CBSE Class 10 Mathematics Real Numbers MCQs Set 26

Answer: (c) 0.12012001200012...

Explanation:
1. In 0.12012001200012... the number of zeros keeps increasing, so there is no repeating block and the decimal does not end. It is irrational. 53.123456789 terminates, 0.123123... repeats and 3.25 terminates, so they are rational.

Teacher's Note:
1. A terminating decimal is always rational.
2. A repeating block of fixed length also means rational.

Answer: (b) 1679

Explanation:
1. By Euclid's division lemma, dividend = divisor × quotient + remainder = 61 × 27 + 32 = 1647 + 32 = 1679.

Teacher's Note:
1. Use a = bq + r with 0 ≤ r < b.
2. Check the remainder is smaller than the divisor.

Answer: (d) 45

Explanation:
1. Apply the algorithm: 2520 = 405 × 6 + 90; 405 = 90 × 4 + 45; 90 = 45 × 2 + 0. The last non-zero remainder is the divisor of the final step, so HCF = 45.

Teacher's Note:
1. Divide the larger number by the smaller and repeat with the divisor and remainder.
2. Stop when the remainder becomes zero; the last divisor is the HCF.

Answer: (c) 36

Explanation:
1. Apply the algorithm: 1188 = 504 × 2 + 180; 504 = 180 × 2 + 144; 180 = 144 × 1 + 36; 144 = 36 × 4 + 0. The last non-zero remainder is the divisor of the final step, so HCF = 36.

Teacher's Note:
1. Divide the larger number by the smaller and repeat with the divisor and remainder.
2. Stop when the remainder becomes zero; the last divisor is the HCF.

Answer: (b) 15

Explanation:
1. Apply the algorithm: 1575 = 960 × 1 + 615; 960 = 615 × 1 + 345; 615 = 345 × 1 + 270; 345 = 270 × 1 + 75; 270 = 75 × 3 + 45; 75 = 45 × 1 + 30; 45 = 30 × 1 + 15; 30 = 15 × 2 + 0. The last non-zero remainder is the divisor of the final step, so HCF = 15.

Teacher's Note:
1. Divide the larger number by the smaller and repeat with the divisor and remainder.
2. Stop when the remainder becomes zero; the last divisor is the HCF.

Answer: (a) HCF = 18, LCM = 1584

Explanation:
1. Prime factorisation: 144 = 2⁴ × 3²; 198 = 2 × 3² × 11. HCF is the product of the smallest powers of the common primes, which is 18. LCM is the product of the greatest powers of all primes, which is 1584.

Teacher's Note:
1. HCF uses the least power of each common prime, LCM uses the greatest power of every prime.
2. Check that HCF divides every number and LCM is divisible by every number.

Answer: (c) HCF = 128, LCM = 14976

Explanation:
1. Prime factorisation: 1152 = 2⁷ × 3²; 1664 = 2⁷ × 13. HCF is the product of the smallest powers of the common primes, which is 128. LCM is the product of the greatest powers of all primes, which is 14976.

Teacher's Note:
1. HCF uses the least power of each common prime, LCM uses the greatest power of every prime.
2. Check that HCF divides every number and LCM is divisible by every number.

Answer: (b) HCF = 4, LCM = 360

Explanation:
1. Prime factorisation: 24 = 2³ × 3; 36 = 2² × 3²; 40 = 2³ × 5. HCF is the product of the smallest powers of the common primes, which is 4. LCM is the product of the greatest powers of all primes, which is 360.

Teacher's Note:
1. HCF uses the least power of each common prime, LCM uses the greatest power of every prime.
2. Check that HCF divides every number and LCM is divisible by every number.

Answer: (a) HCF = 6, LCM = 2160

Explanation:
1. Prime factorisation: 30 = 2 × 3 × 5; 72 = 2³ × 3²; 432 = 2⁴ × 3³. HCF is the product of the smallest powers of the common primes, which is 6. LCM is the product of the greatest powers of all primes, which is 2160.

Teacher's Note:
1. HCF uses the least power of each common prime, LCM uses the greatest power of every prime.
2. Check that HCF divides every number and LCM is divisible by every number.

