8/20/2019 2nd PU Maths 2014.pdf

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++r

8/20/2019 2nd PU Maths 2014.pdf

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)

(-+

-\/-

-+\

-

l-l

2l .lf

a

is

a

unit

vector such

that

[r-

,

Jlr*

o

)=

f,

U*

l'l

22.\f

theposition

vectors

of

the

points

A

and

B

are

?+

Z

i-li

&

i-lc

respectively,

then

find

the

direction

10X3-30

(o,b)*(c,d)=(a+c,b+d),

prove that

*

is

both

'

cosines

of.

AB .

23.

Find

the

equation

the

plane

passing througlr

the

intersection

of the

planes x+

y + z

=

6

and

2x+3y+42-5=0 andthepoint(1,1,1).

+:..

24.

A

dieis

thrown.

If E is the

event

'the

number

appea{rng

is

a

multiple

of

3'

and

F is the

event

'

-,,qr1.p1 aOpeanne

is

even'

then

find

whether

E

and

F

are

independent?

i

PART.

C

Answer

any

TEN

questions:

25.\f

x

is abinary

operation

defined

on

A=N XN'

by

commutative

and

associative.

Find

the

identity

if

it

exists'

26. Prove

that

2tan-r(;).,*-,[})

=,*-,(lJ

lr

21

27. Exeress

L;

Ol

*

*.

sum

of

a symmetric

and

skey

symmetric

matrices'

28.rf ,=,;-,ri-"\

1 I

''dv

'

(1

'=T?)*({E

'nndfi'

zg

.

tf

x

=

a(0+

sin d)

ar.rd

y

=

a(l

-cos

d)

,

prove

*"

*=

,*(;)

30' Find

the

intervals

in which

the

function

'f

given

by

f

(x)

=

4xt

-

6x2

-72x

+ 30

is

(i)

strictly

increasing

(ii)

strictly

decreasing.

31. Find l---dtt-

J

x

* xlogx

32.

Find

I,*-'

x.dx.

33' Find

the

are

bounded by

the

parabola

y2

=

5x

and

the

line

y

-

r'

34.

Form

the

differential equation

representing

the

family

of curves

/ =

c sin(-x

+

b)

,

where

a,

b

are

arbitrary

constants.

'+

'+ -'

-t

-+ -')

-)

'

-)

'-t

-)

-+

-+

.+

35.If

i,bordc

arethreeunitvectorssuchthat

a+b+c=0,findthevalueof

a-b+b.c+c'a'

36.

Find

a

unit

vector

perpendicular to

each

of the

vectors

(i+i)ana(;-b,

where

+ A-+ A

A

a

=

i+

j+k,b

=

i+2

j+3k

.

37. Find

the shortest

distance

between

thelines

/,

and

/,

whose

vector

equations

are;

1

^

\ '+

A A

A

A

-A

-

T

i

=

i+

j+

1(2i-7+k)

and

r

=2i+

j-k+

p(3i-5

j+Tk).

38.

A bag

contains

4 red

and

4 black balls,

another

bag contains

2

red

and

6

black

balls.

One

of

the

two

I

bags is

setected at

random

and

a ballis

Oru*n

tori

ttre bag

which

is

found

to

be

red'

Find

the

probability that the ball

is

drawn

from

the

firstbag'

2

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8/20/2019 2nd PU Maths 2014.pdf

3/3

PART

_

D

Answer

any

Six questions:_

6X5=30

"

::::,.t1*"-Y:':::,

-{

'

^*.:

[4,

""]

defined

bv

/(x)

=

)c' +4

,

where

R+ being

the

set

of

al

non

-

egative

realfl*b:T

is

invertible. Also

write

the inverse

of

(x).

oo

o

n=1f,

?],'=[?

:)*,,=l:

l],;;;;;;;;@+c)

Veri&

that

A(B

#-C)

=

AB

+

rrI

tll.Sclvethefeilowtngsystgili..rf

equar.-.rtrudlru*_-lX:rrlthoo,

x+-,.,+7=u

i

;,*sz.=ii

airr.,

x--2y*

z

=

0.

42.If

y

=

(tan-rx)2

then

show

that

(*,

+gt +2x(x2

+1.)g

43.lstone

is dropped

into

a

quiet

lake

*;f;"r.";,";*"r;;

x2

instant,*r,.niri"u

jil;;il"ffi

ffiXffi

tr;:ffiTltHffi1,::,.'I;1il

j,".Il,lrro'*.

44'

Find

the

integral

of

+

with

respect

to

x

and

hence

evaluate

[g

,tx-

-d'

I

i*z

q5

-r*

w,

45'

Find

the

area

of

the

region

in

the

first

quadrant

encrosed

by

the

x_axis,

the

rine

y:

x and

the

circle

x2 +

yz

_

32.

46.

Solve

the

differential

equ4io

n,

*9*2y

=

xr.logx.

-'

U;Hjle

eeuatlon

of

a

plane

purfne

through

three

non

-

colrinear

points

both

in

the

vector

and

-t

*i::J,::T?,i'"f,:ff1,ll.::$li

an

odd

number"

is

a success,

what

is

the

probability

or(i)5

"i

Answer

any

One

question:

PART

_

E

a

trXI0:trO

2a

ag.

@)Prove

that

-i

a'

.,.1^-

t,n

n

1f

@)dr

=

z

lf

G)dx,

when

[tzr

-

x)

=

f

(x)

and

hence

evaruate

jl*.rFr.

ogo

(b)

Determine

the

value

of

K,

ir

f(x)

=ffi,*

**r,*=

*i

is continuou,

o

,

=+

50' (a)

solve

the

following

linear

programming

problem

graphically;

H,]|*andMaximiie

z=x+2"y,

subjr.tio.or,rtilr,ir";;';;>100,

Z*_ys0,

Zx+y