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e

m

a

tr

ix

o

f

T

w i

th

r

es

p

ect

to

t

he

b

as

es

{

1,

x

x

2 }

an

d

{

1,

x

1

x

2

, 1

x

3

}

of

P

2

a

n

d

P

3

r

es

p

ec

ti v

e

ly

. A

ls

o

, f

in

d

th

e

nu

ll

s

pa

c

e o

f

T

.

1

0

2

.( a

)(

ii

)

L

e

t

V

be

a

-d

im

en

s

io

na

l

ve

ct

or

s

pa

c

e

an

d

T

:

V

V

b

e

an

i

nv

er

ti

b l

e

li n

ea

r

op

er

at

o

r.

f

=

{X

1

, X

2

,

•••

Xn}

i

s a

b

as

is

o

f

V

, s

ho

w

th

a

t{

J'

=

{

TX

1

,

TX

2

, •

•• ,

T

X

n}

is also

a

ba

si

s

of

V

.

8

2

.(b

)(

i)

L

et

]

h

e

r

e ro

(

' I

) i

s

a

cu

be

r

o

ot

o

f

u

n

ity

.

f

A

1

A

2

,

A

3

d

en

o

te

1

m

m

2

t

he

e

ig

en

v

al

ue

s

of

A

2

s

ho

w

t

ha

t

I

A

1

1

A

2

l

+ IA

3 1

9.

8

2

.( b

)(

ii

)

F i

n

d t

he

r

an

k

o

f

th

e

m

at

ri x

I

2

3

4

5

2

3

5

8

1

2

A

=

5

8

1

2

1

7

5

8

12

1

7

2

3

8

12

17

2

3

3

0

8

2

.(

c)

(i )

Le

t

A

b e

a

H

e

rm

e

ti

an

m

a

tr

ix

h

av

in

g

al

l

di

sti

nc

t

ei

ge

n

va

lu

es

A

.b

A

.

2

,

.

..

A

n

.

f

X

1

,

X

2

, •

•. ,

X

n a

re

c

or

re

sp

on

d

in

g

eig

e

nv

ec

to

r s

t

he

n

sh

o

w

th

a

t t

he

n

x

n

m

a

tr i

x

C

w

h

os

e

kth

co

lu

m

n

co

n

sis

ts

o

f

th

e

ve

ct

or

X

k is

n

o

n

si

ng

ul

ar

.

8

2

.(

c)

(i i

)

S

ho

w

t

ha

t

th

e vect

or

s X

1

=

1 ,

1

i

i),

X

2

=

i

.

- i.

1- i) and X

3

=

0.

1-2i

,

2-

i

in

C

3

a

re

l

in

ea

rly

i

nd

e

pe

nd

e

nt

o

ve

r

th

e f

ie

ld

o

f r

ea

l

nu

m

b

er

s

bu

t ar

e l

in

ea

rl

y

de

pe

n

de

nt

o

ve

r

the

f

ie

ld

o

f

c

o

m p

le

x

n

um

b

er

s.

8

3

.( a

)

U

si

ng

L

a

gr

an

g

e's

m

u

lt

ip

li e

r

m

et

ho

d

, f

in

d

th

e

sh

o

rte

st

d

is

ta

nc

e

be

tw

e

en

th

e

lin

e

x

2

y

2

y =

1

0

-

x

an

d

th

e

e

ll i

ps

e

-

+ -

=

.

20

4

9

a-

br

l-

m

-n

bu

a

6


7/19

\

t

j

I

i

3

.

b

)

C

o

m

p

u

t

e

f

x

y

0

,

0

)

a

n

d

f

y

x

0

,

0

)

f

o

r

t

h

e

f

u

n

c

t

i

o

n

{

x

y

3

(

-

2

,

x

,

y

;

t

{

0

0

)

,

y

-

x

+

y

0

x

,

y

)

=

O

,

O

)

.

A

l

s

o

,

d

i

s

c

u

s

s

t

h

e

c

o

n

t

i

n

u

i

t

y

o

f

f

x

y

a

n

d

/

y

x

a

t

0

,

0

)

.

15

. c) Evaluate

f

xy

dA

,

w

h

e

r

e

D

i

s

t

h

e

r

e

g

i

o

n

b

o

u

n

d

e

d

b

y

t

h

e

l

i

n

e

y

=

x

-

1

a

n

d

t

h

e

4

.

a

)

4

.

b

)

4 . c)

D

p

a

r

a

b

o

l

a

y

2

=

x

+

6

.

