Riaziat Payeh Daraee[Www.riazisara.ir]

www.riazisara.ir

oUoMzkA

(RBBnnBkBB],B\)

:

AnAjkeA

www.riazisara.ir

:k"

jnmoM zkA oUoM q joi B] kAki B M

/jo Ao qa ,k BLT{A v k BLT{A ,k nB v k aBQB

,kAU JBT A SwA Bn nj unkU B ,U ,Mo\U pA A{ ,kAi JBT A nj An `C

B\A nj An `C A pA /k{BM /// ,oAi ,njAoM ,Swj ,kpo. M kB jB M Ji Ak

B1M ,Sv n] BjC l o ,kz Ao ,Sv . MB9 {n BTAoM A RB ,k M An ApBU qa o ,oj ioM Be nj ,ko jB dw HLBE SeAn jp An pBU KB

/ko

,kkz >. Un nj kpAjoPM C e M ji kAhM An JBT k{ jAj e An C pA yQ k w ,Aqq

/kMBM C e AoM {n BUji k xU ,kkj JBT nj An Fv e An C pA uQ kLM An JBT cU

o SwA A nj RBBn s ,jnC Swj M An \T oTM S.oSwj nj 9 kBM ,Bn BMBT kAi AoM

An BC nBMj SzoM L9 RB M kBM KEA ,S M ,SwA L9 RB oM T C pBU R

/jo nAoU

KeB An AkzAj pA nBvM k{BM AoTA vAoSj JBve AnAq BQ pA ,U >dwA

,Un a M Aj > :k a ,SwA T{Aj j] ozM jAs nj poA BU kATvAj o. oTnqM

M Bnj eBw nj Tv Bwjoi j ` kC a ji o

/koM o

KBSwoA ~.

1 / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / /U A] j BjBdUAj ~.

18 / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / RjBw ~.

46 / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / C Bs > nk9nBa ~.

83 / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / cd q]\Q ~.

103 / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / (MBve kB~U) jk kB~Uz{ ~.

114 / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / wk kB~UTS ~.

126 / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / TnBTz ~.

146 / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / /RBXX ~.

216 / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / omQ yhMj ~.

233 / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / MBUjpB ~.

303 / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / YYe jAkA BTwjjpAj ~.

328 / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / BLjjqw ~.

363 / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / owjnBa ~.

394 / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / kejqBQ ~.

469 / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / TwQjqB{ ~.

501 / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / /KB\kS ~.

525 / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / />Tzk\ ~.

675 / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / /AoTA

1 . oUoMzkA UA] jBjBdUA

A~%

UA]jBjBdUA

a2+ b2 = (a + b)2 - 2ab1)a4 + b4 = (a2 + b2)2 - 2a2b2

a3 + b3 = (a + b)3 - 3ab (a + b)2)

a6 + b6 = (a2 + b2)3 - 3a2b2(a2 + b2)

///

a3 - b3 = (a - b)(a2 + ab + b2)3)

an - bn = (a - b)(an-1 + an-2b + ... + abn-2 + bn-1)

x3 - 1 = (x - 1)(x2 + x + 1)

x4 - 1 = (x - 1)(x3 + x2 + x + 1)

xn - 1 = (x - 1)(xn-1 + xn-2 + ... x + 1)

a3 + b3 = (a + b)(a2 - ab + b2)4)

a5 + b5 = (a + b)(a4 - a3b + a2b2 - ab3 + b4)///an + bn = (a + b)(an-1- an-2b + ... - abn-2 + bn-1) (jo. n)

x3 + 1 = (x + 1)(x2 - x + 1)

x5 + 1 = (x + 1)(x4 - x3 + x2 - x + 1)///xn+ 1 = (x + 1)(xn-1 - xn-2 + ... + x2 - x + 1)

5)

(ax + by)2 + (bx - ay)2 = (a2+ b2)(x2 + y2)6)

x4 + x2y2 + y4 = (x2 + xy + y2)(x2 - xy + y2)

x4 + 4y4 = (x2 + 2xy + 2y2)(x2 - 2xy + 2y2)

www.riazisara.ir

A ~ oUoMzkA . 2

(a + b + c)3 = a3 + b3 + c3 + 3a2b + 3a2c + 3b2a + 3b2c + 3c2a + 3c2b + 6abc7)

(a + b + c)3 - a3 - b3 - c3 = 3(a + b)(b + c)(c + a)

a3 + b3 + c3 - 3abc = (a + b + c)(a2 + b2 + c2 - ab - ac - bc) , (sAo jBdUA)

a + b + c = a3 + b3 + c3 = 3abc , (oA jBdUA)Ba = b = c

?SwA Ak (2x - 7)3 + (x + 6)3 + (1 - 3x)3 = jB Bzn :\ :SvU

(4172

(3-136

(2176

(1

(2x - 7) + (x + 6) + (1 - 3x) = :e

(2x - 7)3 + (x + 6)3 + (1 - 3x)3 = 3(2x - 7)(x + 6)(1 - 3x) =

x = __72 x + x + x = ____-136x = -6

x = __138) :jBw BjAn j M BA Rn nj C kLU Ko BjAn

?k JBve An Be :BX

a = 7

b = 48

:oj e An

9)

ZjqjBdUA joMnB%

: Bo

1)

2)

:BX

www.riazisara.ir

3.oUoMzkA UA] jBjBdUA

?knC SwkM An Be : BX

/SwA k{ jBSTwA nBM ka (2) o. pA Aw A nj

Bo. pA kATM SwA A SwA RBBn nj qa ,SwA nBvM (2) (1) o. joMnB !B`M

jnAj kAn BA SwA AkAn kB kAj An xjoMnB kAj o. ~h{ ,k jBSTwA kAj

/kpBM An o. o joMnB S k w ,kTv An AkTMA nj A uQ ,Sv kM kAn

:BM SwA oMAoM ApA M RnBL Be :SvU

-1 (2 1 (1(3(4

:e

: o

:BX1)

2)

3)

k e kAU An (2) (1) BX k{BM Se o. o A pBM SwB\A Ajo vYU 5 oM Aoa k HTe

k{BM (1) SLv nk9 k ]U (2) (1) BX M o A kAi B{ pA An (3) BX n X UBBdTA nj Un nj

/SwA ,5 SLv nk9 ;(3) BX nj Un nj

kAkjo bo] ,TvU RAw nj BeAo] rwU BjBdUA WdL nj ,An co ov 6 B\A nj M KwB A

kp An A coe RBBn nj qa jnAk B{ AoM joMnB ov 6 A jM Se kj kAi pBM ,j AnA

1) /SwA S2)

3) 4)

5) 6)

www.riazisara.ir

A ~ oUoMzkA . 4

nj FU nk9 o A BA k{BM BjBdUA B{ R A kC oL] = 1

2) = 4 o. >L] = 1

3) = 2 o. >L] = 52

?Sva MBU nA (4

/SwA Y.A i xnAj Am y = 4 * 1 = 4 AoMBM SwA k{ Jo 4jk nj SwA 2 o. B A :e

:BM SwA oMAoM (x3-3x) BC , o A :SvU

2 (1(24 (4 (3

Bwn 3 AU M An .o]:

?SwA Ak RnBL Be ,k{BM x2 + x = 3 Bo :SvU

-9 (4 9 (3 -3 (2 3 (1oA jBdUA >L]

:e

?SwA nkKa RnBL Be ,k{BM Bo :SvU

9 (4 3 (3 (213

27 (1

oA jBdUA >L]:e

:C*vM U A] j

op nj /k A] j C rvM An C BM pnA A] ka U A] j An (a + b)nRn M A] j o

/SwA kC n = 1 2 3 ApA M U A] j

www.riazisara.ir

5 .oUoMzkA UA] jBjBdUA

(a + b) = 1

(a + b)1 = a + b

(a + b)2 = a2 + 2ab + b2

(a + b)3 = a3 + 3a2b + 3ab2 + b3

:vM op KUoU M An BA ] ka A KAo o A

1

1 1

1 2 1

1 3 3 1

j n kj An kM ow BLo pA ow o jk j o R\ j{ Be WX L{ {

rvM KAo ) kM Bow kM ow nj 9A jAkA ,p. WX M /jnAj nAo9 1 SMBY jk ,WX pBw

BX A M ,kC SwkM n L] jk o AoM A] j rvM KAo \T nj kC Swj M (/// (a+ b)4

1 4 6 4 1 :pA SvUnBL \Q ow jAkA

(a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b3 :Am

:nAj SBe nj

:BX

:A] j*vMKAo/LwBdBwCxn

:k ]U op o. M

rvM ] o Ko = L9 B ] nj a AU * L9 B ] KoL9B R] jAkU

:BX

www.riazisara.ir

A ~ oUoMzkA . 6

= a4 + 4a3b + 6a2b2 + 4ab3 + b4

?k JBve An (2x + 3y)7 Be :BX

(2x + 3y)7 = (2x)7 + 7(2x)6(3y)2 + 21(2x)5(3y)2 + 35(2x)4(3y)3+ 35(2x)3(3y)4

+ 21(2x)2 (3y)5 + 7(2x)(3y)6 + (3y)7 = ....

:SwA op Bs AnAj U A] j rvM

(AU pA oTzM ) n + 1 BM SwA oMAoM rvM R] jAkU (1

/SwA n ]nj pA b aKveoM SwAn Sw A] ka (2

(nAj q ab2 ,nAj a2bX nB ) /SwA nBYT b aKveoM A] ka (3

/SwA n oMAoM R] BU nj b aBAU R\ ,SwA A] ka SwAn Sw A] ka (4

/SwA j SwAn M Oa pA bBAU q SwAn M Oa pA aBAU (5

/j{ .BA b AU M keA k{ a AU pA keA ,L9 ] BM vBY nj ] o nj (6

/knAj BvT KAo ,kAB. M co] j pA BA] (7

(/SwA 2 oMAoM , w ] nj b AU X ) /SwA rvM nj ] LUo pA oT keA ,b AU z (8

(/kC Swj M rvM KAo R\ BU j nAo9 ,b aB] M SwA .B) 2n BM SwA oMAoM rvM KAo R\ (9

:

:BM SwA oMAoM (x + y)4 LvM KAoO :\ :SvU

12 (4 2 (3 18 (2 16 (1

x = y = 1 (1 + 1)4 = 24 = 16

:nCjB(x + y)4= x4 + 4x3y + 6x2y2 + 4xy3 + y4

1 + 4 + 6 + 4 + 1 = 16

,k{BM oT (a + b)2n LvM KAoO :\ pA keA 24 , LvM jk KAoO :\ o A :SvU

?SwA Ak n nAkK

3 (4 5 (3 4 (2 6 (1

22n = 2n + 24

(2n)2 - 2n - 24 =

(2n + 15)(2n - 16) =

2n = -15 oE

2n = 16 n = 4

www.riazisara.ir


:BC ,k{BM oMAoM rvM Ar ] Ko BM Ap ] Ko Bo (1

p + r = n + 2

?SwA Ak n ,k{BM oMAoM ]KoO BM kR ]KoO ,(x + y)nLvM nj o A :SvU

26 (4 24 (3 22 (2 3 (1

17 + 9 = n + 2 n = 24

/k{ LwBd op Bo. pA rvM A(k + 1) AkR] (11

Ak ]

A(k + 1) ]

?SwA Ak (3x + 5y)9 LvM z{ ] :SvU

(1(2(3(4

z{ ]

/k{ S SLX Bnj ,BT ,j co] nj ,k{BM S ,rvM A co] nj ,b aM Bo (12

: ,SwA oS ,(a -b)nrvM KAo R\ (13

RnBL M k{BL xB{ SwA A] j] Rn nj ,(a + b)n rvM nj x pA YTv ] :x pA YTv ] (14

B Ak ] o. SwA .B ,(a + b)n rvM nj x pA YTv ] T.B AoM /k{BM oS oMAoM ] A nj x AU oj

] ,] Ak j{ }hz kC Swj M k BU j nAo9 oS Bv An ] A nj x AU ,T{ An A(k+ 1)

/SwA x pA YTv

?SwA C ] Ak , LvM njx pA KTv ] :SvU

nBa ] (4 TS ] (3 \Q ] (2 z{ ] (1

A(k + 1) ] =

/k{BM 3 q ,cd q AoMBM 18 - 3k = k = 6

/k{BM T{Ak j] ,x pA YTv ] ,SwA ,rvM nj : omU

:pA SvUnBL , rvM nj x pA YTv ] , SBe nj (15

(k{BM L] jk kBMm) m = ____ + 1nqq-p

?Aoa , B jnAj j]x pA KTv ] , LvM nj BC :BX

www.riazisara.ir

A ~ oUoMzkA . 8

/Sv L] :Aop ,oi

/k{BM T{Aj j]xp C nj A] LwBd (16

B\C pA k BU j nAo9 p Bv C nj An x AU ,T{ An A(k+ 1) ] o. AkTMA ,A] a LwBd AoM

/kC Swj M (j] Rn nj) SwA xpB{ A]

?SwA Ak (x4 - __)15 LvM njx23 KoO :SvU1x3

-7*13*15 (4 7*13*15 (3 -6*14*13 (2 6*14*13 (1

A(k+ 1) ]

6 - 7k = 32 k = 4

:pA SvUnBL ] A nj x32 ]Ko jnAj j]x32 C nj SvA] ,\Q ( A(4 + 1) = A(k+ 1)) ] Am

:(a + b)nrvM nj rwR] rw ] (17

/SwA A(__+ 1) ] ,rw ] BC ,k{BM Zp L] jk n ,(a + b)nA] j nj o A (tAn2:pA kUnBL rwR] BC ,k{BM jo. L] jk n ,(a + b)n A] j nj o A (J

A(____) + 1 A(____)R]n+ 12n+ 12

?kMBM An (_ + _)1LvM Lw ] :BXax

x

a

rw ] = + 1 = 6 z{ ]12

n = 1 Zp

rw ] = z{ ] =

:U A] j rvM nj B R] (18

/k{BL A Rn M BjAn op coe ,BC nj kTv U] ,B R]

?kMBM An B R] jAkU , LvM nj :BX

q TvBM ,k{BM B ,kC SwkM ] AhM o A ,SwA Rn M UnBL B{ ,rvM pA ] o

24 Bv z ,p+ q BAU R\ kBM H (kCnj BjAn op pA y x BU) ,k{BM 2 Jo1 p 3 Jo1

/nC Swj M An op jB SB cd BMA] jAkU kBM ,9A nj Am ,k{BM

k = 3 6 9 12 :uQ k{BM 3 Jo1 TvBM q k j{ \T ,jB A pA

www.riazisara.ir


B o. M ]U BM knAj j] rvM nj B ] [Q Am

:pA kUnBL

:(A] j rvM U) A] ka rvM (18

,(a+ b+ c)n o

A ~ oUoMzkA . 10

ok BM RnBL j , o. >L] k{BM T{Aj ]U

/kABv

:pA SvUnBL (a + b + c)2 LvM R] jAkU :BX

?Aoa , B jnAj j]x3y4z2 RnM A] ,(x + y + z)1LvM nj BC :BX

3 + 4 + 2 1 Aop ,oi

?SwA Ak (a + b + c)7 LvM nj a2b4c ]KoO :SvU

96 (4 1 (3 15 (2 92 (1

?kj LvM A] ka LvM o M An (a + b + c)3 :BX

n1 + n2 + n3 = 3

:pA kUnBL SwA SB cd JA] AnAj p. jB

A Twj (3,,) (,3,) (,,3)

j Twj (2,1,) (2,,1) (1,2,) (1,,2) (,2,1) (,1,2)

w Twj (1,1,1)

:nAj AoMBM

?SwA Ak (4a + b - 3c)1LvM KAoO :\ :SvU

-512 (4 (3 124 (2 512 (1

/j nAo9 ,c b aB] M SwA .B

(4 + 1 - 3)1 = 21 = 124

:pA SvUnBL x1+ x2+ ... + xn = k jB (L]) SLX cd BMA] jAkU :T

?jnAj LV JA]kax1+ x2 + x3 = 8 jB :SvU

21 (4 9 (3 45 (2 42 (1

/// (5,2,1) (1,1,6) (1,6,1) (6,1,1) :pA kUnBL jB BMA] pA jAkU

www.riazisara.ir

11 . oUoMzkA UA] jBjBdUA

:pA SvUnBL xk c

k/// x2 c2 x1 c1 o{ BMx1 + x2 + ... + xn = n jB cd BMA] jAkU :T

:pA SvUnBL Xo{ BM x1+ x2 + x3 = 8 jB cd BMA] jAkU :SvU

(4 6 (3 3 (2 4 (1

:pA kUnBL BMA] A

(1,2,5) (1,3,4) (1,4,3) (2,2,4) (2,3,3) (3,2,3)

