IIT JEE , AIEEE Forum - Mathematics , Algebra - Quadratic for jee comprehension type question.

Sushmi

Let x ⁴ + 4 x ³ - 8 x ² + p = 0

Quesn :

1.) Values of p when all roots are real :

a.) 0

p

3 b.) p > 3 and p < 0

c.) 0

p

3 d.) none of these

2.) Values of p when 2 roots are real

a.) p < 0 and p > 3 b.) p< 0 and 3 < p < 128

c.) 0

p

3 d.) p < 3 and p > 128

3.) Value of p when no roots are real

a.) p < 32 b.) p > 128

c.) p < 3 d.) None of these

Please provide the solution .

Sushmi

Come on friends some one please answer. Nobody? Should I ask the experts?

hsbhatt

1. It is easy to see that all roots cannot be real. For if they are all real (call them a,b,c and d), then we must have (a+b+c+d)²

3

ab.

Here

a = -4,

ab = 8 and so this condition is not satisfied. So for no value of p will we have all roots real.

3. Suppose all roots are not real. Then since, the leading coefficient is 1, f(x)

0 for all x.

Now look at the variable portion which is g(x) = x⁴+4x³-8x².

Now g'(x) = 4x(x²+3x-4). g'(x) = 0 when x=0, x=-4, x=1.

The minimum occurs when x=-4, where f(x) = -128.

Hence p-128

0 and so p

128.

Sushmi

Thank you very much sir.

hsbhatt

2. Looking at the behaviour of g(x) for p<0 or for 3<p<128, we get two real roots.

Just draw a tentative graph, keeping in mind that the function attains minima at x=-4(this is global minimum) and x=1 (local) and local maximum at x=0. Then it becomes clear.

hsbhatt

could someone post the graph of x⁴+4x³-8x²required in my solution. It will be easier then to see how the answer is got.

elastiboysai

Sir
downld the img

hsbhatt

thank you elasti.

ok, at x=-4, g(x) =-128. Its gone off the graph. So, if p =0, f(x) = g(x) will have 3 roots. So just tug the graph down by making p<0. That way you remove the solution x=0 and so, we get two real roots.

Again, if you increase p just beyond 3, then for all x>0, f(x) becomes +ve, but a new root emerges near x=-4. But you cant increase p beyond 128 as then as I have written in my previous post, there are no real solutions

Sushmi

You are correct sir answer must be wrong first answer is given a.It should be d.

Thank you once again.

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