1

5

h

o

w

t

h

a

t

t

h

r

e

e

m

u

t

u

a

l

l

y

p

e

r

p

e

n

d

i

c

u

l

a

r

t

a

n

g

e

n

t

l

i

n

e

s

c

a

n

b

e

d

r

a

w

n

t

o

t

h

e

s

p

h

e

r

e

2

+

y

2

+

z

2

=

r

2

f

r

o

m

a

n

y

p

o

i

n

t

o

n

t

h

e

s

p

h

e

r

e

2

x

2

+

y

2

+

z

2

=

r

2

.

1

5

c

o

n

e

h

a

s

f

o

r

i

t

s

g

u

i

d

i

n

g

c

u

r

v

e

t

h

e

c

i

r

c

l

e

x

2

+

y

2

+

2

a

x

+

2

b

y

=

0

,

z

=

0

a

n

d

a

s

s

e

s

t

h

r

o

u

g

h

a

f

i

x

e

d

p

o

i

n

t

0

,

0

,

c

)

.

f

h

e

s

e

c

t

i

o

n

o

f

t

h

e

c

o

n

e

b

y

t

h

e

p

l

a

n

e

y

=

s

a

r

e

c

t

a

n

g

u

l

a

r

h

y

p

e

r

b

o

l

a

,

p

r

o

v

e

t

h

a

t

t

h

e

v

e

r

t

e

x

l

i

e

s

o

n

t

h

e

f

i

x

e

d

c

i

r

c

l

e

z

+

y

2

+

z

2

+

2

a

x

+

2

b

y

=

0

2

a

x

+

2by

+

cz

=

.

1

5

v

a

r

i

a

b

l

e

g

e

n

e

r

a

t

o

r

m

e

e

t

s

t

w

o

g

e

n

e

r

a

t

o

r

s

o

f

t

h

e

s

y

s

t

e

m

t

h

r

o

u

g

h

t

h

e

e

x

t

r

e

m

i

t

i

e

s

a

n

d

B

1

o

f

t

h

e

m

i

n

o

r

a

x

i

s

o

f

t

h

e

p

r

i

n

c

i

p

a

l

e

l

l

i

p

t

i

c

s

e

c

t

i

o

n

o

f

t

h

e

h

y

p

e

r

b

o

l

o

i

d

y

2

I

I

I

+

2

-

z

2

c

2

=

1

i

n

P

a

n

d

P

.

P

r

o

v

e

t

h

a

t

B

P

·

B

P

=

a

2

+

c

2

.

a

b

S

E

C

T

I

O

N

B

2

0

5

.

A

n

s

w

e

r

a

l

l

t

h

e

q

u

e

s

t

i

o

n

s

:

5

.

a

)

y

i

s

a

f

u

n

c

t

i

o

n

o

f

x

,

s

u

c

h

t

h

a

t

t

h

e

d

i

f

f

e

r

e

n

t

i

a

l

c

o

e

f

f

i

c

i

e

n

t

:

i

s

e

q

u

a

l

t

o

cos x

+

y

+

sin x

+

y

.

F

i

n

d

o

u

t

a

r

e

l

a

t

i

o

n

b

e

t

w

e

e

n

x

a

n

d

y

,

w

h

i

c

h

i

s

f

r

e

e

f

r

o

m

a

n

y

e

r

i

v

a

t

i

v

e

/

d

i

f

f

e

r

e

n

t

i

a

l

.

1

0

.

b

)

O

b

t

a

i

n

t

h

e

e

q

u

a

t

i

o

n

o

f

t

h

e

o

r

t

h

o

g

o

n

a

l

t

r

a

j

e

c

t

o

r

y

o

f

t

h

e

f

a

m

i

l

y

o

f

c

u

r

v

e

s

e

p

r

e

s

e

n

t

e

d

b

y

r

=

a

s

i

n

n

e

r

,

0

b

e

i

n

g

t

h

e

p

l

a

n

e

p

o

l

a

r

c

o

o

r

d

i

n

a

t

e

s

.

1

0

.

c

)

A

b

o

d

y

i

s

p

e

r

f

o

r

m

i

n

g

S

.

H

.

