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Exam: Maharashtra HSC Class 12
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Q1026
#1026
Mathematics → Vectors → Cross Product of Vectors
·
Moderate
Add
If
A
(
3
,
2
,
−
1
)
A(3,2,-1)
A
(
3
,
2
,
−
1
)
,
B
(
−
2
,
2
,
−
3
)
B(-2,2,-3)
B
(
−
2
,
2
,
−
3
)
,
C
(
3
,
5
,
−
2
)
C(3,5,-2)
C
(
3
,
5
,
−
2
)
,
D
(
−
2
,
5
,
−
4
)
D(-2,5,-4)
D
(
−
2
,
5
,
−
4
)
then (i) verify that the points are the vertices of a parallelogram and (ii) find its area.
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[QMisc II Q.37 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1027
#1027
Mathematics → Vectors → Cross Product of Vectors
·
Hard
Add
Let A, B, C, D be any four points in space. Prove that
∣
A
B
→
×
C
D
→
+
B
C
→
×
A
D
→
+
C
A
→
×
B
D
→
∣
=
4
(
area of
△
A
B
C
)
|\overrightarrow{AB}\times\overrightarrow{CD}+\overrightarrow{BC}\times\overrightarrow{AD}+\overrightarrow{CA}\times\overrightarrow{BD}| = 4(\text{area of } \triangle ABC)
∣
A
B
×
C
D
+
B
C
×
A
D
+
C
A
×
B
D
∣
=
4
(
area of
△
A
B
C
)
.
Show model answer
[QMisc II Q.38 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1028
#1028
Mathematics → Vectors → Cross Product of Vectors
·
Hard
Add
Let
a
^
,
b
^
,
c
^
\hat{a}, \hat{b}, \hat{c}
a
^
,
b
^
,
c
^
be unit vectors such that
a
^
⋅
b
^
=
a
^
⋅
c
^
=
0
\hat{a}\cdot\hat{b} = \hat{a}\cdot\hat{c} = 0
a
^
⋅
b
^
=
a
^
⋅
c
^
=
0
and the angle between
b
^
\hat{b}
b
^
and
c
^
\hat{c}
c
^
be
π
/
6
\pi/6
π
/6
. Prove that
a
^
=
±
2
(
b
^
×
c
^
)
\hat{a} = \pm 2(\hat{b}\times\hat{c})
a
^
=
±
2
(
b
^
×
c
^
)
.
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[QMisc II Q.39 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1029
#1029
Mathematics → Vectors → Scalar and Vector Triple Product
·
Hard
Add
Find the value of 'a' so that the volume of parallelopiped formed by
i
^
+
a
j
^
+
k
^
\hat{i}+a\hat{j}+\hat{k}
i
^
+
a
j
^
+
k
^
,
j
^
+
a
k
^
\hat{j}+a\hat{k}
j
^
+
a
k
^
and
a
i
^
+
k
^
a\hat{i}+\hat{k}
a
i
^
+
k
^
becomes minimum.
Show model answer
[QMisc II Q.40 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1030
#1030
Mathematics → Vectors → Scalar and Vector Triple Product
·
Hard
Add
Find the volume of the parallelepiped spanned by the diagonals of the three faces of a cube of side
a
a
a
that meet at one vertex of the cube.
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[QMisc II Q.41 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1031
#1031
Mathematics → Vectors → Scalar and Vector Triple Product
·
Hard
Add
If
a
⃗
,
b
⃗
,
c
⃗
\vec{a}, \vec{b}, \vec{c}
a
,
b
,
c
are three non-coplanar vectors, then show that
a
⃗
⋅
(
b
⃗
×
c
⃗
)
(
c
⃗
×
a
⃗
)
⋅
b
⃗
+
b
⃗
⋅
(
a
⃗
×
c
⃗
)
(
c
⃗
×
a
⃗
)
⋅
b
⃗
=
0
\dfrac{\vec{a}\cdot(\vec{b}\times\vec{c})}{(\vec{c}\times\vec{a})\cdot\vec{b}} + \dfrac{\vec{b}\cdot(\vec{a}\times\vec{c})}{(\vec{c}\times\vec{a})\cdot\vec{b}} = 0
(
c
×
a
)
⋅
b
a
⋅
(
b
×
c
)
+
(
c
×
a
)
⋅
b
b
⋅
(
a
×
c
)
=
0
.
