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Standard XII
Maths
Question
If
f
(
x
)
satisfies the relation
f
(
x
+
y
)
=
f
(
x
)
+
f
(
y
)
for all
x
,
y
∈
R
, and
f
(
1
)
=
5
, then
f
(
x
)
is an odd function
m
∑
r
=
1
f
(
r
)
=
5
m
(
m
+
2
)
3
f
(
x
)
is an even function
m
∑
r
=
1
f
(
r
)
=
5
m
+
1
C
2
A
m
∑
r
=
1
f
(
r
)
=
5
m
+
1
C
2
B
m
∑
r
=
1
f
(
r
)
=
5
m
(
m
+
2
)
3
C
f
(
x
)
is an even function
D
f
(
x
)
is an odd function
Open in App
Solution
Verified by Toppr
Let
x
=
y
=
0
, then the given equation is :
f
(
0
)
=
2
×
f
(
0
)
⇒
f
(
0
)
=
0
Now for
y
=
−
x
, we have
0
=
f
(
0
)
=
f
(
x
)
+
f
(
−
x
)
∴
f
(
x
)
=
−
f
(
−
x
)
Hence the function is an odd function.
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2
Similar Questions
Q1
If
f
(
x
)
satisfies the relation
f
(
x
+
y
)
=
f
(
x
)
+
f
(
y
)
for all
x
,
y
∈
R
and
f
(
1
)
=
5
, then
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Q2
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)
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(
y
)
,
∀
x
,
y
∈
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and
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)
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, then
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=
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)
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Q3
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satisfies the relation
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)
=
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)
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for all
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,
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∈
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Q4
If
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)
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(
x
)
+
f
(
y
)
for all
x
,
y
∈
R
and
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(
1
)
=
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, then
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∑
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=
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Q5
If
f
:
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→
R
satisfies f(x+y) = f(x) + f(y), for all x, y
∈
R
and f(1) = 7, then
∑
n
r
=
1
f
(
r
)
is equal to