Boost Your Performance in CBSE Class 10 Maths Exam Chapter 1 with These Important Questions
Important questions for Class 10 Maths Chapter 1, Real Numbers is prepared by experts of Vedantu with a purpose of creating important questions for the chapter is to enable the students to pivotal concepts of the chapter that has been introduced in Maths NCERT Solutions Class 10. The important questions pdf version is prepared to give a better conceptual understanding to the students and help them to perceive what questions they can expect from this chapter in exams. There are pdf versions of important questions for other subjects also. You can download them anytime on any device and practice them at your convenient time. The important questions will surely give an insight about what questions can be expected in exams.
Chapter 1, Real numbers for class 10, continues with the real number operations that you studied in earlier grades and also introduces two important properties of positive integers namely Euclid's Division Lemma and algorithm and the fundamental theorem of arithmetic. The important questions are based on the topics that are discussed in this chapter. Let us have a quick glance at the summary of the chapter so that you can solve the important questions for Maths Chapter 1, Real Numbers for Class 10.
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Important Questions for CBSE Class 10 Maths Chapter 1 - Real Numbers 2024-25
Study Important Questions for Class 10 Mathematics Chapter 1 - Real Numbers
Very Short Answer Questions (1 Mark)
1. Use Euclidβs division lemma to show that the square of any positive integer is either of form
(Hint: Let
Ans: Let
Clearly, in the expression (1) we are dividing
Therefore from (1) and (2) we can get,
Squaring both sides of equation (3) we get,
Taking
Equation (5) could be written as
Hence, it can be said that the square of any positive integer is either of the form
2. Express each number as product of its prime factors.
(i).
Ans: The product of prime factors of number
(ii).
Ans: The product of prime factors of number
(iii).
Ans: The product of prime factors of number
(iv).
Ans: The product of prime factors of number
(v).
Ans: The product of prime factors of number
3. Given that HCF
Ans: It is given that the HCF of the two numbers
We know that, LCM
LCM
4. Check whether
Ans: If any number ends with the digit
Let us check this for natural number
Clearly,
5. Prove that
Ans: We will prove this by contradiction. To solve this problem, suppose that
Subtracting
Since we know that
Hence,
6. The number
Composite Number
Whole Number
Prime Number
None of these
Ans: (a) and (b) both
7. For what least value of
No value of
is possible
Ans: (c) 1
8. The sum of a rational and an irrational number is
Rational
Irrational
Both (a) & (b)
Either (a) & (b)
Ans: (b) Irrational
9. HCF of two numbers is
Ans: (b)
10. A lemma is an axiom used for providing
other statement
no statement
contradictory statement
none of these
Ans: (a) other statement
11. If HCF of two numbers is
prime, co-prime
composite, prime
Both (a) and (b)
None of these
Ans: (a) prime, co-prime
12. The number
a terminating decimal number
a rational number
an irrational number
Both (a) and (b)
Ans: (b) a rational number
13. The number
a rational number
a non-terminating decimal number
an irrational number
Both (a) & (c)
Ans: (b) a non-terminating decimal number
14. The smallest composite number is
Ans: (d)
15. The number
an integer
an irrational number
a rational number
None if these
Ans: (c) a rational number
16. The number
a rational number
an irrational number
Both (a) and (b)
neither rational nor irrational
Ans: (b) an irrational number
17. The number
a rational number
an irrational number
an integer
not real number
Ans: (b) an irrational number
Short Answer Questions (2 Marks)
1. Show that any positive odd integer is of the form
Ans: Let
Clearly, in the expression (1) we are dividing
Therefore from (1) and (2) we can get,
And therefore, any odd integer can be expressed in the form
2. An army contingent of
Ans: To solve this problem, we have to find the HCF of two numbers
To find the HCF, we can use Euclidβs algorithm which states that if there are any two integers
For
For
Therefore, from (1) and (2), the HCF
3. Use Euclidβs division lemma to show that the cube of any positive integer is of the form
Ans: Let
Clearly, in the expression (1) we are dividing
Therefore from (1) and (2) we can get,
Cubing both sides of equation (3) we get,
Case 1: When
Where
Case 2: When
Where
Case 3: When
Where
Therefore, from case 1,2 and 3 we conclude that the cube of any positive integer is of the form
4. Find the LCM and HCF of the following pairs of integers and verify that
(i).
