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*UXG]LHñ

Próbny Egzamin Gimnazjalny z OPERONEM

&]ÚĂÊPDWHPDW\F]QRSU]\URGQLF]D

1

Numer zadania 2. 6. 7. 8.

3RSUDZQDRGSRZLHGě D B A D

Zasady przyznawania punktów:

SNWļNDĝGDSRSUDZQDRGSRZLHGě

SNWļEïÚGQDRGSRZLHGěOXEEUDNRGSRZLHG]L 3R]RVWDïH]DGDQLD

UWAGA:

=DNDĝGHSRSUDZQHUR]ZLÈ]DQLH]DGDQLDRWZDUWHJRLQQHQLĝSU]HGVWDZLRQHSU]\]QDMHVLÚPDNV\PDO- QÈOLF]EÚSXQNWöZ

-HĂOL XF]Hñ QD GRZROQ\P HWDSLH UR]ZLÈ]\ZDQLD ]DGDQLD SRSHïQLï MHGHQ OXE ZLÚFHM EïÚGöZ UDFKXQ- NRZ\FK MHGQDN ]DVWRVRZDQH PHWRG\ E\ï\ SRSUDZQH ZöZF]DV RFHQÚ FDïHJR UR]ZLÈ]DQLD REQLĝD VLÚ

RSXQNW

Numer

zadania 3RSUDZQDRGSRZLHGě Liczba

punktów Zasady przyznawania punktów 1. 1.1. P

1.2. P 1.3. F

0–3 SNWļWU]\SRSUDZQHRGSRZLHG]L

SNWļGZLHSRSUDZQHRGSRZLHG]L

SNWļMHGQDSRSUDZQDRGSRZLHGě

SNWļEUDNRGSRZLHG]L

3. NC 0–1 SNWļGZDSRSUDZQHGRSDVRZDQLD

SNWļMHGQRSRSUDZQHGRSDVR- ZDQLHOXEEUDNSRSUDZQ\FK

GRSDVRZDñ 4.