M

.

i

n

a

s

t

r

a

i

g

h

t

l

i

n

e

O

P

Q

.

I

t

s

v

e

l

o

c

i

t

y

i

s

z

e

r

o

a

t

p

o

i

n

t

s

f

u

1

d

Q

w

h

o

s

e

d

i

s

t

a

n

c

e

s

f

r

o

m

0

a

r

e

x

a

n

d

y

r

e

s

p

e

c

t

i

v

e

l

y

a

n

d

i

t

s

v

e

l

o

c

i

t

y

i

s

v

a

t

h

e

m

i

d

-

p

o

i

n

t

b

e

t

w

e

e

n

P

a

n

d

Q

.

F

i

n

d

t

h

e

t

i

m

e

o

f

o

n

e

c

o

m

p

l

e

t

e

o

s

c

i

l

l

a

t

i

o

n

.

1

0

.

d

)

T

h

e

b

a

s

e

o

f

a

n

i

n

c

l

i

n

e

d

p

l

a

n

e

i

s

4

m

e

t

r

e

s

i

n

l

e

n

g

t

h

a

n

d

t

h

e

h

e

i

g

h

t

i

s

3

m

e

t

r

e

s

.

f

o

r

c

e

o

f

8

k

g

a

c

t

i

n

g

p

a

r

a

l

l

e

l

t

o

t

h

e

p

l

a

n

e

w

i

l

l

ju st prevent a weight

of

20 kg

from sliding down. Find

the

coe

f

f

i

c

i

e

n

t

o

f

f

r

i

c

t

i

o

n

·

b

e

t

w

e

e

n

t

h

e

p

l

a

n

e

a

n

d

t

h

e

e

i

g

h

t

.

1

0

.

e

S

h

o

w

t

h

a

t

t

h

e

c

u

r

v

e

A

1

+

t

A

1

-

t

2

J

x

t

)

=

t

i

+

-

1

-

j

+

t

k

l

i

e

s

i

n

a

p

l

a

n

e

.

1

0

7

a

-

b

r

l

-

m

-

n

b

u

a

··


8/19

6

.( a

)

S

ol

ve

th

e d

iff

ere

nti

al

equ

at

ion

5.i

3

+ 1

2x

2

+

6y

2

)

dx

6x

yd

y

= 0

.

6

.(b

)

Us

in

g t

he

me

th

od

o

f va

ri a

tio

n

o

f par

am

ete

rs ,

so

lv

e t

he

dif

fe r

en

tia l

eq

ua

tio

n

6

.(c

)

d

2

{

a

2

y =

sec

ax

.

x

Fin

d

the

g

ene

ra l

so

lu

tio n

of

t

he

eq

ua

tio n

2

x

2

d

; +

x

d

y +

y

=

ln

x s in

(ln

x)

.

x

dx

6

.(d

)

By

u

sin

g

La

pla

ce

tr a

ns

for

m

me

th o

d,

so

lve

th

e

dif

fer

ent

ia l

eq

ua

tio

n

2

+ n

2

)x =

a

s

in(

nt

+a

),

2

=

r s

ub

je c

t t

o t

he

in

itia

l c

on

dit

ion

s

x

=

0

a

nd

:

=

0

, a

t t

=

0 ,

in

wh

ich

a

, n

an

d

a

are

co

ns

tan

ts .

10

10

15

15

7.(

a)

A

pa

rti

cle

of

m

as

s 2

·5

kg

ha

ng

s a

t th

e

end

of

a

st r

ing

, 0

·9

m

lo

ng,

th

e o

th

er

end

of

wh

ic h

is

at

tac

he

d t

o a

fi

xed

p

oin

t.

Th

e p

ar

tic l

e is

p

ro

jec

ted

ho

riz

on

ta l

ly

wi

th

a

vel

oc

ity

8 m

/s

ec

. F

ind

th

e v

elo

ci

ty

o

f

t

he

pa

rtic

le

an

d t

ens

io

n in

th

e s

tr i

ng

wh

en

the

s

trin

g

is

i)

ho

riz

ont

al

ii

) v

ert

ica

lly

up

wa

rd

.

