Soal untuk matematika cepat tepat

(

)

(

)

2

1

1

y

=

x

+

p

−

x

+

p

+

p !

" " # $ #%

& ' " x2 −2px+

(

p2−4p

)

=0 " " ! (

" p ! " " # $ #%

) ' $ ( "

1

1

1

...

2

1

2

+

+

=

−

* '

(

a b c

, ,

)

$ # $ ! " " $ +

• −2x+ − = −y z 3

• x+2y+ =z 8

• x−y= −1

" a+ + =b c ...

, '

x

₁

x

2 $ # - "

(

5 2 log

−

x

)

log

x

=

log1000

( "

2 2

1 2 ... x +x =

. " 2

(

)

(

)

1

5

0

x

+

a

−

x

−

a

−

=

" " ! -

x

₁

x

2

' a>0 2 2

1 2 1 2 12

x x +x x = ( "

a

=

...

/ $ "

(

x

2

+

1

)

2

−

12

(

x

2

+

1

)

+

20

≤

0

$ #%

0 " " ! 1 $ $ 2 3 $ # "

3 $ $ "1 ( # 3 $ # ! (

4 $ " 5 z=3x+5y ! " " # ! x+2y≤10( x+y≤6( x≥0(

0

y≥ $ #%

$ "

(

)

(

)

2

2

2

6

0

20

x

x

x

x

x

−

+ −

>

+ −

$ #%

' 6

2

1

a

P

b

=

−

$ #

1

3 6

3

P

c

−

₌

−

( "

c

$ #%

& '

1

3

2

4

A

=

2 3

0 1

B

=

( " " "

(

A+B

)

2

$ #%

) - 1 " $ #

U

_n ' U₄ = p2+1( U₁₀ =2p2+4(

7

16

U

=

( "

U

₁

=

...

* ' p>0,q>0(

p p

,

+

q

, 4

(

p

+

q

)

" "1 " ( " q=...

, $ " $ $ 7 ! 3 $ # 8

$ ! " "1 $ $ " $ $ # 3

5 ' $

" $ ) $ $ ( " $ ! " "1 & $

$ #%

. 2 1 ) ! $ " , 1

1 . ' 1 !

1 ( " 1 ! ! 9 $ " 1

$ #%

/ '

α

" " # cos2 2 sin

(

)

sin2

(

)

11

2

α

+

π α

− =

π α

+ + ( " sin

α

=...

0 2 $ " ∆ABC( 3 : 0( 8 :

4 2

0

45

ABC

4 " y= x+ x $ # y'=...

&

2

2 4

16

lim

...

5

9

x

x

x

→

−

=

−

+

& '

y

=

g x

( )

$ # 6

( )

2

2

f x

=

x

+

( x≥0( " # $

g $ #%

&&

θ

! " " # cos2 cos4 6 ... 1

3

cos

θ

−

θ

+

θ

+ =

$ #%

&) ! 3 $ " p # ( 1 ! # !

1500

4

p

40

p

+

−

3 # ' 1 ! " " " ! 1 $ # R 3

#( " R=...

&* "

A

1

a

b

c

=

( 3 1 $ 1, ,a c " "1 1 "

1 3 "$ # ) 1 $ 1, ,b c " "1 1 " ( "

detA=...

&, 8 #

2 3 2 1 8

1 5 4 6 4 2

7

27

54

84

x y z

xy z

x y z

x y z

− −

− −

×

− − $ #%

&. $ k ! " " # "

x

a

(

x

a+1

) ( )

a

x

a 1−a

=

x

k−1 $ #%

&/ 2 1 " ax2 +bx+ =c 0 ! "

$ * ! $ a b c, , " " $ # 1 %

&0 # & &( " " 1 !

&4 ' y=bx−a " " 1 $

y

=

ax

2

+

bx

+

(

a

−

2

b

)

( )

1,1

(

x y

0

,

0

)

( "

x

0

+

y

0

=

...

) ₂ 3 ₂ 5

3 2 4 3

x − x+ < x − x+ 1 $ 6 $%

) " $

x

! " " #

1 2

−

x

≥ −

2

x

$ #%

)& ABCD EFGH. $ # 1 # 1 $ - $ ! 1 1

8 : ) 9"( : . 9"

θ

$ # 1 =

8 2 sin 2

θ

=...

