Important questions for class 10 maths chapter 1 real numbers with solutionsExpertClass
ย
1) The document contains a class notes on important questions and solutions for Class 10 Maths Chapter 1 on Real Numbers.
2) It includes short answer type questions with 2 marks, short answer type questions with 3 marks, and long answer type questions with 4 marks.
3) The questions cover topics like finding HCF and LCM using different methods, determining if numbers are rational or irrational, and properties of real numbers.
The document discusses several key concepts regarding polynomials:
- Euclid's division algorithm states that any polynomial can be divided by a non-zero polynomial to obtain a quotient and remainder.
- The zeroes of a polynomial are the x-values where the graph crosses the x-axis.
- For a quadratic polynomial, the sum and product of its zeroes are related to its coefficients.
- Similar relationships exist between the zeroes and coefficients of cubic polynomials.
- Multiple choice and short answer questions are provided to test understanding of these concepts.
500 most asked apti ques in tcs, wipro, infos(105pgs)PRIYANKKATIYAR2
ย
This document provides 100 numerical aptitude questions and solutions that are commonly asked in campus recruitment drives by companies like Infosys, TCS, CTS, Wipro and Accenture. The questions cover topics such as number systems, permutations, combinations, time and work problems, percentages, profit and loss, and geometry. Shortcuts and tips are provided to solve problems more quickly. The questions are divided into parts for each company and an index provides the topic distribution of questions for each company.
This document provides 100 numerical aptitude questions asked in campus placements by companies like Infosys, TCS, CTS, Wipro and Accenture, along with their solutions. It aims to help students target their learning and know more than their competitors. Some key topics covered include number systems, time and work problems, percentages, and geometry. The author provides contact information for students who have additional doubts.
The document defines and describes different types of real numbers including natural numbers, whole numbers, integers, rational numbers, and irrational numbers. It provides examples of each type of number. Real numbers consist of all rational and irrational numbers. A Venn diagram shows the relationships between the different subsets of real numbers. Euclid's division algorithm and its application to find the highest common factor of two numbers is also explained in the document.
The document discusses different types of real numbers including rational and irrational numbers. It provides examples and definitions of natural numbers, whole numbers, integers, rational numbers, and irrational numbers. It also includes information on Euclid's division algorithm and its application in finding the highest common factor of two numbers. Examples are provided to illustrate the algorithm.
The document discusses different types of real numbers including rational and irrational numbers. It provides examples and definitions of natural numbers, whole numbers, integers, rational numbers, and irrational numbers. It also includes information on Euclid's division algorithm and its application in finding highest common factors and lowest common multiples. Examples of proving the irrationality of square roots like โ5 are given.
The document discusses different types of real numbers including rational and irrational numbers. It provides examples and definitions of natural numbers, whole numbers, integers, rational numbers, and irrational numbers. It also includes information on Euclid's division algorithm and its application in finding the highest common factor of two numbers. Examples are provided to illustrate the algorithm.
Important questions for class 10 maths chapter 1 real numbers with solutionsExpertClass
ย
1) The document contains a class notes on important questions and solutions for Class 10 Maths Chapter 1 on Real Numbers.
2) It includes short answer type questions with 2 marks, short answer type questions with 3 marks, and long answer type questions with 4 marks.
3) The questions cover topics like finding HCF and LCM using different methods, determining if numbers are rational or irrational, and properties of real numbers.
The document discusses several key concepts regarding polynomials:
- Euclid's division algorithm states that any polynomial can be divided by a non-zero polynomial to obtain a quotient and remainder.
- The zeroes of a polynomial are the x-values where the graph crosses the x-axis.
- For a quadratic polynomial, the sum and product of its zeroes are related to its coefficients.
- Similar relationships exist between the zeroes and coefficients of cubic polynomials.
- Multiple choice and short answer questions are provided to test understanding of these concepts.
500 most asked apti ques in tcs, wipro, infos(105pgs)PRIYANKKATIYAR2
ย
This document provides 100 numerical aptitude questions and solutions that are commonly asked in campus recruitment drives by companies like Infosys, TCS, CTS, Wipro and Accenture. The questions cover topics such as number systems, permutations, combinations, time and work problems, percentages, profit and loss, and geometry. Shortcuts and tips are provided to solve problems more quickly. The questions are divided into parts for each company and an index provides the topic distribution of questions for each company.
