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Exam: Maharashtra HSC Class 12
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Q1001
#1001
Mathematics → Vectors → Dot Product of Vectors
·
Moderate
Add
Find the acute angles between the curves at their points of intersection.
y
=
x
2
y = x^2
y
=
x
2
,
y
=
x
3
y = x^3
y
=
x
3
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[QMisc II Q.26 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Set · 2 questions
Find the direction cosines and direction angles of the vector.
Q1002
#1002
Mathematics → Vectors → Dot Product of Vectors
·
Easy
Add
2
i
^
+
j
^
+
2
k
^
2\hat{i}+\hat{j}+2\hat{k}
2
i
^
+
j
^
+
2
k
^
Show model answer
[QMisc II Q.27 i) · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1003
#1003
Mathematics → Vectors → Dot Product of Vectors
·
Easy
Add
1
2
i
^
+
j
^
+
k
^
\dfrac{1}{2}\hat{i}+\hat{j}+\hat{k}
2
1
i
^
+
j
^
+
k
^
Show model answer
[QMisc II Q.27 ii) · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1004
#1004
Mathematics → Vectors → Dot Product of Vectors
·
Moderate
Add
Let
b
⃗
=
4
i
^
+
3
j
^
\vec{b} = 4\hat{i}+3\hat{j}
b
=
4
i
^
+
3
j
^
and
c
⃗
\vec{c}
c
be two vectors perpendicular to each other in the XY-plane. Find the vector in the same plane having projection 1 and 2 along
b
⃗
\vec{b}
b
and
c
⃗
\vec{c}
c
, respectively.
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[QMisc II Q.28 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1005
#1005
Mathematics → Vectors → Dot Product of Vectors
·
Moderate
Add
Show that no line in space can make angles
π
/
6
\pi/6
π
/6
and
π
/
4
\pi/4
π
/4
with X-axis and Y-axis respectively.
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[QMisc II Q.29 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1006
#1006
Mathematics → Vectors → Dot Product of Vectors
·
Hard
Add
Find the angle between the lines whose direction cosines are given by the equation
6
m
n
−
2
n
l
+
5
l
m
=
0
6mn-2nl+5lm=0
6
mn
−
2
n
l
+
5
l
m
=
0
,
3
l
+
m
+
5
n
=
0
3l+m+5n=0
3
l
+
m
+
5
n
=
0
.
Show model answer
[QMisc II Q.30 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1007
#1007
Mathematics → Vectors → Dot Product of Vectors
·
Moderate
Add
If Q is the foot of the perpendicular from
P
(
2
,
4
,
3
)
P(2,4,3)
P
(
2
,
4
,
3
)
on the line joining the points
A
(
1
,
2
,
4
)
A(1,2,4)
A
(
1
,
2
,
4
)
and
B
(
3
,
4
,
5
)
B(3,4,5)
B
(
3
,
4
,
5
)
, find coordinates of Q.
Show model answer
[QMisc II Q.31 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1008
#1008
Mathematics → Vectors → Cross Product of Vectors
·
Hard
Add
Show that the area of triangle ABC, the position vectors of whose vertices are a, b and c is
1
2
[
a
⃗
×
b
⃗
+
b
⃗
×
c
⃗
+
c
⃗
×
a
⃗
]
\dfrac{1}{2}\left[\vec{a}\times\vec{b}+\vec{b}\times\vec{c}+\vec{c}\times\vec{a}\right]
2
1
[
a
×
b
+
b
×
c
+
c
×
a
]
.
Show model answer
[QMisc II Q.32 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1009
#1009
Mathematics → Vectors → Cross Product of Vectors
·
Moderate
Add
Find a unit vector perpendicular to the plane containing the point
(
a
,
0
,
0
)
(a, 0, 0)
(
a
,
0
,
0
)
,
(
0
,
b
,
0
)
(0, b, 0)
(
0
,
b
,
0
)
, and
(
0
,
0
,
c
)
(0, 0, c)
(
0
,
0
,
c
)
. What is the area of the triangle with these vertices?
Show model answer
[QMisc II Q.33 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Set · 12 questions
State whether each expression is meaningful. If not, explain why? If so, state whether it is a vector or a scalar.
