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Prove that the sum of or the difference ...

Prove that the sum of or the difference between, a rational number `alpha` and an irrational number `beta` is an irrational number.

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IA MARON-INTRODUCTION OF MATHEMATICAL ANALYSIS-Additional Problems
  1. Prove that the sum of or the difference between, a rational number alp...

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  2. Prove the inequalities : (a) n! lt ((n+1)/2)^(n) for a natural n gt ...

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  3. Prove the inequalities : (a) 202^(303) gt 303^(202) (b) 202! Lt 100^...

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  4. Solve the inequalities : (a) |x|-2| le 1, (b) ||2-3x|-1|gt 2, (c) ...

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  5. Can a sum difference, product or quotient of irrational numbers be a r...

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  6. Do the equations (a) |sin x|=sin x+3, (b) |tan x|=tan x+3 have any r...

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  7. Prove the identity ((x+|x|)/2)^(2)+((x-|x|)/(2))^(2) =x^(2)

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  8. Prove that Bernoulli inqeuality (1+x(1)) [1+x(2)).....(1+x(n)) ge 1+x(...

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  9. Find the domains of defination of the following functions: (a) f(x)=...

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  10. Are the following functions identical? (a) f(x)=x/x and phi (x)=1 ...

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  11. In what interval are the following functions identical? (a) f(x)=x a...

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  12. An isosceles triangle of a given perimeter 2p=12 revolves about its ba...

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  13. Investigating the domain of definition of functions. (a) Solve the i...

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  14. Prove that the product of two even or two odd functions is an even fun...

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  15. Prove that if the domain of definition of the function f(x) is symmetr...

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  16. Prove that any function f(x) defined in a symmetrical interval (-l, l)...

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  17. Extend the function f(x)=x^2 + x defined on the interval (0,3) onto th...

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  18. Prove theat the Drichlet function lambda(x) is a periodic one but has ...

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  19. Prove that if the function f(x)=sin x+cos a x is a periodic, than a is...

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  20. Prove that the sum of two functions increasing on a certain open inter...

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  21. Give an examole of non-monotonic functions that has an inverse.

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