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Exam: Maharashtra HSC Class 12
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Q876
#876
Mathematics → Line and Planes → Coplanarity of Two Lines
·
Hard
Add
Show that lines
r
ˉ
=
(
i
^
+
4
j
^
)
+
λ
(
i
^
+
2
j
^
+
3
k
^
)
\bar{r}=\left(\hat{i}+4\hat{j}\right)+\lambda\left(\hat{i}+2\hat{j}+3\hat{k}\right)
r
ˉ
=
(
i
^
+
4
j
^
)
+
λ
(
i
^
+
2
j
^
+
3
k
^
)
and
r
ˉ
=
(
3
j
^
−
k
^
)
+
μ
(
2
i
^
+
3
j
^
+
4
k
^
)
\bar{r}=\left(3\hat{j}-\hat{k}\right)+\mu\left(2\hat{i}+3\hat{j}+4\hat{k}\right)
r
ˉ
=
(
3
j
^
−
k
^
)
+
μ
(
2
i
^
+
3
j
^
+
4
k
^
)
are coplanar. Find the equation of the plane determined by them.
Show model answer
[QMisc II Q.16 · Maharashtra State Board (Class 12) — Line and Planes (Balbharati textbook)]
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Q877
#877
Mathematics → Line and Planes → Distance of a Point from a Plane
·
Moderate
Add
Find the distance of the point
3
i
^
+
3
j
^
+
k
^
3\hat{i}+3\hat{j}+\hat{k}
3
i
^
+
3
j
^
+
k
^
from the plane
r
ˉ
⋅
(
2
i
^
+
3
j
^
+
6
k
^
)
=
21
\bar{r}\cdot\left(2\hat{i}+3\hat{j}+6\hat{k}\right)=21
r
ˉ
⋅
(
2
i
^
+
3
j
^
+
6
k
^
)
=
21
.
Show model answer
[QMisc II Q.17 · Maharashtra State Board (Class 12) — Line and Planes (Balbharati textbook)]
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Q878
#878
Mathematics → Line and Planes → Distance of a Point from a Plane
·
Easy
Add
Find the distance of the point
(
13
,
13
,
−
13
)
(13,13,-13)
(
13
,
13
,
−
13
)
from the plane
3
x
+
4
y
−
12
z
=
0
3x+4y-12z=0
3
x
+
4
y
−
12
z
=
0
.
Show model answer
[QMisc II Q.18 · Maharashtra State Board (Class 12) — Line and Planes (Balbharati textbook)]
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Q879
#879
Mathematics → Line and Planes → Equations of a Plane
·
Hard
Add
Find the vector equation of the plane passing through the origin and containing the line
r
ˉ
=
(
i
^
+
4
j
^
+
k
^
)
+
λ
(
i
^
+
2
j
^
+
k
^
)
\bar{r}=\left(\hat{i}+4\hat{j}+\hat{k}\right)+\lambda\left(\hat{i}+2\hat{j}+\hat{k}\right)
r
ˉ
=
(
i
^
+
4
j
^
+
k
^
)
+
λ
(
i
^
+
2
j
^
+
k
^
)
.
Show model answer
[QMisc II Q.19 · Maharashtra State Board (Class 12) — Line and Planes (Balbharati textbook)]
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Q880
#880
Mathematics → Line and Planes → Equations of a Plane
·
Moderate
Add
Find the vector equation of the plane which bisects the segment joining
A
(
2
,
3
,
6
)
A(2,3,6)
A
(
2
,
3
,
6
)
and
B
(
4
,
3
,
−
2
)
B(4,3,-2)
B
(
4
,
3
,
−
2
)
at right angle.
Show model answer
[QMisc II Q.20 · Maharashtra State Board (Class 12) — Line and Planes (Balbharati textbook)]
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Q881
#881
Mathematics → Line and Planes → Coplanarity of Two Lines
·
Hard
Add
Show that lines
x
=
y
,
z
=
0
x=y,\ z=0
x
=
y
,
z
=
0
and
x
+
y
=
0
,
z
=
0
x+y=0,\ z=0
x
+
y
=
0
,
z
=
0
intersect each other. Find the vector equation of the plane determined by them.
