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Question

# Two sets A and B are given as A={x |x is an integer root of the equation x5−6x4+11x3−6x2=0} B={x |x is a real root of the equation ax5+2ax3+2bx2+b=0,a, b∈R 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
n(A)=n(B)
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C
n(A)<n(B)
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D
cannot comment because n(A) varries as the values of a and b vary
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Solution

## The correct option is A n(A)>n(B)Given, Set A={x |x is an integer root of the equation x5−6x4+11x3−6x2=0} x5−6x4+11x3−6x2=0 ⇒x2(x3−6x2+11x−6)=0 ⇒x2(x2(x−1)−5x(x−1)+6(x−1))=0 ⇒x2(x−1)(x2−5x+6)=0 ⇒x2(x−1)(x−2)(x−3)=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, b∈R 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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