maths formulas for ssc exams

8/13/2019 Maths Formulas for SSC EXAMS

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iii

Contents

1 krj_bo pbqpNKN p f 1NKO p k 5

NKP _~ f 7NKQ ` k 8

2 ^idb_o^OKN c~ c~ 12OKO m c~ 13OKP m 14OKQ o 15

OKR i~ 16OKS b~ 18OKT f~ 19OKU ` f c~ 22

3 dbljbqovPKN o q~ 24PKO f q~ 27PKP b~~ q~ 28PKQ p~ q~ 29PKR p~ 33PKS o~ 34PKT m~~~ 35PKU o 36PKV q~ 37

PKNM f q~ 38PKNN f q~ f ` 40PKNO q~ f ` 41


2/332

iv

PKNP h 42PKNQ ` n~~~ 43

PKNR q~~ n~~~ 45PKNS d~ n~~~ 46PKNT o~ e~ 47PKNU o~ m 48PKNV ` 50PKOM p ~ ` 53PKON p ~ ` 54PKOO ` 55PKOP o~~ m~~ 56PKOQ m 57PKOR o~ q~ 58PKOS o~ m~ 59PKOT c ~ o~ m~ 61PKOU o~~ o t 62PKOV m~ p 63

PKPM o `~ ` 66PKPN o `~ ` ~ l m~ c~ 68PKPO o `~ ` 69PKPP c ~ o `~ ` 70PKPQ p 72PKPR p~ `~ 72PKPS p~ p 73PKPT p~ p 74

PKPU p~ t 75PKPV b 76PKQM `~ q 78

4 qofdlkljbqovQKN o~~ ~ a j~ ^ 80QKO a ~ d~ q c 81

QKP p q c 86QKQ q c ` ^ 87QKR j f~ c~ 88


3/332

v

QKS o c~ 89QKT m q c 90

QKU o~ q c 90QKV ^ ~ p~ c~ 91QKNM a ^ c~ 92QKNN j ^ c~ 93QKNO e~ ^ c~ 94QKNP e~ ^ q~ f 94QKNQ q~ q b m 95QKNR q~ q b p 97QKNS m q c 98QKNT d~ f q c 99QKNU m~ s~ f q c 102QKNV o~ f q c 103QKOM q b~ 106QKON o~ e c 106

5 j^qof`bp ^ka abqbojfk^kqpRKN a~ 107RKO m a~ 109RKP j~ 110RKQ l~ j~ 111RKR p i~ b~ 114

6 sb`qlop

SKN s `~ 118SKO s ^ 120SKP s p~ 122SKQ p~ s 122SKR p~~ m 123SKS s m 125SKT q m 127

7 ^k^ivqf` dbljbqovTKN l -a~ `~ p 130


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vi

TKO q -a~ `~ p 131TKP p~ i m~ 139

TKQ ` 149TKR b 152TKS e~ 154TKT m~~~ 158TKU q -a~ `~ p 161TKV m~ 165TKNM p~ i p~ 175TKNN n~ p~ 180TKNO p 189

8 afccbobkqfî `î`rirpUKN c ~ q d~ 191UKO i c 208UKP a ~ m a~ 209UKQ q~ a~ 211

UKR e l a~ 215UKS ^~ a~ 217UKT a~ 221UKU j~~ c 222UKV a~ l~ 225

9 fkqbdoî `î`rirpVKN f f~ 227

VKO f~ o~~ c 228VKP f~ f~~ c 231VKQ f~ q c 237VKR f~ e c 241VKS f~ b~ ~ i~ c 242VKT o c~ 243VKU a f~ 247

VKV f f~ 253VKNM a f~ 257VKNN q f~ 269


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vii

VKNO i f~ 275VKNP p~ f~ 285

10 afccbobkqf^i bnr^qflkpNMKN c l l~ a~ b~ 295NMKO p l l~ a~ b~ 298NMKP p m~~ a~ b~ 302

11 pbofbpNNKN ^ p 304NNKO d p 305NNKP p c p 305NNKQ f p 307NNKR m ` p 307NNKS ` q 308NNKT ^~ p 310NNKU m p 311

NNKV a~ ~ f~ m p 312NNKNM q~ ~ j~~ p 313NNKNN m p b~ p c 314NNKNO _~ p 316NNKNP c p 316

12 mol_^_fifqvNOKN m~ ~ `~ 318

NOKO m~ c~ 319


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viii

q ~ ~ ~K


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1

Chapter 1

Number Sets

1.1 Set Identities

pW ^I _I `r~ W f` W ^ m W _^ b W r W _^

f W _^ a W _y^

1. f^

2. ^^

3. _^ = _^ ~ ^_ .

4. b p^

5. r p{ }_^_^` ==


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CHAPTER 1. NUMBER SETS

2

Figure 1.

6. `~

^__^ =

7. ^~( ) ( ) `_^`_^ =

8. f p

{ }_~^_^` ==

Figure 2.

9. `~^__^ =

10. ^~( ) ( ) `_^`_^ =


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3

11. a( ) ( ) ( )`^_^`_^ = I

( ) ( ) ( )`^_^`_^ = K

12. f^^^ = I^^^ =

13. a~

=^ Iff^ =

14. f^^ = I

^f^ =

15. `

{ }^f^ =

16. ` f ~ rf^^ = I=^^

17. a j~ i~

( ) _^_^ =

I( ) _^_^ =

18. a p{ }^~_^y_` ==


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4

Figure 3.

19. ( )_^y_^y_ =

20. ^_^y_ =

21. =^y^

22. ^_y^ = =_^ .

Figure 4.

23. ( ) ( ) ( )`_y`^`_y^ =

24. ^yf^ =

25. `~~ m

( ){ }_~^I_^` ==


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5

1.2 Sets of Numbers

k~~ W kt W Mk

fW w

m W +w

k~ W wo~~ W n

o~ W o` W `

26. k~~ k` W { }KIPIOINk = K

27. t k

` ~ W { }KIPIOINIMk M= K

28. ft ~ ~ W

{ }KIPIOINkw ==+ I

{ }NIOIPIw = K I

{ } { }KK IPIOINIMINIOIPIwMww == + K

29. o~~ ko~ ~ ~W

== M~w~w~~

~n K

30. f~~ k

k~ ~ ~ ~K


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6

31. o~ kr ~~ ~ ~~ W oK

32. ` k{ }o~o` += I

~~ K

33. `onwk

Figure 5.


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7

1.3 Basic Identities

o~ W ~I I

34. ^ f~M~ =+

35. ^ f

( ) M~~ =+

36. `~ ^~~ +=+

37. ^~ ^( ) ( )~~ ++=++

38. a p~( )~~ +=

39. j~ f~N~ =

40. j~ f

N~

N~ = I M~

41. j~ q MMM~ =

42. `~ j~

~~ =


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8

43. ^~ j~( ) ( )~~ =

44. a i~( ) ~~~ +=+

45. a aN

~~

=

1.4 Complex Numbers

k~~ W f~~ W

` W o~ ~W ~I f~~ ~W I j ~ W I N I O

^ ~ W I N I O

N

= R

= NQ

=+

N O = N S = N OQ =+ P = T = PQ =+

46.

N Q = N U = N Q =

47. ~ +=

48. ` m~


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9

Figure 6.

49. ( ) ( ) ( ) ( )~~ +++=+++

50. ( ) ( ) ( ) ( )~~ +=++

51. ( )( ) ( ) ( )~~~ ++=++

52.

~

~

~OOOO

+

+++

=++

53. `~ ` k

~~|||||||

=+

54. = ~ I =


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10

Figure 7.

55. m~ m~ ` k( )+=+ ~

56. j ~ ^ ~ ` kf ~ + ~ I OO ~ += EFI

~

~~= E~FK

57. m m~ o~

( ) ( )OOONNNON ++= ( ) ( )[ ]ONONON +++=

58. `~ k m~ o~

( ) ( ) ( )[ ]+=+ |||||||||||||||||||||

59. f ~ ` k m~ o~

( ) ( ) ( )[ ]+=

+

N

N


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11

60. n m~ o~( )

( ) ( ) ( )[ ]ONONON

OOO

NNN

O

N

+=+

+

=

61. m ~ ` k

( )[ ] ( ) ( )[ ]+=+=

62. c~ a j

( ) ( ) ( )+=+

63. k o ~ ` k

( )

++

+=+=

O

O I

NIIOINIM = K K

64. b c~

+=


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12

Chapter 2

Algebra

2.1 Factoring Formulas

o~ W ~I I k~~ W

65. ( )( )~~~ OO +=

66. ( )( )OOPP ~~~~ ++=

67. ( )( )OOPP ~~~~ ++=+

68. ( )( ) ( )( )( )OOOOOOQQ ~~~~~~ ++=+=

69.( )( )

QPOOPQRR

~~~~~~ ++++=

70. ( )( )QPOOPQRR ~~~~~~ +++=+

71. f I

( )( )NOOPON ~~~~~~ +++=+ K K

72. f I ( )( )NOOPON ~~~~~~ +++++= K I


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CHAPTER 2. ALGEBRA

13

( )( )NOOPON ~~~~~~ +++=+ K K

2.2 Product Formulas

o~ W ~I I t W I

73. ( ) OOO ~O~~ +=

74. ( ) OOO ~O~~ ++=+

75. ( ) POOPP ~P~P~~ +=

76. ( ) POOPP ~P~P~~ +++=+

77. ( ) QPOOPQQ ~Q~S~Q~~ ++=

78. ( ) QPOOPQQ ~Q~S~Q~~ ++++=+

79. _~ c~

( ) I`~`~`~`~`~ N

NOO

ON

N

M +++++=+ K

( )>>

>`

= ~ ~ K

80. ( ) O~O~O~~ OOOO +++++=++

81. ( ) ++++++=+++++ OOOOOO

~~ KK ( )~~~~O +++++++++++ KKK


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CHAPTER 2. ALGEBRA

14

2.3 Powers

_~ E ~ FW ~I m E~~ FW I

82. ~~~ +=

83.

~~

~ =

84. ( ) ~~ =

85.

~

~=

86. ( ) ~~ =

87. N~ M = I M~

88. N~ N =

89.

~N~ =

90.

~~ =


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CHAPTER 2. ALGEBRA

15

2.4 Roots

_~W ~I m E~~ FW I

MI~ E O = I k F

91. ~~=

92. ~~ =

93.

~

~= I M

94.

~

~

~

== I M K

95. ( ) ~~ =

96. ( ) ~~ =

97. ~~ =

98.

~~ =

99. ~~ =

100. ( ) ~~ =


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CHAPTER 2. ALGEBRA

16

101.~

~

~

N N

= I M~ K

102.O

~~

O

~~~

OO

+=

103.~

~

~

N

=

m

2.5 Logarithms

m ~ W I I ~I I k~~ W

104. a i~

~= ~ ~= I M~ > I N~ K

105. MN~ =

106. N~~ =

107.

