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NDA 2011
Exam
NDA
Year
2011
Subject
Mathematics
Chapter
Determinants

NDA 2011 · Mathematics · Determinants

What is the value of the determinant | array ccc x+1 & x+2 & x+4...

What is the value of the determinant \left|x+1x+2x+4x+3x+5x+8x+7x+10x+14\begin{array}{ccc}x+1 & x+2 & x+4 \\ x+3 & x+5 & x+8 \\ x+7 & x+10 & x+14\end{array}\right| ?

Options

  1. Ax+2x+2
  2. Bx2+2x^2+2
  3. C2
  4. D-2

Correct answer

D. -2

Step-by-step solution

∣x+1x+2x+4x+3x+5x+5x+7x+10x+14∣c1→c2−c1,c3−c2\begin{aligned} & \left|\begin{array}{ccc}x+1 & x+2 & x+4 \\ x+3 & x+5 & x+5 \\ x+7 & x+10 & x+14\end{array}\right| \\ & c_1 \rightarrow c_2-c_1, c_3-c_2\end{aligned}\left|−1−2x+4−2−3x+8−3−4x+14\begin{array}{ccc}-1 & -2 & x+4 \\ -2 & -3 & x+8 \\ -3 & -4 & x+14\end{array}\right|=∣12x+423x+834x+14∣=R2−R1,R3−R2\begin{aligned} & =\left|\begin{array}{ccc}1 & 2 & x+4 \\ 2 & 3 & x+8 \\ 3 & 4 & x+14\end{array}\right| \\ & =\mathrm{R}_2-\mathrm{R}_1, \mathrm{R}_3-\mathrm{R}_2\end{aligned}∣12x+4114116∣=R1−R2,R2−R3\begin{aligned} & \left|\begin{array}{ccc}1 & 2 & x+4 \\ 1 & 1 & 4 \\ 1 & 1 & 6\end{array}\right| \\ = & \mathrm{R}_1-\mathrm{R}_2, \mathrm{R}_2-\mathrm{R}_3\end{aligned}∣01x00−2116∣=1{(−2)}=−2.\begin{aligned} & \left|\begin{array}{ccc}0 & 1 & x \\ 0 & 0 & -2 \\ 1 & 1 & 6\end{array}\right| \\ = & 1\{(-2)\}=-2.\end{aligned}
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