1 Unit -I : Number System Chapter - : Real Numbers TOPIC- Euclid s Division Lemma and Fundamental Theorem of rithmetic lgorithm : n algorithm is a series of well defined steps which gives a procedure for solving a type of problems. Lemma : lemma is a proven statement used for proving another statement. Euclid s Division Lemma : For given positive integers a and b, there exist unique integers q and r, satisfying a bq + r, 0 r < b. Here a Dividend, b Divisor, q Quotient, r Remainder i.e., Dividend (Divisor Quotient) + Remainder For example, Quotient Divisor ) 5 Remainder ccording to the above formula, 5 ( ) + We state Euclid s division algorithm for positive integers only but it can be extended for all integers except zero, b 0. When a and b are two positive integers such that a bq + r, 0 r < b, then HCF (a, b) HCF (b, r). The steps to find the HCF of two positive integers by Euclid s division algorithm are given below : (i) Let two integers be a and b such that a > b. (ii) Take greater number a as dividend and the number b as divisor. (iii) Now find whole numbers q and r as quotient and remainder respectively. a bq + r, 0 r < b. (iv) If r 0, b is the HCF of a and b. If r 0, then take r as divisor and b as dividend. (v) Repeat step (iii), till the remainder is zero, the divisor thus obtained at last stage is the required HCF. The Fundamental Theorem of rithmetic Every composite number can be expressed as the product of powers of primes and this factorisation is unique, apart from the order in which the prime factors occur. Fundamental theorem of arithmetic is also called a Unique Factorisation Theorem. Composite number Product of prime numbers Or ny integer greater than can either be a prime number or can be written as a unique product of prime numbers. e.g., (i) is the same as. [ignoring the order] (ii) 6 can be written as or, where and are prime numbers. (iii) 5 can be written as 5 or 5, where and 5 are prime numbers. The prime factorisation of a natural number is unique, except to the order of its factors. e.g., made by multiplying the prime numbers, and together, We would probably write it as It is still a unique combination of (, and ). y using Fundamental Theorem of rithmetic, we shall find the HCF and LCM of given numbers (two or more).
2 ] Oswaal CSE Chapterwise, MTHEMTICS, Class-X This method is also called Prime Factorization Method. Prime Factorization Method to find HCF and LCM : (i) Find all the prime factors of given numbers. (ii) HCF of two or more numbers Product of the smallest power of each common prime factor involved in the numbers. (iii) LCM of two or more numbers Product of the greatest power of each prime factor involved in the numbers. (iv) For two positive integers a and b, we have HCF (a, b) LCM (a, b) a b or, HCF (a, b) and LCM (a, b) a b LCM ab, ( ) a b HCF ab, ( ) Q. Show that 6 n can never end with digit 0 for any natural number n. Sol. : Step I : ny number which ends in zero must have at least and 5 as its prime factors. Step II : or, 6 or, 6 n ( ) n n n Hence prime factor of 6 are and. Step III : Since 6 n does not contain 5 as a prime factor. Hence 6 n can never end in zero. TOPIC- Irrational Numbers, Terminating and Non-Terminating Recurring Decimals Rational Numbers : The number in the form p, where p and q are coprime number and q 0, is known as rational q number. For Example :,,, etc rational numbers. 7 5 Irrational Numbers : number is called irrational if it cannot be written in the form p, where p and q are integers q and q 0. For example,, 5, π are irrational numbers. Let p be a prime number. If p divides a, then p divides a where a is a positive integer. Terminating Decimals : If decimal expansion of rational number p q comes to an end, then the decimal obtained from p q is called terminating decimal. Non-terminating Repeating Recurring Decimals : The decimal expansion obtained from p q repeats periodically, then it is called non-terminating repeating or recurring decimal. Just divide the numerator by the denominator of a fraction. If you end up with a remainder of O, you have a terminating decimal otherwise repeating or recurring decimal. The sum or difference of a rational and irrational number is irrational. The product and quotient of a non-zero rational and irrational number is irrational. Let x p q be a rational number, such that the prime factorization of q is of the form m 5 n, where n and m are
3 Oswaal CSE Chapterwise, MTHEMTICS, Class-X [ non-negative integers. Then x has a decimal expansion which terminates after k places of decimals, where k is the largest of m and n. Let x be a rational number whose decimal expansion terminates. Then x can be expressed in the form p q, where p and q are co-prime and the prime factorisation of q is of the form m 5 n, where m and n are non-negative integers. Let x p q be a rational number, such that the prime factorization of q is not of the form m 5 n, where n and m are non-negative integers. Then x has a decimal expansion which is non-terminating repeating. Q. Show that + 5 is an irrational number. Sol. : Step I : Let + 5 be a rational number. rational number can be expressed as a b, where b 0 Step II : or, + 5 a b or, a 5 b or, Here R.H.S a b -5 a b -5 is rational while L.H.S, is irrational which is not possible. Step III : Hence our assumption that + 5 is a rational number is wrong. Hence + 5 is an irrational number. Unit -II : lgebra Chapter - : Polynomials TOPIC- Zeroes of a Polynomial and Relationship between Zeroes and Coefficients of Quadratic Polynomials Polynomial : n algebraic expression of the form a n x n + a n x n a x +a x + a 0, (where n is a whole number and a 0, a, a,..., a n are real numbers) is called a polynomial in the variable x. Value of a Polynomial at a given point : If p(x) is a polynomial in x and α is a real number, then the value obtained by putting x α in p(x) is called the value of p(x) at x α. Zero of a Polynomial : real number k is said to be a zero of the polynomial p(x), if p(k) 0. Geometrically, the zeroes of a polynomial p(x) are precisely the x-co-ordinates of the points, where the graph of y p(x) intersects the x-axis. (i) linear polynomial has at most one zero. (ii) quadratic polynomial has at most two zeroes. (iii) cubic polynomial has at most three zeroes. (iv) In general, a polynomial of degree n has at most n zeroes. Graphs of Different type of Polynomials : Linear Polynomial : The graph of a linear polynomial p(x) ax + b is a straight line that intersects x-axis at one point only. Quadratic Polynomial : (i) Graph of a quadratic Polynomial p(x) ax + bx + c is a parabola open upward like U. If a > 0 and intersects x-axis at maximum two distinct points.