This document provides 100 numerical aptitude questions asked in campus placements by companies like Infosys, TCS, CTS, Wipro and Accenture, along with their solutions. It aims to help students target their learning and know more than their competitors. Some key topics covered include number systems, time and work problems, percentages, and geometry. The author provides contact information for students who have additional doubts.
The document defines and describes different types of real numbers including natural numbers, whole numbers, integers, rational numbers, and irrational numbers. It provides examples of each type of number. Real numbers consist of all rational and irrational numbers. A Venn diagram shows the relationships between the different subsets of real numbers. Euclid's division algorithm and its application to find the highest common factor of two numbers is also explained in the document.
The document discusses different types of real numbers including rational and irrational numbers. It provides examples and definitions of natural numbers, whole numbers, integers, rational numbers, and irrational numbers. It also includes information on Euclid's division algorithm and its application in finding the highest common factor of two numbers. Examples are provided to illustrate the algorithm.
The document discusses different types of real numbers including rational and irrational numbers. It provides examples and definitions of natural numbers, whole numbers, integers, rational numbers, and irrational numbers. It also includes information on Euclid's division algorithm and its application in finding highest common factors and lowest common multiples. Examples of proving the irrationality of square roots like โ5 are given.
The document discusses different types of real numbers including rational and irrational numbers. It provides examples and definitions of natural numbers, whole numbers, integers, rational numbers, and irrational numbers. It also includes information on Euclid's division algorithm and its application in finding the highest common factor of two numbers. Examples are provided to illustrate the algorithm.
Here are the steps to solve these problems using Euclid's division algorithm:
1) Find the HCF of 135 and 225:
225 = 135 * 1 + 90
135 = 90 * 1 + 45
90 = 45 * 2 + 0
Therefore, the HCF of 135 and 225 is 45.
2) Find the HCF of 196 and 38220:
38220 = 196 * 195 + 0
Therefore, the HCF of 196 and 38220 is 196.
3) A sweet seller has 420 kaju barfis and 130 badam barfis. She wants to stack them in equal stacks with the least tray area.
Find the HCF of
The document is about a management aptitude test and social entrepreneurship program. It provides examples of math and logic problems along with their step-by-step solutions. The last problem asks the reader to find the common difference if the first term is 7, last term is 51, and sum is 34.
The document discusses permutations and combinations. It provides examples of calculating permutations and combinations for different scenarios like selecting committees from a group of people and arranging books on a shelf. Formulas for permutations (nPr) and combinations (nCr) are given. Order matters for permutations but not for combinations. The key difference between the two is explained.
The document discusses various number theory concepts including:
- Types of numbers like natural numbers, whole numbers, integers, rational numbers, irrational numbers, prime numbers, and composite numbers.
- Euclid's division lemma and how it can be used to express integers in certain forms.
- Fundamental theorem of arithmetic and prime factorisation of numbers.
- Properties of rational numbers like their representation as fractions and different types of decimal expansions.
- Proofs that some numbers like square roots of 2 and 5 are irrational using contradiction.
The document provides information about a management aptitude test and social entrepreneurship program. It discusses developing change makers and offers a free, comprehensive program in social and spiritual entrepreneurship open to all. It then provides examples of math and reasoning questions along with solutions.
This document discusses least common multiples (LCM) and greatest common factors (GCF). It provides examples of finding the LCM and GCF of various number pairs and sets. It also discusses applications of LCM and GCF in fractions, measurements, and problems involving multiples. Some key points include how to list multiples to find the first number common to both lists for LCM, and finding the largest number that is a factor of both numbers for GCF. Word problems demonstrate using LCM and GCF to determine things like lengths, times, or amounts that are exactly divisible.
Euclid's division lemma states that for any two positive integers a and b, there exist unique integers q and r such that a = bq + r, where 0 โค r < b. This provides an algorithm, known as the Euclidean algorithm, to find the highest common factor (HCF) of two integers by repeatedly dividing the larger number by the smaller number. The fundamental theorem of arithmetic states that every composite number can be uniquely expressed as a product of prime numbers, apart from the order of factors. Some key results discussed are: every positive odd integer is of the form 6q + 1, 6q + 3, or 6q + 5; the square of any positive integer is of the form 3m
Here are the step-by-step workings:
1) 224-1 can be factorized as (212+1)(212-1)
2) 212-1 can be further factorized as (26+1)(26-1)
3) Therefore, 224-1 = (212+1)(26+1)(26-1)
4) The numbers between 60 and 70 that divide (26+1)(26-1) are 65 and 63.