)) 8 > ? $ 3 @ l " "1 0

45

> 300 ? l $ #%

)* 2 # 1 ABCD EFGH. # =( # 2 ( #

8 # 1 $ ∆APS $ ! ∆APS

1 ABCD $ #%

), ' 3 x−1, 2x−1,3 , 5x x−3, 4x+3, 6x+2 $ # 0(

" " ! $ #%

). 2 " # ( 8( ( 2 " ( &( )( * 1 $ "

" 1 $ ! " $ # ) # # "1

$ " " " 1 $ # " "1 )&*( " 1 ! ! $

" ! 1 $ #%

)/ 2 $ " 1 1 ! # " ( &, 9 " $ 1

$ 8 ! ! ! 3 $ #%

)0 ' 0 1

cotan54

x

= ( " 0

)4 $ " 5

12 cos 5sin 16

d

x− x+ $ # ) $ d $ #%

* "1 8 1 $ a b ...

b

+ =

* 8 $ tan

θ

+tany= p p≠0( "

(

)

cos cos

...

sin

y

y

θ

θ

+

=

*& 2 # 1 # $ ! " ! "1 ! 1 3

3

y= x

' $ $ y= 3( " 3 !

$ #%

*) 2 #

( )

2

3

4

h x

=

x

+

x

−

" $ # 9

( )

4 3 2

2

14

g x

=

x

+

x

−

ax

−

x b

−

g x

( )

1 x+1 1 %

** 1 ! x4+ax3+2x2+bx+5 3 1 x−2 1 / 8 $ 1

1 x+3 1 0& $ a2−4ab+4b2 =...

*,

f x

( )

= +

x

2

x>0 g x

( )

15

x

= x>0 2 "

(

1 1

)

( )

1

f− g− x =

x

=

...

*. '

f x

( )

= −

2

x

A

g x

( )

=

x

2

+

1

h x

( )

=

3

x

( "

(

h g

f

)( )

3

=

...

*/ lim 2sin1tan1 ...

x→ x _{x x}=

*0 4

cos sin

lim ...

2 2

x

x x

x

π

π

→

−

= −

*4 8 $

y

=

xe

2x( " dy ...

, 2 #

f x

( )

=

ax b

+

(

f x

( )

<

0

0≤ ≤x 4

f x

( )

≥

0

x≥4 '

( )

8

0

0

f x dx= $ # ! 1 $ #

y

=

f x

( )

,

x

=

0,

x

=

8

"1

x

$ # .( "

f x

( )

=

...

, $ " " " z= +x 3y ! " " # " 3x+2y≥12(

2 8

x+ y≥ ( x+y≥8 y≥0 $ #%

,& ' k " " # "

1

1

0

1

0

1

k

x

y

k

−

=

−

( " x+y=...

,) 8 ! M x y( , ) $ # " ! 1 "

2

1

1

0

−

$ 3

3

2

0

1

−

$ # M'( 50,5)− ( " M

$ #%

,* $

x

! " " # "

2

3

log

1

3

log 81

log 3

x

x

+

>

$ #%

,, " x2−2x+ =3 0 " " ! -

α

β

"

1 -

(

α β

+

)

1

1

α

+

β

$ #%

,. " _x23_+y23 _{=1 ! 1 !}

2 4 B

,/ ! $ + 32x+2+8.3x− =1 0 $ #%

,0 # $ +

1

1

1

...

1

1

+

2

+

2

+

3

+

3

+

4

+

+

99

+

100

B

. @ ! $ 6

y

=

tan

x

,1 4

π

" " "1 y %

. ' + CaC : &( CbC : ) Ca D bC :

5

$ # $ Ca E &bC

.& 1

ln

...

e

xdx

=

.) " ! - ! 1 1 $

-" x2−3ax+2a2 =0 " " "1 y P

$ # P B

.* @ 1 $

y

=

x

2

−

x

(

1, 0

)