C U BUj oTz ] jBdUA (19

(x + a)(x + b) = x2 + (a + b) x + ab1)

(x + a)(x + b)(x + c) = x3 + (a + b + c)x2 + (ab + ac + bc)x + abc2)

(x + a)(x + b)(x + c)(x + d) = x4 + (a + b + c + d)x3 + (ab + ac + ad + bc + bd + cd)x2 +3)

(abc + abd + acd + bcd) x + abcd

o. (x + a1)(x + a2) + ... + (x + an) = xn + (a1 + a2 + ... + an) xn-1 + ... + a1a2 ... an4)

/knBPvM o]Bi M An op RB ,p. jBdUA nj

/SwA oMAoM xn-1 Ko (1

/SwA oMAoM xn-2 Ko (2///

/SwA oMAoM xKo (n - 1

/SwA a1a2 ... an oMAoM ,SMBY nAkY (n

:BM SwA oMAoM (x + 1)2(x - 2)(x + 3)RnBL njx2 KoO :SvU

-12 (4 6 (3 12 (2 -6 (1

(x + 1)(x + 1) (x - 2)(x + 3)

x2Ko =

:nAj BC a1 = a2 = ... an = a p. o. nj Bo :omU

(x + a)(x + a) ... (x + a) = xn+ (a+ a+ ... + a)xx-1 + (a2+ a2+ ... + a2)xn-2 + ... + an=

/k{BM A] j o. B

www.riazisara.ir

A ~ oUoMzkA . 12

?SwA Akx7+ __Be ,x + __ = -2 Bo 11x9

1x

-1 (4 2 (3 -2 (2 (1

?SwA Akx - __Be ,x + __ = a Bo 21x

1x

(1(2(3(4

?SwA Akx4 + __ Be ,x + __ = 4 Bo 31x4

1x

182 (4 196 (3 24 (2 194 (1

?SwA Ak jk j Jo\Be ,SwA 113 AT cd jk j RBMo :\ 4

48 (4 24 (3 112 (2 56 (1

:BM SwA oMAoM ____________ nAkK x + y = 2xy y + z = 3yz ,x + z = 5xzLMAn pA 5xy + yz + zxxyz

2 (4 15 (3 5 (2 1 (1

?SwA Akx3 + __Bex > Xo{ BM ,_____ = ___ Bo 61x3

x2

x4+ 1114

76 (4 36 (3 52 (2 48 (1

:BM SwA oMAoM _______ ov SLX Be ,a , b > Xo{ BM ,k{BM a2 - 8ab + 9b2 = Un nj 7a + 3ba - 3b

(1(2(3(4

:BM SwA oMAoM 8

(1(2(3(4

?SwA Akx3+ y3+ xyBe ,k{BM x + y = __ Bo 913

(413

(3181

(2127

(119

:BM SwA oMAoM RnBL 1

(x + 2)n - (x + 1)n (2 (x + 2)n + (x + 1)n (1

(x + 2)n+1 - (x + 1)n+1 (4 (x + 2)n+1 + (x + 1)n+1 (3

www.riazisara.ir


(2) 1x + __ = -2 x2 + 2x + 1 = (x + 1)2 = 1x

x = -1 x7 + __ = -21x9

(4) 2(x + __)2 - (x - __)2 = 4 (x - __)2 = a2 - 41x

1x

1x

x - __ = 1x

(1) 3x2 + __ = (x + __)2 - 2 = 42 - 2 = 141x2

1x

x4 + __ = (x2 + __)2 - 2 = 142 - 2 = 1941x4

1x2

(1) 4a2 + b2 = (a - b)2 + 2ab (n + 1)2 + n2 = (n + 1 - n)2 + 2n(n + 1)

113 = 1 + 2n(n+ 1) n(n + 1) = 56

(2) 5

/ ] BM An jB w .o]

2(__ + __ + __) = 1 __ + __ + __ = 51x

1y

1z

1x

1y

1z

_________ = __ + __ + __ = 5xy+ yz+ zxxyz

1z

1x

1y

(2) 6

x >

(3) 7

(1) 8

(2) 9

(2) 1jBdUA >L]

www.riazisara.ir

A ~ oUoMzkA . 14

?SwA Ak Be 1

(11 (4 2 (3 (2

:BM SwA oMAoM Be 2

4 (4 3 (3 2 (2 1 (1

?SwA Ak :\ 3

9 (2 1 (1(3(4

?SwA Ak ApA M A = x3 + 3x2 + 3x + 5 RnBL jk nAkK 4

9 (4 6 (3 8 (2 12 (1

?SwA Akx = 11 ApA M x5 - 12x4 + 12x3 - 12x2 + 12x - 1 RnBL jk nAkK 5

12 (4 -11 (3 11 (2 1 (1

?SwA Ak ApA M RnBL jk nAkK 6

(1(2(3(4

:BM SwA oMAoM (a + b + c)3 ,k{BM ,Bo 7

27abc (3 8a3b3c3 (2 9abc (1(4

?SwA AkA = ____________Be ,x + 2y = 3z Bo 8x3+ 8y3-27z3

9xyz

(412

2 (3 -2 (2 1 (1

?SwA Ak ov Be ,k{BM Bo 9

x (1-__ (4x3 -x (3 (2

?SwA Akx3 + (y - 3)3 + z3 Be ,x + y + z3 = 3 Bo 1

xz (y + 3) (4 3xz (y + 3) (3 xz (y - 3) (2 3xz (y - 3) (1

www.riazisara.ir


(3) 1

(2) 2

(2) 3

:nCjB

(3) 4

(1) 5/nAm x+ 1 ,12 B] Mx = 11 x + 1 = 12

x5 - (x + 1)x4 + (x + 1)x3 - (x + 1)x2 + (x + 1)x - 1

= x5 - x5 - x4 + x4 + x3 - x3 - x2 + x2 + x - 1 = x - 1 = 11 - 1 = 1

(4) 6/nAm 2 KUoU M x2 xB] M

P = x5 - (x2 + x)x3 + x2(x2) + 3x - x

= x5 - x5 - x4 + x4 + 2x = 2x =

(3) 7a + b + c = a3 + b3 + c3 = 3abc :nCjB

(a + b + c)3 = 27abc

(2) 8x + 2y - 3z = x3 + 8y3 - 27z3 = 3x(2y)(-3z) A = _______ = -2-18xyz9xyz

(3) 9

(1) 1x + (y - 3) + z = x3 + (y - 3)2 + z3 = 3x (y - 3)z

/TvAT ,TvAj a TvAk TvAU BU

www.riazisara.ir

A ~ oUoMzkA . 16

?SwA op jAkA pAAk BM oMAoM ,(x2 - __)6 LvM njx pA KTv ] 11x

2 (4 15 (3 -15 (2 -2 (1

(a > ) /k{BM 7 oMAoM ,a AU , LvM pA ] Ak nj 2

\Q (4 nBa (3 w (2 TS (1

:BM SwA oMAoM n nAkK ,SwA oTzM keA 24 , LvM KAoO :\ pA (a + b)2nLvM KAoO :\ 3

6 (4 5 (3 4 (2 3 (1

?SwA oMAoM ] AkKoO BM TR ]KoO ,(x + y)25LvM nj 4

TvM (4 k\ (3 jp (2 kS (1

?SwA Ak (x+ y)4Lw ] 5

(1(2(3(4

?SwA Ak n ,k{BM oTzM j ]KoO pA keA 44 ,w ]KoO , LvM nj o A 6

n = 18 (4 n = 13 (3 n = 11 (2 n = 1 (1

?SwA Ak ,x99KoO ,(x + 1)(x + 2)(x + 3) ... (x + 1)LvM nj 7

99! (4 11 (3 55 (2 1! (1

:BM SwA oMAoM KoO oTnqM ,(x + 1)13LvM nj 8

1726 (4 1821 (3 1716 (2 13 (1

:BM SwA oMAoM x2KoO ,(x - 1)(x + 2)(x + 4)(x - 7)LvM nj 9

-34 (4 -33 (3 -31 (2 29 (1

:BM SwA oMAoM (3x2+ x - 2)99 + (x - 5)49LvM KAoO :\ 1

2 (4 299 (3 298 (2 (1

www.riazisara.ir


(3) 1(x2 - x-1)6

:nCjBm = ____ + 1 = _______ + 1 = 5nqq-p

6(2)2-(-1)

/SwA \Q ] ,x pA YTv ]uQ\Q ] =

(2) 2A(k+ 1) ] =

1 - ___ = 7 k = 2 w ]3k2

(2) 322n = 2n + 24 (2n)2 - 2n - 24 =

(2n - 16)(2n + 15) = 2n = 16 n = 4

(4) 4TS ] Ko = k\ ] Ko 7 + 2 = 25 + 2

(x + y)4 __ + 1 = 21 SvM ]n2 (3) 5

(2) 6

_______ = 44 + n n(n - 1) - 2n = 88n(n-1)2

n(n - 3) = 88 11 * 8 n = 11

(2) 7x99Ko = 1 + 2 + 3 + /// + 1 = = 551*112

(2) 8Ko oTnqM = rw ] j pA Ko =

(3) -9x2Ko = (-1)(2) + (-1)(4) + (-1)(-7) + (2)(4) + (2)(-7) + 4(-7) = -33

(2) 1x = 1 299 + (-4)49 = 299 - 298 = 298(2 - 1)=

( oT) /SwA BvA jo d pA oTM ,d jAj RB\

(oT un) /SwA kzkA poA k{ kAi B Ajo ,SwA yi nBA TiBw jC

www.riazisara.ir

j~ oUoMzkA . 18

j~%

RjB

:j yB ax + b = Rn M An \ A ]nj jB o :A ]nj jB 1x = ___ :BM SwA oMAoM jB JA] ,k{BM a o A (tA-b

a:kC yQSBe j k{BM a = o A (J

kAU YYe jk ox ,SBe A nj kC nj x= B x+ = Rn M jB BC ,b = o A :ASBe/SwA L SBe A nj jB k HeA jnAj JA] nBzM SBe A nj jB Am k{BM

,SBe A nj jB Am SwA oE kC nj = -b B x = -bRn M jB BC ,b o A :jSBe/jnAk JA]

:SS AU nB~TiA n] M AoMBMa x = ___-b

a

ax = -b b = /SwA L jBa =

b SwA (T) oE jB

?j{ L jB BU k{BM kam nAkK ,(m - 1)x = m2- 1 jB nj :SvU

1 (4 (3 -1 (2 1 (1oTz zn

?j{ o. jB BU k{BM kam nAkK ,(m2- 4)x = m + 2 jB nj :SvU

Ak` (4 2 (3 -2 (2 2 (1

oTzo JA]

nj /jAj Bz (a ) ax2+ bx + c = Rn M AU SBe nj An j ]nj jB o :j ]nj jB 2

}9B j ]nj jB An jB ,BTBe A pA o nj k{BM oS kAU ,BC j o B c B b jB A

/k

:}B j ]nj RjB e

/j{ e onT B. >o] pA jB BC ,k{BM c = o A (tA

www.riazisara.ir

19 . oUoMzkA RjB

:BC k{BM b = o A (J

:9A nj :pA kUnBL jnAj o9 zn j jB ,k{BM A tTh c a o A (I

/jnAk YYe zn SBe A nj jB ,k{BM (S) AkdT c a o A (II

/SwA x = tB1 zn AnAj jB SBe A nj ,b = c = o A (Z

tB1 zn

(D nTwj B b nTwj) :( SBe nj j ]nj jB e

/ op Rn M jo JBve An jB (D) L , SBe nj j ]nj jB e AoM

D = b2- 4ac

tA) if D > /SwA qBT YYe zn j AnAj jB

J) if D = B Bv zn j AnAj jB /SwA tB1 YYe zn

Z) if D < /jnAk YYe zn jB /(jnAj rTh zn j SBe A nj jB TLA)

pA ,j ]nj jB e AoM AU ,k{BM Zp b ,j ]nj jB xKo o A :D nTwj B b nTwj :omU

/k{ oTBwC RBLwBd Rn A nj (SwA bt~ b) jo jBSTwA D nTwj B b nTwj

tA) if D > /jnAj qBT YYe zn j AnAj jB

J) if D = B Bv zn j AnAj jB /SwA tB1 YYe zn

Z) if D < /jnAk YYe zn jB /(jnAj rTh zn j SBe A nj jB TLA)

BMA] b nTwj pA B e b nTwj pA An jB k 9o. ,k{BM Zp b Bp ,|Bi SBe nj :omU

/kTv SBe j o nj kC SwkM

:j ]nj jB Bzn S jAkU nj WdM

/k{BM Bzn Jo1Be R\ KUoU M op nj p s k{BM T{Aj ]U

www.riazisara.ir

j~ oUoMzkA . 20

1) _ > /kAAkdT zn jca

2) _ < (x< /jnAj qBT YYe zn j jB(x < xo. BM)

3) _ = /SwA ___ oj zn oS zn ca

-ba

s = ___ > < x < x kASLX zn j o-ba1) _ > c

as = ___ < x < x < kAS zn jo-b

a

x < < x| x| > | x|

zn > nk9/SwA oTnqM SLX

x < < x| x| > | x|

zn > nk9/SwA oTnqM S

s = ___ < -ba

2) _ < s = ___ < ca

-ba

s = ___ = {x = -x /kAo9 zn j-ba

s = ___ > x = < x-ba3) _ = c

as = ___ < x < = x-b

a

j SBe: if D BD = x = x = ___ jnAj Bv YYe zn j jB-b2a

s = ___ > < x = x /SwA SLX tB1 zn-ba

s= ___ < x = x < /SwA S tB1 zn-ba

s = ___ = x = x = /SwA oS tB1 zn-ba

w SBe: if D BD < /jnAk YYe zn jB

:j ]nj jB M ]An UB

/jM kAi _ oMAoM oj zn jM 1 oMAoM Bzn pA ,k{BM oS KAo R\ ,j ]nj jB nj o A (1ca

ax2 + bx + c =

if a + b + c =

?kMBM An op RjB Bzn :BX

tA)

J) (sin2a)x2 - x + cos2a = cba

www.riazisara.ir

21 . oUoMzkA RjB

1 jM -1 oMAoM jB Bzn pA BC ,a + c = b :{BM T{Aj ,ax2+ bx + c = j ]nj jB nj Bo (2

/jM kAi -_ oMAoM oj znca

?kMBM An op jB Bzn :BX

tA) 81x2+ 1x + 19 =

J)

