Continuity&Diff Assingment

7/29/2019 Continuity&Diff Assingment

1/21


2/21


3/21

e

)x(f ])x[1(sin ]x[ 0]x[, + 0]x[,0 ==)x(flim 6.If , then isfi0x-(a) -1 (b) 0 (c) 1 (d) none of these

(a) -2 (b) -1 (c) 0 (d) 1

7. If-2x= -)x(f ) 1e(sin-)1x(log)x(flim , then isfi

8.tan)x1(lim 1xL p2 x=

lfi-

(a) -1 (b) 0 (c) 2 p(d) p2

-1

lim-fi1x+ -p1x xcos 9. is given by

10.fi+ + (b)p2 1(c) 1 (d) 0

(a)

0xp 1

xsin1 xtan1limxeccos i

s11.(a) e (b) e-1)bxsinax(coslim +x/1 0xfi(c) 1(d) none of these

b/

a--= (a)fi0x L

1x 2fi

6 x63 xsin x

l- (c) 24-(d) 24

1 is (a) 1 (a) ab (c) eab(d) e

12. If is

limthenx25)x(f1x )1(f)x(f

24 1(b)xxsin lim +-5 1

13. Value of is

(a) 0 (b) 12 1(c)30 1(d) 120 1


4/21

lim -+-+3 0xfix )x1log(xcosxsin1

23. If-3x 2limthen, x18- is (c) -3 1(d) none of thes

exfi L2 l i

s21. Value of is (a) -1/2 (b) 1/2 (c) 0 (d) none of these

1 )x(f=

(a) 0 (b) fi3x )3(f)x(f

9

124.sinx limx 1-|x|1 x

(a) 0 (b) 1 (c) -1 (d) none of these

2= = + then ))x(f(glim is

25. If pZn,nx,xsin )x(f and = otherwise,2)x(g= 2x,5 0x,4 2,0x,1x0x fi

(a) 5 (b) 4 (c) 2 (d) -5

limfi0x

limfi]x[cos1 ]x[cossin+-)ee(log )ax(lo

g30. Iflimfi1x23- +)1x(sin 2xx i

s26.

27.

28. is (a) 1 (b) 0 (c) does not exist (d) none of

these

ax ax is (a) 1 (b) -1 (c) 0 (d) noneof these


5/21

(a) 2 (b) 5 (c) 3 (d) none of these =l -+2x],x[]x[ )x(f, then 'f' is continuous atx = 2 provided l is= 2x,(a) -1 (b) 0 (c) 1 (d)


6/21

2

2


7/21

s

)x(f= x0 + >-+ the

n39. Let 4x,8x2 4x,dy|)2y|3(

(a) f(x) is continuous as well as differentiable everywhere. (b) f(x) is

continuous everywhere but not differentiable at x = 4

(c) f(x) is neither continuous nor differentiable at x = 4. (d) 2)4(fL =40. If f(x) =cos(x2 - 2[x]) for

0 < x < 1, where

[x] denotes the

greatest integer

x , then 2 f isequal tol L p

p(d) none ofthese41. The following functions are continuous on ),0( p

x(a) tan x

(b) 01 sintdt

t (c) 9x23 x0,1p


8/21

LEVEL - 2 (Sub jec t i ve)

1. Let to...L +f+ 2 yx= l2 )y(f)x(f for all real x and y. If )

0(f exists and equals to -1and f(0) = 1, then f(2) is equa

l1 nflim)0(fandR]1,1[:f= fi-nfi 2. If 0)0(fand nL =

l1 cos)1n( 2limfi1 n. Then thevalue of n n+ p L l

is ........

Given that 2n If= fi2bax 1x21x Limxfi --(b) If 0bax1x L+ + l= p -+0x,4])x(16[lim

0xxx2xsinx cxe)x1log(baxe = 2

.x


9/21

limfi0x=f)y(f)x(f for all x, )1xy(,Ry and 2x L 3 1f and )1(f .. Find l1, then find lim 2xfi-2x )x(f2)2(xf(b)f(x) is differentiable functiongiven

derivative of function at x = c then 2lim)1(f)hh1(f )2(f)hh22(ffi .2

0h--+ -++

x2/a1 ,0x 2 2

1 2 1 then find the

value of 'a' and

prove that 64b =

(4 - c)

.12.= sinb )x(f-1 + L2 cxl =


10/21

.

20 A function 'f' is

defined as f(x) = 1

1

2

| |

2

x i f x a x b x c i f

x | | | |

. If 'f'

is differentiable at++ < 12

x = 1 2

and x =

- l1 2

2

2 xx

tanxse

c.sec.