A. 2³OXE1 2

 3











−

OXE

B. 2^2 ; 1 2

 4











−

0–2 SNWļWU]\SRSUDZQHRGSRZLHG]L

SNWļGZLHOXEMHGQDSRSUDZQD

RGSRZLHGě

SNWļEUDNRGSRZLHG]L

5. FP 0–1 SNWļGZLHSRSUDZQHRGSRZLHG]L

SNWļMHGQDSRSUDZQDRGSRZLHGě

OXEEUDNRGSRZLHG]L

9. FP 0–1 SNWļGZLHSRSUDZQHRGSRZLHG]L

SNWļMHGQDSRSUDZQDRGSRZLHGě

OXEEUDNRGSRZLHG]L 10. 10.1. B

10.2. D 10.3. F

0–3 SNWļWU]\SRSUDZQHGRSDVRZDQLD

SNWļGZDSRSUDZQHGRSDVRZDQLD

SNWļMHGQRSRSUDZQHGRSDVRZDQLH

SNWļEUDNGRSDVRZDñOXEEïÚGQH

GRSDVRZDQLD

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2

Numer

zadania 3RSUDZQDRGSRZLHGě Liczba

punktów Zasady przyznawania punktów 11. 3URSR]\FMDUR]ZLÈ]DQLD

'DQH

a b c

360 36

250 25

120 12

mm cm

mm cm

mm cm

r 3 cm 6]XNDQH

OļOLF]EDJDïHNORGöZ]MHGQHJRSRMHP- nika

V V

p p

= ⋅ ⋅

=

36 25 12 10800cm³ V

V

l l

= ⋅

=

0 95 10800 10260 ³

, cm

V r

V

V V

g

g

g

g

=

= ⋅ ⋅

≈ ⋅ ⋅

≈ 4 3 4 3 3 4 3

3 27 108

3

3

3

Q

Q

cm l10260 

108 95

3 3

cm cm

2GSRZLHGě&XNLHUQLN]MHGQHJRSRMHP- QLNDORGöZXWZRU]\RNRïRJDïHN

0–4 SNWļSHïQHUR]ZLÈ]DQLHļREOLF]H- QLHREMÚWRĂFLSRMHPQLNDQD

ORG\REMÚWRĂFLORGöZZSR- MHPQLNXREMÚWRĂFLMHGQHMJDï- NLRUD]LORUD]XW\FKZLHONRĂFL

SNWļSRSUDZQHZ\]QDF]HQLH

REMÚWRĂFLORGöZLMHGQHMJDïNL

ORGöZDOHEïÚGQ\VSRVöE

Z\]QDF]HQLDOLF]E\JDïHN

SNWļEUDNREOLF]HQLDREMÚWRĂFL

ORGöZZSRMHPQLNXSRSUDZ- QHZ\]QDF]HQLHREMÚWRĂFL

SRMHPQLNDLMHGQHMJDïNL

ORGöZDOHEïÚGQ\VSRVöE

Z\]QDF]HQLDOLF]E\JDïHNQD

SRGVWDZLHLORUD]XREMÚWRĂFL

SRMHPQLNDLJDïNLORGöZ OXE REOLF]HQLHW\ONRREMÚWRĂFLOR- GöZZSRMHPQLNXLREMÚWRĂFL

JDïNLORGöZ

SNWļZ\NRQDQLHW\ONRMHGQHJR

HWDSXUR]ZLÈ]DQLD]DGDQLD

REOLF]HQLHREMÚWRĂFLSRMHP- QLNDQDORG\

OXE

REOLF]HQLHREMÚWRĂFLMHGQHM

JDïNLORGöZ

SNWļUR]ZLÈ]DQLHEïÚGQHSU]\- SDGNRZHG]LDïDQLDLQLH- SRSUDZQHREOLF]HQLDQS

]EïÚGQ\PLMHGQRVWNDPLOXE

EUDNUR]ZLÈ]DQLD 12. 3URSR]\FMDUR]ZLÈ]DQLD

]ï⋅1500=3750]ï

100 50 10 5 8

100 73 27

% % % % %

% % %

−

(

)

= − =

]]ï]ïx]ï 2GSRZLHGě=\VNSURGXFHQWDORGöZ

Z\QRVLRNRïR]ï

0–3 SNWļSHïQHUR]ZLÈ]DQLHļREOLF]H- QLHFHQ\JDïHNXG]LDïX

SURFHQWRZHJRZ]\VNXSUR- GXFHQWDRUD]SU]\EOLĝRQHJR

]\VNXSURGXFHQWD]HVSU]H- GDĝ\JDïHNORGöZ

SNWļSRSUDZQHZ\]QDF]HQLH

]\VNXSURGXFHQWDDOHEH]

SU]\EOLĝDQLDZ\QLNX

OXE

Z\]QDF]HQLHSU]\EOLĝRQHJR

]\VNXSURGXFHQWD]MHGQHM

JDïNLORGöZ

SNWļZ\NRQDQLHW\ONRMHGQHJR

HWDSXUR]ZLÈ]DQLD]DGDQLD

REOLF]HQLHNRV]WXJDïHN OXE

REOLF]HQLHXG]LDïXSURFHQWR- ZHJRZ]\VNXSURGXFHQWD

SNWļUR]ZLÈ]DQLHEïÚGQHSU]\- SDGNRZHG]LDïDQLDLQLHSR- SUDZQHREOLF]HQLDOXEEUDN

UR]ZLÈ]DQLD

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3

Numer

zadania 3RSUDZQDRGSRZLHGě Liczba

punktów Zasady przyznawania punktów 13. A. OXEx = −1OXEx^o= −1Z\QLN

(

)

%RGOXEx > −1Z\QLNx ≥−1QLH

MHVWGRSXV]F]DOQ\

&OXEy Z\QLNQLHMHVWGR- SXV]F]DOQ\

0–3 SNWļWU]\SRSUDZQHRGSRZLHG]L

WMZSXQNWDFK$%L&

SNWļGZLHSRSUDZQHRGSRZLHG]L

SNWļMHGQDSRSUDZQDRGSRZLHGě

SNWļEUDNRGSRZLHG]LOXEEïÚGQH

RGSRZLHG]L 14. 3URSR]\FMDUR]ZLÈ]DQLD

:\]QDF]HQLHGïXJRĂFLRGFLQNDDE z tw.

3LWDJRUDVD

DE²=10²−6² DE²=100−36 DE²64 DE=⁸

[ ]

2EOLF]HQLHGïXJRĂFLRGFLQNDAB:

AB 2DE AB=¹⁶

[ ]

2EOLF]HQLHSRODWUöMNÈWDABE:

P AB DA

= ⋅

2 P =16 6⋅

2 P= j

  48 ²

2GSRZLHGě3ROHWUöMNÈWDABEZ\QR- si 48 [j²].

