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8th Term 3 Mathematics - Free download as PDF File .pdf), Text File .txt) or read online for free. 5. The marks obtained by Rani in her twelfth standard exams are tabulated She get more marks in math which 20 marks greater than English . The cost Price of 16 note books is equal to the selling Price of 12 note books. Samaseer kalvi Maths 9th STd - Free download as PDF File .pdf), Text File .txt) such as a collection of books, a group of students, a list of states in a country. Results 1 - 24 of 32 Maruti Suzuki Maruti Std - for aracer.mobi is a well maintained Petrol car that has been less aracer.mobi contact me for further details.
Solution Here the denominator is 8 - 2 5. Write the rationalizing factor of the following. Rationalize the denominator of the following i 3 5 ii 2 3 3 iii 1 12 iv 2 7 11 3 v 33 5 9. Simplify by rationalizing the denominator. Find the values of the following upto 3 decimal places. Given that 3.
A series of well defined steps which gives a procedure for solving a problem is called an algorithm. In this section we state an important property of integers called the division algorithm. As we know from our earlier classes, when we divide one integer by another non-zero integer, we get an integer quotient and a remainder generally a rational number.
We can rephrase this division, totally in terms of integers, without reference to the division operation.
We observe that this expression is obtained by multiplying 1 by the divisor 5. We refer to this way of writing a division of integers as the division algorithm. In the above statement q or r can be zero.
Using division algorithm, find the quotient and remainder of the following pairs. Points to Remember p , q! In this case, the decimal.
W then the rational number will have a terminating decimal. Otherwise, the. A rational number can be expressed by either a terminating or a non-terminating repeating decimal. An irrational number is a non-terminating and non-recurring decimal, p i. Every real number is either a rational number or an irrational number. If a real number is not a rational number, then it must be an irrational number.
The sum or difference of a rational number and an irrational number is always an irrational number The product or quotient of non-zero rational number and an irrational number is also an irrational number. If a is a positive rational number and n is a positive integer such that n a is an irrational number, then n a is called a surd or a radical.
Division Algorithm Main Targets To represent the number in Scientific Notation. To convert exponential form to logarithmic form and vice-versa. To understand the rules of logarithms. To apply the rules and to use logarithmic table. It is easier to express these numbers in a shorter way called Scientific Notation, thus avoiding the writing of many zeros and transposition errors.
Napier is placed within a short lineage of mathematical thinkers. That is, the very large or very small numbers are expressed as the product of a decimal number 1 a 1 10 and some integral power of Key Concept Scientific Notation. A number N is in scientific notation when it is expressed as the product of a decimal number between 1 and 10 and some integral power of To transform numbers from decimal notation to scientific notation, the laws of exponents form the basis for calculations using powers.
Let m and n be natural numbers and a is a real number. The laws of exponents are given below: For a! Step 1: Move the decimal point so that there is only one non - zero digit to its left. Step 2: Count the number of digits between the old and new decimal point. This gives n, the power of Step 3: If the decimal is shifted to the left, the exponent n is positive. If the decimal is shifted to the right, the exponent n is negative. Example 3.
Solution In integers, the decimal point at the end is usually omitted. The decimal point is to be moved 3 places to the left of its original position. So the power of 10 is 3. Solution 0. The decimal point is to be moved four places to the right of its original position. So the power of 10 is 4. To convert scientific notation to integers we have to follow these steps.
Write the decimal number. Move the decimal point the number of places specified by the power of ten: Add zeros if necessary.
Rewrite the number in decimal form. Exercise 3. Represent the following numbers in the scientific notation. Write the following numbers in decimal form. Represent the following numbers in scientific notation. They were designed to transform multiplicative processes into additive ones. Before the advent of calculators, logarithms had great use in multiplying and dividing numbers with many digits since adding exponents was less work than multiplying numbers.
Now they are important in nuclear work because many laws governing physical behavior are in exponential form. Examples are radioactive decay, gamma absorption, and reactor power changes on a stable period.