Show model answer
[QMisc II Q.42 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1032
#1032
Mathematics → Vectors → Scalar and Vector Triple Product
·
Hard
Add
Prove that
(
a
⃗
×
b
⃗
)
⋅
(
c
⃗
×
d
⃗
)
=
∣
a
⃗
⋅
c
⃗
b
⃗
⋅
c
⃗
a
⃗
⋅
d
⃗
b
⃗
⋅
d
⃗
∣
(\vec{a}\times\vec{b})\cdot(\vec{c}\times\vec{d}) = \begin{vmatrix}\vec{a}\cdot\vec{c} & \vec{b}\cdot\vec{c}\\ \vec{a}\cdot\vec{d} & \vec{b}\cdot\vec{d}\end{vmatrix}
(
a
×
b
)
⋅
(
c
×
d
)
=
a
⋅
c
a
⋅
d
b
⋅
c
b
⋅
d
.
Show model answer
[QMisc II Q.43 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1033
#1033
Mathematics → Vectors → Scalar and Vector Triple Product
·
Moderate
Add
Find the volume of the parallelopiped whose coterminus edges are represented by the vector
j
^
+
k
^
\hat{j}+\hat{k}
j
^
+
k
^
,
i
^
+
k
^
\hat{i}+\hat{k}
i
^
+
k
^
and
i
^
+
j
^
\hat{i}+\hat{j}
i
^
+
j
^
. Also find volume of tetrahedron having these coterminous edges.
Show model answer
[QMisc II Q.44 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1034
#1034
Mathematics → Vectors → Scalar and Vector Triple Product
·
Hard
Add
Using properties of scalar triple product, prove that
[
a
⃗
+
b
⃗
b
⃗
+
c
⃗
c
⃗
+
a
⃗
]
=
2
[
a
⃗
b
⃗
c
⃗
]
[\vec{a}+\vec{b}\ \ \vec{b}+\vec{c}\ \ \vec{c}+\vec{a}] = 2[\vec{a}\ \vec{b}\ \vec{c}]
[
a
+
b
b
+
c
c
+
a
]
=
2
[
a
b
c
]
.
Show model answer
[QMisc II Q.45 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1035
#1035
Mathematics → Vectors → Scalar and Vector Triple Product
·
Hard
Add
If four points
A
(
a
⃗
)
A(\vec{a})
A
(
a
)
,
B
(
b
⃗
)
B(\vec{b})
B
(
b
)
,
C
(
c
⃗
)
C(\vec{c})
C
(
c
)
and
D
(
d
⃗
)
D(\vec{d})
D
(
d
)
are coplanar then show that
[
a
⃗
b
⃗
d
⃗
]
+
[
b
⃗
c
⃗
d
⃗
]
+
[
c
⃗
a
⃗
d
⃗
]
=
[
a
⃗
b
⃗
c
⃗
]
[\vec{a}\ \vec{b}\ \vec{d}] + [\vec{b}\ \vec{c}\ \vec{d}] + [\vec{c}\ \vec{a}\ \vec{d}] = [\vec{a}\ \vec{b}\ \vec{c}]
[
a
b
d
]
+
[
b
c
d
]
+
[
c
a
d
]
=
[
a
b
c
]
.
Show model answer
[QMisc II Q.46 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1036
#1036
Mathematics → Vectors → Scalar and Vector Triple Product
·
Hard
Add
If
a
⃗
,
b
⃗
\vec{a}, \vec{b}
a
,
b
and
c
⃗
\vec{c}
c
are three non coplanar vectors, then
(
a
⃗
+
b
⃗
+
c
⃗
)
⋅
[
(
a
⃗
+
b
⃗
)
×
(
a
⃗
+
c
⃗
)
]
=
−
[
a
⃗
b
⃗
c
⃗
]
(\vec{a}+\vec{b}+\vec{c})\cdot\left[(\vec{a}+\vec{b})\times(\vec{a}+\vec{c})\right] = -[\vec{a}\ \vec{b}\ \vec{c}]
(
a
+
b
+
c
)
⋅
[
(
a
+
b
)
×
(
a
+
c
)
]
=
−
[
a
b
c
]
.
