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Question

Two sets A and B are given as
A={x |x is an integer root of the equation x56x4+11x36x2=0}
B={x |x is a real root of the equation ax5+2ax3+2bx2+b=0,a, bR such that the given equation have maximum number of real roots.}
Then, which of the following is correct?

A
n(A)<n(B)
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B
cannot comment because n(A) varries as the values of a and b vary
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C
n(A)>n(B)
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D
n(A)=n(B)
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Solution

The correct option is C n(A)>n(B)
Given,
Set A={x |x is an integer root of the equation x56x4+11x36x2=0}
x56x4+11x36x2=0
x2(x36x2+11x6)=0
x2(x2(x1)5x(x1)+6(x1))=0
x2(x1)(x25x+6)=0
x2(x1)(x2)(x3)=0
A={0,1,2,3}
n(A)=4

Set B={x |x is a real root of the equation ax5+2ax3+2bx2+b=0,a, bR such that the given equation have maximum number of real roots.}

Case 1: ab>0
f(x)=ax5+2ax3+2bx2+b
Using Descartes' rule of Signs
f(x) = No sign change zero positive roots
f(x)=ax5+2ax3+2bx2+b
one sign change
It can have maximum one real root.
As it is odd polynomials, it must have atleast one real root.
It has exactly one root.
Case 2: ab<0
Proceeding in the same way as Case 1.
Set B contains only one real element
n(B)=1
Hence, n(A)>n(B)

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