" "1

"1

x

1 %

., = $ # +12+22+32+42+52+... 20+ 2 B

.. $

x

! " " # 22x+2−2x+3+ =4 0 B

./ 6

y

=

12

−

x

2 $ # $ " 5 3 ABCD 3 8 1 # "

"1

x

2 $ 6 1 B

.0 2

0

cos

...

x

xdx

π

=

.4 " 2

9

0

x

−

x

+

p

=

" " ! -

α

β

'

α

=

β

+1( "

$ # $

α

β

B

/ = " ! $ " + 1₂ 12 27 0

3 x −3x + = $ #%

/ = $ # + 1 1 1 ... 1

/& 1 ABCD EFGH. 9" #- # = $ # 3 82 B

/) @ 6

y

=

x

2

+

5

! 3 3 12x−y=17

" ! 6 %

/* 2 " 1 1 " " -"

3 5 9" $ 1 ! 9" $ 24m2 3

! $ " " $ # 5 ! B

/, 2

3

0

sin

xdx

...

π

=

/. '

f x

( )

=

x

3

−

4

x

2

+

2

x

+

9

(

(

f−1

)

' 2

( )

B

// tan 750−tan150 =...

/0 0

lim

...

1

1

x

x

x

x

→

_{− −}

₊

=

/4 1 # 3 " " $ 3 $ 1 " -" 5

9 5 $ # $ " 5 3 1 B

0 $ # $

x

! " " # + 2

log

(

3

log

(

2

log

x

)

)

=

1

0 $ # $

m

" y= +x m " ! 1 $ 1 2 2

2

y= x − B

0& = $ # + cos 20.cos 40.cos 60.cos 80 B

0) F 1 1 "1 # 3 $ 3

7

cm

s

$ # $ 3

0* $ # $

2

2

2

x dx

−

"

x

$ # 1 $ 1 $ $ 1# 9 $

"

x

B

0,

lim

1 cos

2

...

n

nr

_n

π

→

−

=

0. 2 # $ 2 2

4

8

0

x

+

y

−

px

−

y

+ =

" $ $

(

4, 6

)

= $ # 3 -3

! B

0/ 1 ABCD EFGH. '

α

$ # ! 1

@ 1 82@( " sin

α

=...

00 ' 3 "$ # # + a 1 1 1₂ ...

a a

+ + + + $ # 4a( " $

a

=

...

04 '

(

)

(

)

log 1

log log 2 2

log 1

log 4 1

x

b

a a

a b

− =

− ( "

x

=

...

4 '

...

3

x x

x

x

= ( "

x

=

...

4 $ - " 7 $ # ,, ' 1 $ , 7 (

$ - " 3 ,) = $ # - $ , 7 1 B

4& ' 23a =5( 23b =10(

1

2

b a

c

₌

− 16 2

2

x

=

16

x 9 $ # "

1 " # 5( " # 3 "$ # 1 B

4) = $ #

(

)

2 2

2

5

6

0,

1

x

x

x

x

R

x

−

+

≥

∈

−

B

4* 8 $

f x

( )

=

a x

(

−

1

)(

x

−

3

)

f

' 1

( )

= −

4

( $ # 9

f x

( )

B

4, 8 $

( )

3 2

4 7

x f x

x

− =

− + ( $ # $

( )

4. $ # 1 -1 $ 5 " 00 ≤ ≤x 900 ! " " # " 2

tan

1

1 sec

x

x

=

+

B

4/ 8 $ x=0,999... 0, 666... 0, 444...+ + ( " $ tan 115

π

x $ #%

40

( )

3

f x

=

x

+

ax b

+

a≠b ' 6

x

=

a

3 3

6 x=b( " $

f

( )

1

=

...

44

(

)

2

3 2

0

4

4 cos

lim

...

7 sec

1

x

x x

x

x

x

x

→

−

+

=

+

−

" 1 $ #

96 3

9"2 = 3

" 1 B

8 $ x 1 3

x

+ = ( " x4 1₄ ...

x

+ =

& f x

(

+x−1

)

=x3+x−3( " $ #

f x

( )

B

) '

x x x x x

₁

,

₂

,

₃

,

₄

,

₅" ! $

5 4 1 3 2

2.3 x−3 x+ +3x+5.3 x+2.3x−12=0 $

x

₁

+

x

₂

+

x

₃

+

x

₄

+

x

₅

=

%

* 1 ! - ! , $ 1 # !

