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Let Sn be the sum of first n terms of the arithmetic sequence. This is nothing but the sum of the arithmetic sequence 1, 2, 3, g, n - 1. Now, let us find the sum of the first n positive integers.
We shall use a small trick to find the above sum. There are n terms in each of 1 and 2. We merely added corresponding terms from 1 and 2. The above method was first used by the famous German mathematician Carl Fredrick Gauss, known as Prince of Mathematics, to find the sum of positive integers upto This problem was given to him by his school teacher when he was just five years old.
When you go to higher studies in mathematics, you will learn other methods to arrive at the above formula. Carl Fredrick Gauss Now, let us go back to summing first n terms of a general arithmetic sequence. Find the arithmetic series. That is, the series is Example 2. Here n, being the number of terms needed, cannot be negative. Thus, 26 terms are needed to get the sum - Solution The three digit natural numbers divisible by 8 are , , , g , Let Sn denote their sum.
Now, the sequence , , , g , forms an A. The least measurement in the sequence is 85c. The greatest measurement is c. Find the number of sides in the given polygon. Solution Let n denote the number of sides of the polygon. Now, the measures of interior angles form an arithmetic sequence. From 1 , we have Exercise 2. Find the sum of the first i 75 positive integers ii natural numbers. Find the sum of the first 30 terms of an A.
Find the Sn for the following arithmetic series described. In an arithmetic series, the sum of first 11 terms is 44 and that of the next 11 terms is In the arithmetic sequence 60, 56, 52, 48,g , starting from the first term, how many terms are needed so that their sum is ?
Find the sum of all 3 digit natural numbers, which are divisible by 9. Find the sum of first 20 terms of the arithmetic series in which 3rd term is 7 and 7th term is 2 more than three times its 3rd term. Find the sum of all natural numbers between and which are divisible by Find the sum of all numbers between and which are not divisible by 5. A construction company will be penalised each day for delay in construction of a bridge.
Find the maximum number of days by which the completion of work can be delayed. Calculate the interest at the end of each year. Do these interest amounts form an A. If so, find the total interest at the end of 30 years. The sum of first n terms of a certain series is given as 3n - 2n. Show that the series is an arithmetic series. If a clock strikes once at 1 oclock, twice at 2 oclock and so on, how many times will it strike in a day? The ratio of the sums of first m and first n terms of an arithmetic series is m: Mathematics 2 2 2.
A gardener plans to construct a trapezoidal shaped structure in his garden. The longer side of trapezoid needs to start with a row of 97 bricks. Each row must be decreased by 2 bricks on each end and the construction should stop at 25th row. How many bricks does he need to buy? We want to find the sum of the first n terms of this sequence.
The sum of the first n terms of a geometric series is given by n n a r - 1 a 1 - r , if r! Actually, if - 1 1 r 1 1 , then the following formula holds: Let n be the number of terms required to get the sum.
Find the common ratio and the sum of the first 14 terms. Solution Let a be the first term and r be the common ratio of the given G.
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Y Y 1-r r The sum of the first two terms is 8 and the sum of the last two terms is Find the series. How many saplings are needed to complete the work? Solution The number of saplings to be planted for each of the 25 streets in the town forms a G.
Let Sn be the total number of saplings needed. Thus, the number of saplings to be needed is 2 - 1. How many consecutive terms starting from the first term of the series The second term of a geometric series is 3 and the common ratio is 4. Find the sum 5 of first 23 consecutive terms in the given geometric series.
A geometric series consists of four terms and has a positive common ratio. The sum of the first two terms is 9 and sum of the last two terms is Suppose that five people are ill during the first week of an epidemic, and each sick person spreads the contagious disease to four other people by the end of the second week. By the end of 15th week, how many people will be affected by the epidemic? A gardener wanted to reward a boy for his good deeds by giving some mangoes. He gave the boy two choices.
He could either have mangoes at once or he could get 1 mango on the first day, 2 on the second day, 4 on the third day, 8 mangoes on the fourth day and so on for ten days. Which option should the boy choose to get the maximum number of mangoes?