Show model answer
[QMisc II Q.21 · Maharashtra State Board (Class 12) — Line and Planes (Balbharati textbook)]
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Q882
#882
Mathematics → Line and Planes → Vector and Cartesian Equations of a Line
·
Easy
Add
Verify that point having position vector
4
i
^
−
11
j
^
+
2
k
^
4\hat{i}-11\hat{j}+2\hat{k}
4
i
^
−
11
j
^
+
2
k
^
lies on the line
r
ˉ
=
(
6
i
^
−
4
j
^
+
5
k
^
)
+
λ
(
2
i
^
+
7
j
^
+
3
k
^
)
\bar{r}=(6\hat{i}-4\hat{j}+5\hat{k})+\lambda(2\hat{i}+7\hat{j}+3\hat{k})
r
ˉ
=
(
6
i
^
−
4
j
^
+
5
k
^
)
+
λ
(
2
i
^
+
7
j
^
+
3
k
^
)
.
Show model answer
[Q6.1 SolvedEx.1 · Maharashtra State Board (Class 12) — Line and Planes (Balbharati textbook)]
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Q883
#883
Mathematics → Line and Planes → Vector and Cartesian Equations of a Line
·
Easy
Add
Find the vector equation of the line passing through the point having position vector
4
i
^
−
j
^
+
2
k
^
4\hat{i}-\hat{j}+2\hat{k}
4
i
^
−
j
^
+
2
k
^
and parallel to vector
−
2
i
^
−
j
^
+
k
^
-2\hat{i}-\hat{j}+\hat{k}
−
2
i
^
−
j
^
+
k
^
.
Show model answer
[Q6.1 SolvedEx.2 · Maharashtra State Board (Class 12) — Line and Planes (Balbharati textbook)]
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Q884
#884
Mathematics → Line and Planes → Vector and Cartesian Equations of a Line
·
Moderate
Add
Find the vector equation of the line passing through the point having position vector
2
i
^
+
j
^
−
3
k
^
2\hat{i}+\hat{j}-3\hat{k}
2
i
^
+
j
^
−
3
k
^
and perpendicular to vectors
i
^
+
j
^
+
k
^
\hat{i}+\hat{j}+\hat{k}
i
^
+
j
^
+
k
^
and
i
^
+
2
j
^
−
k
^
\hat{i}+2\hat{j}-\hat{k}
i
^
+
2
j
^
−
k
^
.
Show model answer
[Q6.1 SolvedEx.3 · Maharashtra State Board (Class 12) — Line and Planes (Balbharati textbook)]
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Q885
#885
Mathematics → Line and Planes → Vector and Cartesian Equations of a Line
·
Moderate
Add
Find the vector equation of the line passing through
2
i
^
+
j
^
−
k
^
2\hat{i}+\hat{j}-\hat{k}
2
i
^
+
j
^
−
k
^
and parallel to the line joining points
−
i
^
+
j
^
+
4
k
^
-\hat{i}+\hat{j}+4\hat{k}
−
i
^
+
j
^
+
4
k
^
and
i
^
+
2
j
^
+
2
k
^
\hat{i}+2\hat{j}+2\hat{k}
i
^
+
2
j
^
+
2
k
^
.
Show model answer
[Q6.1 SolvedEx.4 · Maharashtra State Board (Class 12) — Line and Planes (Balbharati textbook)]
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Q886
#886
Mathematics → Line and Planes → Vector and Cartesian Equations of a Line
·
Easy
Add
Find the vector equation of the line passing through
A
(
1
,
2
,
3
)
A(1,2,3)
A
(
1
,
2
,
3
)
and
B
(
2
,
3
,
4
)
B(2,3,4)
B
(
2
,
3
,
4
)
.
Show model answer
[Q6.1 SolvedEx.5 · Maharashtra State Board (Class 12) — Line and Planes (Balbharati textbook)]
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Q887
#887
Mathematics → Line and Planes → Vector and Cartesian Equations of a Line
·
Easy
Add
Find the Cartesian equations of the line passing through
A
(
1
,
2
,
3
)
A(1,2,3)
A
(
1
,
2
,
3
)
and having direction ratios
2
,
3
,
7
2,3,7
2
,
3
,
7
.
Show model answer
[Q6.1 SolvedEx.6 · Maharashtra State Board (Class 12) — Line and Planes (Balbharati textbook)]
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Q888
#888
Mathematics → Line and Planes → Vector and Cartesian Equations of a Line
·
Easy
Add
Find the Cartesian equations of the line passing through
A
(
1
,
2
,
3
)
A(1,2,3)
A
(
1
,
2
,
3
)
and
B
(
2
,
3
,
4
)
B(2,3,4)
B
(
2
,
3
,
4
)
.