22 xx

tansec.

22 xx

tan......

sec. 22

x

, then find the values of a, b & c if possible . 21 Find a function continuous and

derivable for all x and satisfying the functional relation,

f (x + y) . f (x - y) = f(x) , where x & y are independent variables & f (0) 0 .

+ + + + 1 n n 2 3 2 n fi

where x L p

24 Evaluate: Limit 0h-+fi h x)hx(+x hx, (x > 0


11/21

)

L E V E L - 3(Questions asked from previous Engineering Exams)xafi2 is equal to

distinct roots of ax2 +bx + c = 0, then lim a-2++-)x( )cbxax(cos1

(a) -2 2a b-a)( 2(b) b-a)( 22

(c) 2 2)( 2a b-a (d) 0

l1sinx)x(f=the

n=2.If,0x,xL)x(flimfi0x

(a) 1 (b) 0 (c) -1 (d) does not exist3.lim0xfi 2x ) x1log(xcosx= +-

4.(a)

limp fi4 x2 1

-1xcot 1xcos2= -(b) 0

(c) 1 (d) none of thes

e(a) 2 log(b)

- e a (b) 2 1

log(c) 22 1

e(d)

15.(a)

limfi0x2 1

x1x1 1a= -+2 1e a (c) a log

2 (d) none of

thes

elimx6. = -0x2fi 2x xcose

(a) 2 3(b) 2 1(c) 3 2(d) none of these

7.)x(f --+ = x px1px1+


12/21

8.limfi2 0x= -x3sinx x5sin)x2cos1(

5-(b) = 10 3(c) 5 6(d) 6

59.(a) 3 10 32x )x(f2x, 2x= 2x,kis continuous at x = 2, then k

=(a) 16 (b) 80 (c) 32 (d) 8

10. If a, b, c, d are positive, then xfi1lim= l

L bxa 1+ ++dxc(a) ed/b(b) ec/a(c) e(c+d)/ (a+b)(d) e

11.limfi0x-xxtanxxtan ee= -

12. The value

of k which

makes =(a) 1 (b) e

(c) e - 1 (d) 0

continuous at x = 0

is

(a) 8 (b) 1

(c) -1 (d)

none of thes

e13. The value of f(0)

so that the function )

x(fxtanx2 xsinx2= is

continuous at each

point on its domain

is-1-1+ -

(a) 2 (b) 3 1(c) 3 2(d)-3 1

14. If the function 2xfor2x= 2xforx= is continuous at x = 2, thenAx)2A(x )x(f++-2

(a) A = 0 (b) A = 1 (c) A = -1 (d) none of these

(a) )x(f--+x xsin1xsin1

15. Let = . The value which should be

assigned to 'f' at x = 0 so that it is continuous

everywhere is

2 1(b) -2 (c) 2 (d) 1

16. The number of points at which the functionx|log 1)x(f= is discontinuous is

(a) 1 (b) 2 (c) 3 (d) 4

17. The value of 'b' for

which the function

1x0,4x5 )x(f2= 2x1,bx3x4 is

continuous at every

point of its domain is (a) -1 (b) 0 (c) 1 (d) 313


13/21

= -x sin Ll 3x L 1log 42l is continuous everywhere is

18. The value of f(0) sothat )14( )x(f3 x

(a) 3(ln 4)

3

(b) 4 (ln 4)

3

(c) 12 (ln 4)

3

(d) 15 (ln 4)

3

19. Let = 3x2,x22x0,4x3 )x(f. If 'f' is continuousat x = 2, then l is


14/21

28. If)x3(log)x3(log limfi0x= --+,kx then k is

(a) -3 1(b)3 2(c) -3 2(d) 0

29. If1lim L ++b x a2 x22 xfi,e x= l then the values of 'a' and'b' are

(a) Rb,Ra (b) Rb,1a =(c) 2b,Ra = (d) a = 1 and b =

230. If)x(f+ -xcosx xsinx= , then )x(flim2xfi is

(a) 0 (b) (c) 1 (d) none of these

(c)

lim

0x-2/

131.x limfi0x1)x1( 12= -+ (a) log 2 (b) 2 log

232.12log 2 = -fi 3x xsinxtan(d)

0(a) 1 (b) 2 (c) 2 1(d) -1

33. The function=

,xsinbx2cosa2p< px 2)x(f

iscontinuous for,x0

pthen a, barex0,xsin2axp


15/21

36.limfi0x2 1= -x )x2cos1(

(a) 1 (b) -1 (c) 0 (d) none of these

lim37. = p0xfi 22x )xcos(sin(a) p(b)

p(c) 0x-1k)1k()1(

-1kk)1((d) p-x2 p

k k

lim -

-(d) 1 38. The left-hand derivative of f(x) = [x] sin )x(p at x = k, k an integer, is

