CBSE Class 10 Real Numbers Sure Shot Questions Set I

Question. If two positive integers \( a \) and \( b \) are written as \( a = x^4y^2 \) and \( b = x^2y^3 \), where \( x \) and \( y \) are prime numbers, then \( \text{LCM} (a, b) \) is
(a) \( x^8y^6 \)
(b) \( x^6y^5 \)
(c) \( x^4y^3 \)
(d) None of the options
Answer: (c) \( x^4y^3 \)

Question. The prime factorisation of 2475 is
(a) \( 3^2 \times 5 \times 11 \)
(b) \( 3^1 \times 5^2 \times 11 \)
(c) \( 3^2 \times 5^2 \times 11 \)
(d) None of the options
Answer: (c) \( 3^2 \times 5^2 \times 11 \)

Question. Two natural numbers whose difference is 66 and the LCM is 360, are
(a) 180 and 114
(b) 90 and 24
(c) 120 and 54
(d) 130 and 64
Answer: (b) 90 and 24

Question. The HCF and LCM of the smallest composite number and the smallest prime number are respectively.
(a) 2 and 2
(b) 2 and 4
(c) 4 and 4
(d) 8 and 4
Answer: (b) 2 and 4

Question. If HCF of two numbers is 4 and their product is 160, then their LCM is
(a) 40
(b) 60
(c) 80
(d) 120
Answer: (a) 40

Question. If \( \text{LCM} (32, a) = 64 \) and \( \text{HCF} (32, a) = 4 \), then \( a \) is equal to
(a) 16
(b) 8
(c) 20
(d) 10
Answer: (b) 8

Question. Three bells rings at intervals 5, 3 and 15 min. All three rang at 10 am. When will they ring together again?
(a) 10 : 10 am
(b) 10 : 15 am
(c) 10 : 20 am
(d) None of the options
Answer: (b) 10 : 15 am

Question. The ratio of LCM and HCF of second smallest prime number and second smallest composite number is
(a) 2 : 5
(b) 2 : 1
(c) 1 : 2
(d) 5 : 2
Answer: (b) 2 : 1

Question. Find the least number that is divisible by all the numbers from 3 to 7 (both inclusive).
(a) 400
(b) 410
(c) 420
(d) 430
Answer: (c) 420

Question. If \( x^2 = 1 + \frac{2}{36} + \frac{5}{6} \), then \( x \) is
(a) irrational
(b) rational
(c) whole number
(d) integer
Answer: (a) irrational

Question. Prime factors of the denominator of a rational number with the decimal expansion 62.47 are
(a) 2 and 35
(b) 2 and 5
(c) 3 and 5
(d) 4 and 5
Answer: (b) 2 and 5

Question. The decimal expansion of the rational number \( \frac{53}{2^3 \times 5} \), will terminate after how many places of decimal?
(a) 1
(b) 3
(c) 4
(d) 2
Answer: (b) 3

Question. The smallest number by which \( \frac{1}{17} \) should be multiplied so that its decimal expansion terminator after one decimal place is
(a) \( \frac{17}{100} \)
(b) \( \frac{17}{10} \)
(c) \( \frac{100}{17} \)
(d) \( \frac{10}{17} \)
Answer: (b) \( \frac{17}{10} \)

Question. The decimal number of \( \left( \frac{21}{8} + \frac{7}{40} \right) \) will terminate after how many places?
(a) 2
(b) 1
(c) 3
(d) 4
Answer: (b) 1

Question. What smallest number must be multiplied in the denominator so that the decimal number \( \frac{14588}{625} \) will be terminated?
(a) 4
(b) 18
(c) 16
(d) 20
Answer: (c) 16

EXERCISE

Question. State whether the given statement is true or false :
(i) The sum of two rationals is always rational
(ii) The product of two rationals is always rational
(iii) The sum of two irrationals is an irrational.
(iv) The product of two irrationals is an irrational
(v) The sum of a rational and an irrational is irrational
(vi) The product of a rational and an irrational is irrational
Answer: (i) True (ii) True (iii) False (iv) False (v) True (vi) True

Question. Define (i) rational numbers (ii) irrational numbers (iii) real numbers.
Answer: (i) Rational numbers: Numbers that can be expressed in the form \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \). (ii) Irrational numbers: Numbers that cannot be expressed in the form \( \frac{p}{q} \) and have non-terminating, non-repeating decimal expansions. (iii) Real numbers: The set of all rational and irrational numbers combined.

