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NDA 2023
Exam
NDA
Year
2023
Subject
Mathematics
Chapter
Determinants

NDA 2023 · Mathematics · Determinants

Let A X=B be a system of 3 liner equations with 3 -unknowns. Let...

Let AX=BA X=B be a system of 3 liner equations with 3 -unknowns. Let X1X_1 and X2X_2 be its two distinct solutions. If the combination aX1+bX2a X_1+b X_2 is a solution of AX=BA X=B; where a,ba, b, are real numbers, then which one of the following is correct?

Options

  1. Aa=ba=b
  2. Ba+b=1a+b=1
  3. Ca+b=0a+b=0
  4. Da−b=1a-b=1

Correct answer

B. a+b=1a+b=1

Step-by-step solution

AX=B⇒AX1=BA X=B \Rightarrow A X_1=B and A X_2=B Since (aX++bX2)\left(a X_{+}+b X_2\right) is also a solution. Hence, A(aX1+bX2)=B⇒aAX1+bAX2=BaB+bB=BB(a+b)=B∴a+b=1A\left(a X_1+b X_2\right)=B\Rightarrow \quad a A X_1+b A X_2=Ba B+b B=BB(a+b)=B\therefore \quad a+b=1
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