A geometric series consists of even number of terms. The sum of all terms is 3 times the sum of odd terms. Find the common ratio. Y x-1 Note that the left hand side of the above equation is a special polynomial in x of degree n - 1. This formula will be useful in finding the sum of some series. We have already used the symbol R for summation. Let us list out some examples of finite series represented by sigma notation. This can also be obtained 2 using A. Hence, using sigma notation we write it as Let us derive the formulae for i Proof: This is an A.
Extending this pattern to n terms, we get 3 3 3 3. Find the sum of the following series. Find the total area of 12 squares whose sides are 12cm, 13cm, g, 23cm. Find the total volume of 15 cubes whose edges are 16 cm, 17 cm, 18 cm, g , 30 cm respectively. A A sequence is a real valued function defined on N. B Every function represents a sequence. C A sequence may have infinitely many terms.
D A sequence may have a finite number of terms. C neither A. P nor G. C a constant sequence D neither A. P Sequences and series of real numbers The sequence 3, 3, 3,g is A an A. P D both A. If the product of the first four consecutive terms of a G.
P is and if the common ratio is 4 and the first term is positive, then its 3rd term is A 8 B 1 16 C 1 32 D If the n term of an A. A sequence of real numbers is an arrangement or a list of real numbers in a specific order. The formula for the general term of an A. N where r is a constant.
The formula for the general term of a G. An expression of addition of terms of a sequence is called a series. If the sum consists only finite number of terms, then it is called a finite series.
If the sum consists of infinite number of terms of a sequence, then it is called an infinite series. Do you know? If M is a prime, then it is called a Mersenne prime.
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Interestingly, if 2 p - 1 is prime, then p is prime. The largest known prime number ,, - 1 is a Mersenne prime.
Algebra is an important and a very old branch of mathematics which deals with solving algebraic equations. In third century, the Greek mathematician Diophantus wrote a book Arithmetic which contained a large number of practical problems. In the sixth and seventh centuries, Indian mathematicians like Aryabhatta and Brahmagupta have worked on linear equations and quadratic equations and developed general methods of solving them.
The next major development in algebra took place in ninth century by Arab mathematicians. In particular, Al-Khwarizmis book entitled Compendium on calculation by completion and balancing was an important milestone.
There he used the word aljabra - which was latinized into algebra - translates as competition or restoration. In the 13th century, Leonardo Fibonaccis books on algebra was important and influential. Other highly influential works on algebra were those of the Italian mathematician Luca Pacioli , and of the English mathematician Robert Recorde In later centuries Algebra blossomed into more abstract and in 19th century British mathematicians took the lead in this effort.
Peacock Britain, was the founder of axiomatic thinking in arithmetic and algebra. For this reason he is sometimes called the Euclid of Algebra. DeMorgan Britain, extended Peacocks work to consider operations defined on abstract symbols. In this chapter, we shall focus on learning techniques of solving linear system of equations and quadratic equations.
He presented the first systematic solution of linear and quadratic equations. He is considered the founder of algebra.
His work on arithmetic was responsible for introducing the Arabic numerals based on the Hindu-Arabic numeral system developed in Indian Mathematics, to the Western world. Conversely, every solution x, y of the equation is a point on this straight line. A set of finite number of linear equations in two unknowns x and y that are to be treated together, is called a system of linear equations in x and y.
Such a system of equations is also called simultaneous equations. In this section, we shall discuss only a pair of linear equations in two variables.
You will learn this in higher classes. Geometrically the following situations occur. The two straight lines represented by 1 and 2 i may intersect at exactly one point ii may not intersect at any point iii may coincide. Algebra If i happens, then the intersecting point gives the unique solution of the system. If ii happens, then the system does not have a solution. If iii happens, then every point on the line corresponds to a solution to the system.
