Consider the equation 
, where 
.
(i) If equation has at least two distinct positive real roots, then all possible values of 
are
2
1
0
(ii) If equation has at least two distinct negative real roots, then all possible values of
are none of these
(iii) If exactly two roots are positive and two roots are negative, then the number of integral values of
is 3
Text Solution
Verified by ExpertsA
(i) , (ii) , (iii)
Given equation is
where 
. Now,
(i) Now, Eq. (i) will have at least two positive roots, when at least one root of Eq. (ii) will be greater than 2. From Eq. (ii),
Let the roots of Eq. (ii) be 
. If 
, then
and 
and 
and 
Therefore, at least one root will be greater than 2 . Then,
Combining (iii) and (iv), we get
Hence, at least one root will be positive if 
.
(ii) Now, Eq. (i) will have at least two roots negative, when at least one root of Eq. (ii) will be less than -2 . If 
, then
Combining (iii) and (v), at least one root will be less than -2 for Eq. (ii) if
(iii) If exactly two roots are positive, then other two roots are negative. Then -2 and 2 must lie between the roots. So,
and 
Hence, no such values of 
exist.
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