So the two numbers that exactly divide 224-1 and lie between 60 and 70 are 65 and 63.
The document contains 44 math word problems. It provides the problems, solutions, and answers. The problems cover a range of math topics including ratios, percentages, averages, algebra, geometry, and more. For each problem, the full question is stated, the steps to solve the problem are shown, and the final numerical answer is provided.
The document provides information on finding the highest common factor (HCF) and lowest common multiple (LCM) of numbers. It defines HCF as the greatest number that divides two or more numbers, and LCM as the smallest number divisible by all numbers. Methods for finding HCF and LCM include factorization and division. Several examples are given to demonstrate calculating HCF and LCM of integers, fractions, and decimals.
Euclid's division algorithm states that any positive integer a can be divided by another positive integer b with a remainder r that is smaller than b. This algorithm can be used to find the highest common factor (HCF) of two integers. The Fundamental Theorem of Arithmetic states that every composite number can be expressed as a unique product of prime numbers. These concepts are used to prove numbers like sqrt(2) and sqrt(3) are irrational and to explore when decimal expansions of rational numbers like p/q terminate or repeat.
Questions Of Quantitative Aptitude Tests For Competitive ExaminationsDr. Trilok Kumar Jain
ย
The document contains a collection of quantitative aptitude questions often asked in competitive exams. It provides the questions, explains the solutions, and includes links to additional practice resources on quantitative aptitude and entrepreneurship.
Questions of quantitative aptitude tests for competitive examinations Dr. Trilok Kumar Jain
ย
This material is for PGPSE / CSE students of AFTERSCHOOOL. PGPSE / CSE are free online programme - open for all - free for all - to promote entrepreneurship and social entrepreneurship PGPSE is for those who want to transform the world. It is different from MBA, BBA, CFA, CA,CS,ICWA and other traditional programmes. It is based on self certification and based on self learning and guidance by mentors. It is for those who want to be entrepreneurs and social changers. Let us work together. Our basic idea is that KNOWLEDGE IS FREE & AND SHARE IT WITH THE WORLD
Questions Of Quantitative Aptitude Tests For Competitive ExaminationsDr. Trilok Kumar Jain
ย
This document provides a collection of quantitative aptitude questions often asked in competitive exams, along with explanations and links to additional practice resources. The questions cover a range of topics including number series, time, ratios, mixtures, averages, profit and loss, time and work, and data interpretation. Solutions to sample puzzles are also included.
Questions Of Quantitative Aptitude Tests For Competitive ExaminationsDr. Trilok Kumar Jain
ย
This document provides a collection of quantitative aptitude questions often asked in competitive exams, along with explanations and links to additional practice resources. The questions cover a range of topics including number series, time, ratios, mixtures, averages, profit and loss, time and work, and data interpretation. Solutions to sample puzzles are also included.
Questions Of Quantitative Aptitude Tests For Competitive ExaminationsDr. Trilok Kumar Jain
ย
The document contains a collection of quantitative aptitude questions often asked in competitive exams. It provides the questions, explains the solutions, and includes links to additional practice resources on quantitative aptitude and entrepreneurship.
This document contains a summary of cubes and cube roots. It begins with an introduction discussing how the mathematician Ramanujan recognized a number's interesting property. It then defines cubes and cube numbers, provides examples of cubes of numbers from 1 to 10, and discusses patterns with cubes. Finally, it covers finding cube roots through prime factorization and estimation from digit groups of cube numbers. The document contains examples and exercises related to cubes and cube roots.
This document provides examples of business mathematics problems with multiple choice options for solving. Some examples include:
1) A two digit number where the sum of the digits is 10 and subtracting 18 reverses the digits.
2) The ratio of volumes of two cylinders and the ratio of their heights is used to find the ratio of their diameters.
3) A group of 7,300 troops is divided into 4 groups so that halves and thirds of the groups are equal, calculating the size of each group.
The document provides examples of business mathematics problems with multiple choice options to solve. Some key problems discussed include:
- A two digit number where the sum of digits is 10 and subtracting 18 reverses the digits, with the answer being 73.
- A number where half exceeds 1/5th by 15, with the answer being 50.
- Ages of a person and their two sons based on information 5 years ago and now, with the answer being that the person's current age is 50.