4 4 ] Oswaal CSE Chapterwise, MTHEMTICS, Class-X (ii) Graph of a quadratic polynomial p(x) ax + bx + c is a parabola open downward like Ç if a < 0 and intersects X axis at maximum two distinct points. Graph of a cubic polynomial : Graph of cubic polynomial p(x) ax + bx + cx + d intersects x-axis at maximum three distinct points. Relationship between the zeroes and the coefficients of Polynomials : (i) Zero of a linear polynomial - Constant term Coefficientof x If ax + b is a given linear polynomial, then zero of linear polynomial ax + b is b a (ii) In a quadratic polynomial, Sum of zeroes of a quadratic polynomial Coefficient of x Coefficient of x Product of zeroes of a quadratic polynomial Constant term Coefficient of x If α and b are the zeroes of a quadratic polynomial ax + bx + c, then α + β b a and αβ c a (iii) If α, β and γ are the zeroes of a cubic polynomial ax + bx + cx + d, then α + β + γ b a, αβ + βγ + γα c a and αβγ d a Discriminant of a Quadratic Polynomial : For f (x) ax + bx + c, a 0, b 4ac is called its discriminant D. The discriminant D determines the nature of roots/zeroes of a quadratic polynomial. Case I : If D > 0, graph of f (x) ax + bx + c will intersect the X-axis at two distinct points, x-co-ordinates of points of intersection with X-axis is known as zeroes of f (x). Y Y X' O X X' O X Y' Y' f (x) will have two zeroes and we can say that roots/zeroes of the to given polynomials are real and unequal. Case II : If D 0, graph of f (x) ax + bx + c will touch the X-axis at one point only. Y Y X' O X X' O X Y' Y' f (x) will have only one zero and we can say that roots/zeroes of the given polynomial are real and equal. Case III : If D < 0, graph of f (x) ax + bx + c will neither touch nor intersect the X-axis.
5 Oswaal CSE Chapterwise, MTHEMTICS, Class-X [ 5 Y Y X' O X X' O X Y' Y' f (x) will not have any real zeroes. Relationship between the zeroes and the coefficients of a Polynomial : S. No. Type of polynomial General form Maximum Number of zeroes Relationship between zeroes and coefficients. Linear ax + b; a 0 k b a, i.e., k Constant term ( Coefficient of x). Quadratic ax + bx + c; a 0 Sum of zeroes (α + β) Coefficient of x Coefficient of x b a. Cubic ax + bx + cx + d; a 0 Product of zeroes (αβ) Constant term Coefficient of x Sum of zeroes (α + β + γ) c a Coefficientof x Coefficientofx b a Product of sum of zeroes taken two at a time (αβ + βγ + γα) + Coefficientof x Coefficientofx c a Product of zeroes (αβγ) Constant term Coefficient of x d a Q. If α and β are the roots of equation x 5x + 6 0, then find the value of α β. Sol. : Step I : Compare x 5x with ax + bx c 0 to get a, b 5 and c 6. Step II : Sum of roots a + b -b -- ( 5) a or, α + β 5...(i) Step III : Product of roots αβ c 6 a or, αβ 6...(ii) Step IV : α β ( α+ β) -4αβ or, α β (5) ± or
6 6 ] Oswaal CSE Chapterwise, MTHEMTICS, Class-X TOPIC- Problems on Polynomials Degree of a Polynomial : The exponent of the highest degree term in a polynomial is known as its degree. In other words, the highest power of x in a polynomial f (x) is called the degree of the polynomial f (x). e.g., (i) f(x) 5x + is a polynomial in variable x of degree. (ii) g(y) y 5 y + 7 is a polynomial in variable y of degree. Constant Polynomial : polynomial of degree zero is called a constant polynomial. e.g., f (x) 8, g(y) -5 Linear Polynomial : polynomial of degree is called a linear polynomial. e.g., f (x) x + 5, g (y) 7y, h(z) 5z. In general a linear polynomial in variable x with a real coefficient is of the form f (x) ax + b, where a and b are real numbers and a 0. Quadratic Polynomial : polynomial of degree is called a quadratic polynomial. e.g., f (x) 5x, g(y) 7y 5y, and h (z) 8z + 6z + 5 In general, a quadratic polynomial in variable x with real coefficients is of the form f (x) ax + bx + c, where a, b, c are real numbers and a 0. Division lgorithm for Polynomials : If p(x) and g(x) are any two polynomials with g(x) 0, then we can find polynomials q(x) and r(x), such that p(x) g(x) q(x) + r(x), where degree of r(x) < degree of g(x). and r(x) is denoted for remainder. Note : (i) If r(x) 0, then g(x) is a factor of p(x). (ii) Dividend Divisor Quotient + Remainder. To Divide Quadratic Polynomial by Linear Polynomial : Let, p(x) ax + bx + c and g(x) mx + n a m x + m b an m Quotient Divisor mx + n) ax + bx + c Dividend + ax + an m x an b m x + c an b m x + n m an b m c n m b an m Remainder Constant term Step I : To obtain the first term of the quotient, divide the highest degree term of the dividend (i.e., ax ) by the highest degree term of the divisor (i.e., mx). i.e., a m an What remains is b m x + c. x. Then, carry out the division process.
