31 BÀI TOÁN HHKG LỚP 11 ÔN THI HK 2

31 BÀI TOÁN ÔN TẬP HÌNH HỌC KHÔNG GIAN- HỌC KỲ II - LỚP 11
 !"#$%&'(%)*+
Bài 1,- --/012,3/456 -7,389:;<
1 =,3>⊥
2?<<@5A
5BC856 --D7/EF4G:8H-1<1,<6132
FI,I:
, =1>< ,3 =13>< 3 =1,>⊥ ⊥ ⊥
JI,I:
1, =@>⊥
6
 =@>∈
2
I,I:
@ =1,>⊥
8K4/HF
@ ⊥
2
Bài2,- --/012,3/456 -8-,389:;6/
1 1,<1 13= =
2
FI,I:
1; =,3>⊥
2
JI?<@5A5BC8568H4G:EF-<,2,I:H
@ =13>⊥
6
@ 13⊥
2
Bài 3,-8LM;,/'-
;<;<;,

47:N87/O-F2@P
; =,>⊥
82
,I:
FI
; ,<; ,⊥ ⊥
6
;, 
⊥
2
JI568HQ89:EF
,V
2
I
) ) ) )
* * * *
; ; ; ;,
= + +
Bài4,-8LM,3/
 =,3>⊥
2H
,3V
R4BSF"63TU8-F8;2
H:0=,3>R
3@ ,
⊥
8@2?568HQ89:EF
,3V
2
FI,I:

:0=3,> :0=">⊥
<6
:0=3,> :0=3T@>⊥
2
JI,I:
; :0=3,>⊥
2
Bài5F8F:9,63,V:8H):W80-XY--F/--4,8
8LM,32?568H4G:-,2
FI,I:
, 3⊥
2
JI?564BSFEF
3V
2,I:H
 =,3>⊥
2
Bài 6*,- --/012,3/456 -8-,389:</-JVF64BS-Z
3[F2,-
F \
1,
)
=
7/O:0=,3>2,I:
:0=1> :0=13>⊥
2
Bài 7/8,- --/012,3/456 -8-,3-F6/
1 1 1, F= = =
2,I:
FI:0

=,3> :0=13>⊥
JI
13V
7812
Bài8/ 9,-8LM1,/
1 =,>⊥
2?6@5A5BC8568HQ89:EFEF
,V
6
1,V
2,I:H
FI
<1@<,
4]^2
JI
1, =@>⊥
6
=1,> =@>⊥

I
@ =1,>⊥
6
=1,> =@>⊥
Bài 9/10,-8LM1,/'4_-<<,88-6-:N88F:794_-6
, )F
=
`
1 =,>⊥
6
1 F

=
2
FI,I:
:0=1> :0=1,>⊥
JIH:0=1>R
 1
⊥
82,I:
 =1,>⊥
2
Ia-4NL642
LIK8H4G:;EF4,R
;@ =1,>⊥
U8=1,>8@2a-4NL64;@2
Bài 10/12,- --/012,3/456 -8-F,37863`JD8
 )F<3 3, F` 1 =,3><1 F= = = ⊥ =
2
FI,I:
=13> =13,>⊥
6
=1,> =1,>⊥
2
JI?
ϕ
56/bF):W80-X=1,>6=,3>2a-
8F ϕ
2
I?
= >α
56:W80-X-F1367/O:0=1,>2cd4e-

= >α
6d4e-8-D8LM
EF --/012,3O
= >α
2a-LM8a-8-D8LM4/2
Bài 11/13,- --/012,3/456 -7,389:;<
1 =,3>⊥
61[F2?
<f5A5BC8568H4G:EF1,62
FI,I:
; =,3>⊥
2
JIa-
L=`,f>
2
Bài 12/14,-
,V
O[(<,[&<,[%2H4BS8-X7/O:0=,>8
5g4G:;F-;[+2a-
L=;`,>
2
Bài 13/15*,-
,V
78</-[FV:8H:0
= >α
<
, F )=
68O
= >α
:N8

/
#
\#
2
FIa-
[ ]
L ,`= > ,α =
JI,I:H,-,8O
= >α
:N8/
#
+&ϕ =
2
Bài 14/16*,
1 =,3>⊥
61[-23Q68a-4NL647/-EF
FI16,3JI1,63
Bài 15/18,- --/012,3/456 -7,389:;<[F2BSF1;EF
--/07/O:W84=,3>6/1;[F2a-Y-h-bF)4BS8-X-Z
-F1,62
Bài 16/ 20,- --/08F:4i12,/-4JV'F<-JJV)F2?568H?
89:EF
,V
2
FIa-
[ ]
L 1`=,>
JIa-
L=`1>
2

