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Question

$alpha comma blank beta$ are the roots of the equation $x squared minus 3 x plus A equals 0$ and $gamma comma delta$ are the roots of the equation $x squared minus 12 x plus B equals 0. I f alpha comma beta comma gamma comma delta$ form an increasing GP, then $left parenthesis A comma blank B right parenthesis$ is equal to

(3, 12)
(12, 3)
(2, 32)
(4, 16)

hint Hint:

A geometric progression is a special type of progression where the successive terms bear a constant ratio known as a common ratio. It is also commonly referred to as GP. The GP is generally represented in form a, ar, ar^2.... where 'a' is the first term and 'r' is the common ratio of the progression.

The correct answer is: (3, 12)

Given : $alpha comma blank beta$ are the roots of the equation $x to the power of 2 end exponent minus 3 x plus a equals 0$ and $gamma comma delta$ are the roots of the equation $x squared minus 12 x plus b equals 0.$
The GP is generally represented in form of $a space comma space space a r space comma space a r squared space comma space a r cubed$
$L e t space a space equals space alpha space comma space a r space equals space beta space comma space a r squared space equals space gamma space a n d space a r cubed space equals space delta$
$x squared minus 3 x plus A equals 0 space h a v i n g space r o o t s space alpha space a n d space beta space o r space a space a n d space a r S i m i l a r l y space x squared space minus space 12 x space plus space B space equals space 0 space h a v e space r o o t s space gamma space a n d space delta space o r space a r squared space a n d space a r cubed F r o m space f i r s t space e q u a t i o n S u m space o f space r o o t s space left parenthesis b over a right parenthesis space equals space a space plus space a r space equals space 3 space rightwards double arrow a space left parenthesis 1 space plus space r right parenthesis space equals space 3 space...... space left parenthesis 1 right parenthesis P r o d u c t space o f space r o o t s left parenthesis c over a right parenthesis space equals space a space cross times a r space equals space A space....... space left parenthesis 3 right parenthesis F r o m space s e c o n d space e q u a t i o n S u m space o f space r o o t s space left parenthesis b over a right parenthesis space equals space a r squared space plus space a r cubed space equals space 12 space rightwards double arrow space a r squared left parenthesis 1 space plus r right parenthesis space equals space 12 space...... space left parenthesis 2 right parenthesis P r o d u c t space o f space r o o t s left parenthesis c over a right parenthesis space equals space a r squared space cross times a r cubed space equals space B space....... space left parenthesis 4 right parenthesis D i v i d i n g space left parenthesis 1 right parenthesis space a n d space left parenthesis 2 right parenthesis fraction numerator a r squared left parenthesis 1 space plus r right parenthesis over denominator a space left parenthesis 1 space plus space r right parenthesis end fraction space equals space fraction numerator 12 space over denominator 3 end fraction space equals space r squared space equals space 4 space rightwards double arrow space r space equals space 2 S u b s t i t u t e space t h i s space v a l u e space i n space e q u a t i o n space left parenthesis 1 right parenthesis space a space left parenthesis 1 space plus space 2 right parenthesis space equals space 3 space rightwards double arrow space a space equals space 1 S u b s t i t u t i n g space t h e s e space v a l u e s space i n space left parenthesis 3 right parenthesis space a n d space left parenthesis 4 right parenthesis comma space w e space g e t A space equals 2 space a n d space B space equals space 32$

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