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Exam: Maharashtra HSC Class 12
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Q926
#926
Mathematics → Line and Planes → Equations of a Plane
·
Moderate
Add
Find the vector equation of the plane which is at a distance of 6 unit from the origin and to which the vector
2
i
^
−
j
^
+
2
k
^
2\hat{i}-\hat{j}+2\hat{k}
2
i
^
−
j
^
+
2
k
^
is normal.
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[Q6.4 SolvedEx.7 · Maharashtra State Board (Class 12) — Line and Planes (Balbharati textbook)]
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Q927
#927
Mathematics → Line and Planes → Equations of a Plane
·
Moderate
Add
Find the perpendicular distance of the origin from the plane
x
−
3
y
+
4
z
−
6
=
0
x-3y+4z-6=0
x
−
3
y
+
4
z
−
6
=
0
.
Show model answer
[Q6.4 SolvedEx.8 · Maharashtra State Board (Class 12) — Line and Planes (Balbharati textbook)]
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Q928
#928
Mathematics → Line and Planes → Equations of a Plane
·
Hard
Add
Find the coordinates of the foot of the perpendicular drawn from the origin to the plane
2
x
+
y
−
2
z
=
18
2x+y-2z=18
2
x
+
y
−
2
z
=
18
.
Show model answer
[Q6.4 SolvedEx.9 · Maharashtra State Board (Class 12) — Line and Planes (Balbharati textbook)]
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Q929
#929
Mathematics → Line and Planes → Equations of a Plane
·
Moderate
Add
Reduce the equation
r
ˉ
⋅
(
3
i
^
−
4
j
^
+
12
k
^
)
=
8
\bar{r}\cdot(3\hat{i}-4\hat{j}+12\hat{k})=8
r
ˉ
⋅
(
3
i
^
−
4
j
^
+
12
k
^
)
=
8
to the normal form and hence find (i) the length of the perpendicular from the origin to the plane (ii) direction cosines of the normal.
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[Q6.4 SolvedEx.10 · Maharashtra State Board (Class 12) — Line and Planes (Balbharati textbook)]
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Q930
#930
Mathematics → Line and Planes → Equations of a Plane
·
Hard
Add
Find the vector equation of the plane passing through the point
(
1
,
0
,
2
)
(1, 0, 2)
(
1
,
0
,
2
)
and the line of intersection of planes
r
ˉ
⋅
(
i
^
+
j
^
+
k
^
)
=
8
\bar{r}\cdot(\hat{i}+\hat{j}+\hat{k})=8
r
ˉ
⋅
(
i
^
+
j
^
+
k
^
)
=
8
and
r
ˉ
⋅
(
2
i
^
+
3
j
^
+
4
k
^
)
=
3
\bar{r}\cdot(2\hat{i}+3\hat{j}+4\hat{k})=3
r
ˉ
⋅
(
2
i
^
+
3
j
^
+
4
k
^
)
=
3
.
Show model answer
[Q6.4 SolvedEx.11 · Maharashtra State Board (Class 12) — Line and Planes (Balbharati textbook)]
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Q931
#931
Mathematics → Line and Planes → Equations of a Plane
·
Easy
Add
Find the vector equation of a plane which is at 42 unit distance from the origin and which is normal to the vector
2
i
^
+
j
^
−
2
k
^
2\hat{i}+\hat{j}-2\hat{k}
2
i
^
+
j
^
−
2
k
^
.
Show model answer
[QEx 6.3 Q.1 · Maharashtra State Board (Class 12) — Line and Planes (Balbharati textbook)]
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Q932
#932
Mathematics → Line and Planes → Equations of a Plane
·
Easy
Add
Find the perpendicular distance of the origin from the plane
6
x
−
2
y
+
3
z
−
7
=
0
6x-2y+3z-7=0
6
x
−
2
y
+
3
z
−
7
=
0
.
Show model answer
[QEx 6.3 Q.2 · Maharashtra State Board (Class 12) — Line and Planes (Balbharati textbook)]
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Q933
#933
Mathematics → Line and Planes → Equations of a Plane
·
Moderate
Add
Find the coordinates of the foot of the perpendicular drawn from the origin to the plane
2
x
+
6
y
−
3
z
=
63
2x+6y-3z=63
2
x
+
6
y
−
3
z
=
63
.
