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Exam: Maharashtra HSC Class 12
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Q1051
#1051
Mathematics → Vectors → Vectors and Their Types
·
Easy
Add
Determine the values of
c
c
c
that satisfy
∣
c
u
⃗
∣
=
3
|c\vec{u}|=3
∣
c
u
∣
=
3
,
u
⃗
=
i
^
+
2
j
^
+
3
k
^
\vec{u}=\hat{i}+2\hat{j}+3\hat{k}
u
=
i
^
+
2
j
^
+
3
k
^
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[Q5.1 SolvedEx.5 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1052
#1052
Mathematics → Vectors → Vectors and Their Types
·
Easy
Add
Find a unit vector (i) in the direction of
u
⃗
\vec{u}
u
and (ii) in the direction opposite of
u
⃗
\vec{u}
u
, where
u
⃗
=
8
i
^
+
3
j
^
−
k
^
\vec{u}=8\hat{i}+3\hat{j}-\hat{k}
u
=
8
i
^
+
3
j
^
−
k
^
.
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[Q5.1 SolvedEx.6 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1053
#1053
Mathematics → Vectors → Vectors and Their Types
·
Easy
Add
Show that the vectors
2
i
^
−
3
j
^
+
4
k
^
2\hat{i}-3\hat{j}+4\hat{k}
2
i
^
−
3
j
^
+
4
k
^
and
−
4
i
^
+
6
j
^
−
8
k
^
-4\hat{i}+6\hat{j}-8\hat{k}
−
4
i
^
+
6
j
^
−
8
k
^
are parallel.
Show model answer
[Q5.1 SolvedEx.7 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1054
#1054
Mathematics → Vectors → Vectors and Their Types
·
Moderate
Add
The non-zero vectors
a
⃗
\vec{a}
a
and
b
⃗
\vec{b}
b
are not collinear. Find the value of
λ
\lambda
λ
and
μ
\mu
μ
: (i)
a
⃗
+
3
b
⃗
=
2
λ
a
⃗
−
μ
b
⃗
\vec{a}+3\vec{b}=2\lambda\vec{a}-\mu\vec{b}
a
+
3
b
=
2
λ
a
−
μ
b
(ii)
(
1
+
λ
)
a
⃗
+
2
λ
b
⃗
=
μ
a
⃗
+
4
μ
b
⃗
(1+\lambda)\vec{a}+2\lambda\vec{b}=\mu\vec{a}+4\mu\vec{b}
(
1
+
λ
)
a
+
2
λ
b
=
μ
a
+
4
μ
b
(iii)
(
3
λ
+
5
)
a
⃗
+
b
⃗
=
2
μ
a
⃗
+
(
λ
−
3
)
b
⃗
(3\lambda+5)\vec{a}+\vec{b}=2\mu\vec{a}+(\lambda-3)\vec{b}
(
3
λ
+
5
)
a
+
b
=
2
μ
a
+
(
λ
−
3
)
b
Show model answer
[Q5.1 SolvedEx.8 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1055
#1055
Mathematics → Vectors → Vectors and Their Types
·
Moderate
Add
Are the following sets of vectors linearly independent? (i)
a
⃗
=
i
^
−
2
j
^
+
3
k
^
\vec{a}=\hat{i}-2\hat{j}+3\hat{k}
a
=
i
^
−
2
j
^
+
3
k
^
,
b
⃗
=
3
i
^
−
6
j
^
+
9
k
^
\vec{b}=3\hat{i}-6\hat{j}+9\hat{k}
b
=
3
i
^
−
6
j
^
+
9
k
^
(ii)
a
⃗
=
−
2
i
^
−
4
k
^
\vec{a}=-2\hat{i}-4\hat{k}
a
=
−
2
i
^
−
4
k
^
,
b
⃗
=
i
^
−
2
j
^
−
k
^
\vec{b}=\hat{i}-2\hat{j}-\hat{k}
b
=
i
^
−
2
j
^
−
k
^
,
c
⃗
=
i
^
−
4
j
^
+
3
k
^
\vec{c}=\hat{i}-4\hat{j}+3\hat{k}
c
=
i
^
−
4
j
^
+
3
k
^
. Interpret the results.
