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Exam: Maharashtra HSC Class 12
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Set · 2 questions
Find the distance from (4, -2, 6) to each of the following:
Q1076
#1076
Mathematics → Vectors → Vectors and Their Types
·
Easy
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The Y-axis
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[Q5.1 Ex.Q8 e) · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1077
#1077
Mathematics → Vectors → Vectors and Their Types
·
Easy
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The Z-axis
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[Q5.1 Ex.Q8 f) · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Set · 2 questions
Find the co-ordinates of the point which divides the line segment joining the points A(2, –6, 8) and B(–1, 3, –4).
Q1078
#1078
Mathematics → Vectors → Section Formula
·
Moderate
Add
(i) Internally in the ratio 1 : 3.
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[Q5.2 SolvedEx.1 i) · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1079
#1079
Mathematics → Vectors → Section Formula
·
Moderate
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(ii) Externally in the ratio 1 : 3.
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[Q5.2 SolvedEx.1 ii) · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1080
#1080
Mathematics → Vectors → Section Formula
·
Hard
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If the three points A(3, 2, p), B(q, 8, –10), C(–2, –3, 1) are collinear then find (i) the ratio in which the point C divides the line segment AB, (ii) the values of p and q.
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[Q5.2 SolvedEx.2 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1081
#1081
Mathematics → Vectors → Section Formula
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If A(5, 1, p), B(1, q, p) and C(1, –2, 3) are vertices of a triangle and
G
(
r
,
−
4
3
,
1
3
)
G\left(r, -\dfrac{4}{3}, \dfrac{1}{3}\right)
G
(
r
,
−
3
4
,
3
1
)
is its centroid, then find the values of p, q and r.
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[Q5.2 SolvedEx.3 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1082
#1082
Mathematics → Vectors → Section Formula
·
Moderate
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If
a
⃗
\vec{a}
a
,
b
⃗
\vec{b}
b
,
c
⃗
\vec{c}
c
are the position vectors of the points A, B, C respectively and
5
a
⃗
−
3
b
⃗
−
2
c
⃗
=
0
⃗
5\vec{a} - 3\vec{b} - 2\vec{c} = \vec{0}
5
a
−
3
b
−
2
c
=
0
, then find the ratio in which the point C divides the line segment BA.
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[Q5.2 SolvedEx.4 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1083
#1083
Mathematics → Vectors → Section Formula
·
Hard
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Prove that the medians of a triangle are concurrent.
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[Q5.2 SolvedEx.5 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1084
#1084
Mathematics → Vectors → Section Formula
·
Hard
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Prove that the angle bisectors of a triangle are concurrent.
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[Q5.2 SolvedEx.6 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1085
#1085
Mathematics → Vectors → Section Formula
·
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Using vector method, find the incenter of the triangle whose vertices are A(0, 3, 0), B(0, 0, 4) and C(0, 3, 4).
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[Q5.2 SolvedEx.7 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1086
#1086
Mathematics → Vectors → Section Formula
·
Hard
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If
4
i
^
+
7
j
^
+
8
k
^
4\hat{i}+7\hat{j}+8\hat{k}
4
i
^
+
7
j
^
+
8
k
^
,
2
i
^
+
3
j
^
+
4
k
^
2\hat{i}+3\hat{j}+4\hat{k}
2
i
^
+
3
j
^
+
4
k
^
and
2
i
^
+
5
j
^
+
7
k
^
2\hat{i}+5\hat{j}+7\hat{k}
2
i
^
+
5
j
^
+
7
k
^
are the position vectors of the vertices A, B and C respectively of triangle ABC, find the position vector of the point in which the bisector of
∠
A
\angle A
∠
A
meets BC.
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[Q5.2 SolvedEx.8 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1087
#1087
Mathematics → Vectors → Section Formula
·
Easy
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If G(a, 2, –1) is the centroid of the triangle with vertices P(1, 3, 2), Q(3, b, –4) and R(5, 1, c), then find the values of a, b and c.