Q1010
#1010
Mathematics → Vectors → Scalar and Vector Triple Product
·
Easy
Add
a
⃗
⋅
(
b
⃗
×
c
⃗
)
\vec{a}\cdot(\vec{b}\times\vec{c})
a
⋅
(
b
×
c
)
Show model answer
[QMisc II Q.34 (a) · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1011
#1011
Mathematics → Vectors → Scalar and Vector Triple Product
·
Easy
Add
a
⃗
×
(
b
⃗
⋅
c
⃗
)
\vec{a}\times(\vec{b}\cdot\vec{c})
a
×
(
b
⋅
c
)
Show model answer
[QMisc II Q.34 (b) · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1012
#1012
Mathematics → Vectors → Scalar and Vector Triple Product
·
Easy
Add
a
⃗
×
(
b
⃗
×
c
⃗
)
\vec{a}\times(\vec{b}\times\vec{c})
a
×
(
b
×
c
)
Show model answer
[QMisc II Q.34 (c) · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1013
#1013
Mathematics → Vectors → Scalar and Vector Triple Product
·
Easy
Add
a
⃗
⋅
(
b
⃗
⋅
c
⃗
)
\vec{a}\cdot(\vec{b}\cdot\vec{c})
a
⋅
(
b
⋅
c
)
Show model answer
[QMisc II Q.34 (d) · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1014
#1014
Mathematics → Vectors → Scalar and Vector Triple Product
·
Easy
Add
(
a
⃗
⋅
b
⃗
)
×
(
c
⃗
⋅
d
⃗
)
(\vec{a}\cdot\vec{b})\times(\vec{c}\cdot\vec{d})
(
a
⋅
b
)
×
(
c
⋅
d
)
Show model answer
[QMisc II Q.34 (e) · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1015
#1015
Mathematics → Vectors → Scalar and Vector Triple Product
·
Easy
Add
(
a
⃗
×
b
⃗
)
⋅
(
c
⃗
×
d
⃗
)
(\vec{a}\times\vec{b})\cdot(\vec{c}\times\vec{d})
(
a
×
b
)
⋅
(
c
×
d
)
Show model answer
[QMisc II Q.34 (f) · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1016
#1016
Mathematics → Vectors → Scalar and Vector Triple Product
·
Easy
Add
(
a
⃗
⋅
b
⃗
)
⋅
c
⃗
(\vec{a}\cdot\vec{b})\cdot\vec{c}
(
a
⋅
b
)
⋅
c
Show model answer
[QMisc II Q.34 (g) · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1017
#1017
Mathematics → Vectors → Scalar and Vector Triple Product
·
Easy
Add
(
a
⃗
⋅
b
⃗
)
c
⃗
(\vec{a}\cdot\vec{b})\vec{c}
(
a
⋅
b
)
c
Show model answer
[QMisc II Q.34 (h) · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1018
#1018
Mathematics → Vectors → Scalar and Vector Triple Product
·
Easy
Add
(
∣
a
⃗
∣
)
(
b
⃗
⋅
c
⃗
)
(|\vec{a}|)(\vec{b}\cdot\vec{c})
(
∣
a
∣
)
(
b
⋅
c
)
Show model answer
[QMisc II Q.34 (i) · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1019
#1019
Mathematics → Vectors → Scalar and Vector Triple Product
·
Easy
Add
a
⃗
⋅
(
b
⃗
+
c
⃗
)
\vec{a}\cdot(\vec{b}+\vec{c})
a
⋅
(
b
+
c
)
Show model answer
[QMisc II Q.34 (j) · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1020
#1020
Mathematics → Vectors → Scalar and Vector Triple Product
·
Easy
Add
a
⃗
⋅
b
⃗
+
c
⃗
\vec{a}\cdot\vec{b}+\vec{c}
a
⋅
b
+
c
Show model answer
[QMisc II Q.34 (k) · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1021
#1021
Mathematics → Vectors → Scalar and Vector Triple Product
·
Easy
Add
∣
a
⃗
∣
⋅
(
b
⃗
+
c
⃗
)
|\vec{a}|\cdot(\vec{b}+\vec{c})
∣
a
∣
⋅
(
b
+
c
)
Show model answer
[QMisc II Q.34 (l) · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1022
#1022
Mathematics → Vectors → Cross Product of Vectors
·
Hard
Add
Show that, for any vectors
a
⃗
,
b
⃗
,
c
⃗
\vec{a}, \vec{b}, \vec{c}
a
,
b
,
c
:
(
a
⃗
+
b
⃗
+
c
⃗
)
×
c
⃗
+
(
a
⃗
+
b
⃗
+
c
⃗
)
×
b
⃗
+
(
b
⃗
+
c
⃗
)
×
a
⃗
=
2
a
⃗
×
c
⃗
(\vec{a}+\vec{b}+\vec{c})\times\vec{c}+(\vec{a}+\vec{b}+\vec{c})\times\vec{b}+(\vec{b}+\vec{c})\times\vec{a}=2\vec{a}\times\vec{c}
(
a
+
b
+
c
)
×
c
+
(
a
+
b
+
c
)
×
b
+
(
b
+
c
)
×
a
=
2
a
×
c
.
Show model answer
[QMisc II Q.35 · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Set · 3 questions
Suppose that
a
⃗
≠
0
⃗
\vec{a} \ne \vec{0}
a
=
0
.
Q1023
#1023
Mathematics → Vectors → Dot Product of Vectors
·
Moderate
Add
If
a
⃗
⋅
b
⃗
=
a
⃗
⋅
c
⃗
\vec{a}\cdot\vec{b} = \vec{a}\cdot\vec{c}
a
⋅
b
=
a
⋅
c
then is
b
⃗
=
c
⃗
\vec{b} = \vec{c}
b
=
c
?
Show model answer
[QMisc II Q.36 (a) · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1024
#1024
Mathematics → Vectors → Cross Product of Vectors
·
Moderate
Add
If
a
⃗
×
b
⃗
=
a
⃗
×
c
⃗
\vec{a}\times\vec{b} = \vec{a}\times\vec{c}
a
×
b
=
a
×
c
then is
b
⃗
=
c
⃗
\vec{b} = \vec{c}
b
=
c
?
Show model answer
[QMisc II Q.36 (b) · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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Q1025
#1025
Mathematics → Vectors → Cross Product of Vectors
·
Moderate
Add
If
a
⃗
⋅
b
⃗
=
a
⃗
⋅
c
⃗
\vec{a}\cdot\vec{b} = \vec{a}\cdot\vec{c}
a
⋅
b
=
a
⋅
c
and
a
⃗
×
b
⃗
=
a
⃗
×
c
⃗
\vec{a}\times\vec{b} = \vec{a}\times\vec{c}
a
×
b
=
a
×
c
then is
b
⃗
=
c
⃗
\vec{b} = \vec{c}
b
=
c
?
Show model answer
[QMisc II Q.36 (c) · Maharashtra State Board (Class 12) — Vectors (Balbharati textbook)]
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