Bài 17/21*,- --/08F:12,/4568F:4i-(F`
1, =,>⊥
61,[(F2
FIa-/bF16,2JIa-
L=1`,>
Bài 18/22,- -8-,389:;<-F6/
F '
;
'
=
2H4BS8-X7/O
:0=,3>8;8F5g:N84G:1F-1[F2
FI,I:
1,V
568F:76
1, 3⊥
JI,I:
=13> =1> `=1,> =1,3>⊥ ⊥
2
Ia-
L=1`3>
2
LIa-/bF=
α
>6=,3>2
Bài 19/28 :,-8LM;,/
; ; ;, F
= = =
6
·

·
·
# #
; ;, \# `;, $#= = =
2
FI,I8j
,V
56:N88F:72
JI,I:
; ,
⊥
2?<k5A5BC8568H4G:EF;6,2,-8jk564BS7/-
EF;6,2
Bài 20/29H:0
= >α
-4BS8Hl4BSYa-2g:N84G:
1 = >∉ α
F-
1 = >⊥ α
2?
56:N84G:8H4BS8HlY-O62
FI,I:H:0=1>
⊥
:0=1>2
JIH:0=1>R
@ 1
⊥
8@2,I:
@ 1
⊥

2
Bài 21/ 31,- --/012,3/456 -J.--6-O[F<3[)F2
1V
79
82f564G:8H-3=fY-63>2fW80-X
α
^FfO:0=1>U8,<
1,<135A5BC88<<!2
FI,I:f!56 -8-F72
JIW8d[f2a-LM8a-EFf!8-mF6d2
Bài 22/32,-8LM4i,3-JVF`?f568H4G:EF,2a-EF/
·
=<3f>
2
Bài 23/33,-8LM,3/):W8,63,56)8F:9/-4,2
FI,I:H
, 3
⊥
JI?568H4G:EF,<564BSFEF
3V
2,I:
 =,3>⊥
2
Bài 24/34,- --/012,3/456 -8-,389:;2D81[1,<1[132,I:H
FI
1; :0=,3>⊥
JI
, 13⊥
Bài 25/35,- --/012,3/4,356 --b-n8<[F<
3 F '=

2,-J1
7/O461[F2a-
FI/bF
·
=1`,3>
JI/bF
·
=13`=1>>
I/bF
·
==1,3>`=,3>>
Bài 26/37,- --/012,3/456 -7,3-F2
1 :0=,3>⊥
<1[F2o
4e-68a-4NL647/EFW04BS8-X
FI163JI1,63I16,3
Bài 27/38,- --/012,3/456 -7,3-F6
1 1 1, 13 F )= = = =
2
?6k5A5BC8568H4G:EF36,2
FI,I:
:0=1k> :0=1,>⊥
JIa-
L=3`1>
Bài 28/39,-5p8Hq,2rr,r/8g8h-44iJVF2D8/88-6-Js-
J6:W8456
#
\#
6 --DEF4_-5:W80-X=rr,r>8HtO8H4G:EF
-r,r2

FIa-Y-h-bF):W84JIa-8F=,<,r>2
Ia-8FEF/bF:0=rr>6:W842
Bài 29/40,-8LM,3/,568F:4i-F<
3 ,
⊥
<3[F6Y-h-8K3
4D,56F2?568H4G:EF,6568H4G:EF2
FI,I:
, :0=3>⊥
63[F2JI,I:
3 =,>⊥
Io4e-68a-4NL647/-EF)4BS8-X36,2
Bài 30/41,- --/012,3/456 -8-,389:;<-F2/
µ
#
 \#=
6/4BS
F1;[F2
FIa-
[ ]
L ;`=1,>
JIa-
L=3`1>
Bài 31/45*:,-8LM1,/
·
#
, $#=
<[)F<
, F '=
2

1 =,>⊥
<1[)F2?f56
8H4G:EF2
FIa-/bF):W80-X=1,>6=,>2
JIa-4BSF@EF
f,V
2
Ia-
8F ϕ
O
·
==1f,>`=,>>ϕ =
2
LIa-
[ ]
L `=1f,>
2

Chúc các em thành công!