Expert: Wei Chong - 2/14/2008

Question
Q.The sum of a number and its reciprocal is ½ . The sum of their squares is,

a)9/4, b)4/5,c)5/3,d)7/4

The correct answer as given by the author of the choice is 9/4.
however if i calculate then its -7/4
please see the procedure...

Let x be the no. and its reciprocal is 1/x,
according to question,
x 1/x=1/2
then squaring on both side,
(x 1/x)2=1/4

x2 2[{x}(1/x)] 1/x2=1/4

x2 1/x2=(1/4)-2

so sum of their square=-7/4.

However read somewhere in mathematics text book that sum of any no. and its reciprocal is always equal to or greater than 2  

Answer
Hi Paul,

Your calculation is shown below:

----------------------------------
Let x be the no. and its reciprocal is 1/x,
according to question,
x + 1/x=1/2
then squaring on both side,
(x +1/x)2=1/4

x2 +2[{x}(1/x)] +1/x2=1/4

x2 +1/x2=(1/4)-2

so sum of their square=-7/4
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Your calculation is correct if x is a complex number.

Since this question seems to be testing on basic math, the scope for this question should be limited for x > 0 (x is a real number). Hence, the question is wrongly written.

This is because, as you have already mentioned, the sum of the number and its reciprocal is >=2 for all x > 0.

I hope this helps.