This is a Test of Mathematics Solution Subjective 71 (from ISI Entrance). The book, Test of Mathematics at 10+2 Level is Published by East West Press. This problem book is indispensable for the preparation of I.S.I. B.Stat and B.Math Entrance.

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Consider the following simultaneous equations in x and y :

where is a real constant. Show that these equations admit real solutions in x and y.

… (i)

… (ii)

[ replacing x in terms of y. ]

This is a 3rd degree polynomial over y.

Now as we know any odd degree polynomial has at least one real root.That is why y has also at least one real root.Now for that particular root we get also real.

*Conclusion*: & admits real solutions in x and y.