Thus, the system will have infinitely many solutions in this case. Now, we will solve a system of linear equations in two unknowns using the following algebraic methods i the method of elimination ii the method of cross multiplication. The elimination of one unknown can be achieved in the following ways. Then, eliminate by addition if the resulting coefficients have unlike signs and by subtraction if they have like signs.
Example 3. So, we can eliminate y easily. Obtaining equation 3 in only one variable is an important step in finding the solution. We obtained equation 3 in one variable x by eliminating the variable y. So this method of solving a system by eliminating one of the variables first, is called method of elimination. Find the cost of each pencil and each eraser.
Solution Let x denote the cost of a pencil in rupees and y denote the cost of an eraser in rupees. Note It is always better to check that the obtained values satisfy the both equations. In Example 3. Thus, first we shall do some manipulations so that coefficients of either x or y are equal except for sign. Then we do the elimination. However, note that the coefficient of x in one equation is equal to the coefficient of y in the other equation.
In such a case, we add and subtract the two equations to get a new system of very simple equations having the same solution. Thus, the required solution is 2, 3. Observe that the given system is not linear because of the occurrence of xy term. So, 0, 0 is a solution for the system and any other solution would have both x! Thus, we consider the case where x! Let us apply the elimination method for equating the coefficients of y.
Case i a1 b2 - a2 b1! That is, a1 b! In this case, the pair of linear equations has a unique solution. That is, if a2! If c1! Hence, if 74 10th Std. Note Now, we summarise the above discussion.
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There is another method called the cross multiplication method, which simplifies the procedure. Now, let us describe this method and see how it works. The arrows between the two numbers indicate that they are multiplied, the second product upward arrow is to be subtracted from the first product downward arrow. Method of solving a linear system of equations by the above form is called the cross multiplication method.
Hereafter, we shall mostly restrict ourselves to the system of linear equations having unique solution and find the solution by the method of cross multiplication.
Hence, the solution is 0, 5. Thus x is only a notation and it is 0 0 17 - 17 not division by zero. It is always true that division by zero is not defined. If the digits are reversed, the new number is 27 more than the given number.
Find the number. Solution Let x denote the digit in the tenth place and y denote the digit in unit place.. Find the fraction. Hence, the system has a unique solution. Hence, the fraction is Find the number of days taken by one man alone to complete the work and also one boy alone to complete the work. Solution Let x denote the number of days needed for one man to finish the work and y denote the number of days needed for one boy to finish the work.
Clearly, x! So, one man can complete 1 part of the work in one day and one boy can complete x 1 part of the work in one day. Writing the coefficients of 3 and 4 for the cross multiplication, we have a 6 4 Thus, we have - 1 20 - 1 28 b 4 3 1 6 4.
Hence, one man can finish the work individually in days and one boy can finish the work individually in days.
Exercise 3. Solve the following systems of equations using cross multiplication method. Formulate the following problems as a pair of equations, and hence find their solutions: If 4 times the smaller number exceeds the greater by 5, find the numbers. The number formed by reversing the digits is 18 less than the given number. Find the given number. What is the total cost of 2 chairs and 3 tables? If the length is reduced by 1 cm and the 2 breadth increased by 2 cm , then the area increases by 33 cm.
Find the area of the rectangle. Find the distance covered by the train. Real constants are polynomials of degree zero. A polynomial may not have any zero in real numbers at all. Geometrically a zero of any polynomial is nothing but the x-coordinate of the point of intersection of the graph of the polynomial and the x-axis if they intersect. Since we can choose any non zero a, there are infinitely many quadratic polynomials with zeros a and b.
Solution Let a and b be the zeros of a quadratic polynomial. It is a polynomial with zeros zero real number. Find a quadratic polynomial each with the given numbers as the sum and product of its zeros respectively.
We know that when 29 is divided by 7 we get, 4 as the quotient and 1 as the remainder. This is called the Division Algorithm. Division algorithm: Now, we have the following results.
An elegant way of dividing a polynomial by a linear polynomial was introduced by Paolo Ruffin in His method is known as synthetic division.
