CAT Exam Model Paper 6 with solutions for free online practice

Question : 19 of 60

Marks: +1, -0

If the roots of the equation ax

^{3}

+ bx

^{2}

+ cx + d = 0 are in Geometric Progression, then which of the following relations is true?

Solution:

Lets the roots be k, kr, kr

^{2}

.
Now k + kr + kr

^{2}

-b/a

⇒

k ( 1 + r + r ^ { 2 } ) = - { b }/ { a }

...(i)
Again

{ k } . { kr } + { kr } . \ { kr } ^ { 2 } + { k } . { kr } ^ { 2 } = { { c } }/ { { a } }

⇒

k ^ { 2 } r ( 1 + r ^ { 2 } + r ) = { c }/ { a }

....(ii)
and

{ k } . { kr } .{ kr } ^ { 2 } = - { { d } }/ { { a } }

⇒

{ k } ^ { 3 } { r } ^ { 3 } = - { { d } } /{ { a } }

...(iii)
Dividing equation (ii) by equation (i):

{ kr } = - { { c } }/ { { b } }

From equation (iii):

( - { c }/ { b } ) ^ { 3 } = - { d }/ { a }

⇒

a c ^ { 3 } = b ^ { 3 } d

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