M1 Prsolutions08

1. :->

2. _ _ _ _ _ _ _ _ _ :->a

3. Change the order of integration in the integral :->

4. Change the order of integration in the :->

5. In polar coordinates the integral is :->

6. In polar coordinates the integral :->

7. :->

8. :->

9. Change the order of integration in the following integral :->

10. Change the order of integration in the integral :-

>

11. :->

12. :->1

13. :->3

14. :->

15. :->26

16. :->

17. :->

18. :->(e-1)3
19. Evaluate taken over the volume bounded by the planes x=0, x=1,
y=0, y=1 and z=0, z=1. :->

20. :->1

21. :->1

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22. _ _ _ _ _ _ _ _ _ _ _ :->

23. :->

24. _ _ _ _ _ _ _ _ _ _ _ :->9

25. _ _ _ _ _ _ _ _ _ _ _ :->
26. By double integration, the area bying inside the circle r=asin θ and outside the cardioid

r=a(1-cosθ) is :->
27. By double integration, the area lying between the parabola y=4x-x2 and the line y=x is :-

>

28. The area between the parabolas y2 =4ax and x2=4ay is :->
29. The area bounded by the curves and is :->
30. The surface area of the solid generated by the revolution about y-axis , of the arc of the

curve x=f(y) from y=a to y=b is :->
31. The surface area of the solid generated by the revolution about x-axis , of the arc of the

curve y=f(x) from x=a to x=b is :->
32. The volume of the solid generated by the revalution of the area bounded by the curve

r=f(θ) and the radii vectors , , about the line θ= is :->
33. The volume of the solid generated by revolution of the area bounded by the curve r=f(θ),

and the radii vectors , , about the initial line θ =0 is :->
34. The volume of the solid generated by revolution about the y-axis , of the area bounded by

the curve x=f(y), the y-axis and the abscissac y=a, y=b is :->
35. The volume of the solid generated by revolution about the x-axis , of the area bounded by

the curve y=f(x), the x-axis and the ordinates x=a, x=b is :->
36. The length of the curve r=asin θ between θ =0 and is :->πa
37. The length of the arc x=t, y=t from t=0 to t=4 is :->4
38. The length of the arc of the catenary y=c cosh from x=0 to x=a is given by :-

>
39. The length of the arc of the curve x=acosθ, y=a sinθ from θ=0 to is :->


40. The length of the arc of the curve θ=f(r) between the points where r=a and r=b, is :->L=

41. The length of the arc of the curve r=f(θ) between the points where and , is :-

>L=
42. The length of the arc of the curve x=f(t), y=g(t) between the points where t=a and t=b is :-

>L=
43. The length of the are of the curve x=f(y) between the points where y=a and y=b is :->L=

44. The length of the arc of the curve y=f(x) between the points where x=a and x=b is :-

>
45. The length of the curve x=et cost from t=oto t=π/2 is :->
46. The length of the curve y=x from x=0 to x=4/3 is :->
47. The arc of the upper half of the cardioid r=a(1+cos θ) is bisected at θ = _ _ _ _ _ _ _ _ _ _
_ _ _ :->π/2
48. The length of one arch of the cycloid x=a(t-sint), y=a(1-cost) is :->8a
49. The entire length of the cardioid r=a(1+cosθ) is :->8a
50. Where the curve x=a(θ+sin θ), y=a(1+cos θ) is symmetrical about :->y-axis
51. Where the curve x=a(θ+sin θ), y=a(1-cos θ) meet the x-axis? :->(0,0)
52. The curve x= a(t-sint) ; x=a(1-cost) is symmetrical about :->y-axis
53. If x=f(x) is odd and y=g(t) is even then the curve is symmetrical about :->y-axis
54. If x=f(t) is even and y=g(t) is odd then the curve is symmetrical about :->x-axis
55. The curve x=a (cost + log Tan ), y= a sin t is symmetrical about :->x-axis
56. Where the curve x=a(cost + log tan ), y=a sint meet the y-axis ? :->(0, )
57. Where the curve x=acos y=b sin ) meet the y-axis ? :->(0, )
58. The Asymptotes to the curve x=a[cos )], y= a sin θ is :->y=0
59. The Asymptotes to the curve x=a( ), y= a(1+cos θ) is :-> o asymptotes
60. The line θ=0 is tangent at the pole to the following ine of the curve :->r=a(1-cos θ)
61. The Tangents at the pole to the curve r=a(1+cos θ) is :->θ =π
62. The Tangents at origin to the curve r2=a2 cos 2 θ are :->θ =
63. The curve r=a(1+sin θ) is symmetrical about :->θ = π/2
64. If the equation of a curve does not change when θ is replaced by `-θ' then the curve is
symmetrical about the _ _ _ _ _ :->initial line
65. The curve r2 = a2sin 2 θ is symmetrical about :->The pole and the line θ =
66. The curve r=acos2 θ, a 0 consists of _ _ _ _ _ _ _ _ _ _ loops :->4
67. The tangents at the pole to the curve r=a sin 3 θ are :->The Lines
68. The tangents at the pole to the curve r2=a2sin 2θ are :->θ = 0 and θ=
69. The equations of asymptotes to the curve r2=a2sec2θ are :->