To introduce the notation of logarithm, we shall first introduce the exponential notation for real numbers. We have already introduced the notation a x , where x is an integer. We knowpthat a n is a positive number whose nth power is equal to a. Now we can see how to define a q , where p is an integer and q is a positive integer. Notice that p 1. We will not show how a x may be defined for irrational x because the definition of a x requires some advanced topics in mathematics.
Key Concept Logarithmic Notation. Let a be a positive number other than 1 and let x be a real number positive, negative, or zero. In both the forms, the base is same. The Rules of Logarithms 1. Product Rule: The logarithm of the product of two positive numbers is equal to sum of their logarithms of the same base. Quotient Rule: The logarithm of the quotient of two positive numbers is equal to the logarithm of the numerator minus the logarithm of the denominator to the same base.
The logarithm of a number in exponential form is equal to the logarithm of the number multiplied by its exponent. Change of Base Rule: If M, a and b are positive numbers and a! State whether each of the following statements is true or false. Solve the equation in each of the following. Find the value in each of the following in terms of x , y and z. Logarithms to the base 10 are called common logarithms. Therefore, in the discussion which follows, no base designation is used, i.
Consider the following table. So, log N is an integer if N is an integral power of What about logarithm of 3. For example, 3.
Notice that logarithm of a number between 1 and 10 is a number between 0 and 1 ; logarithm of a number between 10 and is a number between 1 and 2 and so on. Every logarithm consists of an integral part called the characteristic and a fractional part called the mantissa. For example, log 3. It is convenient to keep the mantissa positive even though the logarithm is negative.
Scientific notation provides a convenient method for determining the characteristic.
Thus, the power of 10 determines the characteristic of logarithm. The negative sign of the characteristic is written above the characteristics as 1, 2, etc. For example, the characteristic of 0. Hence, i log Note that the mantissas of logarithms of all the numbers consisting of same digits in same order but differing only in the position of decimal point are the same.
The mantissas are given correct to four places of decimals. A logarithmic table consists of three parts. These columns are marked with serial numbers 1 to 9. We shall explain how to find the mantissa of a given number in the following example.
Suppose, the given number is Now Therefore, the characteristic is 1. The row in front of the number 4. The number is 0. Next the mean difference corresponding to 5 is 0. Thus the required mantissa is 0. Hence, log This table gives the value of the antilogarithm of a number correct to four places of decimal.
For finding antilogarithm, we take into consideration only the mantissa. The characteristic is used only to determine the number of digits in the integral part or the number of zeros immediately after the decimal point. The method of using the table of antilogarithms is the same as that of the table of logarithms discussed above. Since the logarithmic table given at the end of this text book can be applied only to four digit number, in this section we approximated all logarithmic calculations to four digits.
From the table, log 4. So, the number contains two digits in its integral part. Mantissa is 0. From the table, antilog 0. So, the number contains one zero immediately following the decimal point. Taking logarithm on both sides, we get 0. Write each of the following in scientific notation: Write the characteristic of each of the following i log iv log 0. The mantissa of log is 0. Find the value of the following. Using logarithmic table find the value of the following. Using antilogarithmic table find the value of the following.
Points to Remember A number N is in scientific notation when it is expressed as the product of a decimal number 1 a 1 10 and some integral power of Product rule: To use Remainder Theorem. To use Factor Theorem. To use algebraic identities. To factorize a polynomial. To solve linear equations in two variables. To solve linear inequation in one variable.
DIophAntus to A. Diophantus was a Hellenistic mathematician who lived circa AD, but the uncertainty of this date is so great that it may be off by more than a century. He is known for having written Arithmetica, a treatise that was originally thirteen books but of which only the first six have survived.
Arithmetica has very little in common with traditional Greek mathematics since it is divorced from geometric methods, and it is different from Babylonian mathematics in that Diophantus is concerned primarily with exact solutions, both determinate and indeterminate, instead of simple approximations. The language of algebra is a wonderful instrument for expressing shortly, perspicuously, suggestively and the exceedingly complicated relations in which abstract things stand to one another.
Algebra has been developed over a period of years. But, only by the middle of the 17th Century the representation of elementary algebraic problems and relations looked much as it is today. By the early decades of the twentieth century, algebra had evolved into the study of axiomatic systems.
