1.)xsinx(cosx

4

xsinxcoslim

4x +

−π

−π→

is

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

2.3log

31x

x)x3(loglim→

is

(a) 1 (b) e (c) e2 (d) none of these

3.xcot

0x)x(coslim

→ is

(a) 0 (b) 1 (c) e (d) does not exist

4. xtanx

15210lim

xxx

0x

+−−→ is

(a) ln 2 (b) 5nl

2nl(c) (ln 2) (ln 5) (d) ln 10

5.1x2

x 2x

1xlim

+

∞→

++ is

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

6.xsin

)x(cos)x(coslim

2

3/12/1

0x

−→

is

(a) 6

1(b)

12

1− (c) 3

2(d)

3

1

7.x/1xxx

0x 3

cbalim

++→

is

(a) abc (b) abc (c) (abc)1/3 (d) none of these

8. xcotxcot2

xcot1lim

3

3

4x −−

−π→

is

(a) 4

11(b)

4

3(c)

2

1(d) none of these

9.22a2x a4x

a2xa2xlim

−−+−

→ is

(a) a

1(b)

a2

1(c)

2

a(d) none of these

LEVEL - 1 (Objective)

10. If

=

≠+=

0]x[,0

0]x[,]x[

])x[1(sin)x(f , then )x(flim

0x −→ is

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

11. If )1x(log

)1e(sin)x(f

2x

−−=

−

, then )x(flim2x→

is

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

12. 20x x

)x1(logxcosxlim

+−→

is

(a) 2

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

13. =

π−

→ 2

xtan)x1(lim

1x

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

π(d)

π2

14.1x

xcoslim

1

1x +−π −

−→ is given by

(a) π1

(b) π2

1(c) 1 (d) 0

15. xsin

|x|limx

π+π−→ is

(a) -1 (b) 1 (c) π (d) none of these

16. If ),Nb,a(,e)ax1(lim 2x/b

0x∈=+

→ then

(a) a = 4, b = 2 (b) a = 8, b = 4 (c) a = 16, b = 8 (d) none of these

17.xeccos

0x xsin1

xtan1lim

++

→ is

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

18.x/1

0x)bxsinax(coslim +

→ is

(a) 1 (a) ab (c) eab (d) eb/a

19. If 1x

)1(f)x(flimthenx25)x(f

1x

2

−−−−=

→ is

(a) 24

1(b)

5

1(c) 24− (d)

24

1

20. Value of xsinx

6x

xxsinlim 6

3

0x

+−

→ is

(a) 0 (b) 12

1(c)

30

1(d)

120

1

21. Value of 30x x

)x1log(xcosxsin1lim

−+−+→

is

(a) -1/2 (b) 1/2 (c) 0 (d) none of these

22. Value of xcosxsinx

xcos1lim

3

0x

−→ is

(a) 5

2(b)

5

3

(c) 2

3(d) none of these

23. If 3x

)3(f)x(flimthen,

x18

1)x(f

3x2 −−

−=

→ is

(a) 0 (b) 9

1− (c) 3

1− (d) none of these

24.

