CBSE Class 10 Mathematics Real Numbers Worksheet Set 05

SECTION A

Question. The sum of exponents of prime factors in the prime factorisation of 196 is
(a) 3
(b) 4
(c) 5
(d) 2
Answer: (b) 4

Question. The LCM and HCF of two rational numbers are equal then the numbers must be
(a) prime
(b) composite
(c) not equal
(d) equal
Answer: (d) equal

Question. The product of a non zero rational and an irrational number is 
(a) always irrational
(b) always rational
(c) rational or irrational
(d) one
Answer: (a) always irrational

Question. If two positive integers \( p \) and \( q \) are written as \( p = a^2b^3 \) and \( q = a^3b \); \( a, b \) are prime numbers then find HCF (\( p, q \)).
Answer: \( a^2b \)

Question. Given that HCF (135, 225) = 45, find the LCM (135, 225). 
Answer: 675

SECTION B

Question. What is the least number that is divisible by all the numbers from 1 to 10?
Answer: 2520

Question. Find the sum of \( 0.\overline{68} + 0.\overline{73} \).
Answer: \( 1.\overline{42} \)

Question. Show that \( 5 + 2\sqrt{7} \) is an irrational number, where \( \sqrt{7} \) is given to be an irrational number. 
Answer: Let us assume, to the contrary, that \( 5 + 2\sqrt{7} \) is rational.
Then, \( 5 + 2\sqrt{7} = \frac{a}{b} \), where \( a \) and \( b \) are coprime and \( b \neq 0 \).
\( \implies 2\sqrt{7} = \frac{a}{b} - 5 \)
\( \implies 2\sqrt{7} = \frac{a - 5b}{b} \)
\( \implies \sqrt{7} = \frac{a - 5b}{2b} \)
Since \( a \) and \( b \) are integers, \( \frac{a - 5b}{2b} \) is rational, which implies \( \sqrt{7} \) is rational.
But this contradicts the fact that \( \sqrt{7} \) is irrational.
Hence, \( 5 + 2\sqrt{7} \) is an irrational number.

Question. Given that \( \sqrt{2} \) is irrational, prove that \( (5 + 3\sqrt{2}) \) is an irrational number. 
Answer: Let us assume, to the contrary, that \( 5 + 3\sqrt{2} \) is rational.
Then, \( 5 + 3\sqrt{2} = \frac{a}{b} \), where \( a \) and \( b \) are coprime and \( b \neq 0 \).
\( \implies 3\sqrt{2} = \frac{a}{b} - 5 \)
\( \implies 3\sqrt{2} = \frac{a - 5b}{b} \)
\( \implies \sqrt{2} = \frac{a - 5b}{3b} \)
Since \( a \) and \( b \) are integers, \( \frac{a - 5b}{3b} \) is rational, which implies \( \sqrt{2} \) is rational.
But this contradicts the fact that \( \sqrt{2} \) is irrational.
Hence, \( 5 + 3\sqrt{2} \) is an irrational number.

Question. Given that \( \sqrt{5} \) is irrational, prove that \( 2\sqrt{5} - 3 \) is an irrational number.
Answer: Let us assume, to the contrary, that \( 2\sqrt{5} - 3 \) is rational.
Then, \( 2\sqrt{5} - 3 = \frac{a}{b} \), where \( a \) and \( b \) are coprime and \( b \neq 0 \).
\( \implies 2\sqrt{5} = \frac{a}{b} + 3 \)
\( \implies 2\sqrt{5} = \frac{a + 3b}{b} \)
\( \implies \sqrt{5} = \frac{a + 3b}{2b} \)
Since \( a \) and \( b \) are integers, \( \frac{a + 3b}{2b} \) is rational, which implies \( \sqrt{5} \) is rational.
But this contradicts the fact that \( \sqrt{5} \) is irrational.
Hence, \( 2\sqrt{5} - 3 \) is an irrational number.

Question. Find the LCM and HCF of 12, 15 and 21 by applying the prime factorisation method.
Answer: LCM = 420; HCF = 3

Question. Find the LCM of \( x^2 - 4 \) and \( x^4 - 16 \).
Answer: \( (x^2 + 4)(x^2 - 4) \)

Question. Show that \( 3\sqrt{2} \) is an irrational number.
Answer: Let us assume, to the contrary, that \( 3\sqrt{2} \) is rational.
Then, \( 3\sqrt{2} = \frac{a}{b} \), where \( a \) and \( b \) are coprime and \( b \neq 0 \).
\( \implies \sqrt{2} = \frac{a}{3b} \)
Since \( a \) and \( b \) are integers, \( \frac{a}{3b} \) is rational, which implies \( \sqrt{2} \) is rational.
But this contradicts the fact that \( \sqrt{2} \) is irrational.
Hence, \( 3\sqrt{2} \) is an irrational number.