It facilitates the division of a polynomial by a linear polynomial with the help of the coefficients involved. Paolo Ruffin , Italy. Let us explain the method of synthetic division with an example. We shall find the quotient s x and the remainder r, by proceeding as follows. Insert 0 for missing terms. Step 2 Find out the zero of the divisor. Step 3 Put 0 for the first entry in the 2nd row.
The zero of the divisor is 3. Find the values of a and b, also the remainder. Thus, the quotient is 2. Find the quotient and remainder using synthetic division. Mathematics 4 2 3 2. In this section, let us learn, how to factorize the cubic polynomial using synthetic division. If we identify one linear factor of cubic polynomial p x , then using synthetic division we get the quadratic factor of p x.
Further if possible one can factorize the quadratic factor into two linear factors. Hence the method of synthetic division helps us to factorize a cubic polynomial into linear factors if it can be factorized. Therefore, we have to search for different values of x by trial and error method.
Thus, x - 2 is a factor of p x. To find the other factors, let us use the synthetic division. The other factor is x - x - Factorize each of the following polynomials.
Consider the simple expressions i a ,a ,a ,a 4 3 5 6 2 3. In i , note that a, a , a are the divisors of all these expressions. Out of them, a is 3 4 3 5 6 the divisor with highest power. Therefore a is the GCD of the expressions a , a , a , a.
In ii , similarly, one can easily see that ab is the GCD of a b , ab c , a b c. If the expressions have numerical coefficients, find their greatest common divisor, and prefix it as a coefficient to the greatest common divisor of the algebraic expressions.
Let us consider a few more examples to understand the greatest common divisor. Mathematics 4 3 4 5 2 2 7. Examples 3. Now let us take the given expressions 15x y z and 12x y z. Here the common divisors of the given expressions are 3, x , y and z. Similar technique works with polynomials when they have GCD. We want to find GCD of f x and g x.
However, the following method gives a systematic way of finding GCD. If the remainder r1 x is 0 , then r x is the required GCD. Step 3 If r1 x is non-zero, then continue the process until we get zero as remainder. Euclids division algorithm is based on the principle that GCD of two numbers does not change if the small number is subtracted from the larger number.
The two original expressions have no simple factors constants. Thus their GCD can have none. Mathematics 4 3 2 4 3 2. Common factor of 3x and 2x is x. Thus, 1. Find the GCD of the following pairs of polynomials using division algorithm. For example, consider the 4 3 6 simple expressions a , a , a. Now, a , a , a , g are common multiples of a , a and a.
Of all the common multiples, the least common multiple is a 4 3 6 6 3 7 3 4 5 2 7 Hence LCM of a , a , a is a. We shall consider some more examples of finding LCM. Let us first find the factors for each of the given expressions. In the same way, we have the following result: Let us justify this result with an example. Find their LCM. Solution 6. Find the LCM of each pair of the following polynomials.
A rational number is defined as a quotient m , of two integers m and n! Let us consider some examples. Multiply the following and write your answer in lowest terms.
Simplify the following as a quotient of two polynomials in the simplest form. Let a! R be a non negative real number. The positive square root of a is denoted by 2 a or a. In the same way, the square root of any expression or a polynomial is an expression whose square is equal to the given expression.
In general, the following two methods are very familiar to find the square root of a given polynomial i factorization method ii division method.
In this section, let us learn the factorization method through some examples for both the expressions and polynomials when they are factorable. Find the square root of the following: Also division method is a convenient one when the polynomials are of higher degrees.
One can find the square root of a polynomial the same way of finding the square root of a positive integer. Let us explain this method with the following examples. Therefore, it is a matter of finding the suitable a, b and c. Algebra 99 4 3 2 2.
Solution Given polynomial is already in descending powers of x. Solution Arrange the polynomial in descending power of x. Mathematics 4 3 2 2 3 4. Find the square root of the following polynomials by division method. Find the values of a and b if the following polynomials are perfect squares.
