NCERT Solutions for Class 10 Maths Chapter 1 – Real Number Exercise 1.1

NCERT Solutions Class 10 Maths Chapter 1 – Real Numbers Exercise 1.1 are provided here to assist students in their Class 10 exam preparations. These solutions are prepared by our subject expert faculty to help students solve problems efficiently while using them as a valuable reference. The solutions are designed to be easy to understand, making it easier for students to tackle divisibility questions involving Euclid’s Division Algorithm.

NCERT Solutions for Class 10 Maths Chapter 1 – Real Number Exercise 1.1

Real Numbers

Medium

Q.

Use Euclid’s division algorithm to find the HCF of:

1. 135 and 225
2. 196 and 38220
3. 867 and 255

View Solution

Real Numbers

Medium

Q.

In a teachers' workshop, the number of teachers teaching French, Hindi and English are 48,80 and 144 respectively. Find the minimum number of rooms required if in each room the same number of teachers are seated and all of them are of the same subject.

[CBSE - 2024]

View Solution

Real Numbers

Medium

Q.

Show that every positive odd integer is of the form (4q + 1) and (4q + 3), where q is some integer.

[CBSE - 2023]

View Solution

Real Numbers

Medium

Q.

Prove that $$\rm 5\ –\ 2\sqrt{3}$$​ is an irrational number. It is given that $$\rm \sqrt{3}$$​ is an irrational number.

[CBSE - 2024]

View Solution

Exercise 1.1 explains Euclid's Division Algorithm and focuses on divisibility of integers, with a detailed explanation of each question in the exercise. The NCERT solutions are written in a clear and step-wise manner to help students understand the concepts more easily.

These solutions follow NCERT guidelines and ensure that students are aligned with the syllabus for effective exam preparation. They provide the right approach to scoring well in exams while reinforcing fundamental mathematical concepts.

NCERT Solutions for Class 10 Maths Chapter 1 – Real Numbers: Exercise 1.1

Q1. Use Euclid’s division algorithm to find the HCF of:

1. 135 and 225

2. 196 and 38220

3. 867 and 255

Answer:

1. For 135 and 225:

   Apply the Euclid’s division algorithm:

   $$225 = 135 \times 1 + 90$$ $$135 = 90 \times 1 + 45$$ $$90 = 45 \times 2 + 0$$

   Therefore, the HCF is 45.

2. For 196 and 38220:

   $$38220 = 196 \times 195 + 0$$

   The remainder is 0, so the HCF is 196.

3. For 867 and 255:

   $$867 = 255 \times 3 + 102$$ $$255 = 102 \times 2 + 51$$ $$102 = 51 \times 2 + 0$$

   Therefore, the HCF is 51.

Q2. In a teachers' workshop, the number of teachers teaching French, Hindi, and English are 48, 80, and 144 respectively. Find the minimum number of rooms required if in each room the same number of teachers are seated and all of them are of the same subject.

Answer:

To find the minimum number of rooms, we need to calculate the HCF of the number of teachers in each subject (48, 80, and 144).

Using the Euclid’s division algorithm to find the HCF of 48, 80, and 144:

• First, find the HCF of 48 and 80:

  $$80 = 48 \times 1 + 32$$ $$48 = 32 \times 1 + 16$$ $$32 = 16 \times 2 + 0$$

  So, the HCF of 48 and 80 is 16.

• Now, find the HCF of 16 and 144:

  $$144 = 16 \times 9 + 0$$

  Therefore, the HCF of 48, 80, and 144 is 16.

Thus, the minimum number of rooms required will be 16.

Q3. Show that the square of any positive integer cannot be of the form (5q + 2) or (5q + 3) for any integer q.

Answer:

We need to show that the square of any positive integer cannot be in the form (5q + 2) or (5q + 3) for any integer q.

Consider any positive integer n. When divided by 5, the remainder can be one of the following: 0, 1, 2, 3, 4. We will now examine the square of n in each case:

• If n = 5q, then n² = (5q)² = 25q², which is of the form 5r, so it is divisible by 5.

• If n = 5q + 1, then n² = (5q + 1)² = 25q² + 10q + 1, which is of the form 5r + 1.

• If n = 5q + 2, then n² = (5q + 2)² = 25q² + 20q + 4, which is of the form 5r + 4.

• If n = 5q + 3, then n² = (5q + 3)² = 25q² + 30q + 9, which is of the form 5r + 4.

• If n = 5q + 4, then n² = (5q + 4)² = 25q² + 40q + 16, which is of the form 5r + 1.

As we see, the square of any integer can only be in the form 5r, 5r + 1, or 5r + 4, and not in the form 5q + 2 or 5q + 3. Hence, the square of any positive integer cannot be of the form (5q + 2) or (5q + 3) for any integer q.

Q4. Show that the number 5 × 11 × 17 + 3 × 11 is a composite number.

Answer:

We need to prove that the number 5 × 11 × 17 + 3 × 11 is a composite number.

First, simplify the expression:

$$5 \times 11 \times 17 + 3 \times 11 = 11(5 \times 17 + 3) = 11(85 + 3) = 11 \times 88$$

This is the product of 11 and 88, which is not a prime number because it has factors other than 1 and itself. Therefore, 5 × 11 × 17 + 3 × 11 is a composite number.

Q5. Prove that every positive odd integer is of the form (4q + 1) and (4q + 3), where q is some integer.

Answer:

To prove that every positive odd integer is of the form (4q + 1) or (4q + 3) for some integer q, consider any odd integer. An odd integer can be written as 2k + 1, where k is an integer.

Now, when we divide any odd integer by 4, the remainder can be either 1 or 3.

• If the odd integer is of the form 4q + 1, then it is a remainder of 1 when divided by 4.

• If the odd integer is of the form 4q + 3, then it is a remainder of 3 when divided by 4.

Thus, every positive odd integer is either of the form (4q + 1) or (4q + 3) for some integer q.

FAQs: Class 10 Maths Chapter 1 – Real Numbers Exercise 1.1

Q1. What is Euclid’s Division Algorithm?
Answer:
Euclid's Division Algorithm is a method used to find the quotient and remainder when one integer is divided by another. It states that for any two positive integers a and b, there exist unique integers q (quotient) and r (remainder), such that
a = b × q + r, where 0 ≤ r < b.

Q2. How can Euclid's Division Algorithm help in solving problems?
Answer:
It helps in finding the gcd (greatest common divisor) of two numbers and proves useful in solving problems involving divisibility.

Q3. What are the main topics covered in Exercise 1.1?
Answer:
Exercise 1.1 covers the divisibility of integers using Euclid’s Division Algorithm and includes problems related to finding remainders and quotients.

Q4. Are there any tips for solving the problems in this exercise?
Answer:

• Always start by identifying the dividend and divisor.

• Apply the division formula a = b × q + r to find the quotient and remainder.

• Double-check your calculations for accuracy, especially for finding remainders.

Q5. How do NCERT Solutions help with exam preparation?
Answer:
These solutions provide clear step-by-step explanations, helping students understand the methods and concepts behind the problems. By practicing these solutions, students can boost their exam performance and gain a solid foundation in number theory.