Most Important Questions for CBSE Class 10 Maths Real number with Solutions

Well, we all know the importance of math in our lives, be it study or any other real-life work. Math is a very basic and essential skill to have. As students move forward in their educational journey, mastering important mathematical concepts becomes increasingly important. Class 10, Chapter 1 Real numbers provide the basics for numerous chapters. Understanding and practicing the important questions from this chapter is crucial for scoring well in exams.

To get a deep understanding and to get every mark that is allocated to this chapter, students should focus on solving Class 10 Math Chapter 1 important questions and extra questions. These questions not only revise your learning but also prepare you to solve the variety of problems you might see in exams. Using Class 10 Maths Ch. 1 important questions can help students identify key concepts, improve problem-solving skills, and ultimately perform better in exams.

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Chapter 1 Real Numbers: Important Questions

1. Let p be a prime number and k be a positive integer.

If p divides k², then which of these is DEFINITELY divisible by p?

a. only k

b. Only k and 7 k

c. only k, 7 k and k³ 

d. all -k/2, 7k, k and k³

Soln.

‍c) only k, 7 k and k³ 

^‍Explanation: 

‍If a prime number divides the square of an integer, then it must divide the integer itself.p | k²  ⇒ p | kp | k ⇒ p | 7kp | k  ⇒ p | k³p may not divide k/2 if k is odd; p may not divide -k/2 if k is odd.

2. √n is a natural number such that n > 1.

Which of these can DEFINITELY be expressed as a product of primes?

i. √n 

ii. n

iii. √n/2

a. only ii)

b. only i) and ii)

c. all - i), ii) and iii)

d. (cannot be determined without knowing n )

Soln.

‍b) only i) and ii)

‍Explanation:

‍Given that, √n is a natural number such that n>1Let us consider n=2, then √n=√2 ∉ NIf n=4, then √n=√4=2∈NIf n=5, then √n=√5∉ NSo, we can say that √n is a natural number if n is a perfect square number.Let us consider n=16, then √n=4=2×2If n=16, it can be expressed as the product of primes as 16=2×2×2×2If n=9, then √n/2=3/2 which cannot be expressed as the product of primes.So, √n and n can be expressed as the product of primes.

3. ẞ and δ are positive integers. HCF of ẞ and 630 is 210. HCF of 6 and 110 is 55.  Find the HCF of ß, 630, 8 and 110 using Euclid's division algorithm. Show your steps.

Soln.

‍HCF of β and 630 is 210
HCF of δ and 110 is 55β

= 210a + β1, where a is a positive integer and β1 is the remainder
630 = 3*210

630 = 210*3 + β1

β1 = 0

HCF(630, 210) = 210

HCF(δ, 110) = 55

HCF(β, 630, δ, 110) = 55

4. Two representations of real numbers are shown below.

Which one is correct?

Soln.

Both representations describe the structure of real numbers, but Representation 1 is more accurate and commonly accepted.

Explanation:

Representation 1 (Correct):

• It clearly shows the hierarchy of number sets:
• Natural numbers ⊆ Whole numbers ⊆ Integers ⊆ Rational numbers.
• Rational numbers and Irrational numbers are disjoint sets, and together, they form the set of Real numbers.
• This matches the standard mathematical classification.

Representation 2 (Incorrect):

• It implies that all subsets (Natural numbers, Whole numbers, Integers, Rational numbers, and Irrational numbers) are overlapping without a proper hierarchy.
• This is misleading since, for example:
• Natural numbers are not part of irrational numbers.

Rational numbers and irrational numbers are mutually exclusive.

Therefore, Representation 1 correctly depicts the relationship among the sets of numbers.

5. GrowMore Plantations have two rectangular fields of the same width but different lengths. They are required to plant 84 trees in the smaller field and 231 trees in the larger field. In both fields, the trees will be planted in the same number of rows but in different numbers of columns.

i) What is the number of rows that can be planted in this arrangement the most? Show your work.

ii) If the trees are planted in the number of rows obtained in part (i), how many columns will each field have?

Soln.

Given that, there are two rectangular fields of the same width but different lengths

They are required to plant 84 trees in the smaller field and 231 trees in the larger field.

In both fields, the trees will be planted in the same number of rows but in a different number of columns.

We need to find the most number of rows that can be planted in this arrangement.

To find the maximum number of rows that can be planted in this arrangement, we need to find the H.C.F of the trees planted in both the fields.

To find the H.C.F, we can use prime factorisation method.

