Question 15 - Week 3 Practice | Prasnya
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Qualifier / Quiz 1
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Week 3
Question 15
00:00
Est. 2 min
Marks:
+2.00
0.00
Passage
Let
V
=
{
(
x
,
y
,
5
)
:
x
,
y
∈
R
}
V=\{(x,y,5):x,y\in\mathbb{R}\}
. Define addition and scalar multiplication by
(
x
1
,
y
1
,
5
)
+
(
x
2
,
y
2
,
5
)
=
(
x
1
+
x
2
,
y
1
+
y
2
,
5
)
(x_1,y_1,5)+(x_2,y_2,5)=(x_1+x_2,y_1+y_2,5)
and
c
(
x
,
y
,
5
)
=
(
c
x
,
c
y
,
5
)
c(x,y,5)=(cx,cy,5)
for
c
∈
R
c\in\mathbb{R}
. Answer the given subquestions with respect to the given information.
Question
Which of the following is/are correct?
Comprehension
Medium Difficulty
25 February 2024
A
V
V
has no zero element with respect to the given addition.
B
(
0
,
0
,
0
)
(0,0,0)
is the zero element of
V
V
with respect to the given addition.
C
(
0
,
0
,
5
)
(0,0,5)
is the zero element of
V
V
with respect to the given addition.
D
For any real number
c
c
, we always have
c
(
0
,
1
,
5
)
=
(
0
,
1
,
5
)
c(0,1,5)=(0,1,5)
.
E
For each element
v
∈
V
v\in V
and for each pair
a
,
b
∈
R
a,b\in\mathbb{R}
,
(
a
+
b
)
v
=
a
v
+
b
v
(a+b)v=av+bv
.