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Class 10 Maths Index Page With All Chapters and Sub-Topics

Foreword iii Preface v 1. Real Numbers 1 1. Polynomials 20 2. Pair of Linear Equations in Two Variables 38 3. Quadratic Equations 70 4. Arithmetic Progressions 93 5. Triangles 6. Coordinate Geometry 7. Introduction to Trigonometry 8. Some Applications of Trigonometry 9. Circles Constructions Areas Related to Circles Surface Areas and Volumes Statistics Probability We continue our discussion on real numbers in this chapter. We begin with two very important properties of positive integers in Sections 1.

Stated simply, it says any positive integer a can be divided by another positive integer b in such a way that it leaves a remainder r that is smaller than b. Many of you probably recognise this mathematics ncert book of 10th class rule the usual long division process.

Although this result is quite easy to state and understand, it has many applications related to the divisibility properties of integers.

We touch Mathematics Ncert Book Of 10th Class Java upon a few of them, and use it mainly to compute the HCF of two positive integers. The Fundamental Theorem of Arithmetic, on the other hand, has to do something with multiplication of positive integers. You already know that every composite number can be expressed as a product of primes in a unique way � this important fact is the Fundamental Theorem of Arithmetic.

Again, while it is a result that is easy to state and understand, it has some very deep and significant applications in the field of mathematics. We use the Fundamental Theorem of Arithmetic for two mathematics ncert book of 10th class rule applications. First, we use it to prove the irrationality of many of the numbers you studied in Class IX, such as 23 and 5. We do so by looking at the prime factorisation of the denominator p q of.

You will see that the prime factorisation of q will completely reveal the nature q of the decimal expansion of p. A trader was moving along a road selling eggs. This grew into a fight, he pulled the basket with eggs and dashed it on the floor. The eggs broke. The trader requested the Panchayat to ask the idler to pay for the broken eggs. The Panchayat asked the trader how many eggs were broken.

He gave the following response: If counted in pairs, one will remain; If counted in threes, two will remain; If counted in fours, three will remain; If counted in fives, four will remain; If counted in sixes, five will remain; If counted in sevens, nothing will remain; My basket cannot accomodate more than eggs.

So, how many eggs were there? Let us try and solve the puzzle. Let the number of eggs be a. If counted in fives, four will remain. If counted in fours, three will remain.

If counted in threes, two will remain. If counted in pairs, one will remain. That is, in each case, we have a and a positive integer b in our example, b takes values 7, 6, 5, 4, 3 and 2, respectively which divides a and leaves a remainder r in our case, r is 0, 5, 4, 3, 2 and 1, respectivelythat is smaller than b. Rampal, and. Getting back to our puzzle, do you have any idea how you will solve it? You must look for the multiples of 7 which satisfy all the conditions. By trial and error, you will find he had eggs.

You may have also realised that this is nothing but a restatement of the long division process you have been doing all these years, and that the integers q and r are called the quotient and remainder, respectively.

A formal statement of this result is as follows :. Theorem 1. An algorithm is a series of well defined steps which gives a procedure for solving a type of problem. The word algorithm comes from the name of the 9th century Persian mathematician al-Khwarizmi. A lemma is a proven statement used for Muhammad ibn Musa al-Khwarizmi proving 10th Class Ncert Mathematics Book Pdf another statement.

Recall that the HCF of two mathematics ncert book of 10th class rule integers a and b is the largest positive integer d that divides both a and b. Let us see how the algorithm works, through an example. Suppose we need to find the HCF of the integers and We start with the larger integer, that is, We claim that the HCF of and 42 is the divisor at this stage, i. You can easily verify this by listing all the factors of and Why does this method work? It works because of the following result.

Step 3 : Continue the process till the remainder is zero. The divisor at this stage will be the required HCF. Since the divisor at this stage is 4, the HCF of and is 4. Remarks : 1. However, we shall not discuss this aspect. We give some examples of these applications below:.

If a is of the mathematics ncert book of 10th class rule 2q, then a is an even integer. Also, a positive integer can be either even or odd. Solution : Let us start with taking a, mathematics ncert book of 10th class rule a is a positive odd integer.

Example mathematics ncert book of 10th class rule : A sweetseller has kaju barfis and badam barfis. She wants to stack them in such a way that each stack has the same number, and they take up the least area of the tray.

What is the maximum number of barfis that can mathematics ncert book of 10th class rule placed in each stack for this purpose? Solution : This can be done by trial and error. But to do it systematically, we find HCF Then this number will give the maximum number of barfis in each stack and the number of stacks will then be the.

The area of the tray that is used up will be the. Therefore, the sweetseller can make stacks of 10 for both kinds of barfi. An army contingent of members is to march behind an army band of 32 members in a parade. The two groups are to march in the same number of columns.

What is the maximum number of columns in which they can march? Now, let us try and look at natural numbers from the other direction. That is, can any natural number be obtained by multiplying prime numbers? Let us see. Take any collection of prime numbers, say 2, 3, 7, 11 and If we multiply some or all of these numbers, allowing them to repeat as many times as we wish, we can produce a large collection of positive integers In fact, infinitely.

Now, let us suppose your collection of primes includes mathematics ncert book of 10th class rule the possible primes. What is your guess about the size of this collection? Does it contain only a finite number of integers, or infinitely many?

Infact, there are infinitely many primes. So, if we combine all these primes in all possible ways, we will get an infinite collection of numbers, all the primes and all possible products of primes. The question is � can we produce all the composite numbers this way? What do you think? Do you think that there may be a composite number which is not the product of powers of primes?

Before we answer this, let us factorise positive integers, that is, do the opposite of what we have done so far.

We are going to use the factor tree with which you are all familiar. Let us take some large number, say,and factorise it as shown :. Let us try another number, say, Of course, you have to check that and are primes!

Chapter-wise links are provided below from where students can download the PDF of any chapter as per their requirement. Chapter � 9 Some Applications of Trigonometry. Class 6. You can select the chapter from the menu on the left side of the screen. By continuing to use our website, you agree to our Privacy Policy and Cookie Policy. Now, at this time when the schools are unable to conduct the classes in the new academic session on the account of the COVID pandemic, students can themselves start their studies at home with the help of NCERT books available online.
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