7 Oswaal CSE Chapterwise, MTHEMTICS, Class-X [ 7 Step II : Now, to obtain the second term of the quotient, divide the highest degree term of the new dividend an ie.., b - m x by the highest degree term of the divisor (i.e., mx). i.e., m b an m. Then carry on the division process. Sol. : Q. What should be added to x 8x + 0 to make it divisible by x? Step I : Let k be added to x 8x + 0 to make it divisible by x Step II : pplying Long division p(x) x 8x k x )x 8x k( x 5 + x + x 5x k + 5x k Step III : Here remainder is 5 + k which must be zero. or, 5 + k 0 \ k 5 Chapter - : Pair of Linear Equations In Two Variables TOPIC- Graphical Solution of Linear Equations in Two Variables Consistency/Inconsistency Linear Equation in two variables : n equation of the form ax + by + c 0, where a, b and c are real numbers and a and b are not both zero, is called a linear equation in two variables x and y. General form of a pair of linear equations in two variables is : a x + b y + c 0 a x + b y + c 0, where a, a, b, b, c, c are real numbers, such that a, b 0 and a, b 0. e.g., x y + 7 0, 7x + y are linear equations in two variables x and y. There are two methods of solving simultaneous linear equations in two variables :. Graphical method,. lgebraic method.. Graphical Method : (i) Express one variable say y in terms of the other variable x, y ax + b, for the given equation. (ii) Take three values of independent variable x and find the corresponding values of dependent variable y, take integral values only. (iii) Plot these values on the graph paper in order to represent these equations. (iv) If the lines intersect at a distinct point, then point of intersection is the unique solution of the two equations. In this case, the pair of linear equations is consistent. (v) If the lines representing the linear equations coincides, then system of equations has infinitely many solutions. In this case, the pair of linear equations is consistent and dependent. (vi) If the lines representing the pair of linear equations are parallel, then the system of equations has no solution and is called inconsistent.
8 8 ] Oswaal CSE Chapterwise, MTHEMTICS, Class-X Parallel Lines : (i) If a a b c, then the pair of linear equations is inconsistent with no solution. b c ax + by c Y X' O Y' ax + by c Intersecting Lines : (ii) If a b, then the pair of linear equations is consistent with a unique solution. a b Y ax + by c X X' O Y' ax + by c Coincident Lines : (iii) If a b c, then the pair of linear equations is consistent with infinitely many solutions. a b c Y X ax + by c ax + by c X' X O Y' If a x + b y + c 0 and a x + b y + c 0, is a pair of linear equations in two variables x and y such that : Possibilities of solutions and Inconsistency : Pair of lines a b c Compare the Graphical a b c ratios representation lgebraic interpretation Conditions for solvability x y 0 x 4y a a b b Intersecting lines Exactly one solution Unique solution System is consistent x + y 9 0 4x + 6y a a b c b c Coincident lines Infinitely many solutions System is consistent x + y 4 0 x + 4y a a b c b c Parallel lines No solution System is inconsistent
9 Oswaal CSE Chapterwise, MTHEMTICS, Class-X [ 9 Q. From the given pair of linear equations x + ay 50 and 9x y 5. find the value of a for them to be parallel Sol. : Step I : x + ay 50 and 9x y 5 a, b a, c 50 and a 9,b,c 5 Step II : For lines to be parallel a b c a b c Step III : 9 a 50-5 or, a ( ) So, a 7 TOPIC- lgebraic Methods to solve pair of Linear Equations and Equations reducible to Linear Equations lgebraic Method : We can solve the linear equations algebraically by substitution method, elimination method and by cross-multiplication method.. Substitution Method : (i) Find the value of one variable say y in terms of the other variable i.e., x from either of the equations. (ii) Substitute this value of y in other equation and reduce it to an equation in one variable. (iii) Solve the equation so obtained and find the value of x. (iv) Put this value of x in one of the equations to get the value of variable y.. Elimination Method : (i) Multiply given equations with suitable constants, make either the x-coefficient or the y-coefficient of the two equations equal. (ii) Subtract or add one equation from the other to get an equation in one variable. (iii) Solve the equation so obtained to get the value of the variable. (iv) Put this value in any one of the equation to get the value of the second variable. Note : (a) If in step (ii), we obtain a true equation involving no variable, then the original pair of equations has infinitely many solutions. (b) If in step (ii), we obtain a false equation involving no variable, then the original pair of equations has no solution i.e., it is inconsistent.. Cross-multiplication Method : If two simultaneous linear equations a x + b y + c 0 and a x + b y + c 0 are given, then a unique solution is given by : x bc - bc or x y ca - ca bc - bc ab - ab Note : To obtain the above result, following diagram may be helpful : x y b c a ab - ab and y ca ab - ca - ab b b c a b The arrows between the two numbers indicate that they are to be multiplied. The product with upward arrows are to be subtracted from the product with downward arrows. Remember :. If the equations a x + b y + c 0 and a x + b y + c 0 are in the form a x + b y c and a x + b y c
10 0 ] Oswaal CSE Chapterwise, MTHEMTICS, Class-X Then, we have by cross-multiplication x bc - bc y ca - ca - ab - ab Equations reducible to a Pair of Linear Equations in two variables : Sometimes, a pair of equations in two variables is not linear but can be reduced to linear form by making some suitable substitutions. Here, first we find the solution of new pair of linear equations and then find the solution for the given pair of equations. Steps to be followed for solving word problems S. No. Problem type Steps to be followed. ge Problems If the problem involves finding out the ages of two persons, take the present age of one person as x and of the other as y. Then a years ago, age of st person was x a years and that of nd person was y a and after b years, age of st person will be x + b years and that of nd person will be y + b years. Formulate the equations and then solve them.. Problems based on Numbers and Digits. Problems based on Fractions Let the digit in unit s place be x and that in ten s place be y. The two-digit number is given by 0y + x. On interchanging the positions of the digits, the digit in unit s place becomes y and in ten s place becomes x. The two digit number becomes 0x + y. Formulate the equations and then solve them. Let the numerator of the fraction be x and denominator be y, then the fraction is x y. 4. Problems based on Distance, Speed and Time 5. Problems based on commercial Mathematics Formulate the linear equations on the basis of conditions given and solve for x and y to get the value of the fraction. We know that Speed Distance Time Þ Distance Speed Time and Time Distance Speed., To solve the problems related to speed of boat going downstream and upstream, let the speed of boat in still water be x km/h and speed of stream be y km/h. Then, the speed of boat downstream x + y km/h and speed of boat upstream x y km/h. For solving specific questions based on commercial mathematics, the fare of full ticket may be taken as ` x and the reservation charges may be taken as ` y, so that one full fare x + y and one half fare x + y. 6. Problems based on Geometry and Mensuration To solve the questions of profit and loss, take the cost price of st article as ` x and that of nd article as ` y. To solve the questions based on simple interest, take the amount invested as ` x at some rate of interest and ` y at some other rate of interest. Make use of angle sum property of a triangle ( + + C 80 ) in case of a triangle. In case of a parallelogram, opposite angles are equal and in case of a cyclic quadrilateral, opposite angles are supplementary. Q. Solve the following system of equations by substitution method. x + y 5, x + y 4 Sol. : Step I : Given equations are x + y 5 and x + y 4 Step II : From x y 5, we get x 5 y Step III : Putting this value of x in other equation x + y 4 we get (5 y) + y 4 0 4y + y 4 y 0 4 y 6 Setp IV : On putting this value of y in (iii) lso x 5 () x x and y.