70. The curve 9 ay2 = (x-2a)(x-5a)2 is semmetrical about :->x-axis
71. The curve a2y2=x3(2a-x) is symmetrical about :->x-axis
72. Where the curve y(x2+4a2) = 8a3 meet y-axis? :->(0, 2a)
73. The tangents at the origin to the curve y2(a+x)=x2(3a-x) are :->y=
74. The curve y2(a+x) =x2(3a-x) is symmetrical about :->y-axis
75. The tangents at origin to the curve y2(x-a)=x2(x+a) are :->y=
76. The tangent at origin to the curve (x2+y2)=a2x is :->y=
77. The asymptotes to the curve a2y2=x3(2a-x) are :-> o asymptotes
78. The Asymptote to the curve xy2=a2(a-x) is :->x=0
79. The Asymptote to the curve y2(a+x)= x2(3a-x) is :->x+a=0
80. The curve x3 +y3 = 3 axy is symmetrical about _ _ _ _ _ _ _ _ _ :->The line y=x
81. The equation of the a symptote of the curve y2(2a-x) =x3 is _ _ _ _ _ _ _ _ _ _ :->x=2a
82. The Tangents at the Origin to the curve x2 y2 = a2(y2-x2) is _ _ _ _ _ _ _ _ _ _ :->
83. For the curve ay2=x3 the origin is _ _ _ _ _ _ _ _ _ _ _ _ :->a cusp
84. The curve ay2 = x2 is symmetrical about _ _ _ _ _ _ _ _ _ _ :->The x-axis
85. The asymptote of the curve (a-x)y2 = x3 is _ _ _ _ _ _ _ _ _ _ :->x=a
86. The Equation of the oblique asymptote of the curve x3+y3 = 3 axy is :->x+y+a=0
87. The curve x3-y3 = 3 axy is symmetrical about _ _ _ _ _ _ _ _ _ _ :->The line y=-x
88. Find the points where the curve meets the x-axis :->x=0, x=-a
89. For the equation y2(a-x)=x2(a+x), the origin is _ _ _ _ _ _ _ :->a node
90. Envelope of y=mx+ is :->y2=8x
91. The envelope of the family of curves y=m2x+am is _ _ _ _ _ _ _ _ _ m being a parameter.
:->a2+4xy=0
92. If a one parameter family of curves is given by A alpha2+Bα+C =0 then its envelope is :-
>B2-4AC=0
93. The envelope of the family of straight lines x cosα+ ysin α = c sec α where α being a
parameter, is _ _ _ _ _ _ _ _ _ :->a parabola
94. Envelope of the family of lines y=mx+a/m is _ _ _ _ _ _ :->y2=4ax
95. If the equation of a family of curves be a quadratic in parameter α, the envelope of the
family is Discriminant _ _ _ _ _ _ _ _ _ _ _ :->=0
96. The Envelope of the familly of curves y= mx+ , is m being a parameter _ _ _ _
_ _ _ _ _ _ _ :->
97. Radius of curvature y=4sinx-sin2x at x= is _ _ _ _ _ _ _ _ _ _ :->
98. Evolute of a curve is the envelope of its _ _ _ _ _ _ _ _ _ :-> ormals
99. Envelope when f(x,y,α) is of the form A cos α+B sin α=C is where A,B,C are functions
of x and y, and α is the parameter :->A2+B2=C2
100. The tangent at the origin to the curve y-x=x2+2xy+y2 is :->y=x
101. The tangent at the origin to the curve x4-y4+x3-y3+x2-y2+y=0 is :->x-axis
102. The coordinates of centre of curvature of the curve y=ex at (0,1) is _ _ _ _ _ _ _ _
:->(-2, 3)


