CBSE Class 10 Maths: Chapter 1 Real Numbers - Part 1: MCQ
Welcome to Part 1 of our new 8-part series on Chapter 1, Real Numbers. This post contains the top 25 Multiple Choice Questions (MCQs) to help you master the fundamental concepts of HCF, LCM, and irrational numbers.
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Top 25 MCQs - Real Numbers
Question 1: HCF & LCM Relation
The HCF of two numbers is 18 and their product is 12960. Their LCM will be:
HCF(a, b) × LCM(a, b) = Product of the two numbers
Given, HCF = 18 and Product = 12960
So, 18 × LCM = 12960
LCM = 12960 / 18
LCM = 720
Question 2: Prime Factorization
The total number of factors of a prime number is:
Question 3: HCF of Smallest Primes
The HCF of the smallest composite number and the smallest prime number is:
The smallest prime number is 2.
The smallest composite number is 4 (since 1 is neither, 2 is prime, 3 is prime).
Prime factors of 2 = 2
Prime factors of 4 = 2 × 2
HCF(2, 4) = 2
Question 4: Decimal Expansion
The decimal expansion of the rational number 14587 / 1250 will terminate after how many decimal places?
Denominator = 1250 = 125 × 10 = 5³ × (2 × 5) = 2¹ × 5⁴
The highest power is 4.
Therefore, the decimal will terminate after 4 places.
Question 5: Irrational Numbers
Which of the following is an irrational number?
(a) 3.1416 is a terminating decimal, so it is rational.
(b) 22/7 is in p/q form, so it is rational. (Note: It is an approximation of π, not π itself).
(c) √9 = 3, which is rational.
(d) π (Pi) is a non-terminating, non-repeating decimal, which is the definition of an irrational number.
Question 6: Fundamental Theorem
The HCF and LCM of 12, 21, and 15 respectively are:
12 = 2² × 3¹
21 = 3¹ × 7¹
15 = 3¹ × 5¹
HCF (Highest Common Factor) = Smallest power of common factors.
HCF = 3¹ = 3
LCM (Lowest Common Multiple) = Highest power of all factors.
LCM = 2² × 3¹ × 5¹ × 7¹ = 4 × 3 × 5 × 7 = 420
Question 7: Product of Primes
The number 140 can be expressed as a product of its prime factors as:
140 = 14 × 10 = (2 × 7) × (2 × 5) = 2 × 2 × 5 × 7 = 2² × 5 × 7
Question 8: Rational/Irrational Sum
The sum of a rational and an irrational number is always:
Question 9: Non-Terminating Decimal
Which of the following rational numbers will have a non-terminating repeating decimal expansion?
(a) 3125 = 5⁵ (Terminating)
(b) 8 = 2³ (Terminating)
(c) 455 = 5 × 7 × 13. Since it has 7 and 13 as factors, it is non-terminating repeating.
(d) 1600 = 16 × 100 = 2⁴ × 10² = 2⁴ × (2² × 5²) = 2⁶ × 5² (Terminating)
Question 10: HCF Application
Two tankers contain 850 litres and 680 litres of kerosene oil respectively. Find the maximum capacity of a container that can measure the kerosene oil of both the tankers when used an exact number of times.
850 = 85 × 10 = (5 × 17) × (2 × 5) = 2 × 5² × 17
680 = 68 × 10 = (4 × 17) × (2 × 5) = (2² × 17) × (2 × 5) = 2³ × 5 × 17
HCF = Smallest powers of common factors = 2¹ × 5¹ × 17¹ = 2 × 5 × 17 = 170
Question 11: LCM Application
The traffic lights at three different road crossings change after every 48 sec, 72 sec and 108 sec respectively. If they all change simultaneously at 7:00:00 a.m., at what time will they change simultaneously again?