Show model answer
[Q5.1 SolvedEx.9 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1056
#1056
Mathematics → Vectors → Vectors and Their Types
·
Easy
Add
If
a
⃗
=
4
i
^
+
3
k
^
\vec{a}=4\hat{i}+3\hat{k}
a
=
4
i
^
+
3
k
^
and
b
⃗
=
−
2
i
^
+
j
^
+
5
k
^
\vec{b}=-2\hat{i}+\hat{j}+5\hat{k}
b
=
−
2
i
^
+
j
^
+
5
k
^
, find (i)
∣
a
⃗
∣
|\vec{a}|
∣
a
∣
, (ii)
a
⃗
+
b
⃗
\vec{a}+\vec{b}
a
+
b
, (iii)
a
⃗
−
b
⃗
\vec{a}-\vec{b}
a
−
b
, (iv)
3
b
⃗
3\vec{b}
3
b
, (v)
2
a
⃗
+
5
b
⃗
2\vec{a}+5\vec{b}
2
a
+
5
b
Show model answer
[Q5.1 SolvedEx.10 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1057
#1057
Mathematics → Vectors → Vectors and Their Types
·
Easy
Add
What is the distance from the point (2, 3, 4) to (i) the XY plane? (ii) the X-axis? (iii) origin (iv) point (-2, 7, 3).
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[Q5.1 SolvedEx.11 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1058
#1058
Mathematics → Vectors → Vectors and Their Types
·
Moderate
Add
Prove that the line segment joining the midpoints of two sides of a triangle is parallel to and half of the third side.
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[Q5.1 SolvedEx.12 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1059
#1059
Mathematics → Vectors → Vectors and Their Types
·
Hard
Add
In quadrilateral ABCD, M and N are the mid-points of the diagonals AC and BD respectively. Prove that
A
B
⃗
+
A
D
⃗
+
C
B
⃗
+
C
D
⃗
=
4
M
N
⃗
\vec{AB}+\vec{AD}+\vec{CB}+\vec{CD}=4\vec{MN}
A
B
+
A
D
+
C
B
+
C
D
=
4
M
N
Show model answer
[Q5.1 SolvedEx.13 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1060
#1060
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 the 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 SolvedEx.14 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1061
#1061
Mathematics → Vectors → Vectors and Their Types
·
Moderate
Add
Show that the three points A(1, -2, 3), B(2, 3, -4) and C(0, -7, 10) are collinear.
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[Q5.1 SolvedEx.15 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1062
#1062
Mathematics → Vectors → Vectors and Their Types
·
Moderate
Add
Show that the vectors
4
i
^
+
13
j
^
−
18
k
^
4\hat{i}+13\hat{j}-18\hat{k}
4
i
^
+
13
j
^
−
18
k
^
,
i
^
−
2
j
^
+
3
k
^
\hat{i}-2\hat{j}+3\hat{k}
i
^
−
2
j
^
+
3
k
^
and
2
i
^
+
3
j
^
−
4
k
^
2\hat{i}+3\hat{j}-4\hat{k}
2
i
^
+
3
j
^
−
4
k
^
are coplanar.
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[Q5.1 SolvedEx.16 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1063
#1063
Mathematics → Vectors → Vectors and Their Types
·
Easy
Add
The vector
a
⃗
\vec{a}
a
is directed due north and
∣
a
⃗
∣
=
24
|\vec{a}|=24
∣
a
∣
=
24
. The vector
b
⃗
\vec{b}
b
is directed due west and
∣
b
⃗
∣
=
7
|\vec{b}|=7
∣
b
∣
=
7
. Find
∣
a
⃗
+
b
⃗
∣
|\vec{a}+\vec{b}|
∣
a
+
b
∣
.
Show model answer
[Q5.1 Ex.Q1 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Set · 3 questions
In the triangle PQR,
P
Q
⃗
=
2
a
⃗
\vec{PQ}=2\vec{a}
P
Q
=
2
a
and
Q
R
⃗
=
2
b
⃗
\vec{QR}=2\vec{b}
QR
=
2
b
. The mid-point of PR is M. Find the following vectors in terms of
a
⃗
\vec{a}
a
and
b
⃗
\vec{b}
b
.