Ans: Let us first find the HCF of the two given numbers.
Writing the prime factorization of both of the numbers we get,
Since
From (1),
Product of two numbers,
From (2) and (3),
Hence from (4) and (5), product of two numbers
(ii).
Ans: Let us first find the HCF of the two given numbers.
Writing the prime factorization of both of the numbers we get,
Since
From (1),
Product of two numbers,
From (2) and (3),
Hence from (4) and (5), product of two numbers
(iii).
Ans: Let us first find the HCF of the two given numbers.
Writing the prime factorization of both of the numbers we get,
Since
From (1),
Product of two numbers
From (2) and (3),
Hence from (4) and (5), product of two numbers
5. Find the LCM and HCF of the following integers by applying the prime factorisation method.
(i).
Ans: Let us first find the HCF of the two given numbers.
Writing the prime factorization of all the numbers we get,
Since
From (1),
(ii).
Ans: Let us first find the HCF of the two given numbers.
Writing the prime factorization of all the numbers we get,
Since
From (1),
(iii).
Ans: Let us first find the HCF of the two given numbers.
Writing the prime factorization of all the numbers we get,
Since
From (1),
6. Explain why
Ans: Numbers are of two types β prime and composite.
Prime numbers can be divided by
It can be observed that
The given expression has
Also,
The given expression has
7. There is a circular path around a sports field. Sonia takes
Ans: It can be observed that Ravi takes lesser time than Sonia for completing one round of the circular path. As they are going in the same direction, they will meet again at the time that will be the
Let us now calculate
Therefore, Ravi and Sonia will meet together at the starting point after
8. Prove that
Ans: We will prove this by contradiction.
Let us suppose that
Squaring both sides of (1), we get
From (2) we can conclude that
From (3) we can write
Substituting value of (4) in (2) we get,
It means that
From
9. Write down the decimal expansion of those rational numbers in Question 1 which have terminated decimal expansion.
Ans:
(i).The decimal expansion of
(ii) The decimal expansion of
(iii) The decimal expansion of
(iv) The decimal expansion of
(v) The decimal expansion of
(vi) The decimal expansion of
10. The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If, they are rational, and of form
(i). Number -
Ans: Since the decimal expansion is terminating so it is a rational number and therefore it can be expressed in
Hence, the factors of
(ii). Number -
Ans: Since the decimal expansion is neither terminating nor non-terminating repeating so it is an irrational number.
