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NDA 2024
Exam
NDA
Year
2024
Subject
Mathematics
Chapter
Quadratic Equation

NDA 2024 · Mathematics · Quadratic Equation

If n is a root of the equation x^2+p x+m=0 and m is a root of the...

If nn is a root of the equation x2+px+m=0x^2+p x+m=0 and mm is a root of the equation x2+px+n=0x^2+p x+n=0, where m≠nm \neq n, then what is the value of p+m+np+m+n ?

Options

  1. A−1-1
  2. B00
  3. C11
  4. D22

Correct answer

C. 11

Step-by-step solution

Given, nn is a root of x2+px+m=0⇒n2+pn+m=0x^2+p x+m=0\Rightarrow \mathrm{n}^2+\mathrm{pn}+\mathrm{m}=0..(i) Given, m is a root of the equation x2+px+n=0⇒m2+pm+n=0\mathrm{x}^2+\mathrm{px}+\mathrm{n}=0\Rightarrow \mathrm{m}^2+\mathrm{pm}+\mathrm{n}=0...(ii) Now subtracting Eq (ii) from Eq (i), we get (n2−m2)+p(n−m)−(n−m)=0⇒(n−m)[(n+m)+p−1]=0⇒n+m+p−1=0 {given, n≠m}⇒n+m+p=1\begin{aligned} & \left(\mathrm{n}^2-\mathrm{m}^2\right)+\mathrm{p}(\mathrm{n}-\mathrm{m})-(\mathrm{n}-\mathrm{m})=0 \\ \Rightarrow & (\mathrm{n}-\mathrm{m})[(\mathrm{n}+\mathrm{m})+\mathrm{p}-1]=0 \\ \Rightarrow & \mathrm{n}+\mathrm{m}+\mathrm{p}-1=0 \text{ \{given, } \mathrm{n} \neq \mathrm{m}\} \\ \Rightarrow & \mathrm{n}+\mathrm{m}+\mathrm{p}=1 \end{aligned}
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