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MATHS-2 (051) (Question Bank)

1

QUESTION BANK

MATHS-2 (051) E

Question : 1

Q.1 (A) (1) Prove that, limsin

x

x

xÆ=

01 (2)

(2) Prove that, lim , | | |n

nr rÆ•

= <0 (2)

(B) Calculate any two (4)

(1) limlog ( )x

x x x

x xÆ

- - ++0

15 5 3 1

1

(2) limsin

x

x

xÆ

- -

-12

16

2 1

p

(3)lim

cosx

x

xÆ -

-1 1 2

1

e j

(C) (1) Find limx

x mmx nx

xÆ

+ - +0 2

1 1b g b g(2)

(2) Calculate any two (2)

(i) Find the limit of the sequence an

n

n= +

-3 1

3 1

(ii) f xkx

xx

x

b g= - π

= =

10

12 0

2

cos,

,

is continous at x=0, then find k.

(iii) Express 5.277777 .............. as a vulgar fraction.

(D) (1) If y e x xx= +cos sinb g then prove that y y y2 12 2 0- + = (2)

(2) find dy

dx if y

x

xx x= - π >-cot , ,1

21

20 0 (1)

Q.2 (A) (1) Define continuity on an interval and prove that sin ,x x RŒ is continousfunction. (2)

(2) Define limit of a sequance and prove that a n a n Nn

n

n= + Œ1 1e j l q, is

bounded sequence. (2)


2

(B) Find limit of the following Any two. (4)

(1) limsin cos

x

x x

xÆ

--1

3

3

3 p

(2) limx x

x

Æ +0 11p

(3) lim ,x m n

m

x

n

xm n N

Æ --

-

FHG

IKJ Œ

1 1 1

(C) (1) If f xe

kxx

x

b g= - >2 1

0, is continous at x =0 find value of k and a (2)

= <|sin |x x 0

= =a x 0

(D) Do as directed any two. (2)

(1) Find limx

x xÆ-

+8

(2) Find lim cos sec

x

xx

Æ+

0

51 3b g(3) Express as a vulgar fraction : 2.312 312 312 .....

(E) (1) State chain rule for finding derivative of a composite function. (1)

(2) If f a b R: ,b gÆ is differentiable function at x a bŒ ,b g then prove that it is also

continues at x. (2)

Q.3 (A) (1) If limx a

f xÆ

b g and limx a

g xÆ

b g exist, then prove that

lim lim limx a x a x a

f x g x f x g xÆ Æ Æ

+ = +b g b gm r b g b g (2)

(2) Prove that lim , | | ,n

nr r n NÆ•

= < Œ0 1 (2)

(B) Evaluate any two limits. (4)

(1) lim tanx

xx

Æ-

11

2b g p

(2) lim .n

rn

r

n

nÆ• =Â 1

31

(3) limy

yy

yÆ

+-

FHG

IKJ0

12 1

1 2


3

(C) (1) Let f xx

x ex e

k x e

b g= --

π

= =

log,

,

1

If f is continous at x e= find k. (2)

(2) Answer any two. (2)

(i) obtain the limit of the sequence 1.7, 1.77, 1.777, ......

(ii) Find the smallest m NŒ such that " ≥ Œ fi - <n n m n N an, , | | .1 0 001

where an

n

n= +2 1

2

(iii) Express x R xŒ

£ + <RSTUVW0 3 1 2|| | in an interval and N a,db g forms.

(D) (1) Prove that f x x ab g= -| | is not differentiable only at x a= . (2)

(2) Find d

dxx

an

nloge j (1)

Q.4 (A) (1) Prove that limsin

q

qqÆ

=0

1 (2)

(2) Prove that lim ; | | .n

nr rÆ•

= <0 1 (2)

(B) Find the following limit (any two) (4)

(1) limsin cos

q p

q q

q pÆ

- -

-42

2

4b g

(2) limcos

tanx

x

xÆ

+1 2

1 pp

(3) limlogx

x x

e

e e

x xÆ

-- ++0

2

1b g

(C) (1) Find lim tanx

x xÆ

-1

12

b g p(2)

(2) Attempt any two. (2)

(1) limn

n

n nÆ•

ÂÂ

2

(2) lim cotx

x xÆ0

5

(3) If N a, ,db g b g= 1 9 then find d and a.


4

(D) (1) Obtain d

dxxsind i; using the definition. (2)

(2) If f x e xb g= - log , then find f x'b g (1)

Q.5 (A) (1) If a read function f is defined every where in some deleted neighbourhood

of a and limx a

f xÆ

b g exists, then prove that this limit is unique.

(2) Define limit of a sequance and prove that limn nÆ•

=10

(B) Solve any two. (4)

(1) limsin sin

x

x x

xÆ

-

0 33

13

(2) limx a

x ae e

x aÆ

--

2 2

(3) limn r

n

rÆ• = -Â 1

4 121

(C) (1) Find the limit lim .n

rn

r

n

nÆ• =Â 1

31

(2)

(2) Fill in the blanks with required calculation (any two) (2)

(1) lim ............ , ,n

n n

n nÆ• +-

+= -L

NMOQP

3 5

3 50 1

1

51

(2) x o x x R/ | | |; ............. , , , , ,< + < Œ = - - - - - - - -2 3 1 2 1 3 2 3 1l q b g l q b g l q b g

(3) lim .............. , ,n

n

n nÆ•

ÂÂ

= LNM

OQP

3

2

2

3

3

42

(D) (1) Differentiable cos x2 by definition w, r, t, x. (2)

(2) Show that f x xb g= | | is not differentiable at 0 (1)

Q.6 (A) (1) P.T. limsin

q

qqÆ

=0

1 and 0 2< <q p from deduce limtan

q

qqÆ

=0

1 (2)

(2) P.T. lim | | |.n

nr r n NÆ•

= < Œ0 (2)

(B) Solve any two (4)

(1) limtan tan

tan tanx a

x a

x aÆ - ---1 1


5

(2) limsin

tanx

x

xÆ

--p

6

2 1

3 1

(3) limx

x x x

xÆ

- - +0 2

6 3 2 1

(C) (1) limx

n mmx nx

xÆ

+ - +0 2

1 1b g b g(2)

(2) Fill up blanks with necessary calculations. (2)

(1) The range of f R k R f x x: , tan- - Æ =2 1 2b go t b gp is .........

z z R, ,- +0l qe j

(2) lim ............ , , ,x

xx

xe

ee eÆ•

-+

FHG

IKJ = F

HIK

1

11 12

2

(D) (1) Find dy

dx if y x x

x x= +d i (2)

(2) Find dy

dx if y xe= log sin 2e j


x

x

xÆ=

01 and then prove 0 2< <x p (2)

(2) Derive lim ,| | |.n

nr r n NÆ•

= < Œ0


(1) limsin

tanx

x

xÆ

--p

6

2 1

3 1

(2) limsin sin

x a

x a

x aÆ

- ---

1 1

(3) limsin

x

x

xÆ

- -

-12

16

2 1

p

(C) (1) Solve any two.

(i) If an

n N n n m n Nnn = Œ " ≥ Œ fi - <Œ1 1

0; . , , then for Œ= 0 03. Find the

minimum value of m.

(ii) limx

xxÆ

-0

5 1d i find the limit


6

(iii) If N N3 5, ,d db g b g« = ∆ then find the maximum value of d

(D) (1) If y xx

a bx=

+FHG

IKJlog then P.T. x

d y

dxx

dy

dxy3

2

2

2

= -FHG

IKJ

(2) If f x f x'b g b g= and f 0 1b g= then find the value of limx

f x

xÆ

-0

1b g


q

qqÆ

=0

1 (2)

(2) Prove that anl q; where ann

n Nn

= +FHG

IKJ Œ -1

11; l q is a bounded sequnce. (2)

(B) Evaluate any two (4)

(1) limsin

tanx

x

xÆ

--p

6

2 1

3 1

(2) limtan tan

tan tanx a

x a

x aÆ - ---1 1

(3) limsin

x x x

x x

e eÆ -- +0 2

(C) (1) If f xe e

e ex

x x

x xb g= -

+π

-

-

1 1

1 10; then prove that lim

xf x

Æ0b g does not exist. (2)

(2) Attempt any two. (2)

(i) If lim .cos lim cosq q

q q q qÆ Æ

=0 0

k es esk then prove that k = ±1

(ii) Find lim! !

! !n

n n

n nÆ•

+ + ++ - +

3 2

3 2

b g b gb g b g

(iii) If limn

nk

ne

Æ•+F

HGIKJ =1

4 3

then find k.

