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Exam: Maharashtra HSC Class 12
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Q976
#976
Mathematics → Vectors → Vectors and Their Types
·
Moderate
Add
In a pentagon ABCDE, show that
A
B
→
+
A
E
→
+
B
C
→
+
D
C
→
+
E
D
→
=
2
A
C
→
\overrightarrow{AB} + \overrightarrow{AE} + \overrightarrow{BC} + \overrightarrow{DC} + \overrightarrow{ED} = 2\overrightarrow{AC}
A
B
+
A
E
+
B
C
+
D
C
+
E
D
=
2
A
C
.
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[QMisc II Q.3 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q977
#977
Mathematics → Vectors → Vectors and Their Types
·
Moderate
Add
If in parallelogram ABCD, diagonal vectors are
A
C
→
=
2
i
^
+
3
j
^
+
4
k
^
\overrightarrow{AC} = 2\hat{i}+3\hat{j}+4\hat{k}
A
C
=
2
i
^
+
3
j
^
+
4
k
^
and
B
D
→
=
−
6
i
^
+
7
j
^
−
2
k
^
\overrightarrow{BD} = -6\hat{i}+7\hat{j}-2\hat{k}
B
D
=
−
6
i
^
+
7
j
^
−
2
k
^
, then find the adjacent side vectors
A
B
→
\overrightarrow{AB}
A
B
and
A
D
→
\overrightarrow{AD}
A
D
.
Show model answer
[QMisc II Q.4 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q978
#978
Mathematics → Vectors → Vectors and Their Types
·
Easy
Add
If two sides of a triangle taken in the same order represented by vectors
i
^
+
2
j
^
\hat{i}+2\hat{j}
i
^
+
2
j
^
and
i
^
+
k
^
\hat{i}+\hat{k}
i
^
+
k
^
, then find the length of the third side.
Show model answer
[QMisc II Q.5 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q979
#979
Mathematics → Vectors → Dot Product of Vectors
·
Moderate
Add
If
∣
a
⃗
∣
=
∣
b
⃗
∣
=
1
|\vec{a}| = |\vec{b}| = 1
∣
a
∣
=
∣
b
∣
=
1
,
a
⃗
⋅
b
⃗
=
0
\vec{a}\cdot\vec{b}=0
a
⋅
b
=
0
and
a
⃗
+
b
⃗
+
c
⃗
=
0
⃗
\vec{a}+\vec{b}+\vec{c}=\vec{0}
a
+
b
+
c
=
0
then find
∣
c
⃗
∣
|\vec{c}|
∣
c
∣
.
Show model answer
[QMisc II Q.6 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Set · 2 questions
Find the lengths of the sides of the triangle and also determine the type of a triangle.
Q980
#980
Mathematics → Vectors → Vectors and Their Types
·
Moderate
Add
A
(
2
,
−
1
,
0
)
A(2,-1,0)
A
(
2
,
−
1
,
0
)
,
B
(
4
,
1
,
1
)
B(4,1,1)
B
(
4
,
1
,
1
)
,
C
(
4
,
−
5
,
4
)
C(4,-5,4)
C
(
4
,
−
5
,
4
)
Show model answer
[QMisc II Q.7 i) · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q981
#981
Mathematics → Vectors → Vectors and Their Types
·
Moderate
Add
L
(
3
,
−
2
,
−
3
)
L(3,-2,-3)
L
(
3
,
−
2
,
−
3
)
,
M
(
7
,
0
,
1
)
M(7,0,1)
M
(
7
,
0
,
1
)
,
N
(
1
,
2
,
1
)
N(1,2,1)
N
(
1
,
2
,
1
)
Show model answer
[QMisc II Q.7 ii) · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Set · 2 questions
Find the component form of
a
⃗
\vec{a}
a
if
Q982
#982
Mathematics → Vectors → Dot Product of Vectors
·
Moderate
Add
It lies in YZ plane and makes
60
∘
60^\circ
6
0
∘
with positive Y-axis and
∣
a
⃗
∣
=
4
|\vec{a}| = 4
∣
a
∣
=
4
Show model answer
[QMisc II Q.8 i) · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q983
#983
Mathematics → Vectors → Dot Product of Vectors
·
Moderate
Add
It lies in XZ plane and makes
45
∘
45^\circ
4
5
∘
with positive Z-axis and
∣
a
⃗
∣
=
10
|\vec{a}| = 10
∣
a
∣
=
10
Show model answer
[QMisc II Q.8 ii) · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q984
#984
Mathematics → Vectors → Vectors and Their Types
·
Moderate
Add
Two sides of a parallelogram are
3
i
^
+
4
j
^
−
5
k
^
3\hat{i}+4\hat{j}-5\hat{k}
3
i
^
+
4
j
^
−
5
k
^
and
−
2
j
^
+
7
k
^
-2\hat{j}+7\hat{k}
−
2
j
^
+
7
k
^
. Find the unit vectors parallel to the diagonals.
