Download - 2013 קיץ שאלון ראשון תשובות
-
: . .
: ", 3102 : 608530, 613
5
. : .
. : .
3 33 1
3 61 2 2#
3 33 1
3 61 2 2#
3 33 1
3 61 2 2#
" 001
. :
(1) . .
.
(2) ().
. :
(1) ; .
(2) . ,
.
, , .
.
(3) .
.
.
!/ /
-
- 2 - , ", ' 608530, 613
1
- 2 - , ", ' 608530, 613 +
! , , .
.
3 33 )1
(
3 61 ). 1-3 ( 2
! , .
1. I II . .
I II . %05 ,
51 I .2
. I II .
. 003 . 6 . I ?
2. na . n : - + + + + + - =2 n2 4 01 6 2 5n n n S]) ( ... [
. na .
. , 201 .
( ).
/ 3/
1.
()
( 1 )
"
x I : x11
.
x5 1
.
x15 1
y II: y11
.
y5 1
.
y15 1
I II :tt
11
T T11
/ 3/
-
- 3 - , ", ' 608530, 613
1.
x y t & = +1 ty tx .I yx
+ =
& = +: :5 1 5 1T y T x1 . . .II x y T) ( .
yx+ =5 1
51x T t .III 2
= -
I II III :x y) ( .
yxx yyx
51 5 1x2
= + - +
0
y. x ( !0 x): =5 1 x
= +1 6 6y x .I .
=5 1 xy. .II
I II : 51 = x
I 003 51 , 51 003
I : 02 =
/ 4/
-
- 4 - , ", ' 608530, 613
2
- 2 - , ", ' 608530, 613 +
! , , .
.
3 33 )1
(
3 61 ). 1-3 ( 2
! , .
1. I II . .
I II . %05 ,
51 I .2
. I II .
. 003 . 6 . I ?
2. na . n : - + + + + + - =2 n2 4 01 6 2 5n n n S]) ( ... [
. na .
. , 201 .
( ).
2 / 3/ . - + + + +2 4 01 6 2n) ( ...
n - = - + + - =:2 2 n2n n n n n n S5 3 ))1 (4 2 2( 5 4 2 , :
0
21 n : -- - - =2 1 n1 5 1 3n n S) ( ) (
- - =1 n n nS S a
0
- =n8 6n a
- =n8 6n a
$1-= - - =1 12 8 6 8 6n a= -= - =: :2 1 5 1 3 S, 1 = n :
0
- =n8 6n a n
/ 5/
-
- 5 - , ", ' 608530, 613
2.
1n201 a .
0
-1201 8 6n
0
3 81 n 1
1
0
81 = n 201:
1n0 a :
0
6 n 8
1
0
1 = n
-= - =12 8 6 a :
= + - = - = + + +: :2 1 81 81 3 2488 2 81 5 81 3 a S a a a... :
/ 6/
-
- 6 - , ", ' 608530, 613
- 3 - , ", ' 608530, 613 + 3
3. " " '
: ' '.
' .
.
'
.
2 .
. '
' . ' 5.0 .
. , ',
? .
. .
, ' ,
8.0 .
' .
/ 4/
3x = )' (P = )' (P , 5.0 = )' (P. :
(P+ )' +' +' (P+)' +' +' (P + )' +' +' (P =) 2
)' +' +' ( P +
0
(P ) 2
+ - + - + =: : : : : : : :5 0 5 0 1 5 0 1 5 0x x x x x x x x. . ) ( . ) ( .
0
(P x = ) 2
0
(P )' (P = ) 2
/ 7/
-
- 7 - , ", ' 608530, 613
3.
)' +' +' (P+)' +' +' (P + )' +' +' (P =( 2 /' (P. ) 2 (P
0
( 2 /' (P x. . . ) ( . =
+ - +2 : : : : :8 0 5 0 5 0 1 5 0x x x x x x
0
26 0 x.
": # 11 6 0x .
/ 8/
-
- 8 - , ", ' 608530, 613
4
- 4 - , ", ' 608530, 613 +
3 33 )1
(3 61 ).
4-6 ( 2 ! , .
4. . , .
. CBA L , M , K BC , AC BA .
P ,
L , M C ( ). MB = LA .
(1) =MCA B .
(2) MA = KA .
5. MLKA . MA . F
( ).
: 03 " = LM , a " = LF KLA 3
MLA. . AL
KLA . . a FK .
. +T TLFK MFA .
. : 5.24 " = FA , 2LMa . a .
/ 5/
B A
C
P
K
ML
A
M
L
K
F
4
2MB LA CB LB CA MA, ,1
21
. : = = =
": =CB CA
2:1 ,
,
MB PB LA PA
MB MP LA LP
32
32
31
31
= =
= =
Z
[
\
]]
]]
: MB = LA
: MP = LP , PB = PA
=B BLPB MPA
: T,T MPALPB ..
