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Section:
Mathematics
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© examsnet.com
Question : 2
Total: 8
The number of integral solutions
x
of
log
(
x
+
‌
7
2
)
(
‌
x
−
7
2
x
−
3
)
2
≥
0
is :
[11-Apr-2023 shift 1]
5
7
8
6
Validate
Solution:
log
x
+
‌
7
2
(
‌
x
−
7
2
x
−
3
)
2
≥
0
Feasible region:
x
+
‌
7
2
>
0
⇒
x
>
−
‌
7
2
And
x
+
‌
7
2
â‰
1
⇒
x
â‰
‌
−
5
2
Taking intersection:
x
∈
(
‌
−
7
2
,
∞
)
−
{
−
‌
5
2
,
‌
3
2
,
7
}
Now
log
a
b
≥
0
if
a
>
1
and
b
≥
1
a
∈
(
0
,
1
)
and
b
∈
(
0
,
1
)
C
−
I
;
x
+
‌
7
2
>
1
and
(
‌
x
−
7
2
x
−
3
)
2
≥
1
‌
x
>
−
‌
5
2
;
(
2
x
−
3
)
2
−
(
x
−
7
)
2
≤
0
‌
(
2
x
−
3
+
x
−
7
)
(
2
x
−
3
−
x
+
7
)
≤
0
‌
(
3
x
−
10
)
(
x
+
4
)
≤
0
‌
x
∈
[
−
4
,
‌
10
3
]
Intersection:
x
∈
(
‌
−
5
2
,
‌
10
3
]
‌
C
−
Î
:
x
+
‌
7
2
∈
(
0
,
1
)
‌
and
‌
(
‌
x
−
7
2
x
−
3
)
2
∈
(
0
,
1
)
‌
0
<
x
+
‌
7
2
<
1
;
(
‌
x
−
7
2
x
−
3
)
2
<
1
‌
−
‌
7
2
<
x
<
‌
−
5
2
;
(
x
−
7
)
2
<
(
2
x
−
3
)
2
‌
x
∈
(
−
∞
,
−
4
)
∪
(
‌
10
3
,
∞
)
No common values of
x
.
Hence intersection with feasible region
We get
x
∈
(
‌
−
5
2
,
‌
10
3
]
−
{
‌
3
2
}
Integral value of
x
are
{
−
2
,
−
1
,
0
,
1
,
2
,
3
}
No. of integral values
=
6
© examsnet.com
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