This axiomatic approach soon came to be called modern or abstract algebra.
Important new results have been discovered, and the subject has found applications in all branches of mathematics and in many of the sciences as well. A constant, we mean an algebraic expression that contains no variables at all.
If numbers are substituted for the variables in an algebraic expression, the resulting number is called the value of the expression for these values of variables.
If an algebraic expression consists of part connected by plus or minus signs, it is called an algebraic sum. Each part, together with the sign preceding it is called a term. For instance, in the term - 4xz , the coefficient of z 2 y y is - 4x , whereas the coefficient of xz is 4. A coefficient such as 4, which involves no y 2 2 variables, is called a numerical coefficient. Terms such as 5x y and - 12x y , which differ only in their numerical coefficients, are called like terms or similar terms.
An algebraic expression such as 4rr can be considered as an algebraic expression consisting of just one term. Such a one-termed expression is called a monomial. An algebraic expression with two terms is called a binomial, and an algebraic expression with three terms is called a trinomial.
An algebraic expression with two or more terms is called a multinomial. A term such as - 1 which 2 2 contains no variables, is called a constant term of the polynomial. The numerical coefficients of the terms in a polynomial are called the coefficients of the polynomial.
The coefficients of the polynomial above are 3, 2 and - 1. In adding exponents, one should regard a variable with no exponent as being power one.
The constant term is always regarded as having degree zero. The degree of the highest degree term that appears with nonzero coefficients in a polynomial is called the degree of the polynomial. For instance, the polynomial considered above has degree 8. Although the constant monomial 0 is regarded as a polynomial, this particular polynomial is not assigned a degree.
Key Concept Polynomial in One Variable. Here n is the degree of the polynomial and a1, a2, g, an - 1, an are the coefficients of x, x , gx 2 n The three terms of the polynomial are 5x2, 3x and - 1. Binomial Polynomials which have only two terms are called binomials. A binomial is the sum of two monomials of different degrees. A trinomial is the sum of three monomials of different degrees.
A polynomial is a monomial or the sum of two or more monomials. Constant polynomial A polynomial of degree zero is called a constant polynomial. General form: Linear polynomial A polynomial of degree one is called a linear polynomial.
Quadratic polynomial A polynomial of degree two is called a quadratic polynomial. Cubic polynomial A polynomial of degree three is called a cubic polynomial. Example 4. State whether the following expressions are polynomials in one variable or not. Classify the following polynomials based on their degree.
Give one example of a binomial of degree 27 and monomial of degree 49 and trinomial of degree If the value of a polynomial is zero for some value of the variable then that value is known as zero of the polynomial. Key Concept Zeros of Polynomial. Number of zeros of a polynomial the degree of the polynomial. Carl Friedrich Gauss had proven in his doctoral thesis of that the polynomial equations of any degree n must have exactly n solutions in a certain very specific sense.
This result was so important that it became known as the fundamental theorem of algebra.
The exact sense in which that theorem is true is the subject of the other part of the story of algebraic numbers. Hence zeros of a polynomial are the roots of the corresponding polynomial equation.
Key Concept Root of a Polynomial Equation. Exercise 4. Verify Whether the following are roots of the polynomial equations indicated against them. Also find the remainder. Find the value of a. An identity is an equality that remains true regardless of the values of any variables that appear within it. We have learnt the following identities in class VIII. Using these identities let us solve some problems and extend the identities to trinomials and third degree expansions.
Using these identities of 4. Using algebraic identities find the coefficients of x2 term, x term and constant term. We have seen how the distributive property may be used to expand a product of algebraic expressions into sum or difference of expressions. In both the terms, ab and ac a is the common factor. Factorize the following expressions: In this section. Split this product into two factors such that their sum is equal to the coefficient of x.
The terms are grouped into two pairs and factorize. The constant term is 2. The factors of 2 are 1, 1, 2 and 2. The constant term is 3. The factors of 3 are 1, 1,3 and 3. Factorize each of the following. Let us consider a pair of linear equations in two variables x and y. The substitution method, the elimination method and the cross-multiplication method are some of the methods commonly used to solve the system of equations. In this chapter we consider only the substitution method to solve the linear equations in two variables.