−

−

∞→ |x|1

xx1

sinxlim

2

x is

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

25. If ∈π≠

=otherwise,2

Zn,nx,xsin)x(f and

==

≠+=

2x,5

0x,4

2,0x,1x

)x(g

2

then ))x(f(glim0x→

is

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

26. ]x[cos1

]x[cossinlim

0x +→ is

(a) 1 (b) 0 (c) does not exist (d) none of these

27. )ee(log

)ax(loglim

axax −−

→ is

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

28.)1x(sin

2xxlim

23

1x −−+

→ is

(a) 2 (b) 5 (c) 3 (d) none of these

29. If

=

≠λ−

=0x,

2

1

0x,xsinx

xcos1

)x(f is continuous at x = 0, then λ is

(a) 0 (b) 1± (c) 1 (d) none of these

30. If

=λ≠−+

=2x,

2x],x[]x[)x(f , then 'f' is continuous at x = 2 provided λ is

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

31. If )0x(,2)32x5(

)x7256(2)x(f 5/1

8/1

≠−+

−−= then for 'f' to be continuous everywhere f(0) is equal to

(a) -1 (b) 1 (c) 16 (d) none of these

32. The function

∞<≤−<≤

<≤

=

x2,x

b4b22x1,a

1x0,a

x

)x(f

2

2

2

is continuous for ∞<≤ x0 , then the most suitable values of a and b

are

(a) a = 1, b = -1 (b) a = -1, 21b += (c) a = -1, b = 1 (d) none of these

33. The function ( )

=

≠−

+= −

0x,a

0x,)1e(xtan

)x31(logxsin)x(f 3 x521

3

is continuous at x = 0 if

(a) a = 0 (b) a = 3/5 (c) a = 2 (d) 3

5a =

34. If f(x) = (x + 1)cot x is continuous at x = 0, then f(0) is

(a) 0 (b) e

1(c) e (d) none of these

35. The function

=

≠+−

=0x,0

0x,1e

1e)x(f x/1

x/1

is

(a) continuous at x = 0 (b) discontinuous at x = 0(c) discontinuous at x = 0 but can be made continuous at x = 0 (c) none of these

36. If the derivative of the function

−≥++−<+

=1x,4axbx

1x,bax)x(f

2

2

is everywhere continuous, then

(a) a = 2, b = 3 (b) a = 3, b = 2 (c) a = -2, b = -3 (d) a = -3, b = -2

37. If ,xegerintgreatestthe]x[and2x1,x]x[2

cos)x(f 3 ≤=<<

−π= then

π′ 3

2f is equal to

(a) 0 (b) 3/2

23

π− (c)

3/2

23

π (d)

3/2

2

π

38. Let

<−≥

=0x}1x,x2{max

0x}x,x{min)x(f

2

2

. Then

(a) f(x) is continuous at x = 0(b) f(x) is differentiable at x = 1(c) f(x) is not differentiable at exactly three points(d) f(x) is every where continuous

39. Let

≤+

>−+= ∫4x,8x2

4x,dy|)2y|3()x(f

x

0 then

(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

π′2

f is equal to

(a) π− (b) π (c) 2

π− (d) none of these

41. The following functions are continuous on ),0( π

(a) tan x (b) ∫x

0

dtt

1sint (c)

π<<π

π≤<

x4

3,

9

x2sin2

4

3x0,1

(d)

π<<π+ππ

π≤<

x2

),xsin(2

2x0,xsinx

42. Let f(x) = 1 - |cos x| for all Rx ∈ . Then

(a)

π′

2f does not exist (b) f(x) is continuous everywhere

(c) f(x) is not differentiable anywhere (d) 1)x(fLt2

x

=+π→

43. Let 1|x|,x12)x(f 2 ≤−+=

1|x|,e22)x1( >= −

The points where f(x) is not differentiable are(a) 0 only (b) -1, 1 only (c) 1 only (d) -1 only

44. If n2

n)x(sinLim)x(f

∞→= , then f is

(a) continuous at 2

xπ= (b) discontinuous at

2x

π=

(c) discontinuous at 2

xπ−= (d) discontinuous at an infinite number of points

45. If

≤≤+<≤−

=2x0if,1x2

0x3if],x[)x(f and g(x) = f(|x|) + |f(x)|, then

(a) 2)x(gLim0x

=+→

(b) 0)x(gLim0x

=−→

(c) g(x) is discontinuous at three points (d) g(x) is continuous at x = 0

46. Which of the following is true about n2

n2

n x1

xsinx)x2ln(Lim)x(f

+−+=

∞→

(a) 3ln)x(fLim1x

=−→

(b) 0)x(fLim1x

=+→

(c) f(1) = ln 3 - sin 1 (d) f(x) has removable discontinuity at x = 1

1. Let 2

)y(f)x(f

2

yxf

+=

+

for all real x and y. If )0(f ′ exists and equals to -1 and f(0) = 1, then f(2) is equal

to...