11 Oswaal CSE Chapterwise, MTHEMTICS, Class-X [ Chapter - 4 : Quadratic Equations TOPIC- Solution of Quadratic Equations quadratic equation in the variable x is of the form ax + bx + c 0, where a, b and c are real numbers and a 0. The values of x that satisfies an equation is called the solution or roots or zeros of the equation. real number a is said to be a solution/root or zero of the quadratic equation ax + bx + c 0, if aa + ba + c 0. quadratic equation can be solved by the following algebraic methods : (i) Splitting the middle term (ii) Making perfect squares (iii) Using quadratic formula. If ax + bx + c 0, a 0 can be reduced to the product of two linear factors, then the roots of the quadratic equation ax + bx + c 0 can be found by equating each factor to zero. Method for splitting the middle term of the equation ax + bx + c 0, where a 0. (i) Form the product ac (ii) Find a pair of numbers b and b whose product is ac and whose sum is b (if you can t find such number, it can t be factorised) (iii) Split the middle term using b and b, that expresses the term bx as b x + b x. Now factor by grouping the pairs of terms. Roots of the quadratic equation can be found by equating each linear factor to zero. Since product of two numbers is zero, if either or both of them are zero. ny quadratic equation can be converted in to the form (x + a) b 0 by adding and subtracting some terms. This method of finding the roots of quadratic equation is called the method of making the perfect the square. Method of making the perfect square for quadratic equation ax + bx + c 0, a 0. (i) Dividing throughout by a, we get x + b a x + c a 0 (ii) Multiplying and dividing the coefficient of x by x + b a x + c a 0 (iii) dding and subtracting b 4a or, or, x + ba x b b c a 4 a + 0 a b x + a b x + a b - 4 ac 4a b - 4ac a If (b 4ac) 0, then by taking square root b or, x + a ± b - 4 ac a or, x -b ± b - 4 ac a
12 ] Oswaal CSE Chapterwise, MTHEMTICS, Class-X Roots of ax + bx + c 0, a 0 are -b + b - 4 ac a and -b- b - 4 ac a, where b 4ac > 0. Roots of ax + bx + c 0, a 0 are - b a Quadratic identities : (i) (a + b) a + ab + b (ii) (a b) a ab + b (iii) a b (a + b) (a b) Know the Facts and - b a, where b 4ac 0 The Old-abylonians (700 C) stated and solved problems involving quadratic equations. The Greek mathematician Euclid developed a geometrical approach for finding out roots, which are solution of quadratic equations. In Vedic manuscripts, procedures are described for solving quadratic equations by geometric methods related to completing a square. rahmagupta (C.E ) gave an explicit formula to solve a quadratic equation of the form ax + bx + c 0 Sridharacharya (C.E. 05) derived the quadratic formula for solving a quadratic equation by the method of completing the perfect square. n rab mathematician l-khwarizmi (about C.E. 800) studied quadratic equations of different types. braham bar Hiyya Ha-nasi, in his book Liber embadorum published in Europe in C.E. 45 gave complete solutions of different quadratic equations. Edden ratio f is the root of quadratic equation x x 0. Q. Two water taps together can fill a tank in hrs. the tap of smaller diameter takes hours more than the larger one to fill the tank separately. Find the time in which each tap can separately fill the tank. Sol. : Step : Let time taken by tap of larger diameter be x hrs. and time taken by tap of smaller diameter be (x + ) hrs. Together they fill the tank in hrs Step : ccording to question, x + x + 5 or, x ++ x xx ( +) 5 x + +x Step : Cross multiplying x 5 5(x + ) (x +x) or, 70x + 70 x + 4x or, x 46x 70 0 or, 6x x 5 0 Together they fill in an hour 5 tank 5 hrs. Tap of larger diameter will fill the tank in one hour x tank and other tap x+ tank Step 4 : Now factorizing by splitting the middle term. 0 Rejecting x x 0x + 7x 5 or, 6x(x 5) + 7(x 5) 0 or, (x 5) (6x + 7) 0 or, x 5 or x -7 6 as time cannot be negative. Therefore larger tap takes 5 hrs and smaller tap takes 7 hrs.
13 Oswaal CSE Chapterwise, MTHEMTICS, Class-X [ TOPIC- Discriminant and Nature of Roots For the quadratic equation ax + bx + c 0, the expression b 4ac is known as discriminant i.e., Discriminant D b 4ac Nature of roots of a quadratic equation : (i)if b 4ac > 0, the quadratic equation has two distinct real roots. (ii)if b 4ac 0, the quadratic equation has two equal real roots. (iii)if b 4ac < 0, the quadratic equation has no real roots. R Q. Find the value of k for which the equation 4x + kx has equal roots. Sol. Step : 4x + kx Comparing above equation with ax + bx + c 0 a 4, b k and c 5 Step : Condition for equal roots is D 0 i.e., b 4ac 0 Step : Substituting values in the above condition. (k ) 4(4) (5) 0 or, k or, k (0) 0 or, (k 0) (k + 0) 0 or, k 0, 0 Chapter - 5 : rithmetic Progression TOPIC- To find n th Term of the rithmetic Progression n arithmetic progression is a sequence of numbers in which each term is obtained by adding a fixed number d to the preceding term, except the first term. The difference between the two successive terms of an.p. is called the common difference. Each number in the sequence of arithmetic progression is called the term of an.p. The arithmetic progression having finite number of terms is called a finite arithmetic progression. The arithmetic progression having infinite number of terms is called an infinite arithmetic progression. list of numbers a, a, a, is an.p., if the differences a a, a a, a 4 a, give the same value i.e., a k+ a k is same for all different values of k. The general form of an.p. is a, a + d, a + d, a + d.. If the.p. a, a + d, a + d,, l is reversed to l, l d, l d,, a, then the common difference changes to negative of original sequence common difference. The general term of an.p. is expressed as : a n a + (n )d. where, a is the first term and d is the common difference. The general term of an.p. l, l d, l d.. a is given by : a l + (n )( d), where, l is the last term, d is the common difference and n is the number of terms.
