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Byju's Answer
Standard XII
Mathematics
Multinomial Expansion
The largest p...
Question
The largest positive integral of
α
for which the inequality
3
−
|
x
−
α
|
>
x
2
is satisfied by at least one negative
x
is
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Solution
3
−
|
x
−
α
|
>
x
2
⇒
3
−
x
2
>
|
x
−
α
|
As we increase the value of
α
,
the graph of
y
=
|
x
−
α
|
moves from left to right.
Extreme Case
I
:
Extreme Case
II
:
Let
f
(
x
)
=
3
−
x
2
and
g
(
x
)
=
|
x
−
α
|
|
x
−
α
|
=
{
α
−
x
,
x
<
α
x
−
α
,
x
≥
α
3
−
x
2
=
x
−
α
⇒
x
2
+
x
−
α
−
3
=
0
Δ
=
0
⇒
α
=
−
13
4
3
−
x
2
=
α
−
x
⇒
x
2
−
x
+
α
−
3
=
0
Δ
=
0
⇒
α
=
13
4
So, if the above inequality has solution (either positive or negative),
then
α
∈
(
−
13
4
,
13
4
)
⋯
(
1
)
For the negative solution, the right
y
−
intercept of
g
(
x
)
should be less than
3.
⇒
α
<
3
⋯
(
2
)
From
(
1
)
and
(
2
)
,
α
∈
(
−
13
4
,
3
)
Least positive integral value of
α
is
2.
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0
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