2. If 0)0(fandn

1nflim)0(fandR]1,1[:f

n=

=′→−

∞→. Then the value of n

n

1cos)1n(

2lim 1

n−

+

π−

∞→ is ........

Given that 2n

1coslim0 1

n

π<

< −

∞→.

3. (a) If ∞=

−−

++

∞→bax

1x

1xLt

2

x, find a and b.

(b) If 0bax1x

1xLim

2

x=

−−

++

∞→, find a and b

4. Let f (x) be defined in the interval [-2, 2] such that

≤≤−≤≤−−

=2x0,1x

0x2,1)x(f

g (x) = f (|x|) + | f (x) |. Test the differentiability of g (x) in (-2, 2).

5. If 1)0(fandRy,x3

)y(f2)x(f

3

y2xf =′∈∀+=

+

. Prove that f(x) is continuous for all Rx∈

6: Find the values of constants a, b and c so that xsinx

cxe)x1log(baxelim

2

xx

0x

−

→

++− = 2.

7:

0x,4])x(16[

x0x,a

0x,x

x4cos1)x(f

2

>−+

=

==

<−=

If possible find the value of a so that the function may be continuous at x = 0.

8. Determine the constants a, b and c for which the function

>−+−+

=<+

=

0x,1)1x(

1)cx(0x,b

0x,)ax1(

)x(f

2/1

3/1

x/1

is continuous at x = 0.

LEV EL - 2 (Subjective)

9. Let

=≠=

+−

0x,0

0x,xe)x(fx1

|x|1

Test whether(a) f(x) is continuous at x = 0 (b) f(x) is differentiable at x = 0.

10. If

−+=+xy1

yxf)y(f)x(f for all x, )1xy(,Ry ≠∈ and 2

x

)x(flim

0x=

→. Find

3

1f and )1(f ′ .

11. (a) If f(2) = 4 and f ′(2) = 1, then find 2x

)x(f2)2(xflim2x −

−→ (b) f(x) is differentiable function given

f ′(1) = 4, f ′(2) = 6, where f ′(c) means the derivative of function at x = c then )1(f)hh1(f

)2(f)hh22(f2

2lim

0h −−+−++

→ .

12.

<<−=

<<−

+

=

−

2

1x0,

x

1e0x,2/1

0x2

1,

2

cxsinb

)x(fx2/a

1

If f(x) is differentiable at x = 0 and |c| < 2

1 then find the value of 'a' and prove that 64b2 = (4 - c2).

13 Find for what values of a and b , f(x) defined by

>+

≤+

+

=0x,

x

b

x

xcota

0x,x3

x21

)x(f

2

2

2

is continuous at x = 0

14 Let [x] stands for the greatest integer not exceeding x and f(x) = sin (x2 + [x]). Examine f(x) for differentiability

at x = 1 and at 1x −π= . Find the derivatives at these points if they exist.

15

≥−<−=

≥−<+=

01

01

01

0121

31

2

3

x,)x(

x,)x()x(g;

x,x

x,x)x(fLet Discuss the continuity of g(f(x)).

16 Evaluate

∞→ nn 2

xcos......

8

xcos

4

xcos

2

xcoslim

17. Evaluate }x/{1}x/{1

0x e

})x{1(lim

+→

, if it exists where }{x represents the fractional part of x.

18 Let 'f' be a function such that +∈∀= Ry,x),y(f).x(f)xy(f and )),x(g1(x1)x1(f ++=+ where 0)x(glim0x

=→

.

Then find the value of x.

19 Let

≥+−<+

=

≥−<+

=,0xifb)1x(

0xif1x)x(gand

,0xif1x

0xifax)x(f 2

where a and b are nonegative real numbers. Determine the composite function g of. if (gof) (x) is continuous for

all real x, determine the values of a and b. Further, for these values of a and b, is g of differentiable at x = 0?

