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Question

Consider the equation x2+a|x|+1=0, where a is a parameter.

List IList II (A)No real roots (P)a<2(B)Two real roots (Q)ϕ(C)Three real roots (R)a=2(D)Four distinct real roots (S)a0(T)a=2

Which of the following is the only CORRECT combination?

A
(A)(P)
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B
(A)(Q)
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C
(B)(T)
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D
(B)(R)
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Solution

The correct option is D (B)(R)
Since |x|2=x2, x R
x2+a|x|+1=0=|x|2+a|x|+1 (1)
Let |x|=t
Then, equation (1) becomes
t2+at+1=0 (2)

(A)
|x|2+a|x|+1=0
Clearly, if a0, then there are no real roots as all the terms in L.H.S. will be positive.

(B)
t=0 is not a root of equation (2).
So, for two real roots of equation (1),
equation (2) should have equal positive roots.
D=a24=0 and a2>0
a=±2 and a<0
a=2

(C)
For three real roots of equation (1), one root of equation (2) should be 0 and other should be positive.
But t=0 is not a root of equation (2).
equation (1) cannot have three real roots for any value of a.

(D)
For four distinct real roots of equation (1), equation (2) should have two positive real solutions in t.
So, a2>0, (x-coordinate of vertex)
and a24>0 (D>0)
a<0, and a(,2)(2,)
a<2

(A)(S),(B)(R),(C)(Q),(D)(P)

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