=

N~

N~M~

108. ( ) ~~~ +=

109.

~~~ =


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CHAPTER 2. ALGEBRA

17

110. ( ) ~~ =

111.

N ~

~ =

112. ~

~

~ == I M > I N K

113. ~

N

~ =

114. ~~=

115. i~ _~ NMNM =

116. k~~ i~ = I

KTNUOUNUOUKO

NN

=

+=

117. QPQOVQKM

NM

N ==

118. PMORURKO

N ==


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CHAPTER 2. ALGEBRA

18

2.6 Equations

o~ W ~I I I I I I pW N I O I N I O I P

119. i~ b~ l s~~

M~ =+ I

~

= K

120. n~~ b~

M~O =++ I~O

~Q

O

OIN

= K

121. a~

~Qa

O

=

122. s c~

f MO =++ I

=

=+

ON

ON K

123. M~O =+ I MN= I~

O = K

124. M~O =+ I~

OIN = K

125. ` b~K `~~ c~K

MP

=++ I

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CHAPTER 2. ALGEBRA

19

N += I ( ) ( ) O

P

O

N PIO ++= I

P

OO

P

O

O

+

+= I POO

P

O

O

+

= K

2.7 Inequalities

s~~W I I

o~ W

PON ~II~I~I~

III~

K I I

a~W aI a I a I a

126. f~I f~ k~ ~ d~

f~ f~ k~ d~~ [ ]I~

~ < ( ]I~

~

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CHAPTER 2. ALGEBRA

20

127. f ~> I ~ < K

128. f ~> I M~ > M~ I ~ +>+ K

130. f ~> I ~ > K

131. f ~> ~ > I ~ +>+ K

132. f ~> ~ > I ~ > K

133. f ~> ~ M > I ~> K

134. f ~> ~ M > I

~> K

135. f ~> ~ M ~ M X ~ ~ ~ ~= K

141. O~

N~ + I M~> X ~ ~ ~ ~ ~ N~= K

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CHAPTER 2. ALGEBRA

21

142.

~~~~~~ ON ON

+++

KK I M~II~I~ ON >K K

143. f M~ >+ ~ M~> I ~

> K

144. f M~ >+ ~ M~ ++

M~> M~ <

Ma>

N ON

Ma<

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CHAPTER 2. ALGEBRA

22

146. ~~ ++

147. f ~ I M~> K

149. f ~O K

150. f ~O

> I ~> I M~> K

151. f( )( )

M

> I

( ) ( )

( )

>

M

MK

152.( )( )

M

< I

( ) ( )

( )


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CHAPTER 2. ALGEBRA

23

154. p ` f c~f ~I

~ W( )N`^ += K

155. ` ` ff ~ E FI

`^ = K


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24

Chapter 3

Geometry

3.1 Right Triangle

i ~ ~W ~I eW ^W j~W ~ I I

^W I o~ W oo~ W ^~W p

Figure 8.

156. =+ VM


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CHAPTER 3. GEOMETRY

25

157. ==

~

158. ==

159. == ~

~

160. == ~~

161. ==

162. == ~

163. m~~ qOOO ~ =+

164. ~ O = I O = I ~ ~ ~ ~ I -

I K

Figure 9.


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CHAPTER 3. GEOMETRY

26

165. O = I ~ ~K

166.Q

~

OOO

~ = IQ

~

OOO

= I

~ ~ ~ ~ ~ ~ K

Figure 10.

167.O

= I

~ K

168. Oo ==

169.~

~

O

~

++=

+=

170. ~ =


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CHAPTER 3. GEOMETRY

27

171.O

O

~p ==

3.2 Isosceles Triangle

_~W ~iW _~ ~W s ~W ^ ~W mW i^~W p

Figure 11.

172.O

VM =

173.Q

~OOO =


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CHAPTER 3. GEOMETRY

28

174. O~i +=

175. == O

O

~pO

3.3 Equilateral Triangle

p ~ ~~ ~W ~^W o~ W oo~ W mW i^~W p

Figure 12.

176.O

P~ =


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CHAPTER 3. GEOMETRY

29

177.P

P~

P

Oo ==

178.O

o

S

P~

P

N ===

179. ~Pi =

180. Q

P~

O

~p

O

==

3.4 Scalene Triangle

E^ ~ ~F

p ~ ~W ~I I

pWO

~

++=

^ ~ ~W II ^ ~I I W ~ II

j~ ~I I W ~ II _ ~ II W ~ IIo~ W oo~ W ^~W p


36/332

CHAPTER 3. GEOMETRY

30

Figure 13.

181. =++ NUM

182. ~ >+ I~ >+ I

~ >+ K

183. ~


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CHAPTER 3. GEOMETRY

31

185. i~ `

+= O~ OOO I

+= ~O~ OOO I

+= ~O~ OOO K

186. i~ p

oO

~=

=

=

I

o ~ K

187.pQ

~

O

~

O

~

O

O

O

O

~o

~

====

=

=

=

188.( )( )( )

~ O

= I

~ N

N

N

N ++= K

189.( )( )

O

=

I

( )

~

O

=

I

( )( )( )~

O~

=

K

190. ( )( )( )~~

O ~ = I

( )( )( )~O

= I

( )( )( )~

O = K


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CHAPTER 3. GEOMETRY

32

191. == ~ I

== ~ I

== ~ K

192.Q

~

O

OOOO~

+= I

Q

O

~

OOOO

+

= I

Q

O~

OOO

O += K

Figure 15.

193.~

P

O^j = I

P

O_j = I

P

O`j = EcKNRFK

194.( )

( )OO~

~Q

+

= I

( )( )O

O

~

~Q

+

= I

( )( )O

O

~

~Q +

= K


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CHAPTER 3. GEOMETRY

33

195.O

O

O

~p ~ === I

O

O~

O~p === I

( )( )( )~p = Ee c~FIp = I

oQ

~p = I

= oOp O

I

O~

O~

O~p O

= K

3.5 Square

p ~ ~W ~a~~W o~ W oo~ W mW i^~W p

Figure 16.


40/332

CHAPTER 3. GEOMETRY

34

196. O~ =

197.O

O~

O

o ==

198.O

~ =

199. ~Qi =

200. O~p =

3.6 Rectangle

p ~ ~W ~I a~~W o~ W omW i^~W p

Figure 17.

201. OO ~ +=


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CHAPTER 3. GEOMETRY

35

202.O

o=

203. ( )~Oi +=

204. ~p =

3.7 Parallelogram

p ~ ~~~W ~I a~~W ON I

` ~W I^ ~~W ^W mW i^~W p

Figure 18.

205. =+ NUM

206. ( )OOOOON ~O +=+


42/332


43/332

CHAPTER 3. GEOMETRY

37

210. =+ NUM

211. OOOON ~Q =+

212.~O

~ ON==

213.O

~

~Q

O

ON

===

214. ~Qi =

215. == ~~p O I

ONO

Np = K

3.9 Trapezoid

_~ ~ ~W ~I jW ^W

^~W p


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CHAPTER 3. GEOMETRY

38

Figure 20.

216.O

~ +=

217. O

~p =

+=

3.10 Isosceles Trapezoid

_~ ~ ~W ~I iW

jW ^W a~~W o~ W o^~W p


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CHAPTER 3. GEOMETRY

39

Figure 21.

218.O

~ +=

219. O~ +=

220. ( )OO ~Q

N =

221.( )( )~O~O

~o

O

++

+=

222. O

~p =

+=


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CHAPTER 3. GEOMETRY

40

3.11 Isosceles Trapezoid with

Inscribed Circle

_~ ~ ~W ~I iW jW ^W a~~W o~ W oo~ W mW i^~W p

Figure 22.

223. O~ =+

224. O

~ =

+=

225. OOO +=


47/332

CHAPTER 3. GEOMETRY

41

226.O

~

O

==

227.~

S

~

U

~

O

~

N

O

Q

O

o OO

O

+++

=+=+===

228. ( ) Q~Oi =+=

229. ( ) OiO ~~O~p ===+=+=

3.12 Trapezoid with Inscribed Circle

_~ ~ ~W ~I i~~ W I jW ^W a~~W ON I

^ ~~W o~ W

o~ W omW i^~W p


48/332

CHAPTER 3. GEOMETRY

42

Figure 23.

230. ~ +=+

231.O

O

~

+=

+=

232. ( ) ( )O~Oi +=+=

233. O

O

~p =

+=

+= I

= O

Np ON K

3.13 Kite

p ~ W ~I a~~W ON I

^W II mW i

^~W p


49/332

CHAPTER 3. GEOMETRY

43

Figure 24.

