True and False
Q1: State whether the given statement is true or false :
(i) The product of two rationals is always rational
(ii) The product of two irrationals is an irrational
(iii) The product of a rational and an irrational is irrational
(iv) The sum of two rationals is always rational
(v) The sum of two irrationals is an irrational.
Very Short Questions
Q.2. Classify the following numbers as rational or irrational:
(i) 3.1416
(ii) 3.142857
(iii) 2.040040004......
(iv) 3.121221222...
(v) 3√3
Q.3. Find the prime factorization of 1152
Q4: Show that the product of two numbers 60 and 84 is equal to the product of their HCF and LCM
Q5: If p and q are two coprime numbers, then find the HCF and LCM of p and q.
Short Questions
Q6: Given that H.C.F. (306, 657) = 9, find L.C.M. (306, 657)
Q7: The H.C.F. and L.C.M. of two numbers are 12 and 240 respectively. If one of these numbers is 48; find the other numbers.
Long Questions
Q8: Use Euclid’s Division Algorithm to show that the cube of any positive integer is either of the 9m, 9m + 1 or 9m + 8 for some integer m.
Q9: Show that any positive integer which is of the form 6q + 1 or 6q + 3 or 6q + 5 is odd, where q is some integer.
Q10: Prove that
(i) √2 is irrational number
(ii) √3 is irrational number
Similarly √5, √7, √11…... are irrational numbers.
You can find Worksheets Solutions here: Worksheet Solutions: Real Numbers
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1. What are real numbers? |
2. How are real numbers different from imaginary numbers? |
3. Can a real number be negative? |
4. Are all integers real numbers? |
5. What is the difference between rational and irrational numbers? |
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