the value of a for which the equation 3x2 + 2(a2+1)x + (a2-3a+2)=0 has roots of opposite sign lies in
Answers
Answer:
search-icon-header
Search for questions & chapters
search-icon-image
Class 11
>>Maths
>>Complex Numbers and Quadratic Equations
>>Quadratic Equations
>>The value of a for which the quadratic e
Question
Bookmark
The value of a for which the quadratic equation 3x
2
+2a
2
x+2x+a
2
−3a+2=0 possesses roots of opposite signs lies in
Medium
Solution
verified
Verified by Toppr
Correct option is C)
3x
2
+2a
2
x+2x+a
2
−3a+2=0
3x
2
+2(a
2
+1)x+(a
2
−3a+2)=0
Condition for real distinct roots ⇒D>0
b
2
−4ac>0
4(a
2
+1)
2
−4×3×(a
2
−3a+2)>0
4(a
4
+1+2a
2
)−(12a
2
−36a+24)>0
4a
4
+4+8a
2
−12a
2
+36a−24>0
4a
4
−4a
2
+36a−20>0
a
4
−a
2
+9a−5>0⟶(i)
If roots are of opposite signs then product of roots <0
coefficientofx
2
constant
<0⇒
3
a
2
−3a+2
<0
a
2
−3a+2<0
a
2
−2a−a+2<0
a(a−2)−1(a−2)<0⇒(a−2)(a−1)<0
(a−2)>0 and (a−1)<0 or (a−2)<0 and (a−1)>0
a>2 and a<1 or a<2 and a>1
a<1 and a>2⇒2<a<1 which is not possible
∴a>2 & a<1 is rejected
∴a∈(1,2)