234. =++ PSMO

235. ( )~Oi +=

236.O

p ON=

3.14 Cyclic Quadrilateralp ~ ~~~W ~I I I a~~W ON I

^ ~~W f~ ~W IIIo~ W omW ipW ^~W p


50/332

CHAPTER 3. GEOMETRY

44

Figure 25.

237. =+=+ NUM

238. m q

ON~ =+

239. ~i +++=

240.( )( )( )( )( )( )( )~

~~~

Q

No

+++

= I

O

i = K

241. = O

Np ON I

( )( )( )( )~p = I

O

i = K


51/332

CHAPTER 3. GEOMETRY

45

3.15 Tangential Quadrilateral

p ~ ~~~W ~I I I a~~W ON I

^ ~~W o~ W mW ipW ^~W p

Figure 26.

242. ~ +=+

243. ( ) ( )O~O~i +=+=+++=

244.( ) ( )

O

~~

OOOO

ON += I

Oi = K


52/332

CHAPTER 3. GEOMETRY

46

245. == O

Np ON

3.16 General Quadrilateral

p ~ ~~~W ~I I I a~~W

ON

I

^ ~~W f~ ~W IIImW i^~W p

Figure 27.

246. =+++ PSM

247. ~i +++=


53/332

CHAPTER 3. GEOMETRY

47

248. = O

Np ON

3.17 Regular Hexagon

pW ~f~ ~W p~ W o~ W o~ W omW ipW ^~W p

Figure 28.

249. = NOM

250.O

P~ ==


54/332

CHAPTER 3. GEOMETRY

48

251. ~o=

252. ~Si =

253.O

PP~p

O

== I

O

i = K

3.18 Regular Polygon

pW ~k W f~ ~W p~ W o~ W o~ W omW ipW ^~W p


55/332

CHAPTER 3. GEOMETRY

49

Figure 29.

254.

= NUMO

O

255. = NUMO O

256.

O

~o

=

257.Q

~o

~O

~

O

O ===

258. ~i =

259.

O

O

op

O = I

Q

~op

OO == I


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CHAPTER 3. GEOMETRY

50

O

i = K

3.19 Circle

o~W oa~W `W ~p~ W I q~ W `~ ~W f ~W mW i^~W p

260.O

oO~

=

Figure 30.


57/332

CHAPTER 3. GEOMETRY

51

261. ONON ~~ =

Figure 31.

262. NN =

Figure 32.

263. NO =


58/332

CHAPTER 3. GEOMETRY

52

Figure 33.

264.O

=

Figure 34.

265. oOi ==

266. O

io

Q

op

OO

=

==


59/332

CHAPTER 3. GEOMETRY

53

3.20 Sector of a Circle

o~ ~ W o^ W `~ ~ E ~~FW `~ ~ E FW mW i^~W p

Figure 35.

267. o =

268.=

NUM

o

269. oOi +=

270.

===PSM

o

O

o

O

op

OO

60/332

CHAPTER 3. GEOMETRY

54

3.21 Segment of a Circle

o~ ~ W o^ W `W ~`~ ~ E ~~FW `~ ~ E FW e W mW i

^~W p

Figure 36.

271. OoOO~ =

272. OO ~oQO

No = I o <

273. ~i +=


61/332

CHAPTER 3. GEOMETRY

55

274. ( )[ ] ( )O

o

NUMO

oo~o

O

Np

OO

=

== I

~P

Op K

3.22 Cube

bW ~a~~W o~ W o~ W p~ ~~W psW s

Figure 37.

275. P~ =

276.O~ =


62/332

CHAPTER 3. GEOMETRY

56

277.O

P~o=

278. O~Sp =

279. P~s =

3.23 Rectangular Parallelepiped

bW ~I I a~~W p~ ~~W psW s

Figure 38.

280. OOO ~ ++=

281. ( )~~Op ++=

282. ~s=


63/332

CHAPTER 3. GEOMETRY

57

3.24 Prism

i~~ W eW i~~ ~~W ip

^~ ~W _p

q~ ~ ~~W psW s

Figure 39.

283. _i pOpp += K

284. i~~ ^~ ~ o m( )~~~~p PONi ++++= K

285. i~~ ^~ ~ l mp i= I

K


64/332

CHAPTER 3. GEOMETRY

58

286. ps _=

287. `~~D md ~~ ~K f ~ ~~ ~ ~ ~~~ I ~ ~K

3.25 Regular Tetrahedron

q~ W ~eW ^~ ~W _p

p~ ~~W psW s

Figure 40.

288. ~P

O

=


65/332

CHAPTER 3. GEOMETRY

59

289.Q

~Pp

O

_=

290. O~Pp =

291.OS

~p

P

Ns

P

_ == K

3.26 Regular Pyramid

p ~W ~i~~ W eW

p~ W k W p ~W o~ ~W ^~ ~W _p

i~~ ~ ~~W ip

q~ ~ ~~W psW s


66/332

CHAPTER 3. GEOMETRY

60

Figure 41.

292.Q

~

OO =

293.

O

~Q

OOO

=

294. ~Q~Q

N~

O

Np OOi ===

295. p _=

296. i_ ppp +=

297. P

Np

P

Ns _ ==


67/332

CHAPTER 3. GEOMETRY

61

3.27 Frustum of a Regular Pyramid

_~ ~ W

PON

PON

IIII

~II~I~I~

K

K

eW p~ W ^~ ~W Np I Op

i~~ ~ ~~W ip

m ~W Nm I Omp~ ~W q~ ~ ~~W psW s

Figure 42.

298.

~

~

~

~

~

P

P

O

O

N

N ====== K


68/332

CHAPTER 3. GEOMETRY

62

299. O

N

O p

p=

300.( )

O

mmp ONi

+=

301. ONi pppp ++=

302. ( )OONN ppppP

s ++=

303. [ ]ONO

N NP

p

~

~

N

P

ps ++=

++=

3.28 Rectangular Right Wedge

p ~W ~I q W eW i~~ ~ ~~W ip

^~ ~W _pq~ ~ ~~W psW s


69/332

CHAPTER 3. GEOMETRY

63

Figure 43.

304. ( ) ( )OOOOi ~Q~O

Np ++++=

305. ~p _=

306. i_ ppp +=

307. ( )~OS

s +=

3.29 Platonic Solids

bW ~o~ W o~ W op~ ~~W p

sW s


70/332

CHAPTER 3. GEOMETRY

64

308. c m~ pq ~ ~ ~ ~

~ ~ K

p k s

k b

k c~

p

q~ Q S Q PKOR` U NO S PKOO

l~ S NO U PKOTf~ NO PM OM PKOT

a~ OM PM NO PKOT

Octahedron

Figure 44.

309.S

S~ =

310. O

O~

o=


71/332

CHAPTER 3. GEOMETRY

65

311. P~Op O=

312.P

O~s

P

=

Icosahedron

Figure 45.

313.)

NO

RPP~

+=

314. ( )RROQ~o +=

315. P~Rp O=

316.)

NO

RP~Rs

P +=


72/332

CHAPTER 3. GEOMETRY

66

Dodecahedron

Figure 46.

317.( )

O

RNNORNM~

+=

318.)

Q

RNP~o

+=

319. ( )RORR~Pp O +=

320.Q

RTNR~s

P +=

3.30 Right Circular Cylinder

o~ ~W oa~ ~W


73/332

CHAPTER 3. GEOMETRY

67

eW ei~~ ~ ~~W ip

^~ ~W _pq~ ~ ~~W psW s

Figure 47.

321. oeOp i =

322. ( )

+=+=+= O

eoeoOpOpp _i

323. eoeps O_ ==


74/332

CHAPTER 3. GEOMETRY

68

3.31 Right Circular Cylinder with

an Oblique Plane Face

o~ ~W oq ~ ~ W N

q ~ W O

i~~ ~ ~~W ip

^~ ~ ~W _p

q~ ~ ~~W psW s

Figure 48.

324. ( )ONi op +=

325.

O

ONOO

_ O

ooop

++=


75/332

CHAPTER 3. GEOMETRY

69

326.

++++=+=O

ONOON_i

O

oooppp

327. ( )ONO

O

os +

=

3.32 Right Circular Coneo~ ~W oa~ ~W eW ep~ W i~~ ~ ~~W ip

^~ ~W _p

q~ ~ ~~W psW s

Figure 49.


76/332

CHAPTER 3. GEOMETRY

70

328. OO oe =

329.O

op i

==

330. O_ op =

331. ( )

+=+=+=O

O

Nooppp _i

332. eoP

Nep

P

Ns O_ ==

3.33 Frustum of a Right Circular Cone

o~ ~W oI eW ep~ W p~ ~W ^~ ~W Np I Op

i~~ ~ ~~Wi

p

q~ ~ ~~W psW s


77/332

CHAPTER 3. GEOMETRY

71

Figure 50.

333. ( )OO oe =

334.

o=

335. OO

O

N

O

o

p

p==

336. ( )op i +=

337. ( )oopppp OOiON +++=++=

338. ( )OONN ppppP

s ++=

339. [ ]ONO

N NP

p

o

oN

P

ps ++=

++=


78/332

CHAPTER 3. GEOMETRY

72

3.34 Sphere

o~W oa~W p~ ~~W psW s

Figure 51.

340. OoQp =

341. poP

N

S

Neo

P

Qs PP ===

3.35 Spherical Cap

o~ W oo~ ~W eW ^~ ~ ~W _p

^~ ~ ~W `p q~ ~ ~~W psW s


79/332

CHAPTER 3. GEOMETRY

73

Figure 52.

342.O

o

OO +=

343. O_ p =

344. ( )OO

` p +=

345. ( ) ( )OOO`_ oOOppp +=+=+=

346. ( ) ( )OOO PS

oPS

s +

=

=

3.36 Spherical Sector

o~ W oo~ ~ ~ ~W eW

q~ ~ ~~W psW s


80/332

CHAPTER 3. GEOMETRY

74

Figure 53.

347. ( )Oop +=

348. oP

Os O=

kW q ~ ~ ~ ~ K

3.37 Spherical Segment

o~ W oo~ ~W N I O

eW ^~ ~ ~W pp

^~ ~ ~W Np I Op

q~ ~ ~~W p

sW s


81/332

CHAPTER 3. GEOMETRY

75

Figure 54.

349. oOp p =

350. ( )OOONONp oOpppp ++=++=

351. ( )OOOON PPS

Ns ++=

3.38 Spherical Wedge

o~W oa~ ~ W a~ ~ ~~W ^~ ~ W ip

q~ ~ ~~W psW s


82/332

CHAPTER 3. GEOMETRY

76

Figure 55.

352. oOVM

op O

O

i =

=

353. oOoVM

oop OO

OO +=

+=

354. oP

O

OTM

os P

P

=

=

3.39 Ellipsoid

p-~W ~I I sW s


83/332

CHAPTER 3. GEOMETRY

77

Figure 56.

355. ~P

Qs =

Prolate Spheroid

p-~W ~I I E ~> Fp~ ~~W psW s

356.

+=

~~Op I

~

~

OO = K

357. ~PQs O=

84/332

CHAPTER 3. GEOMETRY

78

Oblate Spheroid

p-~W ~I I E ~ < Fp~ ~~W psW s

358.

+= ~L ~

~~

Op I

~

OO

= K

359. ~P

Qs O=

3.40 Circular Torus

j~ ~W o

j ~W p~ ~~W psW s


85/332

CHAPTER 3. GEOMETRY

79

Picture 57.

360. oQp O=

361. OO oOs =


86/332

80

Chapter 4

Trigonometry

^W I

o~ E~ ~ FW I t W

4.1 Radian and Degree Measures of Angles

362. ?QRDNTRT

NUM

~N

=

363. ~MNTQRPKM~NUM

N

=

364. ~MMMOVNKM~SMNUM

DN

=

365. ~MMMMMRKM~PSMMNUM

?N

=

366. ^EF

M PM QR SM VM NUM OTM PSM

^

E~~F

M

S

Q

P

O

O

P O


87/332

CHAPTER 4. TRIGONOMETRY

81

4.2 Definitions and Graphs of Trigonometric

Functions

Figure 58.

367.

=

368.

=

369.

~ =

370.

=


88/332


82

371.

=

372.

=

373. p c= I NN K

Figure 59.

374. ` c

= I NN K


89/332


83

Figure 60.

375. q~ c

~= I ( )O

NO + I K~

Figure 61.


90/332


84

376. `~ c= I I K

Figure 62.

377. p~ c

= I ( )O

NO + K


91/332


85

Figure 63.

378. `~ c= I K

Figure 64.


92/332


86

4.3. Signs of Trigonometric Functions

379.n~~

p

`

q~

`

p

`

f H H H H H Hff H H

fff H Hfs H H

380.

Figure 65.


93/332


94/332


88

382.

~ ~

NRNO

Q

OS

Q

OS+ PO PO+

NUNM

Q

NR

Q

RONM+R

ROR ROR+

PS

R

Q

RONM

Q

NR +

NR

RONM

+

RONM

NR

+

RQNM

P

Q

NR+

Q

RONMRONM

NR

+NR

RONM

+

TOR

O

Q

RONM+

Q

NR ROR+ R

ROR

TRNO

R

Q

OS+

Q

OS PO+ PO

4.5 Most Important Formulas

383. N OO =+

384. N~ OO =

385. N OO =

386.

=

~


95/332


89

387.

=

388. N~ =

389.

=

N

390. = N

4.6 Reduction Formulas

391. ~ + ~

VM + + + + ~

+VM + ~

NUM + ~

+NUM + ~ +

OTM + + ~

+OTM + ~

PSM + ~

+PSM + + + ~ +


96/332


90

4.7 Periodicity of Trigonometric Functions

392. ( ) = O I O PSM K

393. ( ) = O I O PSM K

394. ( ) = ~~ I NUM K

395. ( ) = I NUM K

4.8 Relations between TrigonometricFunctions

396. ( ) NQO

OONO

NN OO

===

O~N

O~O

O+

=

397. ( ) NO

OONO

NN OO

=+==

O~N

O~N

O

O

+

=

398.

=+

==

=O

ON

ON

ON

~ O


97/332


91

O~N

O~O

ON

ON

O+

=+

=

399.

=+

==

=ON

O

O

ONN

O

O~O

O~N

ON

ONO

=

+=

400.

O~N

O~N

~N

N

O

O

O

+

=+=

=

401.

O~O

O~N

N

N

O

O

+

=+=

=

4.9 Addition and Subtraction Formulas

402. ( ) +=+

403. ( ) =

404. ( ) =+

405. ( ) +=


98/332


92

406. ( )+

=+~~N

