# real numbers class 10 extra questions

Question 1) Find the nature of the product (√2 -√3) ( √3 + √2) ?

Question 2) Prove that the sum of a rational number and an irrational number is always irrational.

Question 3) Prove that √5 is an irrational number.

Question 4) Show that 3 + 5√2  is an irrational number. Is sum of two irrational numbers always an irrational number?
Question 5) Prove that  √3 is an irrational number and hence show that 2√3  is also an irrational number.
Question 6) Prove that 5 - √3  is an irrational number.
Question 7) Prove that 2√5  is an irrational number.
Question 8) Show that (√3+   √5) 2 is an irrational number.
Question 9) Prove that 4 - √5  is an irrational number.
Question 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.

Question 11) Prove that  √2 + 1/√2 is an irrational number
Question 12) Prove that for any positive integer n, n3 – n is divisible by 6.

Question 13) If n is rational and √m   is irrational, then prove that (n + √m) is irrational.
Question 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

Question 15) Prove that √11 is irrational.
Question 16) Show that 3√2 is irrational.
Question 17) Show that 4n can never end with the digit zero for any natural number n.

Question 18)The product of a non-zero rational and an irrational number is
(A) always irrational
(B) always rational
(C) rational or irrational
(D) one

Question 19) Prove that √p + √q is irrational, where p, q are primes.
Question 20) Prove that one of any three consecutive positive integers must be divisible by 3.