Chapter 1: Real Numbers (বাস্তৱ সংখ্যা)

---

GRADE X MATHEMATICS

Chapter 1: Real Numbers (বাস্তৱ সংখ্যা)

SECTION A – 1 MARK QUESTIONS (১ নম্বৰৰ প্ৰশ্ন)

Multiple Choice Questions (MCQs) – Q1–Q5

Q1. [BOARD 2024]

Question: If two positive integers p and q can be expressed as p = ab³ and q = a²b, where a and b are prime numbers, then LCM (p,q) = ? (প্ৰশ্ন (অসমীয়া): দুটা ধনাত্মক পূৰ্ণসংখ্যা p আৰু p-ক p = ab³ আৰু q = a²b ৰূপত প্ৰকাশ কৰিব পাৰি। ইয়াত a আৰু b মৌলিক সংখ্যা। তেন্তে LCM p, q কিমান ?)

Options / বিকল্প: (a) ab  (b) ab²  (c) a²b³  (d) a³b³

Ans / উত্তৰ: (c) a²b³

Explanation: LCM takes the highest power of each prime factor. The highest power of a is a², and that of b is b³. Therefore, LCM = a²b³

^{===================================================}

Q2. [BOARD 2023]

Question: If p and q are natural numbers and p is a multiple of q, what is the HCF of  p and q ?

Options / বিকল্প: (a) pq  (b) p  (c) q  (d) p + q

Ans / উত্তৰ: (c) q

Explanation: Since p is a multiple of q, q divides both numbers exactly. Therefore, their HCF is q.

Example / উদাহৰণ: p = 12, q = 4

HCF 12,4 = 4

===========================================

Q3. [BOARD 2025]

Question: If x = ab³ and y = a²b, where a and b are prime numbers, then HCF(x,y) − LCM (x,y) is equal to what?

Options / বিকল্প: (a) 1 - ab  (b) ab (1 - ab)  (c) aba²b² (d) ab(1-ab)(1+ab)

Ans / উত্তৰ: None of the given options / দিয়া বিকল্পসমূহৰ এটাও শুদ্ধ নহয়।

Explanation:

HCF(x,y) = ab, LCM(x,y) = a²b³

Therefore, গতিকে,

HCF - LCM = ab - a²b³

Taking ab as the common factor: ab(1-ab²)

=========================================

Q4. [BOARD 2025]

Question: Let  x = a²b²c² and y = a³b^mcⁿ, where a,b,c are prime numbers. If  LCM (x,y) = a³b⁴c³, then what is m + n = ?

প্ৰশ্ন: ধৰা হওক, x = a²b²c²  আৰু y = a³b^mcⁿ, য'ত a,b,c মৌলিক সংখ্যা। যদি LCM LCM (x,y) = a³b⁴c³, তেন্তে m + n -ৰ মান কিমান ?

Options / বিকল্প: (a) 10  (b) 7  (c) 6  (d) 5

Ans / উত্তৰ: (b) 7

Explanation:

For the prime factor b:

max (2,m) = 4 ⇒ m = 4

For the prime factor c:

max (2,n) = 3 ⇒ n = 3

Therefore,

m + n = 4 + 3 = 7

====================================

Q5. [BOARD 2025]

Question: Let A = p²q³rⁿ and b = p³q^mr², where p,q,r are prime numbers. If LCM (a,b) = p³q⁴r³, then what is 3n - 2m ? (প্ৰশ্ন: ধৰা হওক, A = p²q³rⁿ আৰু b = p³q^mr², য'ত p, q, r  মৌলিক সংখ্যা। যদি LCM (a,b) = p³q⁴r³, তেন্তে 3n - 2m -ৰ মান কিমান ?)

Options / বিকল্প: (a) −1  (b) 1  (c) 3  (d) −3

Ans / উত্তৰ: (b) 1

Explanation:

For the prime factor q: max (3,m) = 4 ⇒ m = 4

For the prime factor r: max (n,2) = 3 ⇒  n = 3

Therefore,

3n - 2m = 3(3) - 2(4) = 9  -8 = 1