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[QMisc II Q.9 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q985
#985
Mathematics → Vectors → Section Formula
·
Moderate
Add
If D, E, F are the mid-points of the sides BC, CA, AB of a triangle ABC, prove that
A
D
→
+
B
E
→
+
C
F
→
=
0
⃗
\overrightarrow{AD} + \overrightarrow{BE} + \overrightarrow{CF} = \vec{0}
A
D
+
B
E
+
C
F
=
0
.
Show model answer
[QMisc II Q.10 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q986
#986
Mathematics → Vectors → Vectors and Their Types
·
Moderate
Add
Find the unit vectors that are parallel to the tangent line to the parabola
y
=
x
2
y = x^2
y
=
x
2
at the point
(
2
,
4
)
(2, 4)
(
2
,
4
)
.
Show model answer
[QMisc II Q.11 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q987
#987
Mathematics → Vectors → Vectors and Their Types
·
Hard
Add
Express the vector
i
^
+
4
j
^
−
4
k
^
\hat{i}+4\hat{j}-4\hat{k}
i
^
+
4
j
^
−
4
k
^
as a linear combination of the vectors
2
i
^
−
j
^
+
3
k
^
2\hat{i}-\hat{j}+3\hat{k}
2
i
^
−
j
^
+
3
k
^
,
i
^
−
2
j
^
+
4
k
^
\hat{i}-2\hat{j}+4\hat{k}
i
^
−
2
j
^
+
4
k
^
and
−
i
^
+
3
j
^
−
5
k
^
-\hat{i}+3\hat{j}-5\hat{k}
−
i
^
+
3
j
^
−
5
k
^
.
Show model answer
[QMisc II Q.12 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q988
#988
Mathematics → Vectors → Vectors and Their Types
·
Hard
Add
If
O
A
→
=
a
⃗
\overrightarrow{OA} = \vec{a}
O
A
=
a
and
O
B
→
=
b
⃗
\overrightarrow{OB} = \vec{b}
O
B
=
b
then show that vector along the angle bisector of angle AOB is given by
d
⃗
=
λ
(
a
⃗
∣
a
⃗
∣
+
b
⃗
∣
b
⃗
∣
)
\vec{d} = \lambda\left(\dfrac{\vec{a}}{|\vec{a}|}+\dfrac{\vec{b}}{|\vec{b}|}\right)
d
=
λ
(
∣
a
∣
a
+
∣
b
∣
b
)
, where
λ
\lambda
λ
is a real number.
Show model answer
[QMisc II Q.13 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q989
#989
Mathematics → Vectors → Vectors and Their Types
·
Moderate
Add
The position vectors of three consecutive vertices of a parallelogram are
i
^
+
j
^
+
k
^
\hat{i}+\hat{j}+\hat{k}
i
^
+
j
^
+
k
^
,
i
^
+
3
j
^
+
5
k
^
\hat{i}+3\hat{j}+5\hat{k}
i
^
+
3
j
^
+
5
k
^
and
7
i
^
+
9
j
^
+
11
k
^
7\hat{i}+9\hat{j}+11\hat{k}
7
i
^
+
9
j
^
+
11
k
^
. Find the position vector of the fourth vertex.
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[QMisc II Q.14 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q990
#990
Mathematics → Vectors → Section Formula
·
Moderate
Add
A point P with p.v.
−
14
i
^
+
39
j
^
+
28
k
^
5
\dfrac{-14\hat{i}+39\hat{j}+28\hat{k}}{5}
5
−
14
i
^
+
39
j
^
+
28
k
^
divides the line joining
A
(
−
1
,
6
,
5
)
A(-1, 6, 5)
A
(
−
1
,
6
,
5
)
and B internally in the ratio 3:2 then find the point B.
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[QMisc II Q.15 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q991
#991
Mathematics → Vectors → Section Formula
·
Moderate
Add
Prove that the sum of the three vectors determined by the medians of a triangle directed from the vertices is zero.
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[QMisc II Q.16 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q992
#992
Mathematics → Vectors → Section Formula
·
Hard
Add
ABCD is a parallelogram E, F are the mid points of BC and CD respectively. AE, AF meet the diagonal BD at Q and P respectively. Show that P and Q trisect DB.
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[QMisc II Q.17 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q993
#993
Mathematics → Vectors → Section Formula
·
Hard
Add
If ABC is a triangle whose orthocenter is P and the circumcenter is Q, then prove that
P
A
→
+
P
C
→
+
P
B
→
=
2
P
Q
→
\overrightarrow{PA} + \overrightarrow{PC} + \overrightarrow{PB} = 2\overrightarrow{PQ}
P
A
+
P
C
+
P
B
=
2
P
Q
.