Justify your answer.

20 A function 'f' is defined as f (x) = 1 1

212

2

| | | |

| |

xif x

a x bx c if x

≥

+ + <

. If 'f' is differentiable at

x = 1

2 and x = -

1

2 , 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) = f2 (x) , where x & y are independent variables & f (0) ¹ 0 .

22 Evaluate , Limitn → ∞

++++ −1nn232 2

xsec.

2

xtan......

2

xsec.

2

xtan

2

xsec.

2

xtanxsec.

2

xtan

where x Î

π

2,0

23 Evaluate without using series expansion or L ' Hospital's rule , ∞→x

Limit x - x2 l n

+

x

11

24 Evaluate: 0h

Limit→ h

x)hx( xhx −+ +

, (x > 0)

25 If f (x) = 4x

5xsinxBxcosA −+ (x ¹ 0) is continuous at x = 0, then find the value of A and B . Also find f (0)

. Use of series expansion or L ̀ Hospital's rule is prohibited.

1. Let βα and be the distinct roots of ax2 + bx + c = 0, then 2

2

x )x(

)cbxax(cos1lim

α−++−

α→ is equal to

(a) 22

)(2

a β−α−(b) 2)(

2

1 β−α

(c) 22

)(2

a β−α (d) 0

2. If ,0x,x

1sinx)x(f ≠

= then =

→)x(flim

0x

(a) 1 (b) 0 (c) -1 (d) does not exist

3. =+−→ 20x x

)x1log(xcosxlim

(a) 2

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

4. =−

−π→ 1xcot

1xcos2lim

4x

(a) 2

1(b)

2

1(c)

22

1(d) 1

5. =−+

−→ 1x1

1alim

x

0x

(a) 2 loge a (b)

2

1 log

e a (c) a log

e 2 (d) none of these

6. =−→ 2

x

0x x

xcoselim

2

(a) 2

3(b)

2

1(c)

3

2(d) none of these

7.

≤≤−+

<≤−−−+

=1x0,

2x

1x2

0x1,x

px1px1

)x(f is continuous on [-1, 1], then 'p' is

(a) -1 (b) 2

1− (c) 2

1(d) 1

LEVEL - 3(Questions asked from previous Engineering Exams)

8. =−→ x3sinx

x5sin)x2cos1(lim 20x

(a) 3

10(b)

10

3(c)

5

6(d)

6

5

9.

=

≠−−

=2x,k

2x,2x

32x)x(f

5

is continuous at x = 2, then k =

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

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

++

+

∞→

dxc

x bxa

11lim

(a) ed/b (b) ec/a (c) e(c+d)/(a+b) (d) e

11. =−

−→ xxtan

eelim

xxtan

0x

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

12. The value of k which makes

=

≠

=0x,k

0x,x

1sin

)x(f continuous at x = 0 is

(a) 8 (b) 1 (c) -1 (d) none of these

13. The value of f(0) so that the function xtanx2

xsinx2)x(f

1

1

−

−

+−= is continuous at each point on its domain is

(a) 2 (b) 3

1(c)

3

2(d)

3

1−

14. If the function

=

≠−

++−=

2xforx

2xfor2x

Ax)2A(x)x(f

2

is continuous at x = 2, then

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

15. Let x

xsin1xsin1)x(f

−−+= . The value which should be assigned to 'f' at x = 0 so that it is continuous