Here are the steps to solve these problems using Euclid's division algorithm:
1) Find the HCF of 135 and 225:
225 = 135 * 1 + 90
135 = 90 * 1 + 45
90 = 45 * 2 + 0
Therefore, the HCF of 135 and 225 is 45.
2) Find the HCF of 196 and 38220:
38220 = 196 * 195 + 0
Therefore, the HCF of 196 and 38220 is 196.
3) A sweet seller has 420 kaju barfis and 130 badam barfis. She wants to stack them in equal stacks with the least tray area.
Find the HCF of
The document is about a management aptitude test and social entrepreneurship program. It provides examples of math and logic problems along with their step-by-step solutions. The last problem asks the reader to find the common difference if the first term is 7, last term is 51, and sum is 34.
The document discusses permutations and combinations. It provides examples of calculating permutations and combinations for different scenarios like selecting committees from a group of people and arranging books on a shelf. Formulas for permutations (nPr) and combinations (nCr) are given. Order matters for permutations but not for combinations. The key difference between the two is explained.
The document discusses various number theory concepts including:
- Types of numbers like natural numbers, whole numbers, integers, rational numbers, irrational numbers, prime numbers, and composite numbers.
- Euclid's division lemma and how it can be used to express integers in certain forms.
- Fundamental theorem of arithmetic and prime factorisation of numbers.
- Properties of rational numbers like their representation as fractions and different types of decimal expansions.
- Proofs that some numbers like square roots of 2 and 5 are irrational using contradiction.
The document provides information about a management aptitude test and social entrepreneurship program. It discusses developing change makers and offers a free, comprehensive program in social and spiritual entrepreneurship open to all. It then provides examples of math and reasoning questions along with solutions.
This document discusses least common multiples (LCM) and greatest common factors (GCF). It provides examples of finding the LCM and GCF of various number pairs and sets. It also discusses applications of LCM and GCF in fractions, measurements, and problems involving multiples. Some key points include how to list multiples to find the first number common to both lists for LCM, and finding the largest number that is a factor of both numbers for GCF. Word problems demonstrate using LCM and GCF to determine things like lengths, times, or amounts that are exactly divisible.
Euclid's division lemma states that for any two positive integers a and b, there exist unique integers q and r such that a = bq + r, where 0 โค r < b. This provides an algorithm, known as the Euclidean algorithm, to find the highest common factor (HCF) of two integers by repeatedly dividing the larger number by the smaller number. The fundamental theorem of arithmetic states that every composite number can be uniquely expressed as a product of prime numbers, apart from the order of factors. Some key results discussed are: every positive odd integer is of the form 6q + 1, 6q + 3, or 6q + 5; the square of any positive integer is of the form 3m
Here are the step-by-step workings:
1) 224-1 can be factorized as (212+1)(212-1)
2) 212-1 can be further factorized as (26+1)(26-1)
3) Therefore, 224-1 = (212+1)(26+1)(26-1)
4) The numbers between 60 and 70 that divide (26+1)(26-1) are 65 and 63.
So the two numbers that exactly divide 224-1 and lie between 60 and 70 are 65 and 63.
The document contains 44 math word problems. It provides the problems, solutions, and answers. The problems cover a range of math topics including ratios, percentages, averages, algebra, geometry, and more. For each problem, the full question is stated, the steps to solve the problem are shown, and the final numerical answer is provided.
The document provides information on finding the highest common factor (HCF) and lowest common multiple (LCM) of numbers. It defines HCF as the greatest number that divides two or more numbers, and LCM as the smallest number divisible by all numbers. Methods for finding HCF and LCM include factorization and division. Several examples are given to demonstrate calculating HCF and LCM of integers, fractions, and decimals.
Euclid's division algorithm states that any positive integer a can be divided by another positive integer b with a remainder r that is smaller than b. This algorithm can be used to find the highest common factor (HCF) of two integers. The Fundamental Theorem of Arithmetic states that every composite number can be expressed as a unique product of prime numbers. These concepts are used to prove numbers like sqrt(2) and sqrt(3) are irrational and to explore when decimal expansions of rational numbers like p/q terminate or repeat.
Questions Of Quantitative Aptitude Tests For Competitive ExaminationsDr. Trilok Kumar Jain
ย
The document contains a collection of quantitative aptitude questions often asked in competitive exams. It provides the questions, explains the solutions, and includes links to additional practice resources on quantitative aptitude and entrepreneurship.
