Question

# Let f(x)=(λ2+λ−2)x2+(λ+2)x be a quadratic polynomial. The sum of all integral values of λ for which f(x)<1 ∀ x∈R, is

A
1
Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses
B
3
No worries! We‘ve got your back. Try BYJU‘S free classes today!
C
0
No worries! We‘ve got your back. Try BYJU‘S free classes today!
D
2
No worries! We‘ve got your back. Try BYJU‘S free classes today!
Open in App
Solution

## The correct option is A −1(λ2+λ−2)x2+(λ+2)x<1(λ2+λ−2)x2+(λ+2)x−1<0 Let g(x)=(λ2+λ−2)x2+(λ+2)x−1 As g(x)<0, so the graph of g(x) always lies below the x− axis, therefore the required conditions are (i) leading coefficient<0 and (ii) D<0 (i) leading coefficient<0⇒λ2+λ−2<0⇒(λ+2)(λ−1)<0 ⇒λ∈(−2,1)⋯(1) (ii) D<0⇒(λ+2)2−4(λ2+λ−2)(−1)<0⇒(λ+2)2+4(λ+2)(λ−1)<0⇒(λ+2)(λ+2+4λ−4)<0⇒(5λ−2)(λ+2)<0⇒λ∈(−2,25)⋯(2) From equation (1) and (2), ⇒λ∈(−2,25) Integral values in the interval is −1,0 Sum of integer is −1.

Suggest Corrections
3
Join BYJU'S Learning Program
Related Videos
Nature and Location of Roots
MATHEMATICS
Watch in App
Explore more
Join BYJU'S Learning Program