Answer: (d) HCF = 1, LCM = 1260

Explanation:
1. Prime factorisation: 21 = 3 × 7; 28 = 2² × 7; 36 = 2² × 3²; 45 = 3² × 5. HCF is the product of the smallest powers of the common primes, which is 1. LCM is the product of the greatest powers of all primes, which is 1260.

Teacher's Note:
1. HCF uses the least power of each common prime, LCM uses the greatest power of every prime.
2. Check that HCF divides every number and LCM is divisible by every number.

Answer: (b) 308

Explanation:
1. Product = 11 × 7700 = 84700. Other number = 84700 ÷ 275 = 308.

Teacher's Note:
1. Use HCF × LCM = product of the two numbers.
2. Check that the HCF 11 divides 308 (308 = 11 × 28).

Answer: (a) 7 m

Explanation:
1. The greatest length that divides all three is their HCF. 42 = 2 × 3 × 7, 49 = 7², 63 = 3² × 7. The common factor is 7, so the planks are 7 m long.

Teacher's Note:
1. Greatest equal length that divides each piece means HCF.
2. Least length that all can be divided by means LCM.

Answer: (d) 35 cm

Explanation:
1. Convert to centimetres: 700, 385 and 1295. HCF(700, 385) = 35 and 1295 = 35 × 37, so the HCF of all three is 35 cm.

Teacher's Note:
1. Convert all lengths to the same unit before taking the HCF.
2. The greatest measuring length that divides all given lengths is their HCF.

Answer: (c) 91

Explanation:
1. The number of students must divide both 1001 and 910, and we want the greatest such number, the HCF. 1001 = 7 × 11 × 13 and 910 = 2 × 5 × 7 × 13, so HCF = 91.

Teacher's Note:
1. Equal sharing of two quantities with the maximum number of recipients means HCF.
2. Check: 1001/91 = 11 pens and 910/91 = 10 pencils each.

Answer: (a) 814

Explanation:
1. The room measures 1517 cm by 902 cm. The side of the largest square tile is HCF(1517, 902) = 41 cm. Number of tiles = (1517/41) × (902/41) = 37 × 22 = 814.

Teacher's Note:
1. Fewest square tiles means the largest tile, so use the HCF of length and breadth.
2. Number of tiles = area of room ÷ area of one tile.

Answer: (d) 9.6 m

Explanation:
1. The required length is the LCM of 64, 80 and 96. 64 = 2⁶, 80 = 2⁴ × 5, 96 = 2⁵ × 3, so LCM = 2⁶ × 3 × 5 = 960 cm = 9.6 m.

Teacher's Note:
1. Least length measured exactly by several rods is their LCM.
2. Convert centimetres to metres at the end.

Answer: (c) 8:07:12 a.m.

Explanation:
1. The next common change happens after LCM(48, 72, 108) = 432 seconds. 432 s = 7 min 12 s, so the time is 8:07:12 a.m.

Teacher's Note:
1. Events that repeat together again after the LCM of their periods.
2. Convert seconds to minutes carefully: 432 = 7 × 60 + 12.

Answer: (b) 10:31 a.m.

Explanation:
1. LCM(60, 62) = 1860 seconds = 31 minutes. They next beep together at 10:31 a.m.

Teacher's Note:
1. The next simultaneous event occurs after the LCM of the intervals.
2. Convert 1860 s into minutes by dividing by 60.

Answer: (d) 60

Explanation:
1. Product of the two numbers = HCF × LCM = 12 × 240 = 2880. Other number = 2880 ÷ 48 = 60.

Teacher's Note:
1. Product of two numbers = HCF × LCM.
2. Check that 12 divides 60.

Answer: (c) It equals 13 × 78, which has more than two factors

Explanation:
1. 7 × 11 × 13 + 13 = 13 × (7 × 11 + 1) = 13 × 78 = 13 × 13 × 3 × 2. It has factors other than 1 and itself, so it is composite.

Teacher's Note:
1. A number is composite if it can be written as a product of smaller factors.
2. Take out a common factor to see the structure.

Answer: (b) √6

Explanation:
1. √6 lies between √4 = 2 and √9 = 3, and 6 is not a perfect square, so √6 is irrational. 2.5 is rational, √4 = 2 and √9 = 3 are not strictly between 2 and 3.

Teacher's Note:
1. The square root of a non-square between 4 and 9 lies between 2 and 3.
2. 2.5 is rational, so check the roots of perfect squares carefully.