84=2×2×3×7

231=7×3×11

So, H.C.F is the product of the common prime factors that have the least or smallest power.

H.C.F=21

Hence, the maximum number of rows that can be planted in this arrangement is 21

If the rows are 21, the number of trees planted in the smaller field is 84, then the number of columns is 4

If the rows are 21,  the number of trees planted in the larger field is 231, then the number of columns is 11

Hence, the columns each field has are 4, 11

6. M and N are positive integers such that M = p²q³r and N = p³q², where p, q,r are prime numbers.

Find LCM(M, N) and HCF(M, N).

Soln.

Given, M and N are positive integers such that M=p²q³r and N=p³q² where p, q, r are prime numbers.

Prime Factorisation of M=p×p×q×q×q×r

Prime Factorisation of N=p×p×p×q×q

We know that, The product of the prime factors with the highest powers is the L.C.M of the given numbers.

So, the L.C.M of M and N is p³q³r

We know that, H.C.F of the numbers is the product of common prime factors that have the least or smallest power.

So, the H.C.F of M and N is p²q²

7. The number 58732045 is divided by a number between 3256 and 3701.

State true or false for the below statements about the remainder and justify your answer.

i) The remainder is always less than 3701.

ii) The remainder is always more than 3256.

iii) The remainder can be any number less than 58732045.

Soln. When the number 58732045 is divided by a number between 3256 and 3701, the following statements are true or false:

• Statement i: True, the remainder is always less than 3701
• Statement ii: False, the remainder can be less than 3256
• Statement iii: False, the remainder is limited by the divisor, which is much smaller than 58732045 

In any division, the remainder is always less than the divisor. If the remainder is greater than or equal to the divisor, the division is incorrect. 

The number being divided is called the dividend, the number it is being divided by is called the divisor, and the result of the division is called the quotient.

8. (n² + 3 n - 4) can be expressed as a product of only two prime factors where n is a natural number. Find the value(s) of n for which the given expression is an even composite number. Show your work and give valid reasons.

Soln.

‍Given expression is fx=n²+3n-4

Using the middle-term splitting method, we can write the expression as:

n²+3n-4 = n²+4n-n-4

n²+4n-n-4 = nn+4-1n+4

fx=n -1n+4

It is given that the given expression is an even composite number.We know that, an even composite number is a number that are divisible by 2 and have more than 2 factors.For the given expression to be an even composite number, we need to equate n-1=2 or n+4=2

When n+4=2⇒n=-2 which is not possible.

Then n-1=2⇒n=3

Hence, the value of n for which the given expression is an even composite number is 3

9. The HCF of k and 93 is 31, where k is a natural number.

Which of these CAN be true for SOME VALUES of k?i) k is a multiple of 31. ii) k is a multiple of 93. iii) k is an even number. iv) k is an odd number.

1. only ii) and iii)
2. only i), ii) and iii)
3. only i), iii) and iv)
4. all i), ii), iii) and iv)

Soln. c) only i), iii) and iv)

Explanation

Given, H.C.F of k and 93 is 31

Prime factorisation of 93=3×31

We know that, H.C.F or Highest Common Factor is the greatest number that divides each of the two or more numbers.For example: H.C.F of 24 and 36 is 12

Prime Factorisation of 24=2×2×2×3=12×2

Prime Factorisation of 36=2×2×3×3=12×3

So, we can say that both the numbers will be the multiple of 12

Similarly, in the problem, we can say that, k and 93 will be a multiple of 31Multiples of 31=31, 62, 93, 124,...

So, k can be either an odd or even number.Hence, (i), (iii) or (iv) are true.

10. The prime factorisation of a prime number is the number itself.

How many factors and prime factors does the square of a prime number have?

Soln.We know that, a prime number is the number that have only two factors: one and the number itself.

Let us consider the prime factorisation of 5 which is a prime number.5 has only two factors, 1 and 5

Thus, the only prime factor of 5 is the number itself.

Let us consider the square of a prime number 5 which is 25

Factors of 25=1, 5, 25

Prime Factors of 25=5×5

Hence, the square of a prime number have 3 factors and 1 prime factors.

Understanding Real Numbers and their Importance 

Real numbers being the first chapter creates a basic understanding of the topics for students; they help students understand other chapters that are somehow related to real numbers. 

What are Real Numbers?

Real numbers are all the numbers that are found on the number line. This includes rational numbers, such as fractions and integers, and irrational numbers, such as 2 and 3. In Class 10 Maths Ch 1, students learn about the properties of real numbers, their decimal expansion, and how to classify them into rational and irrational numbers.