~~~

407. ( )+

=~~N

~~~

408. ( )+

=+~~

~~N

409. ( )+=

~~

~~N

4.10 Double Angle Formulas

410. = OO

411. NOONO OOOO ===

412.

=

=~

O

~N

~OO~

O

413.O

~

O

NO

O =

=


99/332


93

4.11 Multiple Angle Formulas

414. == POP PQPP

415. = UQQ P

416. += RP NSOMRR

417. == OPP PPQP

418. NUUQ OQ +=

419. += ROMNSR PR

420.

= OP

~PN

~~PP~

421.+

=QO

P

~~SN

~Q~QQ~

422.

+

+=

QO

PR

~R~NMN

~R~NM~R~

423.NP

PP

O

P

=

424.+

=P

QO

~Q~Q

~~SNQ


100/332


94

425.+

+=

~R~NM~

~R~NMNR

PR

QO

4.12 Half Angle Formulas

426.O

N

O

=

427.O

N

O

+=

428. =

=+

=+

=

N

N

N

N

O~

429. +=+

=

=+

=

N

N

N

N

O

4.13 Half Angle Tangent Identities

430.

O~N

O~O

O+

=


101/332


95

431.

O~N

O~N

O

O

+

=

432.

O~N

O~O

~O

=

433.

O~O

O~N

O

=

4.14 Transforming of TrigonometricExpressions to Product

434.O

O

O +

=+

435.O

O

O +=

436.O

O

O +

=+

437.O

O

O +

=


102/332


96

438.( )

+

=+

~~

439.( )

=

~~

440.( )

+

=+

441.( )

=

442.

+

=

=+Q

OQ

O

443.

+=

=

QO

QO

444.( )

=+

~

445.( )

+=

~

446.O

ON O

=+

447.O

ON O

=


103/332


97

448.

=+OQ

ON O

449.

=OQ

ON O

4.15 Transforming of TrigonometricExpressions to Sum

450.( ) ( )

O

+=

451.( ) ( )

O

++=

452.( ) ( )

O

++=

453.++

=

~~~~

454.++

=~~

455.++

=~

~~


104/332


98

4.16 Powers of Trigonometric Functions

456.O

ON O

=

457.Q

PP P

=

458.U

POQQ Q +=

459.NS

RPRNM R

+=

460.

PO

SQSONRNM S

+=

461.O

ON O

+=

462.Q

PP P

+=

463.U

POQQ Q ++=

464.NS

RPRNM R

++=

465. PO

SQSONRNM

S +++

=


105/332


99

4.17 Graphs of Inverse Trigonometric

Functions

466. f p c

~= I NN IO

~O

K

Figure 66.

467. f ` c~= I NN I ~M K


106/332


100

Figure 67.

468. f q~ c

~~= I IO

~~O


107/332


101

469. f `~ c~= I I


108/332


102

471. f `~ c

( ] [ ) KOIMMIO~IINNII~

=

Figure 71.

4.18 Principal Values of InverseTrigonometric Functions

472.

M O

N

O

O

O

P

N~ M PM QR SM VM

~ VM SM QR PM M

O

N

O

O

O

P N

~PM

QR SM

VM

~ NOM

NPR NRM NUM

109/332


103

473.M

P

PN P

P

P N P

~~ M PM QR SM PM QR

SM

~ VM SM QR PM NOM NPR

NRM

4.19 Relations between InverseTrigonometric Functions

474. ( ) ~~ =

475. ~

O

~

=

476. ON~~ = I NM K

477. ON~~ = I MN K

478.ON

~~~

= I N

O < K

479.N

~~O

= I NM < K

480.

= N

~~O

I MN

110/332


104

482. ~O

~

=

483. ON~~ = I NM K

484. ON~~ = I MN K

485.N

~~~O

= I NM < K

486.N

~~~O

+= I MN

111/332


105

493.N

~~O

~~

= I M> K

494.N

~~O

~~

= I M< K

495.N

~~~ = I M> K

496. = N~~~ I M< K

497. ( ) ~~ =

498. ~~O

~

=

499.ON

N~~

+= I M> K

500.ON

N~~

+= I M< K

501.ON

~~+

=

502.N

~~~ = I M> K

503.N

~~~ += I M< K


112/332


106

4.20 Trigonometric Equations

t W

504. ~ = I ( ) ~~N +=

505. ~ = I O~~ +=

506. ~~ = I ~~~ +=

507. ~ = I ~~ +=

4.21 Relations to Hyperbolic Functions

f~~ W

508. ( ) =

509. ( ) ~~ =

510. ( ) =

511. ( ) =

512. ( ) =


113/332

107

Chapter 5

Matrices and Determinants

j~W ^I _I `

b ~ ~W ~ I I ~ I I a~ ~ ~W ^j ~ ~ W j

`~ ~ ~ W `

q~ ~ ~W q^ I ^

^ ~ ~W ^~

q~ ~ ~W ^

f ~ ~W N^ o~ W o~ ~~W

k~~ W I

5.1 Determinants

513. p l a~

NOON

OO

NN~~

~

~^ ==


114/332

CHAPTER 5. MATRICES AND DETERMINANTS

108

514. q l a~

++== POONNPPNOPNOPPOONN

PPPOPN

OPOOON

NPNONN

~~~~~~~~~

~~~

~~~

~~~

^

PNOONPPPONNOPOOPNN ~~~~~~~~~

515. p~ o E^ oF

Figure 72.

