1. If the HCF of two positive integers a and b is 1, then their LCM is: (1 Mark) (CBSE 2025)
(A) a+b
(B) a
(C) b
(D) ab

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1. (D) ab

2. The number 3+√2 is : (1 Mark) (CBSE 2025)
(A) a rational number
(B) an irrational number
(C) an integer
(D) a natural number

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2. (B) an irrational number

3. Two statements are given, one labelled as Assertion (A) and the other is labelled as Reason (R). Select the correct answer from the codes:
Assertion (A): For any two natural numbers a and b, the HCF of a and b is a factor of the LCM of a and b.
Reason (R): HCF of any two natural numbers divides both the numbers. (1 Mark)(CBSE 2025)
(A) Both A and R are true and R is the correct explanation of A
(B) Both A and R are true, but R is not the correct explanation of A
(C) A is true, but R is false
(D) A is false, but R is true

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3. (A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

4. If HCF(98,28)=m and LCM(98,28)=n, then the value of n – 7m is : (1 Mark) (CBSE 2025)
(A) 0
(B) 28
(C) 98
(D) 198

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4. (C) 98
1. Prime factorization of 98 = 2 × 7²
2. Prime factorization of 28 = 2² × 7
3. HCF = 2¹ × 7¹ = 14 ∴ m = 14
4. LCM = 2² × 7² = 196 ∴ n = 196
5. n – 7m = 196 – 7×14 = 196 – 98 = 98

5. If (-1)ⁿ + (1)⁸ = 0, then n is : (1 Mark) (CBSE 2025)
(A) any positive integer
(B) any negative integer
(C) any odd number
(D) any even number

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5. (C) any odd number

6. Which of the following is a rational number between √3 and √5? (1 Mark) (CBSE 2025)
(A) 1.4142387954012…
(B) 2.326
(C) π
(D) 1.857142

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6. (D) 1.857142

7. Prove that √3 is an irrational number. (3 Mark) (CBSE 2025)

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Proof that √3 is Irrational
Step 1: Assume √3 is rational: √3 = p/q where p,q are co-prime, q≠0
Step 2: Square both sides: 3 = p²/q² ⇒ 3q² = p²
Step 3: p² is divisible by 3 ⇒ p is divisible by 3. Let p = 3m
Step 4: Substitute: 3q² = (3m)² ⇒ 3q² = 9m² ⇒ q² = 3m²
Step 5: q² is divisible by 3 ⇒ q is divisible by 3
Step 6: Contradiction: Both p and q divisible by 3, but assumed co-prime
Conclusion: √3 is irrational

8. The factor tree of a number x is shown. Find the values of x, y, a and b. Hence, write the product of the prime factors of the number x so obtained. (3 Mark) (CBSE 2025)

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8. b = 7
a = 3
y = 420
x = 840
x = 840 = 2³ × 3 × 5 × 7

9. Prove that 1/√5 is an irrational number. (3 Mark) (CBSE 2025)

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Proof that 1/√5 is Irrational
Step 1: Assume 1/√5 is rational: 1/√5 = p/q where p,q are co-prime
Step 2: Rearrange: √5 = q/p
Step 3: Square both sides: 5 = q²/p² ⇒ 5p² = q²
Step 4: q² is divisible by 5 ⇒ q is divisible by 5. Let q = 5a
Step 5: Substitute: 5p² = (5a)² ⇒ 5p² = 25a² ⇒ p² = 5a²
Step 6: p² is divisible by 5 ⇒ p is divisible by 5
Step 7: Contradiction: Both p and q divisible by 5, but assumed co-prime
Conclusion: 1/√5 is irrational