(iii). Number -
Ans: Since the decimal expansion is non-terminating repeating so it is a rational number and therefore it can be expressed in
Hence, the factors of
11. Show that every positive integer is of the form
Ans: Let
Clearly, in the expression (1) we are dividing
Therefore from (1) and (2) we can get,
If
If
So, every positive even integer is of the form
12. Show that any number of the form
Ans: The prime factorization of
For the given number of the form
13. Use Euclidβs Division Algorithm to find the
Ans: Step 1: Since
Step 2: Since the remainder in (1) is not zero so we have to apply the Euclidβs division lemma to
Step 3: Again, the remainder in (2) is not zero so we have to apply the Euclidβs division lemma to
Step 4: Again, the remainder in (3) is not zero so we have to apply the Euclidβs division lemma to
Step 5: Again, the remainder in (4) is not zero so we have to apply the Euclidβs division lemma to
Step 6: Again, the remainder in (5) is not zero so we have to apply the Euclidβs division lemma to
Step 7: Again, the remainder in (6) is not zero so we have to apply the Euclidβs division lemma to
Now the remainder in equation (7) is zero. Therefore,
14. Given that
Ans: It is given that the HCF of the two numbers is
We know that, LCM
15. Show that every positive odd integer is of the form
Ans: Let
Clearly, in the expression (1) we are dividing
Therefore from (1) and (2) we can get,
If
If
If
If
So, every positive odd integer is of the form
16. Show that any number of the form
Ans: The prime factorization of
For the given number of the form
17. Find
Ans: The prime factorization of
The prime factorization of
From (1) and (2) we can see that
From (1) and (2) we can see that
18. The
Ans: It is given that the HCF of the two numbers is
We know that, LCM
19. Prove that the square of any positive integer of the form
Ans: Let
Squaring both sides of equation (1) we get,
Hence, the square of any positive integer of the form
20. Use Euclidβs Division Algorithm to find the
Ans: Step 1: Since
Step 2: Since the remainder in (1) is not zero so we have to apply the Euclidβs division lemma to
Step 3: Again, the remainder in (2) is not zero so we have to apply the Euclidβs division lemma to
Step 4: Again, the remainder in (3) is not zero so we have to apply the Euclidβs division lemma to
Step 5: Again, the remainder in (4) is not zero so we have to apply the Euclidβs division lemma to
Step 6: Again, the remainder in (5) is not zero so we have to apply the Euclidβs division lemma to
Step 7: Again, the remainder in (6) is not zero so we have to apply the Euclidβs division lemma to
Now the remainder equation (7) is zero. Therefore,
21. Find the largest number which divides
Ans. Since we want the remainder to be
Step 1: Since
Step 2: Since the remainder in (1) is not zero so we have to apply the Euclidβs division lemma to
Step 3: Again, the remainder in (2) is not zero so we have to apply the Euclidβs division lemma to
Step 4: Again, the remainder in (3) is not zero so we have to apply the Euclidβs division lemma to
Now the remainder in equation (4) is zero. Therefore,
The largest number which divides
22. A shopkeeper has
Ans: The required greatest capacity is the
Step 1: Since
Step 2: Since the remainder in (1) is not zero so we have to apply the Euclidβs division lemma to
Now the remainder in equation (2) is zero. Therefore,
Let us now find the HCF of
Short Answer Questions (3 Marks)
1. Use Euclidβs division algorithm to find the
(i).
Ans: Step 1: Since
Step 2: Since the remainder in (1) is not zero so we have to apply the Euclidβs division lemma to
Step 3: Again, the remainder in (2) is not zero so we have to apply the Euclidβs division lemma to
Now the remainder in equation (3) is zero. Therefore, the
(ii).
Ans: Since
Since, the remainder in (1) is zero. Therefore, the
(iii).
Ans: Step 1: Since
Step 2: Since the remainder in (1) is not zero so we have to apply the Euclidβs division lemma to
Step 3: Again, the remainder in (2) is not zero so we have to apply the Euclidβs division lemma to
Now the remainder is zero. Therefore, the
2. Find the greatest number of
Ans: To find the greatest number of
The prime factorization of
Hence, from (1), (2) and (3),
Now, we know that the greatest number of
From (4) and (5), using Euclidβs division lemma we get,
Hence, the greatest number of
3. Prove that the square of any positive integer is of the form
Ans: Let
Clearly, in the expression (1) we are dividing
Therefore from (1) and (2) we can get,
Case 1: If
Equation (4) could be written as
Case 2: If
Equation (6) could be written as