Show model answer
[QMisc II Q.47 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1037
#1037
Mathematics → Vectors → Dot Product of Vectors
·
Hard
Add
If in a tetrahedron, edges in each of the two pairs of opposite edges are perpendicular, then show that the edges in the third pair are also perpendicular.
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[QMisc II Q.48 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Set · 2 questions
Find the coordinates of the point which is located :
Q1038
#1038
Mathematics → Vectors → Vectors and Their Types
·
Easy
Add
Three units behind the YZ-plane, four units to the right of the XZ-plane and five units above the XY-plane.
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[Q5.1 Ex.Q9 a) · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1039
#1039
Mathematics → Vectors → Vectors and Their Types
·
Easy
Add
In the YZ-plane, one unit to the right of the XZ-plane and six units above the XY-plane.
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[Q5.1 Ex.Q9 b) · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1040
#1040
Mathematics → Vectors → Vectors and Their Types
·
Moderate
Add
Find the area of the triangle with vertices
(
1
,
1
,
0
)
(1, 1, 0)
(
1
,
1
,
0
)
,
(
1
,
0
,
1
)
(1, 0, 1)
(
1
,
0
,
1
)
and
(
0
,
1
,
1
)
(0, 1, 1)
(
0
,
1
,
1
)
.
Show model answer
[Q5.1 Ex.Q10 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1041
#1041
Mathematics → Vectors → Vectors and Their Types
·
Easy
Add
If
A
B
⃗
=
2
i
^
−
4
j
^
+
7
k
^
\vec{AB} = 2\hat{i} - 4\hat{j} + 7\hat{k}
A
B
=
2
i
^
−
4
j
^
+
7
k
^
and initial point
A
≡
(
1
,
5
,
0
)
A \equiv (1, 5, 0)
A
≡
(
1
,
5
,
0
)
. Find the terminal point B.
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[Q5.1 Ex.Q11 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Set · 2 questions
Show that the following points are collinear :
Q1042
#1042
Mathematics → Vectors → Vectors and Their Types
·
Moderate
Add
A (3, 2, -4), B (9, 8, -10), C (-2, -3, 1).
Show model answer
[Q5.1 Ex.Q12 i) · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1043
#1043
Mathematics → Vectors → Vectors and Their Types
·
Moderate
Add
P (4, 5, 2), Q (3, 2, 4), R (5, 8, 0).
Show model answer
[Q5.1 Ex.Q12 ii) · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1044
#1044
Mathematics → Vectors → Vectors and Their Types
·
Moderate
Add
If the vectors
2
i
^
−
q
j
^
+
3
k
^
2\hat{i} - q\hat{j} + 3\hat{k}
2
i
^
−
q
j
^
+
3
k
^
and
4
i
^
−
5
j
^
+
6
k
^
4\hat{i} - 5\hat{j} + 6\hat{k}
4
i
^
−
5
j
^
+
6
k
^
are collinear, then find the value of
q
q
q
.
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[Q5.1 Ex.Q13 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1045
#1045
Mathematics → Vectors → Vectors and Their Types
·
Moderate
Add
Are the four points A(1, -1, 1), B(-1, 1, 1), C(1, 1, 1) and D(2, -3, 4) coplanar? Justify your answer.
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[Q5.1 Ex.Q14 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1046
#1046
Mathematics → Vectors → Vectors and Their Types
·
Hard
Add
Express
−
i
^
−
3
j
^
+
4
k
^
-\hat{i} - 3\hat{j} + 4\hat{k}
−
i
^
−
3
j
^
+
4
k
^
as linear combination of the vectors
2
i
^
+
j
^
−
4
k
^
2\hat{i} + \hat{j} - 4\hat{k}
2
i
^
+
j
^
−
4
k
^
,
2
i
^
−
j
^
+
3
k
^
2\hat{i} - \hat{j} + 3\hat{k}
2
i
^
−
j
^
+
3
k
^
and
3
i
^
+
j
^
−
2
k
^
3\hat{i} + \hat{j} - 2\hat{k}
3
i
^
+
j
^
−
2
k
^
.