516. k- l a~

ON

ON

OOOOON

NNNONN

~~~~

~~~~

~~~~

~~~~

^

KK

KKKKKK

KK

KKKKKK

KK

KK

=

517. jq j ~ ~ ~ -

~ ^ ( )N - ~ ~ ^ - ~ - K


115/332


109

518. `~

( )

jN` +=

519. i~~ b~ - l a~i~~ ~ -

=

=

N`~^ I IIOIN K= K

i~~ ~ -

==

N`~^ I IIOIN K= K

5.2 Properties of Determinants

520. q ~ ~ ~ ~ ~ ~~ ~ K

OO

NN

ON

ON

~

~

~~=

521. f E F ~ ~I ~ ~K

NN

OO

OO

NN

~~

~~ =

522. f E F ~ ~I ~ ~ K

M~~

~~

OO

NN =


116/332


110

523. f ~ E F ~ ~ ~I ~ ~

~K

OO

NN

OO

NN

~

~

~

~=

524. f ~ E F ~ ~ E~F ~ ~ E FI ~ ~

~K

OO

NN

OOO

NNN

~

~

~

~=

+

+

5.3 Matrices

525. a^ ~ ^ ~ ~~ ~~ E-

F ~ K

[ ]

==

ON

OOOON

NNONN

~~~

~~~

~~~

~^

K

MMM

K

K

526. p~ ~ ~ ~ K

527. ^ ~ ~ ~ ~~ = I KK

~ ~ ~~K

528. ^ ~ ~ ~ - ~~ = K


117/332


111

529. a~~ ~ ~ ~ ~ ~ ~ ~~K

530. r ~ ~ ~~ ~ ~ ~~ ~ ~ K q ~ fK

531. ^ ~ ~ ~ K

5.4 Operations with Matrices

532. q ~ ^ ~ _ ~ ~ I ~ I ~ ~ ~ ~ ~~K

533. q ~ ^ ~ _ ~ ~ E ~F I ~ I ~ ~ ~ K f

[ ]

==

ON

OOOON

NNONN

~~~

~~~

~~~

~^

K

MMM

K

K

I

[ ]

==

ON

OOOON

NNONN

_

K

MMM

K

K

I


118/332


112

+++

+++

+++

=+

OONN

OOOOOOONON

NNNONONNNN

~~~

~~~

~~~

_^

K

MMM

K

K

K

534. f ~ ~~I ~ ~^ = ~ ~I

[ ]

==

ON

OOOON

NNONN

~~~

~~~

~~~

~^

K

MMM

K

K

K

535. j~ q j~q ~ ~ ~

K

f

[ ]

==

ON

OOOON

NNONN

~~~

~~~

~~~

~^

K

MMM

K

K

I

[ ]

==

ON

OOOON

NNONN

_

K

MMM

K

K

I


119/332


113

==

ON

OOOON

NNONN

`^_

K

MMM

K

K

I

=

=+++=

NOONN ~~~~ K

E IIOIN K= X IIOIN K= FK

q

[ ]

==

OPOOON

NPNONN

~~~

~~~~^ I [ ]

==

P

O

N

_ I

=

=

POPOOONON

PNPONONNN

P

O

N

OPOOON

NPNONN

~~~

~~~

~~~

~~~^_ K

536. q~ ~ j~f ~ ~ ~ ~ ~I

~ ~ ~ ~ ~Kf ^ ~ ~I ~ q^

^

K

537. q ~ ^ ~ f^^ q = K

538. f ~ ^_ I

( )

qqq

^_^_ = K


120/332


114

539. ^ j~f ^ ~ ~ ~I ~I ^~ I

~ ~ ~ ` ^W

[ ]q`^~ = K

540. q~ ~ j~f ^ ~ ~ ~I ~I ^ I ~ ~~W

OONN ~~~^ +++= K K

541. f ~ j~f ^ ~ ~ ~ ~ ~ ~

^ I N^

^

^~^ N= K

542. f ~ ^_ I ( ) NNN ^_^_ = K

543. f ^ ~ ~ ~I u ~ ~

u^u = I ~ ~ ~~ ~

Mf^ = K

5.5 Systems of Linear Equations

s~~W I I I N I KIO

o~ W KI~I~II~I~I~ NONNNPON


121/332


115

a~W aI a I a I a

j~W ^I _I u

544.

=+

=+

OOO

NNN

~

~I

a

a = I

a

a

= E`~ FI

NOON

OO

NN ~~~

~a == I

NOON

OO

NN

a == I

NOON

OO

NN

~~~

~a == K

545. f Ma I ~ ~ W

a

a = I

a

a

= K

f Ma= ~ Ma E Ma FI ~

Kf Maaa

=== I ~ ~

K

546.

