DAV PUBLIC SCHOOL POKHARIPUT,

BHUBANESWAR-20
MCQ
1, 10fa)= cosVa,0 <N<l, which of the
following is equal to f')
-1
2. The derivative of xwrL. Nis
(a) 2-1
(b) 2r²*logx (o) 2x2(1 + logr) (d) 2r2* (1 - logx)
3. Iffx)= 2]x| + 3]| sinx + 6, then right hand derivative of
f(x) at x =0 is
(a) 6 (b) 5 (c) 3 (d) 2
4. Iftan -k, thenis equal to
(a) - (o (0) sec? ()- sec*
5. Ify = sin-'x, then (1 -x)y: is equal to
(a) xy1 (b) xy (c) xy2
0st-sinx
(d) x
6. Ify , then dy
cosx+sinx

(@) -se7-x) () see-x)

(c)
7. Ify = log(sine") , then is
à-loglsee (;-x))
dx
(a) cote (b) cosece (c) e'cote
8. The derivative of 2* w.r.t. 3* is (d) e* cosece*
Log2 Logs
log3
log2 log3
log2 log3 log2
9. Ify = sin-'x, thenis
dx2
(a) secy (b) secytany (o) sec'y tany
10. The derivative of sin(x)w.r.t. Xat x = yT is (@) tan"y secy
(a) 1 (b) - 1 (c)-2V7 (d) 2v+
11. Ifex*y = c, then dx?
xex"y
(a) 2y (b) -
12. Derivative of e2x w.r.t e* is
(a) e* (b) e2x (c) 2e*
13. The derivative of tan-'x w..t x is (d) 2e3r

(a) (b)r 2x
1+a4 (c) 1+x4 (a)
14. Ify = a cos(logx) + bsin (logx), then x*y, t xy, is
(a) cot (logx) (b) y (c) -y
15. If tan'(r-y') = a, then is (d) tan (logx)
da
 Ify
=v a

33.If
(2

()
(h) (
l6. If fx)=-21then the corect statement is: 4

ASSERTION AND REASONING QUESTIONS

Questions number given below are Assertion and Reason based questions carrying Imark eaen
Statements are given, one labelled Assertion (A)and the other labelled Reason(R). Select the corTect
answer from the codes (a), (b), (c) and (d) are given below.
(a) Both Assertion (A) and Reason(R) are true and Reason(R) is the correct explanation or
Assertion (A).
(b) Both Assertion (A) and Reason(R) are truebut Reason(R) is not the corect explanaton of A.
(c) Assertion (A) is true but Reason(R) is false.
(d) Assertion (A)is false but Reason(R) is true.
d
17. Assertion(A): esinx
dx
= esinxcosx
d
Reason (R): ex=ex Answer: a
dx
ev
18. Assertion(A): 4Vxev
1
Answer: b
Reason(R):log(logs)] = xlog X>]

19. Assertion (A): If f(x) =logx, then f'(x) =

Reason(R): Ify = x'logx, then, d²y =x(5 + 6logx) Answer: b
dy2

This section comprises Very Short Answer (VSA)type questions of 2mark each.
logx-1
20. Ifx = eV, prove that dx = (logx)?
21. Differentiate Vev2x with respect to eV2x for x>0.
22. If x =y*, then find dx
23. Ifx =ey, then prove that=
dx xlogx
sinx
24. Ify = ycosx + y,prove that dy
dx 1-2y

25. Ify = cos (sec2t), find

logx
26. Ifx=exy, prove that dx (1+logx)2"
27. Iff() = |tan2x|, then find the value of f/(*) at x=
28. If y= cosec(cot'x), then prove that V1 +x2-x=0. dx

29. Ify =X*, then find dx-at x = 1.

30. Ifx = asin2t, y = a(cos2t + 1og tant), then find dx
31. Ify =(r+ Va²-1), then showthat (r² - 1) ( =4y².
62. Ify - va+b.prove thaty ( =0
33. If (x +y'y' = xy, find d
34. Ifxa cos tand y= bsint. then find dx
35. Ifx -Vaan ,y=Vat prove thatx+y=0, dx
36. If xy = e), then show that 2-X-1)
da