(D) (1) Find derivative of sin-1 x using the definition of a derivative, where | |x <1 (2)

(2) If log y x= then find yn

(1)

Q.9 (A) (1) P.T. Sine function is continuous on R. (2)

(2) P.T. seqn

n

. 11+F

HGIKJ

RS|T|

UV|W| is bounded sequence and obtain its limit. (2)


7

(B) Attempt any two. (4)

(1) limsin

sinsin

x

x

x xx

xÆ

-FHG

IKJ0

(2) limtan tan

tan tanx a

x a

x aÆ - ---1 1

(3) limx

x

i

n

x eÆ• =

-FHG

IKJÂ

1

1

1

(C) (1) Find limtan sin

x

x x

xÆ

- --0

1 1

3 (2)

(2) Answer the following (any 2) (2)

(i) Is it true that lim . lim . limx a x a x a


=b g b g b g b g ? Why ?

(ii) P.T. lim sinx

xxÆ

=0

10

(iii) Find limsin

x

x

xÆ -p p

(D) (i) Find dy

dx for y t t= + -cot 1 where t x= +1 2 (2)

(ii) If x y a2

32

32

3+ = then find dy

dx(1)


q

qqÆ

=0

1 (2)

(2) Prove that limn nÆ•

=10 (2)

(B) Solve any (Two) (4)

(1) Find limx

x x

xÆ

+ - +-2

14

14

3

3 3 4 1

8

b g b g

(2) Using definition prove that limx

xÆ-

- = -2

3 1 7

(3) Find, lim , ,x m n

m

x

n

xm n N

Æ --

-

FHG

IKJ Œ

1 1 1


8

(C) (1) Find limn r

n rn

nÆ• =Â 1

171

b g (2)

(2) Solve any 2. (2)

(1) Express the following as a vulgar fraction 2.123 123 123 ---------

(2) If f xx

xxb g= πsin

,2

30 continuous atx =0 find K.

(3) Find limh

x h x

hÆ

+ -0

14

14b g b g

(D) (1) For the functionx y y x a3 2 3 21 1- + - = finddy

dx | | |, | |x y< < 1b g (2)

(2) Finddy

dx for y ax= log (1)

Q-11 (A) (1)Prove that limsin

x

x

xÆ=

01 (2)

(2) Prove that lim ,| | |n

nr rÆ•

= <0 (2)


(1) limx

rx

i

n

x eÆ• =

-FHG

IKJÂ 1

1

(2)lim

sin cos

x

x x

xÆ

-

-p p6

3

6

(3) limlogx

x x x

x xÆ

- - ++0

15 5 3 1

1b g

(C) (1) Solve limsin

sinsin

x

x

x xx

xÆ

-FHG

IKJ0

(2)

(2) Calculate any two (2)

(1) Prove that limsin

x

x

xÆ

-=

0

1

1

(2) Express 3. 12 12 12 ------- as a vulgar function

(3) Examine the continuity of f x x x Rb g= Œ, at x =1


9

(D) (1) Ifsin siny x a y= +b g then prove dy

dx

a y

a=

+sin

sin

2 b g(2)

(2) Find d

dxxlog logb g

Q.12 (A) (1) If limx a

f xÆ

b g and limx a

g xÆ

b g exists then prove that

lim lim limx a x a x a


+ = +b g b gc h b g b g (2)

(2) Prove that lim logh

haa

he a

Æ

- = >0

10b g and hence prove lim

h

he

hÆ

- =0

11 (2)

(B) Attempt any two (4)

(1) Find limsec sec

y

x y x y x x

yÆ

+ + -0

b g b g

(2) Find limlog log

x

x x

xÆ

+ - -0

5 5b g b g

(3) Find limx

rn

r

n

x eÆ• =

-FHG

IKJÂ2 1

1

(C) (1) Solve lim

cos sin

x

x x

xÆ

- --p p

6

2

2 3

6b g (2)

(2) Calculate any two. (2)

(i) Find lim tanx

xxÆ•

1

(ii) Prove that lim sinx

xxÆ

=0

10

(iii) Find limn n nÆ• +Â 1

1b g

(D) (1) If cos log- FHG

IKJ = F

HGIKJ

1 y

b

x

n

x

then prove that x y xy n y22 1

2 0+ + = (2)

(2) Find d

dxxsin 0e j (1)


10

Question : 2

Q.1 (A) (1) If f a b R: ,b gÆ is differentiable at x a bŒ ,b g, then prove that f is continuous

at x. (2)

(2) (i) Obtain d

dxx

xcos- = -

-1

2

1

1e j , where | | |x < (1)

(ii) State the Rolle's theorem (1)


(1) If x t t y t= - =2 23 , then find d y

dx

2

2

(2) If y x xx x= + +1 1

1b g find dy

dx

(3) Find differentiation of f x xb g= sin2 by using the definition.

(C) Calculate any two. (4)(1) A particle projected vertically upwords returns to the earth in 8 seconds.

If S ut t= - 4 9 2. (S is in meters, t in sec) find the initial velocity of the

particle.

(2) A circle lamina expands due to heat in such a way that its radius in-creases from 12.5 cm to 12.65 cm. What is the increase in area ?

(3) Verify Rolle's theorem for f x x x xb g= - Œ -3 4 2 2, ,

(4) Find the equation of the tangent to 9 4 362 2x y+ = which is perpendicular

to 2 3 1 0x y- + =

(D) (i) Prove that for y ex

c= the length of the subtangent is a constant at any point. (3)

(ii) Prove that e x x Rx > + Œ1 ,

(iii) Find the approximate value of sin .-1 0 49b g

Q.2 (A) If d

dxx nxn n= -1 using this find

d

dxx n

1(1)

(B) Do as directed any two (4)

(1) If x y y x a x y1 1 1 12 2- + - = < <, | | ,| | find dydx

(2) If y e x xx= +(cos sin ) prove y y y2 12 2 0- + =

(3) If y x x x= + - <- -cos cos ,| |1 1 21 1 find dydx

(C) (1) If f a f a g a g ab g b g b g b g= = = - =2 1 1 2, ' , , ' then find limx a

g x f a f x g a

x aÆ

--

b g b g b g b g(2)


11

(2) Fill up the blanks with need full calculation. (2)

(i)d

dxex

10 3 101log ..............¥ + -FH IK =

(ii)d

dx

x

xtan ..............-

+=1

2

2

1 8

(D) Attempt any two. (4)(1) A particle projected vertically upwards returns to the earth in 8 second. If

s ut t s= - 4 9 2. , is in meters and t is in seconds, find the initial velocity of

particle.

(2) If the line y x= -4 1 touches the curve y ax b2 3= + at point (2,3) find value

of a and b.

(3) Find two positive real numbers whose sum is 60 and the product of one ofthem with the cube of the other is maximum.

(4) By mean value theorem prove that if x>0 then x

xx x

1 21

+< <-tan .

(E) Prove that tan x

x is an increasing function over 0

2,pF

HGIKJ (2)

Q.3 (A) (1) If is defined on (a,b) and is differentiable at x a bŒ ,b g then prove that f is

continuous at x. (2)

(2) (i) Prove that d

dxx

xcos- = -

-1

2

1

1 where | x | < 1. (1)

(ii) Explain the Geometric interpretation of Rolle's Theorem. (1)

(B) Attempt (Any Two) (4)

(1) If x a y a k k z= - = - π Œ1 cos ; sin , ,q q q q pb g b g find d y

dx

2

2 (4)

(2) If y x x= -FH IK-sin 1 22 1 find dy

dx if | |x < 1

2

(3) xy x y= +sin b g find dy

dx

(C) Answer the following (any two) (4)

(1) Prove that for xx

xx x>

+< <-0

1 21, tan .

(2) Which line passing through (3,4) makes a triangle of minimum area with heaxes in first quadrant.


12

(3) If the line 4 4 5= x / touches the curve y ax b2 3= + at (2,3) find a and b.

(4) Equation of motion of a particle projected vertically upwards is S pt qt= +2

(S in meters, t in seconds). If the maximum height attained by the particleis 4.9m and if its acceleration is -9.8m/sec2, find the height of the particleat t = 1/2.

(D) Answer the following questions. (3)

(1) Find the approximate value of log log .10 1099 0 4343e =b g(2) The equation of linear motion of a particle is S t t t= - + +3 26 9 4 find S,

when a=0.

(3) Obtain the equation of the tangent y x= sin at x = p4

.

Q.4 (A) (1) If f is defined on (a,b) and is differentiable at xŒ (a,b) then P.T. f iscontinous at x. (2)

(2) Using d

dxx

x xxsec

| |,| |- =

->1

2

1

11e j , deduce

d

dxec xcos .........- =1e j (1)

(3) State Rolle's theoram cancellation (1)(B) Solve any two. (4)

(1) If y em x=-tan ,

1b g P.T. 1 2 022 1+ + - =x y x m ye j b g

(2) Find dy

dx if y

x

xx=

+ -FHGG

IKJJ π-tan ,1

21 10

(3) Find d y

dx

2

2 if x a= +FHG

IKJcos log.tanq q

2 and y a= sinq

(C) Evaluate any two (4)(1) Sand is being collected in the form of a cone at the rate of 10 cubic meter

/sec. The radius of the base is always the half of the height. Find rate atwhich the height increases, when the height is 5m.

(2) P.T. 4 9 452 2x y+ = and x y2 24 5- = intersect orthogonally.

