MathDB
Problems
Contests
National and Regional Contests
India Contests
India LIMIT
2020 LIMIT
2020 LIMIT Category 2
2
2
Part of
2020 LIMIT Category 2
Problems
(1)
g:R^4-->R F.E. in 6 var
Source: LIMIT 2020 Cat 2 Obj P2
5/25/2020
The number of functions
g
:
R
4
→
R
g:\mathbb{R}^4\to\mathbb{R}
g
:
R
4
→
R
such that,
∀
a
,
b
,
c
,
d
,
e
,
f
∈
R
\forall a,b,c,d,e,f\in\mathbb{R}
∀
a
,
b
,
c
,
d
,
e
,
f
∈
R
:(i)
g
(
1
,
0
,
0
,
1
)
=
1
g(1,0,0,1)=1
g
(
1
,
0
,
0
,
1
)
=
1
(ii)
g
(
e
a
,
b
,
e
c
,
d
)
=
e
g
(
a
,
b
,
c
,
d
)
g(ea,b,ec,d)=eg(a,b,c,d)
g
(
e
a
,
b
,
ec
,
d
)
=
e
g
(
a
,
b
,
c
,
d
)
(iii)
g
(
a
+
e
,
b
,
c
+
f
,
d
)
=
g
(
a
,
b
,
c
,
d
)
+
g
(
e
,
b
,
f
,
d
)
g(a+e, b, c+f, d)= g(a,b,c,d)+g(e,b,f,d)
g
(
a
+
e
,
b
,
c
+
f
,
d
)
=
g
(
a
,
b
,
c
,
d
)
+
g
(
e
,
b
,
f
,
d
)
(iv)
g
(
a
,
b
,
c
,
d
)
+
g
(
b
,
a
,
d
,
c
)
=
0
g(a,b,c,d)+g(b,a,d,c)=0
g
(
a
,
b
,
c
,
d
)
+
g
(
b
,
a
,
d
,
c
)
=
0
is :(A)
1
1
1
(B)
0
0
0
(C)
infinitely many
\text{infinitely many}
infinitely many
(D)
None of these
\text{None of these}
None of these
[Hide=Hint(given in question)] Think of matrices
limit
functions
counting