If one root of the equation $x^{2} - 12 x + 3 k = 0$ is square of the other, then k is

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Solution

The correct option is A 9
$T h e g i v e n e q u a t i o n i s x^{2} - 12 x + 3 k = 0 3 k i s p o s i t i v e h e r e . S o b o t h t h e r o o t s a r e p o s i t i v e o r b o t h t h e r o o t s a r e n e g a t i v e . L e t o n e r o o t b e α ∴ o t h e r r o o t i s α^{2} . B u t a^{2} i s p o s i t i v e a s i t i s a s q u a r e d t e r m . ∴ a i s a l s o p o s i t i v e . S o a s p e r t h e r e l a t i o n b e t w e e n r o o t s a n d c o e f f i c i e n t s α + α^{2} = 12 - - - - (1) a n d α \times α^{2} = 3 k w h i c h i s p o s i t i v e . - - - - (2) F r o m (1) α^{2} + α - 12 = 0 \Rightarrow (α + 4) (α - 3) = 0 \Rightarrow α = - 4, 3 A s p e r t h e a b o v e d i s c u s s i o n w e t a k e t h e p o s i t i v e v a l u e o n l y . T h e r e f o r e α = 3. S u b s t i t u t i n g f o r α i n e q u a t i o n (2) 3 k = 3 \times 3^{2} \Rightarrow k = 9 (A n s)$

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