(3) Find global and local extreme values of f x x x xb g= + Œsin , ,2 0 p

(D) Answer the following. (3)

(1) P.T. tan x

x is increasing function on 0

2,pF

HGIKJ

(2) Apply mean value theorem, for f x x xb g= Œ -cos , ,1 0

(3) If the line y x= -4 5, touches the curve y ax b2 3= + at (2,3), find a and b.


13

Q.2 (A) (1) If f g a b R, :( , )Æ are both differentiable at x a bŒ ,b g then prove that f g. is

also differentiable at x and d

dxf x g x f x

d

dxg x g x

d

dxf xb g b g b g b g b g b g. .= + (2)

(2) (a) State the rule of the derivative of an inverse function. (1)(b) State the mean value theorem. (1)


(1) If 10 10 10x y x y+ = + then find dy

dx

(2) If y x x x= - < <-sin ;1 33 4 12 1e j then find

dy

dx

(3) If xe et t

= - -

2 and y

e et t

= + -

2 then prove that

d y

dx y

2

2 3

1=

(C) Answer any two. (4)

(1) Prove that if l l1 2π then the curves x

a

y

b

2

21

2

21

1+

++

=l l and

x

a

y

b

2

22

2

22

1+

++

=l l intersect each other orthogonally.

(2) A train starts at 8 O'clock and moves eastward at the speed of 75 km/h.Another train starts from the same place at 9 O'clock and travels southwardat the speed of 100 km/h. Find the rate at which they are moving away fromeach other at 12 O'clock.

(3) The resultant resistance R due to two resistances R1 and R2 is given by the

formula 1 1 1

1 2R R R= + . If R R1 2+ is constant, Prove that R is maximum when

R R1 2= .

(D) (1) Find the rate of increase of the area of a circle with respect to diameter. (1)

(2) Prove that f xx

xb g= tan

is increasing over 02

,pF

HGIKJ (1)

(3) Find an approximate value of e1002. where e = 2 71828. (1)

Q.5 (A) (1) State the rule of derivative of Invererse function and P.T.

d

dxx

x xxsec

| |,| |- =

->1

2

1

11 (2)

(2) (i) State chain Rule (1)(ii) Explain gcometric meaning of M.V. thm. (1)


14


(1) If e xx y y- = then P.T. dy

dx

xy x

xy y

=FH IK -

FH IK -

log

log

1

1

(2) If log log siny x x= - 2 then y xy x y2 124 4 3 0+ + + =e j

(3) If x a= +FHG

IKJcos log tanq q

2 and y a

kk z= π Œsin , ,q q p

2 then find

d y

dx

2

2

(C) Attempt Any two. (4)

(1) Find the local extreme values of f x x x x x Rb g= - + + Œ3 10 6 54 3 2 ; .

(2) The point A -1 1,b g and B 2 4,b g are on the curve y x= 2 Find the co-ordinates

targent point of the tangent parallel to AB

¨ Ææ of this curve.

(3) By cutting equal sequares from the four corners of a 21 x 16 tin sheet, a boxto be constructed what should be the length of each square if the volume ofthe box is to be maximum ?

(4) If x >0, then P.T. log 12

2

+ > -x xxb g

(D) (1) Find the approximate value of 80 794 . (1)

(2) Find the lenght of subnormal and subtangent at the point (4,8) of the curve

y x2 3= (1)

(3) Show that f x x x xb g e j= - Œtan , ,0 2p is an increasing function. (1)

Q-6 (A) (1) Prove that d

dxx

xxsin | |- =

-<1

2

1

11e j (2)

(2) (1) State the rule of composite derivative (2)

(2) State Roll's Theorem.

(B) Attempt any (Two) (4)

(1) Ifx at y at t= = π2 2 0, then prove that yd y

dx

dy

dx

2

2

2

0+ FHGIKJ =

(2) If y e ex x= + - prove 4 22 1xy y y+ =

(3) Using definition find derivative of y e x=-sin 1


15

(C) Attempt any (Two) (4)

(1) Find the radian measure of the angle between the tangents to y ax2 4= and

x ay2 4= at their point of intersection other then the origen.

(2) Sand is being coelected in the form of a cone at the rate of 10 as m/sec Theradius of the base is always half of the hight. Find the rate at which theheight inaeases when the hight is 5m.

(3) Verify Rolles theovem for f x x x xb g= - Œsin sin , ,a p0

(4) Find the global and local extreme values of f x x x x xb g= - - + +3 2 6 6 14 3 2 in

[0,2]

(D) (1) Find the approximate value of log10 999 (1)

(2) Prove that f xx

xb g=

sin is increasing over 02

,pF

HGIKJ take f 0 1b gc h= (1)

(3) Find c, by applying mean value theorem to f x x x x a bb g= + + Œ2 1, , (1)

Q-7 (A) (1) Prove that if f g a b R, : ,b gÆ both differentiable at x then fg is also

differentiable at x and d

dxf x g x g x

d

dxf x f x

d

dxg xb g b g b g b g b g b g. = + (2)

(2) (1) Obtain d

dxx x x k k ztan sec , ,b g b g= π + Œ2 2 1

2

p(1)

(2) State the Roll's theorem (1)


(1) Find dy

dx, if y

x

xx= -

+FHG

IKJ < <-tan

cos

cos,1

121

12p p

(2) Find dy

dx if y

t

tx

t

tt=

+=

->- -sin , tan ,1

21

2

2

1

2

11

(3) If x at y at t= = π2 2 0, , then prove that yd y

dx

dy

dx= + FHG

IKJ =

2

2

2

0

(C) Calculate any two. (4)(1) The equation of motion of a particle moving in a straight line is

s t t t= - + +3 25 3 10 when will it change its direction. During which intervals

of time will it have the same direction.

(2) In the calculation of the area of a triangle from the formula D = 1

2bc A Asin ,


16

was taken as p6

actually an 2 3% error crept into this measure of A. What

is the percentage error in the calculation of area. (b and c are constants)

(3) Apply mean value theorem to f x xb g b g= +log 1 our the interval 0, x and

prove that 01

1

11 0<

+- < >

log x xxb g

(4) A boy flies a kite at a height of l00m. The kite moves away from the boyat a horizontal velocity of 6.5 m/sec. Then show that the kife is 260 m awayfrom the boy when the rate of releasing string is 6m/sec.


(2) Find the length of subnormal and subtangent at the point (4,8) of the carve y x2 3= (1)

(3) Show that f x x x xb g= - ŒFHGIKJtan , ,0

2

p is an increasing function. (1)

Q.8 (A) (1) Prove If f g a b R, : ,b gÆ are both differentiable at x then fg is also

differentiable at x and d

dxf x g x f x

d

dxg x g x

d

dxf xb g b gc h b g b g b g b g= + (2)

(2) (i) Using division Rule of derivative prove :

d

dxx ec x x k k zcot cos ,b g b g= - π Œ2 p (1)

(ii) State the second Derivative test (1)


(1) If sin siny x a y= +b g prove that dy

dx

a y

a=

+sin

sin

2 b g

(2) If y x x an

= + +FH IK2 2 then prove that dy

dx

ny

x a=

+2 2

(3) Let y x x= -FH IK-sin 1 22 1 Find dy

dx if

1

21< <x

(C) Calculate any two. (4)

(1) By cutting equal squares from the four corners of a 16x10 tin sheet, a boxis to be constructed. What should be the length of each square, if the volumeof the box is to be maximum ?

(2) Using mean value theorem, prove that if x >0 then x

xx

1 21

+< -tan


17

(3) Prove that the length of tangent is constant for the curve

x a tt

y a t= +FHG

IKJ =cos log tan sin

2

(4) For the curve y x x= -4 23 5 find all points at which the tangent passes

through the origin.


