According to question , we have
The roots α, β of the equation : x² − 6x + k = 0
On comparing with ax² + bx + c = 0 , we get a = 1 , b = - 6 , c = k
Now , α + β = ( -b/a ) = 6

Correct Option: A

According to question , we have
The roots α, β of the equation : x² − 6x + k = 0
On comparing with ax² + bx + c = 0 , we get a = 1 , b = - 6 , c = k
Now , α + β = ( - b/a ) = 6
α + β = 6 and 3α + 2β = 20
On solving ,
⇒ α = 4, β = 2
Product of the roots = k
So, k = αβ = 4 × 2 = 8.
Hence , required answer will be option A .

According to question , we can say that
Let, the roots be 3α and 2α .

∴	5	=		m		2	⇔	5	=	m2
	72			60				72		3600

Correct Option: A

According to question , we can say that
Let, the roots be 3α and 2α .

∴	5	=		m		2	⇔	5	=	m2
	72			60				72		3600

As per the given above question , we know that
Let, α, β be the roots of the equation x² + kx + 12 = 0
∴ α + β = − k and αβ = 12
Now ( α - β )² = ( α + β )² - 4αβ
( 1 )² = k² - 4(12)

Correct Option: D

As per the given above question , we know that
Let, α, β be the roots of the equation x² + kx + 12 = 0
∴ α + β = −k and αβ = 12
Now ( α - β )² = ( α + β )² - 4αβ
( 1 )² = k² - 4(12)
⇒ k² = 49 ⇒ k = ± 7.

According to question ,
On Comparing x² - 2( 1 + 3k )x + 7 ( 3 + 2k ) = 0 with ax₂ + bx + c = 0, we get
a = 1, b = - 2 ( 1 + 3k ), c = 7( 3 + 2k )
For equal roots D = b² - 4ac = 0
∴ 4( 1 + 3k )² - 4 x 1 x 7( 3 + 2k ) = 0
⇒ 4( 1 + 9k² + 6k ) - 84 - 56k = 0
⇒ 4 + 36k² + 24k - 56k - 84 = 0
⇒ 36k² - 32k - 80 = 0
⇒ 9k² - 8k - 20 = 0

Correct Option: A

According to question ,
On Comparing x² - 2( 1 + 3k )x + 7 ( 3 + 2k ) = 0 with ax₂ + bx + c = 0, we get
a = 1, b = - 2 ( 1 + 3k ), c = 7( 3 + 2k )
For equal roots D = b² - 4ac = 0
∴ 4( 1 + 3k )² - 4 x 1 x 7( 3 + 2k ) = 0
⇒ 4( 1 + 9k² + 6k ) - 84 - 56k = 0
⇒ 4 + 36k² + 24k - 56k - 84 = 0
⇒ 36k² - 32k - 80 = 0
⇒ 9k² - 8k - 20 = 0
By shridharachary method ,
Here , A = 9 , B = -8 , C = -20

Given that :- α, β are the roots of the equation x² - 3λx + λ² = 0
α + β = 3λ ....................( 1 )
and αβ = λ² ...................( 2 )

Correct Option: A

Given that :- α, β are the roots of the equation x² - 3λx + λ² = 0
α + β = 3λ ....................( 1 )
and αβ = λ² ...................( 2 )