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CAT 2023 Slot 1 — QA Question 20

MeansEasy

For some positive and distinct real numbers x, y and z if 1y+z is the arithmetic mean of 1x+z and 1x+y, then the relationship which will always hold true, is?

Answer & solution

Correct answer: y, x and z are in Arithmetic Progression

  • A

    x, y and z are in Arithmetic Progression

  • y, x and z are in Arithmetic Progression

  • C

    √x, √y and √z  are in Arithmetic Progression

  • D

    √y, √x and √z are in Arithmetic Progression

Solution

Easy

"Arithmetic mean" means the middle term is half the sum of the other two — equivalently, twice the middle equals their sum. Write that out, clear the denominators by cross-multiplying, and the square-root terms cancel cleanly to leave a plain relation among x,y,zx,y,z.

1

Write the AM condition (2×middle=sum of the others2\times\text{middle}=\text{sum of the others}):

2y+z=1x+z+1x+y\frac{2}{\sqrt y+\sqrt z}=\frac{1}{\sqrt x+\sqrt z}+\frac{1}{\sqrt x+\sqrt y}
2

Combine the right side over a common denominator:

2y+z=(x+y)+(x+z)(x+z)(x+y)=2x+y+z(x+z)(x+y)\frac{2}{\sqrt y+\sqrt z}=\frac{(\sqrt x+\sqrt y)+(\sqrt x+\sqrt z)}{(\sqrt x+\sqrt z)(\sqrt x+\sqrt y)}=\frac{2\sqrt x+\sqrt y+\sqrt z}{(\sqrt x+\sqrt z)(\sqrt x+\sqrt y)}
3

Cross-multiply and expand both sides:

2(x+z)(x+y)=(2x+y+z)(y+z)2x+2xy+2xz+2yz=2xy+2xz+y+2yz+z\begin{aligned} &2(\sqrt x+\sqrt z)(\sqrt x+\sqrt y)=(2\sqrt x+\sqrt y+\sqrt z)(\sqrt y+\sqrt z)\\ &2x+2\sqrt{xy}+2\sqrt{xz}+2\sqrt{yz}=2\sqrt{xy}+2\sqrt{xz}+y+2\sqrt{yz}+z \end{aligned}
4

Cancel matching terms. The surd terms 2xy, 2xz, 2yz2\sqrt{xy},\,2\sqrt{xz},\,2\sqrt{yz} appear on both sides and vanish:

2x=y+z2x=y+z

So xx is the arithmetic mean of yy and zz — i.e. y, x, zy,\,x,\,z are in Arithmetic Progression.

y, xy,\,x and zz are in Arithmetic Progression.

Watch the ordering: the relation is 2x=y+z2x=y+z, which puts xx (not the original middle index) as the mean. That is why the answer is "y,x,zy,x,z in AP" and not "x,y,zx,y,z in AP."

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CAT 2023 Slot 1 QA Q20: For some positive and distinct real numbers x, y and z if 1 y + z is the arithmetic mean of 1 x + z and 1 x + — Solution | TheCATExam