Questions of quantitative aptitude tests for competitive examinations Dr. Trilok Kumar Jain
ย
This material is for PGPSE / CSE students of AFTERSCHOOOL. PGPSE / CSE are free online programme - open for all - free for all - to promote entrepreneurship and social entrepreneurship PGPSE is for those who want to transform the world. It is different from MBA, BBA, CFA, CA,CS,ICWA and other traditional programmes. It is based on self certification and based on self learning and guidance by mentors. It is for those who want to be entrepreneurs and social changers. Let us work together. Our basic idea is that KNOWLEDGE IS FREE & AND SHARE IT WITH THE WORLD
Questions Of Quantitative Aptitude Tests For Competitive ExaminationsDr. Trilok Kumar Jain
ย
This document provides a collection of quantitative aptitude questions often asked in competitive exams, along with explanations and links to additional practice resources. The questions cover a range of topics including number series, time, ratios, mixtures, averages, profit and loss, time and work, and data interpretation. Solutions to sample puzzles are also included.
Questions Of Quantitative Aptitude Tests For Competitive ExaminationsDr. Trilok Kumar Jain
ย
This document provides a collection of quantitative aptitude questions often asked in competitive exams, along with explanations and links to additional practice resources. The questions cover a range of topics including number series, time, ratios, mixtures, averages, profit and loss, time and work, and data interpretation. Solutions to sample puzzles are also included.
Questions Of Quantitative Aptitude Tests For Competitive ExaminationsDr. Trilok Kumar Jain
ย
The document contains a collection of quantitative aptitude questions often asked in competitive exams. It provides the questions, explains the solutions, and includes links to additional practice resources on quantitative aptitude and entrepreneurship.
This document contains a summary of cubes and cube roots. It begins with an introduction discussing how the mathematician Ramanujan recognized a number's interesting property. It then defines cubes and cube numbers, provides examples of cubes of numbers from 1 to 10, and discusses patterns with cubes. Finally, it covers finding cube roots through prime factorization and estimation from digit groups of cube numbers. The document contains examples and exercises related to cubes and cube roots.
This document provides examples of business mathematics problems with multiple choice options for solving. Some examples include:
1) A two digit number where the sum of the digits is 10 and subtracting 18 reverses the digits.
2) The ratio of volumes of two cylinders and the ratio of their heights is used to find the ratio of their diameters.
3) A group of 7,300 troops is divided into 4 groups so that halves and thirds of the groups are equal, calculating the size of each group.
The document provides examples of business mathematics problems with multiple choice options to solve. Some key problems discussed include:
- A two digit number where the sum of digits is 10 and subtracting 18 reverses the digits, with the answer being 73.
- A number where half exceeds 1/5th by 15, with the answer being 50.
- Ages of a person and their two sons based on information 5 years ago and now, with the answer being that the person's current age is 50.
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mathematics questions practice class 10 real number.pdf
1. KOTA ASPIRE
Educational Research Center, Bolangir
MATHEMATICS PRACTICE QUESTIONS
Class โ 10
Chapter 1 Real Number
Question 1.
The decimal expansion of the rational number 432453 will terminate
after how many places of decimals? (2013)
Solution:
Question 2.
Write the decimal form of 129275775
3. Question 3.
Find the largest number that will divide 398, 436 and 542 leaving
remainders 7, 11, and 15 respectively.
Solution:
Algorithm
4. 398 โ 7 = 391, 436 โ 11 = 425, 542 โ 15 = 527
HCF of 391, 425, 527 = 17
Question 4.
Express 98 as a product of its primes.
5. Solution:
2 ร 72
Question 5.
If the HCF of 408 and 1032 is expressible in the form 1032 ร 2 + 408 ร
p, then find the value of p.
6. Solution:
HCF of 408 and 1032 is 24.
1032 ร 2 + 408 ร (p) = 24
408p = 24 โ 2064
p = -5
Question 6.
HCF and LCM of two numbers is 9 and 459 respectively. If one of the
numbers is 27, find the other number. (2012)
7. Solution:
We know,
1st number ร 2nd number = HCF ร LCM
โ 27 ร 2nd number = 9 ร 459
โ 2nd number = 9ร45927 = 153
14. Question 11.
Find the largest number which divides 70 and 125 leaving remainder 5
and 8 respectively. (2015)
15. Solution:
It is given that on dividing 70 by the required number, there is a
remainder 5.
This means that 70 โ 5 = 65 is exactly divisible by the required number.
Similarly, 125 โ 8 = 117 is also exactly divisible by the required
number.