?SwA Ak ax2+ (a+ b+ c)x + b + c = jB Bzn pA :SvU

-_ (4ca

_ (3ca

-1 (2 1 (1

/SwA ___ oj zn ,k{BM oS ,ax2+ bx + c = j ]nj jB Bzn pA o A 3-ba

/k a TvC ,k SBM Sw M ,j ]nj jB Bzn pA A o{ 4

oj zn k SBM Sw M (m + 2) x2 - x + 2m = j ]nj jB Bzn pA o A :SvU

?SwA Ak jB

-2 (4 -4 (3 4 (2 (1

a = m + 2 = m = -2 nAmB] (-2+ 2)x2 - x - 4 = x = -4 oj zn

jAj yBD p s BM KUoU M An j ]nj jB Bzn BSU > nk9 Bzn Jo1Be ,Bzn R\ 5

:nAj

?SwA Ak nAkK ,k{BM x2 + 2mx - 3 = jB Bzn logb loga o A :SvU___ (42m3

___ (323m___ (23m2 -

___ (12m3

www.riazisara.ir

j~ oUoMzkA . 22

?SwA Ak tan(a+ b) ,k{BM x2 + qx - 3 = jB Bzn tanb tana o A :SvU___ (4-q3

__ (3q4___ (2-q4

__ (1q3

j ]nj jB e pA An jk j C AU (k{BM p s) k{BM jk j Jo1Be R\ o A 6

/jnC SwkM x2- sx + p =

?kMBM An jk j C ,SwA BzMo\Be -1 KKe jk j :\ :BX

: o. :j ]nj jB nj BsiMAn 7

:nAj nA Rn A nj

(*) o RBLYA*xn-2

ax2+ bx + c = axn+ bxn-1+ cxn-2 =

/nAm xB] M An x Be axn+ bxn-1+ cxn-2 =

/nAm xB] M An x Be axn+ bxn-1+ cxn-2 = + -------------------a(xn + xn) + b(xn-1+ xn-1) + c(xn-2+ xn-2) =

asn+ bs

n-1+ csn-2 =

:BM SwA oMAoM s3 nAkK ,k{BM s1 = x1+ x1= n o A x2- 2x - 1 = jB nj :SvU

n3- 3n (4 n3+ 3n (3 n3- n (2 n3- 3 (1/Bwn 3 AU M

x3+ x3+ 3x1x1(x1+ x1) = n3x1+ x1= n

(a + b)3= a3+ b3+ 3ab(a + b)

s3+ 3(xx)1 s1 = n3 s3 = n3+ 3(-1)1(n) = n3- 3n

:j ]nj jB e kM ,Bzn KveoM nBYT RAnBL LwBd 8

www.riazisara.ir

23.oUoMzkA RjB

x + x = s

x.x = p

x2 + x2 = (x + x)2- 2xx = s2- 2p

x3 + x3 = (x + x)3- 3xx(x + x) = s3- 3ps

x4 + x4 = (x2 + x2)2- 2x2x2 = (s2- 2p)2 - 2p2

www.riazisara.ir

j~ oUoMzkA . 24

{BM T{Aj BU k{BM ka m ,k{BM mx2- 2(m+ 1)x + m = j ]nj jB Bzn x x o A :SvU

x+ 2x = 3

-2 (4 2 (3 -4 (2 4 (1

BTwj e pA

:BTwj e a

-

/S{Aj kAi o9 zn j jB ,k{BM A tTh c a k{BM b = j ]nj jB nj Bo 9

?k{BM T{Aj oi zn j op jB BU ,k{BM kam :SvU

-9 (4 9 (3 9 (2 3 (1

9x2+ (m2- 81)x + m - 5 =

L9 MB9

tTh qBT zn j AnAj HTe jB (p = _ < ) k{BM A tTh c a ,j ]nj jB nj Bo 1ca

(j{ SLX nA D = b2- 4acSBe A nj Aop) /jM kAi A

?k{BM T{Aj A jTh zn j nA op j ]nj jB BU kMBM Ba An k :BX

-3x2+ 4x + (k2-1) = _ < c > ca

a <

k2- 1 > k < -1 B k > 1

?k{BM T{Aj A jTh zn j jB BU k{BM kam jke :SvU

m < 6 (4 m > 6 (3 m < 3 (2 m > 3 (1

/jM kAi ok t ,(j] Rn nj) jB Bzn ,k{BM a = c ,j ]nj jB nj Bo 11

?kok t jB Ak Bzn :SvU

3x2+ 2x - 5 = (1(2

2x2+ 7x - 2 = (3(4

www.riazisara.ir

25 .oUoMzkA RjB

:Aop SwA cd (4) q :e

/kok o9 u jB Bzn BC ,a = -c :{BM T{Aj j ]nj jB nj o A 12

:BC ,k{BM oj zn oMAoM k ,j ]nj jB Bzn pA o A 13

B

SwA oMAoM m ,k{BM oj zn oMAoM nBa,Bzn pA ,x2- (m + 5)x + 16 = ,j ]nj jB nj o A :SvU

:BM

5 15 (4 5 -1 (3 5 1 (2 5 -15 (1

:e

D = a2d2 :BC ,k{BM x + d xKUoU M ax2+ bx + c = jB Bzn o A 14

:BM SwA oMAoM oTa zn BC ,k{BM AT cd jk j ax2+ bx + c = jB Bzn o A 15

SBdA nj SwA cA D = a2 H

x=_______+ 1-(a+ b)2a:nAj nA ,k{BM AT cd jk j x2+ px + q = jB Bzn o A :SvU

p2- 4q = 1 (4 p2= -4q (3 p2+ 4q = 1 (2 p2= 4q (1

p2- 4q = 1 :Am D = a2 :e

k oS Sw M q b o A Sw M Bzn pA ,k a ,ax2+ bx + c = jB nj o A 16

/k Sw M jB zn j o BC c

?k Sw M (m3- m)x2+ (m2- m)x + 4 = jB zn LK ,m nAkK Ak ApA M :SvU

1-1 (4 (3 -1 (2 1 (1

oTzo JA]

/jMkAip- iqHTejBoj zn,k{BMax2+ bx+ c=jBrThzn o A17

?SwA Ak ,k{BM 3 + 2i C zn j ]nj jB :SvU

x2- 6x + 13 = (4 x2+ 6x + 13 = (3 x2- 3x + 13 = (2 x2+ 3x + 13 = (1

:Am jM kAi 3-2i oj zn Aj :e

A xn:

www.riazisara.ir

j~ oUoMzkA . 26

j xn:

oj zn BC ,k{BM ax2+ bx + c = jB zn (jnAk B nm] q q Q P Q) o A 18

/jM kAi jB

?SwA Ak ,k{BM C zn ,BKAoO BM j ]nj jB :SvU

x2- 4x + 1 = (4 x2- 4x - 1 = (3 x2+ 4x + 1 = (2 x2- 4x - 1 = (1

:Am jM kAi oj zn SwA cA :e

Bzn jAkU oj RnBL M ,k{ oB Zp jAkU M Sv A Bzn ,A KAo BM ,jB nj 19

/SwA jo. z AjB a A

:pA kUnBL jB Bzn :BX

?k{BM C zn j , kj zU BKAoO BM ]nj oT pA AjB :BX

AnAj TvBM 9Ake ,nMq jB AoMBM ,jM jB oj zn j HAqA ,q SwA cA :e

/jM kAi op Rn M J jB Am , k{BM zn nBa

J jB

:S{ B Mo j R\ Rn M AU An jB C ,k{BM S ,j ]nj jB (BTj) L o A 2

:BXD = 4 - 2 < p = x2 - 2x + 5

p = (x - 1)2 + 4 p = (x - 1)2+ 22

s + p = 1 BC ,a + b = c :{BM T{Aj j ]nj jB nj o A 21

f(x) = ax2+ bx + c :j ]nj BA] wS U q\U 22

/j{ q\U op Rn M SBe A nj SwA x x qBT YYe zn j AnAj jB ,D > o A :A SBe

www.riazisara.ir

27 .oUoMzkA RjB

f(x) = a(x - x)(x - x)

:D > SBe nj S U k]

+xx-x

aS >.AaS tBhaS >.Af(x)

B a(x - x)2 Rn M x = x = ___ jnAj tB1 YYe zn jB ,D = o A :j SBe-b2a/j{ q\U f(x) = a(x - x)2

:D = SBe nj S U k]

+x= x= ___-b2a-x

aS >.Aa S >.Af(x)

/f(x) nA BC o A f(x) nA BC o A :\T

,k{ BM 2 T nj nB BA /j{ q q\U Am ,jnAk YYe zn jB BC ,D < o A :w SBe

/jnC nj B Mo j R\ Rn M SBe A nj AU An jB

:D < SBe nj S U k]

+-x

a S >.Af(x)

A] w Am f(x) < nA BC o A f(x) > nA BC o A : \T

SwA S nA Bp k{BM ("x R: f(x) > )SwA SLX nA Bp j ]nj

/k{BM ("x R: f(x) < )

-x2+ x - 1 < ,xKKe jk o ApA M k SMBY :BX

(m - 1)x2 - 2mx + (m - 2) < :nAj ("x R) nA ,m pA ojBK a ApA M :SvU

m > 1 (4 m < __ (323 m TvC k{BM f(x) = ax2+ bx + c j ]nj jB zn j ZnBi a jk A o{ 36

B Mo Rn M AU An nMq A] w ,k{BM oS ax2+ bx + c = j ]nj jB (D) L o A 37

/S{

Rn M AU An nMq A] w k{BM oS pA oTa ax2+ bx + c = j ]nj jB (D) L o A 38

/S{ B Mo j R\

oMAoM Bzn Jo BM Bzn R\ BC ,k{BM b = -c Bo ,ax2+ bx + c = j ]nj jB nj 39

/j{

s = p = -__ Am b = -c = 1 a 2x2+ x - 1 = jB nj :BX12(k{BM oj zn Mo jB Bzn pA ) x=x2 Bo ,ax2+ bx + c = j ]nj jB nj 4

BC ,x= xn o A SBe nj , BC

www.riazisara.ir

j~ oUoMzkA . 32

:w ]nj jB M ]An o~Th UB

jB A Bzn ,k{BM (a ) ax3+ bx2 + cx + d = Rn M B w ]nj jB o {

]nj MAU nAj M ]U BM /k{BM Bx nd BM ,C yB d jnioM BY ] (AjB o n] M )

k 9 Y w nj oX Ake Y nj 9Ake An Bx nd ,w ]nj jB o nAj SS AU ,w

/jnAj YYe zn w oX Ake zn 9Ake ,w ]nj jB o ,\T nj

:w ]nj jB Bzn jAkU }hzUxn

/ JBve An MBU >Tz AkTMA xn A nj

Gk A B SBe A nj MBU AoMBM ,S nA B SwA SLX nA B Rn A nj ,S{Ak zn MBU >Tz o A (tA

/jnAj zn jB k 9 Y nj An Bx nd SBe A nj MBU nAj Am ,q Gk A B SwA j

?jnAj zn kax3 + x - 7 = jB :BX

y = 3x2 + 1 > SwA j Gk A MBU /jnAj zn 1

?jnAj zn ka -x3 + x2 - 5x - 1 = jB :BX

SwA q Gk A MBU jnAj zn 1

M MBU { Rn A nj ,k{BM oS BY A pA o n] M ,k{BM T{Aj zn j MBU >Tz Bo (J

jBw zn SBe A nj jB AoMBM jM B Rn j pA

q yYTz zn ,SwA jB ji zn A oM ,tB1 zn k S9j ,jnAj tB1 zn

/k{BM

?jnAj zn kax3 - x2 - 8x + 12 = jB :BX

/k{BM x = 2 jB tB1 zn H /jnAj tB1 zn jBw zn SBe A nj jB uQ

{ M ]U BM Rn A nj k{BL oS ,Bzn pA ` o n] M ,k{BM T{Aj zn j MBU >Tz Bo (Z

/S{Aj kAi qBT zn w B zn B MBU ,jnAj Bzn A o M TvM MBU

Rn j pA M MBU { k{BM S j o B SLX j o >Tz Bzn o Bo (1

/jnAj YYe zn rY. MBU Am jM B

Rn j pA M MBU { ,k{BM A tTh >Tz Bzn o Bo (2

www.riazisara.ir

33.oUoMzkA RjB

/jnAj qBT YYe zn 3 SBe A nj jB Am jM B

?jnAj zn ka 2x3 - 9x2 + 12x + 6 = jB :BX

/jnAj zn 1 rY. jB AoMBM jM Rn M MBU { Am

: nBvM RAomU

/SwA x = 1 jB Bzn pA j{ oS jB KAo R\ Bo ,oj jB o kB (1

/SwA x = 1 ,x3 + x2 - 2 = jB zn :BX

/SwA x= -1 jB Bzn pA ,a+ c=b+ d ,{BM T{ Aj ax3+ bx2+ cx + d= w ]nj jB nj Bo (2

/SwA x = -1 ,x3 - 2x - 1 = jB zn :BX

/jo e An jB onT B. kMTwj M AU ,jnA pA ioM nj (3

?k e Anx3 - 3x2 - 4x + 12 = jB :BX

(x2- 4)(x - 3) = x = 2,3

/ jBSTwA op Rn M B K zU pA ,w ]nj jB e AoM ,jnA pA ioM nj (4

?k e An x3+ 3x2 + 3x + 9 = jB :BX

/jo e An w ]nj jB AU ,UBXX BoWT oWU M ,jnA pA ioM nj (5

?k e An 3x - 4x3- 1 = jB :BX

:nAj x = sina o.

jB zn

Aop /SwA jB tB1 zn x = __TLA ,SwA jB oj zn x =-1 Am ,a + c = b + da.o] pA12

www.riazisara.ir

j~ oUoMzkA . 34

/k{BM q >Tz zn ,zn A

?k e An jB :BX

:nAj x = tana o.

:pA kUnBL jB Bzn Am

x = tana = znak

x1= tan__p9

__p9

x2 = tan___4p9

___4p91

x3 = tan___7p9

___7p92

:nAj nA ,k{BM ax3+ bx2+ cx + d = w ]nj jB Bzn g b a Bo (6

:pA kUnBL jB Bzn :SvU

c -b -a (4 c b a (3 a+ b b a (2 c -b a (1

/k{BM 3 q 6 T M ]U BM cd gwBQ

/k{BM oS ,x2Ko x3Ko ,k{BM x3+ px + q = Rn M w ]nj jB o B Rn (7

M x = X-___ oWT oWU pA jBSTwA BM AU An ax3+ bx2+ cx + d = kB B w ]nj jB o :1 omUb3a/jnC nj X3+ pX + q = B Rn

jB M An jB AU ,x3 Ko oM jB vYU BM ,k{BM oS x2 Ko ,w ]nj jB nj Bo :2 omU

/jo kLU B

B Rn M jB Sw M ,x = __ oWT oWU BM ,k{BM oS xKo ,w ]nj jB nj Bo :3 omU1X

/j{ kLU

/jnC nj x3+ px + q = B Rn M SBe nj AU An w ]nj jB o AoMBM

www.riazisara.ir

35.oUoMzkA RjB

/k w ]nj jB L AnD = 4p3+ 27q2RnBL ,x3+ px + q = jB nj :w ]njjB (D)L

B( w ]nj jB e

B AjnB nTwj) op nTwj M AU An zn A jnAj YYe zn rY. jB BC k{BMD > o A (1

/jnC Swj M (BBTA AjBn ,AjnB

?kMBM Anx3+ 6x - 2 = jB zn :BX

jnAj YYe zn rY. jB

) jBw zn ,jnAj tB1 zn jBw zn ,w ]nj jB BC ,k{BM D = Bo (2

(SwA qtBh C S

nTwj pA tB1 zn SBe A nj ,k{BM Rn M ,AjnB nTwj nj ,D B] M nAmB] BM

:Aop ,kC Swj Mx2= x3 tB1zn

x1+ x2+ x3= -__b

atB1 zn

jB eSvMq :BX

AoM /jo jBSTwA AjnB nTwj pA AU ,SBe A nj ,jnAj qBT YYe zn w jB ,k{BM D < Bo (3

/MBM An r q ojBY SwA .B ,xU AoM ,k{BM jB zn ,x = rcosq o. jB zn U

x3+ px + q = r3cos3q + prcosq + q =

r3 oM vYU (1)

:Am cos3q = 4cos3q - 3cosq :Aj

(2)

:nAj (2) (1) vBY pA

www.riazisara.ir

j~ oUoMzkA . 36

(4)

/kC Swj M op Rn M x ojBY \T nj q r ojBY (4) (3) rMAn pA

:n] M SwA p M Ap oTa ,3q C nj

o{ B o{ A TLA ,k{BM P < kBM ,k{BM YYe q r ojBY A AoM

/SwA 4p3+ 27q2<

?kMBM Anx3- 3x + 1 = jB Bzn :BX

D = 4p3+ 27q2= -81 <

/jnAj qBT YYe zn w jB uQ

:pA kUnBL jB Bzn q = ___ Am ,3q= ___uQ2p92p3

/k{BM RBST Bwv AnAj TvBM ,q3 q2 q1 ,j jk k M B :omU

RjB M ]An T ka

/jnAj zn n oX Ake ,YYe jAkA nj ,n ]nj pA A] ka jB o (1

/jnAj zn 9Ake HTe ,YYe jAkA nj ,n jo. ]nj pA A] ka jB o (2

B BUnBU k AjBn w M AkTMA nj ,j BM AjnB B M L9 w ]nj jB e AoM o. (3

,nAo . B M oj kzAj ,kM /k{ ozT BBTA Aj Bn AjnB w M ,1545 Bw nj kC Swj M

RnBj (oA) o A , pA kM /jnC Swj M SBe nj (ax4+ bx2+ c = )nm\ j jB e AoM o.

