Moderate Level Worksheet

Very Short Answer Questions (1 Mark Each)

(1) Write one rational number between 2 and 3. 1.51.31.7521.1

(2) State whether 0 is a prime or composite number. Neither prime nor compositeUndefinedPrimeComposite

Correct! 0 is neither prime nor composite by definition, just like 1.

(3) Find the HCF of 81 and 96 using prime factorization. HCF =

Excellent! HCF is the product of lowest powers of common prime factors.

(4) Express 1.272727... in the form pq.

(5) What is the HCF of two consecutive odd numbers?

Correct! Consecutive odd numbers like 3,5 or 7,9 are always coprime.

Short Answer Questions (2 Marks Each)

Answer each question in 2-3 sentences

(1) Use Euclid's division lemma to find the HCF of 100 and 120. HCF =

Excellent! Euclid's algorithm stops when remainder becomes 0.

(2) Find the LCM of 8, 9, and 12. LCM =

Perfect! LCM uses highest powers of all prime factors.

(3) Prove that 3 is irrational.

(4) Write any two irrational numbers between 3 and 4. 10, 1112, 1314, 15

Great! Any non-perfect square root between 9 and 16 works.

(5) If HCF of two numbers is 5 and their product is 400, find their LCM. LCM =

Perfect! This is a fundamental relationship for any two numbers.

Long Answer Questions (4 Marks Each)

(1) Show that 2 + 5 is an irrational number.

(2) Two positive integers a and b are written as a = 4q + 3, b = 4q + 1. Show that a2 − b2 is divisible by 4.

a2 − b2 = 8(2q + 1)12(2q + 1)

Perfect! Since it equals 4 times an integer, it's divisible by 4.

(3) Find the HCF and LCM of 65 and 117 using the Euclidean Algorithm. HCF = and LCM =

Excellent work with the Euclidean algorithm!

(4) Prove that there are infinitely many prime numbers.

Part B: Objective Questions (1 Mark Each)

Choose the correct answer and write the option (a/b/c/d)

(1) Which of the following is not irrational?

(a) 2 (b) 8 (c) 0.5 (d) 5

Correct! 0.5 = 12 is rational. All the square roots are irrational.

(2) The LCM of 45 and 75 is:

(a) 225 (b) 90 (c) 135 (d) 15

Correct! 45 = 32 × 5, 75 = 3 × 52, so LCM = 32 × 52 = 225.

(3) If HCF of two numbers is 18 and their product is 1296, what is their LCM?

(a) 72 (b) 108 (c) 45 (d) 84

Correct! Using HCF × LCM = Product: 18 × LCM = 1296, so LCM = 72.

(4) The decimal expansion of a rational number is always:

(a) Terminating only (b) Repeating only (c) Either terminating or repeating (d) Non-terminating and non-repeating

Correct! Rational numbers can terminate (like 0.5) or repeat (like 0.333...).

(5) The number 50 is:

(a) Rational (b) Irrational (c) Integer (d) Prime

Correct! 50 = 52, and since 2 is irrational, 50 is irrational.

(6) Which of the following is a composite number?

(a) 1 (b) 2 (c) 3 (d) 4

Correct! 4 = 2 × 2, so it's composite. 1 is neither prime nor composite.

(7) 0.142857142857... is:

(a) Terminating (b) Non-terminating and non-repeating (c) Rational (d) Irrational

Correct! It's repeating (142857 repeats), so it's rational. Actually equals 17.

(8) If p and q are positive integers such that p × q = 180 and HCF(p, q) = 6, then LCM(p, q) = ?

(a) 30 (b) 36 (c) 60 (d) 90

Correct! Using HCF × LCM = p × q: 6 × LCM = 180, so LCM = 30.

(9) Which of these numbers is not a prime number?

(a) 2 (b) 11 (c) 21 (d) 17

Correct! 21 = 3 × 7, so it's composite, not prime.

(10) Euclid's lemma is used to:

(a) Find square roots (b) Find cube roots (c) Find LCM (d) Find HCF

Correct! Euclid's division lemma is the foundation of the algorithm for finding HCF.

Real Numbers Challenge

Determine whether these statements about real numbers are True or False:

Real Numbers Quiz