20

7.(

b)

A

u

nif

orm

lad

de

r re

sts

a

t a

n

ang

le

o

f

4

5°

wi

th

th

e h

or

izo

nta

l w

it

h

its

up

pe

r

e

xtr

em

it y

ag

ain

st

a

rou

gh

ve

rti

cal

w

all

an

d i

ts l

ow

er

ex

tr e

mi

ty

on

th

e g

rou

nd

.

f

J l

nd

J.l

a

re

the

c

oe

ffic

ien

ts

of li

mi

tin

g f

ric

tio

n b

et

we

en

th

e l

ad

der

a

nd

th

e

g

ro

un

d a

nd

wa

ll

res

pec

tiv

ely

, t

he

n f

ind

th

e m

in

im

um

h

ori

zon

ta

l fo

rce

re

qu

ire

d t

o

m

o v

e t

he

lo

we

r e

nd

o

f th

e l

ad

der

to

wa

rd

s th

e

wa

ll.

15

7.

(c)

S

ix

eq

ua

l ro

ds

A

B

BC

C

D

D

E E

F

an

d

FA

ar

e e

ac

h

of

w

ei

gh

t W

and

are

fr

ee l

y

jo

int

ed

at

th e

ir

ex

tre

mit

ies

s

o a

s t

o

for

m

a h

ex

ag on; the rod

AB

is fix ed in a

h

ori

zon

ta

l p

osi

tio

n a

nd

th

e m

id

dl

e p

oin

ts

o

f A

B

an

d

DE

are

jo

in

ed

by

a

st r

ing

.

F

ind

th

e

ten

sio

n i

n

the

st

rin

g.

1

5

S

.(a

)

C

alc

ula

te

V

2

r

n) a

nd

f i

nd

its

ex

pre

ss

ion

n

te

rm

s

of

ra

nd

n

r b

ein

g t

he

dis

ta n

ce

of

a

ny

po

int

x

, y

, z

fro

m

the

or

ig i

n,

n b

ein

g

a c

on

sta

nt a

nd

V

2

b

ei

ng

the

L

apl

ace

op

er

ato

r.

10

S

.(b

)

A

cu

rv

e in

s

pac

e i

s d

efi

ne

d by

t

he

ve

cto

r e

qu

ati

on

1

=

2

i

2

t

J- t

3

k.

D

ete

rm

ine

t

he

an

gle

be

tw

ee

n t

he

tan

ge

nts

to

th

is

cu

rve

at

th

e

po

int

s t

=

1

an

d t

-1

. 1

0

S.(

c)

B

y u

si

ng

Di

ver

ge

nce

T

he

ore

m

o

f

G

aus

s,

eva

lu

ate

th

e s

ur

fac

e i

nte

gra

l

1

fJ (

a

2

x

2

b

2

y

2

c

2

z

2

f

2

dS,

w h

er

e S

is

th

e s

ur

fac

e

o

f

el

lip

soi

d

ai

l+

by

+

cz

2

=

a,

b

n

d c

be

ing

all

po

sit i

ve

co

nst

an

ts.

15

S

.(d

)

U

se

Sto

ke

s

the

or

em

to

ev

alu

ate

th

e l

in e

in

teg

ra

l Jc

-y

3d

x+

x

3

d

y

- z

3dz

,

wh

ere

C

is

th

e i

nte

rs e

ct

ion

o

f

th

e c

yl

ind

er

x

y

2

=

1

an

d

the

pl

an

e

x

y

z =

1.

15

a-

brl

-m

-nb

ua

8


9/19

liNI I

MAT

IIE

MAT

ICS

~

~

I I

Pap

er

fr?.Tffm

(11{Z{ :

rft r

eR

Tim e

al

lowed

: Thr

ee H

ours

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a ~

~

u m

CfiT

TT

R

f1 p

i

f¥:;d

P,f(sm

~ ~

c i t

t4Mlj \Cf4

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~ :

w#

J110 JTR

;W

c) "

l fUS1 fcrmfJJ?

r ~ rrm

f t 3/k

c);ii

€W ~ 1

r:rftm2/f

o

il

Ji f

? r:ifw

w;f f

~

~

1

I

A BR

L M

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B

I

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25

M

aximu

m

Marks

:

25

JT R

m

r

3 k

5

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rrm

8

JRitq; (5

j7lg 8 c

nrr-if-Cf

ill 'f C1i JlR

:flct>< fcn::t

l rfR

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ff

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J

T R 'J

J7T

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li "8

J"lR ~

1fli

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w; ff ~

Tr f

r Y

fft 1

1 ~ ffr&

\if"R

;mrr#

; JT


10/19

(c)

R

f x)

=

~

SE

CT

IO

N

x2

=

- -

2

~

x<

O

2

CRn

f

, [- 1

2

]

3'jo

:('l(l(

1

t ~

fll l

let'><

:1 1l

4 ?

