably in terms of cycles).
5
+ )1)2
I2 f
) (0, 0)
x
+
y-
1
, if (x, y) = (0, 0).
f f
Show that — and — exist at (0, 0) though
ax
y
f(x, y)
is not continuous at (0, 0).
5
(c) Use Cauchy integral formula to evaluate
,3z
f
where
c
is the circle I z I = 2.
c (z +1)
4
15
b) Let
f x, y) =
F-DTN-M-NUIB
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(r) Raft* 315N9
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t
Tro,o)tmarr
itio3
ax ay
ftx,
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71
t (0, 0)
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I
5
3 z
4
z,
, <
1
tz+0
Ents“th4TFA9C77,-OcrIz1=2t I
15
F - D T N - M - N U I B Contd.)
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(d) Find the minimum distance of the line given
by the planes 3x +4y + 5z =7 and x — z= 9
from the origin, by the method of Lagrange's
multipliers.
5
3. (a) Is the ideal generated by 2 and
X
in the poly-
nom ial ring
z [X] of polynomials in a single
variable X
w ith c oe fficie nts in the r ing of inte -
gersz, a principal ideal ? Justify your answer.
1 5
(b) Let f(x) be
differentiable on [0. 1] such that
f(l) = f(0) = 0 and I f 2 (x) dx = 1. Prove that
0
x f (x) f s(x) dx =
5
(c) Expand the function f(z)—
•
(z +1 )
1
(z +3)
in
Laur e nt se r ie s val id for
(i) 1
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8/19/2019 Ias Main 2012 Math Paper II
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8/19/2019 Ias Main 2012 Math Paper II
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4.
(a) Describe the maximal ideals in the ring
of
Gaussian integers z [i]
-=
{a +
bi I a, b
E Z}.
20
(b)
Give an example of a function
f(x), that is
not Riemann integrable but I A.01 is Riemann
integrable. Justify.
0
(c)
By the method of Vogel, determine an initial
basic feasible solution for the following trans-
portation problem :
P r o d u c t s
P
l.
P
2,
P
3
and P
4
have
to be sent to
dest inat ions D I , D
2
and
D3
.
The cost of send-
ing product P
i
to destinations D
i
is
C, where
the matrix
1C0[
10
7
0
0
3
11
15
6
9
5 -
15
13
The to al requirements of destinations
D
,
D
2
a n d
D
3
are given by 45, 45, 95 respectively
and the availability of the products
P1
, P
2
, P
3
a n d
P
4
are respectively 25, 35, 55 and 70.
20
Se c tion 'IV
5 .
(a) Solve the partial differential equation
(D — 2 E1 )(D — D 1
2
z = e x •
2
Use Newton-Raphson method to find the real
root of the equation 3x = cos
x + 1 correct to
four decimal places.
2
F-DTN-M-NUIB
Contd.)
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4. (T) 4[74)4
9JTr1
r Eij={a+bila, be z}
al4
P4T3
TIAN mg' W tire
Q1 77 I
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. f(x)
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er
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471 * 31
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-479.
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fa
4i 61t1,
ar t
-
kr4 al
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raft 4rt4
iuftur
Teud P
1 , P
2 , P
3 .
4 0 1
1dan D
I
, D2
3 1 1 7
D
3 ffT 47
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.
t I d c 4 - 1 1 q
P
i
dc4lD
i
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fr e lliic t
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[10 0 15 5
[C
u
l= 7 3 6 15
0 11 9 13
1i
-
aft D
I , D
2 AK D
3 Q
1 1 W 3 q o h d I s b 4 - P T :
45, 45, 95 t, an
-t 3?:fTd P
i
, /
31, P
3
A t< P
4
ft
71
-
di sto-RT: 25, 35, 55 3rF 70 t I
0
07
5.
(T) slifiTT 31 7
-
*-
ff tili chtul
D
- 1)
)(D - D9
2 z =
e
A ±
Y
iA 7
I
2
(V) titilcbtui 3x = cos x + 1 W, q
r< T414-Icti PiT4
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2
F-DTN-M-NUIB
Contd.)
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(c)
Provide a computer algorithm to solve an
o rd ina ry d if fe ren tia l equa t io n C I --
) in
dx
the interval
[a, b]
for n number of discrete
points, where the initial value is
y a) =
a.
using E ule r 's method .
2
(d)
Obta in the eq ua t io ns go ve rning the m o t io n o f
a spher ical pendulum.
2
(e)
A r ig id sphere o f rad ius
a
is p la ced in a st ream
o f f lu id w ho se velo ci ty in the und isturbed state
is
V.