Show model answer
[QEx 6.3 Q.3 · Maharashtra State Board (Class 12) — Line and Planes (Balbharati textbook)]
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Q934
#934
Mathematics → Line and Planes → Equations of a Plane
·
Moderate
Add
Reduce the equation
r
ˉ
⋅
(
3
i
^
+
4
j
^
+
12
k
^
)
=
78
\bar{r}\cdot(3\hat{i}+4\hat{j}+12\hat{k})=78
r
ˉ
⋅
(
3
i
^
+
4
j
^
+
12
k
^
)
=
78
to normal form and hence find (i) the length of the perpendicular from the origin to the plane (ii) direction cosines of the normal.
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[QEx 6.3 Q.4 · Maharashtra State Board (Class 12) — Line and Planes (Balbharati textbook)]
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Q935
#935
Mathematics → Line and Planes → Equations of a Plane
·
Easy
Add
Find the vector equation of the plane passing through the point having position vector
i
^
+
j
^
+
k
^
\hat{i}+\hat{j}+\hat{k}
i
^
+
j
^
+
k
^
and perpendicular to the vector
4
i
^
+
5
j
^
+
6
k
^
4\hat{i}+5\hat{j}+6\hat{k}
4
i
^
+
5
j
^
+
6
k
^
.
Show model answer
[QEx 6.3 Q.5 · Maharashtra State Board (Class 12) — Line and Planes (Balbharati textbook)]
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Q936
#936
Mathematics → Line and Planes → Equations of a Plane
·
Easy
Add
Find the Cartesian equation of the plane passing through
A
(
−
1
,
2
,
3
)
A(-1, 2, 3)
A
(
−
1
,
2
,
3
)
, the direction ratios of whose normal are
0
,
2
,
5
0, 2, 5
0
,
2
,
5
.
Show model answer
[QEx 6.3 Q.6 · Maharashtra State Board (Class 12) — Line and Planes (Balbharati textbook)]
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Q937
#937
Mathematics → Line and Planes → Equations of a Plane
·
Moderate
Add
Find the Cartesian equation of the plane passing through
A
(
7
,
8
,
6
)
A(7, 8, 6)
A
(
7
,
8
,
6
)
and parallel to the XY plane.
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[QEx 6.3 Q.7 · Maharashtra State Board (Class 12) — Line and Planes (Balbharati textbook)]
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Q938
#938
Mathematics → Line and Planes → Equations of a Plane
·
Moderate
Add
The foot of the perpendicular drawn from the origin to a plane is
M
(
1
,
0
,
0
)
M(1, 0, 0)
M
(
1
,
0
,
0
)
. Find the vector equation of the plane.
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[QEx 6.3 Q.8 · Maharashtra State Board (Class 12) — Line and Planes (Balbharati textbook)]
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Q939
#939
Mathematics → Line and Planes → Equations of a Plane
·
Moderate
Add
Find the vector equation of the plane passing through the point
A
(
−
2
,
7
,
5
)
A(-2, 7, 5)
A
(
−
2
,
7
,
5
)
and parallel to vectors
4
i
^
−
j
^
+
3
k
^
4\hat{i}-\hat{j}+3\hat{k}
4
i
^
−
j
^
+
3
k
^
and
i
^
+
j
^
+
k
^
\hat{i}+\hat{j}+\hat{k}
i
^
+
j
^
+
k
^
.
Show model answer
[QEx 6.3 Q.9 · Maharashtra State Board (Class 12) — Line and Planes (Balbharati textbook)]
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Q940
#940
Mathematics → Line and Planes → Equations of a Plane
·
Moderate
Add
Find the Cartesian equation of the plane
r
ˉ
=
(
5
i
^
−
2
j
^
−
3
k
^
)
+
λ
(
i
^
+
j
^
+
k
^
)
+
μ
(
i
^
−
2
j
^
+
3
k
^
)
\bar{r}=(5\hat{i}-2\hat{j}-3\hat{k})+\lambda(\hat{i}+\hat{j}+\hat{k})+\mu(\hat{i}-2\hat{j}+3\hat{k})
r
ˉ
=
(
5
i
^
−
2
j
^
−
3
k
^
)
+
λ
(
i
^
+
j
^
+
k
^
)
+
μ
(
i
^
−
2
j
^
+
3
k
^
)
.