CRff

?

CR

n ~

~

g

Cf)T

3 l f

f dfC

I

f

h g

x

) =

f

x)

m

~

~

Cf;T

acfl

~

q

I

d

)

~

~

~

b ea

+

<

1,

~

a

f

2ff

b 'tH

klOJ

Cf>

3fu

: C

II'Rtf

c et>

t

ffi ~

zn

e-

a -b

ez

~

~

~

q

n

~ ~

~

I

(e)

~

c n a

C

f l

\

z

=

2x

1

3x

2

-

5

x

3

~

fcn

x

1

x

2

x

3

=

7

A B

R L

M

N B

U B

··'


11/19

(a

)

Sh

o

w

t

h

at

th

e

s

et

of

m

a

tr

ic

es

S

{

[:

-

:

a

,

b E

R

}

is

a

fi

el

d

un

d

er

th

e

u

su

a

l

b

in

ar

y

op

e

ra

ti

o n

s

o

f

m

at

ri

x

a

d d

it

io

n

a

n

d

m

at

ri

x

m

u

lti

p

lic

a

tio

n

.

W

h

at

ar

e

t

he

a

d

di

ti

v

e

an

d

m

ul

ti

p l

ic

a t

iv

e

id

e

n t

i t

ie

s

an

d

(1

1]

hat

is

the

in verse of

1

1

? C ons ide r

the map

f:

C S defin ed

(

a

- b

J

b

y

f(a

+

ib

) =

b

a

.

S

h

ow

t

ha

t

f

i

s a

n

i

so

m

o

rp

h

is

m

. (

H

er

e

R

i

s t

h

e

s

e

t o

f

r

ea

l n

u

m

be

r

s

an

d

C

i

s

th

e

se

t

of

c

o

m

pl

ex

n

u

m

b

er

s .

)

1

(b

)

G

iv

e

an

ex

am

pl

e o

f

a

n

in

f

in

ite

gr

o

up

in

w

hi

c

h

ev

e

ry

el

em

e

n

t

ha

s

f

in

it

e

o

rd

er

.

1

x

2

-

4

f

x

~

c

)

L

e

t

f

x

)

=

2

x

2

-

-

2

i

fx

<

O

2

Is

f

R

ie

m

a

nn

in

te

g

ra

b

le i

n

t

h

e

in

te

rv

a

l [

-

1

,

2

]

?

W

h

y

?

D

o

es

t

h

er

e

e

x

is

t a

fu

n

ct

io

n

g s

uc

h

th

a

t g

(

x)

=

f

(x

)?

J

us

ti

fy

y

ou

r

a

n

sw

e

r.

1

(

d

)

P

ro

v

e

th

a

t if

b

e

a +

1

<

1 w

h

e

re

a

a

n

d

b

a

re

p

os

it

iv

e a

n

d

r

ea

l,

th

en

t

h

e

fu

n

ct

io

n

z

n

e-

a

-

b

ez

h

as

n

ze

r

oe

s

i

n th

e

un

i

t c

ir c

le

.

(e) M axim ize z

=

2x

1

+

3x

2

-

5x

3

su

b

je

ct

t o

x

1

x

2

x

3

=

7

an

d

2

x

1

-

5x

2

x

3

10

,

Xj_

?:

0 .

Q

2

a)

8

1

0

~


12/19

Q

3

.

d)

f

co

R

~

f

cr

¥

fc

r¥

l c l i

~

JIO

I;f)<

J

:ef

U M

"'

~

a)

W

ha

t

ar

e t

he

or

der

s

o

f

th

e

f

oll

ow

in g

pe

rm

ut

at

ion

s i

n

8

1

0

?

[1

2

3

4

5

6

7 8

9

10

J

9

a

nd

1

2

3

4 5

)

6 7

).

1

8

7 3

1

5 4 2

b

)

W

h

at

is

th

e

m

ax

im

al p

os

sib

le

or

de

r of

an

el

em

en

t i

n

8

10

?

W

hy

?

Gi

ve

an

e

xam

p

le o

f

su

ch

an

e

lem

e

nt .

