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Standard XII
Mathematics
Nature of Roots
50. F(x)=ax+b...
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50. F(x)=ax+bx+c, a>0 F(x)=f(4-x) xR f(f(x))=0. Has2distinct real roots then (1) At least one root must be positive (2)f(2)f(1) (3)minimum value of f(x)is negative (4) all of these
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Q.
Statement 1 : If
f
(
x
)
=
a
x
2
+
b
x
+
c
, where
a
>
0
,
c
<
0
and
b
∈
R
, then roots of
f
(
x
)
=
0
must be real and distinct .
Statement 2 : If
f
(
x
)
=
a
x
2
+
b
x
+
c
,
where
a
>
0
,
b
∈
R
,
b
≠
0
and the roots of
f
(
x
)
=
0
are real and distinct, then
c
is necessarily negative real number .
Q.
Assertion :If
a
,
b
,
c
∈
R
and
2
a
+
3
b
+
6
c
=
0
, then the equation
a
x
2
+
b
x
+
c
=
0
has at least one real root in
(
0
,
1
)
. Reason: If
f
(
x
)
is a polynomial which assumes both positive and negative values, then it has at least one real root.
Q.
Let f (x) = sinx + ax + b. Then f(x) = 0 has
Q.
If
f
(
x
)
=
a
x
2
+
b
x
+
c
,
a
,
b
,
c
∈
R
and equation
f
(
x
)
−
x
=
0
has non-real roots
α
,
β
. Let
γ
,
δ
be the roots of
f
(
f
(
x
)
)
−
x
=
0
(
γ
,
δ
are not equal to
α
,
β
). Then
∣
∣ ∣
∣
2
α
δ
β
0
α
γ
β
1
∣
∣ ∣
∣
is
Q.
Assertion :If equation
a
x
2
+
b
x
+
c
=
0
and
x
2
−
3
x
+
4
=
0
have exactly one root common, then at least one of
a
,
b
,
c
is imaginary. Reason: If
a
,
b
,
c
are not all real, then equation
a
x
2
+
b
x
+
c
=
0
can have one real root and one one root imaginary.
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