14 4 ] Oswaal CSE Chapterwise, MTHEMTICS, Class-X Know the Terms sequence is defined as an ordered list of numbers. The first, second and third terms of a sequence are denoted by t, t, t respectively. sequence is said to be a finite sequence if it has finite number of terms. s :,, 5, 7. sequence is said to be an infinite sequence if it has infinite number of terms. s :,, 5, If the terms of sequence are connected with plus (+) or minus ( ), then the numbers are called a series. Example : is a series. rithmetic sequence were used by abylonians 4000 years ago. The sequence of numbers,,,, 5, 8,,... was discovered by a famous Italian Mathematician Leonasalo Fibonacci, when he was dealing with the problem of rabbit population. sequence in which the difference between any two consecutive terms is the same constant, is called an arithmetic progression. If the terms of a sequence or a series are written under specific conditions, then the sequence or series is called a progression. If a constant is added or subtracted from each term of an.p. the resulting sequence is also an.p. If each term of an.p. is multiplied or divided by an constant, the resulting sequence is also an.p. If the n th term is in linear form i.e., an + b a n then the sequence is in.p. If the terms are selected at a regular interval then the given sequence is in.p. If three consecutive number a, b and c are in.p., then the sum two numbers is twice the middle number i.e., b a + c. U Q. Which term of the.p. 6,, 0, 7,... is 98 more than its 4 th term? Sol. Step I : The given.p. is 6,, 0, 7,... here first term a 6 Common difference d 6 7 Step II : The 4 th term a 4 a + (4 )d or, a Step III : Now according to question, a a n a + (n )d (n )7 59 (n ) n 7 n or n 8 Hence 8 th term. Quick review TOPIC- Sum of n terms of an rithmetic Progression Sum of n terms of an.p is given by : S n n [a + (n ) d] where, a is the first term, d is the common difference and n is the total number of terms. Sum of n terms of an.p when first and last term is given : S n n [a + l] where, a is the first term and l when first and last term in given. The n th term of an.p is the difference of the sum of first n terms and the sum to first (n ) terms of it. i.e., a n S n S n.
15 Oswaal CSE Chapterwise, MTHEMTICS, Class-X [ 5 U Q. Find the number of terms in the.p 54, 5, whose sum is 5. lso given the reason of double answer. Sol. Step I : The given.p. is 54, 5, here a 54, d 5 54 Sum required is 5. Step II : pplying the sum formula n S n [a + (n )d] n 5 [ 54 + (n )( )] 06 n[08 n + ] 06 n[ n] 06 n n or, n n or, [n 7n + 4] 0 or, n 7n Step III : Factorizing the quadratic equation n 9n 8n n(n 9) 8(n 9) 0 or, (n 9)(n 8) 0 or, n 8 or 9 Since 9th term of given.p. is O Hence S 8 S 9 Unit -III : Coordinate Geometry Chapter - 6 : Lines (In Two-Dimensions) TOPIC- Distance between two points and section formula Two perpendicular number lines intersecting at point zero are called co-ordinate axes. The horizontal number line is the x-axis (denoted by X OX) and the vertical line is the y-axis (denoted by Y OY). y-axis x'-axis Quadrant II (,+) Quadrant I (+,+) O Quadrant III (, ) Quadrant IV (+, ) y'-axis x-axis
16 6 ] Oswaal CSE Chapterwise, MTHEMTICS, Class-X The point of intersection of x-axis and y-axis is called origin and denoted by O. Cartesian plane is a plane obtained by putting the co-ordinate axes perpendicular to each other in the plane. It is also called co-ordinate plane or xy-plane. The x-co-ordinate of a point is its perpendicular distance from y-axis. The y-co-ordinate of a point is its perpendicular distance from x-axis. The point where the x-axis and the y-axis intersect, is co-ordinate point (0, 0). The abscissa of a point is the x-co-ordinate of the point. The ordinate of a point is the y-co-ordinate of the point. If the abscissa of a point is x and the ordinate of the point is y, then (x, y) are called the co-ordinates of the point. The axes divide the Cartesian plane into four parts called the quadrants (one fourth part), numbered I, II, III and IV anti clockwise from OX. The co-ordinates of a point on the x-axis are of the form (x, 0) and that of the point on y-axis are (0, y). Sign of co-ordinates depicts the quadrant in which it lies. The co-ordinates of a point are of the form (+, +) in the first quadrant, (, +) in the second quadrant, (, ) in the third quadrant and (+, ) in the fourth quadrant. Three points, and C are collinear if the distances, C, C are such that the sum of two distances is equal to the third. Three points, and C are the vertices of an equilateral triangle if C C. The points, and C are the vertices of an isosceles triangle if C or C C or C. C Three points, and C are the vertices of a right triangle, if + C C. For the given four points,, C and D : D C. C CD D; C D or, CD is a square.. C CD D; C D or, CD is a rhombus.. CD, C D; C D or, CD is a rectangle. 4. CD, C D; C D or, CD is a parallelogram. Diagonals of a square, rhombus, rectangle and parallelogram always bisect each other. Diagonals of rhombus and square bisect each other at right angle. ll given points are collinear, if the area of the obtained polygon is zero. Centroid is the point of intersection of the three medians of a triangle. In the figure, G is the centroid of the triangle. E G F Centroid divides each median of a triangle in a ratio of :. D C
17 Oswaal CSE Chapterwise, MTHEMTICS, Class-X [ 7 The incentre is the point of intersection of bisector of the internal angles of a triangle. It is also the centre of the circle touching all the sides of a triangle. Here, I is the incentre of the triangle. E I F C D Circumcentre is the point of intersection of the perpendicular bisectors of the sides of the triangle. E O F D C Orthocentre is the point of intersection of perpendicular drawn from the vertices on opposite sides (called altitudes) of a triangle and can be obtained by solving the equations of any two altitudes. Here, O is the orthocentre. E O F D If the triangle is equilateral, the centroid, incentre, orthocentre, circumcentre coincides. If the triangle is right angled triangle, then orthocentre is the point where right angle is formed. If the triangle is right angled triangle, then circumcentre is the mid-point of hypotenuse. Orthocentre, centroid and circumcentre are always collinear and centroid divides the line joining the orthocentre and circumcentre in the ratio of :. In an isosceles triangle, centroid, orthocentre, incentre, circumcentre lie on the same line. ngle bisector divides the opposite sides in the ratio of remaining sides. Three given points are collinear, if the area of triangle is zero. If x y, then (x, y) (y, x) and if (x, y) (y, x), then x y. C The distance between two points i.e., P(x, y ) and Q(x, y ) is ( x x ) + ( y y ). The distance of a point P(x, y) from origin is x + y To plot a point P(, 4) in the cartesian plane. (i) distance of units along X-axis. (ii) distance of 4 units along Y-axis..