H

ow

m

a

ny

e

lem

e

nts

wi

ll th

ere

b

e

i

n

8

10

o

f

th

a

t

o

rd

er

?

CX l

- l

)n

-

1

Sh

ow

th

at

the

se

rie

s

L

c)

-

-

2-

,

i

s un

if

o rm

ly

c

on

ve

rg

en t

b

ut

no

t

1

n +

x

ab

solu

t

ely

fo

r a

ll re

a

l v

al

ue

s ofx

.

d)

S

ho

w

tha

t

ev

ery

o

pen

s

u

bs

et o

f

R

is

a c

ou

n t

ab

le

uni

on

of

d

is

jo i

n t open

in

t

er

va

ls .

a

)

l

iR

J

={

a+

bi

I a

, bE

Z}

J

l1

3f

fl l

l

~

C ~

J

qq

C1

< f)

~

I

J

f?

l=i

R fu

l

d i

t

en

t :r

-m

~

:

(

>

rR

r)

,

: 01

\ill

Cl cll

s1

iR

(

>rR

r),

: O

I"'

@sO

i

(

>

rRr

)

?

~

4

>1M

t

>

I

b

)

l

iR

R

C =

[0

, 1

]

CR

tNT

C

II'R1

f¢lC

f> l H

iRf

>

d

B f

fif

~

~

~

f.:

n:;r

B

f9t>

<

l3"

l

~

.

f

+ g) x = f x) + g x)

fg)

x =

f

x) g

x)

.

l

l

M

=

{ r

e

R

c

I i

~ o

}

M

, R

<

tt

:

OI\l1

1Ci

ci1

~ ?

~

dC

fi

I

c

)

l

lR

f x

, y

) =

y

2

+ 4

x

y +

3

x

2

+

x

3

+

1

~

I m

~

_

m

e n :

f

x,

y)

~

3lf'E

fC

f)(fl

1 l ~

d

q

?

(d )

liT-1

[x ]

C::tl'k' f"ClCfl

x

CfiT

mJT

mRrn

Cf) {ffi

t

n s

X


13/19

a) Let

J =

{

a b

i I

a

, b

E Z} be t

he ring

of

Gau

ss ian

in teg

ers su

brin

g of

C).

W

hich

of the

follo

wing i

s J : Eucl

idean

doma

in , p

rincip

al ide

al doma

in,

u

nique

fac

tor iza t

ion do

m ai .n?

Jus

tify yo

ur

answ

er.

b

) Let RC

=

ring o

f all real

value

d co

ntinuo

us fu

n c t ion

s on [0

, 1], unde

r the

op

eration

s

f +g)

x = f

x) g

x)

fg) x

=

x)

g x).

IsM

a

maximal

ideal

ofR?

Justify

your

answer.

c) L

et

f x, y )

=

y

2

+ 4xy 3

x

2

x

3

+

1 A

t w

hat

point

s wil l f

x, y h

ave a

max

imum

or min

imum

?

d)

Let

[x] denot

e

the

in

teger

part

of the re

al nu

mber

x, i

.e., if

n

x <

n 1

wh

ere

n is an

inte

ger, then

[x]

=

n. Is

t

he

fu

nc t ion

f x)

= [x

]

2

3

Rie

mann

inte

grable

in

[- 1

,

2]

? I

f not ,

expla

in w h

y .

f

it

i s

i

ntegra

bl

e,

2

15

15

1

c o m p

ute

J [x]

2

3 dx.

1

-

1

Q

4

a)

rl{ ldJ:l

f mcR

Cf

i

chl fG \ :

M

l

M z

M3

M

4

l

3

12 5

14

Jz

7

9

8

1

2

J

3

5

1

1

1

12

J4

6

14

4

11

A

BRL M

N BU

B


14/19

IT

I = in

4

8 de

0

Cf f RCfllf IQO

I

c) 4._ i


15/19

~

S

E

CT

IO

N

B

Q

5

(a

)

z =

y

f (x

)

x

g

(y )

: f l

~ f

~

g

~

fcl

0'14

' 1

fp:f)

Cf)

{Ul

I

b

)

a

2

z

a

z

a

z

fp:f)

Cf)

{Uf

y

-

+

(X

+

y

) -

-

+

X

-

= 0

Ox

O

x O

y

Oy

2

(C

h' i

lRC

h


16/19

c)

