Chapter Exam

Chapter-1 - Real Numbers — Online MCQ Test

Q: What is the name of the theorem that states every composite number can be uniquely expressed as a product of primes?

Answer: B. Fundamental Theorem of Arithmetic — The Fundamental Theorem of Arithmetic states that every composite number can be uniquely expressed as a product of primes, apart from the order of factors.

Q: Numbers having non-terminating, non-repeating decimal expansion are known as:

Answer: C. Irrational numbers — Irrational numbers have non-terminating and non-repeating decimal expansions, which distinguishes them from rational numbers.

Q: A proven statement used as a stepping stone towards the proof of another statement is known as:

Answer: D. Lemma — A lemma is a proven statement that is used as an intermediate step in proving a larger theorem.

Q: Which algorithm is used to find the HCF of two positive numbers?

Answer: B. Euclid's Division Algorithm — Euclid's Division Algorithm is specifically used to find the HCF (Highest Common Factor) of two positive numbers.

Q: Let x = 13/(2² × 5⁴). How many decimal places will x have when expanded as a terminating decimal?

Answer: C. 4 — The number of decimal places in a terminating decimal equals the highest power among 2 and 5 in the denominator. Here max(2, 4) = 4.

Q: Two tankers contain 850 litres and 680 litres of petrol. What is the maximum capacity of a container that can measure the petrol of either tanker an exact number of times?

Answer: A. 170 litres — The maximum capacity is the HCF of 850 and 680. HCF(850, 680) = 170 litres.

Q: If the HCF of two numbers is 1, which of the following must be true about their LCM?

Answer: C. LCM equals the product of the numbers — When HCF(a,b) = 1, the numbers are coprime, and LCM × HCF = product of numbers, so LCM × 1 = a × b, meaning LCM = a × b.

Q: If p and q are distinct primes, what is HCF(p², q²)?

Answer: C. 1 — Since p and q are distinct primes, p² = p×p and q² = q×q share no common prime factors. Therefore HCF(p², q²) = 1.

Q: Let a = 2³ × 3² × 5 and b = 2 × 3³ × 7. What is LCM(a, b) / HCF(a, b)?

Answer: A. 2² × 3 × 5 × 7 — HCF(a,b) = 2¹×3² = 18 and LCM(a,b) = 2³×3³×5×7 = 7560. LCM/HCF = 7560/18 = 420 = 2²×3×5×7.

MATHS · CLASS 10th · Karnataka State Board

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Chapter-1 - Real Numbers — Important Questions & Answers (FAQ)

Frequently asked questions from Karnataka State Board CLASS 10th MATHS — Chapter-1 - Real Numbers, with answers and explanations. These are sample questions; the exam has its own separate question set.

What is the name of the theorem that states every composite number can be uniquely expressed as a product of primes?

A. Euclid's Division Lemma
B. Fundamental Theorem of Arithmetic ✓
C. Fundamental Theorem of Algebra
D. Prime Factorization Theorem

Answer: B. Fundamental Theorem of Arithmetic
The Fundamental Theorem of Arithmetic states that every composite number can be uniquely expressed as a product of primes, apart from the order of factors.

Numbers having non-terminating, non-repeating decimal expansion are known as:

A. Rational numbers
B. Natural numbers
C. Irrational numbers ✓
D. Integers

Answer: C. Irrational numbers
Irrational numbers have non-terminating and non-repeating decimal expansions, which distinguishes them from rational numbers.

A proven statement used as a stepping stone towards the proof of another statement is known as:

A. Theorem
B. Corollary
C. Axiom
D. Lemma ✓

Answer: D. Lemma
A lemma is a proven statement that is used as an intermediate step in proving a larger theorem.

Which algorithm is used to find the HCF of two positive numbers?

A. Sieve of Eratosthenes
B. Euclid's Division Algorithm ✓
C. Fundamental Algorithm
D. Prime Factorization Algorithm

Answer: B. Euclid's Division Algorithm
Euclid's Division Algorithm is specifically used to find the HCF (Highest Common Factor) of two positive numbers.

Let x = 13/(2² × 5⁴). How many decimal places will x have when expanded as a terminating decimal?

A. 2
B. 3
C. 4 ✓
D. 5

Answer: C. 4
The number of decimal places in a terminating decimal equals the highest power among 2 and 5 in the denominator. Here max(2, 4) = 4.

Two tankers contain 850 litres and 680 litres of petrol. What is the maximum capacity of a container that can measure the petrol of either tanker an exact number of times?

A. 170 litres ✓
B. 85 litres
C. 340 litres
D. 680 litres

Answer: A. 170 litres
The maximum capacity is the HCF of 850 and 680. HCF(850, 680) = 170 litres.

In the proof of irrationality of √2, a contradiction is reached because:

A. 2 cannot be expressed as a fraction
B. Both a and b turn out to be divisible by 2, contradicting the assumption that they are coprime ✓
C. a² cannot equal 2b² for any integers
D. b must equal zero

Answer: B. Both a and b turn out to be divisible by 2, contradicting the assumption that they are coprime
The proof shows that both a and b must be divisible by 2, which contradicts the initial assumption that a/b is in lowest terms (coprime).

If the HCF of two numbers is 1, which of the following must be true about their LCM?

A. LCM equals 1
B. LCM equals the sum of the numbers
C. LCM equals the product of the numbers ✓
D. LCM equals the larger number

Answer: C. LCM equals the product of the numbers
When HCF(a,b) = 1, the numbers are coprime, and LCM × HCF = product of numbers, so LCM × 1 = a × b, meaning LCM = a × b.

If p and q are distinct primes, what is HCF(p², q²)?