jBp Bz{ ,oUBM 4 ]nj pA RjB e AoM /kjnC Swj M Bo. ,nm\ j RjB e AoM

AjBn ,1824 Bw nj BA ,knC Swj M jB KAo KveoM ,Bzn T.B AoM oL] o. SwA k{

www.riazisara.ir

37 .oUoMzkA RjB

SwA ,|Bi SBe nj ka o ,jnAk j] o. a jo SMBY (182 - 1829) MC o ,ro

/jo e An A]nj o pA A] ka jB

/jnAj rTh jAkA nj zn 9Ake , Bv B oTnqM ]nj pA A] ka MBU o :(oL] wBwA 19) (4

/k{ RBLYA Bw 22 w nj BC tynko. nB BC nz AjBn wM 19 A

/jnAj zn n HY9j rTh jAkA nj ,n ]nj pA A] ka jB o (5

| a| + | b| + ... + | z| = a = b = ... = z = :nAj nA (6

?SwA Ak | x4- 16| + | 2x - 4| + | x3- 8| = jB zn :SvU

-2 (4 2 (3 -1 (2 1 (1

(> nk9 w o oTz zn) /k{BM 3 q cd gwBQ

:nAj nA (7Zp Zp Zp

:nAj nA (8Zp Zp Zp

a + b + ... + z = a = b = ... = z =

:pA SvUnBL jB zn :SvU

2 (4 -1 (3 (2 1 (1

(RnBL w o oTz zn) k{BM 1 q cd gwBQ

?SwA Ak y nAkK jB nj :SvU

9 (3 -9 (2 3 (1(4

oj zn q -a BC ,k{BM f(x) = jB zn a (nBYT Df, f(-x) = f(x)) k{BM Zp MBU f Bo 9

/SwA oS z AjB a KAo R\ SwA jB

KKe jk c) ?SwA Akxoj zn ,k{BMx = 2 ,x4+ 3x2 - | x| - c= jB zn o A :SvU

(SwA SMBY

-2 (3 1 (2 -1 (1(4

/SwA -2 ,jB oj zn ,p. T >L]

q __ BC ,k{BM fzn a "x Df:f(x) = f(__) n] M k{BM MBU f Bo :w jB (11

a1x

/k w jB An f(x) = jB SBe A nj /SwA C zn

/kTv C Bznx = __ x = 2 Aop ,SwA w AjB ,2x2 - 5x + 2 = jB :BX12

www.riazisara.ir

j~ oUoMzkA . 38

:1 SvU

:BM SwA oMAoM ax2+ (a+ b+ c)x + b + c= jB Bzn pA 1

-_ (4ca

_ (3ca

+1 (2 -____ (1b+ ca

:BM SwA oMAoM Be ,k{BM 2x2 - 4x - 1 = jB Bzn b a 2

(418

- (318

(2-14

(114

?SwA Akm ,SwA nAoioM qMAn ,x2 - 3x + m + 1 = jB nj 3

2 (4 (3 -1 (2 1 (1

:j{ \T (x6-7)2+ (x12-y+ 1)14 = jB e pA 4

y = 5 (4 y = 59 (3 y = 57 (2 y = 9 (1

?SwA Ak jB Bzn pA 5

-4 (4 4 (3 -1 (2 1 (1

?SwA oj zn Mo x2 + mx + 27 = jB Bzn pA ,m pA nAkK a ApA M 6

21 (4 -16 (3 -12 (2 18 (1

?SwA Ak 2a - 3b ,a2+ b2+ 18 = 6(a+ b) Bo 7

-6 (4 3 (3 -3 (2 6 (1

:BM SwA oMAoM x + y - z ,4x2+ y2 + 9z2+ 4x - 2y + 12z + 6 = Bo 8

- (476

(376

- (216

(116

?SwA Ak a+ b ,a2 + b2 + c2 + 3 = 2(a+ b+ c) Bo 9

2 (4 -2 (3 (2 1 (1

?SwA Ak b2+ 1 a2+ 1 Bzn BM AjB ,k{BM x2 + px + 1 = jB Bzn b a Bo 1

x2- p2x - p2 = (4 x2- p2x + p = (3 x2- p2x + p2 = (2 x2- px + p = (1

www.riazisara.ir

39 .oUoMzkA RjB

(1) 1

(3) 2

(1) 3

(4) 4

SwA oS jB KAo R\ (4) 5

x = -4 x + 5 = 1 B\C pA SwA y=1 jB Bzn pA Am

(2) 6

(2) 7

(3) 8

x + y - z = __76

(4) 9

(2) 1

( woA) /SwA Bw pnoTM poA kvM jiKi nj

(RoMr) /k :V o BwC nj nBL LK SwA AnBTw A]

www.riazisara.ir

j~ oUoMzkA . 40

:2 SvU

?Sva k jke ,k{BM SLX nA A] w A o M 1

-__ < k < 1 (472 -2 < k < 1 (3 k > -2 (2 k < -2 (1

?jnAj KKe zn ka jB 2

4 (4 2 (3 1 (2 (1

?j{ T{ B Mo j :\ RnM ,4x2- 4x + mA] w ,m pA ojBK a ApA M 3

m > -3 (4 m > 1 (3 m < -3 (2 m < 2 (1

?jnAj KKe zn kax4 - x2 - 6 = jB 4

4 (4 2 (3 1 (2 (1

:BC ,a + b = __ k{BM x2 - 2mx - 3 = jB Bzn tanb tana o A 5p6m = __ (223 (1m = -1 (4 (3

Ak a ,k{BM jB Bznu , jB Bzn o A 6

?SwA

2 (4 - (312

1 (2 3 (1

?j{ BOA nkKa BC Jo\Be M , BOA keA ,x2 - x - 1 = jB BMA] pA o M o A 7

2 (2 3 (11 (4 (3

?SwA Ak _____Be ,k{BM x2- 2mx + m = j ]nj jB Bzn cota tana o A 82sin2x

(412

2 (3 2 (2 1 (1

:nAj BC ,k{BM x = -1 jB\ zn AnAjx3+ px + q = jB o A 9

p = 3, q= 2 (4 p = -3, q= -2 (3 p = -3, q= 2 (2 p = 3, q= -2 (1

?SwA Ak ,k{BM C Bzn j ]nj jB 1

x2- ax + 4 = (4 x2+ 4x - a = (3 x2+ ax - 4 = (2 x2- 4x + a = (1

www.riazisara.ir

41. oUoMzkA RjB

(3) 1P = (2 + k)x2 + 6x + 2k + 1

/j{ -2 < k < 1 ,o AoT{A

(4) 2

zn 2

zn 2

(3) 3

(3) 4

(1) 5

(2) 6

/SwA .B Am jnAj JA] p. BTwj ,k{ jAj Bq M ]U BM aL9 MB9

L9MB9oE ____ = ____ ___3-2-1

12-3

-6

:A An (2) 7

:Am kLU x-1 M An x :j An

(2) 8

(3) 9

(1) 1

www.riazisara.ir

j~ oUoMzkA . 42

:3 SvU

?SwA Ak k ok BTwj njyBzn j ]nj jB 1

5x2- 17x - 4 = (4 5x2- 17x + 2 = (3 5x2+ 17x - 4 = (2 5x2- 17x - 2 = (1

?SwA Ak Bek{BM x2 - 2x - 1 = jB Bznx x Bo 2

(4-17

(337

(2-37

(117

:BM SwA oMAoM m ,k{BM oj zn oMAoM 3 ,Bzn pA ,x2 - 8x + m + 8 = j ]nj jB nj o A 3

(4 5 (3 -5 (2 4 (1

:pA kUnBLx2+ (tana+ cota) x + 1 = jB Bzn 4

-cosa , -sina (4 cosa , sina (3 cota , tana (2 -cota , - tana (1

:pA kUnBL (cos2a)x2+ x + sin2a = jB BMA] 5

tana 1 (4 cosa sina (3 - tan2a -1 (2 tan2a 1 (1

AT R]KUoU M x m x k{BM x2+ 2(m + 1)x + 2m - 1 = jB Bzn x x o A 6

:BM SwA oMAoM m ,k{BM wk kB~U

1 (4 2 (3 -1 (2 -2 (1

:BM SwA oMAoM mk{BM nAoioM log(x+ x) - logx= logx qMAnx2+ (m - 1)x + 4 = jB nj o A 7

-3 (4 -2 (3 2 (2 -1 (1

?jnAj oi KKe zn j , jB ,m AnkK a ApA M 8

-1 (4 (312

- (212

1 (1

?SwA Ak ,k{BM x2 + x - 2 = jB Bzn pA oT keA 3 KUoU M yBzn j ]nj jB 9

x2- 7x - 1 = (4 x2+ 7x - 1 = (3 x2- 7x + 1 = (2 x2+ 7x + 1 = (1

?SwA nAoioM x2 - 5x + k - 1 = jB Bzn M x+ 3x= 7 qMAn ,k nAkK a ApA M 1

5 (4 4 (3 3 (2 2 (1

www.riazisara.ir

43.oUoMzkA RjB

(4) 1

(2) 2s = 2 , p = -1

(1) 3__ = _______ _____ = ___ m = 4b2

ac

(k+ 1)2k

64m+ 8

163

(1) 4

(2) 5

(4) 6

(4) 7

A tTh c a/k{ A tTh c a ,m= -1 ApA M rY. (4) 8

/ kLUx + 3 M An x (1) 9(x + 3)2+ (x + 3) - 2 =

x2 + 7x + 1 =

(4) 1

( nT) /BkoQ AoM BM ,jnAj SA nkKC ,ozM Bkp nj kA

(uBU) / nAjji , ypow C oVBi M An Aoj `C B\A pA

www.riazisara.ir

j~ oUoMzkA . 44

:4 SvU

?SwA Ak a jke ,k{BM jo qi By nd oV nj An Bx nd ,y = ax2+ 2x + 1 - ad Unnj 1

(-,) (1,+) (4 (-3,3) (3 (1,+) (2 (-,) (1

?SwA ka qTv A oqi ,SwA 1 C SeBv qTv Ld 2

(1(2(3(4

?SwA Ak ov Be ,k{BM x2 - 3x + 1 = Bo 3

4 (2 (118

(3(4

?SwA Ak ,Bznt :\ oMAoM j ,k{BM 3x2 + 4x - 9 = jB Bznx x o A 4

- (449

8 (3 (289

- (189

?SwA Ak Be ,x2 - 4x + 1 = jB nj 5

3 (4 2 (3 (214

4 (1

?SwA Ak Be ,k{BM x2 - 26x - 27 = jB Bznx x Bo 6

-2 (4 2 (3 1 (2 (1

?jnAj KKe zn kax3 + x - 2 = jB 7

3 (4 2 (3 1 (2 (1

?jnAj KKe zn ka 2x3 - 3x2 - 36x + 5 = jB 8

3 (4 2 (3 1 (2 (1

?jnAj KKe zn kax3 - 9x2 + 24x - 2 = jB 9

3 (4 2 (3 1 (2 (1

?jo nAoix2+ (m-1)x- m = jB Bzn M 2 jk ,m pA ojBK a ApA M 1

m > -2 (4 m > 2 (3 m < 2 (2 m < -2 (1

www.riazisara.ir

45.oUoMzkA RjB

:k{BM T{Aj A tTh zn j TvBM k{ jAj jB SwA cA (4) 1

a < B a > 1 B\C pA ____ < : xx= _ < :Am1-aa

c

a

(1) 2

x

yD

o9 =

(1) 3

(2) 4

(1) 5

(3) 6

/jnAj zn rY. jB Am k 9 Y nj rY. An Bx nd MBU uQ (2) 7y = 3x2 + 1 >

(4) 8

/SwA zn 3 AnAj jB ,q] nj k{ BM T >L] uQ

(3) 9

f(2) = /tB1 oj jBw SwA zn 2 AnAj jB q] nj k{ BM T >L] uQ

(1) 1

/SwA w y]oi tvA SwA Mi wnk Mo\U

www.riazisara.ir

w~ oUoMzkA . 46

w~%

CBs &nk(

/ toU op Rn M AoC jAj yB | x| jB BM An xYYe jk o > nk9

:SS AU oUjBw wnB. RnBL M , B

| SLX jk o| = xji| | = xji| S jk o| = xAo9

:k ]U op BBX M

1) | 4| = 4

2) | | =

3) | x2 + 1 | = xji = x2 + 1

4) xAo9 =

5) xji =

6) xAo9

7) xAo9

8) | -x4-1| xAo9 = -(-x4-1) = x4+ 1

9) | x2| = xji = x2

1) xji =

11) | +| = + | | = | -| = +

12) | cos15| = xAo9 = -cos15 =

13)

14) | sinx - 1| = xAo9 = -(sinx - 1) = 1 - sinx , (sinx 1)

15) | x - a| + | x - b| = xji + xAo9 , (a < x < b)

www.riazisara.ir

47.oUoMzkA CBsEF nkG

= x - a - (x - b) = b - a16) | x - y| + | x| - | y| = xji + xji - xAo9 , (y < < x)

SLX SSLX

= x - y + x - (-y) = 2x

:EFnkGBs

1) "x R : | x|

2) "x R : | x| x

3) | x| = x x

4) | x| = -x x

5) | x| > x x < , (SwA oTnqM B{ji pA ,S jAkA > nk9)

6) | x| < x x { } /jnAk JA] ,(Sv oTa xji pA jk _ > nk9)

:BM SwA oMAoM | x2-4| + x2 = 4 jB JA] \ :SvU

-2 x (4 x 2 (3 -2 x 2 (2 x -2 Bx 2 (1

:e

7) | x| x x R (SwA Bv B oTnqM xji pA z jk o > nk9)

8) | x| x | x| = x x

9) | x| = a , a > x = + a B x = -a

1) | x| = a , a < oE

11) | x| = | -x| ,(SwA oMAoM xAo9 > nk9 BM jk o > nk9)

12) x = y | x| = | y|

13) | x| = | y| x = + y B x = -y

:EFnkG njA nBaB\MB]SBi

14)

/jnAk j] SBe nj > nk9 nj vYU B\MB] SBi :omU

:> nk9 nj A nBa jo SU SBi

www.riazisara.ir

w~ oUoMzkA . 48

15) | x + y| | x| + | y|

| x - y| | x| - | y|

| xy| = | x| | y|

| _| = ___ , (y )xy

| x|

| y|

16) | x| 2 = | x| 2 = x2

| x| 3 = | x| 3 x3

| x| n = | xn|

k{BM BC pA 9Ake B k{BM S y x xy 17) | x + y| = | x| + | y|

?SwA Ak | x - 1| + | x + 4| = | 2x + 3| jB JA] \ :SvU

x - Bx 1 (432

x -4 Bx 1 (3 - x 1 (232

-4 x 1 (1

:{BM T{Aj TvBM 17 T >L] :e

(x - 1)(x + 4) x -4 Bx 1

18) | x + y| < | x| + | y| /k{BM A tTh y x xy <

?SwA nAoioM x pA ojBK a ApA M ,| 2x - 1| + | 5 - x| > | 3x - 6| jBB :SvU__ x 5 (412 x

__ Bx 5 (312__ < x < 5 (212 x 5 (112:e2x - 1 + x - 5 = 3x - 6 (2x - 1)(x - 5) < S U __ < x < 512

:nCjB| x - y| = | y - x|

19) | x + y| | x| + | y| B | x| + | y| | x + y|

/k XX BvB An BvB A :omU

2) | x - y| | x - z| + | z - y| B | x - z| + | z - y| | x - y|

?SwA Swnj ,MBU A M ]An q Ak ,SwAoR f(x) = | x + 1| + | x - 9| MBU :SvU

-1 f(x) 9 (4 f(x) 1 (3 8 f(x) 1 (2 f(x) 8 (1

| x + 1| + | x - 9| | -1-9| = 1 f(x) 1 :nAj 16 T >L] :e

21) | x - y| | x| - | y|

22) | x| - | y| | x - y| | x| + | y|

23)B

- | x - y| | x| - | y| | x - y|

www.riazisara.ir

49 .oUoMzkA CBsEF nkG

?SwA Ak f(x) = | x + 1| - | x + 5| MBU joM :SvU

f(x) 6 (4f(x) -4 B f(x) 4 (3 -4 f(x) 4 (2 -6 f(x) 6 (1

:nAj 18 T >L] :e

- | x + 1 - x - 5| | x + 1| - | x + 5| | x + 1 - x - 5|

-4 f(x) 4

24) | x| | x| = x =

25) | x| a , (a > ) -a x a x2 a2

26) | x| a , (a > ) x -a Bx a x2 a2

27) | x| a , (a < ) x R

28) | x| a , (a < ) oE

29) a | x| b , (a , b > , a < b) -b x -a B a x b

3)

31)