-2

2 4 0

x + x− = B

, 2 # 2

3 0

x − x+ =n " " $ -

x

₁

x

2 '

(

)

(

)

2 2 2 2

1 2 1 2 1 2 1 2

1 3

x +x = − x +x − x x +x x " $ # " !

1 " $ $ ;&( 0< B

.

(

)

2

(

)

1

2

1

12

0

p

+

x

−

p

+

x

+

=

" " $ -

α

β

'

2 3 3 2

18

/ 20− 20− 20 ...− =...

0 2

2 3 0

x −ax+ a− = " " $ -

α

β

' 2 2

35

α β αβ

+

=

("

a

:%

4 $ # ! - ! $ 1 # ! + x2+2x− =1 0

' 3 "$ # - " x2−3x+ =n 0 "

3 "$ # - " 2

0

x + − =x n ( " $

n

:%

2 # x2+bx+ =8 0 " " $ -

x

₁

x

₂

x

₁

>

x

₂

.

'

1

,

2

, 2

1

x x

x

" "1 1 " ( $ # $ bB

& ' 3sinx−4 cosx=0( " $ sin 2x:%

) '

sin

sin

3

2

A

+

B

=

cos

cos

1

2

A

+

B

=

A≥0 B≤180

" E8:%

* '

α

β

$ 9 ( cos

(

)

1 3

2

α β

− = cos .cos

α

β

=1( "

(

)

(

)

cos cos

α

β

α β

+

− :%

, 2 $ " 8 ( (1 9 $ # - ! ' tan 3

4

a=

4 tan

3

b= ( " 9 : %

. ' sin 2 1

4

α

= ( sin 2 1 4

α

=

(

sin

α

+

cos

β

)(

cos

α

−

sin

β

)

=

0

( " $

(

)

/ 2 # 2 sin3x−cos2 x−2 sinx=0 0 ≤ ≤x 2

π

( " 3 "$ #

! $ " 1 $ #%

0 ' "

20

6

12

2

5

3

2

4

A

=

a

+

a

−

a

−

$ # " $ ( "

a

:%

4 ' ;&( &( <( G; ( ( &<( F; ( *( )< " $ GF:%

& ' a=1, 6666.... b=2, 7777....( " $ # $ a

log

b

B

& 2nlog1944₌n log(486 2)

( " n6:%

&& 5 + 2log log 22

(

x+1+3

)

= +1 2logx B

&)

0

sin 2 sin 6 sin10 sin18 lim

3sin sin 3

x

x x x x

x x

→

+ + −

= −

&*

[

]

sin 3 (1 cos ) 0

lim sec

x

x x

x

x

−

→

=

&,

0

1 cos sin lim

1 cos sin

x

x x

x x

→

− +

=

− −

&.

2

1 cos sin lim

1 cos sin

x

x x

x x

π →

+ −

=

− −

&/ ₃

0

tan sin lim

x

x x

x

→

−

=

&0 ₂

0

1 2 cos cos 2

lim ...

x

x x

x

→

− +

=

&4 # +

4 4 4 4

4 4 4 4

(61

324)(73

324)(85

324)(97

324)

...

(55

324)(67

324)(79

324)(91

324)

+

+

+

+

=

+

+

+

+

) # + 3

2.4.8 4.8.16 8.16.32 16.32.64 ...

_...

1.3.9 2.6.18 4.12.36 8.24.72 ...

−

−

−

−

=

−

−

−

−

) = $ # + 1 3 5 7 ...

)& 3 "$ # ₂1 ₂1 ₂1 ... 3 +1+4 +2+5 +3+ B

)) = $ # + 1 1 1

cos 40 +cos 80 −cos 20

)* 3 "$ # + ₂3_{2 2}5 _{2 2}7 ₂ ... ₂29 ₂

1 .2 +2 .3 +3 .4 + +14 .15 B

), ' 111 1

2

x= − ( $

(2

x

5

+

2

x

4

−

53

x

3

−

57

x

+

54)

2004 B

). The expression4 3 ₊21 ₋6

x

x has the same remainder when divided by (x−a)or by(x+b)where

a

≠

−

b

. Find the value ofa2 +b2 −ab

!

