ias main 2012 math paper ii

8/19/2019 Ias Main 2012 Math Paper II

1/14


2/14

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.)

IMS(Institute of Mathematical Sciences)

IMS(Institute of Mathematical Sciences) 2

http://www.ims4maths.com/coaching/download


3/14

1.

CO

a

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ft

aft4P.

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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.)





4/14

(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

Contd.)





5/14

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(x

+ )

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rq

(x

,

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o)

f(x, v

2

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,

trft

(x, y ). (0, o).

R 1 . I W 7 W

t

Tro,o)tmarr

itio3

ax ay

ftx,

tidd

71

t (0, 0)

C R

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.)





6/14

(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


7/14


8/14

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.)





9/14

4. (T) 4[74)4

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r Eij={a+bila, be z}

al4

P4T3

TIAN mg' W tire

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0

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. f(x)

t*

(7

er

47,

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towboi

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-479.

f14-Itql*r97, a*Th

fa

4i 61t1,

ar t

-

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-

raft 4rt4

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1 , P

2 , P

3 .

4 0 1

1dan D

I

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3 1 1 7

D

3 ffT 47

- 1

.

t I d c 4 - 1 1 q

P

i

dc4lD

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dctMa

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1 -1-

7

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[C

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2 AK D

3 Q

1 1 W 3 q o h d I s b 4 - P T :

45, 45, 95 t, an

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71

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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

ffT

altdkch

r

-ri

a9

-

kcFr9

fdfF7 W T T44[6

- QN-7 I

2

F-DTN-M-NUIB

Contd.)





10/14

(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.)





11/14

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;

n v i

. 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

1 T9T t

1341

3

I 41 q

fit19voir it

ir

t

4,4iii ar t W T R *c irIT y(x.

t)

4 1 1 e k e r

CAI

0

4

ti-MaT

c h

fiT

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

(Contd.)





12/14

(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

Contd.)





13/14

ti) thcbit OF* R71k

r = a

71917 f 0)

Tur.ticti t

tTratheurft

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ditml fl

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4, d 4 4 - 1 4

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4 )

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3-

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Pr 4 E 4 )4 g m' shiErf f

kW AM i t 14 k ,14-1-+1qc,K f t 4

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50% t ir no t

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f t

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50% zIT thtt 3fr 'art' Tet k

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4 4

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0

F-DTN-M-NUIB

3

Contd.)





14/14

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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