0
LB = MA
2CA MA1
2CB LB , =1
: =
0
CB = CA / 9/
B A
C
P
ML
-
- 9 - , ", ' 608530, 613
4.
.(1) I CB = CA ,
0
CL = CM M L
KC K CB 0
KC BBCA 0
PC LMC 0
PC TLMC 0
=CM MP o09
II CB = CA ,
0
CL = CM M L
MP = LP
PC 0
3,3 PMCPLC . . .
0
=B BPLC PMC .I
+= B Bo081 PLC PMC .II o081
I II : = Bo09 PMC
/ 01/
-
- 01 - , ", ' 608530, 613
4.
(2) (1): MB CA 0
CB = BA ,
(1): CB = CA
: BA = CA
2CA MA BA KA,1
21
: = =
: MA = KA
/ 11/
-
- 11 - , ", ' 608530, 613
5
- 4 - , ", ' 608530, 613 +
3 33 )1
(3 61 ).
4-6 ( 2 ! , .
4. . , .
. CBA L , M , K BC , AC BA .
P ,
L , M C ( ). MB = LA .
(1) =MCA B .
(2) MA = KA .
5. MLKA . MA . F
( ).
: 03 " = LM , a " = LF KLA 3
MLA. . AL
KLA . . a FK .
. +T TLFK MFA .
. : 5.24 " = FA , 2LMa . a .
/ 5/
B A
C
P
K
ML
A
M
L
K
F
5
. = Bo09 MLA
0
2LA S T=: : MLA03 1
3S S1
: T T=MLA KLA
0
2LA LA h03 ( h AL)1
31
21
=: : : : :
0
01 " = h
/ 21/
-
- 21 - , ", ' 608530, 613
5.
A
M
L
K01
03
aE F
.
=B BLKFE MF
= =B Bo09 FLM FEK
0
T+T MFLKFE ..
0
a EF03 01 =3a EF & =
3 01 3a a FK) ( KFE : + = + =2 2 2009 1
. =B BAFM KFL
=B BFMA FKL
0
+T TAFM KFL ..
MF FKa .I . :FA
=
3a FK .II :+ =2009 1
+ =2 203 a MF .III MFL :
5.24 " = FA .VI :
= + -20 009 5 721a a. I , II , III VI : 0
/ 31/5.7 " = a
-
- 31 - , ", ' 608530, 613
6- 5 - , ", ' 608530, 613 +
6. O CBA .
OA CB E .
OC BA F
( ).
: =a BCAB , =b BCBA .
FC .EA
. a b
FC , =bo06 .EA
21
. : =
2CB .1
BCA
,
,
3 33 )1
(
3 61 ). 7-9 ( 2
! , .
3 0x .7
3x x gnis) ( ) ( r # #r2
7. - =
. )x(g .
. )x(g
=x x fnis ) ( .
. A )x(g B )x(f
BA y .
(1) BA .
(2) BA ? ./ 6/
A
C
B
F
E
O
6
B B2 EAB EAC a
. = =
2 09 FCB FCA B Bo
b a- = =
+
nis nis) ( ) ( EAC :EA AC
2a
+b a b=
+) ( &
) (nis
AC EAnis
2a
b
b a=
+
+
AFC:nis nis) (
AC FC2 09
bb a ao=
- ++
& nis) (
AC FCnis2 09
o
aa b
=+
-
) ( :) ( ) (
nisnis
nissoc
FCEA
2
2a:
b
b aa
a b
=+
+-
/ 41/
-
316 ,035806 ' ," , - 14 -
.6
: . ( )( 60 ) (30 )sin sin
sin cos21
2 602
o
o o
:
:
aa
aa
=+
+ -
0
( )( 60 ) (30 )cos sin
sin cos21
30 22
o
o o
:
:
aa
aa
=-
+ - : ( )sin cos2 60 90 2 60o o o
a a+ = - +a k9 C
0
( )sin
sin21 60o
aa
=+
0
)60+(asin o
: sin sin cos cos sin2 60 2 60o oa a a= +
0
cos 0a= & 90oa=
0
ABC BC
0
BC21
=
/15 /
-
- 51 - , ", ' 608530, 613
7
- 5 - , ", ' 608530, 613 +
6. O CBA .
OA CB E .
OC BA F
( ).
: =a BCAB , =b BCBA .
FC .EA
. a b
FC , =bo06 .EA
21
. : =
2CB .1
BCA
,
,
3 33 )1
(
3 61 ). 7-9 ( 2
! , .
3 0x .7
3x x gnis) ( ) ( r # #r2
7. - =
. )x(g .
. )x(g
=x x fnis ) ( .
. A )x(g B )x(f
BA y .
(1) BA .