18 8 ] Oswaal CSE Chapterwise, MTHEMTICS, Class-X Y P (,4) -7 X' O X Y' Co-ordinates of point (x, y) which divides the line segment joining the points (x, y ) and (x, y ) in the ratio m : n internally are mx + nx my + ny x m+ n and y m+ n Co-ordinates of mid-point of the line segment joining the points (x, y ) and (x, y ) are x x + x and y y + y Co-ordinates of a point which divides the line segment joining the points (x,y ) and (x,y ) in the ratio m : n externally are : x mx - nx and y my - ny m- n m- n m n (x, y ) ( x, y ) ( x, y ) If (x, y ), (x, y ) and C(x, y ) are vertices of a triangle, then the co-ordinates of centroid are G x + x + x y + y + y, If (x, y ), (x, y ) and C(x,y ) are vertices of a triangle and a, b and c are the lengths of sides C, C and respectively. then the co-ordinates of incentre are ax + bx + cx ay + ay + ay I, a+ b+ c a+ b+ c b c a C If (x, y ), (x, y ) and C(x, y ) are vertices of a triangle, then the area of triangle C is given by the numerical value of rea of DC [x (y y ) + x (y y ) + x (y y )] rea of quadrilateral [(x y x y ) + (x y x y ) + (x y 4 x 4 y ) + (x 4 y x y 4 )]
19 Oswaal CSE Chapterwise, MTHEMTICS, Class-X [ 9 Know the Facts Co-ordinate geometry is the system of geometry where the position of points on the plane is described using an ordered pair of numbers. Cartesian plane was discovered by Rene Descarte. The other name of co-ordinate geometry is nalytical Geometry. Co-ordinate Geometry acts as a bridge between the lgebra and Geometry. Medians of a triangle are concurrent. The point of concurrency is called the centroid. Trisection of a line segment means dividing it into equal parts, so points are required. Centroid of a triangle divides its median in the ratio of :. Q. Find the point of trisection of the line segment joining the points (5, 6) and ( 7, 5). Sol. Step I : Diagrammatic representation. (, x y ) ( x, y) (5, 6) P Q ( 7, 5) Points of trisections divides line segment into three equal parts. i.e., P PQ Q Step II : The ratio of P : PQ : Q : : or, P P and Q Q Step III : finding co-ordinates of P using section formula. (5, 6) P (, x y ) ( 7, 5) : x ( 7 ) + 5 ( ) + y 5 ( ) + ( 6 ) + \ P(x, y ) is, 7. Step IV : Finding co-ordinates of Q using section formula (5, 6) x y : Q 7 ( x, y ) ( 7, 5) ( 7) + 5 ( ) ( ) + ( 6) + 4 \ Q (x, y ) is, 4. TOPIC- rea of Triangle U Q. Find k, if the point (, ), (5, K) and C(7, 9) are collinear Sol. Step I : If points,, and C are collinear then the area of triangle formed by them is zero. Step II : rea of triangle formed by the vertices (x, y ), (x, y ) and (x, y ) is given by [x (y y ) + x (y y ) + x (y y )] Step III : ccording to question 0 [(k 9) + 5(9 ) + 7( k)] or, k k 0 or, 5k + 0 or, 5k Or k 5
20 0 ] Oswaal CSE Chapterwise, MTHEMTICS, Class-X Unit -IV : Geometry Chapter - 7 : Triangles triangle is one of the basic shapes of geometry. It is a polygon with sides and vertices /corners. Two figures are said to be congruent if they have the same shape and the same size. Those figures which have the same shape but not necessarily the same size are called similar figures. Hence, we can say that all congruent figures are similar but all similar figures are not congruent. Similarity of Triangles : Two triangles are similar, if : (i) their corresponding sides are proportional. (ii) their corresponding angles are equal. If C and DEF are similar, then this similarity can be written as C ~ DEF. Criteria for Similarity of Triangles : L P In LMN and DPQR, if (i) L P, M Q, N R then LMN ~ DPQR. M N Q (ii) R LM PQ MN LN, QR PR (i) -Criterion : In two triangles, if corresponding angles are equal, then The triangles area similar and hence their corresponding sides are in the same ratio. Remark : If two angles of a triangle are respectively equal to the two angles of another triangle, then by the angle sum property of a triangle their third angles will also be equal. Therefore, similarity criterion can also be stated as follows : -Criterion : If two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar. (ii) SSS-Criterion : In two triangles if the sides of one triangle are proportional to the sides of another triangle, then their corresponding angles are equal and the two triangles are similar and hence corresponding angles are equal. (iii) SS-Criterion : If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are in the same ratio, then the two triangles are similar. Some theorems based on similarity of triangles : (i) If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio. It is known as asic Proportionality Theorem or Thales Theorem. (ii) If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side. It is the Converse of asic Proportionality Theorem. (iii) If two triangles are similar, then the ratio of areas of these triangles is equal to the ratio of squares of their corresponding sides. Theorems ased on Right ngled Triangles : (i) If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse, then the triangles on both sides of the perpendicular are similar to the whole triangle and also to each other. (ii) In a right angled triangle, the square of hypotenuse is equal to the sum of the squares of the other two sides. It is known as Pythagoras Theorem. In any triangle, the sum of the squares of any two sides is equal to twice the square of half of the third side together with the twice of the square of the median which bisects the third side. Three times the sum of the squares of the sides of a triangle is equal to four times the sum of the squares of the median of the triangle. Three times of the square of any side of an equilateral triangle is equal to four times the square of the altitude.