103. The circle with centre at the centre of curvature and radius equal to the radius of
curvature is called _ _ _ _ _ _ _ _ _ :->circle of curvature
104. The coordinates of the centre of curvature at any point p(x,y) on the curve y=f(x)
is _ _ _ _ _ _ _ _ _ _, where y1= , y2 = :->
105. The locus of centre of curvature of a curve is called _ _ _ _ _ _ _ _ :->Evolute
106. The curvature at any point of a circle of radius `r' is _ _ _ _ _ _ _ _ :->1/r
107. If the circle of curvature is (a+b) (x2+y2)= 2(x+y) then find radius of curvature :-
>
108. The coordinates of the centre of curvature of the curve y=x2 at is _ _ _ _ _ _
_ _ _ :->(-1/2, 5/4)
109. The coordinates of the centre of curvature of the curve xy=2 at (2,1) is _ _ _ _ _ _
_ _ _ :->
110. Find the radius of curvature at P= on the curve x3+y3=3axy. Given
:->
111. The radius of curvature at Origin for y4+x3+a(x2+y2)-a2y=0 is _ _ _ _ _ _ _ :->a/2
112. The radius of curvature at the origin for x2-y2-2x-2y=0 is _ _ _ _ _ _ _ _ _ _ _ _ _
_ :->
113. The radius of curvature at the origin for x4-y4+x3-y3+x2-y2+y=0 is _ _ _ _ _ _ _ _
_ _ _ _ _ _ :->
114. If the y-axis is tangent to the curve at the origin O then radius of curvature at

origin is given by ρ = _ _ _ _ _ _ _ _ :->
115. If the x-axis is tangent to the curve at the origin O then radius of curvature at

origin is given by ρ = _ _ _ _ _ _ _ _ _ :->
116. Find the radius of curvature at P= ( ) on the curve x2+y2=4 :->2
4 4 2 2
117. Find ρ at (0,0) for 2x +3y +4x y+xy-y +2x=0 (ρ= Radius of curvature) :->1
118. The radius of curvature at origin for y= x4-4x3-18x2 is _ _ _ _ _ _ _ _ _ _ :->
119. The radius of curvature at origin for x3+y3-2x2+6y=0 is _ _ _ _ _ _ _ _ _ _ :->3/2
120. Find the radius of curvature at any point 't' of the curve x=a(cost+t sint), y=a (sint-
t cost) given = tant, = :->at
121. For the curves x=f(t); y=g(t), the formula for the radius of curvature is P= _ _ _ _
_, where x1 = :->
122. For the curve r=f(θ), the formula for the radius of curvature is P= _ _ _ _ _ _ _ _ _
_ _, where r = :->
123. For the curve y=f(x), the formula for the radius of curvature is P= _ _ _ _ _ _ _ _,
Where , :->