Determ ine the ve loc ity o f the f luid a t a ny
po in t o f the d is turbed s trea m.
2
6.
(a) Solve the partial differential equation
px + qy =
3z .
0
(b)
A string of length
I
is fixed at its ends. The
string from the mid-point is pulled up to a
height
k
and then re lea sed f rom res t. Find the
def lect ion
y x, t)
of the vibrating string. 20
(c)
Solve the following system of simultaneous
equations, using Gauss-Seidel iterative
m etho d :
3 x + 20y — z = — 18
20 x +
y —
2z = 17
2x — 3 y + 20z = 25 .
0
dy
7.
(a) Find
dx
at
x =
0.1 from the following data :
x : 0.1
0.2
03
0 . 4
y
0.9975
0.9900
0 .97 7 6
0 . 96 0 4
20
F-DTN-M-NUIB
10
(Contd.)
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I T ) * I T E R I I T
4 4 W
-4 4i4kbeli
f(x
•
y)
W ‘ T
s i c k i + 1
dx
[a, hi 4, siticin
n L IT
e t e l l ,
OITITR
etKi
F,
b c ; i
r w
FTt4hIcur
5.7
4tr-47,
mtrITT 41
-4
Y(d)
=
a
t I
2
(IT)ThAtIl
• 410*
4WW°T q,4r4
t 1 1 1 ,
mu1
in Sim of g I
2
() 1 ;1 4v-ti a
k
4;
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. T R T
4 K w r
, 1
44
-
4
r thcrea anwr 4 44 vt
uff tw i R r €44pu Erru Phki t
r r
k i r
z f i r
ccir4r;
I
2
6.
id alifiTW 3r4W 7 Iika c htut
1q
ft :
px+9Y
=3z
0
(T4) ei“
twr
rt
3 T 9 4
a il tr r itrr Fr t
,1 t4 w a r
1tict:i Et
41
,
1
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3
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ir
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4,4iii ar t W T R *c irIT y(x.
t)
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ti-MaT
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4 7 1 T 4 1 1 1 7
. 4 - A . Z i
trg,
Cr
-47
3x + 20y z =
-1S
20x + y – 2z = 17
2x – 3y + 20z = 25.
0
d
7.
(W)
1 4 - 1
ti
t q c r 3 r iT
-
t — }
X
= 0 1 Hier f
:
dx
x
0.1
0.2
0.3
0.4
y :
0.9975
0.9900
0.9776
0.9604
20
F-DTN-M-NUIB
1 1
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(b)
The edge r =
a of a
circular plate is kept
at temperature
f(0).
The plate is insulated so
that there is no loss of heat from either surface.
Find the temperature distribution in steady
state.
0
(c)
In a certain examination, a candidate has to
appear for one major and two minor subjects.
The rules for declaration of results are : marks
fo r m ajo r a r e de no te d by M
1
and for minors by
and
M
3 .
If the candidate obtains 75% and
above marks in each of the three subjects, the
candidate is declared to have passed the
examination in first class with distinction. If
the candidate obtains 60% and above marks in
each of the three subjects, the candidate is
declared to have passed the examination in
first class. If the candidate obtains 50% or
above in major, 40% or above in each of the
two minors and an average of 50% or above in
all the three subjects put together, the
candidate is declared to have passed the
examination in second class. All those
candidates, who have obtained 50% and above
in major and 40% or above in minor, are
declared to have passed the examination. If the
candidate obtains less than 50% in major or
less than 40% in any one of the two minors,
the candidate is declared to have failed in the
examinations. Draw a flow chart to declare the
results for the above.
0
F-DTN-M-NUIB
2
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ti) thcbit OF* R71k
r = a
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F-DTN-M-NUIB
3
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8. (a) A pendulum consists of a rod of length 2a and
mass
in;
to one end of which a spherical bob of
r ad iu s a / 3 an d ma s s 15 m i s a t tac h e d . F in d the
moment of inertia of the pendulum :
(i)
abo ut an ax is thro ugh the o the r end o f the
rod and at right angles to the rod.
5
(ii)
a bout a pa ra lle l a x is th ro ugh the cent re o f
mass of the pendulum.
[Given : The centre of mass of the
pend ulum is
a/ 12
a bove the centre o f the
sphere.]
5
(b) Show that
0
= x f r)
is a possible form for the
velocity potential for an incompressible fluid
motion. If the fluid velocity
7/
as r
find the surfaces of constant speed.
0
F-DTN-M-NUIB
4
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