?SwA Ak jBBJA] \ :SvU

-3 x-1 B 1 x3 (4 1 x 3 (3 -1 x 1 (2 -3 x 3 (1

:e

1 | x| 3 -3 x -1 B 1 x 3

32) "x R : - | x| x | x|

33) | x1+ x2 + ... + xn| | x1| + | x2| + ... + | xn| B

/k{BM SBe nj ,XX BvB B ,BvB A :omU

34) y = A| x - a| + B| x - b| + ... + Z| x - z| : Rn M BUnBL nj

: /j{ Be ,RnBL nj > nk9 iAj Bzn ApA M MBUMin ,A B ... Z > o. BM

Miny = Min {f(a) , f(b) , ... , f(z)}

joM = [Min(y) , + ) :j{ MBU joM \T nj

www.riazisara.ir

w~ oUoMzkA . 50

?SwA Ak ,y = | x-1| + | x+ 2| + | x+ 3| MBUMin :SvU9(4 6 (3 4 (2 5 (1

35) | x| < a | x| a

36) | a| + | b| + ... + | z| = a = b = c = ... = z =

?SwA AkB = | a + b| + | a| + | b| RnBL Be ,| a| > | b| b < < a o A :SvU

2a (4 2b (3 -2a (2 -2b (1

:e

B = a + b + a - b = 2a

:nAm jk :j xn

?jnAj zn ka | x - 1| = | 3 - x| jB :SvU

3 (4 1 (3 (2 2 (1

oE :e

oTz Y ,MBU j nAj a :nAj wn >o] pA :j xn

pA zn A ,j{ kj nB ,jnAj zn MBU Am knAj

/SwA 2 oMAoM q { n

?jnAj zn ka jB :SvU

(4 1 (3 2 (2 4 (1x - 2 = 5x - 2 = -5x - 2 = 1x - 2 = -1

:e

?k e An | x - 1| + 2x = 5 jB :BX

if x 1 x - 1 + 2x = 5 x = 2 L9 MB9

if x < 1 -x + 1 + 2x = 5 x = 4 L9 MB9oE

www.riazisara.ir


?k e An | x - 1| + | x - 2| = 3 jB :BX

if x < 1 -x + 1 - x + 2 = 3 x = L9 MB9

(jnAk JA] [1,2) B. nj MBU Am)if 1 x < 2 x - 1 - x + 2 = 3 1 = 3 oE

if 2 x x - 1 + x - 2 = 3 x = 3 L9 MB9

JA] \ = {,3}

/jo Ai BM p. jB Bzn T.B AoM oTow {n ,kM BTv9 nj :omU

?k e An op RjBB :BX

tA) | 2x - 3| 5 -5 2x-3 5 -2 2x 8 -1 x 4

J) | x + 1| | x - 2| | x + 1| 2 | x - 2 | 2

x2 + 2x + 1 x2 - 4x + 4 x __12Z)

Bj)

| x + 1| -1 oE

B B| x + 1| 9

?k jBw An op RAnBL :BX

tA)

| x| + | x - 3| = - x - (x - 3) = 3 - 2x

J)

= xji + xAo9

Z)

= xji + xAo9 =

www.riazisara.ir

w~ oUoMzkA . 52

/Bwn 2 AU M An .o] :j e An

j)

______ + _______1

xji1

xAo9


2 x < 3 (4 -1 x < 3 (3 1 x < 3 (2 x 2 Bx > 3 (1

x - 2 D = [2,+) :e

/k{BM S TvBM Zoh ,Am SwA SLX nA ,jBB Oaco] Rn aAj M ]U BM 2 x < 3| x - 1| - 2 < -2 < x - 1 < 2 -1 < x < 3


x 4 (4 x 4 (3 x 1 Bx 4 (2 x Bx 4 (1

:ex D = [ , +)2 AU M

x2 - 4x + 4 x x2 - 5x + 4

+41-x

x 1 Bx 4+-+

Aj M ]U BM x 1 Bx 4

(U) /jM _ jL Bq JBT o A

( vjA) /j{ qo SwA AoQ , B

www.riazisara.ir


(Ak MBU) (a < b) y = |x-a| + |x-b| MBU nAj wn

9A nj SwA k{ zU B] nd pA rinBQ ri j pA { >MB MAU A wk yB

:nAj

MBU A { M /k q Ak MBU An MBU A k{BM ( ) Ak Rn M MBU A nAj a

/k ]U

?k wn An op MAU nAj :BX

tA) y = | x-1| + | x-4|

J) y = | x - 2| + | x + 2|

Ak MBU Bs

1) Df= R Aj

2) Rf= [| a - b| , +) B y | a - b| joM

3) Min (y) = | a - b|

4) x = ____ nBYU nda+ b25) y = | x - a| + | x + a| Zp MBU x = _____ = nBYU nd-a+ a2

/j{ Zp MBU ,k{BM o9 b a Bo 9A nj

6) /Sv omQ>Tz b a nj MBU

www.riazisara.ir

w~ oUoMzkA . 54

7)

8)

/Rw iB B qM voeSwnj B

:(k MBU) Rn M MAU nAj wn

(tA) SBe nj 9A nj /SwA k{ zU rinBQ B] nd pA ri j pA MAU A wk yB

:nAj

/k q k MBU An MBU A ,k{BM ( B ) k Rn M MBU A nAj a

/k ]U (J) (tA) SBe nj MBU A { M

(J) (tA)

?k wn An op MAU nAj :BX

tA) y = | x - 2| - | x - 4|

J) y = | x - 1| - | x + 1|

www.riazisara.ir


(A { nj)kMBU Bs

1) Df= R

2) Rf= [- | a - b| , + | a - b| ] B - | a - b| y | a - b|

3) Min(y) = - | a - b| , Max (y) = | a - b|

4) nBYU qo

nBYU qo5) y = | x - a| - | x + a| jo. MBU

6) /Sv omQ>Tz b a nj MBU

B7)

8)

:woU >o pA jB Bzn jAkU S ,j] nj WdM

/jo ,op Rn j pA o M AU xn A nj /SwA woU xn ,jB e B{n pA

BM d jnioM BY B] ,jo wn An f MBU nAj SwA .B ,f(x) = jB Bzn T.B AoM :A xn

nj Bzn LoYU nAkY AU xn A nj TLA jM kAi jB Bzn B (j] Rn nj) Bx nd

/jnC Swj M An jB Bzn ji ,jnA pA 1M

?woU woV pAx2 - 1 = jB Bzn S jAkUSvMq :BX

j AnAj jB A j{ kj y = x2-1 nAj M ]U BM :e

/SwA S oj SLX ,zn

Swj M An jB j jnioM BY B] joM SwAn co] M An f(x) = jB pA Tv9 xn A nj :j xn

/k{BM f(x) = jB Bzn B] A ,jnC

?woU woV pAx2 - 1 = jB Bzn S jAkUSvMq :BX

,zn j AnAj jB j{ kj ,nAj M ]U BM :e

/SwA S oj SLX

www.riazisara.ir

w~ oUoMzkA . 56

? ex- sinx = jB Bzn S jAkUSvMq :BX

:e

/jnAj S zn nBzM jB j{ kj nAj M ]U BM

/kj tnj kM k BdTA A ,SwA oThw Mo\U

| x - a| + | x - b| = k (k > ) jB Bzn jAkU j] nj WdM

BY jAkU tBwA oM jo wn An MBU j nAj SwA .B p. j xn >L]oM

/SwA a WdM i , WdM B{jnioM

SBe A nj S{Aj kAi zn j jB k{BM k > | a - b| o A (1

/SwA a pA oTa oj b pA oTnqM zn j pA

/kC Swj M op o. pA zn j A nAkY

x = _______ (b pA oTnqM zn)a+ b+ k2x = ______ (a pA oTa zn)a+ b-k2

pA kUnBL jB Bzn SBe A nj S{Aj kAi JA] nBzM jB { >MB ,k{BM k = | a - b| Bo (J

[a , b] pBM jAkA

/k ]U { M /S{Aj kAh zn jB k{BM k < | a - b| Bo (Z

?jnAj zn ka | x - 1| + | x + 9| = 1 jB :SvU

nBzM (4 1 (3 2 (2 (1

[-9,1] pBM jAkA pA kUnBL jB A Bzn ,jnAj zn nBzM Am a :e

www.riazisara.ir


?jnAj zn ka ,| x| + | x - 1| - | x - 4| = | x - 3| jB :SvU

2 (4 nBzM (3 1 (2 (1

| x| + | x - 1| = | x - 3| + | x - 4| ,{ M ]U BM :e

/jnAj zn 1 jB j{ kj

/k{BM 2 jk zn A ,j{ kj nB TLA

(a < b), | x - a| - | x - b| = k jB Bzn jAkU j] nj WdM

/SwA a WdM i , nAj wn BM q SBe A nj

/jnAk zn jB ,k < - | a - b| B k > | a - b| (1

/kC Swj Mx=_______nTwj pA jB znSBeA nj ,jnAj zn 1 jB , - | a - b| < k < | a - b| o A (2a+ b+ k2

jAkA :pA kUnBL Bzn ,k = | a-b| SBe nj /jnAj JA] nB{M jB ,k = - | a - b| Bk = | a - b| o A (3

/(-, a] pBM jAkA pA kUnBL Bzn ,k = - | a-b| SBe nj [b,+) pBM

?jnAj zn ka | x - 2| - | x - 4| = 1 jB :SvU

nBzM (4 2 (3 (2 1 (1

/jnAj zn 1 jB uQ ,-2 < 1 < 2 a :e

?SwA Ak | x| 3 B nj ,y = | x| - | x - 1| MBU nAj Tv{ LiV :SvU

(1(2(3(4

ABCD Tv{ ri ] = :e

www.riazisara.ir

w~ oUoMzkA . 58

?SwA Ak k ,k{BM wLq oM y = | x-2| - | x+ 5| nAj pA Tvi y = kKA Li o A :SvU

-7 < k < 7 (4 k = -7 B 7 (3 -3 < k < 3 (2 k = 3 B -3 (1

k = 7 B -7 SwA .B Am , | a-b| = | 2-(-5) | = 7 a :e

?jnAj zn ka | x - 2| = | x - 3| + | x - 4| + | x - 5| jB :SvU

2 (4 (3 1 (2 nBzM (1

pA kUnBL jnAj JA] nBzM jB ,nAj M ]U BM :e

| x - 2 | - | x - 3 | = | x - 4| + | x - 5| [4,5] pBM jAkA

/ :SwBA AoM RB oUkM

jB A Bs ,SwA Mo jB ,(k > ),| x - a| + | y - b| = kRn M jB o :Mo nAj

:pA kUnBL

/Sv MBU Mo (1

/SwA RB~Th Bnd pA ,Mo A nB9A (2

/SwA Mo nBYU qo 9A nj Mo qo (3

/SwA 2k oMAoM Mo o9 Am SwA Mo o9 t~ k jk (4

(VnEBX. 19) /SwA Mo , jk (5

:pA SvUnBL Mo rd (6

:pA SvUnBL Mo SeBv (7

/kTv C nBYU Bnd y = (x-a)+ b y = b x = ai (8

:pA kUnBL Mo joM Aj Am y x jke ,p. { M ]U BM (9

Aj D = [a - k , a + k]

joM R = [b - k , b + k]

/SwA CMin Y , Y Max Y , Y (1

:BM koMAoM y q B (11

www.riazisara.ir


nk9 iAj pA An RAnBL ,BC M ]U BM uPw S.o o

/ wn An Mo

RB~Th Bpd pA oM j ri j qo nj uPw ,jnC Swj M An xqo AkTMA ,Mo oTBwC wn AoM BA

/j{ wn Mo kC Swj M Mo tn BU ,jo Ak] k pAkA M co] o pA ,jo wn

?k wn An | x - 1| + | y + 3| = 2 Mo :BX

?SwA Ak | x - 2| + | y + 3| = 4 qMAn nj y nAkKMax :SvU-7 (4 7 (3 -1 (2 1 (1

Max (y) = b + k = - 3 + 4 = 1 :e/jM kAi (a,b)RB~Th M Y nAj BC ,k{BM k = o A ,| x-a| + | y-b| = k MAn nj :T

| x - a| - | y - b| = k MAn nAj wnoM :TSBe A nj (k ]U { M) knAk oTz tEn SwA B9 Ap j ,p. MAn nAj ,k{BM k > o A

] M qo nj Mo zU ,Ap j RA jAkTA/k{BM RB~Th Bnd pA ,C nB9A kj 2k oMAoM o9,x = b x = a i nBYU qo SBe A nj

/k{BM p. MAn nAj nBYU Bnd/j{ oM j ri j ,nAj k{ >L oM A ABY ,k{BM k = o A :omU

| y - b| - | x - a| = k MAn nAj wnoM :TSBe A nj ,(k ]U { M) knAk oTz tEn SwA B9 Ap j ,p. MAn nAj ,k{BM k > o A

pA xnB9A kj Mo zU Ap j RA jAkTAxo9 ] 2k qo jM RB~Th Bndy = x = ai nBYU qo SBe A nj /k{BM

/k{BM p. MAn nAj nBYU Bnd ,b

www.riazisara.ir

w~ oUoMzkA . 60

/j{ oM j ri j ,nAj k{ >L oM A ABY ,k{BM k = o A :omU

jB A Bs ,SwA p jB (k > ) , | ax - b| + | cy - d| = kRn M jB o :p nAj

:pA SvUnBL

/Sv MBU p (1

/SwA RB~Th Bnd pA ,p A nB9A (2

/SwA p nBYU qo 9A nj qo (3

___ , ___ :pA SvUnBL C nB9A ] (42k| c|

2k| a|

S = __________________ = ___ :pA SvUnBL p SeBv (5a o9 * nqM o922k2ac

/k{BM C nBYU Bnd y = __ x = __i (6dc

b

a

?SwA Ak ,| 2x - 1| + | y - 1| = 4 { M jkd SeBv :SvU

64 (4 4 (3 8 (2 16 (1

S = ___ = 162k2

ac:e

:DEA pAT nAj

SwA RA pAT jB Rn M jB o

/k{BM ixnB9A RjB

:H

/j{ p nAj ,p. nAj ,k{BM nAo9oM __ = -__o{ Bo (1ba

ab

/j{ Tv nAj ,p. nAj ,k{BM nAo9oM a2 + b2 = a2+ b2 o{ Bo (2

/j{ Mo nAj ,p. nAj ,k{BM nAo9oM B o{ Bo (3

/

www.riazisara.ir


:nCjBThBFMAn MAU nAj BM B{C

2) y = -x21) y = x2

3) x = y2 4)

5) 6)

8) y = x37)

9) y = -x3 1) x = y3 B

12) y = | x|11) x = - y3 B

14) | x| = | y|13) x = | y|

www.riazisara.ir

w~ oUoMzkA . 62

16) y = ___-1x

15) y = __1x

17) 18)

2) y = ___| x|x

19) | y| = ___1| x|

22) y = sign (| x| )21) y = sign(x)

24) x2+ y2= R223) y = sign(x2)

/SwA oAj RB{ R oAj qo

25) (x - a)2+ (y - b)2 = R2 26)

www.riazisara.ir


27) 28)

/SwA oAj RB{ R qo /SwA 1M a o9 t~ b nqM o9 t~ a,1M qo

3) y = sinx29)

b nqM o9 t~ a ,1M qo

/SwA 1M a o9 t~

32) y = tanx31) y = cosx

34) y = secx = ____1cosx

33) y = cotx

-1 cosx 1 secx -1 B secx 1

36) y = Arc sinx35) y = cosecx = ____1sinx

-1 sinx 1 cosecx -1 B cosecx 1

www.riazisara.ir

w~ oUoMzkA . 64

38) y = Arctanx37) y = Arccosx

__p2

4) y = ax , (a > 1)39) y = Arccot x

Gk A SBe A nj MBU AoMBM ,j{ oTnqM q y ax ,j{ oTnqM x ao Am ,SwA a > 1 a k S9j :T

: SwA j

41) y = ax , ( < a < 1)

SBe A nj MBU AoMBM ,j{ oTa y ax ,j{ oTnqM x ao Am ,SwA < a < 1 a k S9j :T

: /SwA q Gk A

42)

A nj AoMBM j{ oTnqM q y ,j{ oTnqM x ao Am ,SwA a > 1 a k{BM T{Aj ]U :T

: /SwA j Gk A SBe

43)

www.riazisara.ir


nj MBU AoMBM ,j{ oTa y ,j{ oTnqMx ao Am ,SwA < a < 1 a k{BM T{Aj ]U :T

: /SwA q Gk A SBe A

/kok t MAU ,y = ax y = logxaMAU k{BM T{Aj ]U :T

/

y = f(x)oG MBU nAj n pA/// MBo BITA xn M JTh BnAj wn M]An UB

/j BYTA BM co M k pAkA M An f nAj SwA .B ,(k > ), y = f(x) + k nAj wn AoM (1

:BXy = | x|y = | x| + 2

/j BYTA BQ co] M k pAkA M An f nAj SwA .B ,(k > ) ,y = f(x) - k nAj wn AoM (2

:BXy = x2y = x2 - 1

(SwAn Sw M HBLT{A ) OaSw M k pAkA M An f nAjSwA .B ,(k > ) ,y = f(x + k) nAj wn AoM (3

/j BYTA

:BXy = x3y = (x + 1)3

www.riazisara.ir

w~ oUoMzkA . 66

(Oa Sw M HBLT{A ) SwAn Sw M k pAkA M An f nAjSwA .B ,(k > ) ,y = f(x - k) nAj wn AoM (4