124. If the curvature of curve is K, the radius of curvature is _ _ _ _ _ _ _ :->1/K
125. The radius of curvature of the curve r=a(1+cosθ) at θ=0 is _ _ _ _ _ _ :-> a
126. The radius of curvature of the curve x=et+e ; y=et-e at t=0 is _ _ _ _ _ _ :->2
127. The radius of curvature of the curve y=ex at the point where it crosses the y-axis is
_ _ _ _ _ _ :->2
128. The radius of curvature at any point of the catenary y=c cosh is _ _ _ _ _ _ _ _
_ :->
129. The radius of curvature of the curve r=aθ at (r,θ) is _ _ _ _ _ _ _ _ _ :->
130. If rt-s2=0 at a point p=(a,b) then the case is _ _ _ _ _ _ _ _ _ where
:->failure
131. If rt-s2 0 at a point p=(a,b) then P is a _ _ _ _ _ _ _ _ _ where
:->saddle point
132. A function f(x,y) has a minimum value at (a,b) if _ _ _ _ _ _ _ _ where
r= :->rt-s2 0,r 0
133. A function f(x,y) has a maximum value at (a,b) if _ _ _ _ _ _ _ _ where
r= :->rt-s2 0,r 0
134. The necessary conditions for a function f(x,y) to have an extreme value are :-
>
135. If f(s,y) =xy, the stationary point (0,0) is _ _ _ _ _ _ _ _ _ :->saddle point
136. If f(x,y) = 1-x2-y2 then the stationary point is _ _ _ _ _ _ _ _ _ _ :->(0,0)
137. If f(x,y) = xy+(x-y) then the critical points of f are _ _ _ _ _ _ _ _ _ :->x=1, y=-1
138. If A=f (a,b), B=f (a,b), c=f (a,b) , then f(x,y) will have a maximum at (a,b) if
_ _ _ _ _ _ :->fx=0, fy=0, AC B2 and A 0
139. If f(x,y) = x2+y2, and (0,0) is stationary point. then the stationary point (0,0) is _ _
_ _ _ _ _ _ :->Minimum point
140. If u=x+ , v= then = _ _ _ _ _ _ _ _ _ :->
141. If u,v are 'functionally related' functions of x,y Then = _ _ _ _ _ _ _ :->= 0
142. If u=ax+by and v = cx+dy find :->ad-bc
143. If u= v= Tan x+Tan y are functionally dependent find the relation
between them :->v=Tan u
144. If u= , v= Tan x +Tan y then = _ _ _ _ _ _ _ _ :->0
145. If u=xsiny, v=ysinx then = _ _ _ _ _ _ _ _ _ :->sinx siny - xycosx cosy
146. The functions u=xy+yz+zx, v=x2+y2+z2, w=x+y+z are functionally dependent.
Find a relation between them :->w2= v+2u
147. If u=x+y+z, v=x2+y2+z2, w=x3+y3+z3-3xyz find :->0
u u 2u
148. If x=e cos v, y=e sin v then = _ _ _ _ _ _ _ :->e


149. The functions u= xeysinz, v=xeycosz, w=x2e2yare functionally related. Find the
relation between them :->u2+v2=w
150. If x=r cos θ, y= r sin θ then = _ _ _ _ _ _ _ _ _ _ :->
151. If x=r cos θ, y= r sin θ then = _ _ _ _ _ _ _ _ _ _ :->r
152. If u,v,w are 'functionally related' functions of x, y, z then =_________
_ _ _ :->= 0
153. If u, ϑ are functions of r, s and r, s re in turn functions of x,y then __
_ _ _ _ _ _ _ :->
154. If = _ _ _ _ _ _ _ _ _ _ _ _ _ :->=1
x y
155. If u=e , v=e then = _ _ _ _ _ _ _ _ :->uv
156. If x=r cos θ, y=r sin θ, z=z then = _ _ _ _ _ _ _ _ _ _ :->r
x+y
157. If u= e then J = _ _ _ _ _ _ _ _ _ _ _ :->2e2y
158. If x=u(1-v), y=uv then = _ _ _ _ _ _ _ _ _ _ _ _ _ :->1
159. If x=rsin θ cos Ø, y= r sin θ sin Ø, z=r cos θ then = _ _ _ _ _ _ _ _ _ :-
>r2sinθ
160. In Taylor's theorem, the schlomilah and Roche form of remainder is :-
>
161. f(a+h)= f(a)+ ..................+ frac{{h^{n - 1} }}{{ left| !{
underline { , {n - 1} ,}} right. }}f^{n - 1} (a) + R where Rn = fn(a+θh), is called
_ _ _ _ _ _ _ _ _ _ _ _ :->Taylor's theorem with Schlomileh - Roche's form of
remainder
162. In the Taylor's theorem the Lagrange's form of remainder is :->
163. In the Taylor's theorem cauchy's form of remainder is :->
164. If (a+h) = f(a) +h f1(a)+ f"(a) +......+ fn (a+θh), 0 θ 1 is called _ _ _ _ _ _ _ _
_ _ _ :->Taylor's theorem with Langrange form of remainder
165. f(x) = f(0)+ is
called _ _ _ _ _ _ _ _ _ _ _ _ :->Maclaurin's theorem with Lagrange's form of
remainder

166. Maclaurin's expansion of cosx is _ _ _ _ _ _ _ _ _ _ _ _ :->
167. Maclaurin's expansion for log(1+x) is _ _ _ _ _ _ _ _ :-
> ......................
168. The expansion of sinx in powers of is _ _ _ _ _ _ _ _ _ :-
>


