5x2-10X+7=0,alpha+beta =? and alpha×beta=?
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Step-by-step explanation:
Correct option is B)
To determine:
2a
11
a
12
−7a
10
Given: α & β are roots of x
2
+10x−7=0
a
n
=α
n
+β
n
[n≥1]
So, x=α is solution of given eq
n
.
Thus, α
2
+10α−7=0 →⟨1⟩
Multiplying ⟨1⟩ by α
10
we get:
α
12
+10α
11
−7α
10
=0
⇒α
12
−7α
10
=−10α
11
→⟨2⟩
Similarly, β
12
−7β
10
=−10β
11
[∵β is also root of given eq
n
]
Now,
a
12
=α
12
+β
12
a
10
=α
10
+β
10
a
11
=α
11
+β
11
⇒a
12
−7a
10
=(α
12
+β
12
)−7(α
10
+β
10
)
⇒=(α
12
−7α
10
)+(β
12
−7β
10
)
=−10(α
11
+β
11
) [Using eq
n
⟨2⟩]
So,
⇒
2a
11
a
12
−7a
10
=
2(α
11
+β
11
)
−10(α
11
+β
11
)
⇒−5
Hence, Option (B) is correct.
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