/j BYTA

:BXy = | x|y = | x - 2|

?k wn An y = | x - 1| + 2 nAj :LoU BX

(1) y = | x| (2) y = | x - 1| (3) y = | x - 1| + 2

/ wn Bynd M SLv An f nAj o9 SwA .B ,y = f(-x) nAj wn AoM 5

:BXy = x3

y = (-x)3= -x3 y = -x3

/ wn Bxnd M SLv An f nAj o9 SwA .B ,y = -f(x) nAj wn AoM 6

y = x2- 1

:BXy = - (x2- 1) = -x2 + 1

/ wn RB~Th EkL M SLv An f nAj o9 SwA .B ,y = -f(-x) nAj wn AoM 7

:BX

www.riazisara.ir


ri) w A Mn pBv ri M SLv An f nAj o9 SwA .B ,(ft B nA) ,x = f(y) nAj wn AoM 8

/ wn (y = x

y = x3

:BXx = f(y) x = y3 B

nBa j Mn pBv ri M SLv An f nAj o9 SwA .B ,x = -f(-y) B -x = f(-y) nAj wn AoM 9

/ wn (y = -xri)

y = x2

:BX-x = (-y)2 x = -y2

/

o9 ,SwA | f(x)| A M ]U BM uPw jo wn An f nAj ,SwA .B ,y = | f(x)| nAj wn AoM 1

/ wn (Bx nd BM nj) Bx nd M SLv An kAT.o nAo9 Bx nd op An BTv9

y = x2- 1 :BXy = | x2- 1|

y = | x| - 1 :BX

y = | x2- 1| - 2 :BX

www.riazisara.ir

w~ oUoMzkA . 68

(2) y = | x2- 1| - 2 (1) y = | x2- 1 |(3)

:A M ]U BM ,y = f(| x| ) nAj wn AoM 11

Sw) kB9BM BTv9 o9 uPw ,jo cme An y nd Oa Sw BTv9 ,jo wn An f nAj SwA .B

/ .BA C M ,By nd M SLv An (SwAn

y = x3+ 1 :BX

:A M ]U BM , | y| = f(x) nAj wn AoM 12

B

(Bx nd BM) kB9BM BTv9 o9 uPw ,jo cme An Bxnd op BTv9 ,jo wn An f nAj SwA .B

/ .BA C M Bx nd M SLv An

By = x2- 1 :BX

/o o nk9 |Ai M ]U BM | y| = f(| x| ) nAj wn AoM 13

1) x , y y = f(x)

2) x , y -y = f(x) B y = -f(x)

3) x , y y = f(-x)

4) x , y -y = f(-x) B y = -f(-x)

www.riazisara.ir


/SwA A Mn nj JBhTA An BTv9 ,y = f(x) nAj pA SwA .B , | y| = f(| x| ) nAj wn AoM Be

nAj pA ,(x , y ) /SwA nBa Mn nj JBhTA An BTv9 ,y = -f(x) nAj pA ,(x , y )

nj JBhTA An BTv9 ,y = -f(-x) nAj pA oiBM SwA j Mn nj JBhTA An BTv9 ,y = f(-x)

/SwA w Mnx + y = 1 B y = -x + 1 :BX| y| = - | x| + 1

B| x| + | y| = 1

/

,k{BM y = f(x) nAj n Y o A k{BM T{Aj ]U AkTMA (k > ) ,y = kf(x) nAj wn AoM 14

oo SwA .B ,y = kf(x) nAj wn AoM Am /uBM SwA y = kf(x) nAj n Y BC

MBU joM ,k{BM [c,d] pBM ,y = f(x) MBU joM o A SBe A nj , Jo k jk nj (C ] Se BM) An f nAj pA Y

/jM kAi [kc , kd] pBM ,y =k f(x)

MBU joM ) /j{ (oU]) oUkz B9 S] nj ,Aj B nj nAj ,k{BM k > 1 o A (tA

/(j{ oTnqM : omU

(/j{ oTa ,MBU joM ) /j{ oUpBM ,B9 S] nj ,Aj B nj nAj ,k{BM < k < 1 o A (J

y = x2 , (-2 x 2) :BXAj

joMtA)

Aj

joMJ)

www.riazisara.ir

w~ oUoMzkA . 70

y = f(x) nAj n Y o A k{BM T{Aj ]U AkTMA ,(k > ) , y = f(kx) nAj wn AoM 15

SwA .B ,y = f(kx) nAj wn AoM Am /uBM SwA y = f(kx) nAj n Y BC ,k{BM

,k{BM [a,b] pBM ,y = f(x) MBU Aj o A SBe A nj , vYU k jk oM (C o Se BM) An fd Y o ]

/jM kAi [__ , __] pBM ,y = f(kx) MBU Ajak

b

k

/j{ oUTvM ,joM B BM nAj Aj ,k{BM k > 1 o A (tA : omU

/j{ oUpBM ,joM B BM nAj Aj ,k{BM < k < 1 o A (J

y = | x| , (-2 x 2) :BXAj

joMtA)

Aj -2 2x 2 -1 x 1

Aj

joMJ)

Aj -2 x 2 -4 x 41

2

Aj

joM:BXtA)

Aj

joMJ)

www.riazisara.ir


:DT BX ka

1) y = sin| x| , (-2p x 2p)

2)

4) y = Ln(x - 1)3) y = x2- 4x + 5 = (x - 2)2+ 1

5) 6)

7)

kz #oU8) :nCjB

www.riazisara.ir

w~ oUoMzkA . 72

y = x9)

1)

2

11)

/AjAj BYTA SwAn Sw M keA 2 An y = sign(x) nAj ,9A nj

B

12)

1-1

/

/j{ nB{C A] nj oTzM SwA qa jAkTwA>

/jnC j]M An So C ji ,k{BM jikp nj JiSo kwn nB~TA nj A B] M BAj BvA

www.riazisara.ir


:1 SvU

?jM kAi Akx2- 2| 3 - x| < 2 jBBJA] \ ,k{BM x 3 o A 1

x 5 (4 3 x (372

3 x < 5 (2 2 x < 4 (1

?SwA Ak | x2 - 9| + | x - 3| < jBBJA] \ 2

R (4 R- (3 (2 {3} (1

?SwA Ak RnBL Be ,k{BM < x < 1 Bo 3

2(x + __) (41x

(3 __ (22x

2x (1

?SwA Akx + | x - 5| = 5 jB LV Bzn Jo\Be 4

12 (4 6 (3 84 (2 24 (1

?SwA Ak | x - 2| + | x - 5| = 9 jB Bzn :\ 5

6 (4 7 (3 12 (2 9 (1

:BM SwA oMAoM | cosx - sinx| Be ,__ < x < __ o A 6p4p2

(4 sinx-cosx (3 cosx-sinx (2 cosx+ sinx (1

?SwA Ak | x - 1| + | x - 5| < 8 jBBJA] \ 7

(-,-1) (4 (7,+) (3 (-,7) (2 (-1,7) (1

?SwA Ak jBBJA] \ 8

| x| 1 (4 [,1] (3 (2 | x| 1 (1

?SwA Ak | x| - | x - 2| < 4 jBBJA] \ 9

(-,2] (4 (-,) (3 R (2 [,2] (1

?SwA Ak BTwj JA] \ 1

-3x -2 B 2 x 3 (4 -3 x -2 (3 -3 x 3 (2 2 x 3 (1

www.riazisara.ir

w~ oUoMzkA . 74

(2) 1x 3 | x - 3| = | 3 - x| = x - 3

x2 - 2| 3 - x| - 21 < x2- 2(x - 3)- 21 < x2- 2x - 15 < x 3

(x - 5)(x + 3) < -3 < x < 5 3 x < 5

(2) 2| x - 3| (| x + 3| + 1) < JA] \ = { }

SLX S

< x < 1 __ > 1 __ > x (1) 31x

1x

(2) 4| x - 5| = 5 - x | x - 5| = -(x - 5) x - 5

xN x 5 x = 1 B 2 B 3 B 4 B 5 Bzn Jo1Be = 5!

zn j (3) 5

(3) 6

-1

(1) 7| x - 1 | + | x - 5| = 8

BY o ,7 -1 M Bx ApA M j{ kj { M ]U BM

/SwA oT y = 8 ri BYo pA y = | x-1| + | x-5| nAj

(1) 8

(2) 9| x - | - | x - 2| < | | + | 2| = 2 < 4

"x R : | x| - | x - 2| < 4

nAj ,xYYe jk o ApA M j{ kj q { wn BM

/SwA oTBQ y = 4 nAj pA y = | x| - | x - 2 |

AoT{AB B (4) 1

www.riazisara.ir


:2 SvU:BM SwA oMAoM RnBL Be , < x < __o BM 1p2

-sin__ (4x2 2sin__ (3x2 2cos

__ (2x2 cos__ (1x2

?SwA k{ T{ SwnjB q Ak nj nBKU qo 2

(1(2

(3(4

?SwA Ak 4| x| - | x2- x| - 5| x - 1| + 2 jBBJA] \ 3

-3 x 5 (4 x R (3 -5 x 5 (2 -4 x 4 (1

?SwA Akx2+ y2 = 2| xy| jB nAj 4

oM j ri j (4 p (3 1M (2 oAj (1

?SwA Ak 2| x| + | x - 1| = | x + 2| jB Bzn jAkU 5

(4 2 (3 nBzM (2 1 (1

:BM SwA oMAoM RB~Th Bnd BM nAj M n~d eB SeBv 6

(432

2 (3 (252

1 (1

?SwA Ak ( qMAn A Aj) xRAoU jke ,| x| + | y-1| = 2 o A 7

2 x 4 (4 x 2 (3 x -2 B x 2 (2 -2 x 2 (1

:pA SvUnBL | x3+ 1| < x2- x + 1 jBBJA] \ 8

-1 < x < 1 (4 -2 < x < (3 -1 < x < (2 < x < 1 (1

?SwA Ak RnBL Be 9

2 (2 1 (1(3(4

?SwA Ak | 4 - x2| < 2| x - 2| jBBJA] \ 1

(-2,2) (4 (-4,) (3 (-4,4) (2 [-4,4] (1

www.riazisara.ir

w~ oUoMzkA . 76

(2) 1

:oj xn

/k{BM RB~Th EkL ynBYU qo Am SwA jo. MBU x3+ y3= x MBU (4) 2

(4) 3

(4) 4

(3) 5| x| + | x - 1| = | x + 2| - | x|

/jnAj zn 2 ,{ M ]U BM

(4) 6

S = ____ + ____ = __1*12

2*12

32

(1) 7

oj xn| x - a| + | y - b| = k > Mo x jke = [a-k , a+ k] = [-2,2]

SLX nA

(3) 8

(1) 9

(3) 1

www.riazisara.ir


:3 SvU

?SwA Ak ,| y - x| = x- | x| qMAn nAj 1

nBa j Mn pBv (4 w A Mn pBv (3 j Mn pBv (2 A Mn pBv (1

?SwA Akx + y + 3 Be , Bo 2

(1(2-1 B 1 (4 B (3

?SwA Ak | x - 1| + | x - 2| > xjBBJA] \ 3

(-,1) (3,+) (4 [,+) (3 (1,3) (2 (-,3) (1

:BC , Bo 4

x = Bx = 8 (4 x = -4 Bx = 8 (3 -2 < x 2 (2 x < -2 Bx 2 (1

?SwA Swnj z q Ak ,k{BM a x b Bo 5

| x| Max{| a| ,| b| } (4 | x| | a| (3 | x| | b| (2 | x| b (1

?jnAj JA] ka jB 6

nBzM (4 2 (3 1 (2 (1

?SwA AkJA]A ,jnAj JA] (,1)B nj - | 2x| - | 2x-2| + | 4x - 5| = 2 jB 7

(435

(314

(234

(115

:SwA p ,k{BM nAoioM | x| > x2 jBB C AoM 8

x , -1 < x < 1 (4 < x < 1 (3 -1 < x < (2 x < -1 Bx > 1 (1

?SwA Ak | x - 2| - | x - 3| Be , Bo 9

2x+ 1 (4 1 (3 -1 (2 -2 (1

?SwA Ak (x - 2)2- 3| x - 2| - 4 < jBBJA] \ 1

-2 < x < 6 (4 -2 < x < 4 (3 -4 < x < 2 (2 < x < 8 (1

www.riazisara.ir

w~ oUoMzkA . 78

oE*

(1) 1

:Am k{BM oS TvBM ,j{ oS SB RnBL ka R\ Bo (3) 2

x < 1 B x > 3 B. nj j{ kj { M ]U BM (4) 3

y = x ri BY o pA y = | x-1| + | x-2| nAj BY o

/SwA oTzM

JA] \ B :oj e An

(2) 4

(4) 5

(4) 6

/jnAj j] xAoM \T nj yAoM JA] nB{M* *SLX

(3) 7

-2x + 2x - 2 - 4x + 5 = 2 x = __14

(4) 8

(2) 9

(4) 1

www.riazisara.ir


:4 SvU

:pASvUnBL jBBJA] \ 1

(-,)-{-1} (4 (,+)-{1} (3 (-,] (2 [,+) (1

?SwA AkMax(y) , MBU nj 2

(452

5 (3 (253

5 (1

?SwA Akx3+ 6xy + y3 Be ,y = | a+ 1| x = | a-1| | a| < 1 Bo 3

8 (4 6 (3 a2+ a + 1 (2 a2- 1 (1

?SwA op e Ak jBx [3,15]RnBL 4

| x-9| > 6 (4 | x-9| 6 (3 | x-9| 6 (2 | x-9| < 6 (1

?Sv Swnj ,x R o ApA M op e Ak 5

-1 | x-1| - | x+ 9| 1 (2 | x - 2| + | x + 2| 4 (1

(3(4

?SwA Ak | x| (x2-3x+ 2) jBBJA] \ 6

[-2,-1] {} (4 [1,2] {} (3 [-2,-1] (2 [1,2] (1

?SwA Ak jBBJA] \ 7

[-11,-1] (4 (-11,-1) (1,11) (3 (-11,1) (2 (1,11) (1

?SwA Ak jB JA] \ ,| x - 3| + | x + 3| = | 2x| Bo 8

[6,+) (4 (-,-3] [3,+) (3 [,+) (2 [-3,3] (1

?SwA Ak Be ,-1 < x < Bo 9

2| x| + 9 (4 1 - 2x (3 2| x| (2 11 (1

?jnAj cd JA]ka ,| x2 - 225| +x2- 225 = jB 1

jnAk cd zn (4 27 (3 31 (2 29 (1

www.riazisara.ir

w~ oUoMzkA . 80

(3) 1

(2) 2

| x - a| + | x - b| | b - a| :nCjB

(4) 3

(4) 4

(4) 5

:nCjB

(3) 6

JA] \ = [1,2] {}

(3) 7B 1 < x < 11

(3) 8

Bx 3

(1) 9

= x + 1 - x + 1 + 9 = 11

(1) 1

/jnAj cd JA] 29 , (x = -14 -13 /// /// 13 14)

/SwA kU BM tBwA JBT

www.riazisara.ir


:6 SvU

?SwA Ak y = | x + 1| + | x - 1| + | x - 3| MBU nAj nBKU nd jB 1

y = x (4 x = 3 (3 y = 1 (2 x = 1 (1

?SwA MBK { L{ ,op MAU pAAk nAj 2

y = x + | x - 1 | (1

y = x - 1 + | x-1| (2

y = x - 1 + | x| (3

y = x - 1 - | x+ 1| (4

?SwA Ak y = | x + 5| + | x - 15| + 3 MBU joM 3

(5,+) (4 [5 ,+) (3 [4,+) (2 R (1

?SwA Ak Kq nkijB Bzn :\ 4

2 (4 -1 (3 1 (2 (1

?SwA Ak | x + 3| + | 2y - 4| = 1 qMAn nAj nBKU qo 5

(3,2) (4 (2,-3) (3 (-3,4) (2 (-3,2) (1

?jnAj zn ka B nj , jB 6

nBzM (4 3 (3 2 (2 1 (1

?SwA op jBB AkJA] \ (7,8)pBM 7

| x + ___| < __ (415212 | 2x - 15| < 1 (3 | x -

___| < 1 (2152 | x - 1 | < 15 (1

ka nj An Bxnd ,y = f(| x| ) MBU nAj ,k{BM RnM ,y = f(x) MBU nAj Bo 8

?k qi qK

5 (4 4 (3 3 (2 2 (1

?SwA Ak k ,k{BM wLq oM ,y = | x + 1| + | x - 5| yB d pA Tvi y = kLi Bo 9

-6 (4 5 (3 4 (2 6 (1

?SwA Ak | x + y| = 1 nAj 1

ri j (4 ]BYT ri j (3 oM j ri j (2 pA ri j (1

www.riazisara.ir

w~ oUoMzkA . 82

/SwA MBU nBYU nd jB ,x = 1 ri ,{ M ]U BM (1) 1

Bo ,y = | x-a| + | x-b| + | x-c| MBU nj n] M :T

/SwA MBU nBYU nd jB x = bri a+ c=2b

/k pk (2) q jB nj BU (1,)Y (2) 2

(3) 3| x - a| + | x - b| | a - b| :nCjB

y | -5-15 | + 3 y 5

(3) 4

(1) 5| ax + b| + | cy + d| = k , k > nBYU qo

(4) 6| x + y| = | x| + | y| xy"

jB RnBL M ,k{BM oS BC pA 9Ake B S cosx sinx SwA nAo9oM T9 k{ jAj BvU

,j{ SLX BzMo1Be Am ,kAS j o tv tw ,[p,___]B. nj a Be k{BM sinxcosx 3p2/jnAj JA] nBzM jB AoMBM