48 = 16 × 3 = 2⁴ × 3¹
72 = 8 × 9 = 2³ × 3²
108 = 27 × 4 = 3³ × 2²
LCM = Highest powers of all factors = 2⁴ × 3³ = 16 × 27 = 432 seconds
Now, convert 432 seconds to minutes:
432 / 60 = 7 minutes and 12 seconds
They will change again after 7 min 12 sec.
New time = 7:00:00 + 0:07:12 = 7:07:12 a.m.
Question 12: Irrational Product
The product of two different irrational numbers is always:
Case 1 (Irrational): √2 × √3 = √6 (Irrational)
Case 2 (Rational): √2 × √8 = √16 = 4 (Rational)
Case 3 (Rational): (2 + √3) × (2 - √3) = 2² - (√3)² = 4 - 3 = 1 (Rational)
Since the result can be both, this is the correct choice.
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Question 13: Nature of a Number
The number (5 - 3√2) is:
Question 14: n² - 1 Divisibility
If 'n' is any odd positive integer, then n² - 1 is always divisible by:
If n = 1, n² - 1 = 1 - 1 = 0 (Divisible by 8)
If n = 3, n² - 1 = 9 - 1 = 8 (Divisible by 8)
If n = 5, n² - 1 = 25 - 1 = 24 (Divisible by 8)
Algebraic Proof: An odd integer can be written as (2k+1) or (4q+1) or (4q+3).
n² - 1 = (n-1)(n+1). If n=2k+1, then (2k)(2k+2) = 4k(k+1). Since k(k+1) is always even, k(k+1)=2m. So, 4(2m) = 8m. It is always divisible by 8.
Question 15: Rational Form
The p/q form of 0.23 (with a bar over 23) is:
Multiply by 100 (since 2 digits are repeating):
100x = 23.232323...
Subtract x from 100x:
100x - x = 23.2323... - 0.2323...
99x = 23
x = 23 / 99
Question 16: HCF of 2ⁿ × 3² and 2² × 3³
If p = 2³ × 3² and q = 2² × 3³, then HCF(p, q) is:
p = 2³ × 3²
q = 2² × 3³
Common factors are 2 and 3.
Smallest power of 2 is 2².
Smallest power of 3 is 3².
HCF = 2² × 3²
Question 17: LCM of 2³ × 3¹ and 2² × 3²
If a = 2³ × 3¹ and b = 2² × 3², then LCM(a, b) is:
a = 2³ × 3¹
b = 2² × 3²
Factors are 2 and 3.
Highest power of 2 is 2³.
Highest power of 3 is 3².
LCM = 2³ × 3²
Question 18: Nature of (√2 - √3)(√2 + √3)
The number (√2 - √3)(√2 + √3) is:
Here, a = √2 and b = √3
(√2)² - (√3)² = 2 - 3 = -1
Since -1 can be written as -1/1, it is a rational number.
Question 19: Ending Digit 0
For any natural number 'n', the number 6ⁿ cannot end with the digit:
The prime factorization of 6ⁿ is (2 × 3)ⁿ = 2ⁿ × 3ⁿ.
Since the prime factors do not contain 5, 6ⁿ can never end with the digit 0.
Question 20: HCF of Co-primes
Two numbers are co-prime if their HCF is:
Question 21: LCM of Co-primes
The LCM of two co-prime numbers is always:
HCF × LCM = Product of numbers
1 × LCM = Product of numbers
LCM = Product of numbers
Question 22: Smallest Prime Factor
The smallest prime factor of the sum of the two smallest prime numbers is:
The smallest prime number is 2.
The second smallest prime number is 3.
Their sum is 2 + 3 = 5.
The number 5 is a prime number, and its smallest prime factor is 5 itself.
Question 23: Decimal Expansion of √3
The decimal expansion of √3 is:
Question 24: Terminating Decimal Condition
A rational number p/q has a terminating decimal expansion if the prime factorization of q is of the form:
Question 25: HCF × LCM
The HCF × LCM for the numbers 50 and 20 is:
Product = 50 × 20 = 1000.
Therefore, HCF × LCM must also be 1000.