=++

=++

=++

PPPP

OOOO

NNNN

~

~

~

I

a

a

= I a

a

= I aa

= E`~

FI


122/332


116

PPP

OOO

NNN

~

~

~

a= I

PPP

OOO

NNN

a = I

PPP

OOO

NNN

~

~

~

a = I

PPP

OOO

NNN

~

~

~

a = K

547. f Ma I ~ ~ W

a

a = I

a

a

= Ia

a = K

f Ma= ~ Ma E Ma Ma FI

~ Kf Maaaa ==== I ~

~ K

548. j~ c ~ p i~ b~ rq ~ ~

=+++

=+++

=+++

OONN

OOOOONON

NNONONNN

~~~

~~~

~~~

K

KKKKKKKKKKKK

K

K

~ ~

=

O

N

O

N

ON

OOOON

NNONN

~~~

~~~

~~~

MM

K

MMM

K

K

I

KK_u^ = I


123/332


117

=

ON

OOOON

NNONN

~~~

~~~

~~~

^

K

MMM

K

K

I

=

O

N

uM

I

=

O

N

_M

K

549. p ~ p i~ b~ _^u N= I

N

^

^K


124/332

118

Chapter 6

Vectors

sW r

I r

I r

I r

I

^ _ I

s W r I r I

r W r

I r

I r

k W Mr

`~

rW NNN wIvIu

`~ r

W OOO wIvIu

p~~W I

a W I I ^ W

6.1 Vector Coordinates

550. r s

( )MIMIN=r

I

( )MINIM=r

I

( )NIMIM=r

I

N === rrr

K

551. ( ) ( ) ( ) ^_ MNMNMNrrrr

++==


125/332

CHAPTER 6. VECTORS

119

Figure 73.

552. ( ) ( ) ( )OMNO

MN

O

MN ^_ ++==r

553. f ^_ r=

I _^ r=

K

Figure 74.

554. = u r

I

= v r I= w

rK


126/332

CHAPTER 6. VECTORS

120

Figure 75.

555. f ( ) ( )NNNN wIvIuwIvIu rr

= I

Nuu = I Nvv= I Nww = K

6.2 Vector Addition

556. rrr+=

Figure 76.


127/332

CHAPTER 6. VECTORS

121

Figure 77.

557. PON r

Krrrr

++++=

Figure 78.

558. `~ i~

rrrr+=+

559. ^~ i~( ) ( ) rrrrrr ++=++

560. ( )ONONON wwIvvIuu +++=+rr


128/332

CHAPTER 6. VECTORS

122

6.3 Vector Subtraction

561. rrr=

rrr=+ K

Figure 79.

Figure 80.

562. ( ) rrrr

+=

563. ( )MIMIMM == rrr

564. MM=r

565. ( )ONONON wwIvvIuu =rr

I

6.4 Scaling Vectors

566. rr=


129/332

CHAPTER 6. VECTORS

123

Figure 81.

567. rr=

568. ( )wIvIu =r

569. = rr

570. ( ) rrr+=+

571. ( ) ( ) ( ) rrr

==

572. ( ) rrrr

+=+

6.5 Scalar Product

573. p~~ m s r

~r

= rrrr

I

~ r

~ r

K


130/332

CHAPTER 6. VECTORS

124

Figure 82.

574. p~~ m `~ cf ( )NNN wIvIu =r I ( )OOO wIvIu=r I

ONONON wwvvuu ++=rr

K

575. ^ _ q sf ( )NNN wIvIu =

rI ( )OOO wIvIu=r

I

O

O

O

O

O

O

O

N

O

N

O

N

ONONON

wvuwvu

wwvvuu

++++

++= K

576. `~ m

rrrr=

577. ^~ m( ) ( )

rrrr=

578. a m( )

rrrrrrr+=+

579. M =rr

r

Ir

~ ~ EO

= FK

580. M >rr

OM


131/332

CHAPTER 6. VECTORS

125

581. M


132/332

CHAPTER 6. VECTORS

126

Figure 83.

588.

OOO

NNN

wvu

wvu

rrr

rrr==

589.

==OO

NN

OO

NN

OO

NN

vu

vuI

wu

wuI

wv

wv

rrr

590. == p rrrr

EcKUPF

591. ^ _ q s EcKUPF

rr

rr

=

592. k~ m( )

rrrr=

593. ^~ m( ) ( )

rrrr=


133/332

CHAPTER 6. VECTORS

127

594. a m( )

rrrrrrr+=+

595. Mrrr

= r

~ r

~ ~~ E M= FK

596. Mrrrrrrr

===

597. rrr

= I rrr

= I rrr

=

6.7 Triple Product

598. p~~ q m[ ] ( ) ( ) ( )

rrrrrrrrrrrr===

599. [ ] [ ] [ ] [ ] [ ] [ ] rrrrrrrrrrrrrrrrrr =====

600. ( ) [ ] rrrrrr=

601. p~~ q m `~ c

( )PPP

OOO

NNN

wvuwvu

wvu

=

rrr

I

( )NNN wIvIu =

rI ( )OOO wIvIu=r

I ( )PPP wIvIu=r

K

602. s m~~( )s rrr=


134/332

CHAPTER 6. VECTORS

128

Figure 84.

603. s m~

( )S

Ns

rrr=

Figure 85.

604. f ( ) M = rrr

I r

I r

I ~ r

~ ~

I rrr+= ~~ ~ K

605. f ( ) M rrr

I r

I r

I ~ r

~ ~

K


135/332

CHAPTER 6. VECTORS

129

606. s q m( ) ( ) ( )

rrrrrrrrr=


136/332

130

Chapter 7

Analytic Geometry

7.1 One-Dimensional Coordinate System

m ~W M I N I O I M I N I O

o~ W a~ W

607. a~ _ q m

ONNO ^_ ===

Figure 86.

608. a ~ i p o~

++

=N

ONM I

`_

^`= I N K

Figure 87.


137/332

CHAPTER 7. ANALYTIC GEOMETRY

131

609. j ~ i p

O

ON

M

+

= I N= K

7.2 Two-Dimensional Coordinate System

m ~WM

IN

IO

IM

IN

IO

m~ ~W Io~ W m ~ W ~I I Ia~ W ^~W p

610. a~ _ q m( ) ( )ONO

O

NO ^_ +==

Figure 88.


138/332


132

611. a ~ i p o~

+

+

= N

ON

M I +

+

= N

ON

M I

`_

^`= I N K

Figure 89.


139/332


133

Figure 90.

612. j ~ i p

O

ONM

+= I

O

ONM

+= I N= K

613. ` Ef j~F ~ q~

P

PONM

++= I

P

PONM

++= I

( )NN I^ I ( )OO I_ I ~ ( )PP I` ~ ~ ^_` K


140/332


134

Figure 91.

614. f Ef ^ _F ~ q~

~

~

PONM ++

++= I ~

~

PONM ++

++= I

_`~= I `^ = I ^_ = K

Figure 92.


141/332


135

615. ` Ef p m~_F ~ q~

N

N

N

O

N

N

N

PP

OO

NN

POP

OP

OOO

OO

NON

ON

M

+

++

= I

N

N

N

O

N

N

N

PP

OO

NN

OP

OPP

OO

OOO

ON

ONN

M

+

++

=

Figure 93.


142/332


136

616. l Ef ^F ~ q~

N

N

N

N

N

N

PP

OO

NN

OPONP

OONPO

ONPON

M

+

+

+

= I

N

N

N

N

N

N

PP

OO

NN

PONOP

ONPOO

NPOON

M

+

+

+

=

Figure 94.

617. ^~ ~ q~

( ) ( )NPNP

NONO

PP

OO

NN

O

N

N

N

N

O

Np

==


143/332


137

618. ^~ ~ n~~~

( ) ( )( ) ( )( )[ ++++= POPOONON O

N

p( )( ) ( )( )NQNQQPQP ++++

Figure 95.

kW f ~ SNTI SNU EHF EF ~ ~ ~ ~~K

619. a~ _ q m m~ `~

( )NOONOO

ON O^_ +==


144/332


138

Figure 96.

620. ` o~~ `~ m~ `~= I = K

Figure 97.

621. ` m~ `~ o~~ `~

OO += I

~ = K


145/332


139

7.3 Straight Line in Plane

m ~W uI vI I M I N I M I N I N~ I O~ I

o~ W I ~I I I I ^I _I `I N^ I O^ I

^W I ^ W k~ W

r

m W r

I ~r

I r

622. d~ b~ ~ p~ iM`_^ =++

623. k~ s ~ p~ iq ( )_I^

r ~ M`_^ =++ K

Figure 98.

624. b b~ ~ p~ i Ep-f cF += K


146/332


140

q ~ = ~ K

Figure 99.

625. d~ ~ i

NO

NO

~

==

Figure 100.


147/332


141

626. b~ ~ i d ~ m ~ d~( )MM += I

~I ( )MM Im ~ K

Figure 101.

627. b~ ~ i q~ m~ q q m

NO

N

NO

N

=

MNN

N

OO

NN = K


148/332


142

Figure 102.

628. f c

N

~

=+

Figure 103.


149/332


143

629. k~ cM =+

Figure 104.

630. m a c

v

u

NN =

I

( )vIu ~ ( )NNN Im K


150/332


144

Figure 105.

631. s~ i~=

632. e~ i=

633. s b~ ~ p~ i

~rrr

+= I

l ~Iu ~ ~~ I~r

~ ^ I

r

~ I ~~ I ~ ~~I

= lur

~ u K


151/332


145

Figure 106.

634. p~ i m~~ c

+=

+=

OO

NN

~

~

I( )I ~ ~ ~ I( )ON ~I~ ~ ~ ~ I( )ON I ~ ~ ~ ~~ I ~ ~~K


152/332


146

Figure 107.

635. a~ c ~ m q ~ iq ~ ( )I~m

M`_^ =++

OO _^

`_^~

+++= K

Figure 108.


153/332


147

636. m~~ iq NN += ~ OO += ~ ~~

ON = Kq M`_^ NNN =++ ~ M`_^ OOO =++ ~~~

O

N

O

N

_

_

^

^= K

Figure 109.

637. m~ iq

NN

+= ~OO

+= ~ ~

N

O

N = I ~I N ON = K

q M`_^ NNN =++ ~ M`_^ OOO =++ ~~

M__^^ ONON =+ K


154/332


148

Figure 110.

638. ^ _ q i

ON

NO

N

~

+

= I

OO

OO

ON

ON

ONON

_^_^

__^^

++

+= K


155/332


149

Figure 111.

639. f q if M`_^ NNN =++ ~ M`_^ OOO =++ -I ~ ~

NOON

NOONM

_^_^

_`_`

+

= INOON

NOONM

_^_^

`^`^

+

= K

7.4 Circle

o~W o

` W ( )I~m ~W I I N I N I o~ W ^I _I `I aI bI cI


156/332


150

640. b~ ~ ` ` ~ l Ep~~cF

OOO o =+

Figure 112.

641. b~ ~ ` ` ~ ^ m ( )I~

( ) ( ) OOO o~ =+

Figure 113.


157/332


151

642. q m c

M

N

N

N

N

PPOP

OP

OOOO

OO

NNON

ON

OO

=

+

+

+

+

Figure 114.643. m~~ c

=

=

o

o

I OM K

644. d~ c

Mcba^^ OO =++++ E^ I ^cQba OO >+ FKq ~ ~ ( )I~ I

^O

a~ = I

^O

b = K

q ~


158/332


152

Ô

^cQbao

OO += K

7.5 Ellipse

p~ ~W ~

p ~W cW ( )MIcN I ( )MIcO a~ W ObW o~ W Î _I Ì aI bI cI mW i^~W p

645. b~ ~ b Ep~~ cF

N

~

O

O

O

O

=+

Figure 115.


159/332


153

646. ~O ON =+ I NI O ~ ~ ~ ( )Im K

Figure 116.

647. OOO ~ +=

648. b

N~


160/332


154

651. d~ c

Mcba`_^ OO =+++++ I

M^`Q_O K

653. `

( )~bQi = I b ~ K

654. ^~ c~ `

( ) ~~RKNi += I

( )OO ~Oi += K

655. ~p =

7.6 Hyperbola

q~ ~W ~`~ ~W cW ( )MIcN I ( )MIcO a~ W ObW ^W I o~ W ^I _I `I aI bI cI I


161/332


155

656. b~ ~ e~ Ep~~ cF

N

~

O

O

O

O

=

Figure 117.

657. ~O ON = I N I O ~ ~ ~ ( )Im

~ K


162/332


156

Figure 118.

658. b~ ^

~

=

659. OOO ~ +=

660. b

N~

>=

661. b~ a

~

~

O

==

163/332


157

662. m~~ b~ o _~ ~ e~

=

=

~

I OM K

663. d~ c

Mcba`_^ OO =+++++ I

M^`Q_O > K

664. d~ c ^ m~~ `~ ^

Mcba`^ OO =++++ I M^` < K

665. ^ c

Q

O

= I

= I

Q

O

= K

f ~ I ~ ~ ~ M= ~M= K


164/332


158

Figure 119.

7.7 Parabola

c~ ~~W cW c

sW ( )MM Ijo~ W ^I _I `I aI bI cI I ~I I

666. b~ ~ m~~~ Ep~~ cF

OO =


165/332


166/332


160

O

= I

`~

O

IMc I

`~ ( )MIMj K

Figure 121.

669. d~ cI ^ m~~ -~Mcba^O =+++ E^I b FI

~ O ++= I~O

N = K

b~

O

M= I

`~


167/332


161

+

O

Ic MM I

`~

~O

M = I

~Q

~Q~

O

MOMM

=++= K

Figure 122.

7.8 Three-Dimensional Coordinate System

m ~W M I M I M I N I N I N I

o~ W a~ W

^~W psW s


168/332


162

670. a~ _ q m

( ) ( ) ( )ONOO

NO

O

NO ^_ ++==

Figure 123.

671. a ~ i p o~

+

+=

N

ONM I +

+=

N

ONM I +

+=

N

ONM I

`_^`= I N K


169/332


163

Figure 124.

Figure 125.


170/332


164

672. j ~ i p

O

ON

M

+

= I O

ON

M

+

= I O

ON

M

+

= I N= K

673. ^~ ~ q~q ~~ ~ ~ ( )NNNN IIm I

( )OOOO IIm I ~ ( )PPPP IIm O

PP

OO

NN

O

PP

OO

NN

O