Show model answer
[Q5.1 Ex.Q15 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1047
#1047
Mathematics → Vectors → Vectors and Their Types
·
Easy
Add
In Fig. 5.30, five vectors
a
⃗
,
b
⃗
,
c
⃗
,
d
⃗
,
e
⃗
\vec{a}, \vec{b}, \vec{c}, \vec{d}, \vec{e}
a
,
b
,
c
,
d
,
e
are shown as directed line segments:
a
⃗
\vec{a}
a
and
b
⃗
\vec{b}
b
point to the right and are parallel to each other with
b
⃗
\vec{b}
b
drawn below
a
⃗
\vec{a}
a
;
c
⃗
\vec{c}
c
and
d
⃗
\vec{d}
d
point to the left and are parallel to each other with
d
⃗
\vec{d}
d
drawn below
c
⃗
\vec{c}
c
;
e
⃗
\vec{e}
e
is a short vector at the bottom pointing to the right, parallel to
a
⃗
\vec{a}
a
and
b
⃗
\vec{b}
b
. State the vectors which are: (i) equal in magnitude (ii) parallel (iii) in the same direction (iv) equal (v) negatives of one another.
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[Q5.1 SolvedEx.1 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1048
#1048
Mathematics → Vectors → Vectors and Their Types
·
Moderate
Add
In the diagram (Fig. 5.31, quadrilateral KLNM with diagonals KN and LM intersecting at T)
K
L
⃗
=
a
⃗
\vec{KL}=\vec{a}
K
L
=
a
,
L
N
⃗
=
b
⃗
\vec{LN}=\vec{b}
L
N
=
b
,
N
M
⃗
=
c
⃗
\vec{NM}=\vec{c}
N
M
=
c
and
K
T
⃗
=
d
⃗
\vec{KT}=\vec{d}
K
T
=
d
. Find in terms of
a
⃗
,
b
⃗
,
c
⃗
\vec{a}, \vec{b}, \vec{c}
a
,
b
,
c
and
d
⃗
\vec{d}
d
: (i)
L
T
⃗
\vec{LT}
L
T
(ii)
K
M
⃗
\vec{KM}
K
M
(iii)
T
N
⃗
\vec{TN}
T
N
(iv)
M
T
⃗
\vec{MT}
M
T
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[Q5.1 SolvedEx.2 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1049
#1049
Mathematics → Vectors → Vectors and Their Types
·
Easy
Add
Find the magnitude of the following vectors: (i)
a
⃗
=
i
^
−
2
j
^
+
4
k
^
\vec{a}=\hat{i}-2\hat{j}+4\hat{k}
a
=
i
^
−
2
j
^
+
4
k
^
(ii)
b
⃗
=
4
i
^
−
3
j
^
−
7
k
^
\vec{b}=4\hat{i}-3\hat{j}-7\hat{k}
b
=
4
i
^
−
3
j
^
−
7
k
^
(iii) a vector with initial point
(
1
,
−
3
,
4
)
(1,-3,4)
(
1
,
−
3
,
4
)
; terminal point
(
1
,
0
,
−
1
)
(1,0,-1)
(
1
,
0
,
−
1
)
.
Show model answer
[Q5.1 SolvedEx.3 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1050
#1050
Mathematics → Vectors → Vectors and Their Types
·
Moderate
Add
A(2, 3), B(-1, 5), C(-1, 1) and D(-7, 5) are four points in the Cartesian plane. (i) Find
A
B
⃗
\vec{AB}
A
B
and
C
D
⃗
\vec{CD}
C
D
. (ii) Check if
C
D
⃗
\vec{CD}
C
D
is parallel to
A
B
⃗
\vec{AB}
A
B
. (iii) E is the point
(
k
,
1
)
(k, 1)
(
k
,
1
)
and
A
C
⃗
\vec{AC}
A
C
is parallel to
B
E
⃗
\vec{BE}
B
E
. Find
k
k
k
.
Show model answer
[Q5.1 SolvedEx.4 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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