Question. Classify the following numbers as rational or irrational :
(i) \( \frac{22}{7} \)
(ii) 3.1416
(iii) \( \pi \)
(iv) 3.142857
(v) 5.636363......
(vi) 2.040040004......
(vii) 1.535335333....
(viii) 3.121221222...
(ix) \( \sqrt{21} \)
(x) \( \sqrt[3]{3} \)
Answer: (i) Rational (ii) Rational (iii) Irrational (iv) Rational (v) Rational (vi) Irrational (vii) Irrational (viii) Irrational (ix) Irrational (x) Irrational

Question. Prove that each of the following numbers is irrational :
(i) \( \sqrt{6} \)
(ii) \( (2 - \sqrt{3}) \)
(iii) \( (3 + \sqrt{2}) \)
(iv) \( (2 + \sqrt{5}) \)
(v) \( (5 + 3\sqrt{2}) \)
(vi) \( 3\sqrt{7} \)
(vii) \( \frac{3}{\sqrt{5}} \)
(viii) \( (2 - 3\sqrt{5}) \)
(ix) \( (\sqrt{3} + \sqrt{5}) \)
Answer: To prove a number is irrational, we typically use the method of contradiction, assuming the number is rational (expressible as \( \frac{a}{b} \)) and showing this leads to a logical fallacy (e.g., \( \sqrt{6} \) having a common factor or being equal to a rational fraction).

Question. Prove that \( \frac{1}{\sqrt{3}} \) is irrational.
Answer: Assume \( \frac{1}{\sqrt{3}} \) is rational, so \( \frac{1}{\sqrt{3}} = \frac{a}{b} \) for coprime integers \( a, b \). This implies \( \sqrt{3} = \frac{b}{a} \). Since the RHS is rational and \( \sqrt{3} \) is known to be irrational, this is a contradiction.

Question. Without actual division, show that each of the following rational numbers is a non-terminating repeating decimal :
(i) \( \frac{11}{(2^3 \times 3)} \)
(ii) \( \frac{73}{(2^3 \times 3^3 \times 5)} \)
(iii) \( \frac{9}{35} \)
(iv) \( \frac{32}{147} \)
(v) \( \frac{64}{455} \)
(vi) \( \frac{77}{210} \)
(vii) \( \frac{29}{343} \)
(viii) \( \frac{129}{(2^2 \times 5^7 \times 7^5)} \)
Answer: A rational number \( \frac{p}{q} \) has a non-terminating repeating decimal expansion if the prime factorization of the denominator \( q \) (in simplest form) contains prime factors other than 2 or 5.

Question. Without actual divison, show that each of the following rational numbers is a terminating decimal. Express each in decimal form :
(i) \( \frac{23}{(2^3 \times 5^2)} \)
(ii) \( \frac{24}{125} \)
(iii) \( \frac{17}{320} \)
(iv) \( \frac{171}{800} \)
(v) \( \frac{15}{1600} \)
(vi) \( \frac{19}{3125} \)
Answer: (i) 0.115 (ii) 0.192 (iii) 0.053125 (iv) 0.21375 (v) 0.009375 (vi) 0.00608