65 = 5 ร 13
117 = 32
ร 13
HCF = 13
Required number = 13
16. Question 12.
Find the prime factorisation of the denominator of rational number
expressed as 6.12ยฏ in simplest form. (2014)
Solution:
Let x = 6.12ยฏ โฆ(i)
100x = 612.12ยฏ โฆ(ii)
โฆ[Multiplying both sides by 100]
Subtracting (i) from (ii),
99x = 606
x = 60699 = 20233
17. Denominator = 33
Prime factorisation = 3 ร 11
Question 13.
Complete the following factor tree and find the composite number x.
(2014)
20. Solution:
Let us assume, to the contrary, that 2 + 3โ5 is rational.
So that we can find integers a and b (b โ 0).
Such that 2 + 3โ5 = ab, where a and b are coprime.
Rearranging the above equation, we get
Since a and b are integers, we get a3bโ23 is rational and so โ5 is
rational.
But this contradicts the fact that โ5 is irrational.
So, we conclude that 2 + 3โ5 is irrational.
23. Solution:
Let us assume, to the contrary, that 3โ7 is rational.
That is, we can find coprime a and b (b โ 0) such that 3โ7 = ab
Rearranging, we get โ7 = a3b
Since 3, a and b are integers, a3b is rational, and so โ7 is rational.
But this contradicts the fact that โ7 is irrational.
So, we conclude that 3โ7 is irrational.
Question 16.
Explain why (17 ร 5 ร 11 ร 3 ร 2 + 2 ร 11) is a composite number?
(2015)
25. = 2 ร 11 ร 256
Number (i) is divisible by 2, 11 and 256, it has more than 2 prime
factors.
Therefore (17 ร 5 ร 11 ร 3 ร 2 + 2 ร 11) is a composite number.
Question 17.
Check whether 4n can end with the digit 0 for any natural number n.
(2015)
26. Solution:
4n
= (22
)n
= 22n
The only prime in the factorization of 4n
is 2.
There is no other prime in the factorization of 4n
= 22n
(By uniqueness of the Fundamental Theorem of Arithmetic).
5 does not occur in the prime factorization of 4n
for any n.
Therefore, 4n
does not end with the digit zero for any natural number n.
Question 18.
Can two numbers have 15 as their HCF and 175 as their LCM? Give
reasons. (2017 OD)
27. Solution:
No, LCM = Product of the highest power of each factor involved in the
numbers.
HCF = Product of the smallest power of each common factor.
We can conclude that LCM is always a multiple of HCF, i.e., LCM = k
ร HCF
We are given that,
LCM = 175 and HCF = 15
175 = k ร 15
โ 11.67 = k
28. But in this case, LCM โ k ร HCF
Therefore, two numbers cannot have LCM as 175 and HCF as 15.
Question 19.
Prove that โ5 is irrational and hence show that 3 + โ5 is also irrational.
(2012)
Solution:
Let us assume, to the contrary, that โ5 is rational.
So, we can find integers p and q (q โ 0), such that
โ5 = pq, where p and q are coprime.
Squaring both sides, we get
5 = p2q2
โ 5q2
= p2
โฆ(i)
โ 5 divides p2
29. 5 divides p
So, let p = 5r
Putting the value of p in (i), we get
5q2
= (5r)2
โ 5q2
= 25r2
โ q2
= 5r2
โ 5 divides q2
5 divides q
So, p and q have atleast 5 as a common factor.
But this contradicts the fact that p and q have no common factor.
So, our assumption is wrong, is irrational.
โ5 is irrational, 3 is a rational number.
So, we conclude that 3 + โ5 is irrational.
Question 20.
Prove that 3 + 2โ3 is an irrational number. (2014)
Solution:
Let us assume to the contrary, that 3 + 2โ3 is rational.
So that we can find integers a and b (b โ 0).
Such that 3 + 2โ3 = ab, where a and b are coprime.
Rearranging the equations, we get
30. Since a and b are integers, we get a2bโ32 is rational and so โ3 is
rational.
But this contradicts the fact that โ3 is irrational.
So we conclude that 3 + 2โ3 is irrational.
Question 21.
Three bells toll at intervals of 9, 12, 15 minutes respectively. If they start
tolling together, after what time will they next toll together? (2013)
Solution:
9 = 32
, 12 = 22
ร 3, 15 = 3 ร 5
LCM = 22
ร 32
ร 5 = 4 ร 9 ร 5 = 180 minutes or 3 hours
They will next toll together after 3 hours.
