8TH STD SAMACHEER KALVI MATHS BOOK

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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 volwarmdilanmi.ga is a well maintained Petrol car that has been less volwarmdilanmi.ga contact me for further details.

A man can complete a work in 4 days, whereas a woman can complete it in only 12 days. If they work together, in how many days, can the work be completed?

Two boys can finish a work in 10 days when they work together. The first boy can do it alone in 15 days. Find in how many days will the second boy do it all by himself? Three men A, B and C can complete a job in 8, 12 and 16 days respectively.

A and B work together for 3 days; then B leaves and C joins. In how many days, can A and C finish the work? A tap A can fill a drum in 10 minutes. A second tap B can fill in 20 minutes. A third tap C can empty in 15 minutes. If initially the drum is empty, find when it will be full if all taps are opened together? A, B and C can do a work in 12, 24 and 8 days respectively. They all work for one day.

Then C leaves the group. In how many days will A and B complete the rest of the work? A tap can fill a tank in 15 minutes. Another tap can empty it in 20 minutes. Initially the tank is empty. If both the taps start functioning, when will the tank become full? Geometry Exercise 2. Choose the correct answer i The Point of concurrency of the medians of a triangle is known as A In centre B circle centre C orthocenter D centroid Ans: C Sol: The side opposite to smallest angle is smallest.

BC is smallest side. Q and R of a triangle PQR are 25 and Is DPQR a right angledtriangle? Moreover PQ is 4cm and PR is 3 cm. Find QR. A 15 m long ladder reached a window 12m high from the ground. On Placing it Against a wall at a distance x m.

Find x. A Painter sets a ladder up to reach the bottom of a Second store window 16 feet above the ground. The base of the ladder is 12 feet from the house. While the Painter mixes the Paint a neighbors Dog bumps the ladder which moves the base 2 Feet farther away from the house.

How far up side Of the house does the ladder reach? Segment of a Circle A chord of a circle divides the circular region Into two Parts. Each Part is called as segment of The circle. Sector of a Circle The circular region enclosed by an arc of a circle and the two radii at its end Points is known as Sector of a circle.

AB is a chord. The chord AB divides the Circle into two Parts. Secant of a Circle A line Passing through a circle and intersecting the circle at two Points is called The secant of the circle Tangent Tangent is a line that touches a circle at exactly one Point, and the Point is Known as Point of contact. Practical Geometry Exercise 4. Draw a rough diagram and mark the measurements.

Flag for inappropriate content. Related titles. Jump to Page. Search inside document. How long is the whole journey? Find the amount. Let the amount be x 3. Find the total time. What is the amount of sale? What is her salary? The marks obtained by Rani in her twelfth standard exams are tabulated Below.

Express these marks as Percentages.

Find the C. Marked Price ii If M. VAT v If the S. I and S. No of carpenters No of hours in a day No of days 12 10 18 15 6 x Step 1: No of machines No of Mobiles No of hours 80 6 1 x 1 25 y 5 Step 1: Area of land No of workers No of days sqm 12 10 x 18 Step1: Time taken by first boy to finish the work 15 days.

A can finish a job in 20 days and B can complete it in 30 days. They work together and finish the job. Geometry Geometry Exercise 2. Which is the shortest side.

PQR is a triangle right angled at P. Find the altitude of an equilateral triangle of side 10 cm. Are the numbers 12, 5 and 13 form a Pythagorean Triplet?

If the sum of the two diameters is mm, find the radius of the circle in cm. Define the circle segment and sector of a circle.

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Sector of a Circle The circular region enclosed by an arc of a circle and the two radii at its end Points is known as Sector of a circle 4. Define the arc of a circle. Define the tangent of a circle and secant of a circle. Now the concentric circle is drawn.

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More From ppats. Madalina Pauna. Santhoshi Priya. Anuja Jayaram. Popular in Mathematics. Santhosh Kumar. JamesBond GuitarPlayer. Otherwise the surds are called unlike surds. For example, i 5 , 4 5 , - 6 5 are like surds. A mixed surd can be converted into a pure surd and a pure surd may or may not be converted into a mixed surd.

Identify whether the following numbers are rational or irrational. Hence, 2 is irrational. When the order of the surds are different, we convert them to the same order and then multiplication or division is carried out. Result n. Among the irrational numbers of same order, the greatest irrational number is the one with the largest radicand.

If the order of the irrational numbers are not the same, we first convert them to the same order. Then, we just compare the radicands. Solution The orders of the given irrational numbers are 3 and 4.

We have to convert each of the irrational number to an irrational number of the same order. Identify which of the following are surds and which are not with reasons.

Simplify the following. Rationalization of surds Rationalization of Surds When the denominator of an expression contains a term with a square root or a number under radical sign, the process of converting into an equivalent expression whose denominator is a rational number is called rationalizing the denominator. If the product of two irrational numbers is rational, then each one is called the rationalizing factor of the other.

Let a and b be integers and x , y be positive integers. Then Remark. Its conjugate is 5 - 3 or the rationalizing factor is 5 - 3. 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. The invention of analytical geometry was the beginning of modern mathematics. In this chapter we learn how to represent points using cartesian coordinate system and derive formula to find distance between two points in terms of their coordinates.

Conversely, a point P on a number line can be specified by a real number x called its coordinate. The two number lines intersect at the -6 zero point of each as shown in the Fig. Generally the Y horizontal number line is called the x-axis and Fig. The x coordinate of a point to the right of the y-axis is positive and to the left of y-axis is negative.