Greek mathematician Euclid developed a geometrical approach for finding out lengths which in our present day terminology, are solutions of quadratic equations. Solving quadratic equations in general form is often credited to ancient Indian Mathematicians. In fact, Brahma Gupta A. Later Sridhar Acharya A. D derived a formula, now known as the quadratic formula, as quoted by Bhaskara II for solving a quadratic equation by the method of completing the square.
In this section, we will learn solving quadratic equations, by various methods. We shall also see some applications of quadratic equations. Algebra 2 2 2. Given a product, if any factor is zero, then the whole product is zero. Conversely, if a product is equal to zero, then some factor of that product must be zero, and any factor which contains an unknown may be equal to zero. Thus, in solving a quadratic equation, we find the values of x which make each of the factors zero.
That is, we may equate each factor to zero and solve for the unknown.
But when we simplify the equation, it will reduce to a quadratic equation. Mathematics 2. To solve radical equation like the above, we rely on the squaring property: Unfortunately, this squaring property does not guarantee that all solutions of the new equation are solutions of the original equation.
Such a solution is called an extraneous solution. Thus, the above example shows that when squaring on both sides of a radical equation, the solution of the final equation must be checked to determine whether they are solutions of the original equation or not. This is necessary because no solution of the original equation will be lost by squaring but certain values may be introduced which are roots of the new equation but not of the original equation.
Such an addition is usually known as completing the square. In this section, we shall find the solution of a quadratic equation by the method of completing the square through the following steps. If the coefficient of x2 is 1, go to step 2. If not, divide both sides of the equation by the coefficient of x2. Get all the terms with variable on one side of equation. Find half the coefficient of x and square it. Add this number to both sides of the equation.
To solve the equation, use the square root property: For a! Now, let us solve some quadratic equations using quadratic formula. Solution Note that the given equation is not in the standard form of a quadratic equation.
Solve the following quadratic equations using quadratic formula.
First we shall form an equation translating the given statement and then solve it. Finally, we choose the solution that is relevant to the given problem. The base of a triangle is 4 cm longer than its altitude. If the area of the triangle is 48 sq. Solution Let the altitude of the triangle be x cm. Thus, the altitude of the triangle is 8 cm and the base of the triangle is 12 cm. Find its usual speed. Speed Let T1 and T2 be the time taken in hours by the car to cover the given distance in scheduled time and decreased time as the speed is increased respectively.
The sum of a number and its reciprocal is The square of the smaller number is four times the larger number. Find the numbers. A farmer wishes to start a sq. Since he has only 30 m barbed wire, he fences the sides of the rectangular garden letting his house compound wall act as the fourth side fence.
Find the dimension of the garden. A rectangular field is 20 m long and 14 m wide. There is a path of equal width all around it having an area of sq. Find the width of the path on the outside. A train covers a distance of 90 km at a uniform speed. Find the original speed of the train. It goes 30 km upstream and return downstream to the original point in 4 hrs 30 minutes. My kids are in grade 6 and 8. Matrices 56 3.
We have compiled the solution of CBSE Class 6 English, Hindi, Math, Sanskrit, and Science Syllabus to ensure students to understand and develop key concepts on various topics, which will also help them to grasp easily. As part of the curriculum revision the CCE pattern covering Formative and Summative Assessment has already been suggested for classes 1 to 8.
Physics is one of the most important branches of science that concerned with the nature and properties of matter and energy.
Download scert hindi guide class 9 Read Book Online online right now by as soon as colleague below. Sudhish, SCERT, Chhattisgarh Genesis: The quest for innovations in the public sector has been on-going for quite sometime, but with the need for the world to achieve the commitments made by its leaders in the Millennium Declaration, the World Summit and many other global and regional conferences, it has attained a NCERT Solutions Complete Chapter wise and detailed.
Describing Motion 2. Reading is a hobby to open the data windows. The intuitive notion that light travels in a straight line seems to contradict what we have learnt in Chapter 8, that light is an electromagnetic wave of wavelength belonging to the visible part of the spectrum.
