Chapter-1 - Real Numbers — Online MCQ Test
MATHS · Grade 10 · CBSE(NCERT)
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This chapter covers: First Chapter.
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Chapter-1 - Real Numbers — Important Questions & Answers
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.
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.
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.
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.
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.
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.
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).
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.
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)?
- A. pq
- B. p²q²
- C. 1
- D. p²
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.
Since p and q are distinct primes, p² = p×p and q² = q×q share no common prime factors. Therefore HCF(p², q²) = 1.
Let a = 2³ × 3² × 5 and b = 2 × 3³ × 7. What is LCM(a, b) / HCF(a, b)?
- A. 2² × 3 × 5 × 7
- B. 2³ × 3³ × 5 × 7
- C. 2² × 3² × 35
- D. 2³ × 3³ × 35 / (2 × 3²)
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.
HCF(a,b) = 2¹×3² = 18 and LCM(a,b) = 2³×3³×5×7 = 7560. LCM/HCF = 7560/18 = 420 = 2²×3×5×7.