It is then substituted in the other equation and solved. Find the cost of each. The cost of three mathematics books is the same as that of four science books. Find the cost of each book. From Dharmapuri bus stand if we download 2 tickets to Palacode and 3 tickets to Karimangalam the total cost is Rs 32, but if we download 3 tickets to Palacode and one ticket to Karimangalam the total cost is Rs Find the fares from Dharmapuri to Palacode and to Karimangalam.
Find the numbers. The number formed by reversing the digits is 9 less than the original number. Find the number. Solution Let the tens digit be x and the units digit be y.
There is only one such value for x in a linear equation in one variable. We represent those real numbers in the number line.
Unshaded circle indicates that point is not included in the solution set. The real numbers less than or equal to 3 are solutions of given inequation.
Shaded circle indicates that point is included in the solution set. Solve the following equations by substitution method.
A number consists of two digits whose sum is 9. The number formed by reversing the digits exceeds twice the original number by Find the original number. Kavi and Kural each had a number of apples. Kavi said to Kural If you give me 4 of your apples, my number will be thrice yours.
Kural replied If you give me 26, my number will be twice yours. How many did each have with them?. Solve the following inequations. Remainder Theorem: Factor Theorem: Main Targets To understand Cartesian coordinate system To identify abscissa, ordinate and coordinates of a point To plot the points on the plane To find the distance between two points Descartes D e s c a r t e s has been called the father of modern philosophy, perhaps because he attempted to build a new system of thought from the ground up, emphasized the use of logic and scientific method, and was profoundly affected in his outlook by the new physics and astronomy.
Descartes went far past Fermat in the use of symbols, in Arithmetizing analytic geometry, in extending it to equations of higher degree. The fixing of a point position in the plane by assigning two numbers - coordinates giving its distance from two lines perpendicular to each other, was entirely Descartes invention.
Coordinate Geometry or Analytical Geometry is a system of geometry where the position of points on the plane is described using an ordered pair of numbers called coordinates. He proposed further that curves and lines could be described by equations using this technique, thus being the first to link algebra and geometry. In honour of his work, the coordinates of a point are often referred to as its Cartesian coordinates, and the coordinate plane as the Cartesian Coordinate Plane.
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No Items M. Exercis 1. Find the Amount and Compound Interest in the following cases: What interest did Sangeetha Pay to Alex if the interest is compounded annually? Find i the amount standing to her credit at the end of second year and ii the interest for the third year.
What amount will he Pay at the end of 2 years and 4 months to clear the loan? The difference between S. I and C. The difference between C. Find the sum of money lent. Who Pays more interest and by how much? Find the value of n. The number of students enrolled in a school is What will be the worth of the car after three years? Find the value after one year.
In a Laboratory, the count of bacteria in a certain experiment was increasing at the rate of 2. Find the bacteria at the end of 2 hours if the count was initially 5,06, From a village People started migrating to nearby cities due to unemployment Problem.
The Population of the village two years ago was 6, Find the Present Population. If the Present Population of the village is 11,, what was the Population two years ago? Poorani wants to download it in 5 instalments. Simple Interest, find the E.
Find the E. Venkat wants to download it in 10 installments. If the company offers it for a S. Twelve carpenters working 10 hours a day complete a furniture work in 18 days. No of carpenters No of hours in a day No of days 12 10 18 15 6 x. Step 1: Consider carpenter and days The multiplying factor is Step 2: Consider no of hour Per day and no of days.
Eighty machines can Produce 4, identical mobiles in 6 hours. How many mobiles can one machine Produce in one hour? How many mobiles would 25 machines Produce in 5 hours?
No of machines No of Mobiles No of hours 80 6 1 x 1 25 y 5. If 14 compositors can compose 70 Pages of a book in 5 hours, how many compositors will compose Pages of this book in 10 hours? If 2, sq.
Area of land No of workers No of days sqm 12 10 x Working 4 hours daily, Swati can embroid 5 sarees in 18 days. How many days will it take for her to embroid 10 sarees working 6 hours daily?
A man can complete a work in 4 days, whereas a woman can complete it in only 12 days.