Understanding the decimal representation of real numbers becomes essential for mathematics because real numbers contain all rational and irrational elements. The behavior of these numbers in decimal format is addressed in this exercise. We use this method to identify non-terminating decimals while understanding how rational numbers form repeating patterns. Learning these concepts establishes our ability to correctly identify numbers while linking theoretical number concepts to their actual decimal expressions.
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Students obtain maximum benefit from these concepts when they consult with NCERT Solutions. The NCERT Books related solutions provide step-by-step explanations to simplify difficult proofs. Students who refer to these educational resources can develop fundamental number theory knowledge that helps them succeed in advanced mathematics studies as well as competitive exams.
Q1 Prove that $\sqrt 5$ is irrational.
Answer:
Let us assume $\sqrt{5}$ is rational, which means it can be written in the form $\frac{p}{q}$ where p and q are co-primes and $q\neq 0$
$\\\sqrt{5}=\frac{p}{q}$
Squaring both sides, we obtain
$\\\left ( \sqrt{5} \right )^{2}=\left (\frac{p}{q} \right )^{2}\\$
$5=\frac{p^{2}}{q^{2}}\\$
$p^{2}=5q^{2}$
From the above equation, we can see that p2 is divisible by 5, therefore, p will also be divisible by 5, as 5 is a prime number. $(i)$
Therefore, p can be written as 5r
p = 5r
p2 = (5r)2
5q2 = 25r2
q2 = 5r2
From the above equation, we can see that q2 is divisible by 5, Therefore, q will also be divisible by 5 as 5 is a prime number. $(ii)$
From (i) and (ii), we can see that both p and q are divisible by 5. This implies that p and q are not co-primes. This contradiction arises because our initial assumption that $\sqrt{5}$ is rational was wrong. Hence proved that $\sqrt{5}$ is irrational.
Q2 Prove that $3 + 2 \sqrt 5$ is irrational.
Answer:
Let us assume $3 + 2 \sqrt 5$ is rational, this means it can be written in the form $\frac{p}{q}$ where p and q are co-prime integers.
$\\3+2\sqrt{5}=\frac{p}{q}$
$2\sqrt{5}=\frac{p}{q}-3$
$\sqrt{5}=\frac{p-3q}{2q}$
As p and q are integers $\frac{p-3q}{2q}\\$ would be rational, which contradicts the fact that $\sqrt{5}$ is irrational. This contradiction arises because our initial assumption that $3 + 2 \sqrt 5$ is rational was wrong. Therefore $3 + 2 \sqrt 5$ is irrational.
Q3 Prove that the following are irrationals :
(i) $\frac{1}{\sqrt 2}$
Answer:
Let us assume $\frac{1}{\sqrt{2}}$ is rational, this means it can be written in the form $\frac{p}{q}$ where p and q are co-prime integers.
$\frac{1}{\sqrt{2}}=\frac{p}{q}$
$\sqrt{2}=\frac{q}{p}$
Since p and q are co-prime integers $\frac{q}{p}$ will be rational, which contradicts the fact that $\sqrt{2}$ is irrational. This contradiction arises because our initial assumption that $\frac{1}{\sqrt{2}}$ is rational was wrong. Therefore $\frac{1}{\sqrt{2}}$ is irrational.
Q3 (2) Prove that the following are irrationals :
(ii) $7 \sqrt 5$
Answer:
Let us assume $7 \sqrt 5$ is rational, this means it can be written in the form $\frac{p}{q}$ where p and q are co-prime integers.
$7\sqrt{5}=\frac{p}{q}$
$\sqrt{5}=\frac{p}{7q}$
As p and q are integers $\frac{p}{7q}\\$ would be rational, which contradicts the fact that $\sqrt{5}$ is irrational. This contradiction arises because our initial assumption that $7 \sqrt 5$ is rational was wrong. Therefore $7 \sqrt 5$ is irrational.
Q3 (3) Prove that the following are irrationals : $6 + \sqrt 2$
Answer:
Let us assume $6 + \sqrt 2$ is rational, this means it can be written in the form $\frac{p}{q}$ where p and q are co-prime integers.
$6+\sqrt{2}=\frac{p}{q}$
$\sqrt{2}=\frac{p}{q}-6$
$\sqrt{2}=\frac{p-6q}{q}$
As p and q are integers $\frac{p-6q}{q}$ would be rational, which contradicts the fact that $\sqrt{2}$ is irrational. This contradiction arises because our initial assumption that $6 + \sqrt 2$ is rational was wrong. Therefore $6 + \sqrt 2$ is irrational.
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Students must check the NCERT solutions for class 10 of Mathematics and Science Subjects.
Students must check the NCERT Exemplar solutions for class 10 of the Mathematics and Science Subjects.
Frequently Asked Questions (FAQs)
Sum and difference of a rational and irrational number is irrational.
“Fundamental theorem of Arithmetic” given in the Class 10 Maths chapter 1 states that “Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur”.
The proof is based on a most common technique called ‘proof by contradiction.
important in board exams, you can check previous year papers for better understanding.
On Question asked by student community
To get the previous year question papers you can visit the official website of your board and search under the section of previous year question papers. You can also search on Google for the papers or visit the educational platforms like Careers 360 for the papers. They also provide with the papers and answer key also.
You must be at least 14 years old by December 31st of the year 2027
Since your date of birth is 29 January 2013, you will turn 14 in January 2027, which is before the December 31st deadline for the 2027 exam.
Hence you are eligible..
Good luck!!
Hello,
If you want to get your 10th marksheet online, you just need to visit an official website like https://www.cbse.gov.in/ or https://results.cbse.nic.in/ for the CBSE board, and for the state board, you can check their website and provide your roll number, security PIN provided by the school, and school code to download the result.
I hope it will clear your query!!
Hello, if you are searching for Class 10 books for exam preparation, the right study material can make a big difference. Standard textbooks recommended by the board should be your first priority as they cover the syllabus completely. Along with that, reference books and guides can help in practicing extra questions and understanding concepts in detail. You can check the recommended books for exam preparation from the link I am sharing here.
https://school.careers360.com/ncert/ncert-books-for-class-10
https://school.careers360.com/boards/cbse/cbse-best-reference-books-for-cbse-class-10-exam
Hello Dinesh !
As per CBSE board guidelines for internal assessment for class 10th you will have to give a 80 marks board exam and 20 marks internal assessment. The internal assessment will be at the end of your year.
For knowing the definite structure of the internal assessment you will have to ask your teachers or your seniors in the school as CBSE has provided flexibility in choosing the methods of internal assessment to schools. For more details related to assessment scheme for class 10 given by CBSE you can visit: Assessment scheme (http://cbseacademic.nic.in/web_material/CurriculumMain2Sec/Curriculum_Sec_2021- 22.pdf)
I Hope you have understood it!
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