(a) p-- )1k()1((b) p--k)1( 39. The integer 'n' for

which finx )ex)

(cos1x(cos is a finite

non-zero number i

s(a) 1 (b) 2 (c) 3 (d) 4

nxsin)xtannx)na(( lim= --2 0xfi40. If0 x, where 'n' is non-zero real number, then 'a' is equal t

o(a) 0 (b) n 1n +

(c) n (d) n +n 1

2 1l 0x 1 ,x]x2[x4 - L - L L 2 11 ,2 , 2

- L

2= axbx,

l x0


16/21

1

43. Let )

x(f ( )

2 1- 1

sinx2sin

= l

L xand0xwhen,x2 11 ,

0 x w he n , 0 2 =

. Then f(x)

is

(a) discontinuous at x = 0 (b)

continuous in -1 , 221 L -1 2 1, 2 but differentiable in l

(c) continuous in -1 , 22 1 L ,02 1 but notdifferentiable at x = 0 (d)

differentiable only in l

44. The set of points where f(x)= |x|9 |x|+ is differentiable is

(a) )0,(- (b) ),0( (c) ),0()0,(- (d) ),(-45. At the point x = 1, the function f(x) =x3 - 1, 1 < x


17/21

)

(

x

)

&

(

f

o

g

)

(

x

)

ar

e

b

o

t

h

c

on

t

.

f

o

r

a

l

l

x

x

+f

o

r

a

l

l

x

&

0

x

f

o

r

x

)

x(

g

=0

x

f

o

r

x

x

R (b) (gof)(x)

& (fog)(x) are

unequal

functions (c)

(gof)(x) is not

differentiable at

p>p

],0[x},xt0:)t(f{imummin)x(g,then=x,1xsin

(a)g(x)isdiscontinuousatp=x(b)g(x)iscontinuousfor ),0[x(c) g(x)isdifferentiableat p=x(d) g(x)isdifferentiablefor ),0[x+ = , ([.] denotesthe greatest

integer function),

4x1


18/21

LEVEL - 1 (Objective)1. b 2. b 3. b 4. c 5. b 6. b 7.

c 8. b 9. b 10. b 11. d 12. a

13. d 14. b 15. d 16. d 17. c

18. c 19. d 20. d 21. a 22. c

23. dANSWERKEY

24. a 25. a 26. b 27. a 28. b 29. b 30. a 31.

d 32. c 33. b 34. c 35. b 36. a 37. a 38.

a,c,d 39. b,d 40. c 41. b,c 42. a,b,d 43. b44. b,c,d 45. a,d 46.


19/21

a

ANSWER KEY

LEVEL -2 (Subjective)

1. f(2) = -1

2. 0

3. (a) 1a and b can have any value

(b) a = 1, b = -1

4. Not differentiable at x = 0 and 1

5.

6. a = 3, b = 12, c = 914.

15. Discontinuous at x = -1, 0, 1

16. sinx / x

17. Limit does not exist

)x(f x =18.)x(f19. a = 1, b = 0 g(f(x)) is differentiabl

e7. a = 8

8.20. a = -4, b =

0, c = 3 +ekx ==)0(fkwheree)x(f21. )

0(f L 2 b, L

3 2=loga = l=l1c,

322. tan x

23. 1/

2xx Lx2 xlnl +x 19. (a) Continuous (b) Not differentiable f= p L 3 3 1= l10. 1)1(fand24.11. (a) 2 (b) 3 12. a = 1 13. a = -1

and b = 125

A = 5, B = 2

5 and f(0) =-24


20/21

5

ANSWER KEYLEVEL - 3 (Questions asked from previous Engineering Exams)1. c 2. b 3. a 4. b 5. a 6. a 7. b 8.

a 9. b 10. a 11. a 12. d 13. b 14. a15. d 16. c 17. a 18. c 19. c 20. a

21. b 22. d 23. b 24. c 25. a26. d

27. a 28. b 29. b 30. c 31. b 32. c

33. c 34. c 35. c 36. d 37. b 38. a39. c 40. d 41. d 42. c 43. b 44. c

45. b 46. a 47. a 48. b 49. a,b,c


21/21

Date post:	14-Apr-2018
Category:	Documents
Upload:	raju-sharma
View:	214 times
Download:	0 times

Continuity&Diff Assingment

Documents