Question. 144 cartons of coke cans and 90 cartons of pepsi cans are to be stacked in a canteen. If each stack is of the same height and is to contain carton of same drink. What would be the greatest number of cartons in each stack?
Answer: 18

Question. 105 donkeys, 140 cows and 175 goats have to be taken across a river. There is only one boat which will have to make many trips in order to do so. The lazy boatman has his own conditions for transporting them. He insists that he will take the same number of animals in every trip and they have to be of the same kind. He will naturally like to take the largest possible number each times, find how many animals went in each trip?
Answer: 35

Question. Prove that \( \sqrt{7} \) is an irrational number.
Answer: Let us assume, to the contrary, that \( \sqrt{7} \) is rational.
Then, \( \sqrt{7} = \frac{p}{q} \), where \( p \) and \( q \) are coprime integers and \( q \neq 0 \).
Squaring both sides, \( 7 = \frac{p^2}{q^2} \implies p^2 = 7q^2 \).
This means \( 7 \) divides \( p^2 \), so \( 7 \) divides \( p \).
Let \( p = 7k \) for some integer \( k \).
\( \implies (7k)^2 = 7q^2 \implies 49k^2 = 7q^2 \implies 7k^2 = q^2 \).
This means \( 7 \) divides \( q^2 \), so \( 7 \) divides \( q \).
Therefore, \( 7 \) is a common factor of both \( p \) and \( q \).
But this contradicts the assumption that \( p \) and \( q \) are coprime.
Hence, \( \sqrt{7} \) is an irrational number.

Case Study-based Questions

Each of the following questions are of 4 marks.

Read the following and answer any four questions from (i) to (v). A Mathematics Exhibition is being conducted in your School and one of your friends is making a model of a factor tree. He has some difficulty and asks for your help in completing a quiz for the audience. Observe the following factor tree and answer the following:

Question. What will be the value of x?
(a) 15005
(b) 13915
(c) 56920
(d) 17429
Answer: (b) 13915

Question. What will be the value of y?
(a) 23
(b) 22
(c) 11
(d) 19
Answer: (c) 11

Question. What will be the value of z?
(a) 22
(b) 23
(c) 17
(d) 19
Answer: (b) 23

Question. According to Fundamental Theorem of Arithmetic 13915 is a
(a) Composite number
(b) Prime number
(c) Neither prime nor composite
(d) Even number
Answer: (a) Composite number

Question. The prime factorisation of 13915 is
(a) \( 5 \times 11^3 \times 13^2 \)
(b) \( 5 \times 11^3 \times 23^2 \)
(c) \( 5 \times 11^2 \times 23 \)
(d) \( 5 \times 11^2 \times 13^2 \)
Answer: (c) \( 5 \times 11^2 \times 23 \)

Read the following and answer any four questions from (i) to (v). To enhance the reading skills of grade X students, the school nominates you and two of your friends to set up a class library. There are two sections- section A and section B of grade X. There are 32 students in section A and 36 students in section B. 

Question. What is the minimum number of books you will require for the class library, so that they can be distributed equally among students of Section A or Section B?
(a) 144
(b) 128
(c) 288
(d) 272
Answer: (c) 288

Question. If the product of two positive integers is equal to the product of their HCF and LCM is true then, the HCF (32, 36) is
(a) 2
(b) 4
(c) 6
(d) 8
Answer: (b) 4

Question. 36 can be expressed as a product of its primes as
(a) \( 2^2 \times 3^2 \)
(b) \( 2^1 \times 3^3 \)
(c) \( 2^3 \times 3^1 \)
(d) \( 2^0 \times 3^0 \)
Answer: (a) \( 2^2 \times 3^2 \)

Question. \( 7 \times 11 \times 13 \times 15 + 15 \) is a
(a) Prime number
(b) Composite number
(c) Neither prime nor composite
(d) None of the options
Answer: (b) Composite number

Question. If p and q are positive integers such that \( p = ab^2 \) and \( q = a^2b \), where \( a, b \) are prime numbers, then the LCM (\( p, q \)) is
(a) \( ab \)
(b) \( a^2b^2 \)
(c) \( a^3b^2 \)
(d) \( a^3b^3 \)
Answer: (b) \( a^2b^2 \)

Read the following and answer any four questions from (i) to (v). A seminar is being conducted by an Educational Organisation, where the participants will be educators of different subjects. The number of participants in Hindi, English and Mathematics are 60, 84 and 108 respectively. 

Question. In each room the same number of participants are to be seated and all of them being in the same subject, hence maximum number of participants that can accommodated in each room are
(a) 14
(b) 12
(c) 16
(d) 18
Answer: (b) 12

Question. What is the minimum number of rooms required during the event?
(a) 11
(b) 31
(c) 41
(d) 21
Answer: (d) 21

Question. The LCM of 60, 84 and 108 is
(a) 3780
(b) 3680
(c) 4780
(d) 4680
Answer: (a) 3780

Question. The product of HCF and LCM of 60, 84 and 108 is
(a) 55360
(b) 35360
(c) 45500
(d) 45360
Answer: (d) 45360

