limit continuity differential assignment for iit-jee
DESCRIPTION
Apex institute for IIT-JEE is the institution of making IITians in the Ghaziabad. It is the Institute in Indirapuram to making IITians (Eng..).Its mission is to upgrade the teaching profession by providing high quality education and training to students who will be the industry's future engineers.ObjectivesThe Institute performs the five basic functions of teaching, fulfilling the following objectives:1. Highly experienced & highly qualified recruitments and specialized training of faculty members.2. Providing study material (specially in house designed) as per requirement3. Teaching methodology and conduct periodical exams.4. Framing of test papers based on Latest Examination pattern.5. Course Co-ordination.TRANSCRIPT
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