NCERT Class 10 Mathematics

PROOFS IN MATHEMATICS 333

EXERCISE A1.6

1. Suppose a + b = c + d, and a < c. Use proof by contradiction to show b d.


2. Let r be a rational number and x be an irrational number. Use proof by contradiction to
   show that r + x is an irrational number.


3. Use proof by contradiction to prove that if for an integer a, a^2 is even, then so is a.
   [Hint : Assume a is not even, that is, it is of the form 2n + 1, for some integer n, and then
   proceed.]


4. Use proof by contradiction to prove that if for an integer a, a^2 is divisible by 3, then a is
   divisible by 3.


5. Use proof by contradiction to show that there is no value of n for which 6n ends with the
   digit zero.


6. Prove by contradiction that two distinct lines in a plane cannot intersect in more than
   one point.

A1.8 Summary

In this Appendix, you have studied the following points :

1. Different ingredients of a proof and other related concepts learnt in Class IX.


2. The negation of a statement.


3. The converse of a statement.


4. Proof by contradiction.