Show model answer
[QEx 6.3 Q.10 · Maharashtra State Board (Class 12) — Line and Planes (Balbharati textbook)]
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Q941
#941
Mathematics → Line and Planes → Equations of a Plane
·
Moderate
Add
Find the vector equation of the plane which makes intercepts
1
,
1
,
1
1, 1, 1
1
,
1
,
1
on the co-ordinate axes.
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[QEx 6.3 Q.11 · Maharashtra State Board (Class 12) — Line and Planes (Balbharati textbook)]
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Q942
#942
Mathematics → Line and Planes → Angle Between Planes and Line-Plane Angle
·
Easy
Add
Find the angle between planes
r
ˉ
⋅
(
i
^
+
j
^
−
2
k
^
)
=
8
\bar{r}\cdot(\hat{i}+\hat{j}-2\hat{k})=8
r
ˉ
⋅
(
i
^
+
j
^
−
2
k
^
)
=
8
and
r
ˉ
⋅
(
−
2
i
^
+
j
^
+
k
^
)
=
3
\bar{r}\cdot(-2\hat{i}+\hat{j}+\hat{k})=3
r
ˉ
⋅
(
−
2
i
^
+
j
^
+
k
^
)
=
3
.
Show model answer
[Q6.5 SolvedEx.12 · Maharashtra State Board (Class 12) — Line and Planes (Balbharati textbook)]
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Q943
#943
Mathematics → Line and Planes → Angle Between Planes and Line-Plane Angle
·
Moderate
Add
Find the angle between the line
r
ˉ
=
(
i
^
+
2
j
^
+
k
^
)
+
λ
(
i
^
+
j
^
+
k
^
)
\bar{r}=(\hat{i}+2\hat{j}+\hat{k})+\lambda(\hat{i}+\hat{j}+\hat{k})
r
ˉ
=
(
i
^
+
2
j
^
+
k
^
)
+
λ
(
i
^
+
j
^
+
k
^
)
and the plane
r
ˉ
⋅
(
2
i
^
−
j
^
+
k
^
)
=
8
\bar{r}\cdot(2\hat{i}-\hat{j}+\hat{k})=8
r
ˉ
⋅
(
2
i
^
−
j
^
+
k
^
)
=
8
.
Show model answer
[Q6.5 SolvedEx.13 · Maharashtra State Board (Class 12) — Line and Planes (Balbharati textbook)]
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Q944
#944
Mathematics → Line and Planes → Coplanarity of Two Lines
·
Moderate
Add
Show that lines
r
ˉ
=
(
i
^
+
j
^
−
k
^
)
+
λ
(
2
i
^
−
2
j
^
+
k
^
)
\bar{r}=(\hat{i}+\hat{j}-\hat{k})+\lambda(2\hat{i}-2\hat{j}+\hat{k})
r
ˉ
=
(
i
^
+
j
^
−
k
^
)
+
λ
(
2
i
^
−
2
j
^
+
k
^
)
and
r
ˉ
=
(
4
i
^
−
3
j
^
+
2
k
^
)
+
μ
(
i
^
−
2
j
^
+
2
k
^
)
\bar{r}=(4\hat{i}-3\hat{j}+2\hat{k})+\mu(\hat{i}-2\hat{j}+2\hat{k})
r
ˉ
=
(
4
i
^
−
3
j
^
+
2
k
^
)
+
μ
(
i
^
−
2
j
^
+
2
k
^
)
are coplanar. Find the equation of the plane determined by them.
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[Q6.6 SolvedEx.14 · Maharashtra State Board (Class 12) — Line and Planes (Balbharati textbook)]
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Q945
#945
Mathematics → Line and Planes → Distance of a Point from a Plane
·
Moderate
Add
Find the distance of the point
4
i
^
−
3
j
^
+
2
k
^
4\hat{i}-3\hat{j}+2\hat{k}
4
i
^
−
3
j
^
+
2
k
^
from the plane
r
ˉ
⋅
(
−
2
i
^
+
j
^
−
2
k
^
)
=
6
\bar{r}\cdot(-2\hat{i}+\hat{j}-2\hat{k})=6
r
ˉ
⋅
(
−
2
i
^
+
j
^
−
2
k
^
)
=
6
.