(2) Prove that f xx

b g= +37 is decreasing for x RŒ - 0l q (1)

(3) Apply Rolle's theorem to f x x x xb g= + - ŒLNMOQPsin cos ,1 0

2

p(1)


18

Question : 3Q.1 (A) (1) State and prove the method of substitation of integration. (2)

OR

Find a x dxx

a xa x

ac2 2 2 2

21

2 2- = - + F

HGIKJ +z -sin , where 0 < <| |x a

(2) If the function f is continous on 0 2, a then prove

then f x dx f x dx f a x dxa aa

b g b g b g= + -z zz0

2

00

2 (2)


(1)2 15

12

x

x xdx

-

+ +z

(2)1

1

--z sin

cos

x

xe dxx

(3)1

13x xdx

+z e j(C) Find the value of any two (4)

(1) obtain 31

2x dxz In as a limit of sum

(2)dx

x x3 20

2

+ +z sin cos

p

(3) Prove that x x dxsin3

0

2

3=z pp

(D) (1) Solve : x

x xedx

x

++z 1

1e j (2)

(2) If f xb g>0 and f f, ' are contiouns functions and f x'b gπ 0 then find

f x f x dxf x

nc

nn

b gc h b g b gc h' ;=

++

+

z1

1; where n π -1 (1)

Q.2 (A) (1) State and prove the rule of integration by substitution. (2)

OR


19

Prove that : e bx c dxe

a ba bx b bx cax

ax

z + =+

- +sin sin cosb g b g2 2

(2) If f is continous on 0 2, a , prove that f x dx f x dx f a x dxa aa

b g b g b g= + -z zz0

2

00

2 (2)

(B) Evaluate any two. (4)

(1)log 1

1 20

1 ++z x

xdx

b g

(2) Evaluate as the limit of sum : cos x dxa

b

z

(3) Prove sin

sin coslog

2

0

2 1

22 1

x

x xdx

+= +z d i

p

(C) Evaluate any two. (4)

(1) log x x dx+ +FH IKz 2 1

(2)sin

sin

x

xdx

3z

(3)dx

x x- -z

1 23

21

2b g b g

(D) (1) Evaluate : tan x dxz (2)

(2) Prove : e f x f x dx e f x cx xb g b gm r b g+ = +z ' (1)

Q.3 (A) (1) If f x dx F xb g b gz = then prove that f ax b dxa

F ax b a+ = + πz b g b g10 (2)

ORProve that

sec log

log sec tan , ,

x dx x c

x x c x k k z

= + +

= + + π + Œ

z p

p4 2

2 1 2

e jb g

(2) If f is continuous on 0,a , then prove f x dx f a x dxaa

b g b g= -zz00

(2)


20

(B) Obtain the following integrals (Any Two) (4)

(1)

1

1 31

2x x

dx

-z

e j

(2)1

1- +z cos sinx xdx

(3)2 3

3 2

x

xdx

++z


(1) e dxx

a

b

z (as a limit of a sum)

(2)1

2 3 20

4

+z cos xdx

p

(3)log 1

1 20

1 ++z x

xdx

b g

(D) (1) Obtain 1

3cos cosx xdx

+z (2)

(2) Prove that f x f x e dx e f x cx xb g b g b g+ = +z ' (1)

Q.4 (A) (1) Prove that cos log tan

log cos

ex dx x c

ecx catx c

= +

= - +z 2

Where, x k k zπ Œp,

OR

If f, 1 are integrable functions over interval ICR then prove that

(i) f x g x dx f x dx g x dxb g b g b g b g+ = +z z z(ii) kf x dx k f x dx k Rb g b g= Œz z ; .

(2) If function f is continuous over interval [0, a] then prove

f x dx f a x dxa a

b g b g= -z z0 0

(2)


21


(1)2 3

3 2

x

xdx

++z

(2)x

x xdx

2 29 4+ -z e je j

(3)x

x xdx

2

4 2 1+ +z(C) Evaluate any two. (4)

(1) sin x dxa

b

z (as a limit of a sum)

(2) | |2 12

5

x dx+-z

(3)x x

x xdx

tan

sec cos+z0

p

(D) (1) Find 1

3cos cosx xdx

+z (2)

(2) Prove that ten x dx x c= +z log |sec | where x k k zπ - Œ2 1 2b gp ; (1)

Q.5 (A) (1) P.T. e bx dx ebx

a bcax ax.cos

cos=

-

++z qb g

2 2 (2)

(2) If f is an even continous function on -a a, then P.T.

f x dx f x dx a Ra

a a

b g b g e j= Œ-

+z z20

, (2)


(1)x

xe dxx

2

2

1

1

++z b g

.

(2)x

x xdx

2

2 29 16- -z e je j

(3)25

02

2

->z x

xdx x


22


(1)sin

sin cos.

2

0

2 x

x xdx

+zp

(2)x

xdx

2

41

21

1

++z .

(3)dx

x a x

a

+ -z 2 20

(D) (1)1

1 2cos sin.

x xdx

+z b g (2)

(2) If f is contiuous and differentiable and f' is contiuous and non zero

f xb gπ 0 f x

f xdx f x ce

'log | |

b gb g b g= +z (1)

Q.6 (A) (1) State and prove the rule of integration by parts. (2)

ORState and prove the rule of change of variable in integration

(2) State the fundamental principle of definite Integral and if the functions is

continuous at [0, a] then prove that f x dx f a x dxa a

b g b g= -z z0 0


(1)dx

x3 2+z tan

(2)dx

x a x bsin sin- -z b g b g

(3)x

xdx

2 1-z(C) Solve any two (4)

(1) log tan10

4

+z x dxb gp

(2) x x dx+ -z tan 1

0

1


23

(3) sin x dxa

b

z (obtain as limit of sum)

(D) (1) tan x dxz (2)

(2) cot log sin , ;x dx x c xn

n z= + π Œz p2

(1)

Q.7 (A) (1) State and prove the method of substitution for indefinite intergration. (2)

OR

Prvoe that e bx dxe

a ba bx b bx cax

ax

cos cos sin=+

+ +z 2 2e jb g

(2) If f is continous on [0, 2a] then prove that,

f x dx f x dx f a x dxa a a

b g b g b g= + -z z z0

2

0 0

2 (2)

(B) Evaluate any two (4)

(1)1

1 23

21

2x xdx

- -z b g b g

(2)x

xe dxx

2

2

1

1

++z b g

(3) tan-z 1 x dx

(C) Answer any two (4)

(1) Evaluate : log 1

1 20

1 ++z x

xdx

b g

(2) Prove that, e

e edx

x

x x

sin

sin cos+=z p

p

p

126

3

(3) Find cos x dxa

b

z as the limit of a sum.

(D) (1) Evaluate : 1

1 4+z xdx (2)


24

(2) Prove that, f x f x dxf x

nc n

nn

b g b g b g. ' ;=

++ π -

+

z1

11 (1)

Where f xb g>0 and f f, ' continous f x'b gπ 0.

Q.8 (A) (1) State and prove rule of integration by parts. (2)

OR

(1) Prove x a dxx

x aa

x x a C2 2 2 22

2 2

2 2+ = + + + + +z log

(2) If f is continuous over [0, 2a] then prove. (2)

f x dx f x dx f a x dxa a a

b g b g b g= + -z z z0

2

0 0

2

(B) Evaluate any two. (4)

(1)x

xdx

1 3-z

(2)log

log

x

xdx

1 2+z b g

(3)dx

x xdx

5 2 2 3- -z cos sin

(C) Evaluate any two, (4)

(1)x x

xdx

sin

sin10

+zp

(2)log 1

1 20

1 ++z x

xdx

b g

(3) x x x dx2 5+-z e jsinp

p

(D) (1) Obtain e dxx

e

e

log

log

2

5

z as the limit of the sum (2)

(2) Find x x dx3 2003

2004

2004

+-z sine j (1)


25

Q.9 (A) (1) State and prove the method of substitation of integration. (2)

OR

(1) P.T. e bx dxe

a ba bx b bx cax

ax

cos cos sin=+

+ +z 2 2 b g

=+

- +e

a bbx c

ax

2 2cos qb g

where cos ,sin , ,q q=+

=+

πa

a b

b

a ba b

2 2 2 20 (2)

(2) If f is continous on [a,b] then

P.T. f x dx f a b x dxa

b

a

b

b g b g= + -z z (2)


(1)2 2

1 2

++

FHG

IKJz sin

cos

x

xe dxx

(2)x

xdx

2 1-z(3) x x dxcosb g2z

(C) Attempt any two. (4)

(1)log x

xdx

1 20

1

-z

(2) sin-+z 1

0

1

1

x

xdx

(3) x x dx15

2

0

1

+z b g

(D) (1) Solve : tan x dxz (2)

(2) Derive : tan log|sec | , , ,x dx x c x k x k z= = π π ¨ Œz p p21 1

2b g (1)


26

Q.10 (A) (1)Prove that (2)

sec log tan log sec tanx dxx

c x x c= +FHG

IKJ + = + +z p

4 2

sec tan sinx x x x n n z+ π fi + π fi π - Œ0 1 0 4 12

b g p

ORState and prove the rule of integration of by parts for indefinite integration.

