Class 10 Maths Standard Question Bank

1. 252 = 2² × 3² × 7; 198 = 2 × 3² × 11.
2. Take minimum prime powers for HCF: 2 × 3² = 18.
3. Take maximum powers for LCM: 2² × 3² × 7 × 11 = 2772.
4. Check: 18 × 2772 = 252 × 198 = 49896.

Common mistake: Use minimum powers for HCF, maximum powers for LCM.

1. The number of packs must divide all three quantities.
2. 144 = 2⁴ × 3²; 180 = 2² × 3² × 5; 252 = 2² × 3² × 7.
3. HCF = 2² × 3² = 36.
4. Divide each colour quantity by 36: 4, 5 and 7.

Common mistake: The question asks for the greatest number of identical packs, so use HCF rather than LCM.

1. 18 = 2 × 3²; 24 = 2³ × 3; 30 = 2 × 3 × 5.
2. LCM = 2³ × 3² × 5 = 360 seconds.
3. 360 seconds = 6 minutes, so the next common flash is at 9:06 a.m.

Common mistake: Convert seconds to minutes before adding the interval to the clock time.

1. Let the other integer be n. Then 12 × 420 = 60n.
2. n = 5040/60 = 84.
3. 60 = 2² × 3 × 5 and 84 = 2² × 3 × 7.
4. Their HCF is 12 and LCM is 420, as required.

Common mistake: The product identity gives a candidate; verify its HCF and LCM.

1. 6ⁿ = 2ⁿ × 3ⁿ, so its only prime factors are 2 and 3.
2. A positive integer ending in 0 is divisible by 10 = 2 × 5.
3. There is no factor 5, so 6ⁿ cannot end in 0.

Common mistake: A factor 2 alone does not make an integer divisible by 10.

1. Suppose √3 = p/q, where p and q are coprime integers and q ≠ 0.
2. Then p² = 3q². Since 3 divides p², it divides p. Write p = 3k.
3. Substitution gives q² = 3k², so 3 also divides q.
4. This contradicts coprimality. Therefore √3 is irrational.

Common mistake: State that p and q are coprime; otherwise the contradiction is incomplete.

1. Suppose 5 + 2√3 = r is rational.
2. Then √3 = (r − 5)/2 would be rational, because rational numbers are closed under these operations.
3. This contradicts the irrationality of √3.

Common mistake: A decimal approximation is not a proof of irrationality.

1. The HCF must divide each integer.
2. Therefore every common multiple, including the LCM, must be divisible by the HCF.
3. 240/18 is not an integer, so these values are impossible.

Common mistake: Checking only the product HCF × LCM is not sufficient to prove a pair exists.

1. LCM(12,15,18) = 2² × 3² × 5 = 180.
2. For a perfect square, every prime exponent must be even.
3. Multiply by one more factor 5: 2² × 3² × 5² = 900.
4. 900 is divisible by all three numbers and equals 30².

Common mistake: The LCM is not automatically a perfect square.

1. The tile side must divide both lengths, so use HCF(378,294).
2. 378 = 2 × 3³ × 7; 294 = 2 × 3 × 7². HCF = 42.
3. There are 378/42 = 9 tiles along one side and 294/42 = 7 along the other.
4. Total = 9 × 7 = 63.

Common mistake: Divide each length by the tile side before multiplying the counts.