Show model answer
[Q6.1 SolvedEx.7 · Maharashtra State Board (Class 12) — Line and Planes (Balbharati textbook)]
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Q889
#889
Mathematics → Line and Planes → Vector and Cartesian Equations of a Line
·
Moderate
Add
Find the Cartesian equations of the line passing through the point
A
(
2
,
1
,
−
3
)
A(2,1,-3)
A
(
2
,
1
,
−
3
)
and perpendicular to vectors
b
ˉ
=
i
^
+
j
^
+
k
^
\bar{b}=\hat{i}+\hat{j}+\hat{k}
b
ˉ
=
i
^
+
j
^
+
k
^
and
c
ˉ
=
i
^
+
2
j
^
−
k
^
\bar{c}=\hat{i}+2\hat{j}-\hat{k}
c
ˉ
=
i
^
+
2
j
^
−
k
^
.
Show model answer
[Q6.1 SolvedEx.8 · Maharashtra State Board (Class 12) — Line and Planes (Balbharati textbook)]
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Q890
#890
Mathematics → Line and Planes → Vector and Cartesian Equations of a Line
·
Moderate
Add
Find the angle between lines
r
ˉ
=
(
i
^
+
2
j
^
+
3
k
^
)
+
λ
(
2
i
^
−
2
j
^
+
k
^
)
\bar{r}=(\hat{i}+2\hat{j}+3\hat{k})+\lambda(2\hat{i}-2\hat{j}+\hat{k})
r
ˉ
=
(
i
^
+
2
j
^
+
3
k
^
)
+
λ
(
2
i
^
−
2
j
^
+
k
^
)
and
r
ˉ
=
(
i
^
+
2
j
^
+
3
k
^
)
+
λ
(
i
^
+
2
j
^
+
2
k
^
)
\bar{r}=(\hat{i}+2\hat{j}+3\hat{k})+\lambda(\hat{i}+2\hat{j}+2\hat{k})
r
ˉ
=
(
i
^
+
2
j
^
+
3
k
^
)
+
λ
(
i
^
+
2
j
^
+
2
k
^
)
.
Show model answer
[Q6.1 SolvedEx.9 · Maharashtra State Board (Class 12) — Line and Planes (Balbharati textbook)]
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Q891
#891
Mathematics → Line and Planes → Vector and Cartesian Equations of a Line
·
Hard
Add
Show that lines
r
ˉ
=
(
−
i
^
−
3
j
^
+
4
k
^
)
+
λ
(
−
10
i
^
−
j
^
+
k
^
)
\bar{r}=(-\hat{i}-3\hat{j}+4\hat{k})+\lambda(-10\hat{i}-\hat{j}+\hat{k})
r
ˉ
=
(
−
i
^
−
3
j
^
+
4
k
^
)
+
λ
(
−
10
i
^
−
j
^
+
k
^
)
and
r
ˉ
=
(
−
10
i
^
−
j
^
+
k
^
)
+
μ
(
−
i
^
−
3
j
^
+
4
k
^
)
\bar{r}=(-10\hat{i}-\hat{j}+\hat{k})+\mu(-\hat{i}-3\hat{j}+4\hat{k})
r
ˉ
=
(
−
10
i
^
−
j
^
+
k
^
)
+
μ
(
−
i
^
−
3
j
^
+
4
k
^
)
intersect each other. Find the position vector of their point of intersection.
Show model answer
[Q6.1 SolvedEx.10 · Maharashtra State Board (Class 12) — Line and Planes (Balbharati textbook)]
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Q892
#892
Mathematics → Line and Planes → Vector and Cartesian Equations of a Line
·
Hard
Add
Find the co-ordinates of points on the line
x
+
1
2
=
y
−
2
3
=
z
+
3
6
\frac{x+1}{2}=\frac{y-2}{3}=\frac{z+3}{6}
2
x
+
1
=
3
y
−
2
=
6
z
+
3
, which are at 3 unit distance from the base point
A
(
−
1
,
2
,
−
3
)
A(-1,2,-3)
A
(
−
1
,
2
,
−
3
)
.
Show model answer
[Q6.1 SolvedEx.11 · Maharashtra State Board (Class 12) — Line and Planes (Balbharati textbook)]
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Q893
#893
Mathematics → Line and Planes → Vector and Cartesian Equations of a Line
·
Easy
Add
Find the vector equation of the line passing through the point having position vector
−
2
i
^
+
j
^
+
k
^
-2\hat{i}+\hat{j}+\hat{k}
−
2
i
^
+
j
^
+
k
^
and parallel to vector
4
i
^
−
j
^
+
2
k
^
4\hat{i}-\hat{j}+2\hat{k}
4
i
^
−
j
^
+
2
k
^
.