Show model answer
[Q6.7 SolvedEx.15 · Maharashtra State Board (Class 12) — Line and Planes (Balbharati textbook)]
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Q946
#946
Mathematics → Line and Planes → Angle Between Planes and Line-Plane Angle
·
Easy
Add
Find the angle between planes
r
ˉ
⋅
(
i
^
+
j
^
+
2
k
^
)
=
13
\bar{r}\cdot(\hat{i}+\hat{j}+2\hat{k})=13
r
ˉ
⋅
(
i
^
+
j
^
+
2
k
^
)
=
13
and
r
ˉ
⋅
(
2
i
^
−
j
^
+
k
^
)
=
31
\bar{r}\cdot(2\hat{i}-\hat{j}+\hat{k})=31
r
ˉ
⋅
(
2
i
^
−
j
^
+
k
^
)
=
31
.
Show model answer
[QEx 6.4 Q.1 · Maharashtra State Board (Class 12) — Line and Planes (Balbharati textbook)]
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Q947
#947
Mathematics → Line and Planes → Angle Between Planes and Line-Plane Angle
·
Moderate
Add
Find the acute angle between the line
r
ˉ
⋅
(
i
^
+
2
j
^
+
2
k
^
)
+
λ
(
2
i
^
+
3
j
^
−
6
k
^
)
\bar{r}\cdot(\hat{i}+2\hat{j}+2\hat{k})+\lambda(2\hat{i}+3\hat{j}-6\hat{k})
r
ˉ
⋅
(
i
^
+
2
j
^
+
2
k
^
)
+
λ
(
2
i
^
+
3
j
^
−
6
k
^
)
and the plane
r
ˉ
⋅
(
2
i
^
−
j
^
+
k
^
)
=
0
\bar{r}\cdot(2\hat{i}-\hat{j}+\hat{k})=0
r
ˉ
⋅
(
2
i
^
−
j
^
+
k
^
)
=
0
.
Show model answer
[QEx 6.4 Q.2 · Maharashtra State Board (Class 12) — Line and Planes (Balbharati textbook)]
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Q948
#948
Mathematics → Line and Planes → Coplanarity of Two Lines
·
Moderate
Add
Show that lines
r
ˉ
=
(
2
j
^
−
3
k
^
)
+
λ
(
i
^
+
2
j
^
+
3
k
^
)
\bar{r}=(2\hat{j}-3\hat{k})+\lambda(\hat{i}+2\hat{j}+3\hat{k})
r
ˉ
=
(
2
j
^
−
3
k
^
)
+
λ
(
i
^
+
2
j
^
+
3
k
^
)
and
r
ˉ
=
(
2
i
^
+
6
j
^
+
3
k
^
)
+
μ
(
2
i
^
+
3
j
^
+
4
k
^
)
\bar{r}=(2\hat{i}+6\hat{j}+3\hat{k})+\mu(2\hat{i}+3\hat{j}+4\hat{k})
r
ˉ
=
(
2
i
^
+
6
j
^
+
3
k
^
)
+
μ
(
2
i
^
+
3
j
^
+
4
k
^
)
are coplanar. Find the equation of the plane determined by them.
Show model answer
[QEx 6.4 Q.3 · Maharashtra State Board (Class 12) — Line and Planes (Balbharati textbook)]
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Q949
#949
Mathematics → Line and Planes → Distance of a Point from a Plane
·
Moderate
Add
Find the distance of the point
4
i
^
−
3
j
^
+
k
^
4\hat{i}-3\hat{j}+\hat{k}
4
i
^
−
3
j
^
+
k
^
from the plane
r
ˉ
⋅
(
2
i
^
+
3
j
^
−
6
k
^
)
=
21
\bar{r}\cdot(2\hat{i}+3\hat{j}-6\hat{k})=21
r
ˉ
⋅
(
2
i
^
+
3
j
^
−
6
k
^
)
=
21
.
Show model answer
[QEx 6.4 Q.4 · Maharashtra State Board (Class 12) — Line and Planes (Balbharati textbook)]
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Q950
#950
Mathematics → Line and Planes → Distance of a Point from a Plane
·
Moderate
Add
Find the distance of the point
(
1
,
1
,
−
1
)
(1, 1, -1)
(
1
,
1
,
−
1
)
from the plane
3
x
+
4
y
−
12
z
+
20
=
0
3x+4y-12z+20=0
3
x
+
4
y
−
12
z
+
20
=
0
.
Show model answer
[QEx 6.4 Q.5 · Maharashtra State Board (Class 12) — Line and Planes (Balbharati textbook)]
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