PP

OO

NN

NN

N

NN

N

NN

N

O

N

p ++= K

674. s ~ q~q ~ ~ ( )NNNN IIm I

( )OOOO IIm I ( )PPPP IIm I ~ ( )QQQQ IIm

N

N

N

N

S

Ns

QQQ

PPP

OOO

NNN

= I

QPQPQP

QOQOQO

QNQNQN

S

Ns

= K

kW t EHF EF ~ ~ ~ K


171/332


165

Figure 126.

7.9 Plane

m ~W I I I M I M I M I N I N I N I

o~ W ^I _I `I aIN

^ IO

^ I ~I I IN

~ IO

~ I I I I k~ W r I N

r I Or

a W I I a~ ~W

675. d~ b~ ~ m~Ma`_^ =+++


172/332


166

676. k~ s ~ m~q ( )`I_I^

r ~ ~

Ma`_^ =+++ K

Figure 127.

677. m~~ `~ b~ ~ m~Ma`_^ =+++

f M^ = I ~ ~~ -~Kf M_ = I ~ ~~ -~Kf M` = I ~ ~~ -~Kf Ma = I ~ K

f M_^ == I ~ ~~ -~Kf M`_ == I ~ ~~ -~K

f M`^ == I ~ ~~ -~K


173/332


167

678. m a c( ) ( ) ( ) M`_^ MMM =++ I

( )MMM IIm ~I ~ - ( )`I_I^ ~ ~K

Figure 128.

679. f c

N

~ =++


174/332


168

Figure 129.

680. q m c

M

POPOPO

PNPNPN

PPP

=

I

M

N

N

NN

PPP

OOO

NNN = K


175/332


169

Figure 130.

681. k~ cM =++ I

~ ~ ~ I ~ I I ~ ~ ~ ~K


176/332


170

Figure 131.

682. m~~ c

++=

++=

++=

~~

ONN

ONN

ONN

I

( )II ~ ~ ~ I ( )NNN IIm ~I ( )NNN II~ ~ ( )OOO II~ ~ ~~ ~K


177/332


171

Figure 132.

683. a~ ^ _ q m~f ~ ~

Ma`_^ NNNN =+++ IMa`_^ OOOO =+++ I

~ ~

OO

OO

OO

ON

ON

ON

ONONON

ON

ON

`_^`_^

``__^^

++++

++=

= rr

rr

K


178/332


172

Figure 133.

684. m~~ m~q ~ Ma`_^ NNNN =+++ ~

Ma`_^ OOOO =+++ ~ ~~

O

N

O

N

O

N

`

`

_

_

^

^== K

685. m~ m~q ~ Ma`_^ NNNN =+++ ~

Ma`_^ OOOO =+++ ~ ~ M``__^^ ONONON =++ K

686. b~ ~ m~ q ( )NNN IIm ~ m~~ q s ( )NNN II~ ~ ( )OOO II~ EcKNPOF


179/332


173

M

~

~

OOO

NNN

NNN

=

687. b~ ~ m~ q ( )NNNN IIm ~ ( )OOOO IIm I~ m~~ q s ( )II~

M

~

NONONO

NNN

=

Figure 134.

688. a~ c ~ m q ~ m~q ~ ( )NNNN IIm ~

Ma`_^ =+++


180/332


174

OOO

NNN

`_^

a`_^

++

+++= K

Figure 135.

689. f q m~f ~ Ma`_^ NNNN =+++ ~

Ma`_^ OOOO =+++ I ~

+=

+=

+=

~

N

N

N

I

~

NNN =

=

I


181/332


175

OO

NN

`_

`_~= I

OO

NN

^`

^` = I

OO

NN

_^

_^ = I

OOO

OO

NN

OO

NN

N~

_a

_a

à

à

++

= I

OOO

OO

NN

OO

NN

N~

à

à~

â

â

++

= I

OOO

OO

NN

OO

NN

N~

â

â

_a

_a~

++

= K

7.10 Straight Line in Space

m ~W I I I N I N I N I

a W I I o~ W Î _I Ì aI ~I I I N~ I O~ I I

a ~ W r

I Nr

I Or

k~ ~ ~W r

^ W

690. m a c b~ ~ i

~

NNN =

=

I

( )NNNN IIm I ~ ( )II~ K


182/332


176

Figure 136.

691. q m c

NO

N

NO

N

NO

N

=

=

Figure 137.


183/332


177

692. m~~ c

+=+=

+=

N

N

N

I

( )NNNN IIm ~ I I I ~

I ~~ ~ ~ K

Figure 138.

693. ^ _ q p~ i

OO

OO

OO

ON

ON

ON

ONONON

ON

ON

~~

~~

++++

++=

= rrrr


184/332


178

Figure 139.

694. m~~ i

q ~ ~~ ON

rrI

O

N

O

N

O

N

~

~== K

695. m~ i

q ~ ~~ M ON =

rrI

M~~ ONONON =++ K

696. f q i

q N

N

N

N

N

N

~

=

=

~


185/332


179

O

O

O

O

O

O

~

=

=

M

~

~

OOO

NNN

NONONO

=

K

697. m~~ i ~ m~

q ~

~

NNN =

=

~ ~

Ma`_^ =+++ ~ ~~ M =

rrI

M`_^~ =++ K

Figure 140.


186/332


180

698. m~ i ~ m~

q ~

~

NNN

=

=

~ ~Ma`_^ =+++ ~ ~

rr

I

`_

~

^== K

Figure 141.

7.11 Quadric Surfaces

m ~ ~ ~W I I o~ W ^I _I `I ~I I I PON II I

187/332


181

699. d~ n~~ b~

MaoOnOmOeOdOcO`_^ OOO =+++++++++

700. `~~ n~ p~

`~ o~EF o~EbF q p~N P Q M< p~ o~ bO P Q M> p~ f~~ bP P Q M> a e N pQ P Q M< a e O pR P P a o~ n~ `S P P p~ f~~ n~ `T O Q M< p~ b m~~U O Q M> a e m~~V O P p~ o~ b

NM O P p~ f~~ b `NN O P a e `NO O O a o~ f m~

NP O O p~ f~~ f m~NQ N P m~~ `NR N O o~ m~~ m~NS N O f~~ m~~ m~NT N N ` m~

e

=

`cd

c_e

de^

I

=

aonm

o`cdnc_e

mne^

b I ( )b= I

PON II ~ ~I

M

`cd

c_e

de^

=

K


188/332


182

701. o~ b E`~ NF

N

~

O

O

O

O

O

O

=++

Figure 142.

702. f~~ b E`~ OF

N

~

O

O

O

O

O

O

=++

703. e N p E`~ PF

N

~

O

O

O

O

O

O

=+


189/332


183

Figure 143.

704. e O p E`~ QF

N

~

O

O

O

O

O

O

=+

Figure 144.


190/332


184

705. o~ n~ ` E`~ RF

M

~

O

O

O

O

O

O

=+

Figure 145.

706. f~~ n~ ` E`~ SF

M

~

O

O

O

O

O

O

=++

707. b m~~ E`~ TF

M

~

O

O

O

O

=+


191/332


185

Figure 146.

708. e m~~ E`~ UF

M

~

O

O

O

O

=

Figure 147.


192/332


186

709. o~ b ` E`~ VF

N

~

O

O

O

O

=+

Figure 148.

710. f~~ b ` E`~ NMF

N

~

O

O

O

O

=+

711. e ` E`~ NNFN

~

O

O

O

O

=


193/332


187

Figure 149.

712. o~ f m~ E`~ NOF

M

~

O

O

O

O

=

713.f~~ f m~ E`~ NPF

M

~

O

O

O

O

=+

714. m~~ ` E`~ NQF

M~

O

O

=


194/332


188

Figure 150.

715. o~ m~~ m~ E`~ NRF

N~

O

O

=

716. f~~ m~~ m~ E`~ NSF

N

~

O

O

=

717. ` m~ E`~ NTF

MO =


195/332


189

7.12 Sphere

o~ ~ W om ~W I I I N I N I N I

` ~ W ( )II~ o~ W ^I aI bI cI j

718. b~ ~ p ` ~ l Ep~~cF

OOOO o =++

Figure 151.719. b~ ~ ` ` ~ ^ m ( )II~

( ) ( ) ( ) OOOO o~ =++

720. a~ c

( )( ) ( )( ) ( )( ) M ONONON =++ I


196/332


190

( )NNNN IIm I ( )OOOO IIm ~ ~ ~K

721. c m c

M

N

N

N

N

N

QQQ

O

Q

O

Q

O

Q

PPPOP

OP

OP

OOOOO

OO

OO

NNNON

ON

ON

OOO

=

++

++

++

++

++

722. d~ c

Mjcba^^^ OOO =++++++ E^ FKq ~ ~ ( )II~ I

Ô

a~ = I

Ô

b = I

Ô

c = K

q ~

Ô

j^Qcbao

OOOO ++= K


197/332

191

Chapter 8

Differential Calculus

cW I I I I

^ E ~~FW o~ W ~I I I k~~ W ^W f W N

8.1 Functions and Their Graphs

723. b c( ) ( ) =

724. l c( ) ( ) =

725. m c( ) ( )q =+

726. f c

( )= ~ I ( )= ( ) N= K


198/332

CHAPTER 8. DIFFERENTIAL CALCULUS

192

Figure 152.

727. ` c( )= I ( ) = I ( )( )= ~ K

728. i~ c~ += I o I = ~~ I

-K


199/332


193

Figure 153.

729. n~~ cO= I o K

Figure 154.


200/332


194

730. ~ O ++= I o K

Figure 155.

731. ` cP= I o K


201/332


195

Figure 156.

732. ~ OP +++= I o K


202/332


196

Figure 157.

733. m c= I k K


203/332


197

Figure 158.

Figure 159.


204/332


198

734. p~ o c

= I [ ) IM K

Figure 160.

735. b~ c~= I M~> I N~ I= ~= I KSTNUOUNUOUQKO =

Figure 161.


205/332


199

736. i~ c ~= I ( ) IM I M~> I N~ I

= ~= I M> K

Figure 162.

737. e p c

= IO

= I o K


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200

Figure 163.

738. e ` c= I

O

+= I o K

Figure 164.


207/332


201

739. e q~ c

~= I

~

+

=== I o K

Figure 165.

740. e `~ c

= I

+

=== I o I M K


208/332


202

Figure 166.

741. e p~ c

= I

O

N

+=== I o K

Figure 167.


209/332


203

742. e `~ c

= I

O

N

=== I o I M K

Figure 168.

743. f e p c~= I o K


210/332


204

Figure 169.

744. f e ` c~= I [ ) IN K

Figure 170.

745. f e q~ c~~= I ( )NIN K


211/332


205

Figure 171.

746. f e `~ c~= I ( ) ( ) INNI K


212/332


206

Figure 172.

747. f e p~ c~= I ( ]NIM K


213/332


207

Figure 173.

748. f e `~ c

~= I o I M K

Figure 174.


214/332


208

8.2 Limits of Functions

cW ( ) I ( ) ^W o~ ~W ~I

749. ( ) ( )[ ] ( ) ( )~~~

+=+

750. ( ) ( )[ ] ( ) ( )~~~

=

751. ( ) ( )[ ] ( ) ( )~~~

=

752.( )

( )

( )

( )

~

~

~

= I ( ) M

~

K

753. ( )[ ] ( )~~

=

754. ( )( ) ( )~~

=

755.( ) ( )~~

= I ( ) ~ ~

= K

756. N

M

=

757. N~

M

=

758. N

N

M=


215/332


209

759. N~

N

M=

760.( )

NN

M

=+

761.

NN

=

+

762.

N =

+

763. N~ M

=

8.3 Definition and Properties of the Derivative

cW I I I I f ~~W o~ ~W ^W

764. ( ) ( ) ( )

MM=

=

+=


216/332


210

Figure 175.

765. = ~

766.( )

+=

+

767.( )

=

768.( )

=

769. m o( )

+=


217/332


211

770. n o

O

=

771. `~ o( )( )= I ( ) = I

= K

772. a~ f c

N

= I

( ) ( ) K

773. o~ o

O

N

=

774. i~ a~( )= I ( ) = I

( ) ( )[ ]

= K

8.4 Table of Derivatives

f ~~W o~ ~W `I ~I I k~~ W


218/332


212

775. ( ) M`

=

776. ( ) N

=

777. ( ) ~~

=+

778. ( ) ~~ O +=++

779. ( ) N

=

780. ( ) N

+

=

781.O

N

N

=

782. ( )O

N

=

783. ( ) N

N

=

784. ( ) N

=

785. ( )~

N

~ = I M~> I N~ K


219/332


213

786. ( ) ~~~

= I M~> I N~ K

787. ( )

=

788. ( )

=

789. ( ) =

790. ( )

N~

OO

==

791. ( )

N

OO

==

792. ( ) ~

=

793. ( )

=

794. ( )ON

N~

=

795. ( )ON

N~

=

796. ( ) ONN~~

+=


220/332


214

797. ( )ON

N~

+=

798. ( )N

N~

O

=

799. ( )N

N~

O

=

800. ( )

=

801. ( )

=

802. ( ) N~ OO ==

803. ( )

N

OO

==

804. ( ) ~

=

805. ( )

=

806. ( )N

N~

O +

=

807. ( )N

N~

O

=

221/332


215

808. ( )ON

N~~

= I N< K

809. ( )N

N~

O

= I N> K

810. ( )

N +=

8.5 Higher Order Derivatives

cW I I I f ~~W k~~ W

811. p ~

( )O

O

=

=

==

812. e-l ~

( ) ( ) ( )( ) === N

813. ( )( ) ( ) ( ) +=+

814. ( )( ) ( ) ( ) =

815. i c~

( ) O ++=


222/332


216

( ) PP +++=

( )( ) ( ) ( ) ( ) ( ) ( )ON ON N ++++= K

816. ( )( )( )

>

>

=

817. ( )( ) > =

818. ( )( ) ( ) ( )

~

>NN

N

~

=

819. ( )( ) ( ) ( )

N >NN

=

820. ( )( )

