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Byju's Answer
Standard XII
Mathematics
Algebra of Limits
For x,y,z>0 a...
Question
For
x
,
y
,
z
>
0
and
x
>
m
,
y
>
n
,
z
>
r
,
if
∣
∣ ∣
∣
x
n
r
m
y
r
m
n
z
∣
∣ ∣
∣
=
0
,
then the greatest value of
27
x
y
z
(
x
−
m
)
(
y
−
n
)
(
z
−
r
)
is
Open in App
Solution
Given:
∣
∣ ∣
∣
x
n
r
m
y
r
m
n
z
∣
∣ ∣
∣
=
0
Applying
R
1
→
R
1
−
R
2
and
R
2
→
R
2
−
R
3
∣
∣ ∣
∣
x
−
m
n
−
y
0
0
y
−
n
r
−
z
m
n
z
∣
∣ ∣
∣
=
0
⇒
(
x
−
m
)
(
y
−
n
)
z
+
n
(
x
−
m
)
(
z
−
r
)
+
m
(
n
−
y
)
(
r
−
z
)
=
0
⇒
m
x
−
m
+
n
y
−
n
+
z
z
−
r
=
0
⇒
x
x
−
m
+
y
y
−
n
+
z
z
−
r
=
2
Let
x
x
−
m
,
y
y
−
n
and
z
z
−
r
be three positive numbers.
Now,
A
.
M
.
≥
G
.
M
.
⇒
x
x
−
m
+
y
y
−
n
+
z
z
−
r
3
≥
(
x
y
z
(
x
−
m
)
(
y
−
n
)
(
z
−
r
)
)
1
/
3
⇒
x
y
z
(
x
−
m
)
(
y
−
n
)
(
z
−
r
)
≤
(
2
3
)
3
⇒
27
x
y
z
(
x
−
m
)
(
y
−
n
)
(
z
−
r
)
≤
8
Hence, the greatest value of
27
x
y
z
(
x
−
m
)
(
y
−
n
)
(
z
−
r
)
is
8
Suggest Corrections
12
Similar questions
Q.
If
n
>
m
,
y
>
n
,
z
>
r
,
(
x
,
y
,
z
>
0
)
such that
∣
∣ ∣
∣
x
n
r
m
y
r
m
n
z
∣
∣ ∣
∣
=
0
then greatest value of
27
x
y
z
(
x
−
m
)
(
y
−
n
)
(
z
−
r
)
is
Q.
If
n
>
m
,
y
>
n
,
z
>
r
,
(
x
,
y
,
z
>
0
)
such that
∣
∣ ∣
∣
x
n
r
m
y
r
m
n
z
∣
∣ ∣
∣
=
0
then greatest value of
27
x
y
z
(
x
−
m
)
(
y
−
n
)
(
z
−
r
)
is
Q.
If
l
(
m
y
+
n
z
−
l
x
)
=
m
(
n
z
+
l
x
−
m
y
)
=
n
(
l
x
+
m
y
−
n
z
)
,
prove
y
+
z
−
x
l
=
z
+
x
−
y
m
=
x
+
y
−
z
n
.
Q.
If
x
>
m
,
y
>
n
,
z
>
r
(
x
,
y
,
z
>
0
)
such that
∣
∣ ∣
∣
x
n
r
m
y
r
m
n
z
∣
∣ ∣
∣
=
0
The value of
x
x
−
m
+
y
y
−
n
+
z
z
−
r
is ?
Q.
(
1
+
x
)
I
(
1
+
y
)
m
(
1
+
z
)
n
≥
1
I
x
+
m
y
+
n
z
for I , m , n > 0 and
ϵ
N , x , y ,z > 0
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