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NDA 2011
Exam
NDA
Year
2011
Subject
Mathematics
Chapter
Determinants

NDA 2011 · Mathematics · Determinants

If 5 and 7 are the roots of the equation | array ccc x & 4 & 5 7 &...

If 5 and 7 are the roots of the equation \left|x457x758x\begin{array}{ccc} x & 4 & 5 \\ 7 & x & 7 \\ 5 & 8 & x \end{array}\right|=0 then what is the third root?

Options

  1. A-12
  2. B9
  3. C13
  4. D14

Correct answer

A. -12

Step-by-step solution

\left|x457x758x\begin{array}{lll}x & 4 & 5 \\ 7 & x & 7 \\ 5 & 8 & x\end{array}\right|=0⇒x(x2−56)−7(4x−40)+5(28−5x)=0⇒x3−56x−28x+280+140−25x=0⇒x3−109x+420=0\begin{aligned} & \Rightarrow x \left(x^2-56\right)-7(4 x-40)+5(28-5 x)=0 \\ & \Rightarrow x^3-56 x-28 x+280+140-25 x=0 \\ & \Rightarrow x^3-109 x+420=0\end{aligned}$Since5and7aretherootsofthegivenequationsothatSince 5 and 7 are the roots of the given equation so thatx^2-12 x+35isafactorofthegivenequation,whenwedividethegivenequationbyis a factor of the given equation, when we divide the given equation byx^2-12 x+35itgivesit givesx+12=0\Rightarrow \quad x=-12$.
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