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[Q5.2 SolvedEx.9 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1088
#1088
Mathematics → Vectors → Section Formula
·
Easy
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Find the centroid of the tetrahedron with vertices A(3, –5, 7), B(5, 4, 2), C(7, –7, –3), D(1, 0, 2).
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[Q5.2 SolvedEx.10 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1089
#1089
Mathematics → Vectors → Section Formula
·
Hard
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Find the ratio in which point P divides AB and CD where A(2, –3, 4), B(0, 5, 2), C(–1, 5, 3) and D(2, –1, 3). Also, find its coordinates.
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[Q5.2 SolvedEx.11 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1090
#1090
Mathematics → Vectors → Section Formula
·
Hard
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In a triangle ABC, D and E are points on BC and AC respectively, such that BD = 2DC and AE = 3EC. Let P be the point of intersection of AD and BE. Find BP/PF using vector methods.
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[Q5.2 SolvedEx.12 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Set · 2 questions
Find the position vector of point R which divides the line joining the points P and Q whose position vectors are
2
i
^
−
j
^
+
3
k
^
2\hat{i}-\hat{j}+3\hat{k}
2
i
^
−
j
^
+
3
k
^
and
−
5
i
^
+
2
j
^
−
5
k
^
-5\hat{i}+2\hat{j}-5\hat{k}
−
5
i
^
+
2
j
^
−
5
k
^
in the ratio 3 : 2.
Q1091
#1091
Mathematics → Vectors → Section Formula
·
Easy
Add
(i) Internally.
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[Q5.2 Ex.Q1 i) · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1092
#1092
Mathematics → Vectors → Section Formula
·
Easy
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(ii) Externally.
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[Q5.2 Ex.Q1 ii) · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1093
#1093
Mathematics → Vectors → Section Formula
·
Easy
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Find the position vector of mid-point M joining the points L(7, –6, 12) and N(5, 4, –2).
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[Q5.2 Ex.Q2 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Set · 2 questions
If the points A(3, 0, p), B(–1, q, 3) and C(–3, 3, 0) are collinear, then find
Q1094
#1094
Mathematics → Vectors → Section Formula
·
Moderate
Add
(i) The ratio in which the point C divides the line segment AB.
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[Q5.2 Ex.Q3 i) · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1095
#1095
Mathematics → Vectors → Section Formula
·
Moderate
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(ii) The values of p and q.
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[Q5.2 Ex.Q3 ii) · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1096
#1096
Mathematics → Vectors → Section Formula
·
Moderate
Add
The position vectors of points A and B are
6
a
⃗
+
2
b
⃗
6\vec{a}+2\vec{b}
6
a
+
2
b
and
a
⃗
−
3
b
⃗
\vec{a}-3\vec{b}
a
−
3
b
. If the point C divides AB in the ratio 3 : 2 then show that the position vector of C is
3
a
⃗
−
b
⃗
3\vec{a}-\vec{b}
3
a
−
b
.
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[Q5.2 Ex.Q4 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1097
#1097
Mathematics → Vectors → Section Formula
·
Hard
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Prove that the line segments joining mid-points of adjacent sides of a quadrilateral form a parallelogram.
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[Q5.2 Ex.Q5 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1098
#1098
Mathematics → Vectors → Section Formula
·
Moderate
Add
D and E divide sides BC and CA of a triangle ABC in the ratio 2 : 3 respectively. Find the position vector of the point of intersection of AD and BE and the ratio in which this point divides AD and BE.
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[Q5.2 Ex.Q6 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1099
#1099
Mathematics → Vectors → Section Formula
·
Hard
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Prove that a quadrilateral is a parallelogram if and only if its diagonals bisect each other.
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[Q5.2 Ex.Q7 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1100
#1100
Mathematics → Vectors → Section Formula
·
Hard
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Prove that the median of a trapezium is parallel to the parallel sides of the trapezium and its length is half the sum of the parallel sides.
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[Q5.2 Ex.Q8 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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