21 Oswaal CSE Chapterwise, MTHEMTICS, Class-X [ Q. In the given triangle, find x if PQ C x x + Step II : x x+ x+ x+ Sol. : P Q x + x + Step I : Given that PQ C Hence by using asic Proportionality theorem P P Q QC C or, x (x + ) (x + ) (x + ) or, x + x x + 5x + 6 or, x 4x 6 0 or, x x 0 or, (x ) (x + ) 0 or, x or x or, x. Chapter - 8 : Circles tangent to a circle is a line that intersects the circle at one point only. The common point of the circle and the tangent is called the point of contact. The length of the segment of the tangent from the external point P and the point of contact with the circle is called the length of the tangent. Pythagoras theorem : In a right triangle, square of the hypotenuse is equal to the sum of the squares of the other two sides. tangent to a circle is a special case of the secant when the two end points of the corresponding chord are coincide. There is no tangent to a circle passing through a point lying inside the circle. There are exactly two tangents to a circle through a point outside the circle. t any point on the circle there can be one and only one tangent. The tangent at any point of a circle is perpendicular to the radius through the point of contact. Tangent Line Radius Point of Contact The lengths of the tangents drawn from an external point to a circle are equal. P In the figure, P P.
22 ] Oswaal CSE Chapterwise, MTHEMTICS, Class-X Know the Facts The word tangent comes from the Latin word tangere, which means to touch and was introduced by the Danish mathematician Thomas Fincke in 58. The line containing the radius through the point of contact is also called the normal to the circle at that point. In two concentric circles, the chord of the larger circle, which touches the smaller circle, is bisected at the point of contact. Q. P O 5 cm Q cm In the given figure if PR cm, OP 5 cm, OQ 4 cm. find RQ. Sol. Step I : OP RP R (as radius is to tangent at point of contact) In right triangle OPR, OR OP + PR OR 5 + OR OR 69 or, OR 69 Step II : Similarly in right triangle OQR OR OQ + QR QR OR OQ QR 4 or, (QR) 69 6 QR 5 or, QR 5 cm. 7 cm Chapter - 9 : Constructions TOPIC- Division of a Line Segment in a Given Ratio. The ratio of the sides of the triangle to be constructed with the corresponding sides of the given triangle is known as scale factor. To divide a line segment internally Tally is given in a ratio m : n, where both m and n are positive integers, we follow the following steps : X m m+ m+n P Step. Draw a line segment of given length by using a ruler. Step. Draw any ray X making an acute angle with. Step. long X mark off (m + n) points,,, m, m+,, m+n, such that m+n m+n. Step 4. Join m+n.
23 Oswaal CSE Chapterwise, MTHEMTICS, Class-X [ Step 5. Through the point m draw a line parallel to m+n by making an angle equal to m+n at m. i.e., Ð m P. This line meets at point P. The point P is the required point which divides internally in the ratio m : n. Q. Draw a line segment 6 cm and divide it in the ratio of :. Sol. Step I. C 4 P 5 Step II. Steps of construction : (i) Draw a line segment 6 cm. (ii) t draw an acute angle C. (iii) Mark,,, 4, 5 at equal distance. (iv) Join 5 to. (v) Draw a line P parallel to 5. (vi) P is the required point on which divides it in the ratio :. : 6 cm TOPIC- Tangents to a Circle from a Point Outside It To draw the tangent to a circle at a given point on it, when the centre of the circle is known. Given : circle with centre O and a point P on it. To construct : the tangent to the circle at the point P. Steps of construction : (i) Join OP. (ii) Draw a line segment ^ OP at the point P. P is the required tangent at P. O P 90 Fig. (9.) Steps of construction : (i) Draw any chord PQ and join P and Q to a point R. (ii) Draw QP equal to PRQ on opposite side of chord PQ. The line segment P is the tangent to the circle at P. (Fig. 9.) Q R Fig. (9.) To draw the tangent to a circle from an external point when its centre is known. Given : circle with centre O and a point P outside it. To construct : the tangents to the circle from P. P
24 4 ] Oswaal CSE Chapterwise, MTHEMTICS, Class-X Steps of construction : (i) Join OP and bisect it. Let M be the mid-point of OP. (ii) Taking M as centre and MO as radius, draw a circle to intersect C (O, r) in two points, say and. (iii) Join P and P. These are the required tangents from P to the circle. Fig. (9.) P M O Fig. (9.) To draw tangents to a circle from a point outside it (when its centre is not known) Given : P is a point outside the circle. Required : To draw tangents from the point P. Steps of construction : (i) Draw a secant P to intersect the circle at and. (ii) Produce P to a point C, such that P PC. (iii) With C as a diameter, draw a semi circle. (iv) Draw PO C, intersecting the semi circle at O. (v) Taking PO as radius and P as centre, draw an arc to intersect the circle at T and T. (vi) Join PT and PT, then PT and PT are the required tangents. (Fig. 9.4) O T C P T' Fig. (9.4) Q. Draw a pair of tangents to a circle of any convenient radius, which are inclined to the line joining the centre of the circle and the point at which they intersect at an angle of 45. OR Draw a circle of radius.5 cm. Draw two tangents to the circle which are perpendicular to each other. Sol. Steps of construction Step I. Draw a circle of any convenient radius with O as centre. Step II. Take a point on the circumference of the circle and join O. Draw a perpendicular to O at point. Step III. Draw a radius O, making an angle of 90 with O. Step IV. Draw a perpendicular to O at point. Let both the perpendiculars intersect at point P. Step V. Join OP. P and P are the required tangents, which make an angle of 45 with OP. O 90 P