Importance of Real Numbers in Mathematics

The concept of real numbers is crucial in mathematics because it forms the basis of more advanced topics like algebra, calculus, and geometry. Understanding real numbers helps in solving equations and real-world problems where approximation is very necessary. Therefore, practicing Class 10 Maths Chapter 1 extra questions is important.

Key Topics in Class 10 Math, Chapter 1: Real Numbers

Fundamental Theorem of Arithmetic

The Fundamental Theorem of Arithmetic states that every composite number can be expressed as a product of primes, and this factorisation is different, except for the order of the factors. Understanding this theorem is essential for solving problems related to prime factorisation, HCF, and LCM.

Irrational Numbers

In class 10 Maths Chapter 1 extra questions, students learn to prove the irrationality of numbers like 2, 3, and 5 using contradiction. These proofs are fundamental to understanding the concept of irrational numbers and their place in the number system. 

Examples for Class 10 Maths Ch 1

The class 10 maths exam includes various types of questions such as MCQs, VSAs, Short answer type questions, and long answers. Here are some examples of them:

Multiple Choice Question

It is proposed to build a new circular park equal in area to the sum of areas of two circular parks of diameters 16 m and 12 m in a locality. The radius of the new park is:

(a) 10m (b) 15m (c) 20m (d) 24m

Very Short Answer Type Questions

Find the value of x if:

2cosec²30° + Xsin²60° -(3/4)tan²30°=10

Short Answer type question

The sum of a two-digit number and the number obtained by reversing the digits is 66. If the digits of the number differ by 2, find the number. How many such numbers are there?

Long Answer Type Questions

A motor boat whose speed is 18 km/h in still water takes 1 hour more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream. 

Case Studies Questions

Manpreet Kaur is the national record holder for women in the shot-put discipline. Her throw of 18.86m at the Asian Grand Prix in 2017 is the maximum distance for an Indian female athlete. Keeping her as a role model, Sanjitha is determined to earn gold in the Olympics one day. Initially, her throw reached 7.56m only. Being an athlete in school, she regularly practised both in the mornings and in the evenings and was able to improve the distance by 9cm every week. During the special camp for 15 days, she started with 40 throws and every day kept increasing the number of throws by 12 to achieve this remarkable progress. 

(i) How many throws did Sanjitha practice on the 11th day of the camp?

(ii) What would be Sanjitha’s throw distance at the end of 6 weeks? (or) When will she be able to achieve a throw of 11.16 m?

(iii) How many throws did she do during the entire camp of 15 days? 

Why to Practice Class 10 Maths Chapter 1 Important Questions

Real Numbers Lesson 10 important questions are very crucial for practice. Let us see a few benefits of using course 10 math CH 1 important questions:

Improving Conceptual Understanding of Questions

Practising Class 10 Ch 1 important questions help students pick up a deeper understanding of concepts. By working through different issues, understudies can discover different exam formats and different concepts, improving their overall grasp of the subject.

Boosts Problem-solving Skills

The standard practice of additional questions for Lesson 10 Maths Ch 1 permits students to create and improve their problem-solving skills. These questions frequently incorporate a mix of essential, intermediate, and advanced problems that challenge students to apply their information differently.

Exam Preparation 

Focussing on Real Numbers Class 10 important questions with solutions is a viable way to get ready for exams. These questions are frequently comparable to those enquired in past year's papers, making them an important resource for students aiming to score well.

Strategies for Solving Class 10 Math Chapter 1 Important Questions

Well, just using and implementing the class 10 chapter 1 questions are different things. Here are some ways to help you implement your tips correctly. 

• Before attempting to solve a question, it’s essential to understand what is being asked. Students should carefully read the problem, identify the key information, and determine which mathematical concepts are applicable.
• Once the problem is understood, students should recall and apply the relevant formula or theorem. For example, if the problem involves finding the HCF of two numbers, the correct formula and correct way should be applied.
• Consistent practice is key to mastering Class 10th Maths Chapter 1 important questions. Students should set aside time each day to work through a set of problems, gradually increasing the difficulty level as they progress.
• After solving a set of questions, it’s very essential to review any mistakes made. Understanding where and why errors occur helps students avoid similar mistakes in the future.

Mastering the concepts in Class 10 Maths Chapter 1 is important for building a solid mathematical foundation. By focussing on Class 10 Maths Ch 1 important questions and extra questions, students can enhance their understanding, improve problem-solving skills, and prepare effectively for their exams. Regular practice and review are the main points for success in mathematics.