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NDA 2016
Exam
NDA
Year
2016
Subject
Mathematics
Chapter
Determinants

NDA 2016 · Mathematics · Determinants

Which of the following are correct in respect of the system of...

Which of the following are correct in respect of the system of equations x+y+z=8,x−y+2z=6x+y+z=8, x-y+2 z=6 and 3x−y+5z=k3 x-y+5 z=k ? 1. They have no solution. if k=15\mathrm{k}=15. 2. They have infinitely many solutions, if k=20k=20. 3. They have unique solution, if k=25\mathrm{k}=25 Select the correct answer using the code given below:

Options

  1. A1 and 2 only
  2. B2 and 3 only
  3. C1 and 3 only
  4. D1,2 and 3

Correct answer

A. 1 and 2 only

Step-by-step solution

Here \left[1111−123−15\begin{array}{ccc}1 & 1 & 1 \\ 1 & -1 & 2 \\ 3 & -1 & 5\end{array}\right]\left[xyz\begin{array}{l}x \\ y \\ z\end{array}\right]=\left[86k\begin{array}{l}8 \\ 6 \\ k\end{array}\right] or AX=BA X=B. ∣A∣=0(∵|A|=0(\because the system does not have a unique solution )) Now, (\operatorname{Adj} A)=\left[−3−6312−124−2\begin{array}{ccc}-3 & -6 & 3 \\ 1 & 2 & -1 \\ 2 & 4 & -2\end{array}\right] and B=\left[86k\begin{array}{c}8 \\ 6 \\ k\end{array}\right] For k=15,(\operatorname{Adj} A) B=\left[−24−36+158+12−1516+24−30\begin{array}{c}-24-36+15 \\ 8+12-15 \\ 16+24-30\end{array}\right]=\left[−45510\begin{array}{c}-45 \\ 5 \\ 10\end{array}\right] \neq 0 ( ∵\because system is inconsistent i.e., it has no solution) For k=20,(\operatorname{Adj} A) B=\left[−24−36+608+12−2016+24−40\begin{array}{c}-24-36+60 \\ 8+12-20 \\ 16+24-40\end{array}\right]=\left[000\begin{array}{l}0 \\ 0 \\ 0\end{array}\right]=0 ( ∵\because system has infinitely many solutions)
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