In

a

n e

xa

min

ati

on ,

th

e n

um

be

r

of stu

den

ts wh

o

ob

tai

ned

m

ar

ks

bet

we

en

ce

rtai

n lim

it

s w

er

e

gi

ven

in

th

e fo l

low

ing

t

abl

e :

Ma

rks

30

-

40

1 40

-

5

0

5

0 -

6

0 60

-

70

70

-

80

N

o.

ofS

tud

en

ts

31

I

42

51

35

31

U

sin

g N

ew

ton

fo

rw

ard

in t

erp

ola

tio

n for

mu

la,

fm

d

th

e n

um

be

r

of

stu

den

ts wh

os

e m

ark

s lie

bet

we

en 45

a

nd

50

.

d)

Pr

ove

tha

t

t

he

n

ece

s

sar

y an

d su f

f ic i

en t

on

diti

on

th

at th

e

vo

rte

x li

nes

ma

y

be

at ri

ght

ang

les

t o

th e

s

tre

am

li

nes

a

re

u,

v

,

w

= J l

Ckp

,

c3q

>,

o

P

f.1

W

;

fe

nm

t

(f )

f

com

m

t T

R

f

W

it fcnm

m

x T

R i

rtJ

q:)

f

~ ~ c f i M q :

I

a

)

So

lve

CD

2

+ DD

- 6

D

2

z

=

x

2

sin

x

+

y)

·

a

a

wh

er

e

D

and

D

d

en

ote

-

a

nd

-

.

ax

ay

A

R

M

NB

UB

10

10

1

5


17/19

(b

)

Find the sur

face w

hich

inte

rsects

t

he su

rfaces

of

the sy

stem

z(x + y

)

=

C(3

z

+

1)

, (C b

eing a

c

onstan

t)

orth

og ona

l ly and

whi

ch pa

sses

thr

ough the

circl

e x

2

+ y

2

= 1

z

= 1

5

(c)

A

tigh

tl

y

stretc

hed strin

g

w

ith fix

ed

end p o in

ts x

= 0

an

d

x

=

IS

in

itially

at res

t in

equili

brium p

osit ion

. If

it is s

et v

ibratin

g

b

y g iv in

g

eac

h poin

t

a

veloci

ty A

. x

l - x

),

find

the displa

ceme

nt of t

he

string

at

any

distan

ce

x

from o

ne

end at

any

time

t.

Q7

(a)

o ~

~ f(x)

=

0

cn ~

~

~

-

fu

fu ~

W

~

~ c - -

f { ~ fcl

fBd ~

n

I ~ R 1 l l l

efa

ep

s f.:rm

ftf ~ 8 J

~ C1

2IT f (

x)

n fu Q:

f.:rm

fur

W

;r

~

~ I

(b )

l i fll

iflll

y = x

(y

+

x

) - 2

y O)

=

2

~

~

l i f l C I f

~ ~

fltt,

y 0·6)

CfiT flblCh(

lfR

q

f{f;hfe;ld

if>

~ ~

3WfT1=J

Rtr ~

h

=

0

·15

ferftr Cfi1

I

c)

fcm

11

~ -mi'l:f

~

{ f l l 11:$)

CfiT ~

q

fiR

Uft

q Tfm

~

I

~ q ~ C 1 2 I T ~

k ~ - q ~

I

t

2

4

6

8

10

12

14

16

18 2

0

v

16

28·8

40

4

6·4 5

1·2 3

2·0

17 ·6

8

3 ·

2

0

(a )

D ev

elop an

algor

ithm for N

ewton

-

Raph

son me

thod

to

so

lve f(x

) =

0

start

ing w

ith ini

tial it

erate x

0

n

be

the

numb

er of i

teratio

ns

al

lowed,

ep s

be th

e presc

ribed

relativ

e

error an

d d

elta b

e the pres

cribed

l ower

2

b

ound for f (

x) .