Question. 108 can be expressed as a product of its primes as
(a) \( 2^3 \times 3^2 \)
(b) \( 2^3 \times 3^3 \)
(c) \( 2^2 \times 3^2 \)
(d) \( 2^2 \times 3^3 \)
Answer: (d) \( 2^2 \times 3^3 \)

Objective Type Questions: 

Question. Choose and write the correct option in each of the following questions.
(i) For some integer a, every odd integer is of the form
(a) 2 a + 1
(b) 2 a
(c) a + 1
(d) a
Answer: (a) 2 a + 1

Question. If the LCM of p and 18 is 36 and the HCF of p and 18 is 2 then p is equal to
(a) 2
(b) 3
(c) 4
(d) 1
Answer: (c) 4

Question. Which of the following is an irrational number?
(a) \( \frac{\sqrt{2}}{\sqrt{8}} \)
(b) \( \frac{\sqrt{63}}{\sqrt{7}} \)
(c) \( \frac{\sqrt{5}}{\sqrt{20}} \)
(d) \( \frac{\sqrt{3}}{3\sqrt{5}} \)
Answer: (d) \( \frac{\sqrt{3}}{3\sqrt{5}} \)

Question. Is \( 9 + \sqrt{2} \) an irrational number?
(a) Yes, because if \( 9 + \sqrt{2} = \frac{a}{b} \), where a and b are integers and \( b \neq 0 \), then \( \sqrt{2} = \frac{a - 9b}{b} \), but \( \sqrt{2} \) is an irrational number. So, \( 9 + \sqrt{2} \neq \frac{a}{b} \).
(b) Yes, because if \( 9 + \sqrt{2} = \frac{a}{b} \), where a and b are integers and \( b \neq 0 \), then \( \sqrt{2} = \frac{9b + a}{b} \), but \( \sqrt{2} \) is an irrational number. So, \( 9 + \sqrt{2} \neq \frac{a}{b} \).
(c) No, because if \( 9 + \sqrt{2} = \frac{a}{b} \), where a and b are integers and \( b \neq 0 \), then \( \sqrt{2} = \frac{9b - a}{b} \), but \( \sqrt{2} \) is an irrational number. So, \( 9 + \sqrt{2} \neq \frac{a}{b} \).
(d) No, because if \( 9 + \sqrt{2} = \frac{a}{b} \), where a and b are integers and \( b \neq 0 \), then \( \sqrt{2} = \frac{9b + a}{b} \), but \( \sqrt{2} \) is an irrational number. So, \( 9 + \sqrt{2} \neq \frac{a}{b} \).
Answer: (a) Yes, because if \( 9 + \sqrt{2} = \frac{a}{b} \), where a and b are integers and \( b \neq 0 \), then \( \sqrt{2} = \frac{a - 9b}{b} \), but \( \sqrt{2} \) is an irrational number. So, \( 9 + \sqrt{2} \neq \frac{a}{b} \).

Question. The LCM of smallest two digit composite number and smallest composite number is 
(a) 12
(b) 4
(c) 20
(d) 44
Answer: (c) 20

Very Short Answer Questions: 

Question. Arnav has 40 cm long red and 84 cm long blue ribbon. He cuts each ribbon into pieces such that all pieces are of equal length. What is the length of each piece?
Answer: 4 cm

Question. The LCM of two numbers is 9 times their HCF. The sum of LCM and HCF is 500. Find the HCF of two numbers. 
Answer: HCF = 50

Question. Write whether \( \frac{2\sqrt{45} + 3\sqrt{20}}{2\sqrt{5}} \) on simplification gives an irrational or a rational number. 
Answer: Rational

Question. Find a rational number between \( \sqrt{2} \) and \( \sqrt{3} \). 
Answer: 1.7, Any rational number between 1.41 and 1.73

Short Answer Questions-I:

Question. Write whether every positive integer can be of the form \( 4q + 2 \), where q is an integer. Justify your answer.
Answer: No, because an integer can be written in the form \( 4q, 4q + 1, 4q + 2, 4q + 3 \).

Question. Can the numbers \( 4^n \), n being a natural number end with the digit 5? Give reasons.
Answer: No, because \( 4^n = (2 \times 2)^n = 2^n \times 2^n \), so the only primes in the factorisation of \( 4^n \) are 2 only, and not 5.

Question. The HCF and LCM of two numbers are 9 and 360 respectively if one number is 45, find the other number. 
Answer: 72

Question. On a morning walk, three persons step out together and their steps measure 30 cm, 36 cm and 40 cm respectively. What is the minimum distance each should walk so that each can cover the same distance in complete steps? 
Answer: 360 cm

Short Answer Questions-II: 

Question. If the HCF (210, 55) is expressible in the form \( 210 \times 5 - 55y \), find y.
Answer: 19

Question. Three bulbs red, green and yellow flash at intervals of 80 seconds, 90 seconds and 110 seconds. All three flash together at 8:00 am. At what time will the three bulbs flash altogether again?
Answer: 10:12 AM

Question. Find the greatest number that will divide 445, 572 and 699 leaving remainders 4, 5 and 6 respectively.
Answer: 63