Click on the following images corresponding to your subject to download textbook Clipping Magic The demand of a high-quality Clipping Magic is pretty high lately. Class 8 part English hand book for class x scert new 1. Teachers can best use these resources to make the classroom live and interesting.
As part of the curriculum revision the CCE Continuous Comprehensive Evaluation pattern covering Formative and Summative Assessment has already been suggested for classes 1 to 8.
Teachers' Text class2. The students from cbse will need to have check out the book of Kerala Padavali Class 10 Answers In Textbooks written by stats. Besides, it can provide the inspiration and spirit to face this life. Most of the question paper is covered in the books but I would also like for CBSE to raise the level of books one notch high. Sateesh wrote all numbers between and which leave a remainder 3 when divided by 8.
Computers are seen everywhere around us, in all spheres of life. Meritnation offers you an extensive list of NCERT textbooks in downloadable format, making it easier for students to study for CBSE exams, and have efficient homework sessions. The massive training programmes are mandatory for teachers as they are usually linked to curriculum and textbook revision process. In this chapter we will cover: 1.
NCERT textbooks come in very handy for the students. World of sounds 4. Our colorful world 5. The massive teacher training programmes are conducted during the vacation time immediately before new curriculum and textbooks are introduced.
We have tried to organise As students of Class X, the highest class in a high school you are at the portals of the higher secondary education.
But it has a problem only a few papers are English others all are Malayalam. This scene is real to digital marketing, graphic designing or simply others that require the practitioners to look professional, reliable and careful about details. Please select the book to view. Determinants 4. This will be continued for the classes 9 and In some industries, perfection is not compulsory, but highly needed.
No agency or individual may make electronic or print copies of these books and redistribute them in any form whatsoever. As virtual education is an emerging concept. A As a part of an assignment Smitha wrote all multiplies of 8 between and Notification No. This is especially true in developing countries where it Class 8 Social important questions in PDF format.
Class 12 part I love this Maths Blog. List of textbooks for Classes I-V. The speed of light in vacuum is the highest speed attainable in nature. Previously, the textbooks in use were drawn from Department of Education, Gujarat. More than a million users visit aglasem. From time to time to SCERT puts out notification in newspapers for contacting teachers who are interested in taking part in curriculum and textbook preparation list of bio data thus received is maintained in SCERT and suitable teacher selected from the list for curriculum preparation workshops.
This year, ie. Class 10 part Scert Textbooks For Class 10 downloads at Ebookmarket. The Complete list of Std. Un-aided Higher Secondary school principals are requested to pay the price of text books to the C-apt by way of In September , NCERT constituted expert groups to review and recommend changes in these textbooks. Remaining few tittles will be uploaded as soon as possible. Ncert stands for National Council of Educational Research And Training which is publishing the standard textbooks for cbse.
Get these very important questions of Class 8th Social subject right away!! SCERT has developed Gujarati textbooks for class 8, 9 and 10 for the first time in the history of school education in Kerala. Students; Teachers; Educators; Parents; Programmes. The KBPE is the official governing authority which is responsible for designing the curriculum, syllabus, and textbooks for various classes. Class 6 part The curriculum committee decided to introduce state own textbooks in Gujarati. I am 8th standard student who need to improve his maths abilities so I request you please translate Malayalam papers to English.
Not much culture stuff given. These handbooks have been designed by experts from the relevant fields. This 'Chemistry' text has accordingly, been designed to cater to the demands of the next academic level.
Look no further, your search now ends here! SCERT is responsible for preparing the curriculum, prescribing syllabi, course of study, academic calendar for these Courses.Solving quadratic equations by quadratic formula.
As of August , six of the problems remain unsolved. Understanding the value and difficulty level of these exams all students do their best to score well in these exams. The sum of three terms of a geometric sequence is 39 and their product is 1.
To determine the sum of some finite series. Similar triangles theorems without proof.
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