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NDA 2017
Exam
NDA
Year
2017
Subject
Mathematics
Chapter
Sequences and Series

NDA 2017 · Mathematics · Sequences and Series

If x_1 and x_2 are positive quantities, then the condition for the...

If x1x_1 and x2x_2 are positive quantities, then the condition for the difference between the arithmetic mean and the geometric mean to be greater than 1 is

Options

  1. Ax1+x2>2x1x2\mathrm{x}_1+\mathrm{x}_2 > 2 \sqrt{\mathrm{x}_1 \mathrm{x}_2}
  2. Bx1+x2>2\sqrt{\mathrm{x}_1}+\sqrt{\mathrm{x}_2} > \sqrt{2}
  3. C∣x1−x2∣>2\left|\sqrt{\mathrm{x}_1}-\sqrt{\mathrm{x}_2}\right| \gt \sqrt{2}
  4. Dx1+x2<2(x1x2+1)\mathrm{x}_1+\mathrm{x}_2 \lt 2 \left(\sqrt{\mathrm{x}_1 \mathrm{x}_2}+1\right)

Correct answer

C. ∣x1−x2∣>2\left|\sqrt{\mathrm{x}_1}-\sqrt{\mathrm{x}_2}\right| \gt \sqrt{2}

Step-by-step solution

Arithmetic mean of x1,x2=x1+x22x_1, x_2=\frac{x_1+x_2}{2} Geometric mean of x1,x2=x1x2\mathrm{x}_1, \mathrm{x}_2=\sqrt{\mathrm{x}_1 \mathrm{x}_2} Given, x1+x22−x1x2>1⇒x1+x22>x1x2+1\frac{\mathrm{x}_1+\mathrm{x}_2}{2}-\sqrt{\mathrm{x}_1 \mathrm{x}_2} > 1\Rightarrow \frac{\mathrm{x}_1+\mathrm{x}_2}{2} > \sqrt{\mathrm{x}_1 \mathrm{x}_2}+1⇒x1+x2>2x1x2+2⇒x1+x2−2x1x2>2⇒(x1)2+(x2)2−2x1x2>2\begin{aligned} & \Rightarrow \mathrm{x}_1+\mathrm{x}_2 > 2 \sqrt{\mathrm{x}_1 \mathrm{x}_2}+2 \\ & \Rightarrow \mathrm{x}_1+\mathrm{x}_2-2 \sqrt{\mathrm{x}_1 \mathrm{x}_2} > 2 \\ & \Rightarrow\left(\sqrt{\mathrm{x}_1}\right)^2+\left(\sqrt{\mathrm{x}_2}\right)^2-2 \sqrt{\mathrm{x}_1} \sqrt{\mathrm{x}_2} > 2\end{aligned}$\begin{aligned} & \Rightarrow\left(\sqrt{\mathrm{x}_1}-\sqrt{\mathrm{x}_2}\right)^2 > 2 \\ & \Rightarrow\left|\sqrt{\mathrm{x}_1}-\sqrt{\mathrm{x}_2}\right| > \sqrt{2}\end{aligned}
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