~~~

=

821. ( )( ) =

822. ( )( ) ~~~ =

823. ( )( ) += O

824. ( )( )

+=

O


223/332


217

8.6 Applications of Derivative

cW I I m ~ W sW ^~W f ~~W qW k~~ W

825. s ~ ^~( ) = ~ ~ ~

~ ~ ~ I( ) == ~~ I

( ) === ~~ ~~

K

826. q~ i( )( )MMM =

224/332


218

Figure 176.

827. k~ i

( )( )M

M

M

N

= Ec NTSF

828. f~ ~ a~ cK

f ( ) M M > I EF ~ ~ MK Ec NTTI N< IO< FI

f ( ) M M


225/332


219

Figure 177.

829. i~ ~^ EF ~ ~ ~ ~~ N ~

~ ~ N ~( ) ( ) N ~ ~ EcKNTTFK

^ EF ~ ~ ~ ~ O ~

~ ~ O ~

( ) ( ) O ~ ~ EcKNTTFK

830. `~ m^ ~ EF ~ M ~

( )M ~ K

831. c a~ q i~ b~Kf EF ~ E ( ) M > F ~ ~( ]NI~ ~ EF ~ E ( ) M

226/332


220

832. f EF ~ E ( ) M F ~ ~ [ )IO I EF ~ ~ ~ ~ O KEcKNTTFK

833. p a~ q i~ b~Kf ( ) M N = ~ ( ) M N I EF ~ ~ ~

~ O K EcKNTTF

834. `~KEF ~ ~ ~ M ~ ( ) ~ ~ M EcKNTTI P< FKEF ~ ~ ~ M ~ ( ) ~ ~ MK EcKNTTI P< FK

835. p a~ q `~Kf ( ) M M > I EF ~ ~ ~ M Kf ( ) M M


227/332


221

8.7 Differential

cW I I f ~~W a~ ~ W ( ) I ( ) o~ ~W `a~ ( )= W a~ W p~ ~ W p~ ~ W

838. =

839. ( ) ( ) ( ) +=+

Figure 178.


228/332


222

840. p~ `~ ( ) ( ) +=

841. ( ) +=+

842. ( ) =

843. ( ) `` =

844. ( ) +=

845.O

=

8.8 Multivariable Functions

c ~~W ( )I I ( )I I ( )I I ( )I^W I I p~ ~ I I I W I I K

846. c l m~~ a~q ~~ ~

=

E~

=

FI

q ~~ ~

=

E~

=

FK


229/332


223

847. p l m~~ a~

O

O

=

=

I

O

O

=

=

I

O

=

=

I

O

=

=

K

f ~ ~ I

OO

=

K

848. `~ o

f ( ) ( )( )II = E ~ ~~ FI ( )( )

I

=

I ( )( )

I

=

K

f ( ) ( ) ( )( )I = I ( )

+

= K

f ( ) ( )( )III= I

+

=

I

+

=

K

849. p~ `~

+

230/332


224

850. i~ j~~ ~ j~( )I ~ ~ ~ ~~ ( )MM I ( ) ( )MM II

~ ( )I ( )MM I K

( )I ~ ~ ~ ~ ( )MM I ( ) ( )MM II ~ ( )I ( )MM I K

851. p~~ m

M

=

=

K

i~ ~~ ~ ~ ~ ~ ~~ K

852. p~ m^ ~~ ~ ~ ~ ~ ~

853. p a~ q p~~ m

i ( )MM I ~ ~~ E M

=

=

FK

( ) ( )

( ) ( )MMMM

MMMM

II

IIa = K

f Ma > I ( ) MI MM > I ( )MM I ~ ~ ~Kf Ma > I ( ) MI MM < I ( )MM I ~ ~ ~~Kf Ma < I ( )MM I ~ ~ Kf Ma = I ~K

854. q~ m~q ~ ~ ~ ~ ( )I= ~ ( )

MMM

II

( )( ) ( )( )MMMMMMM II += K


231/332


225

855. k~ p~q ~ ~ ~ ( )I= ~

( )MMM II

( ) ( ) N

I

I

M

MM

M

MM

M

=

=

K

8.9 Differential Operators

r ~ ~ ~W r

I r

I r

p~~ E~~ FW ( )II I ( )ON III K d~ ~ ~~ W ~ I

a~ ~W

s E FW ( )oInImcr

a ~ W cr

I cr

` ~ W cr

I cr

i~~~ ~W O

856. d~ ~ p~~ c

==

I

I

~ I

==ON

II

I

~ K K

857. a~ a~

+

+

=

I


232/332


226

( ) IIr

I N OOO =++ K

858. a ~ s c

o

n

mcc

+

+

== rr

859. ` ~ s c

onm

cc

==

rrr

rr

m

n

o

m

n

o rrr

+

+

=

860. i~~~ l~

O

O

O

O

O

OO

+

+

=

861. ( ) ( ) Mcc = rr

862. ( ) ( ) M~ =

863. ( ) ( ) ~ O==

864. ( ) ( ) ( ) cccc~c OO rrrrr

==


233/332

227

Chapter 9

Integral Calculus

cW I I I

f ~~W I I f ~ ~ W ( ) I ( ) I a~ ~ W ( ) I ( ) I ( )c I o~ ~W `I ~I I I I k~~ W I I I

9.1 Indefinite Integral

865. ( ) ( ) `c += ( ) ( )c = K

866. ( )( ) ( ) =

867. ( ) ( ) =

868. ( ) ( )[ ] ( ) ( ) +=+

869. ( ) ( )[ ] (

maths formulas for ssc exams

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