0–2 SNWļSHïQHUR]ZLÈ]DQLHļREOLF]H- QLHSRODWUöMNÈWDABE

SNWļSRSUDZQHZ\]QDF]HQLHGïX- JRĂFLRGFLQNDDERUD]AB

DOHEïÚGQHZ\]QDF]HQLHSROD

WUöMNÈWDABE

SNWļUR]ZLÈ]DQLHEïÚGQHSU]\- SDGNRZHG]LDïDQLDLQLHSR- SUDZQHREOLF]HQLDOXEEUDN

UR]ZLÈ]DQLD

15. 3URSR]\FMDUR]ZLÈ]DQLD

2EOLF]HQLHSRODWUöMNÈWDABC]HZ]RUX

QDSROHWUöMNÈWDUöZQRERF]QHJRWUöMNÈW

ABCVWDQRZLSRïRZÚWUöMNÈWDUöZQR- ERF]QHJR

P_ABC

∆ = ⋅1 ⋅

2

6 3

4

2

P_ABC

∆ =36 3

8 P_∆ABC= j

  4 5 3, ²

:\]QDF]HQLHNÈWöZZWUöMNÈFLHABD RUD]REOLF]HQLHGïXJRĂFLRGFLQNDBD

∠ABD =30°

∠ADB =60°

∠DAB =90°

1LHFK BD 2bZöZF]DV ADb :LDGRPRĝH AB  3]DWHP]ZïDVQRĂFL

WUöMNÈWDUöZQRERF]QHJRWUöMNÈWABD VWDQRZLSRïRZÚWUöMNÈWDUöZQRERF]QHJR

>PRĝQDUöZQLHĝVNRU]\VWDÊZWZ3LWDJR- rasa: 2

( )

0–3 SNWļSHïQHUR]ZLÈ]DQLHļREOLF]H- QLHVWRVXQNXSöOWUöMNÈWöZ

WUöMNÈWöZSRGREQ\FK

SNWļSRSUDZQHZ\]QDF]HQLHSöO

WUöMNÈWöZABC i ABDDOH

EïÚGQ\VSRVöEZ\]QDF]HQLD

VWRVXQNXSöO

OXE ]DXZDĝHQLHĝHWUöMNÈW\ABC i ABDVÈSRGREQHRUD]

Z\]QDF]HQLHVNDOLSRGRELHñ- stwa k

OXE ]DXZDĝHQLHĝHWUöMNÈW\ABC i ABDPDMDMHGQDNRZÈ

SRGVWDZÚABZ\VRNRĂÊ

AD1 AD

3 OXESRSUDZ- QH]DSLVDQLHZ]RUöZQDSROD

REXWUöMNÈWöZ]XZ]JOÚGQLH- QLHPĝH AD1 AC

3 DOH

EUDNGDOV]\FKREOLF]HñOXE

ZQLRVNXSURZDG]ÈFHJRGR

XGRZRGQLHQLDWH]\

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4

Numer

zadania 3RSUDZQDRGSRZLHGě Liczba

punktów Zasady przyznawania punktów ABb 3F]\OL

b 33

b = ⋅ = =

3 3

3 3

3 3 3

3

[ ]

[ ]

2EOLF]HQLHSRODWUöMNÈWDABD]HZ]RUX

QDSROHWUöMNÈWDUöZQRERF]QHJRWUöMNÈW

ABDVWDQRZLSRïRZÚWUöMNÈWDUöZQR- ERF]QHJR

P∆ABD= ⋅¹

( )

2

2 3 3 4

2

P∆ABD= ⋅1 ⋅ 2

12 3 4 P∆ABD=12 3

8 P_∆ABD= j

  1 5 3, ²

:\]QDF]HQLHVWRVXQNXSRODWUöMNÈWD

ABCGRSRODWUöMNÈWDABD:

P P

j j

ABC ABD

∆

∆

= 

 

 

 

 4 5 3 1 5 3

2 2

, , P

P

ABC

ABD

∆

∆

= 3 Lub

LQQ\VSRVöEZ\]QDF]HQLDVWRVXQNXSöO

WUöMNÈWöZļ]ZïDVQRĂFLSRGRELHñVWZD

WUöMNÈWöZABC i ABD7UöMNÈW\ABC i ABDVÈSRGREQHZVNDOLkUöZQHM k 6 

2 3 3 FRR]QDF]DĝH

P

P^ABC k

ABD

∆

∆

= ²=3 Lub

LQQ\VSRVöEZ\]QDF]HQLDVWRVXQNXSöO

WUöMNÈWöZEH]NRQLHF]QRĂFLZ\NRU]\VWD- QLDGïXJRĂFLERNXBC

7UöMNÈW\ABC i ABDPDMÈMHGQDNRZÈ

SRGVWDZÚABDZ\VRNRĂÊAD1 AC 3 

]DWHPSROHWUöMNÈWDABCMHVWWU]\UD]\

ZLÚNV]HRGSRODWUöMNÈWDABD:

P

AB AC

∆ABC = ⋅

2 P

AB AD AB AC

P

AB AC

ABD

ABD

∆

∆

= ⋅

=

⋅

= ⋅

2

1 3 2

6 P

P

AB AC

AB AC

ABC

ABD

∆

∆

= ⋅

⋅ ⋅ =

2

6 3

b.d.d.u.

SNWļSRSUDZQHZ\]QDF]HQLHSROD

WUöMNÈWDABCRUD]GïXJRĂFL

RGFLQNDBDOXEAB

SNWļUR]ZLÈ]DQLHEïÚGQHSU]\- SDGNRZHG]LDïDQLDLQLHSR- SUDZQHREOLF]HQLDOXEEUDN

UR]ZLÈ]DQLD

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