Case 3: If
Equation (8) could be written as
Case 4: If
Equation (10) could be written as
Therefore, from equations (5), (7), (9) and (11) it can be said that the square of any positive integer is either of the form
4. There are
Ans: Given
Using Euclidβs division algorithm,
Step 1: Since
Step 2: Since the remainder in (1) is not zero so we have to apply the Euclidβs division lemma to
Step 3: Again, the remainder in (2) is not zero so we have to apply the Euclidβs division lemma to
Step 4: Again, the remainder in (3) is not zero so we have to apply the Euclidβs division lemma to
Now the remainder in (4) is zero. Therefore, the
So, greatest number of cartoons is
5. Prove that product of three consecutive positive integers is divisible by
Ans: Let us consider three consecutive positive integers
Using Euclidβs lemma, let us write
Clearly, in the expression (1) we are dividing
Therefore from (1) and (2) we can get,
Case 1: If
Case 2: If
which is divisible by
Case 3: If
which is divisible by
Case 4: If
which is divisible by
Case 5: If
which is divisible by
Case 6: If
which is divisible by
From equations (3) to (8) we can conclude that the product of three consecutive positive integers is divisible by
6. Prove that
Ans: We will prove this by contradiction. To solve this problem, suppose that
Subtracting
Since we know that
7. Prove that if
Ans: Let
Squaring
Where
From (2), it is clear that
8. Show that one and only one out of
Ans: Let
Clearly, in the expression (1) we are dividing
Therefore from (1) and (2) we can get,
Case 1: If
Case 2: If
Case 3: If
9. Use Euclidβs Division Lemma to show that the square of any positive integer of the form
Ans: Let
Clearly, in the expression (1) we are dividing
Therefore from (1) and (2) we can get,
Squaring both sides of equation (3) we get,
Taking
Equation (5) could be written as
Hence, it can be said that the square of any positive integer is either of the form
10. Prove that if
Ans: We will prove this by contradiction.
Let us suppose that
Squaring both sides of (1), we get
From (2) we can conclude that
From (3) we can write
Substituting value of (4) in (2) we get,
It means that
From
11. Prove that the difference and quotient of
Ans: Difference of
We will prove this by contradiction.
Let us suppose that
Squaring both sides of (1), we get
From (2) we can conclude that
From (3) we can write
Substituting value of (4) in (2) we get,
It means that
From
Now, the quotient of
Similarly, using the approach used above, we can prove that
12. Show that
Ans: Let
Then from (1),
Taking
Hence, from (3) we can conclude that
13. Use Euclid Division Lemma to show that cube of any positive integer is either of the form
Ans: Let
Clearly, in the expression (1) we are dividing
Therefore from (1) and (2) we can get,
Cubing both sides of equation (3) we get,
Taking
Equation (5) could be written as
Hence, it can be said that the square of any positive integer is either of the form
Long Answer Questions (4 Marks)
1. Prove that the following are irrationals.
(i).
Ans: (i) We will prove this by contradiction. Let us suppose that
Since we know that
(ii).
Ans: We will prove this by contradiction. Let us suppose that
Since we know that
(iii).
Ans: We will prove this by contradiction. To solve this problem, suppose that
Subtracting
Since we know that
2. Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating decimal expansion.
(i).
Ans: Theorem: Let
For rational number
From (2) we can see that the denominator
Therefore, it follows from theorem (1), the rational number
(ii).
Ans: Theorem: Let
For rational number
From (2) we can see that the denominator
Therefore, it follows from theorem (1), the rational number
(iii).
Ans: Theorem: Let
For rational number
From (2) we can see that the denominator
Therefore, it follows from theorem (1), the rational number
(iv).
Ans: Theorem: Let
For rational number
From (2) we can see that the denominator
Therefore, it follows from theorem (1), the rational number
(v).
Ans: Theorem: Let
For rational number
From (2) we can see that the denominator
Therefore, it follows from theorem (1), the rational number
(vi).
Ans: Theorem: Let
For rational number
From (2) we can see that the denominator
Therefore, it follows from theorem (1), the rational number
(vii).
Ans: Theorem: Let
For rational number
From (2) we can see that the denominator
Therefore, it follows from theorem (1), the rational number
(viii).
Ans: Theorem: Let
For rational number
From (2) we can see that the denominator
(ix).
Ans: Theorem: Let
For rational number
From (2) we can see that the denominator
(x).