We use the same scale that is, the same unit Y 8 distance on both the axes. To obtain these number, we draw two lines through the point P parallel and hence perpendicular to the axes. We are interested in the coordinates of the. There are two coordinates: The x-coordinate is called the abscissa and the y-coordinate is called the ordinate of the point at hand. These two numbers associated with the point P are called coordinates of P.

They are usually written as x, y , the.

8th Term 3 Mathematics

In an ordered pair a, b , the two elements a and b are listed in a specific order. So the ordered pairs a, b and b, a are not equal, i. The terms point and coordinates of a point are used interchangeably. To find the x-coordinate of a point P: To find the y-coordinate of a point P: The coordinate axes divide the plane into four parts called quadrants, numbered counter-clockwise for reference as shown in Fig.

The signs of the coordinates are shown in parentheses in Fig. Let us now illustrate through an example how to plot a point in Cartesian coordinate system. The intersection of these two lines is the position of 5, 6 in the cartesian plane. That is we count from the origin 5 units along the positive direction of x-axis and move along the positive direction of y-axis through 6 units and mark the corresponding point.

This point is at a distance of 5 units from the y-axis and 6 units from the x-axis. Thus the position of 5, 6 is located in the cartesian plane.

Example 5. The intersection of these two lines is the position of 5, 4 in the Cartesian plane. Thus, the point A 5, 4 is located in the Cartesian plane. The intersection of these two lines is the position of - 4, 3 in the Cartesian plane.

Thus, the point B - 4, 3 is located in the Cartesian plane. The intersection of these two lines is the position of - 2, - 3 in the Cartesian plane. Thus, the point C - 2, - 3 is located in the Cartesian plane. The intersection of these two lines is the position of 3, - 2 in the Cartesian plane Thus, the point D 3, - 2 is located in the Cartesian plane. Observe that if we interchange the abscissa and ordinate of a point, then it may represent a different point in the Cartesian plane.

What can you say about the position of these points? When you join these points, you see that they lie on a line which is parallel to x-axis. Discuss the type of the diagram by joining all the points. Find the coordinates of the points shown in the Fig. Solution Consider the point A. Hence the coordinates of A are 3, 2. For any point on the x-axis its y coordinate is zero.

Write down the abscissa for the following points. Write down the ordinate of the following points. Plot the following points in the coordinate plane. How is the line joining them situated? The ordinates of two points are each - 6. How is the line joining them related with reference to x-axis?

The abscissa of two points is 0. How is the line joining situated? With rectangular axes plot the points O 0, 0 , A 5, 0 , B 5, 4. What are the coordinates of C?

Distance between any Two Points One of the simplest things that can be done with analytical geometry is to calculate the. The distance between two points A and B is usually denoted by AB. Consider the two points A x1, 0 and B x2, 0 on the x-axis. Consider two points A 0, y1 and B 0, y2.

These two points lie on the y axis. Draw AP and BQ perpendicular to x-axis. Draw AP and BQ perpendicular to y-axis. The distance between A and B is equal to the distance between P and Q. The distance between two points on a line parallel to the coordinate axes is the absolute value of the difference between respective coordinates. We shall now find the distance between these two points. AR is drawn perpendicular to BQ. Given the two points x1, y1 and x2, y2 , the distance between these points is given by the formula: Hence Aliter: Let A and B denote the points 6, 0 and 0, 8 and let O be the origin.

The point 6, 0 lies on the x-axis and the point 0, 8 lies on the y-axis. Solution Let the points be A 4, 2 , B 7, 5 and C 9, 7. Hence the points A, B, and C are collinear.

8th Term 3 Mathematics

Hence ABC is a right angled triangle since the square of one side is equal to sum of the squares of the other two sides. Solution Let the points be represented by A a, a , B - a, - a and C - a 3 , a 3. The opposite sides are equal. Hence ABCD is a parallelogram. That is, all the sides are equal. That is, the diagonals are equal. Hence the points A, B, C and D form a square. If the abscissa and the ordinate of P are equal, find the coordinates of P.

Solution Let the point be P x, y. Therefore, the coordinates of P are x, x. Let A and B denote the points 2, 3 and 6, 5. Find also its radius. Solution suppose C represents the point 4, 3. Let P, Q and R denote the points 9, 3 , 7, - 1 and 1, - 1 respectively. Hence the points P, Q, R are on the circle with centre at 4, 3 and its radius is 5 units. Solution Let P deonte the point a, b. Let A and B represent the points 3, - 4 and 8, - 5 respectively. It is known that the circum-center is equidistant from all the vertices of a triangle.

Find the distance between the following pairs of points. Show that the following points form a right angled triangle. Show that the following points taken in order form the vertices of a parallelogram.

Show that the following points taken in order form the vertices of a rhombus. Examine whether the following points taken in order form a square. Examine whether the following points taken in order form a rectangle.

If the distance between two points x, 7 and 1, 15 is 10, find x. Show that 4, 1 is equidistant from the points - 10, 6 and 9, - If the length of the line segment with end points 2, - 6 and 2, y is 4, find y. Find the perimeter of the triangle with vertices i 0, 8 , 6, 0 and origin ; ii 9, 3 , 1, - 3 and origin Find the point on the y-axis equidistant from - 5, 2 and 9, - 2 Hint: A point on the y-axis will have its x coordinate as zero.

Find the radius of the circle whose centre is 3, 2 and passes through - 5, 6. Prove that the points 0, - 5 4, 3 and - 4, - 3 lie on the circle centred at the origin y with radius 5.