(2) If f is continuous over [a,b] then prove that f x dx f a b x dxa

b

a

b

b g b g= + -z z (2)

(B) Solve any (Two) (4)

(1)dx

x x5 4 3+ +z cos sin

(2)dx

x x- -z

1 23

21

2b g b g

(3) 2 2x x dxcosz(C) Solve any Two (4)

(1) Obtain sin x dxa

b

z as the limit of a sum

(2) Prove that x dx

x xsin coslog

+= +z p

p

2 21 2

0

2

d i

(3) Obtain x x x dx2

0

5+z e jsinp

(D) (1) Obtain x x dx2 cos logb gz (2)

(2) State rule of subsituation for indefinite integration (1)Q.11 (A) (1)State and prove the method of substitution of integration (2)

OR

(1) Prove that cos log tan log cos cot ,ec x dxx

c ec x x c x k k z= + = - + π Œz 22 p

(2) If the function f is continues and even in [-a, a] then prove that

f x dx f x dxa

a a

b g b g=-z z2

0(2)


27

(B) Calculate any two. (4)

(1) cos-z -+

FHG

IKJ >1

2

2

1

10

x

xdx x

(2)2 2

1 2

++

FHG

IKJz sin

cos

x

xe dxx

(3)sin

sin

x

xdx

4z(C) Calculate any two. (4)

(1)x

x

2

41

21

1

++z

(2)cos

sin cos

2

0

2 x

x xdx

+zp

(3) s dxx

2

3

z (as a limit of sum)

(D) (1) Solve x

x x xdx

2

2sin cos+z b g (2)

(2) If f is continuous and differential function f' is also a continuous and

f xb gπ 0, f x x a b' , , ,b gπ Œ0 then prove that f x

f xdx f x c

'log | |

b gb g b g= +z (1)

Q.12 (A) (1) State the rule of integration by parts and prove it. (2)

OR

(1) State and prove the method of substitution of integration. (2)


(1)cos

sin

x

xdxz

(2)1

23sin sin, ,

x xdx n n z

+µπ Œz b g

p

(3)sin

sin

x

xdx

4z


28


(1) tan cotx x dx+z d i0

2p

(2) 2 11

3

x dx--z

(3)x x

xdx

sin

sin10

+zp

(D) (1) Solve : log loglog

xx

dxb gb g

+RS|T|

UV|W|

z 12 (2)

(2) Prove : f x f x dxf x

nc n

nn

b gc h b g b g=

++ π -

+

z1

11, where f xb g>0 and f f, ' are

continuous and f x'b gπ 0 (1)


29

Question : 4Q.1 (A) Calculate any two (4)

Solve the differential equations.

(1)dy

dxx y= + +4 1 2b g

(2) 2 2dy

dxy e

x- =

(3) 2 3 2 0x y dx y dy+ + + =b g b g(4) Rate of decay of uranium is proportional to its mass. If the initial mass is

m0 and half of its mass has decayed in 1200 years, how much of uranium

will be left at the end of 2400 years ?


(1) Find the area of the region bounded by y x=5 2 and 2 9 02x y- + =

(2) Find the volume of the solid revolution generated when the region bounded

by y x y x x= = -2 24, is rotated about x - axis.

(3) Correct the following statements with necessery calculation.

(i) "The area of the regoin bounded by y x= 4 and y x= 4 2 is 4"

(ii) The volumn of a solid generated by upper part of the region bounded

by the parabola y x2 = , line x =1 is rotated about x-axis is p unit.


(1) The mean and standard deviation of a random variable x are 10 and 5

respectively. Find E x E x x2 1e j b g, + and Ex -F

HGIKJ

10

5 and E

x -FHG

IKJ

10

5

2

.

(2) In a city of some western country 70 percent of the married persons takedivorce. What is the probabilty that at least three among four persons willtake divorce ?

(3) Three out of 10 screws in a box are defective. Four screws are selected atrandom from the box find the probability that out of 4 screws (i) None and(ii) One screw is detective.

(D) (1) Fill in the blanks with the necessary calculation. (1)

cos

sin..............

x

xdx

2 +=z

(2) (i) Define : Linear differential equation. (1)

(ii) Write the general form of differential equation of the first order and the

first degree. (1)


30

Q.2 (A) (1) State and prove the rule of method of substiution for an inclefinite integral. (3)

(2) Intergrate 1

2

+z x e

xedx

x

x

b ge jcos (1)

(B) Integrate any two. (4)

(1)dx

x ax x2 2-z

(2)3 1

8 2 2

x

x xdx

+

- -z

(3)x dx

x

2

41+z(C) (1) Prove that f x f x e dx e f x cx xb g b gc h b g+ = +z (1)

(2) Integrate sin

sin

x

xdx

4z OR dx

e ex x1 2+ +z e je j (2)

(3) Find intergral of x

xdx

2

4 16-z(D) Find integral of any two. (2)

(1) log x dxz(2)

e

edx

x

x

2

2

1

1

-+z

(3)dx

x x2 5 2 5+ +z b g log

Q.3 (A) Answer any two. (4)

(1) The vertex of a parabola is -a,0b g and its latus rectum is 4a. Prove that the

differential equation is 122

- FHGIKJ =dy

dx

x

y

dy

dx.

(2) Solve dy

dxx y x y= + + +sin cosb g b g

(3) y - intercept of tangent at any point x y,b g on a curve is 2xy . Find the

equation of the curve.


31

(4) A body having mass 60 kg. slides on the top of a table under a force

54 2.sin t Newtons. Force of friction is 60 times its velocity and initially thevelocity is zero. Express velocity of the body as a function of time.

(B) Answer any two. (4)

(1) Find the area of the region enclosed by y x2 8= and x y+ = 0

(2) Find the volume of the right circular cone having semi-vertical angle x andradius of base eqial to r.

(3) The line x = C divides the area of the region bounded by y x2 4= and x =16

in two regions having eqial areas. Find C.


(1) Khushboo tosses a balanced die five times. It she gets 5,4,3,2 amd 1 heads,she receives Rs. 10, 8, 6, 4 and 2. respectively and loses Rs. 16 if no headis obtained. Find the expected gain of khushboo.

(2) A random variable x follows binomial distribution whose mean and variance

are 103 and 10

9 respectively. Find the parameters n and p of this binomial

distribution. Also find P x >0b g .(3) Four faces of a balanced die are marked with integers 1,2,3,4 and the re-

maining two faces are marked with 0 each. If X denotes the integer obtainedon tossing the die. Find the mean and variance of the random variable x.

(D) (1) Obtain a e dxa x x3 1log + -z e j (1)

(2) (i) Write the general form of differential equation of first order and firstdegree. (1)

(ii) Determine the order and degree of y xdy

dx

dy

dx= + + FHG

IKJ5 1

2

(1)

Q.4 (A) Answer any two. (4)

(1) Solve cos x ydy

dx+ =b g 1

(2) Solve x ydx x y dy2 3 3 0- + =e j(3) Solve 2 4 2 1 0x y dy x y dy+ - + - =b g b g(4) The rate of melting a piece of ice is proportional to its quantity at any

moment. In 30 minutes, half of the ice melted. Prove that 1/8 th originalquantity will remain after 90 minutes.

(B) Answer any two of the following. (4)

(1) Find the area of the region bounded by the curve y x x= - +2 5 4 and x axis.


32

(2) Find the volume of the right circular cone having semi vertical angle a andradius of the base equal to r.

(3) Obtain the area of the region bounded by, y x2 8= and x y+ = 0.

(C) Solve any two of the following. (4)

(1) The mean and the variance of a binomial random variable with parameter nand p are 4 and 2 respectively. Find n and p.

(2) 25% of the mangoes in a box of dozen mangoes are rotten. Three mangoesare selected at random from the box. What is the probability that one of thethree mangoes is rotten.

(3) The mean and standard deviation of a random variable x are 5 and 3

respectively. Find E X E X2 23 2e j b g, +

(D) (1) log sin cot .............x x dx =z (1)

(2) Define : Differential equation.

(3) State general formula for differential equation of first order and first degree.(1)

Q.5 (A) Evaluate any two. (4)

(1) Find differential equation of circles having radius r and having centreon x axis.

(2) Solve the differential equation x ydy

dxx y+ = + -b g b g2 22 3

(3) Solve : y dx x dy x y dx- + - =2 2 0

(4) The mass of a boat and sailor together is 150 kg. The sailor applies force of70 Newton in the direction of imotion. If the repellent force is 30 times thevelocity in m/s find the velocity of boat at the end of t seconds. Intially theboat is at rest.

(B) Answer the following (any 2) (4)

(1) Find the area of region bounded by x

a

y

ba b

2

2

2

21+ = >, b g

(2) Find volume of solid generated by revolving the region bounded by y x= +2 1

and y x= +2 1 about x axis.

(3) The region bounded by y x= 2 2 X-axis, Y-axis and x =5 is rotated about

Y-axis. Find volume of the solid generated.


(1) Probability distribution of a discrete rendom X is defined below.

P(x) = 0.2, for x = 0

= kx, for x = 1,2

= k (6-x), for x = 3,4


33

(i) Find the coustant k

(ii) Draw the graph of P(x)

(iii) Find the value of P X ≥3b g and P 0 3£ ¥ £b g.(2) It has been found from an experiment that 40 percent of rats get stimulated.

On administering a particular drug. If 5 rats are given this drug. What is theprobability that (i) exactly three (ii) all rats get stimulated ?

(3) A player playing a game of tossing a balanced die receives Rs. 10 from hisopponent if he throws 1 or 2 or 5 or 6, how much should he pay to hisopponent so that the game becomes fair ?

(D) (1) Obtain x x dxe2 logz (1)

(2) Define the Homogeneous differential equation and explain the method of it. (2)Q.6 (A) Calculate any two. (4)

(1) A steamer having mass 45 106¥ starts motion with force of propeller 9,00,000

N from steady state. The force of resistance is 1,50,000 v is Newtons, wherev is the velocity of the steamer in meter/sec. Express the velocity as afunction of time.