everywhere is

(a) 2

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

16. The number of points at which the function |x|log

1)x(f = is discontinuous is

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

17. The value of 'b' for which the function

<<+≤<−

=2x1,bx3x4

1x0,4x5)x(f 2

is continuous at every point of its domain is

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

13

18. The value of f(0) so that

+

−=

3x

1log4x

sin

)14()x(f

2

3x

is continuous everywhere is

(a) 3(ln 4)3 (b) 4 (ln 4)3 (c) 12 (ln 4)3 (d) 15 (ln 4)3

19. Let

≤<λ+≤≤−

=3x2,x2

2x0,4x3)x(f . If 'f' is continuous at x = 2, then λ is

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

20. If the function

=≠

=0x,k

0x,)x(cos)x(f

x/1

is continuous at x = 0, then value of 'k' is

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

21. If ,5005x

5xlim

kk

kx=

−−

→ then positive value of 'k' is

(a) 3 (b) 4 (c) 5 (d) 6

22. =

−

−π→ xsin21

xtan1lim

4x

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

23.x

)bx1(log)ax1(log)x(f

−−+= is not defined at x = 0. The value which should be assigned to 'f' at x = 0 so that

it is continuous at x = 0 is

(a) a - b (b) a + b (c) log a + log b (d) 2

ba+

24. The values of A and B such that the function

π≥

π<<π−+

π−≤−

=

2x,xcos

2x

2,BxsinA

2x,xsin2

)x(f is continuous everywhere, are

(a) A = 0, B = 1 (b) A = 1, B = 1 (c) A = -1, B = 1 (d) A = -1, B = 0

25. =−−

π→ xcosxcot

aaLt

xcosxcot

2/x

(a) ln a (b) ln 2 (c) a (d) ln x

26. =−→ 20x x

1)xcos(sinLim

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

1(d)

2

1−

27. =

++++

∞→

x

2

2

x 2xx

3x5xlim

(a) e4 (b) e2 (c) e3 (d) e

28. If ,kx

)x3(log)x3(loglim

0x=−−+

→ then k is

(a) 3

1− (b) 3

2(c)

3

2− (d) 0

29. If ,ex

b

x

a1lim 2

x2

2x=

++

∞→ then the values of 'a' and 'b' are

(a) Rb,Ra ∈∈ (b) Rb,1a ∈=

(c) 2b,Ra =∈ (d) a = 1 and b = 2

30. If xcosx

xsinx)x(f 2+

−= , then )x(flimx ∞→

is

(a) 0 (b) ∞ (c) 1 (d) none of these

31. =−+

−→ 1)x1(

12lim 2/1

x

0x

(a) log 2 (b) 2 log 2

(c) 2log2

1(d) 0

32. =−→ 30x x

xsinxtanlim

(a) 1 (b) 2 (c) 2

1(d) -1

33. The function

π≤<π−

π≤≤π+

π<≤+

=

x2

,xsinbx2cosa

2x

4,bxcotx2

4x0,xsin2ax

)x(f is continuous for ,x0 π≤≤ then a, b are

(a) 12

,6

ππ(b)

6,

3

ππ(c)

12,

6

π−π(d) none of these

34.

>−+

=

<−

=

0x,4x16

x0x,a

0x,x

x4cos1

)x(f2

. If the function be continuous at x = 0, then a =

(a) 4 (b) 6 (c) 8 (d) 10

35. The function ,2

1x2cos]x[)x(f π

−= where [.] denotes the greatest integer function, is discontinuous at

(a) all x (b) all integral points (c) no x (d) x which is not an integer

36. =−

→ x

)x2cos1(21

lim0x

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

37. =π→ 2

2

0x x

)xcos(sinlim

(a) π− (b) π

(c) 2

π(d) 1

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

(a) π−− )1k()1( k (b) π−− − )1k()1( 1k

(c) π− k)1( k (d) π− − k)1( 1k

39. The integer 'n' for which n

x

0x x

)ex)(cos1x(coslim

−−→

is a finite non-zero number is

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

40. If 0x

nxsin)xtannx)na((lim

20x=−−

→, where 'n' is non-zero real number, then 'a' is equal to

(a) 0 (b) n

1n +

(c) n (d) n

1n +

41. The function f(x) = sin (loge |x|), 0x ≠ , and 1 if x = 0

(a) is continuous at x = 0 (b) has removable discontinuity at x = 0(c) has jump discontinuity at x = 0 (d) has oscillating discontinuity at x = 0

