2nd puc mathematics mid term nov 2015.pdf

8/20/2019 2nd PUC Mathematics Mid Term Nov 2015.pdf

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RDPUCPA

.

.

E

DISTRTCT

LEVEL

MID-TERM

EXAMINATION,

November

2015

Time

:

3%

Hours

II

PU.C.

MATHEMATTCS

(3s)

Max.

Marks : 100

'l'r

i

Instructions

: The

question

paper

has

giveparts

narnely

A,B,C,E

and E.

Answer

all

the parts,

PART:A

I.

Answer all

the

questions.

1.

Define

one

-one

function.

2.

Give

an example

of a

relation

which

is

reflexive,

transitive

but not symmetric.

3.

Wirte

the

domain of

-f

(x):

Cosec''x.

4.

Find

the

principal

value

of

Cot'

(+

)

s

rfA:[

] l]

then

nnd

l3Al

6. Find

the

value

of

'x'if

l?il:|,|/-

il

7.

Define

a

scalar

matrix

8. If

y:

Cos

(ax

+

b), find

ff

.

9. Find

the

derivative

of

tan

(6)

with

respect to

x.

10'

r'indJ(3x'i.nrn:.

pARr_B

III.

Answer

any

tefi1qu".&rrr.

^Ar\r-u

l0

x2

:20

11.

Find *hdth&

th"

bTnary

operation'*'

defined

by a;6b

:

+on

the

set

e

of rational

numbers

is

commutative.and

associative.

l?.

Arelation'Ris

definedonthe

setA:

{1,2,3,4,5,6}

byR:

{(ay):yisdivisiblebyr}.

Verify

whether R is symrnetric

and

reflexive.

13.

If

Sin

{Sin

'

(}

)

+

Cos-'x}

:

I, them

find

x.

14*

Find

Cos'

{Cos

(+)}

15.

Prove

that

,,,

[+]+ran'

t+]

:tan't+]

16_

Findxandy,i12[r

i].[l

g]:Ii

S]

17

. Find

the

area

of the

triangle

whose

*,e*i"",

u..

(2,

7),(1,

l

)

and (i

0,

8)

using

determinants.

.

18.

Find

the

equation

of

line

joining

(3,

l)

and

(9,

3)

using

determinants.

19.

Check the

continuity

of

the

function

/(x)

:

3x - 4

at

x

:

-1.

20.

If

y:

Sin

(log".),

prove

that

4-

@_

21.

If

the

radiu,

qf

1

sphere

l, *"f;Jr."d

uf

q

cm.

With an'effor

0.03cm,

then

find

rhe

approximate

error

in

calculating

its volume

l

22.

Find points

on the curve

x'

*

y'

:

i

at

which

the tangents

drawn to

the

curve

are

parallel

to

x_

axis.

9

lU-

23.

Find

the

local maximum

value

of

the

function/(x)

:

xr-3.t.

24.

Find flog

x

.

dx

J

l0Xt:10

P.T.O.

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2/3

lII.

Answer

any ten

questions.

3X10--30

25.

Verify

whether

the

functiony':.N

+,N

defined

by

,

(n+1

'

:c-':-

'tt

r-;- .

ifrisodd

f(n\:1

z

is

one-one

and

onto or

not.

t n

ifriseven

(-,

)

26.

Show

that the

relation 'R'

in

the

set

of all

intezess

'Z'

defined

by

R:{(a,

b) :

2

divides

(a-b)}

is an

eqvivalence

relation.

27.

28.

29.

30.

31.

32.

33.

34.

35.

36.

37.

Express,"r'(-_qffi),

+(

x

(

*

rnthe

simptest

fonn.

Using

elemcnrary

transformations,

find

the

inverse

",

[?

_5]

Express

[;

;]

as

the sum

of

a

symmetric

and

skewiymetric

matrices.

If

x:

a

(O+sing;

and;,

:

a

(7-Cos0),

prove

,t,u,

$:

tan

(-- -).

Verifo

mean

value

theorem

for

f

(x)

:

x' -4x-3

in the

interval

[1

,

4].

Find two

positive

numbers

x

and

y

such

that

x+y:60

and

xli

is maximum.

Veriff

Rolle's

theorem

for

the

function

f

(x)

:

yr+2x-g,

xel_4,

2).

lf

x2+par':100,

then

find

4'

Differentiate

(log

xy

-

1-.i"=4to-.r.t.

'-r,.

Ifx:\Fl-l'

,

/:.,r;s,.-u

,

then

show tbat

dY

:

+

Find

fSin

(ax+b)

.

Cos(ax+b)

.

dr.

)

pART-D

IV

Ansrver

anv

six questions.

5 X

6:30

39.

If

R-

is the

set

of

all

non-negative

real

numbers.

Prove

that

the

function/:

R-

--+

[-5,

*)

defined

by

-f

(x)

:

9x'+6v-5

is invertible.

Also

find

f

'

.

40.

Define

invertible

function

and

let/:

N

-+

y

be

a

function

defined

as

-f

(x):

4x*3,

where-y

:

fveN;

t':4x+3,

for

some

xeN).

Show

that'f

is

invertible.

Find

the

inverse

of

'-f

'

.

on':.o

"(;

1)'

:(?

3)'

t:

(;

I

)'

carcurateAB'

BC'4(8+61

Verify

that

AB+AC

:

A(B+C).

'

42y'

lf

A:

find

A

4

Solve the

system

of

linear

equations

using

matrix

method:

2x+y*z:l;

x-2y-z:

3

;3/>5y:9.

44.

Solve

:

x*y*z:6;

x-2y+322:

6;

x-y+z:2byusing

matrix

method.

38.

P.T.O.



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3/3

::3::

45. If

y:

3 Cos (iog

x)

+

4

Sin

(log

x), show

that

x'y,+),,*y

:

O.

46.

If

y

:

a'

"n-'',

prove

that

(l-x'?)

z-x r

-

*'y

:

o.

.

47

.

The volume of

a

cube

is increasing

at the rate

of

9CClsec.

Find the

rate

atwhich

its

length

of a

side and

its

surface

area

increases

when the

length

of its

edge

is 10

cm.

48.

Find

the

integral

of

ul

,

w.r.t.

'x'

and hence

evaluate

[___Z__

ax

x -a

J*o-16

PART.E

V.

Answer

any

one

question.

lz

-1

r'l

49.

a) IfA:l

-i

2

-1

|

,

thenverify

that43-6A'+9A-4I:

O

and

findA-'.

Lr

-r

2)'

(

s

;if

x

2nd puc mathematics mid term nov 2015.pdf

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