(2) BA ? ./ 6/
A
C
B
F
E
O
7
2 0 g) ( & 0 = x . y:3
2 0) , ( & =3
3k x & 0 = )x(g x:r2
(k ) = -r
0
3k x r2
- =r
0
3x x, 2
3r r5
= =
0
3) , ( , ) , ( 3 0 2
r r0 5
/ 61/
-
- 61 - , ", ' 608530, 613
7.
3x xnis nis) ( . r2= -
0
0 3 3 xsoc nis) (r r
: = -
0 3 x nis) ( r
3k x & = -r
+ =r
0
3 3x x x, , 4
3r r r7
= = =
0
2 3) , ( , ) , ( , ) , ( :3
34
23
37
2r r r3
-
- = =x g x f x d BA) ( ) ( ) ( . (1)
3 3x :4
# #r r
3x x x x d BAnis nis nis) ( ) ( ) ( 2
3r r
- = - - = =
3 dx xsoc ') ( ) ( r
- =
0 3 x soc) ( & = dx '0 ) ( r
6k x & = -r5
6 x & + =rr5
=
3 x x dnis '') ( ) ( :
r6 d''0 ) ( & - -=
51 3
r r-
# #3 0x :r
7
3x4
# #3r r
3x x x x d BAnis nis nis) ( ) ( ) ( 2
3r r
- = - - = =
3 dx xsoc ') ( ) ( r
-- =
6 x & = dx '0 ) ( r11
=
6 3 3d x x dnis'' ''0 ) ( ) ( ) ( :11
1 &r r r
- - -=
6d d) ( ) ( BA:5
6= =r r1 11
2 BA , =0 BA . ) ( =0 x =11 3
6 x .r11
6 x =r5
(2) (1), , =
/ 71/
-
- 71 - , ", ' 608530, 613
8- 6 - , ", ' 608530, 613 +
8. : + + =24b x x x f) ( + - =2c x x g) (
b c 0. P .
. b ( ) P . . )x(f
)x(g , 24 b .
a = x D , )x(f A )x(g B (D , A B ).
. DP BAP . . )x(f ,
=a x - =a x , S . S )x(f , )x(g
=a x - =a x .
/ 7/
8
.
)x('g = )x('f P , : 0
-= +2 4 2x x
0 -=1 x x P:
- = -3 1b f) ( y P:
- -3 1b P) , ( :
/ 81/
-
- 81 - , ", ' 608530, 613
8.
x
. y
( )x(f - -4 2b) , ( y . )x(g 0 = x . x -1 .)
= - = -g f'2 )1 ( )1 ( ' . - -3 1b P) , ( :- + =1 2b x y P:
0
- + =D1 2b a y y D:
+ + = =2 A4b a a a f y) ( y A:
- = -1 1g f) ( ) (
0 - =2 b c c:
0
- + -= =2 B2 b a a g y) ( y B :
D A1 2a a y y DA + + = - =2
B D1 2a a y y BD + + = - =2
0
DB = DA DP BAP .
1 2 1 2xd x x xd b x x f S) ( ]) ( ) ( [ .I . a
a
a
a+ + = - + - =2
--
##
1 2 2xd x x xd x g x f) ( ]) ( ) ( [ S .II :a
a
a
a2
+ + = - =:
--
##
=:S 2 S I II :
/ 91/
-
- 91 - , ", ' 608530, 613
- 7 - , ", ' 608530, 613 + 9
!
9. + + - =2 8c xb xa x f) ( -# #2 2x .
a , b c , 20 c. . a b .
a b , -. . )x(f =2 x ,
- =2 x .
2 .2 94
x c .
. -# #2 2x )x(g : - =x f x g) ( ) ( . )x(g =2 x ,
- =2 x .
)x(f )x(g ? .
9.
: + -$20 8xb xa -# #2 2x
= + -: :2 b a0 2 2 8 .I :
= + +: :20 2 2 8b a .II
I II : -=2 b , 0 = a
/ 02/
-
- 02 - , ", ' 608530, 613
9.
+ - =2 2 8c x x f) ( .
0
x f) ( ' x
x
2 82
- =2
-
0
= -2 2 f) ( ' , -=2 2 f) ( ' :
+ = = -2 2 2c f f) ( ) ( y :
+ + -=4 2c x y +2 2c A) , ( :
c x A x:2
4=
+
+ + =4 2c x y + -2 2c B) , ( :
c x B x :2
4-=
+
C )c + 4 , 0( ( ) :
CBA 20 c 2BA y S ( ) :
1=:C
0
2c c c S) ( 4 1
24
24
+ =::+
+m c+
0
) (
,
cc
24
20 2 94
22
+=
0
3 = c
/ 12/
y
BOAx
C
-
- 12 - , ", ' 608530, 613
9.
-=x f x g) ( ) ( .
0
-=fx x g' ') ( ) ( 0
=2 2 g') (
-= -2 2 g') (
)x(g =2 x )x(f -=x2 , )x(g -=2 x )x(f =2 x ,
.
A B ( ). A , B C , , AC = BC , .