Question

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★ STANDARD QUESTION 17 ★

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Question is -

The product of two of the four roots of the equation x⁴ - 18x³ + kx² + 200x - 1984 = 0 is - 32 , Evaluate the value of " k "


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Answer 1

Let α,β,∞,∅ be the roots
α+β+∞+∅=18                             -----------------------i
αβ+β∞+∞∅+α∞+α∅+β∅=k         -----------------------ii
αβ∞+β∞∅+∞∅α+αβ∅=200         -----------------------iii
αβ∞∅=-1984                              -----------------------iv
       Given 
αβ=-32
From iv , we get 
∞∅=62
From iii ,we get
-32∞+68β+62α-38∅=-200
62α+63β-32∞-32∅=-200
31α+31β-16∞-16∅=-100           ----------------------v
solving i and iv, we get
α+β=4                                        ---------------------vi
From i , we get 
∞+∅=14
(α+β)²=(α+β)²-4αβ
(α-β)²=4²-4(-32)
(α+β)²=144
α+β=12                                       --------------------vii

Solving vi and vii
α=8  and    β=-4
subsituting the value
In ii , we get
αβ+β∞+∞∅+α∞+α∅+β∅=k
-32+β(∞+∅)+α(∞+∅)+62 = k 
-32-56+112+62=k
k =86



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Answer 2

Hi friend here is your answer

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Plz refer to the attachment


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Hope it helps you..........!!