(2) Solve xy

xy

y

xdx

y

xdysin cos cos-F

HGIKJ + = 0

(3) Obtain the general solution x yx dy y xy dy2 2 2 2 2 0+ + - =e j e j

(4)dy

dxe x ex y y= +- -2 and find the particular solution subject to initial condition

x y= =1 1&


(1) Obtain the area of the region bounded by the circle x y2 2 4+ = and y x2 3= .

(2) The region bounded by x y y2 2 16 0- = =, and x =8 is rotated about x axis.

Find the volume of the solid generated.

(3) Correct and rewrite the following statements with necessary calculation.

(a) The area of the region bounded by y x= 2 and y x= is 13 unit.

(b) The region bounded by the parabola y x y= =2 4, and the y axis is

rotated about the y axis, then the volume generated is p unit.


(1) The mean and standard deviation of a binomial random variable x are 10 and

5 respectively, then find E x x +1b g and Ex +F

HGIKJ

10

5

2


34

(2) Probability distribution of a random variable is given below

X x

P x K K K

= 0 1 2 3

0 3 5 2b gFind the acceptable value of K. Also obtain the mean of X.

(3) 400 out of a lot of 1000 screws produced in a factory are defective. Fromthis lot 5 screws are selected at random. Find the probability that

(1) 2 screws (2) at most 1 screw is defective

(D) (1) Calculate x

xdx

sin2z (1)

(2) (1) Define : The differential equation (1)

(2) State the applications of the differential equation (1)Q.7 (A) Answer any two. (4)

(1) Obtain the general solution of xdy

dxy x y

x= + FH IKcos2

(2) Obtain the general solution of dy

dxy e x xx- = +2 3 2tan sece j

(3) Obtain the general solution of dy

dxx y x+ - =tanb g 1

(4) According to Newton's law of cooling the rate of cooling of a body is equalto the difference between temperature of the body and temperature of air At200C air temperature the body cools from 1000C to 600C in 20 minutes.When will the body temperature become 300C ?


(1) Find the volume of the right circular cone having semi vertical angle a andradius of base equal to r.

(2) Prove that the area of the region enclosed by ellipse x

a

y

ba b

2

2

2

21+ = >b g is p ab

(3) Line x c= divides the area of the region bounded by y x2 4= and x =16 in

two regions having equal area. Find c.


(1) A die is constructed in such a way that the probability of getting an integeron its face is proportional to that integer A random variable X defined on thesample space U associated with the random experiment of tossing a die isas follows :

X (U) = -2, u=1,2

= 4, u=3,4

= 8, u=5,6

Obtain the probability distribution of the random variable x.


35

(2) Probability distribution of a random variable x is given by

P x P X xm

x mb g b g= = = =11 2 3; , , ....... . Where m is a positive integer, find

the mean and variance of the random variable x.

(3) Rachana Participate in a shooting competition. The probability of her shoot-ing a target is 0.2 What is the probability of shooting the target exactly threetimes out of five trials ?

(D) (1) Evaluate : log x dxz (without using Integration by parts) (1)

(2) (a) Write the form of 'Linear differential Equation' and obtain its general

solution. (1)

(b) Define : homogeneous function. (1)

Q.8 (A) Attempt any two. (4)

(1) Find the general solution of xdy

dxy x

y

x= + F

HGIKJcos2

(2) Find the general solution of xy xdy ydx y dy+ =b g 2 3 and also find the particular

solution when x y= =2 1&

(3) Solve dy

dx

y x

y x= - +

+ +1

1

(4) The population of a town was in 1970 AD and 5000 in 1980 AD. If the rateof increase S of population is proportional to population present at that timewhat will be the population in 2010 AD ?


(1) The region bounded by y x x x x= - = =4 1 32 , , and x axis is divided in two

parts with equal area by the line x c= find c.

(2) The right side of the y axis of region bounded by y x= 4 2 and y =16 is

evoled about y axis. Find the volume of the solid generatal.

(3) Correct the following statements with calculation.

(i) The area of the region bounded by y x x x= = =cos , ,0 p and x axis b 1

unit.

(ii) The volume of a solid generated by upper part of the region bounded

by y x2 = line x =1 is rotated about x axis is p unit.

(C) Attempt any two. (4)

(1) Rohan tosses a balanced die 4 time. We say that a success occurs if inteqers1 or 3 obtained on any toss. What is the probability of obtaining at most onesuccess in 4 tosses of a die ?


36

(D) (1) Fill in the blank with calculation. tan sec ..............3 2x x dx c= +z (1)

(2) (i) Write the form of a honogeneous differential eqiation and state its general

solution. (1)(ii) Define the lines differetial equation and state its general solution. (1)

Q.9 (A) Solve any (Two) (4)

(1) Prove the differential equation of family of circle heaving centers on y axis

and touching x-axis is x ydy

dxxy2 2 2- =e j

(2) Solvedy

dxy ex+ = and if x y= fi =0 1 find the particular solution.

(3) Solve dy

dx

y x

y x= - +

+ +1

5

(4) y - intercept of tangent at any point (x,y) on curve is 2 2xyb g Find the equation

of the curve.

(B) Attempt any Two (4)

(1) Find the area of the region between the circle x y2 2 4+ = and x y x2 2 4+ =

(2) the region bounded by y x x x x= - = =4 1 32 , , and x axis divided in two parts

with equal area by x=c Find c

(3) Find the volume of a sphere with radius r

(C) Solve any Two (4)(1) A player playin a game of loosing a balanced die receives Rs 10 from his

opponent if he throws an integer 3 or 4. If he throws 1 or 2 or 5 or 6 howmuch should he pay to his opponent or that the game becomes fair ?

(2) The mean and s.d of a random variable X are 10 and 5 respectively. Find

E x V x E x x Ex2 1

10

5e j b g b g, , ,+ -F

HGIKJ

(3) A balanced die is tossed five times. What is the probability the integer 1appears at least once and exactly once.

(D) (1) Find 5000

x xdx

x+zloge j (1)

(2) Discuss the method of solving a linear differential eqn. Define Differential

equation (2)Q.10 (A) Calculate any two. (4)

(1) Solve dy

dxx y x y= + + +sin cos( )b g


37

(2) Solve xdy

dxy y x= - +log log 1b g

(3) Solve 2 2dy

dxy e

x- =

(4) Rate of decay of uranium is proportional to its mass at any moment. If theinitial mass is m and half of its mass has decayed in 1200 years how muchof uranium will be left at the end of 2400 years.


(1) Find the area of region bounded by y x x= =tan ,p4

and x axis.

(2) Upper part of the region bounded by y x2 8= from its wertex X to x=2 is

rotated about x axis, find the volume of the solid generated.

(3) Correct the following statement with necessary calculation.

(i) The area of the region bounded by y x x= =cos , 0 x = p and x axis is 0 unit.

(ii) The volume of a solid generated by upper part of the region bounded

by the parabola y x2 = line x = 1 is rotated about x axis is p unit.


(1) A random variable x U R: Æ where U is the sample space, associated withthe experiment of a tossing of a balanced coin twice is defined as follows.

for every u U x uŒ =, b g number of heads in u. If the elementary events of the

U are equally likely then find probability distribution of x.

(2) A random variable x, assumes values -2, -1, 0, 1, 2 with the probabilities 0.1,0.2, 0.4, 0.1, 0.2 respectfully find E(x).

(3) 3 out of 10 screws in a box are defective 4 screws are selected at randomfrom the box. Find the probability that out of 4 screw none is defective.

(D) (1) Fill in the blanks with necessary calculation. x

xe dx cx-F

HGIKJ = +z 1

2................ (1)

(2) (i) Define : Order and degree of the differential equation. (1)(ii) State the method of solation of the equation of the variable separable

type and give its standard from (1)

Q.11 (A) Calculate any two. (4)

(1) Solve : ydx x dx x y dx x- + - = >2 2 0 0b g

(2) Solve the differential equation : dy

dx

y x

y x= - +

+ +1

5

(3) y intercept of tangent at any point x y,b g an curve is 2 2xy . Find the equ. of

the curve.


38

(4) At any instant rate of decay of radium is proportional to its mass present. Ifthe masses at time t

1 and t

2 are m

1 and m

2 then the time required to make

the mass half of its original mass is

t t e

em

m

2 12

1

2

-b g log

log


(1) Find the area of the region enclosed by y x2 8= and x y+ = 0.

(2) Find the volume of the solid obtained by revolution of portion of the ellipseon right hand semiplane of y axis about y axis.

(3) Correct following statements :

(i) Volume of solid formed when region bounded by y x= 2 between

(0,0) and (2,2) is rotated about y axis is 8p .

(ii) Area of the region bounded by xy c x a x b a b= = = < <2 0, , b g is c ab2 log


(1) Probability of a product of a machine being defective is P. Find the probabilitythat the number of defective products is greater than the number of non-

defective products out of 5 products selected at random. If p = 1

2 obtain the

probability of the event.

