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KVPY SX SB Exam 05-Nov-2017 Question Paper
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© examsnet.com
Question : 2
Total: 120
The number of solution pairs (x, y) of the simultaneous equations
log
1
∕
3
(
x
+
y
)
+
log
3
(
x
−
y
)
=
2
and
2
y
2
=
512
x
+
1
0
1
2
3
Validate
Solution:
We have,
log
1
∕
3
(
x
+
y
)
+
log
3
(
x
−
y
)
=
2
2
y
2
=
512
x
+
1
⇒
log
3
−
1
(
x
+
y
)
+
log
3
(
x
−
y
)
=
2
⇒
−
log
3
(
x
+
y
)
+
log
3
(
x
−
y
)
=
2
⇒
x
−
y
x
+
y
=
3
2
=
9
⇒
x
−
y
=
9
x
+
9
y
⇒
−
8
x
=
10
y
⇒
−
4
x
=
5
y
and
2
y
2
=
2
9
(
x
+
1
)
⇒
y
2
=
9
(
x
+
1
)
From Eqs. (i) and (ii), we get
(
−
4
x
5
)
2
=
9
(
x
+
1
)
⇒
16
x
2
=
225
x
+
225
⇒
16
x
2
−
225
x
−
225
=
0
⇒
(
16
x
+
15
)
(
x
−
15
)
=
0
⇒
x
=
15
,
−
15
16
When
x
=
15
,
y
=
−
12
x
=
−
15
16
,
y
=
3
4
(
not possible
)
∴
Only one solution
(
15
,
−
12
)
© examsnet.com
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