37. Iff)-tanyE, then find f ()

38. Let Cbe acurve defined paransetricnlly as x acos'e.y " asin'0:0s0s4Detemine the poimt P
on C, wherethe tangentto Cis parallel to the chord joining the points (a.0) and 0,a
39. 1fx = at', y 2at, then find
40, Differentiate 2c03* with respect to cos'x.
41. If tan-1(²+y²) =a', then find
42. Ditferentiate ()with respect to x
43. If-2x'- 5xy + y' = 76, then find
This section comprises Short Answer (SA) type questions of 3 mark each.
44. Ifx cos(p +y) + cosp sin(p +y) = 0, prove that cosp=
dx
-cos²(p+y), where pisa
constant.

45. Differentiate y=logsin (-1) with respect to x

46. 1y =log() ,then showthat x(x+ 1)°y, + (x+ 1)y, =2.

47. IfV1-x+1-y =a(*-y), prove that

48. Ify = (tanx)", then find
constants, find
49. Given that x -y = a', where a, bare positive
50. Given that y = (sinr)". x"iht + a, find
S1. Show that |==0. ylogx
prove that
52. Ifx = e0s3t, y =ein3t. xlogy
cOSX
that= (1-six)
53. Ify =tan x + secx,then prove
sin-(2rv1-x).
54. Diierentiate sec-(ith respect to
55. If x+yma', hen find
56. Ife(x+l) =1, then showthat (
 ely

.
84.Ifxsin(a
If
x
x '

57. Ify =x,then prove that d'y L(-(0

dx? - - 0 .
I f

ith respect tosinwhen x= 0.

N1+x2-1
58. Differentiate tan 8 5I
. f

39. Ifx=asin21(1+cos2t) and y=bcos21(1-cos20) ,then find the value ofat

dx
t=and t=
60. If xcos(aty) =cosy , then prove that da cOs(e22 Hence show that sinad²y sin2(a+y) =0
sina dx2 dx

(6x-4V1-4x2
61. Findif
dx
y=sin( 5

(3x+4V1-x?
62. Find if
dx y= cos 5

63. Findif
dx
y=cos (2x-3V1-x
V13
64. Ify =excos* + (cosx) , then find
65. Ifx =sint and y=sinpt , then prove that (1-x)y=0.
66. Ify =emsin"x, then showthat (1- x)-x-my
dx2 d =0.
67. Ify =eacos tx,-1<x<1then showthat (1 x)-xa'y
dy2 dx
=0.
V1+x2+V1-x?
68. If y =tan! (V1+a2-V1-xs1,then f)nd dxy
69. Ifx = acos0 + bsin, y = asin bcose , then prove that y ' --X+y=0.
x.

70. Ify = (logx)" +xcosY then find dx

dy
71. Ify = (logx)" + xogx , then find dx

72. Differentiate (sinx) anx +xcosx with respect to x .

73. Differentiate (sinx cosx +xalIN with respect to x.
74. If (cosx) = (cosy) ,then find dx
dy
75. Ify = (cosx) + (sinx)l, then find dx
76. Ify = (sinx) +sin'vz, then find dx
x-1 then find dy
77. If y =xsinN-cosx x2+1 dx
2r?-3
78. If y =xc0D + x2+x+2 , then find dx
x+1
79. Differentiate xXCOSX 4 x2-1 with respect to x.
1

80. Ifx/1+y+yv1+x=0,(x *y), then provethat dx =, (1+x)2

that (1+logy)²
81. If y= ex then prove dx logy
logx
82. If x' = ey, then prove that dy dx (1+logx)2
 re,then prove that dx logx
(logxe)
A. I NSnaty) =siny, then prove that sn'(o4y)
sna
8s If xsin(aty) +sina.cos(aty), then prove dy sn'(a4y)
that dx Sha
N6. lfy -3cos(logx) +4sin(logx) , then show that x2
+x+y =0.
dx2 dx

S7. Ify = xlog)then prove that x -(x-y).

