For what values of 
, then equation
(i) both roots are smaller than 2 ?
(ii) both roots are greater than 2?
(iii) both roots lie in the interval 
?
(iv) exactly one root lie in the interval 
?
(v) one root is smaller than 1 and the other root is greater than 1 ?
(vi) one root is greater than 3 and the other root is smaller than 2?
(vii) atleast one root lies in the interval 
?
(viii) atleast one root is greater than 2 ?
(ix) atleast one root is smaller than 2?
(x) roots 
and 
, such that both 2 and 3 lie between 
and 
?
Text Solution
Verified by Experts(iv) (iii); (viii) (II)
or 
and let
(i) Both roots are smaller than 2 .
Consider the following cases:
Case I 
[from Eq. (i)]
Case II 
-coordinate of vertex 
.
[from Eq. (ii)]
or 
Case III 
Combining all cases, we get
(ii) Both roots are greater than 2 .
Consider the following cases:
Case I 
[from Eq. (i)]
Case II 
-coordinate of vertex 
[from Eq. (ii)]
Case III 
from part (i) 
Combining all cases, we get
(iii) Both roots lie in the interval 
.
Consider the following cases:
Case I 
[from Eq. (i)]
Case II 
[from part (i)]
Case III 
or 
Case IV 
-coordinate of vertex 
or 
or 
Combining all cases, we get
(iv) Exactly one root lie in the interval 
.
Consider the following cases:
Case I D > 0
[from Eq. (i)]
Case II 
Combining all cases, we get
(v) One root is smaller than 1 and the other root is greater than 1.
Consider the following cases:
Case I 
[from Eq. (i)]
Case II 
[from Eq. (iii)]
Combining both cases, we get
(vi) One root is greater than 3 and the other root is smaller than 2.
Consider the following cases:
[from Eq. (i)]
Case II 
Case III 
Combining all cases, we get
(vii) Atleast one root lies in the interval 
.
i.e. part (iv) 
(iii)
(viii) Atleast one root is greater than 2 .
i.e. (Exactly one root is greater than 2 
(Both roots are greater than 2)
or 
Exactly one root is greater than 2 
b)…(i)
Consider the following cases:
Case I 
[from Eq. (i)]
Case II 
Combining both cases, we get
Finally from Eqs. (I) and (II), we get
(ix) Atleast one root is smaller than 2.
i.e. (Exactly one root is smaller than 2 
(Both roots are smaller than 2) or part (viii) (II) 
(i)
We get, 
(x) Both 2 and 3 lie between 
and 
.
Consider the following cases:
[from Eq. (i)]
Case II 
Case III 
Combining all cases, we get
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