Ans: Theorem: Let
For rational number
From (2) we can see that the denominator
Therefore, it follows from theorem (1), the rational number
Maths Chapter 1- Real Numbers
Introduction
All numbers can be classified into two categories -- Real and Imaginary. A real number can be actually perceived and can be represented in a number line. It includes all integers -- positive and negative, all natural numbers (1, 2, 3, β¦.., ) all whole numbers (0, 1, 2, 3, β¦.., ), fractions, rational and irrational numbers.
An imaginary number on the other hand cannot be perceived. Square root of all negative numbers is imaginary.
A real number can be represented as an infinite decimal expansion. Ex: 5 = 5.0000β¦.., β = 0.3333β¦β¦
A real number can be rational or irrational.
All rational numbers and all irrational numbers together make a group of real numbers.
(i) A group of rational numbers is denoted by Q.
(ii) A group of irrational numbers is denoted by Qβ.
(iii) A group of real numbers is denoted by R.
A rational number an be written in the form pq, both p and q being integers (q 0) while an irrational number can not be impressed in such a way.
The decimal expansion of a rational number say x can terminate after a finite number of digits.
Ex: β = 0.625 or eventually begins to repeat the same finite sequence of digits continuously.
Ex: 56/99 = 0.565656β¦β¦β¦.
On the other hand in case of an irrational number, the decimal expansion continues without repeating. Ex: 1.7320508β¦., π= 3.141592β¦.)
Euclidβs Division Lemma and Algorithm
First, let us understand the meaning of Lemma and Algorithms.
A Lemma is a minor proven statement which is used to prove other statements. The Euclidβs division Lemma is actually another statement of the common long division process.
For example, on dividing 36 by 5, we get quotient = 7 and remainder = 1 which is less than 5.
If there are positive integers a and b, then there are unique integers existing q and r that satisfy a = bq + r, 0 rb.
Example: If a =47 and b = 9, then we can write
47 = 9 x 5 + 2, where q = 4 and r = 15 b
Note:
(i) The unique characteristics of qand r are nothing but the quotient and remainder respectively.
(ii) Although Euclidβs division algorithm is expressed for only positive integers but it can be extended for all integers except 0.
(iii) When aand bare positive integers, then qand rcan take values only from whole numbers.
An algorithm means a series of well defined steps, which provides a procedure of calculations repeatedly successfully on the results of earlier steps. With the help of defined procedure the desired result can be obtained.
Euclidβs Division Algorithm
The Euclidβs Division Algorithm is also an algorithm to determine the H.C.F (Highest Common Factor) of the given positive integers.
Ordinarily, if we want to find the HCF of two positive integers, we first find the factors of the two integers.
For ex: For two integers 24 and 42, the common factors are 1, 2, 3 and 6. As 6 is the highest of other numbers, 6 is the HCF of 24 and 42.
In Euclidβs Division Algorithm, the largest integer is divided by the smaller one and a remainder is obtained.
Next the smaller integer is divided by the above remainder and the process is repeated till zero is obtained. The remainder in the last but one division is the required HCF.
The Fundamental Theorem of Arithmetic
According to this theorem every positive integer is either a prime number or it can be expressed as the product of primes. It is also known as Unique Factorization Theorem. Thus, any integer that is greater than 1 can be either a prime number or can be expressed as a product of prime factors.
Example: 8 = 2 x 2 x 2, where 2 is a prime factor.
Note:
(i) When a number has no factors other than 1 and the number itself, the number is called a prime number.
(ii) 1 is neither prime nor composite.
(iii) A number is a composite number if it has at least one factor other than 1 and the number itself.
(iv) 2 is the smallest prime number and an even number. It is the only number that is both prime and even.
(v) Two numbers are co-prime if they have no common factors other than 1, i.e, their HCF = 1.
This theorem is termed as βFundamentalβ because of its importance in the development of number theory.
Factor Tree
A chain of factors that is demonstrated in the form of a tree, is called a factor tree.
HCF and LCM using Prime Factorization
HCF is the product of the smallest power of each common prime factor present in the numbers.