SS. Ife' +e= ety prove that dx +e=0.
$9. Ifx"y" = (xtymen , prove that dy y
dx

90. Differentiate y = lan V1+x2-V1-x2\

Vi+r?4V-~) With respect to cos'x
91. Ify =log x+ va +1,then prove that (x²+1)+=0.
92. Ify =loglx + vr +a,then prove that(+a)=0. dx
93. Iflog Vi+x-x =y V1+x²,then provethat (1+x)+xy +1 =0.
2
94. Ify =asinx +beosx , then prve that y? +( =a'+b?.
95. Ify =sinfxV1-*-V+VI-x} and 0<x <1 ,hen find
96. Differentiate sin-1 li+(36)*J with respect to x.
97. Ifx =vasintand y=yacos't , then showthat dx =2
98. Ify =|x +v1 +x , then show that (1-+x+ =ny
99. Ify = Petk +Qex, then show that(atb)+
dx2 da
aby =0.
100.If y=3e2x+ 2e, then show that d²y + 6y =0.
dx2 dx

d²y9+
101.If y= e(sinx+cosx),then show that dx2 dx
2y =0.
sin 'x
102.Ify =:V1-x7 ,then show that (1-x?)3x
dx2 dy.y=0.
dx

103.Ify = e'sinx , showthat dy -2+2y =0.

dx2dx

104. Find dx if (r² + y²)2 =xy .

105. Ify=(cot-'x)? , then showthat (r? +1)2+
dx? 2x(x? +1)-2.
dx

106. Ify=(tan-'x)? ,then showthat (*? +1)2+

dx2 2x(x² +1) dx =2

107.1f y=cosec-'x, x>1,then show that x(r² 1)+(22-1) =0. dx

d²y
108.Ify = sin x ,then show that (1 - x) dx
=0.
 7eroller

109.Ifx =a(cose +logtan )and y=asine then find the value ofa =
110.Ifx =alcost +logtan and y=asint ,then find and dt
111.lfx =a(8 - sine) and y=a(1+ cose) , then find
the value of
112.If x=a(cose + sine) and y=a(sin® - cos@) . then
find the value of
113.Ify =(sinx - cosx)inecos).<<then find
dx
114.If x=acos³0 and y= asin³0 ,then find the value
ofat 8 = -6
115.1f x =asec'8 and y= atan³0 , then findthe
value ofat
dx?
8=
116.1f x =cos0 and y= sin'0 ,then
prove that =3sin'0(5cos8 -1).
117. Ifx =tan logy). then prove that (1 +x*)at (2r-a) 0.
dx
118, Prove that Va-x 4sin-Va-x
119.Ifx = ae (sine - cose) andy= ae" (sin8 +
cos8) , then find dax at
120. If x=2cos0 - cos20 and y =2sin9 -
sin28, then prove that=
d tan
121.If (tan-'r) +yotx = 1, then find
122. Ify =tan (+ log. xta Prove that 2a?
d x-at

123. If (x-y)er-y =a, prove that ydx +x=2y

124.If x=cost(3-2cost) and y= sint(3-2sin-t), then find at t =
4
125. Differentiate tan-(ith respect to sin(2rv1-x).
126. Differentiate tan1 V1-x?
with respect to cos-(2rv1-x)
127. If x=a(20 - sin20) and y =a(1 - cos26) , then find at = .
128.If x=a(0 - sin0) and y = a(1- cose) , a> 0 then find
129.If y= sin(sinx), prove that dy2 +tanx+ycos'x =0.
130. Differentiate y = sin'(3r 4x°) w.r.t.xwhen x E
131. Differentiate y= cosSw.r.L. XWhen x E(0,1).
132. Show that the derivative of tan (secx + tanx).- with respect to xis equal to
133. Diferentiate log (x* + cosec*x) with respect x.
CASE BASED QUESTIONS
The equation of thepath traced by aroller-coaster is given by the polynomial f() =a(x+9)X(*+1)\(-3)
the rolleT-C0aster Crosses y-axis
at a point (0. -1), answer the
following questions

(i) Find the value of 'a'.

