Friday, 28 October 2011

Algebra-12 (CAT-2002)

If x2+5y2+z2=2y(2x+z), then which of the following statements are necessarily true?
I. x=2y  II. x=2z  III. 2x=z
(1)only I   (2)only II   (3)only III   (4)both I and II
Solution follows here:

This is an algebraic manipulation and simplification problem;
x2+5y2+z2=2y(2x+z) => x2+5y2+z2=4xy+2yz
x2+5y2+z2-4xy-2yz=0 => x2+4y2+y2+z2-4xy-2yz=0
observe 4xy can be written as 2(x)(2y) and 2yz can be written as 2(y)(z)
x2+(2y)2-2(x)(2y)-2(x)(y)+y2+z2-2yz =0
(x-2y)2+(y-z)2 = 0
This specifies sum of squares of two terms is zero
=> both the terms are inevitably zero (it is not possible if the terms are +ve or –ve,
both the terms must be zero)
=> x-2y = 0; y-z = 0
=> x=2y; y=z
Only x=2y is given in the options.
Answer (1)

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