169. The expansion of ex in power of (x-1) is _ _ _ _ _ _ _ _ _ _ :->
170. The c of the cauchy's mean value theorem for the pair of functions f(x) = sinx,
g(x) = cos x for all x in [ -π/2, 0] is _ _ _ _ _ _ _ _ _ :->-π/4
171. The value of c of cauchy's mean value theorem for f(x)= log x, g(x) = in [1,e] is
:->
172. The value of c of cauchy's mean value theorem for f(x)= x3 and g(x) = x2 in [1,2]
is :->
173. The value of c of cauchy's mean value theorem for f(x)= and g(x) = in [1,4]
is :->2
174. The value of c of cauchy's mean value theorem for f(x) = sinx and g(x) = cos x in
[0, π/2] is :->π/4
175. Lagrange's mean-value theorem for f(x) = sec x in (0, 2 π) is :->not applicable
due to discontinuity
176. If f and g are differentiable on [0, 1] such that f(0) =2 and g(0) = 0 ; f(1) =6 and
g(1)=2 then there exists Cε (0,1) such that _ _ _ _ _ _ _ _ _ _ _ :->f1(c) = 2 g1(c)
177. The value of c of cauchy's mean-value theorem for the functions f(x) = x2, g(x) =
x4 in [1,2] is _ _ _ _ _ _ _ _ :->
178. The value of c of cauchy's mean-value theorem for the functions f(x) = 1/x2, g(x)
= 1/x in [ a,b], 0 a b is :->
179. The value of c of cauchy's mean-value theorem for the functions f(x) = ex and g(x)
= e defined on [ a,b], 0 a b is _ _ _ _ _ _ _ _ _ _ :->
180. The value of c of lagrange's mean value theorem for f(x)= in [2,4] is :->
181. The value of c in lagrange's mean value theorem for f(x) = (x-2) (x-3) in [0,1] is :-
>0.5
182. The value of c in lagrange's mean-value theorem for f(x)= cosx in [ 0, ] is :->sin
( 2/π)
183. The value of c in lagrange's mean-value theorem for f(x) = log x in [1,e] is :->e-1
184. The value of c in lagrange's mean-value theorem for f(x) = ex in (0,1) is :->log (e-
1)
185. If f(x) = x2, find θε(0, 1) such that f (x+h) = f(x) +h f1 (x+θh) :->
186. Lagrange's mean value theorem is not applicable to the function defined on [-1, 1]
by f(x) = sin , (x ) and f(0) = 0, because :->f is not derivable in (-1, 1)
187. Lagrange's mean value theorem is not applicable to the function f(x) =x in [-1,
1] because :->f is not derivable in (-1, 1)
188. Find c of Lagrange's mean value theorem for f(x) = x(x-1) (x-2) in [ 0, 1/2] :-
>
189. Find c of Lagrange's mean value theorem for f(x) = (x-1) (x-2) in [1,3] :->2
190. The value of c in Rolle's theorem for f(x)= sinx in (0, π) is :->
191. The value of c in Rolle's theorem for f(x) = x2 in (-1, 1) is :->0
192. The value of c in colle's theorem for f(x) = sinax in (0, πa) is :->


193. If a fuction is such that (i) it is continuous in [a,a+h], (ii) it is derivable in (a,a+h)
(iii) f(a)=f(a+h) then there exists at least one number such that _ _ _ _ _ _ _ _ :-
>f1(a+θh)=0
194. If F:[a,b] R is (i) continuous in [a,b] (ii) derivable in (a,b) (iii) f(a) = f(b) then
there exist at least one point c in (a,b) such that _ _ _ _ _ _ _ _ _ _ _ :->f1(c) =0
195. The value of c in Rolle's theorem f(x) = ex sin x in [0,π] is :->
196. Rolle's theorem is not applicable to the function f(x) = x in[-1, 1] because :->f is
not derivable at x=0ε(-1,1)
197. Rolle's theorem is not applicable to the function f(x) = sinx in [0, ] because :-
>f(0) ≠ f ( )
198. The value of c in Rolle's theorem for f(x) = log [ ] in [a,b] is _ _ _ _ _ _ _ _ _
_ :->
199. The value of c in Rolle's theorem for f(x)= frac {sinx} {ex} in (0, π) is :->

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