42. Let [x] stands for the greatest integer function and f(x) = 0x2

1,x]x2[x4 2 <≤−+ and

= ax2 - bx, 2

1x0 <≤ . Then

(a) f(x) is continuous in

−

2

1,

2

1 iff a = 4 and b = 0

(b) f(x) is continuous and differentiable in

−

2

1,

2

1 iff a = 4,b = 1

(c) f(x) is continuous and differentiable in

−

2

1,

2

1 for all a, provided b = 1

(d) for no choice of a and b, f(x) is differentiable in

−

2

1,

2

1

43. Let ( )

=

−∈≠

=−

0xwhen,0

2

1,

2

1xand0xwhen,

x

1sinx2sin

)x(f

21

. Then f(x) is

(a) discontinuous at x = 0 (b) continuous in

−

2

1,

2

1 but differentiable in

−

2

1,

2

1

(c) continuous in

−

2

1,

2

1 but not differentiable at x = 0 (d) differentiable only in

2

1,0

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 < ∞

= x - 1, -∞ < x < 1, is(a) continuous and differentiable (b) continuous and not differentiable(c) discontinuous and differentiable (d) discontinuous and not differentiable

46. Consider f(x) = ( )( )

−−−

−+−

xsinxsinxsinxsin2

xsinxsinxsinxsin233

33

, x ≠2

πfor x∈(0,π ), f( π /2) = 3

where [ ] denotes the greatest integer function then,

(a) f is continuous & differentiable at x = 2

π (b) f is continuous but not differentiable at x =

2

π

(c) f is neither continuous nor differentiable at x = 2

π (d) none of these

47. Let f & g be two functions defined as follows ; f(x) = 2

xx + for all x &

≥<

=0xforx

0xforx)x(g 2 then

(a) (gof)(x) & (fog)(x) are both cont. for all x∈ R (b) (gof)(x) & (fog)(x) are unequal functions(c) (gof)(x) is not differentiable at x = 0 (d) (fog)(x) is not differentiable at x = 0

48. Let f(x) = cos x,

π>−π∈≤≤

=x,1xsin

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

(a) g(x) is discontinuous at π=x (b) g(x) is continuous for ),0[x ∞∈

(c) g(x) is differentiable at π=x (d) g(x) is differentiable for ),0[x ∞∈

49. If ]x[

1x)x(f

2 += , ([.] denotes the greatest integer function), 4x1 <≤ , then

(a) Range of f is }5{~3

17,2

(b) f is discontinuous at exactly two points

(c) f is not differentiable at x = 3 (d) none of these

ANSWER KEY

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. d

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. b

44. b,c,d

45. a,d

46. a

LEVEL - 1 (Objective)

ANSWER KEY

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 = 9

7. a = 8

8. 1c,3

2b,

3

2loga =

=

=

9. (a) Continuous (b) Not differentiable

10. 1)1(fand33

1f =′π=

11. (a) 2 (b) 3

12. a = 1

13. a = -1 and b = 1

14.

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

16. sinx / x

17. Limit does not exist

18. )x(f

)x(fx

′=

19. a = 1, b = 0 g(f(x)) is differentiable

20. a = -4, b = 0, c = 3

21. )0(f

)0(fkwheree)x(f ekx ′

== +

22. tan x

23. 1/2

24.

+

x

1

x2

xlnx x

25 A = 5, B = 2

5 and f(0) =

24

5−

LEVEL -2 (Subjective)

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. a

15. d

16. c

17. a

18. c

19. c

20. a

21. b

22. d

23. b

24. c

25. a

26. d

27. a

28. b

29. b

30. c

31. b

32. c

33. c

34. c

35. c

36. d

37. b

38. a

39. c

40. d

41. d

42. c

43. b

44. c

45. b

46. a

47. a

48. b

49. a,b,c

LEVEL - 3 (Questions asked from previous Engineering Exams)

ANSWER KEY