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Question

Let p(x) be cubic polynomial 7x 3 −4x 2 +K. Suppose the three roots of p(x) form an arithmetic progression. Then the value of K, is

A
421
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B
16147
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C
16441
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D
1281323
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Solution

The correct option is D 1281323
Given cubic expression p(x)=7x34x2+K
Comparing with the standard cubic expression p(x)=ax3+bx2+cx+d, we get
a=7,b=4,c=0,d=K
The roots of the equation are in arithmetic progression.
Let them be mn,m and m+n.
The sum of the roots =ba = (4)7 = 47=(mn)+m+(mn)=3m ....(1)
Sum of bi-product of roots =ca = 07 = 0 =(mn)m+(mn)(m+n)+m(m+n)= 2m2+m2n2 .....(2)
Product of roots =da = K7 = (mn)m(m+n) = m(m2n2) ........(3)
From equations (1),(2) and (3), we get
m=421 ........(4)
2m2+m2n2=0 .....(5)
m2n2=3K4 .....(6)
Substitute equation (4) and (6) in equation (5), we get
2(421)23K4=0
K=(83)(421)2 = K=(83)(16441)
K=1281323
Hence, option (D) is the correct answer.

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