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NDA 2017
Exam
NDA
Year
2017
Subject
Mathematics
Chapter
Determinants

NDA 2017 · Mathematics · Determinants

If | array ccc x & y & 0 0 & x & y y & 0 & x array |=0 , then which...

If \left|xy00xyy0x\begin{array}{ccc}\mathrm{x} & \mathrm{y} & 0 \\ 0 & \mathrm{x} & \mathrm{y} \\ \mathrm{y} & 0 & \mathrm{x}\end{array}\right|=0$, then which one of the following is correct?

Options

  1. Axy\frac{x}{y} is one of the cube roots of unity
  2. Bx is one of the cube roots of unity
  3. Cy is one of the cube roots of unity
  4. Dxy\frac{x}{y} is one of the cube roots of -1

Correct answer

D. xy\frac{x}{y} is one of the cube roots of -1

Step-by-step solution

∣xy00xyy0x∣=0⇒x(x2−0)−y(0−y2)+0=0⇒x3+y3=0\begin{aligned} & \left|\begin{array}{lll}\mathrm{x} & \mathrm{y} & 0 \\ 0 & \mathrm{x} & \mathrm{y} \\ \mathrm{y} & 0 & \mathrm{x}\end{array}\right|=0 \Rightarrow x\left(\mathrm{x}^2-0\right)-\mathrm{y}\left(0-\mathrm{y}^2\right)+0=0 \\ & \Rightarrow \mathrm{x}^3+\mathrm{y}^3=0\end{aligned}⇒x3=−y3⇒x3y3=−1⇒(xy)3=−1⇒xy=−13\begin{aligned} & \Rightarrow x^3=-y^3 \\ & \Rightarrow \frac{x^3}{y^3}=-1 \Rightarrow\left(\frac{x}{y}\right)^3=-1 \Rightarrow \frac{x}{y}=\sqrt[3]{-1}\end{aligned}$
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