2

A B

RL M

NBUB


18/19

(b

)

U

s

e

E

u

le

r

s

m

e

th

o

d

w

i

t

h

s

t

e

p

s

iz

e

h

=

0

·1

5

t

o

co

m

p

u

te

th

e

a

p

p

ro

x

i

m

at

e

v

a

l

u

e

of

y

(0

·

6

),

c

o

rr

e

c

t

u

p

t

o

f

iv

e

d

ec

i

m

a

l

p

la

c

es

f

r

o

m

t

h

e

i

n

it

i a

l

v

al

u

e

p

ro

b

l

e

m

y

=

x

(y

+

x

) -

2

y O)

=

2

(

c

)

T

h

e v

e

l

oc

i t

y

o

f

a

tr

ai

n

w

h

ic

h

s

ta

rt

s

f

ro

m

re

s

t

is

gi

v

en

i

n

t

h

e

f

o l

lo

w

i

n

g

ta

b

l

e .

T

h

e

t

im

e

is

in

m

i

nu

t

e

s

a

n d

v

el

o

c

it

y

i

s i

n

k

m

/h

o

u

r.

t

2

4

6

8

1

0

12

1

4

1

6

18

20

v

1

6

2

8

·

8

4

0

4

6

·4

5

1

·2

3

2

·0

17

·

6

8

3

·

2

0

E

s

t

im

at

e

a

pp

r

o

x

im

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ly

t

h

e

to

t

a

l d

is

t

an

c

e

r

u

n

i

n

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0 m

i

n

ute

s

b

y

u

si

n

g

5

c

om

p

osite

Simpson

s

. _

rul

e.

5

3

Q

S

(

a)

l

m

~

A

B

B

C

,

B

t

n:

fif

c

qo

f

~

A

~

3

fn

:

A

-B

d i

~

h

n

C

R

W

~

I

fc

1

S

l("

lllO

J

H

I

C

flK

1

~

n

2

=

3

±

~

)

7

(

b)

~

(

f {

ff

,

x

-

31

1

t

H

klO

JC

f>

fu

n

T

R

3

'1

C

lC

f>l

t

l

a

bf

"

Cf>

1 tR

d

T

t

it

fc

l1

~

q

fH

f1

lO

ll

~

f

~

~

~

(

0

,

a

)

"Y

l: ~

m

e

t

m

~ 3

fn

:

0

,

b

)

TR

m

ff

r

n

~

f

~

~

=;f

fU

II r

JO

Cf)

"Y

l:

l

TG

ff

~

3

1-

R

f

tR

~

~

t

d

l

fc

1

qf

<

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fto

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s:

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m

2

(

b

)

2

t

p

C

flT

t

:f.1

fC

J

~

1

{

2

a

b

(

a

+

b

)}

c )

~

m

~

K

n

("

JU

;f

{@

l

< l

a

~

~

("

lt:t

f?

l(1

~

~

~

ef

t

~

2

3

~

~

3

fu

:

B

S

n

a

~

1"

ffu

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I

f

cn

m

n

)

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t

fi

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Yl:

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w

rn

I

A

B

R

L

M

N

B

U

B

1


19/19

a)

Two

e

qu l

rods

AB

a

nd

BC, e

ch o

f len g

th l sm

oothl

y

jo

inte

d

at

B , are

su

spend

ed fro

m A and

osc

illate in

a ve

r t ical p

l

ane

thro

ugh A

. Sh

ow tha

t

the

per

iods of

no

r

mal

osci

llation

s

ar

e

2

wher

e n

2

=

3

± ~

g .

5

n

~

b)

I f

luid fills

the

region of space on

the

positive side of

the

x -axis, which is

a

rig id

bound

ry and

if

th

ere

b

e a so

urce m

at

the

poin

t 0, a

) and a

n

equ l sink

a

t 0,

b)

an

d if

the pre

ssure

on

the

ne

gative

s ide b

e

the

sam

e

s the

pre

ssure at

i

nf inity

, show

th

at the re

sul tan

t

pressu

re

on

th

e

r

rpm

2

a - b)

2

b ou

n d ary

is

wh

ere p

is the

den

sity of th

e flu

id.

5

{2ab

a+ b

)}

c

) If

n

rec ti li

near v

ortice

s o f th

e same str

ength

K

a

re

sym

m etr i

cal ly

arran

ged as

g

ener

tors o

f a cir

cular

cylind

er of ra

dius

a

in a

n

in fin

ite

li quid, pro

ve

that

the

v

ortice

s w ill m

ove roun

d the

cyl

inder

uniform

ly

in

. 2

3

time

Srr a

. Find

the

veloc

ity

at

any

po

int of the

liq

uid.

20

n

-1

K

A BRL

M

N

BUB

Documents

IAS Mains Mathematics 2013