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(1)

ARKUSZ PRÓBNEJ MATURY Z OPERONEM

MATEMATYKA

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LISTOPAD 2013

**3(6(/='$-k&(*2**

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x x

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Matematyka Poziom rozszerzony

1 Listopad 2013

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MHVW]JRGQ\]SRGDQ\PVFKHPDWHPRUD]LQQHSRSUDZQHRGSRZLHG]LZQLPQLHSU]HZLG]LDQH

Numer

zadania 0RGHORZHHWDS\UR]ZLÈ]\ZDQLD]DGDQLD Liczba punktów

1. 3RVWÚS

]DSLVDQLHW\ONRZDUXQNöZx x₁· ₂>0 i x₁+x₂>0

SNW

,VWRWQ\SRVWÚS

]DSLVDQLHZDUXQNöZ% >0 i x x₁· ₂>0 i x₁+x₂>0

SNW

3RNRQDQLH]DVDGQLF]\FKWUXGQRĂFL

]DVWRVRZDQLHZ]RUöZ9LHWHĳDLZ\]QDF]HQLH

% =12m²+5, x x₁· ₂=m⁴+1, x₁+x₂=2m²+3

SNW

5R]ZLÈ]DQLHEH]EïÚGQH

]DXZDĝHQLHĝHZV]\VWNLHZDUXQNL

% =12m²+5>0, x x₁· ₂=m⁴+1>0, x₁+x₂=2m²+3>0 ]DFKRG]ÈGODm R

4 pkt

,VWRWQ\SRVWÚS

SRSUDZQHQDU\VRZDQLHNDĝGHMF]ÚĂFLZ\NUHVXQLHNRQLHF]QLHXZ]JOÚG- QLDMÈFG]LHG]LQÚ

SNWSRSNW]D

NDĝGÈF]ÚĂÊ

VSRU]ÈG]HQLHFDïHJRZ\NUHVXIXQNFMLy=f x

( )

1 –1 –2 –3 –4 –5

–6 2 3 4 5 6 7 8

1

–1 –2 –3 2 3 4

0 x

y

SNW

]DSLVDQLH

UR]ZLÈ]DñGODm ∈ −∞

(

^{, 0}

)

UR]ZLÈ]QLHGODm ∈

(

^4,+ ∞

)

UR]ZLÈ]DQLDGODm ∈

{

^{0 4}^,

}

UR]ZLÈ]DQLDGODm ∈ ^{2 4}^,

)

UR]ZLÈ]DQLDGODm ∈

(

^{0 2}^,

)

^.

5 pkt (SNWMHĂOLSR- SHïQLRQRMHGHQEïÈG

(16)

w w w. o p e r o n . p l

2

3. 3RVWÚS

]DSLVDQLHW x

( )

=2

(

x−1

)

²

(

x+2

)

SNW

,VWRWQ\SRVWÚS

XSRU]ÈGNRZDQLHSRVWDFLLORF]\QRZHMLSRUöZQDQLH

2x³+ax²+bx+c=2x³−6x+4 Z\]QDF]HQLHa 0b= −6c 4

SNW

]DSLVDQLHZLHORPLDQX

W x

(

+1

)

=2x³+6x²

SNW

UR]ZLÈ]DQLHQLHUöZQRĂFLL]DSLVDQLH]ELRUXUR]ZLÈ]Dñ −∞ −

(

^, ³

)

4 pkt

4. 3RVWÚS

]DSLVDQLHa³−b³=

(

a−b a

) (

²+ab+b²

)

SNW

SU]HNV]WDïFHQLHGUXJLHJRF]\QQLNDa³−b³=

(

a−b

) ( (

a+b

)

²−ab

)

SNW

VWZLHUG]HQLHQDSRGVWDZLH]DïRĝHQLDĝHMHĝHOLOLF]E\ a b

(

+

)