137. The equation x3 +ax2 +bx+c=0has three distinct integral solutions and

c

is a prime number. Find the largest possible value of

a

.

138. Given that

a

is a root of the quadratic equationx2 −3x+1=0. Find the value of

1

8

2

2 4

+

−

a

a

a

139. If

a

a

a

...

=

a

+

a

+

a

+

...

, find the value of

a

140. For positive real

x

, 2 . . .

x

x

xx

=

.Find all possible values of

x

For question 39, 40, and 41

Let a,b,cbe the roots of the equationx3 −2007x2 +2008x−2009=0. Find the value of

141.

(

)(

)

)

(

)

)(

(

)

(

)

)(

(

)

(

b

c

a

c

b

a

ab

a

b

c

b

a

c

ca

c

a

b

a

c

b

bc

−

−

+

+

−

−

+

+

−

−

+

142.

)

)(

(

1

)

)(

(

1

)

)(

(

1

b

c

a

c

c

a

b

c

b

b

c

a

b

a

143.

Evaluate the following limits

,. $ " $ 3 2

x

+

px

+

qx

+

r

- !

(

x a

−

)(

x

+

a

)(

x b

−

)

(

# 1 ( H( B

,/ "1 x1000−1 $ # x+1 $ #%

,0 "1 x1000−1 $ #x2 −1 $ #%

,4 ' ( 1( 9 $ # - " 3x3−x2−5x+ =6 0( "

$ +

(

2 2 2

)

a b c abc a b c

+ +

+ +

. = $ # + 26 6 13 4 8 2 6 2 5+ − + − E 26 6 13 4 8 2 6 2 5− + − + B

. = $ # + 3

45 29 2

+

+

3

45 29 2

−

B

.& $

x

! " " # 3 3

2

x

+

x

− =

4

2

B

.) '

4

3

1

2

₋

−

x

x

: ( $ # $

.* 2 # - $ $ " $ : .()

" 1 ! : (& ' $ # $ $ " "1 #

$ $ $ " - $ ! 1 " 3 0(.

" 1 ! " 3 &(* 8 # $ E

., ' ; < : E

....

2

1

2

1

2

1

+

+

+

x

x

x

I $ # $

( )

99 . ' 99

( )

f

f

.. ' ; < :

3 6

+

x ; < :

./ $ # 3 "$ # D 3 1 + ; E & D <._%%

.0 2 1 # " # 1 #7 3 "$ # * " ( * :

/ 0 : ,0 ( " $ # 1 "

.4 $ # 1 D 1 $ ! " " # " +

) D ) E ) &_{D )} )_{E % J .}

/ $ # $ " " " + : 3 ) D 13 9 ) E 0

/ 2 # " + :

3

4

2

1

:

1

0

0

1

$ ! " ; D < " " $

/&

1

1

sin

lim

0

₋

₋

→

_x

x

x : %

/) $ # $ ! " " # +&_$ *_{$ E}*_$ &_{$ : &}

/* 2 # ; < :

x

1

( ; < " ! ; <

$ # $ ; <_{; <}

/, ' & $ # D " +&$ ;4 D & < : ) D (

$ # $ E &

/. $ # 3 * D ) D / : 0 D . E :

/0 # $ # "1 1

' # +

+ 8 : & + ,

8G + G : ) + *

8 # F + F K

/4 $ # 1 D 1 $ ! " " # " +

._{$ ;} &_{D < <}

0 ' : L )_{E .} &_{E ) ( " $ # $ +} & _{M D & N E &}

0 3 D 3 1

$ " ∆ 8 ∆ 8 " :

8 ' 8 : &

3

$ # $ # !

0& 2 # )_E )_{: * :}

3 2

" $ # 1 $ $(

$ # $ E

8 G

0) $ # $ D 9 3 # & :

2003 2002

4 5

π

J J

4 9

π

0* 6 &_{D ; D <}&_{: " " ! 1 #}

I ' 1 " $ $ ; ( < ! ; (

<( " $ # $ D

b a

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