(2) From the following probability distribution of a random variable x :

X x

P x

= -1 0 1 2 3

1

6

1

3

1

6

1

6

1

6b g

find i E x ii Exb g b g b g 2 3

4

2+FHG

IKJ

(3) Three out of 10 screws in a box are defective. Four screws are selected atrandom from the box. Find the probability that out of 4 screws (i) none and(ii) one screws is defective.

(D) (1) Fill in the blanks with necessary calculation. tan

tan.............

x

xdx c

1 2+= +z (1)

(2) (i) Define : Homogenous Differential equation. (1)

(ii) List the applications of differential equations. (1)


39

Q.12 (A) Calculate any two. (4)

(1) Solve 6 7 14 02 2x y dx xy- - =e j

(2)dy

dxy e x+ = - has solution y x e e x= + -b g (c is arbitrany constant)

(3) According to Newton's law of cooling the rate of cooling of a body isproportional to difference between temperature of the body and thetemparature of air, At 200C air, temperature of the body cools from 1000Cto 600C in 20 minutes when will the body temperature becomes 300C ?

(4) Solve dy

dx

y x

y x= - +

+ +1

5


(1) Find the area of the rigin bounded by x y= -3 92 the y axis and the line y = 0

and y =1.

(2) Find the volume of the solid formal when the rigin bounded by x x2 4 4+ =and the x axis is rotated about x axis.

(3) Correct and rewrite the following statements with necessary calculation.

(i) The area of the rigion bounded by xy c x a= =2 and x b a b= < <0b gis c ab2 log

(ii) The area of the rigin bounded by y x x= =cos 0 and x = p is 12


(1) Three faces of a balanced die is marked with integers 1,2 and 3 and theremaining sides are marked with integers 0 each. If x denotes the number &intergers obtained on tossing the die, find the mean and variance of therandom variable x.

(2) 3 out at 10 screws in a box are defective 4 screws are selected at randomform the box. Find the probability that out of four screws (i) none and (ii)1 screw is defective.

(3) The probalility function of a binomial distribution is

P xx

P q xx xb g= FHGIKJ =-6

0 1 2 66 , , , ......

If 3 2 3P x Pb g b g= then find the value of p.

(D) (1) Find 1

2 -z cos xdx (1)

(2) (i) Difine : Homogeneous function. (1)(ii) Explain : the general solution of a differential equation. (1)


40

Question : 5

Q.1 (A) (1) (i) For event A and B prove that P A B P A P A B« = - «'b g b g b g (4)

(ii) Define : Additive set function

OR

What is equally likely events ? In usual notations prove that the prob-

ability of the event is P A rnb g=

(2) If A1 and A

2 and A

3 are mutally indepentent events then prove that A

1 and

A A2 3» are independent events.


(1) Two cards selected a random from pack of 52 cards. Find the probability thatboth the cards are (i) of black colour (ii) face cards (iii) of diamond.

(2) A box contains 5 white and some black balls. A ball is drawn at randomfrom the box. If A denotes the event that the selected balls is black and if

P Ab g= 0 9. , then find the number of black balls in the box.

(3) For event A and B prove that P A B P A P B« ≥ + -b g b g b g 1

(C) (1) Four faces of a die are of red colour and the remaining faces are of black

colour. If this die is tossed twice, find the probability that (2)

(i) face with red colour is obtain twice

(ii) face with black colour is obtained twice.

(2) (i) If P A P Bb g b g= =0 4 0 6. , . and P A B« =b g 0 1. then find P A B« 'b g (2)

(ii) Find P A B'b g if P A P Bb g b g= =0 25 0 50. , . and P A B« =b g 0 035. .

(D) (1) What we mean by the kilobyte and Megabyte ? (1)

(2) Convert (216.444)8 into decimal number. (2)

Q.2 (A) Prove that

(1) For A S P A P AŒ = -'b g b g1 (1)

(2) If A BÃ then P B A P B P A A B S- = - Œb g b g b g , (1)

(3) A and B are independent events then A' an B' are also independent events. (2)

(B) Attempt any three. (6)

(1) 5 persons N, Q, R, S, T stands in a queue at the booking windows of acinema house Find the probability of the event that

(i) Person N and Q stand adjacent to each other and

(ii) person Q is not at a first position in the queue.

(2) For two events A and B, P A B P A B» = « =b g b g0 9 0 4. , ' . and P A B' .« =b g 0 3

Find value of P A P B P A Bb g b g b g, ' '»


41

(3) Box I contain 7 white and 3 black balls and box II contains 3 white and 7black balls. Two balanced coins are tossed. If two heads are obtained box Iis chosen and a ball is drown at random from it otherwise box II is chosenand a ball is drawn from it. What is the probability that the ball drawn iswhite ?

(4) If A, B, C are independent events and P A P B P Cb g b g b g= = =2 4 0 4. , find

P A B C P B C» » »b g b g, and P A B C« «b g.(C) Define : (1) Classical definition of probability.

(2) Pair wise independent events. (2)

(D) Attempt any three. (3)

(1) A,B,C,D are primary events of some sample space U. The allocation of

probability given is P A P B P Cb g b g b g= = =0 29 0 36 0 16. , . , . and P Db g= 0 19. .

Is it possible ?

(2) For U a b c P a b P a c= = =, , , , ,l q l qc h l qc h23

13 and P b c,l qc h= 1

3distribution of probability of events possible ?

(3) If A A1 2 are mutually exclusive events then prove that P A AP A

P A1 21

21/ 'b g b g

b g=-

(4) It A and B are exhaustive and independent events and if P Ab g= 0 2. find

P Bb g.Q.3 (A) (1) State and prove the rule of addition of probability. (2)

OR

Define : (i) Additive set function.

(ii) Classical definition of probability.

(2) State and prove Baye's rule of probability.


(1) A die is constracted in such a way that the probability that integer obtainedon its face when tossed, is propertional to the square of that integer. Find theprobability that an interger on the face of a die is even.

(2) If for the independent events A B S P A, ,Œ =b g 34 and P Bb g= 5

8 then prove

that 1 8 3£ « £. 'P A Bb g .

(3) Suppose that 3 out of 100 men and 3 out of 1000 women in a city sufferfrom colourblindness. A randomly selected person of the city is found to becolour-blind. It the person is a man or woman is assumed to be equallylikely, what is the probability that the selected person is a woman ?


42

(C) (1) The probability that 60 years old man will be alive at the age of 70 years is

34 and the probability that his 50 years old wife will be alive at 60 years

is 23 . Find the probability that

(i) Only wife will be alive 10 years hence &

(ii) none of them will be alive 10 years hence. (2)

(2) (a) Find the probability of 53 sundays in a leap year. (1)

(b) A,B,C are independent events. If P A P B P Cb g b g b g= ◊ = ◊ =2 4 0 4. find

P A B C» »b g (1)

(D) (1) Distingaish between bit and byte. (1)

(2) Convert 25 1875 10.b g in octal form. (2)

Q.4 (A) (1) Define Axiomatic definition of probability and pair wise independent events.(2)OR

If A and B are two events prove P A B P A P B P A B» = + - «b g b g b g b g and

write the formula for P A B C» »b g(2) If A and B are independent events, then prove that A' and B' are also

independent events, also prove that A and B' are also independent. (2)

(B) Answer any Two of the following. (4)

(1) 13 cards are selected at random from a pack of 52 cards. Find the probabilitythat 3 out of 13 cards are kings.

(2) For two events A and B, P A B P A B» = « =b g b g0 9 0 4. , . and P B A« =b g 0 3.

Find the value of P A P Bb g b g, and P A B' '»b g.(3) If E and F are indepenent events and if G denotes the event that only one

out of events E and F occurs then show that -

P G P E P F P E P Fb g b g b g b g b g= + - 2 .

(C) (1) Suppose that 3 out of 100 men and 3 out of 1000 women in a city sufferfrom colour blindness. A random selected person of the city is found to becolour blind. If the person is a man or a woman is assumed to be equallylikely, what is the probability that the selected person is a woman.

(2) (i) From a pack of 52 cards, four cards are selected without replacement.Find the probability that they all may have different colour.

(ii) Prove P A B P A B' / /b g b g= -1 (1)

(D) (1) 1 kilo byte = ............ bytes. (1)(2) Convert decimal number (39.625)

10 into binary form. (2)

Q.5 (A) (1) If A and B are events then P.T. P A B P A P B P A B» = + - «b g b g b g b g (2)

(2) State and prove Bayes Rule


43


(1) For events A and B, P A P Bb g b g= = 12 and P A B» =b g 2

3 find the values

of the following.

( ) ( ) ' ( ) ' ' ( ) 'i P A B ii A B iii P A B iv P A B« « « »b g b g b g b g(2) A card is selected at random from a pack of 52 cards if A denotes the event

that the card drawn is a card of heart, find the value of P(A)

(3) A box has 10 black and 4 white balls. If 3 balls are drawn at random fromthe box, what is the probability of the event that

(i) All are white ?

(ii) Atleast one is white ?

(iii) At most one is white ?

