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Question

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

A
1
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B
3
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C
0
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D
2
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Solution

The correct option is A 1
(λ2+λ2)x2+(λ+2)x<1(λ2+λ2)x2+(λ+2)x1<0
Let g(x)=(λ2+λ2)x2+(λ+2)x1

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<0λ2+λ2<0(λ+2)(λ1)<0
λ(2,1)(1)

(ii) D<0(λ+2)24(λ2+λ2)(1)<0(λ+2)2+4(λ+2)(λ1)<0(λ+2)(λ+2+4λ4)<0(5λ2)(λ+2)<0λ(2,25)(2)

From equation (1) and (2),
λ(2,25)
Integral values in the interval is
1,0
Sum of integer is 1.

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