Question 22.
Two tankers contain 850 liters and 680 liters of petrol. Find the
maximum capacity of a container which can measure the petrol of each
tanker in the exact number of times. (2012)
31. Solution:
To find the maximum capacity of a container which can measure the
petrol of each tanker in the exact number of times, we find the HCF of
850 and 680.
850 = 2 ร 52
ร 17
680 = 23
ร 5 ร 17
HCF = 2 ร 5 ร 17 = 170
Maximum capacity of the container = 170 liters.
32. Question 23.
The length, breadth, and height of a room are 8 m 50 cm, 6 m 25 cm and
4 m 75 cm respectively. Find the length of the longest rod that can
measure the dimensions of the room exactly. (2015)
33. Solution:
To find the length of the longest rod that can measure the dimensions of
the room exactly, we have to find HCF.
L, Length = 8 m 50 cm = 850 cm = 21
ร 52
ร 17
B, Breadth = 6 m 25 cm = 625 cm = 54
H, Height = 4 m 75 cm = 475 cm = 52
ร 19
HCF of L, B and H is 52
= 25 cm
Length of the longest rod = 25 cm
Question 24.
Three alarm clocks ring at intervals of 4, 12 and 20 minutes respectively.
If they start ringing together, after how much time will they next ring
34. together? (2015)
Solution:
To find the time when the clocks will next ring together,
we have to find LCM of 4, 12 and 20 minutes.
4 = 22
12 = 22
ร 3
20 = 22
ร 5
LCM of 4, 12 and 20 = 22
ร 3 ร 5 = 60 minutes.
So, the clocks will ring together again after 60 minutes or one hour.
35. Question 25.
In a school, there are two Sections A and B of class X. There are 48
students in Section A and 60 students in Section B. Determine the least
number of books required for the library of the school so that the books
can be distributed equally among all students of each Section. (2017
OD)
36. Solution:
Since the books are to be distributed equally among the students of
Section A and Section B. therefore, the number of books must be a
multiple of 48 as well as 60.
Hence, required numยฌber of books is the LCM of 48 and 60.
48 = 24
ร 3
60 = 22
ร 3 ร 5
LCM = 24
ร 3 ร 5 = 16 ร 15 = 240
Hence, required number of books is 240.
37. Question 26.
By using Euclidโs algorithm, find the largest number which divides 650
and 1170. (2017 OD)
Solution:
Given numbers are 650 and 1170.
1170 > 650
1170 = 650 ร 1 + 520
650 = 520 ร 1 + 130
520 = 130 ร 4 + 0
HCF = 130
The required largest number is 130.
38. Question 27.
Find the HCF of 255 and 867 by Euclidโs division algorithm. (2014)
Solution:
867 is greater than 255. We apply the division lemma to 867 and 255, to
39. get
867 = 255 ร 3 + 102
We continue the process till the remainder is zero
255 = 102 ร 2 + 51
102 = 51 ร 2 + 0, the remainder is zero.
HCF = 51
Question 28.
Using Euclidโs division algorithm, find whether the pair of numbers 847,
2160 are coprime or not.
To find out the minimum (least) time when the bells toll together next,
we find the LCM of 9, 12, 15.
41. Solution:
Let us assume, to the contrary, that 3 + 2โ5 is rational
So that we can find integers a and b (b โ 0), such that
3 + 2 โ5 = ab, where a and b are coprime.
Rearranging this equation, we get
Since a and b are integers, we get that a2b โ 32 is rational and so โ5 is
rational.
But this contradicts the fact that โ5 is irrational.
So we conclude that 3 + 2โ5 is irrational.
Question 30.
There are 104 students in class X and 96 students in class IX in a school.
In a house examination, the students are to be evenly seated in parallel
rows such that no two adjacent rows are of the same class. (2013)
(a) Find the maximum number of parallel rows of each class for the
42. seating arrangeยฌment.
(b) Also, find the number of students of class IX and also of class X in a
row.
(c) What is the objective of the school administration behind such an
arrangement?
Solution:
104 = 23
ร 13
96 = 25
ร 3
HCF = 23
= 8
43. (a) Number of rows of students of class X = 1048 = 13
Number maximum of rows class IX = 968 = 12
Total number of rows = 13 + 12 = 25
(b) No. of students of class IX in a row = 8
No. of students of class X in a row = 8
(c) The objective of school administration behind such an arrangement is
fair and clean examination, so that no student can take help from any
other student of his/her class.