25 Oswaal CSE Chapterwise, MTHEMTICS, Class-X [ 5 TOPIC- Construction of a Triangle Similar to a Given Triangle Construction of triangles similar to a given triangle : (a) Steps of constructions, when m < n : Step I. Construct the given triangle C by using the given data. Step II. Take any one of the three sides of the given triangle as base. Let be the base of the given triangle. Step III. t one end, say, of base. Construct an acute X below the base. Step IV. long X mark off n points,,,, n such that n n Step V. Join n. Step VI. Draw m parallel to n which meets at. Step VII. From draw C C meeting C at C. m Triangle C is the required triangle each of whose sides is n th of the corresponding side of C. (b) Steps of construction, when m > n : Step I. Construct the given triangle by using the given data. Step II. Take any one of the three sides of the given triangle and consider it as the base. Let be the base of the given triangle. Step III. t one end, say, of base. Construct an acute angle X below the base i.e., on the opposite side of the vertex C. Step IV. long X mark off m (larger of m and n) points,,, m such that m m. Step V. Join n to and draw a line through m parallel to n, intersecting the extended line segment at. Step VI. Draw a line through parallel to C intersecting the extended line segment C at C. D C so obtained is the required triangle. Q. Construct a right triangle whose hypotenuse and one side measure 5 cm and 4 cm respectively. Then construct another triangle whose sides are times of the corresponding sides of this trian- 5 gle. Sol. ' cm 5 cm 4 cm C' C Steps of Construction : Step I. (i) Draw a line segment C 4 cm. (ii) Draw an angle of 90 at. (iii) Taking C as centre and radius equal to hypotenuse 5 cm draw an arc to intersect at Joint to C. Step II. Draw a line making an acute angle with side C and divide it in five equal sections and mark,,, 4, 5. Step III. Join 5 to C, and draw a parallel line C' to 5 C which meets C at C'. Step IV. Draw a parallel line 'C' to C Hence C is the required triangle. 4 5
26 6 ] Oswaal CSE Chapterwise, MTHEMTICS, Class-X Unit -V : Trigonometry Chapter - 0 : Introduction To Trigonometry nd Trigonometric Identities In fig., a right triangle C right angle at is given and C q is an acute angle. Here side which is adjacent to is base, side C opposite to is perpendicular and the side C is hypotenuse which is opposite to the right angle. C Hypotenuse 90 ase The trigonometric ratios of in right triangle C are defined as Perpendicular sine of sin θ Perpendicular Hypotenuse C C cosine of cos θ ase Hypotenuse C tangent of tan θ Perpendicular ase cosecant of cosec θ secant of sec θ Hypotenuse ase cotangent of cot θ C Hypotenuse Perpendicular C C sinθ C cosθ ase Perpendicular C tanθ It is clear from the above ratios that cosecant, secant and cotangent are the reciprocals of sine, cosine and tangent respectively. lso, tan θ sin θ cosθ and TOPIC- Trigonometric Ratios and Trigonometric Ratios of Complementary ngles cot θ cos θ sinθ The trigonometric ratios of an acute angle in a right triangle express the relationship between the angle and length of its sides. The value of trigonometric ratio of an angle does not depend on the size of the triangle but depends on the angle only. Complementary ngles : Two angles are said to be complementary if their sum is 90. Thus (in fig.) and C are complementary angles. 90 C
27 Oswaal CSE Chapterwise, MTHEMTICS, Class-X [ 7 Trigonometric Ratios of Complementary ngles : We have, C ase, Perpendicular, C Hypotenuse, with respect to q. sin θ C, cos θ, tan θ C C C gain, cosec θ C C C, sec q, cot θ C. with respect to the angel (90 q) C Perpendicular, ase C Hypotenuse therefore sin (90 θ) C C cos θ cos (90 θ) C sin θ tan (90 θ) C cot θ cosec (90 θ) C C sec θ sec (90 θ) C cosec θ cot (90 θ) C tan θ sin 0 cos tan 0 cosec sec cot 0 0 Q. If sin q 0, then Prove that sin q sin q 4 sin q. Sol. Step I : Given sin q 0 or, sin q or, sin q...(i) Step II : sin 0 from (i) and (ii), q 0 Step III : L.H.S. sin q sin ( 0 ) sin 90...(ii) R.H.S. sin q 4 sin q sin 0 4(sin 0) 4 L.H.S.
28 8 ] Oswaal CSE Chapterwise, MTHEMTICS, Class-X TOPIC- Trigonometric Identities n equation is called an identity if it is true for all values of the variable(s) involved. n equation involving trigonometric ratios of an angle is called a trigonometric identity if it is true for all values of the angle. In C, right-angled at, y Pythagoras Theorem. + C C...(i) Dividing each term of (i) by C, or C C C + C C + C C C or (cos ) + (sin ) C C or cos + sin...(ii) This is true for all values of such that So, this is a trigonometric identity. now divide eqn.() by. C + C or C + C or + tan sec...(iii) Is this equation true for 0? Yes, it is. What about 90? Well, tan and sec are not defined for 90. So, eqn. (iii) is true for all values of such that 0 < 90. dividing eqn. (i) by C. C + C C C C or C C + C C C or cot + cosec...(iv) Note that cosec and cot are not defined for all 0. Therefore eqn. (iv) is true for all value of such that 0 < 90. Using these identities, we can express each trigonometric ratio in terms of other trigonometric ratios, i.e., if any one of the ratios is known, we can also determine the values of other trigonometric ratios. C Q. Prove that. sec θ- tan θ + sin θ cos θ Sol. : Step I : L.H.S. secθ tan θ Rationalising with sec q tan q secθ + tan θ L.H.S. secθ tan θ secθ + tan θ secθ + tan θ [(a + b) (a b) a b ] sec θ tan θ sec q + tan q [sec q tan q ] sin θ + cos θ cos θ + sin θ R.H.S. cos θ
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Maths Higher Prelim Content Straight Line Gradient of a line A(x 1, y 1 ), B(x 2, y 2 ), Gradient of AB m AB = y 2 y1 x 2 x 1 m = tanθ where θ is the angle the line makes with the positive direction of
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