The product of the largest power of each prime factor present in the numbers is the LCM.
Important Note:
(i) LCM of two or more numbers is the smallest number which is the smallest number and divisible by all the given numbers.
(ii) If there is no common prime factor, then the HCF of the given number is 1.
Relationship Between Numbers and Their HCF and LCM
For given positive integers aand b, the relation between these numbers and their HCF and LCM is
HCF (a, b) =
For given three positive integers a, b and c, the relation between these numbers and their HCF and LCM is HCF (a, b, c) =
or LCM (a, b,c) =
Method of Proving Irrationality of Numbers
We use the method of contradiction in order to prove irrationality of the numbers. i.e., first we assume that the given number is rational and after reaching a contradiction we prove that the given number is irrational.
Theorem
If a prime number p divides a2, then p divides a, where ais a positive integer.
Let us understand how to write the decimal expansion of those rational numbers with terminating decimal expansion
Let a rational number in the lowest form be p/q such that the prime factorization of q is of the form 2mx 5n, where m, n are non-negative integers. To write decimal expansion of p/q, convert p/q to an equivalent rational number of the form c/d, where dis a power of 10.
The above notes will be a huge benefit for solving the Important Questions Of Chapter 1 of Maths for Class 10, Real Numbers. You can now master the Euclidβs Division Lemma and Algorithm. The important questions for this chapter is basically a compilation of higher and more advanced techniques a student can expect in Class 10 examinations. The questions of the chapter and the notes related to the topic provided by Vedantu will not only help you to understand the concept better but also solve the questions successfully. If you harbour any doubts then you can get answers to all your queries with our experienced teachers by registering with Vedantu and gain expertise in the subject.
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FAQs on Important Questions for CBSE Class 10 Maths Chapter 1 - Real Numbers 2024-25
1. What do you mean by HCF and LCM?
Ans: HCF stands for "Highest Common Factor," also known as the Greatest Common Divisor (GCD). It refers to the largest number that divides two or more integers without leaving a remainder. In other words, the HCF is the largest common factor shared by a set of numbers.
LCM stands for "Least Common Multiple." It refers to the smallest positive integer that is divisible by two or more integers without leaving a remainder. In other words, the LCM is the smallest common multiple shared by a set of numbers.
2. How will Important Questions for Chapter 1 for Class 10 help in exams?
Ans: Important Questions for Chapter 1 in Class 10 will help in exams by focusing on the key concepts, topics, and problem-solving techniques. These questions are carefully selected to cover the important aspects of the chapter, allowing students to practice and gain confidence in their understanding. They serve as a valuable tool for exam preparation and improving overall performance.
3. What does Chapter 1, real number for Class 10 discuss?
Ans: Chapter 1, "Real Numbers," for Class 10 discusses the concept of real numbers and their properties. It covers topics such as rational numbers, irrational numbers, decimal representation of rational numbers, fundamental theorem of arithmetic, Euclid's division algorithm, and the concept of prime and composite numbers. The chapter also introduces the concept of HCF (Highest Common Factor) and LCM (Least Common Multiple) of numbers. It provides a foundation for understanding number systems and lays the groundwork for further mathematical concepts in higher grades.
4. Which is the most important chapter in maths class 10?
Ans: The most important chapter in maths are:
Chapter 3 - Linear Equations
Chapter 5 - Arithmetic Progression
Chapter 6 - Triangles
Chapter 7 - Coordinate Geometry
Chapter 8 - Introduction to Trigonometry
Chapter 9 - Applications of Trigonometry
Chapter 13 - Surface Areas & Volumes
Chapter 14 - Statistics
5. What is the weightage of marks in class 10 maths 2024-25?
Ans: CBSE Class 10 Mathematics in the academic year 2024-25 includes a theory paper worth 80 marks and internal assessment carrying 20 marks. The theory paper consists of 30 questions divided into four sections, with different marks assigned to each question. The total duration of the examination is three hours.