Show model answer
[QMisc II Q.18 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q994
#994
Mathematics → Vectors → Section Formula
·
Hard
Add
If P is orthocenter, Q is circumcenter and G is centroid of triangle ABC, then prove that
Q
P
→
=
3
Q
G
→
\overrightarrow{QP} = 3\overrightarrow{QG}
QP
=
3
QG
.
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[QMisc II Q.19 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q995
#995
Mathematics → Vectors → Section Formula
·
Hard
Add
In triangle OAB, E is the midpoint of BO and D is a point on AB such that AD : DB = 2 : 1. If OD and AE intersect at P, determine the ratio OP : PD using vector methods.
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[QMisc II Q.20 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q996
#996
Mathematics → Vectors → Dot Product of Vectors
·
Moderate
Add
Dot-product of a vector with vectors
3
i
^
−
5
k
^
3\hat{i}-5\hat{k}
3
i
^
−
5
k
^
,
2
i
^
+
7
j
^
2\hat{i}+7\hat{j}
2
i
^
+
7
j
^
and
i
^
+
j
^
+
k
^
\hat{i}+\hat{j}+\hat{k}
i
^
+
j
^
+
k
^
are respectively -1, 6 and 5. Find the vector.
Show model answer
[QMisc II Q.21 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q997
#997
Mathematics → Vectors → Dot Product of Vectors
·
Moderate
Add
If
a
⃗
,
b
⃗
,
c
⃗
\vec{a}, \vec{b}, \vec{c}
a
,
b
,
c
are unit vectors such that
a
⃗
+
b
⃗
+
c
⃗
=
0
⃗
\vec{a}+\vec{b}+\vec{c}=\vec{0}
a
+
b
+
c
=
0
, then find the value of
a
⃗
⋅
b
⃗
+
b
⃗
⋅
c
⃗
+
c
⃗
⋅
a
⃗
\vec{a}\cdot\vec{b}+\vec{b}\cdot\vec{c}+\vec{c}\cdot\vec{a}
a
⋅
b
+
b
⋅
c
+
c
⋅
a
.
Show model answer
[QMisc II Q.22 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q998
#998
Mathematics → Vectors → Dot Product of Vectors
·
Hard
Add
If a parallelogram is constructed on the vectors
a
⃗
=
3
p
⃗
−
q
⃗
\vec{a} = 3\vec{p} - \vec{q}
a
=
3
p
−
q
,
b
⃗
=
p
⃗
+
3
q
⃗
\vec{b} = \vec{p} + 3\vec{q}
b
=
p
+
3
q
and
∣
p
⃗
∣
=
∣
q
⃗
∣
=
2
|\vec{p}| = |\vec{q}| = 2
∣
p
∣
=
∣
q
∣
=
2
and angle between
p
⃗
\vec{p}
p
and
q
⃗
\vec{q}
q
is
π
/
3
\pi/3
π
/3
show that the ratio of the lengths of the sides is
7
:
13
\sqrt{7} : \sqrt{13}
7
:
13
.
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[QMisc II Q.23 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q999
#999
Mathematics → Vectors → Dot Product of Vectors
·
Moderate
Add
Express the vector
a
⃗
=
5
i
^
−
2
j
^
+
5
k
^
\vec{a} = 5\hat{i}-2\hat{j}+5\hat{k}
a
=
5
i
^
−
2
j
^
+
5
k
^
as a sum of two vectors such that one is parallel to the vector
b
⃗
=
3
i
^
+
k
^
\vec{b} = 3\hat{i}+\hat{k}
b
=
3
i
^
+
k
^
and other is perpendicular to
b
⃗
\vec{b}
b
.
Show model answer
[QMisc II Q.24 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1000
#1000
Mathematics → Vectors → Dot Product of Vectors
·
Hard
Add
Find two unit vectors each of which makes equal angles with
u
⃗
,
v
⃗
\vec{u}, \vec{v}
u
,
v
and
w
⃗
\vec{w}
w
.
u
⃗
=
2
i
^
+
j
^
−
2
k
^
\vec{u} = 2\hat{i}+\hat{j}-2\hat{k}
u
=
2
i
^
+
j
^
−
2
k
^
,
v
⃗
=
i
^
+
2
j
^
−
2
k
^
\vec{v} = \hat{i}+2\hat{j}-2\hat{k}
v
=
i
^
+
2
j
^
−
2
k
^
and
w
⃗
=
2
i
^
−
2
j
^
+
k
^
\vec{w} = 2\hat{i}-2\hat{j}+\hat{k}
w
=
2
i
^
−
2
j
^
+
k
^
.
Show model answer
[QMisc II Q.25 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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