(3) 7

(3) 8

(1) 9

(1) 1

/{BL B C yiK pA SwA C K oTnqM

www.riazisara.ir

83.oUoMzkA cdq]

nBa~

cdq]

n M ,P kB oj jk (n z) n cd jk \ Rn M AU An x kB ""e jk o

/k& 'oU op Rn M AoC jAj yB jB BM An xkB ""e jk o cd q] /S{ , p < 1

/S4 AU jBw i BMp M xBv B oTa& B

nBa~ oUoMzkA . 84

/kj yB [x] Bx jB BM AoC & (xBv B oTa& B

85.oUoMzkA cdq]

-1 B

Lc MBcoA

/k{BM Swnj ,(3) q AoMBM

15) x = y x = y

16) x = y x = y

/Sv M MBU ,f(x) = x MBU & kk Bz T A

2. 5 = 2. 7 = 2 2. 5 2. 7 BX A M

17)

: SBe nj cd q] nj A nBa B\MB] SBi (18

'A) x + y = y + x

J) x - y y - x

Z) xy = yx

j)

: SBe nj A nBa nj cd q] jo ,U SBi (19

'A) x + y x + y

J) x - y x - y

Z) xy xy

j)

2) " x , y R : x + y x + y

:BX A M SwA nAocoM BvU ,jz oTzM 1 pA BzBjni ] B k{BM cd y x pA cAke o A H\

2 + 3. 5 = 2 + 3. 5

3. 8 + 4. 1 = 3. 8 + 4. 1

4. 5 + 7. 9 4. 5 + 7. 9

/j{ 11 SwAn Sw Be Be nj j{ 12 OaSw Be ,cA nj

21) B

www.riazisara.ir

nBa~ oUoMzkA . 86

/SwA oTzM keA oX& Ake ,jk j C Bdd q] \ pA jk j \ cd q]

:RBLYAx x < x + 1

y y < y + 1+ -----------------

?SwA Ak Be ,k{BM Bo :SvU

2 B 4 (4 3 B 4 (3 5 B 4 (2 4 (1

/k{BM (2) q cd q ,h< T iL :e

:dozU e An

+ B 5

22)

23) x + n = x + n

x - n = x - n

nx nx , (n z)

/kC oM cd q] pA ,io4U ] nj j"< cd jk :omU

:BXx + 2 = x + 2

3 - x = 3 + -x

2x 2x

?SwA Ak ,y = x + x MBU t :SvU

y = x + 2x (4 y = x + __x (312 y = x -__x (212 y = x - x (1

www.riazisara.ir

87 .oUoMzkA cdq]

:A e An

nj An x= 4 jk ,Am j{ 4 A MBU nj y nAk" ,x = 2 ApA M a :Aj :j e An

,cd q AoMBM ,SwA 2 q ,j{ 2 oMAoM y C nj & Aq BU ,j nAoc xB] M Bq U U

/k{BM 2 q

?SwA Ak Be :SvU

-8 (4 -16 (3 -2 (2 -12 (1

:eA = 4-3. 7 = -16

/kwn JA] M pBM ,jo& JBveU U An Bdd q] nAk" ,iAj pA o A TLA

?SwA Ak jB JA] \ :SvU__ x < __ (454

94

__ x < __ (37494

__ x < 1 (214__ x < __ (134

74

:e

(n z)24) " x R : x n x n

25) " x R , " n N :

26) " x R+ {} , " n N : nx x

27) " x R- {} , " n N : nx x

28) " x R+ {} , " n Z- : nx + x

29) " x R- {} , " n Z- : nx + x

3) " x R : x Z

?jnAj JA] ka x + 2x + 7x = 3 jB :SvU

1 (4 nB{M (3 2 (2 (1

/k{BM cd kBM q x HAqA Am ,kTv cd 7x 2x 3 a :e

/S{AjoM An cd q] S AU AoMBM ,7x Z 2x Z \T nj ,x Z AoMBM

x + 2x + 7x = 3 x = 3 Lc MBc :

31) " x R :x2 x x2 = x x

32)

www.riazisara.ir

nBa~ oUoMzkA . 88

?SwA Ak Be , Bo :SvU

1 (4 1 (3 5 (2 25 (1

:e

/SwA cd (2) q ,h< T iL Am

33)

:nAj |BiSBe nj

?SwA Ak , MBU Aj :SvU

x __ Bx > __ (41232 x [m]VMAn ,m nAkJ Ak ApA M 8

m = -1 (4 -1 < m 2 (3 m < -1 (2 m -2 (1

?SwA nkJa UBn VMAn nAjSeBv 9

9p (4 5p (3 2p (2 p (1

:BM SwA oMAoM | x| + | y| = 4 UBn VMAn nAjSeBv 1

36 (4 12 (3 32 (2 18 (1

www.riazisara.ir

nBa~ oUoMzkA . 96

(2) 11 . 85 M SLX jk + 1 . 85 M SLX jk + 1 . 85 M SLX jk = 2.Ajni = 2

:& k& ]U

/jnAj JA] j"< ,{ M ]U BM (1) 2

(2) 3x + -x = x z :nCjB

__ z x = __ , k zJA] nBzM1x

1k

B B (2) 4

(3) 5x + 1 + -x + 2 + x = 7

(4) 6

:nCjB

cd cdoA(4) 7/jnAk JA]

(2) 8x2- 2x - m >

S = 5p - 4p = p (1) 94 x2+ y2 < 5

S= 5 - 32 = 18 (1) 14 | x| + | y| < 5

| x - a| + | x - b| = k S = 2k2 Mo SeBv :nCjB

/kAka poQ Sv{ pA HLBb ,JBB Ajo

www.riazisara.ir

97 .oUoMzkA cdq]

:3 SvU

?SwA Ak , VMAn pA yRAodU jke 1

< y 2 (4 1 y < 2 (3 y < 1 (2 y < 2 (1

:BM SwA oMAoM ,x ojBJ ] ApA M RnBL 2

(4 3 (3 2 (2 1 (1

?SwA nkJaRB~Th Bnd A eB M n~d ,x + y = 5 UBn VMAn SeBv 3

(4112

11 (3 36 (2 25 (1

?SwA op RnAk M , nAj 4

(4 (3 (2 (1

?SwA Ak (x + 1)2 BC ,k{BM 2(x2 + 1) = 4x Bo 5

(4x)2 (3 4x (2 x2 + 1 (1(4

?SwA Ak x + yBe ,k{BM y = 4 x = -3 Bo 6

2 (4 4 B 3 (3 2 B 1 (2 1 (1

:BM SwA oMAoM Be 7

3 (4 9 (3 1 (2 2 (1

?jnAj JA] ka ,__ = __jB 8x3x

2

4 (4 3 (3 2 (2 1 (1

?jnAj JA] ka 6x = 5xjB 9

7 (4 5 (3 3 (2 1 (1

:BM SwA oMAoM Be ,-sinx = 1 Bo 1

kz 'oU (4 -1 (3 1 (2 (1

www.riazisara.ir

nBa~ oUoMzkA . 98

(1) 1

(1) 2

(4) 3

x + y = 5 5 x + y < 6

jnin{B Svc SeBv = - = 6*62

5*52

112

(2) 4

kz oU :nCjB

(2) 5

:nCjB

(2) 6x + y = x + y B x + y = x + y + 1

x + y = -3+4 B -3 + 4 + 1 x + y = 1 B 2

:T (1) 7

2.Ajni = 2

(3) 8

k = -2 B -1 B x = 2k = -4 , -2 , jnAj JA] w

(3) 9

x < 5 k = B 1 B 2 B 3 B 4

x = , , , , JA] 565

125

185

245

(j{ 1 pA oTzM B` sin a) (3) 1-sinx = 1 1 -sinx < 2 -sinx = 1

www.riazisara.ir

99 .oUoMzkA cdq]

:4 SvU

:BM SwA oMAoM ,| x| 2 BL nj ,Bx nd nAj M SeBv 1

2 (4 (332

4 (2 3 (1

?SwA Ak , MBU goU Aj 2

Z- (4 N (3 Z (2 R (1

?SwA Ak f(__)Be ,k{BM Bo 3p4

3 (4 2 (3 1 (2 (1

?SwA nkJa cd q] BC ,k{BM ,Bo 4

726 (4 723 (3 724 (2 725 (1

?SwA Ak ,f(x) = x+ 4 + 3-x + 1 MBU joM 5

{7,8} (4 [8,9) (3 [7,8) (2 {8,9} (1

?SwA Ak MBU joM 6

[2,6) (4 [2,4) (3 [2,4] (2 [2,6] (1

?SwA MBU h a f(x) = sin(px) MBU 7

jo< Zp (4 jo< Zp (3 jo< (2 Zp (1

?SwA Ak fofMBU joM BC ,k{BMf(x) = x + -x Bo 8

{1} (4 {} (3 {,1} (2 {,-1} (1

?SwA Ak , jB JA] \ 9

[2,3] (4 [2,3) (3 [__,2) (253 [__,2] (152

?jnAj JA] ka , jB 1

jnAk JA] (4 nB{M (3 2 (2 1 (1

www.riazisara.ir

nBa~ oUoMzkA . 100

kz oU(2) 1

tTv SeBv = 4 * 1 = 4

kz oU

(2) 2

:nCjB

(2) 31.Ajni = 1

(3) 4= 724 - .Ajni = 723.Ajni

(4) 5f(x) = x + 4 + 3 + -x + 1 = x + -x + 8 = (-1B) + 8 = 7 B 8

(3) 6

A

/SwA jo< Zp MBU (3) 7x = k z f(x) = sin(kp) = (3) 8

(2) 93x = 5 5 3x < 6 __ x < 253

(4) 1

/SwA oA h< BvU Am

/koM [n qa pAyM jo oL pA jo

www.riazisara.ir

101 . oUoMzkA cdq]

:5 SvU

?SwA Ak , Be ,k{BM __ < x < p Bo 1p2

1 - __ (41a

1 + __ (31a

a - __ (21a

a + __ (11a


[9,27] (4 [___,__) (312719 [

__,__] (21913 (

___,__] (112719


[-1,] [1,2) (4 [-1,) [1,2) (3 [-1,1] [2,3) (2 (-1,) [1,2) (1

:BM SwA oMAoM (fofof)() Be ,f(x) = x - 1.5 Bo 4

-4. 5 (4 -5. 5 (3 -5 (2 -6 (1

:pA SvUnBL VMAn nAj 5

tTv (4 Mo j (3 tTv j (2 ji j (1

?SwA Ak -6xBe ,RnC nj ,k{BM Bo 6

-3 (4 -2 (3 2 (2 1 (1

?SwA { a [-1,2)BL n MBU nAj 7

ji j (4 ji nBQ w (3 ji nBQ j (2 ji nBQ (1

?SwA Ak ,x + .4 + x - 2.6 = 1 jB JA] \ 8

. 4 x < 1. 4 (4 2 x < 3 (3 1. 4 x < 2. 4 (2 1. 6 x < 2. 6 (1

?jnAj JA] ka , jBB 9

2 (4 1 (3 (2 nB{M (1


[-__,__] (45313 [-

___,___) (313-73 [-3,

___) (2-83 [-___,____) (1133

-13

www.riazisara.ir

nBa~ oUoMzkA . 102

(3) 1

(3) 2

B B (3) 3

(3) 4f() = -1. 5 f(-1. 5) = -3. 5 f(-3. 5) = -5. 5

(fofof)() = f(f(f())) = -5. 5

(2) 53! + 1! = 7 x = 3 , y = 1

x! + y! = 7 1! + 3! = 7 y = 3 , x = 1

3! + ! = 7 y = 3 , x =

x! + y! = 7 ! + 3! = 7 x = 3 , y = 1

(4) 6

(2) 7-1 x < y = ___jinBQx-2 jinBQ j x < 1 y = -xjinBQy = ______x

x-11 x < 2 y = kz 'oU

(1) 8

(1) 9

(1) (2) " x R : 5x + 1 3x + 2 + 2x - 1 /jnAj JA] nB{MuQ

(3) 1

www.riazisara.ir

103.oUoMzkA (MBvekB~U)jkkB~U

\Q~

(MBvekB~U)jkkB~U

jk A /kC Swj M yLcB ] M TMBY jk jq

\Q~ oUoMzkA . 104

:nAj ,jk kB~U o nj (5

m + n = p + q tm+ t

n= t

p+ t

q

m - n = p - q tm- t

n= t

p- t

q

A jpB ] ,jM 47 25 13 jAkA KUoU M MBve kB~U jnBa Tz \Q R] :SvU

:BM SwA oMAoM kB~U

37 (4 33 (3 39 (2 35 (1

5 + 14 = 8 + 11 t5+ t14= t8+ t11 t11 = 35

:nAj ,MBve kB~U o nj (6

m + n = 2k tm+ t

n= 2 t

k

\ BMSwA oMAoM ,tw ] oMAoM j BC k{BM jo< jk ,MBve kB~U R] jAkU o A ,oj RnBL M

/oiC A ] j

TvC (k{BM jk kB~U AT ] w ) kkM jk kB~U zU c b a jk w A so{ (7

b = ____ B a + c = 2ba+ c2/k c a jk j (MBve B) ,(MBve twA) jk twA An bSBe A nj

?SwA Akx ,k{BM jk kB~U AT ] w 4x + 5 2x + 1 x - 1 o A :SvU

-2 (4 -3 (3 2 (2 3 (1

(x - 1)+ (4x + 5) = 2(2x + 1) x = -2

x2 - (2a + 3)x + a = j ]nj jB Bzn UB?U Jo4Be h\ ,a pA nAkJ a ApA M :SvU

?kj MBve kB~U zU

-3 (4 3 (3 -2 (2 2 (1

2p = s + D 2xx = (x+x) + (x- x) 2xx = 2x x = 1

1 - 2a - 3 + a = a = -2 :Am k& hk jB nj TvBM zn A

:BM SwA oMAoM cosa + cosc ,k{BM qSLv nkP BM MBve kB~U AT ] w c b a Bo :SvU

2sinbcosq (4 2cosbcosq (3 2cosbsinq (2 2sinbsinq (1

a , b , c a + c = 2b

a - c = -2q , cosa + cosc = 2cos____ cos ____ = 2cosbcosqa+ c2a-c2

/kC Swj M op tMAn pA kB~U SLv nkc (j nAoc) & Znj MBve twAm ,b a jk j M Bo (8

d = ______b - am + 1

/jjo MBve kB~U ,Be T{n jAj nAoc b aM An BC BU jo jk m BLj B ,oj RnBL M

www.riazisara.ir


?SwA Ak kB~U SLv nkP ,Ajo Znj jk VwA [Q ,25 7 jk j M :SvU

(452

4 (3 3 (2 2 (1

d = ______ = _____ = 3b - am + 1

25-767 , , , , , , 25

B t7= t1 + 6d 25 = 7 + 6d d = 3t7t1/k q An kB~U ,k{BM d < o A j An kB~U ,k{BM d > o A ,jk kB~U nj (9