Show model answer
[QEx 6.1 Q.1 · Maharashtra State Board (Class 12) — Line and Planes (Balbharati textbook)]
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Q894
#894
Mathematics → Line and Planes → Vector and Cartesian Equations of a Line
·
Easy
Add
Find the vector equation of the line passing through points having position vectors
3
i
^
+
4
j
^
−
7
k
^
3\hat{i}+4\hat{j}-7\hat{k}
3
i
^
+
4
j
^
−
7
k
^
and
6
i
^
−
j
^
+
k
^
6\hat{i}-\hat{j}+\hat{k}
6
i
^
−
j
^
+
k
^
.
Show model answer
[QEx 6.1 Q.2 · Maharashtra State Board (Class 12) — Line and Planes (Balbharati textbook)]
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Q895
#895
Mathematics → Line and Planes → Vector and Cartesian Equations of a Line
·
Easy
Add
Find the vector equation of line passing through the point having position vector
5
i
^
+
4
j
^
+
3
k
^
5\hat{i}+4\hat{j}+3\hat{k}
5
i
^
+
4
j
^
+
3
k
^
and having direction ratios
−
3
,
4
,
2
-3,4,2
−
3
,
4
,
2
.
Show model answer
[QEx 6.1 Q.3 · Maharashtra State Board (Class 12) — Line and Planes (Balbharati textbook)]
Report
Q896
#896
Mathematics → Line and Planes → Vector and Cartesian Equations of a Line
·
Moderate
Add
Find the vector equation of the line passing through the point having position vector
i
^
+
2
j
^
+
3
k
^
\hat{i}+2\hat{j}+3\hat{k}
i
^
+
2
j
^
+
3
k
^
and perpendicular to vectors
i
^
+
j
^
+
k
^
\hat{i}+\hat{j}+\hat{k}
i
^
+
j
^
+
k
^
and
2
i
^
−
j
^
+
k
^
2\hat{i}-\hat{j}+\hat{k}
2
i
^
−
j
^
+
k
^
.
Show model answer
[QEx 6.1 Q.4 · Maharashtra State Board (Class 12) — Line and Planes (Balbharati textbook)]
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Q897
#897
Mathematics → Line and Planes → Vector and Cartesian Equations of a Line
·
Easy
Add
Find the vector equation of the line passing through the point having position vector
−
i
^
−
j
^
+
2
k
^
-\hat{i}-\hat{j}+2\hat{k}
−
i
^
−
j
^
+
2
k
^
and parallel to the line
r
ˉ
=
(
i
^
+
2
j
^
+
3
k
^
)
+
λ
(
3
i
^
+
2
j
^
+
k
^
)
\bar{r}=(\hat{i}+2\hat{j}+3\hat{k})+\lambda(3\hat{i}+2\hat{j}+\hat{k})
r
ˉ
=
(
i
^
+
2
j
^
+
3
k
^
)
+
λ
(
3
i
^
+
2
j
^
+
k
^
)
.
Show model answer
[QEx 6.1 Q.5 · Maharashtra State Board (Class 12) — Line and Planes (Balbharati textbook)]
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Q898
#898
Mathematics → Line and Planes → Vector and Cartesian Equations of a Line
·
Easy
Add
Find the Cartesian equations of the line passing through
A
(
−
1
,
2
,
1
)
A(-1,2,1)
A
(
−
1
,
2
,
1
)
and having direction ratios
2
,
3
,
1
2,3,1
2
,
3
,
1
.
Show model answer
[QEx 6.1 Q.6 · Maharashtra State Board (Class 12) — Line and Planes (Balbharati textbook)]
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Q899
#899
Mathematics → Line and Planes → Vector and Cartesian Equations of a Line
·
Easy
Add
Find the Cartesian equations of the line passing through
A
(
2
,
2
,
1
)
A(2,2,1)
A
(
2
,
2
,
1
)
and
B
(
1
,
3
,
0
)
B(1,3,0)
B
(
1
,
3
,
0
)
.
Show model answer
[QEx 6.1 Q.7 · Maharashtra State Board (Class 12) — Line and Planes (Balbharati textbook)]
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Q900
#900
Mathematics → Line and Planes → Vector and Cartesian Equations of a Line
·
Moderate
Add
A
(
−
2
,
3
,
4
)
A(-2,3,4)
A
(
−
2
,
3
,
4
)
,
B
(
1
,
1
,
2
)
B(1,1,2)
B
(
1
,
1
,
2
)
and
C
(
4
,
−
1
,
0
)
C(4,-1,0)
C
(
4
,
−
1
,
0
)
are three points. Find the Cartesian equations of the line AB and show that points A, B, C are collinear.
Show model answer
[QEx 6.1 Q.8 · Maharashtra State Board (Class 12) — Line and Planes (Balbharati textbook)]
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