(ii) Find fll () at x =1
Answer:
(0, -1) so, -]=a (0 + 9)X0 +1)(0- 3)=’a=
() As the roller-coaster crosses y-axis at a point
(G) fl() =6x +14)
 DAV PUBLIC SCH00L
POKHARIPUT
CONTINUITY AND DIFERENTIABILITY
MCQ
e-2x
1. A
function f: R-R is defined by f(x) =
4 nsx<0 Which of the following
x>0
statements is true about the function at the point x = lni
(a) f() is not continuous but differentiable
(b) f(x) is continuous but not differentiable
(c) f() is neither continuous nor differentiable
(d) f() is both continuous and differentiable
2. The function f(x)= |x|x is
(a) Contin uous but not differentiable at x = 0
(b) Continuous and differentiable at x = 0
(c) Neither continuous nor differentiable at x = 0
(d) Differentiable but not continuous at x =0.
3. For what value ofk may the function f) = Jk(3x - 52) x< 0 is continuous at x =0
(a) 0
1 COSX x>0
(b) 1 (c) 3 (d) no value
4. For what value of kmay the function fx) = 3x +5 x > 2is continuous at x =2
kx? x<2
(c) 11
1-cos4x
5. The value of 'k' for which the function f(x) = 8x2 if x*0 is continuous at x=0 is
if x = 0
(a)0 (b) 1 (c) 1 (d) 2
6. The function f(x) = x\x| is differentiable
(a) Only at x=0 (b) only at x=1 (c) in R () in R- {0}
7. The function f(x) =xx| is
(a) Continuous and differentiable at x =0
(b) Continuous but not differentiable atx = 0
(c) Differentiable but not.continuous at x=0
(d) Neither differentiable nor continuous at x = 0
8 The value ofk for which f(x) = x' x> 0 is differentiable at X=0 is
kx x0
(a) 1 (b) 2 (c) any real number (d) 0
9 The function f (x) = |x| is
(a) continuous and differentiable everywhere
(b) continuous and differentiable nowhere
(c) continuous everywhere, but differentiable everywhere except at x = 0
(d) continuous everywhere, but differentiable nowhere
10. The function f(x) =[x], where [x] denote the greatest integer less than or equal
to x, is
continuous at
(a) x=1 (b) x =1.5 (©) x=-2 (d) x =4
 24. The number of pomts where ( ) - N 0 N g e
ditlerentable is
(a)1 (b)2
25. The value of constant e that akes the function Fdeld by

Cx +20 fa4
()-2 (b-1

/(0)w-2rE-s
26. The nunber of points of discontinulty of
(
(a) 0 (b)1
ofk, the functio ieL blow iso kal
21 lor What value

(0
(a) 0 ()

28. Afunetion f()1-xthllis

(a) discontinuous atxlonly
(b) discontinuous atx 0only
(c) discontinuous atX01
(d) continuous everywhere
qorevt?
Iff()=xl+lx-1, then
which of the followg is
29, and ditterentiableat 01
(a) f0 is both contiùnuous x=0.1
but notcontinuous al
(b) f)is dilerentiable dittereutiable at a 01
f ) is continuous but not 01
(c) at
continuous nor dierentiable
(d) f ) is neither
log(1*av)tlog (t-b) then k is
is continous at x 0
30. I0F) = 0
(0-b
(b)e+d
(a) e
ae
1
if3<x<Sisconiuous ba A, theu th¹ vales of aand b
31. IffQ)= ar + b
(0) a=3,b -8
(b) a=3, b=8
-8
(c) a 3 , b
(d) a=-3,b 8 diterentiable futioas
denotes theset of continuous functions and 2 denotes the set of
32, IfA ad
the corectrelation vween sets A
then which ofthefollowing depiets

(A)

(stn'ax
33. If)= X0j eontinuous ata0, then the valhe of aix
1 X0
(0)0
(a) t1 (b)-1
 This sectin omprise Shrt Aswer (Svpe questits ef Smark cack.

>
is cotinsat

somtinosa
Hence chec the dierenaby of atx0

Show that thefanctien 2

=0 is continuos atxthen fnd the vahe ofa

S2. Find the value ofk for which N is continaUs atx 0

r620
S3. Find the value ofk fer which tx) is continuoussatx2
a

koasx
S4. Find the value ofk for which R)= is continuous atx
3 x=

S5. Find the value of afor which fx) = asin(+1) xs0 is continuoUs at x 0

S6. Find the value ofk for which fx) S is continuous atx0.

57. Find the value of kfor which fx) = is continuous at x*