/kC Swj M op Bo< pA A ] n \ ,jk kB~U nj (1

/k{BM \ B M sum & A moe s

L jAkA \ Zp L jAkA \ ,n BU 1 pA L jAkA \ Bo< AU h< o

\Q~ oUoMzkA . 106

A ] oL ,kC Swj M sn= n2 + 2n nTwj pA A ] n h\ ,jk kB~U nj :SvU

?SwA Ak kB~U

tn= 2n + 4 (4 t

n= 2n + 2 (3 t

n= 2n + 1 (2 t

n= 2n + 3 (1

:e

?SwA Ak kB~U SLv nkP ,(R] h\) sn= n2 + 3n ,jk kB~U nj :SvU

4 (4 1 (3 2 (2 3 (1

:A An

:j An

oMAoM kB~UR] \ BC ,k{BM tw ] o A ,jo< BT R] jAkU BM jk kB~U nj (15

sn= R] jAkU * tw ] :BM SwA

:pA SvUnBLR] h\ ,-2 3 8 13 18 BT jk kB~U nj :BX

S5 = 5 * 8 = 4

:BM SwA oMAoM S2n+ 3 ,k{BM tn+2 = 1 Bo k{BM 2n + 3 jk kB~U R] jAkU :SvU

1n + 3 (4 1n + 1 (3 2n + 3 (2 2n + 1 (1

/SwA cd 2 q ,h< T iL

}iBM , tm= p + q - m a = p + q - 1 d = -1 BC ,(p q) , t

q= p t

p= q o A ,jk kB~U o nj (16

/tp+ q

=

:pA kUnBL jpAj ] SLv nkP , t7 = 5 t5 = 7 jk kB~U nj :SvU

(1(2(3(4

/SwA Swnj 2 q ,Lc T iL

t12 = & j{ kj jBvM ,d = -1 Am - 2d = 2 /& & pA An jB j

(p q) sp+ q

= BC k{BM sp= s

qBo ,jk kB~U nj (17

[Q \ B AT ] w \ X) MBve kB~U AT R] pA jo< jAkU \ Bo (18

M An R] A SwA oTM ,kAhM An kB~U pA /// jw ] ,A ] ,SLv nkc k{BM (AT ]

www.riazisara.ir

107 .oUoMzkA (MBvekB~U)jkkB~U

a - d , a , a + d AT ] 3 :Rn

a - 2d , a - d , a , a + d , a + 2d AT ] 5

/SwA jw ] a & k{BM T{Aj oBi M ,SBe A nj /oM o~ nj

?SwA Ak kB~U pw ]SwA 35 oMAoM MBve kB~U AT ][Q h\ :SvU

22 (4 11 (3 7 (2 8 (1

(a - 2d) + (a - d) + a + (a + d) + (a + 2d) = 35

5a = 35 a = 7

/& JBhTA a - 3d , a - d , a + d , a + 3dRn M An BC SwA oTM ,k{BM MBve kB~U M jk nBa Bo (19

4d 3d { M BC & Rn ,Rn A nj ,kkM MBve kB~U zU AqA Bc WX \A Bo (2

/jM kAi 5d

o4 kB~U A(p+ q)] BC ,k{BM oMAoM Aq] oMAoM q BM Ap] oMAoM p Bo MBve kB~U nj (21

ptp= qt

q t

p+ q= : /SwA

SLv nkc B BM MBve kB~U pBM ,\T ,&

\Q~ oUoMzkA . 108

:1 SvU

?SwA Ak A ] 19 h\ ,SwA nAoPoM t3 + t9 + t11 + t17 = 4 VMAn MBve kB~U nj 1

19 (4 18 (3 17 (2 16 (1

:BM SwA oMAoM a2 ,Ajo Znj -2 SLv nkP BM MBve VwA w a-1 4 jk j M 2

16 (4 9 (3 4 (2 3 (1

?SwA Ak pw ] ,6 7 8 /// 54 BLj nj 3

3 (4 28 (3 26 (2 24 (1

R] h\ RnA nj ,SwA 18 C pw ] w h\ 39 MBve kB~U R] jAkU 4

?SwA nkJa

351 (4 468 (3 72 (2 234 (1

?SwA Ak ,kB~U A nBa] ,Sn= 2n2- n :nAj jk kB~U nj 5

14 (4 15 (3 13 (2 12 (1

:BM SwA oMAoM t6+ t1+ t15 nAkJ ,SwA nAoPoM tn+2= 3n + 6 VMAn MBve kB~U nj 6

111 (4 93 (3 83 (2 73 (1

?SwA AkA = 12- 992+ 982- 972 + ... + 22-12 Be 7

45 (4 45 (3 55 (2 55 (1

?SwA Ak 9 pA oTnqM ]Tvh ,SwA 7 SLv nkP 17 A ]MBve kB~U nj 8

129 (4 128 (3 126 (2 127 (1

?j{ 78 oMAoM tn= 4n - 1 ] BM jk kB~U pA ] ka h\ 9

8 (4 7 (3 6 (2 5 (1

h\ ,k{BM oTTBw 15 WX A oU 6 o A ,kj jk kB~U zU ,AqA BP WX hUA 1

?SwA Ak C oj U j 6

25 (4 23 (3 21 (2 19 (1

www.riazisara.ir

109 . oUoMzkA (MBvekB~U)jkkB~U

(4) 1

(3) 2 t5 = t1 + 4d a - 1 = 4+ (-8)t1= 4

a = -3 a2 = 9t5 = a - 1

(4) 3tn= t1 + (n - 1)d 54 = 6 + (n - 1)(1) n = 49

jw ] = \Q SvM ] = t1 + (n - 1)d = 6 + 24(1) = 3

(1) 4a + b + c = 18 3b = 18 b = 6 jw ]

S39 = 39 * 6 = 234

(2) 5t4 = s4 - s3 = 28 - 15 = 13

(3) 6

(2) 7

= 1 + 99 + 98 + 97 + /// + 2 + 1 = = 551*112

(3) 8

n > 127.Ajni n 128 Jt n = 128

(2) 9

j{ 78 A BU Bz\ LwBd ,kB~U R] T{ BM AU ,Sv nqM i jk ,78 a :j xn

3 + 7 + 11 + 15 + 19 + 23 = 78 /kwn JA] M ,q

(2) 13d , 4d , 5d 5d = 15 d = 3

3d + 4d = 7d = 21

/SBe T{m pA C qo{ M SB T{C SwA c SM BM

www.riazisara.ir

\Q~ oUoMzkA . 110

:2 SvU

w w h\ RnA nj ,k{BM 1 oMAoM ,T? SvM T? R] h\ ,MBve kB~U nj 1

:BM SwA oMAoM kB~U A ]

265 (4 156 (3 175 (2 165 (1

:BM SwA oMAoM d + xRnA nj ,kTv jk kB~U M dSLv nkP BM XX BAp 2

(SwA WX Ap oTa x)

4 (4 7 (3 6 (2 5 (1

?SwA Ak Bzn pA ,kj MBve kB~U zU ax2+ 2bx + c = jB nj c b aKAoU 3

___ (4-ba

_ (3ca

-_ (2ca

1 (1

:BM SwA oMAoM (1 + x)+ (4 + x)+ (7 + x) + ... + (28 + x) = 155 jB JA] 4

1 (4 9 (3 3 (2 (134

?SwA nkJa jp j R] UB?U ,5 SLv nkP BM jk kB~U nj 5

55 (4 45 (3 5 (2 35 (1

?SwA Ak kB~U SLv nkP , t9 = 11 nAj MBve kB~U nj 6

3 (4 2 (3 6 (2 4 (1

oMAoM SLv nkP M A ]SLv ,SwA nBa] oMAoM w ,z{ ] oMAoM j ,MBve kB~U nj 7

:BM SwA

(423

2 (3 (212

1 (1

?SwA Swnj q Ak ,kTv dSLv nkP BM jk kB~U Am AnR] h\ KUoU M tm tn8

tm+ n= t

n- md (4 t

m+ n= t

n+ md (3 2 t

m+ n= t

m+ t

n(2 t

m+ n= t

n+ t

m(1

?SwA Ak ,kB~U A R] h\ ,SwA 43 oMAoM pw ] ,jnAj ] 21 ,jk kB~U nj 9

256 (4 39 (3 93 (2 83 (1

?SwA Ak C An ] , tn+ 1= tn+ 3 t1= 4 MBve kB~U nj 1

5n - 1 (4 2n + 2 (3 3n + 1 (2 n + 5 (1

www.riazisara.ir

111. oUoMzkA (MBvekB~U)jkkB~U

(1) 1t7+ t27 = t1 + t33 t1 + t33 = 1

S33 =___(t1 + t33) = 165332

(2) 23d , 4d , 5d 12d = 18 d = 15

x = 3d = 45 d + x = 6

(2) 3

/SwBU 1 R] jAkU & SwA c\A (4) 4

tn= __(t1 + tn) 155 = (1 + x + 28 + x) x = 1n

212

(3) 5t19 - t1 = 9d = 9(5) = 45

(3) 6

(1) 7__ = 1ad

2 t6 = 3 t4 2a + 1d = 3a + 9d a = d

(3) 8tm+ n

= a + (m + n - 1)d = a + md + (n - 1)d = tn+ md

(2) 9S21 = 21 * 43 = 93

(2) 1tn+ 1 = tn + 3 tn+ 1- tn = 3 d = 3

2 q ,cd q BU Am ,SwA d = 3 a SwA d B ,nKo\ , tn An ] o< nj ,Aj

/SwA

/jo kAi nB BLU SwA TP AjoL

/k BM An j~J ,BLoe BU pAyM Rw B

/SwA Swj ,B] nj Bqa oUJBB

/Sv BV pA oUBM U m

www.riazisara.ir

\Q~ oUoMzkA . 112

:3 SvU

,SwA 4 oMAoM oiC ]1 h\ 1 oMAoM A ]1 h\ ,]5 B{MBvekB~U nj 1

?SwA Ak oiC A R] h\

25 (4 5 (3 4 (2 1 (1

Ap oTnqM o A ,kj zU MBve kB~U BAp pAkA ,SwA 54U [Q BAp h\ 2

?SwA Ak oTa Ap pAkA ,k{BM 136

84 (4 8 (3 78 (2 94 (1

?jo Znj AU SLv nkP BM MBve VwA ka 51 11 jAkA M 345

32 (4 35 (3 5 (2 49 (1

?SwA Ak ( 8*8) JoU k] jAkA h\ 4

1496 (4 1296 (3 1246 (2 1396 (1

h\ ,k{BM 69 oiC ] w h\ 15 oMAoM A ] w h\ ,A]12 MBvekB~U nj 5

:BM SwA oMAoM R] BU

168 (4 2 (3 186 (2 366 (1

?SwA nkJa RnA nj ,kj zU jk kB~U ,1 logx2 -2 jAkA 6

(413

- (316

(216

- (113

?kj jk kB~U zU x3- 3x2 + (2m - 1)x - 5 = jB Bzn ,m pA nAkJ a ApA M 7

m = -4 (4 m = 4 (3 m = __ (212 m = 5 (1

An oj MBve kB~U zU BeR] , LBUA keA 5 MBve kB~U R] pA o M o A 8

/SwA A SLv nkP ///C SLv nkP kj

BM Bv (4 pA oT& keA 5 (3 oMAoM 5 (2 pA oTzM keA 5 (1

?SwA Ak (x + 4)RnA nj ,SwA 546 oMAoM o? q] M ,xjk A Jo4 13 h\ 9

12 (4 1 (3 8 (2 6 (1

?SwA Ak ,T{n A jk j ,SwA 327 oMAoM QnjQ jk 143 jk oTnqM 1

18 (4 194 (3 195 (2 185 (1

www.riazisara.ir


& ](3) 1

:T nCjBt2 + t49= t1+ t5X , m + n = p + q tm+ tn= tp + tq

(3) 2

(1) 3BtwA jAkU

:j And = _____ __ = ______ m = 49b-am+ 1

45

51-11m+ 1

(3) 4(1 + 2 + 3 + /// +8) (1 + 2 + 3 + /// + 8) = * = 12968*92

8*92

(4) 5

]

3(t1+ t12) = 84 t1+ t12 = 28

S12 =___(t1 + t12) = 6 * 28 = 168122

(2) 6

(3) 7x1 + x2+ x3 = -__ = 3 3x2 = 3 x2 = 1 jB znb

a

1 - 3 + (2m - 1) - 5 = m = 4

(4) 8

(3) 9x + 2x + 3x + /// + 13x = 546 x(1 + 2 + 3 + /// + 13) = 546

x (______) = 546 x = 6 x + 4 = 113*142

(2) 1327 - 143 = 184 A ] 184 + 9 = 193

:nAj cA njt1= t1+ (n - 1)d t1= 184 + 9(1) = 193

/ B ,Bqa {nSvP M

www.riazisara.ir

z{~ oUoMzkA . 114

z{~

wkkB~U

Swj M yLcB ] nj TMBY jk k{ Jo\ BM xA] o & ,k{BM ""e jAkA pA T{n wk kB~U

/kj Bz r B q BM AoC k kB~U SLv nkc ,k{BM o4 'Bh & An SMBY jk A /kC

a , aq , aq2 , ... , aq9 , ... , aqn-1 , ...

...,tn

,...t1, ...t3,t2 ,t1

/k kB~U ( An ]) ] An tn

/knAocoM op jMAn Bo< wk kB~U nj

1)

tn= t1q

n-1 B tn= aqn-12)

?SwA Ak kB~U z{ ] tn= 3(2n-1)wk kB~U nj :BX

t6 = 3(25) = 96

?SwA Ak kB~U \Q ] , wk kB~U nj :SvU

(354

(2564

(1532

(4

n = 1 t5 =__54 :e

M op tMAn pA kB~U SLv nkc ,k{BM tn tmo A oj RnBL M :nAj wk kB~U o nj (3

/kC Swj

,(k{BM jo< ,(m-n) Bo)

,(k{BM Zp ,(m-n) Bo)

?SwA jk Ak q k{BM KUoU M wk kB~U T? w R] Bo :SvU814

14

3 (4 2 (3 2 (2 3 (1

www.riazisara.ir

115.oUoMzkA wkkB~U

m + n = p + q tm.tn= t

p.tq

:nAj wk kB~U o nj (4

m + n = 2k tm.tn= t2

k:nAj wk kB~U o nj (5

BM SwA oMAoM tw ] nm\ BC ,k{BM jo< jk ,wk kB~U R] jAkU o A ,oj RnBL M

/oiC A ] j Jo`Be

(k{BM wk kB~U AT ] w ) kkM wk kB~U zU c b a jk w A so{ (6

/k c a jk j wk B B twA An bSBe A nj B ac = b2 :TvC

?SwA Ak jAkA wk VwA :SvU

(232

(1 (434

(3

/kC Swj M op tMAn pA kB~U SLv nkc (j nAoc) & Znj MBve twAm , b a jk j M Bo (7

,(k{BM jo< ,(m+ 1) o A)

,(k{BM Zp ,(m+ 1) o A)

3 , , , ,48 ?kMBM An BC ,Ajo Znj wk VwA w ,48 3 jk j M :BX

/kC Swj M op Bo< pA A ] n \ ,wk kB~U o nj (8

B

nkc (Zp n) k{BM Snwk kB~U nj o A }iBM :nAj wk kB~U o nj (9

/j{ JBve op tMAn pA kB~U SLv

oMAoM kB~USLv nkP ,SwA 126 A ]y{ h\ 112 A ] w h\ ,wk kB~U nj :SvU

:BM SwA

(414

(323

(212

2 (1

www.riazisara.ir

z{~ oUoMzkA . 116

Sn= na BC ,k{BM q = 1 Bo wk kB~U nj (1

:j{ LwBd op tMAn pA R] \ ke ,k{BM BTB ,kB~U R] jAkU | q| < 1 o A (11

:BM SwA oMAoM 1 + + + + //// LoJU Be :SvU12

14

18

(487

(343

2 (2 (132

?SwA Ak C R] h\ ke ,k{BM wk kB~U ] :SvU

- (443

(343

- (232

(132

LwBd tMAn pA R] RBMo \ ke ,k{BM BTB kB~U R] jAkU | q| < 1 o A (12

/j{j

knAj xjo nj j"< & BC oB~ Rn M Bov& M :jBw JBT nBzA ov

/k jBw JBT nBzA ov&

nj T{Aj xjo nj & ,BC oB~ ,. 27 Rn M Bov& M :Ko JBT nBzA ov

/k K& o JBT nBzA ov& ,xjooA

& v"U Zoh oM An Rn , ov& nj Bo Aj :Ko B jBw JBT nBzA ov k LnBT ov23

. 6 jBw JBT nBzA ov& (k& kU) k ov& An

117.oUoMzkA wkkB~U

JBT ov& ,Be nBzA ov& ,k{BM T{Aj j] 5 2 oA BB j"< ,k{ jAj

z{~ oUoMzkA . 118

?SwA Ak nBzA ov k LnBT ov :SvU

(413721

(315111

(24333

(11411

:e

tn= S

n- S

n-1 /SwA LwBd MBc op tMAn pA tn ,MBve kB~U kB ,wk kB~U nj (15

/j{ JBve op tMAn pA ,wk kB~U o A ] nJo`Be (16

?SwA Ak C A ]S? Jo4Be ,k{BM 16 wk kB~U nBa] :S

Documents

Riaziat Payeh Daraee[Www.riazisara.ir]