(C) Box I contains 4 red and 6 white balls and box II contains 3 red and 7 whiteballs. A ball is drawn at random from the box I and is transferred to box II. Nowa ball is drawn at random from box II. What is the probability that the selected

ball is white ? (2)

(D) (1) Find probability of getting 5 Mondays in month of April.

(2) Find the probability of geting 72 marks in paper in your hand. (1)(E) (1) State the main parts of computer (1)

(2) Write the binary form of the decimal number, (1000)10

(3) Subtract 0.4891 E-07 from 0.1645-05. (1)

Q.6 (A) (1) For event A and B If A BÃ then (2)

(i) P B A P B P A- = -b g b g b g(ii) P A P Bb g b g£

OR

P.T. P A B C» » =b gP A P B P C P A B P B C P A C P A B Cb g b g b g b g b g b g b g+ + - « - « - « + « «

(2) Prove that the set function P A B/b g treated as a function of event A for fixed

event B is aprobability function on S.


(1) If P(E)=0.4, P(F) = 0.2 and P(E»F) = 0.5 find P(E/F), P(F/E) and P(E'/F).

(2) 10 student are given find the probability two paticular persons sited to gather.

(3) A A A A1 2 3 4, , , are independent event then prove that A1 and A A2 3»b g are

also independent.

(C) (1) For event A and B P.T. P A B P A P B« ≥ + -b g b g b g 1 and

P B P A Bb g b g= » =0 3 0 7. , .

and P B A/ .b g= 0 2 then find P(A) (2)


44

(2) (i) P A P B P A Bb g b g b g= = » =0 1 0 2 0 25. , . , . then find P A B/b g (1)

(ii) What is probability to you get 74 marks in this paper. (1)(D) (1) Give a difference between RAM and ROM (1)

(2) Convert the number 32 34 8.b g into decimal and binary system. (2)

Q.7 (A) (1) If A and B are events and if A BÃ then prove that P B A P B P A- = -b g b g b gand P A P Bb g b g£ (2)

OR

If A is some event then prove that,

P A P A'b g b g= -1 and hence show that

P ∆ =b g 0

(2) State and prove 'Bayes rule' for two events. (2)


(1) If A and B are independent events and if C denotes the event that onlyone out of events A and B occurs then show that.

P C P A P B P A P Bb g b g b g b g b g= + - 2 .

(2) Suppose that 3 out of 100 men and 3 out of 1000 women in a city sufferfrom colour blindness. A random selected person of the city is found to becolour blind. If a person is a man or woman is assumed to be equally likely,what is the probability that the selected person is a woman ?

(3) A box contain 10 balls amon xg which 3 are black and 7 are white. Nowfollowing game is played.

'At each trail a ball is selected at random, its colour is noted and then it isreplace back into the box with 2 more balls of the same colour. Find theprobability of first three balls to be black.

(C) (1) For two events A and B P A B P A B» = « =b g b g0 9 0 4. , ' . and P A B' .« =b g 0 3

find the value of P Ab g and P A B' '»b g (2)(2) (a) If A and B are mutually exclusive events then prove

that P AB

P A

P B'e j b gb g=

-1 (1)

(b) If A and B exhaustive and independent events and if P Ab g= 0 2.

find P Bb g (1)

(D) (1) Distinguish between bit and byte. (1)

(2) Convert (32.34)8 into binary numbers and decimal numbers.

Q.8 (A) (1) For event A, B of A BÃ then

P.T. ( )i P B A P B P A- = -b g b g b g ii P A P Bb g b g b g£ (2)

OR


45

(1) (i) P.T. P ∆ =b g 0

(ii) Define : Mutually exclusive and exhaustive events. (1)

(2) State and prove : Bayes Rule (2)(B) Attempt any two. (4)

(1) Two faces of balanced die are marked with interger 2 and 3 and other sidesare blank. If we toss blanced die five times then what the probability ofgetting sum of intergers exactly 12 ?

(2) If E and F are indepedent events and if G denotes the event that only one

out of E and F occurs then P.T. P G P E P F P E P Fb g b g b g b g b g= + - 2

(3) There are 3 black, 6 while and 4 red balls in a box. 3 balls are drawn atrandom without replacement. Find the porbability that all of them are ofsame colour.

(C) (1) If from pack of 52 playing card. Two card are selected and keep a side. Then

what is the probability of getting a card of ace from remaining cards ? (2)

(2) (i) If P A B» =b g 11

12 and P Ab g= 1

3 then P B A« =' ............b g (1)

(ii) A box contains 5 white and some black ball. A ball is drawn at randomfrom the box. It A denotes the event that the selected ball is balck and

if P(A)=0.9, find the number of black balls in the box. (1)(D) (1) What are main components if a computers ? (1)

(2) Convert (39.625)10

in binary and octal form. (2)

Q.9 (A) (1) Define : (i) Additive set function. (1)

(ii) Partition of the sample space. (1)OR

(1) For events A and B prove that P A B P A P B P A B» = + - «b g b g b g b g (2)

(2) State and prove the rule of buyes' for two events. (2)


(1) 5 men and 5 women are seated at random in a row. Find the probability ofthe event that in a row (1) 5 women are seated together. (2) No. 2 Womenare seated together.

(2) A box contains 5 white 4 black and 6 red balls. 3 balls are selected atrandom from it. Find the probability that there is a ball of each colour.

(3) If, A1, A2 and A3 are independent event, then prove that A1' and A A2 3« 'are independent events.

(C) (1) A and B are two independent witness of some incident and the probabilitiesof speaking true of A and B are respectively x and y. If for same incidentA and B are agree to each other then show that the probability of that

incident happens is xy

x y xy1 2- - + (2)


46

(2) (i) P A P B P A Bb g b g b g= = » =0 1 0 2 0 25. , . . the find P A B/b g (1)

(ii) P A B P A B» = « =b g b g0 8 0 3. , ' . and P A B' .« =b g 0 2 then find P A B' '»b g (1)

(D) (1) State the difference between RAM and ROM (1)(2) Convert (1101.011)

2 into decimal and octal form. (2)

Q.10 (A) (1)(i) State the axiomatic definition of probability. (1)

(ii) For events A and B prove that P A B P A P B» £ +b g b g b g (1)

OR

(1) Prove : If A and B are events then if A BÃ then P B A P B P A- = -b g b g b gand P A P Bb g b g£ (2)

(2) State and prove the rule of baye's for two events. (2)


(1) Two cards are drawn from a pack of 52 cards. What is the probability thateither both are red or both are kings ?

(2) If A and B are independent events associated with a random experiment thenshow that (i) A', B (ii) A', B' are also independent events.

(3) A factory has two machines A and B part records show that the machine Aproduced 60% of the items of output and machine B produced 40% of theitems. Further 2% of the items produced by machine A were defective and1 % produced by machine B were defective. It an item is drown at random,what is the probability that at is defective ?

(C) (1) A word consists of 9 letters; 5 consonants and 4 vowels. Three letter arechosen at random. What is the probability that more than one vowel is

selected ? (2)

(2) (i) If P A P B' . , .b g b g= =0 7 0 7 and P BAe j= 0 5. then find P A B»b g (1)

(ii) A problem in mathematics is given to 3 students whose chances of solving

it are 1

2

1

3

1

4, , what is the probability that the problem is solved ? (1)

(D) (1) How many types of arithmetic are available in a computer ? Name them. (1)

(2) Convert (101100.0011)2 into octal form.

Q.11 (A) (1) (i) For impossible event ∆ prove that P ∆ =b g 0 (1)

(ii) For every event A prove that 0 1£ £P Ab g (1)

OR

(1) Define (i) Equiprobable Primary events. (1)

(ii) Intersection events.

(2) If A and B are independent events then prove that A and B', A' and B and

A' and B' are also independent events. (2)


47

(B) Calculate any two. (4)(1) The percentage of students passing in three subjects A B and C are show

below A : 50%, B : 40%, C : 30%, A and B : 35% B and C : 20%, A andC : 25%, A and B and C : 15%, Find the percentage of students passing inat least one subject.

(2) If A1, A

2 and A

3 one mutualy independent events then prove that A

1', A A2 3« '

are independent events.

(3) If for the events P A P B P A Bb g b g b g= = » =1

42

3 then find P A B'«b g and

P A B'»b g(C) (1) A card is drawn at randow from a pack to well shuffled 52 cards. If A and

B denote the events that the card draw in black colour and the card drawn

is a face card respectively then find P A B»b g (2)

(2) Fill in the blanks by choosing the proper alternative (with necessary

calculations) from those given in the brackets. (2)

(1) The probability of setting 100 marks in the question paper of board

which is in your hand is ....... (0 34 1, , not given)

(2) On tossing 3 coins once the probability of getting 2

consicative T is ............. 1

4

1

2

3

8

3

4, , ,F

HGIKJ

(D) (i) State the names of electrinic components used in the third and forth generation

of computers. (1)

(ii) Substract .4891 07Œ- from .1645 05E - (2)

Question : 1gujarat-education.gov.in/education/e-questionpdf/12_sci_eng/maths-… · QUESTION BANK...

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