Q1064
#1064
Mathematics → Vectors → Vectors and Their Types
·
Moderate
Add
P
R
⃗
\vec{PR}
P
R
Show model answer
[Q5.1 Ex.Q2 i) · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1065
#1065
Mathematics → Vectors → Vectors and Their Types
·
Moderate
Add
P
M
⃗
\vec{PM}
P
M
Show model answer
[Q5.1 Ex.Q2 ii) · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1066
#1066
Mathematics → Vectors → Vectors and Their Types
·
Moderate
Add
Q
M
⃗
\vec{QM}
QM
Show model answer
[Q5.1 Ex.Q2 iii) · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1067
#1067
Mathematics → Vectors → Vectors and Their Types
·
Hard
Add
OABCDE is a regular hexagon. The points A and B have position vectors
a
⃗
\vec{a}
a
and
b
⃗
\vec{b}
b
respectively, referred to the origin O. Find, in terms of
a
⃗
\vec{a}
a
and
b
⃗
\vec{b}
b
, the position vectors of C, D and E.
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[Q5.1 Ex.Q3 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1068
#1068
Mathematics → Vectors → Vectors and Their Types
·
Hard
Add
If ABCDEF is a regular hexagon, show that
A
B
⃗
+
A
C
⃗
+
A
D
⃗
+
A
E
⃗
+
A
F
⃗
=
6
A
O
⃗
\vec{AB}+\vec{AC}+\vec{AD}+\vec{AE}+\vec{AF}=6\vec{AO}
A
B
+
A
C
+
A
D
+
A
E
+
A
F
=
6
A
O
, where O is the centre of the hexagon.
Show model answer
[Q5.1 Ex.Q4 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1069
#1069
Mathematics → Vectors → Vectors and Their Types
·
Moderate
Add
Check whether the vectors
2
i
^
+
2
j
^
+
3
k
^
2\hat{i}+2\hat{j}+3\hat{k}
2
i
^
+
2
j
^
+
3
k
^
,
−
3
i
^
+
3
j
^
+
2
k
^
-3\hat{i}+3\hat{j}+2\hat{k}
−
3
i
^
+
3
j
^
+
2
k
^
and
3
i
^
+
4
k
^
3\hat{i}+4\hat{k}
3
i
^
+
4
k
^
form a triangle or not.
Show model answer
[Q5.1 Ex.Q5 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1070
#1070
Mathematics → Vectors → Vectors and Their Types
·
Moderate
Add
In Fig. 5.34,
Δ
P
Q
R
\Delta PQR
Δ
P
QR
is shown with S a point on side QR.
P
Q
⃗
=
a
⃗
\vec{PQ}=\vec{a}
P
Q
=
a
,
P
R
⃗
=
b
⃗
\vec{PR}=\vec{b}
P
R
=
b
,
P
S
⃗
=
c
⃗
\vec{PS}=\vec{c}
P
S
=
c
,
S
Q
⃗
=
−
d
⃗
\vec{SQ}=-\vec{d}
S
Q
=
−
d
and
S
R
⃗
=
d
⃗
\vec{SR}=\vec{d}
S
R
=
d
(so S is the midpoint of QR). Express
c
⃗
\vec{c}
c
and
d
⃗
\vec{d}
d
in terms of
a
⃗
\vec{a}
a
and
b
⃗
\vec{b}
b
.
Show model answer
[Q5.1 Ex.Q6 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1071
#1071
Mathematics → Vectors → Vectors and Their Types
·
Easy
Add
Find a vector in the direction of
a
⃗
=
i
^
−
2
j
^
\vec{a}=\hat{i}-2\hat{j}
a
=
i
^
−
2
j
^
that has magnitude 7 units.
Show model answer
[Q5.1 Ex.Q7 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Set · 4 questions
Find the distance from (4, -2, 6) to each of the following:
Q1072
#1072
Mathematics → Vectors → Vectors and Their Types
·
Easy
Add
The XY-plane
Show model answer
[Q5.1 Ex.Q8 a) · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1073
#1073
Mathematics → Vectors → Vectors and Their Types
·
Easy
Add
The YZ-plane
Show model answer
[Q5.1 Ex.Q8 b) · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1074
#1074
Mathematics → Vectors → Vectors and Their Types
·
Easy
Add
The XZ-plane
Show model answer
[Q5.1 Ex.Q8 c) · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1075
#1075
Mathematics → Vectors → Vectors and Their Types
·
Easy
Add
The X-axis
Show model answer
[Q5.1 Ex.Q8 d) · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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