Check the validity of the statements given below by the method given against it.
(i) p: The sum of an irrational number and a rational number is irrational (by contradiction method).
(ii) q: If n is a real number with `n gt 3`, then `n^(2)gt9` (by contradiction method).

Verify by the method of contradiction that sqrt7 is irrational number

Find any five rational numbers between 2 and 3 using mean method.

Find the sum of: (i) the first 1000 natural numbers, (ii) the first n natural numbers.

Solve 3x+8 gt2 , when (i) x is an integer. (ii) x is a real number.

State whether the following statements are true or false. Justify your answers. (i) Every irrational number is a real number. (ii) Every point on the number line is of the form where m is a natural number. (iii) Every real number is an irrational number.

Show by using the principle of mathematical induction that for all natural number n gt 2, 2^(n) gt 2n+1

Check whether (i) 5sqrt2 (ii) (5)/(sqrt2) (iii) 21+sqrt3 (iv) pi+sqrt3 are irrational numbers or not?

Given below are two pairs of statements. Combine these two statements using ''if and only if ''. (i) p: If a rectangle is a square, then all its four sides are equal. q: If all the four sides of a rectangle are equal, then the rectangle is a square. (ii) p: If the sum of digits of a number is divisible by 3, then the number is divisible by 3. q: If a number is divisible by 3, then the sum of its digits is divisible by 3.

Let p be the statement ''x is an irrational number'', q be the statement'' y is a transcendental number'' and r be the statement'' x is a rational number iff y is a transcendental number.'' Statement-1 : r is equivalent to either q or p Statement-2 : r is equivalent to ~(p iff~q) .