1) Find the nature of the product (√2 -√3) ( √3 + √2) ?
2) Prove that the sum of a rational number and an irrational number is always irrational.
3) Prove that √5 is an irrational number.
4) Show that 3 + 5√2 is an irrational number. Is sum of two irrational numbers always an irrational number?
5) Prove that √3 is an irrational number and hence show that 2√3 is also an irrational number.
6) Prove that 5 - √3 is an irrational number.
7) Prove that 2√5 is an irrational number.
8) Show that (√3+ √5) 2
is an irrational number.
9) Prove that 4 - √5 is an irrational number.
10) Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9m, 9m + 1 or 9m + 8.
11) Prove that √2 + 1/√2 is an irrational number
12) Prove that for any positive integer n, n3
– n is divisible by 6.
13) If n is rational and √m is irrational, then prove that (n + √m) is irrational.
14) Show that one and only one out of n, n + 4, n + 8, n + 12 and n + 16 is divisible by 5, where n is any positive integer
15) Prove that √11 is irrational.
16 Show that 3√2 is irrational.
17) Show that 4n
can never end with the digit zero for any natural number n.
18) The product of a non-zero rational and an irrational number is
(A) always irrational
(B) always rational
(C) rational or irrational
19) Prove that √p + √q is irrational, where p, q are primes.
20) Prove that one of any three consecutive positive integers must be divisible by 3.
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