AP ICET 2017 Shift 2 Paper

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Question : 132 of 200

Marks: +1, -0

If a,b and care such that

ak^{2}+bk+c≠0

for some k ad if the determinant of the matrix

[\table a,b,ak+b;b,c,bk+c;ak+b,bk+c,0]

is zero then a,b,c are in

ail arithmetic progression
a harmonic progression
an arithmetic - geometric series
a geometric progression

Solution:

Given :

|\table a,b,ak+b;b,c,bk+c;ak+b,bk+c,0|=0

(ak+b)[b(bk+c)–c(ak+b)]–(bk+c)[a(bk+c)–b(ak+b)]=0

⇒(ak+b)[b^2k+bc–ack–bc]–(bk+c)[abk+ac–abk–b^2]=0

⇒(ak+b)[(b^2–ac)k]–(bk+c)(ac–b^2)]=0

⇒(b^2–ac)[k(ak+b)+(bk–c)]=0

⇒(b^2–ac)[ak^2+2bk+c]=0

We can write

b^2–ac=0⇒b=√{ac}

Hence, a,b and c are in G.P.

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