Question. Express each of the following as a fraction in simplest form :
(i) \( 0.\overline{8} \)
(ii) \( 2.\overline{4} \)
(iii) \( 0.\overline{24} \)
(iv) \( 0.1\overline{2} \)
(v) \( 2.2\overline{4} \)
(vi) \( 0.3\overline{65} \)
Answer: (i) \( \frac{8}{9} \) (ii) \( \frac{22}{9} \) (iii) \( \frac{8}{33} \) (iv) \( \frac{11}{90} \) (v) \( \frac{101}{45} \) (vi) \( \frac{181}{495} \)

Question. Decide whether the given number is rational or not :
(i) 53.123456789
(ii) \( 31.\overline{123456789} \)
(iii) 0.12012001200012...
Give reason to support your answer.
Answer: (i) Rational, since it is a terminating decimal (ii) Rational, since it is a repeating decimal (iii) Not rational, since it is a non-terminating and non-repeating decimal

Question. What do you mean by Euclid's division algorithm.
Answer: Euclid's division algorithm is a technique used to find the Highest Common Factor (HCF) of two given positive integers using repeated applications of Euclid's division lemma (\( a = bq + r \)).

Question. A number when divided by 61 gives 27 as quotient and 32 as remainder. Find the number.
Answer: 1679

Question. By what number should 1365 be divided to get 31 as quotient and 32 as remainder ?
Answer: 43

Question. Using Euclid's algorithm, find the HCF of
(i) 405 and 2520
(ii) 504 and 1188
(iii) 960 and 1575
Answer: (i) 45 (ii) 36 (iii) 15

Question. Using prime factorisation, find the HCF and LCM of
(i) 144, 198
(ii) 396, 1080
(iii) 1152, 1664
Answer: (i) HCF = 18, LCM = 1584 (ii) HCF = 36, LCM = 11880 (iii) HCF = 128, LCM = 14976

Question. Using prime factorisation, find the HCF and LCM of
(i) 24, 36, 40
(ii) 30, 72, 432
(iii) 21, 28, 36, 45
Answer: (i) HCF = 4, LCM = 360 (ii) HCF = 6, LCM = 2160 (iii) HCF = 1, LCM = 1260

Question. The HCF of two numbers is 23 and their LCM is 1449. If one of the numbers is 161, find the other.
Answer: 207

Question. The HCF of two numbers is 11 and their LCM is 7700. If one of the numbers is 275, find the other.
Answer: 308

Question. Three pieces of timber 42 m, 49 m and 63 m long have to be divided into planks of the same length. What is the greatest possible length of each plank ?
Answer: 7 m

Question. Find the greatest possible length which can be used to measure exactly the length 7 m, 3 m 85 cm and 12 m 95 cm.
Answer: 35 cm

Question. Find the maximum number of students among whom 1001 pens and 910 pencils can be distributed in such a way that each student gets the same number of pens and the same number of pencils.
Answer: 91

Question. Three sets of English, Mathematics and Science books containing 336, 240 and 96 books respectively have to be stacked in such a way that all the books are stored subject wise and the height of each stack is the same. How many stacks will there be ?
Answer: 14

Question. Find the least number of square tiles required to pave the ceiling of a room 15 m 17 cm long and 9 m 2 cm broad.
Answer: 814

Question. Three measuring rods are 64 cm, 80 cm and 96 cm in length. Find the least length of cloth that can be measured an exact number of times, using any of the rods.
Answer: 9.6 m

Question. The traffic lights at three different road crossings change after every 48 seconds, 72 seconds and 108 seconds respectively. If they all change simultaneously at 8 hours, then at what time will they again change simultaneously ?
Answer: 8 : 7 : 12 hrs

Question. An electronic device makes a beep after every 60 seconds. Another device makes a beep after every 62 seconds. They beeped together at 10 am. At what time will they beep together at the earliest ?
Answer: 10 : 31 hrs

Question. Six bells commence tolling together and toll at intervals of 2, 4, 6, 8, 10, 12 minutes respectively. In 30 hours, how many times do they toll together ?
Answer: 16 times