Question 31.
Dudhnath has two vessels containing 720 ml and 405 ml of milk
44. respectively. Milk from these containers is poured into glasses of equal
capacity to their brim. Find the minimum number of glasses that can be
filled. (2014)
Solution:
1st vessel = 720 ml; 2nd vessel = 405 ml
We find the HCF of 720 and 405 to find the maximum quantity of milk
to be filled in one glass.
45. 405 = 34
ร 5
720 = 24
ร 32
ร 5
HCF = 32
ร 5 = 45 ml = Capacity of glass
No. of glasses filled from 1st vessel = 72045 = 16
No. of glasses filled from 2nd vessel = 40545 = 9
Total number of glasses = 25
Question 32.
Amita, Sneha, and Raghav start preparing cards for all persons of an old
age home. In order to complete one card, they take 10, 16 and 20
minutes respectively. If all of them started together, after what time will
they start preparing a new card together? (2013)
Solution:
To find the earliest (least) time, they will start preparing a new card
together, we find the LCM of 10, 16 and 20.
10 = 2 ร 5
16 = 24
20 = 22
ร 5
LCM = 24
ร 5 = 16 ร 5 = 80 minutes
They will start preparing a new card together after 80 minutes.
46. Question 33.
Find HCF of numbers 134791, 6341 and 6339 by Euclidโs division
algorithm. (2015)
47. Solution:
First, we find HCF of 6339 and 6341 by Euclidโs division method.
6341 > 6339
6341 = 6339 ร 1 + 2
6339 = 2 ร 3169 + 1
2 = 1 ร 2 + 0
HCF of 6341 and 6339 is 1.
Now, we find the HCF of 134791 and 1
134791 = 1 ร 134791 + 0
HCF of 134791 and 1 is 1.
Hence, the HCF of the given three numbers is 1.
48. Question 34.
If two positive integers x and y are expressible in terms of primes as x =
p2
q3
and y = p3
q, what can you say about their LCM and HCF. Is LCM a
multiple of HCF? Explain. (2014)
49.
50. Solution:
x = p2
q3
and y = p3
q
LCM = p3
q3
HCF = p2
q โฆ..(i)
Now, LCM = p3
q3
โ LCM = pq2
(p2
q)
โ LCM = pq2
(HCF)
Yes, LCM is a multiple of HCF.
Explanation:
Let a = 12 = 22
ร 3
b = 18 = 2 ร 32
HCF = 2 ร 3 = 6 โฆ(ii)
LCM = 22
ร 32
= 36
LCM = 6 ร 6
LCM = 6 (HCF) โฆ[From (ii)]
Here LCM is 6 times HCF.
51. Question 35.
Show that one and only one out of n, (n + 1) and (n + 2) is divisible by
3, where n is any positive integer. (2015)
52. Solution:
Let n, n + 1, n + 2 be three consecutive positive integers.
We know that n is of the form 3q, 3q + 1, or 3q + 2.
Case I. When n = 3q,
In this case, n is divisible by 3,
but n + 1 and n + 2 are not divisible by 3.
Case II. When n = 3q + 1,
In this case n + 2 = (3q + 1) + 2
= 3q + 3
= 3(q + 1 ), (n + 2) is divisible by 3,
but n and n + 1 are not divisible by 3.
Case III.
When n = 3q + 2, in this case,
n + 1 = (3q + 2) + 1
= 3q + 3 = 3 (q + 1 ), (n + 1) is divisible by 3,
but n and n + 2 are not divisible by 3.
Hence, one and only one out of n, n + 1 and n + 2 is divisible by 3.
53. Question 36.
Find the HCF and LCM of 306 and 657 and verify that LCM ร HCF =
Product of the two numbers. (2016 D)
55. R.H.S. = Product of two numbers = 306 ร 657 = 201042
L.H.S. = R.H.S.
56. Question 37.
Show that any positive odd integer is of the form 41 + 1 or 4q + 3 where
q is a positive integer. (2016 OD)
57. Solution:
Let a be a positive odd integer
By Euclidโs Division algorithm:
a = 4q + r โฆ[where q, r are positive integers and 0 โค r < 4]
a = 4q
or 4q + 1
or 4q + 2
or 4q + 3
But 4q and 4q + 2 are both even
a is of the form 4q + 1 or 4q + 3.