² iabVÈ

SRG]LHOQHSU]H]k, WRLFKUöĝQLFDMHVWSRG]LHOQDSU]H]kRUD]a b MHVW

OLF]EÈFDïNRZLWÈ OXE]DSLVDQLH

a³−b³=

(

a−b

) ( (

a+b

)

²−ab

)

⁼

⁽

a⁻b k p

⁾ ⁽

² ²⁻kq

⁾

⁼k a

⁽

⁻b kp

⁾ ⁽

²⁻q

⁾

^JG]LH

pLqVÈOLF]EDPLFDïNRZLW\PLRUD]a b Lkp² VÈOLF]EDPLFDïNRZLW\PLq

3 pkt

5. 3RVWÚS

]DSLVDQLHZDUXQNöZ

1 1 0

2 1 0

3 1 0

1 3

2 1

3

( )

+ >

( ) (

+

)

>

( )

^

(

+

)









 ≥





 x

x

x log

log log











SNW

UR]ZLÈ]DQLHMHGQHJR]ZDUXQNöZOXE log1 log

3

1 3

1 1 0 1 1 1 0

x+ x x

( )

> ⇔ < + < ⇔ − < <

log1 3

1 1 0 1 1

3 1 2

x+ x x 3

( )

≥ ⇔ < + ≤ ⇔ − < ≤ −

SNWSNW

MHĂOLUR]ZLÈ]DQR

MHGHQZDUXQHN

UR]ZLÈ]DQLHXNïDGXZV]\VWNLFKZDUXQNöZ

x x x

> −

− < <

− < ≤ −













 1

1 0

1 2

3

L]DSLVDQLH

D = − −



 1 2 , 3

4 pkt

pobrano z www.sqlmedia.pl

(17)

w w w. o p e r o n . p l 3

6. ,VWRWQ\SRVWÚS

]DSLVDQLHb_n₊₁=5^an⁺¹=5^{an r}⁺ =5^an⋅5^r=b_n⋅5^r, n∈N₊ i 5^rļOLF]ED

SNWSNWMHĂOL

QLHZ\MDĂQLRQR

ĝH5^rMHVWOLF]EÈ 3RNRQDQLH]DVDGQLF]\FKWUXGQRĂFL

]DSLVDQLHb b b1 2 3 b_n â â ân

1 2

⋅ ⋅ ⋅...⋅ =5 ⁺ ⁺^...⁺

SNW

5R]ZLÈ]DQLHSUDZLHFDïNRZLWH

]DVWRVRZDQLHZ]RUöZQDnWÈVXPÚF]ÚĂFLRZÈ

SNW

Z\]QDF]HQLHb b b b_n

n n

1 2

32 2

3 5

⋅ ⋅ ⋅…⋅ =

−

5 pkt

7. 3RVWÚS

]DSLVDQLHDOWHUQDW\Z\XNïDGöZ

cos

sin cos x

x x

≥

=











 0

2 1

2

OXE

cos sin cos

x

x x

<

− =











 0

2 1

2

SNW

,VWRWQ\SRVWÚS

]DVWRVRZDQLHZ]RUXQDsin2x cos

sin x

x

≥

=











 0

2 1

2

OXE

cos sin

x x

<

− =











 0

2 1

2

SNW

UR]ZLÈ]DQLHUöZQDñGODx 0 2, Q 

sin2 1

x2

x Q x Q x Q x Q

12

5 12

13 12

17 lub , lub , lub 12

sin2 1

x = −2

x7 x x x

12

11 12

19 12

23 12

Qlub Q, lub Q, lub Q

SNW

5R]ZLÈ]DQLHSUDZLHFDïNRZLWH

SRSUDZQHUR]ZLÈ]DQLHNDĝGHJR]XNïDGöZ

cos

, , ,

x x

≥

∈





























 0

12 5 12

13 12

17 12 Q Q Q Q OXE

cos

, , ,

x x

<

∈





























 0 7 12

11 12

19 12

23 12

Q Q Q Q

x ∈



















Q Q

12 5

,12 OXEx ∈

















 7 12

11 12 Q, Q

SNW

]DSLVDQLHUR]ZLÈ]DQLD

റ

^{x ∈}





















Q Q Q Q

12 5 12

7 12

11 , , ,12

5 pkt

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w w w. o p e r o n . p l

4

zadania

8. 3RVWÚS

Z\NRQDQLHU\VXQNX

r

R P

M

N

k

B A

a

OXERSLVR]QDF]Hñ

PļSXQNWSU]HFLÚFLDSURVWHMk]SURVWÈAB MļSXQNWVW\F]QRĂFLo A r

(

)

]SURVWÈk

NļSXQNWVW\F]QRĂFLo B R

(

)

]SURVWÈk

SNW

,VWRWQ\SRVWÚS

]DVWRVRZDQLHWZLHUG]HQLD7DOHVD

BN BP

AM AP

R R r a

r

= a

+ +

= JG]LH APa

SNW

Z\]QDF]HQLH AP a R r r R r

= =

(

+

)

−

SNW

Z\]QDF]HQLH]WUöMNÈWDAMP

റ

sinB = = − + r AP

R r R r

4 pkt

9. 3RVWÚS

R]QDF]HQLHZLHU]FKRïNöZWUöMNÈWDA x y B x y C x y=

(

^A ^A

)

=

(

^B ^B

)

=

(

^C ^C

)

LZ\NRU]\VWDQLHZ]RUXQDZVSöïU]ÚGQHĂURGNDRGFLQND

K x x y y

L x x y y

M x

A B A B B C B C A

= + +









 = + +









 =

2 , 2 , 2 , 2 i  ++ +











x_C y_A y_C

2 , 2

SNW

,VWRWQ\SRVWÚS

]DSLVDQLHRGSRZLHGQLFKXNïDGöZUöZQDñ

x x

y y y y

A B

B C

A C

A B

B

+ =

+ = −











+ =

+

2 2

2 1

2 2

i ^C^C

A C

y y

2 1

=

+ = −











SNW

UR]ZLÈ]DQLHXNïDGöZUöZQDñL]DSLVDQLHZVSöïU]ÚGQ\FKSXQNWöZ

A=

(

^{3 0}^,

)

^,B=

(

^{1 4}^,

)

^,C= −

(

⁵^,−²

)

SNW

Z\]QDF]HQLHREUD]öZSXQNWöZA B C, , V\PHWULLĂURGNRZHMZ]JOÚGHP

SRF]ÈWNXXNïDGXZVSöïU]ÚGQ\FK

റ

^A^{′ = −}

⁽

³^,⁰

⁾

^,^B^{′ = −}

⁽

¹^,⁻⁴

⁾

^,^C^{′ =}

⁽

⁵^,²

⁾

4 pkt

pobrano z www.sqlmedia.pl

(19)

w w w. o p e r o n . p l 5

10. 3RVWÚS

]DVWRVRZDQLHWZLHUG]HQLDVLQXVöZGRWUöMNÈWDABCLREOLF]HQLH

sin

(

)ABC

)

=3 5

SNW

,VWRWQ\SRWÚS

REOLF]HQLHcos )ABC

( )

=4

5)ABCļNÈWRVWU\

SNW

]DVWRVRZDQLHWZLHUG]HQLDFRVLQXVöZGRWUöMNÈWDABK

AK² 10² 2² 2 2 10 4

= + − ⋅ ⋅ ⋅5

SNW

REOLF]HQLH AK 6 2

4 pkt

11. 3RVWÚS

REOLF]HQLHSUDZGRSRGRELHñVWZDZ\ORVRZDQLD]]LHORQHJRSXGHïND

]ïRUD]]ï

P B1 P B2

2 3

1

( )

=

( )

=3

SNW

,VWRWQ\SRVWÚS

REOLF]HQLHSUDZGRSRGRELHñVWZSU]\ORVRZDQLX]ELDïHJRSXGHïND

p1 p2

3 1

3 1 6 2

3 5

2 1

=





































= =





































= 4 1 6 2

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