IX Mathematics (English Medium) Support Material 2023-2024 PDF

DIRECTORATE OF EDUCATION Govt. of NCT, Delhi SUPPORT MATERIAL (2023-24) CLASS : IX MATHEMATICS (ENGLISH MEDIUM) Under the Guidance o...

DIRECTORATE OF EDUCATION Govt. of NCT, Delhi SUPPORT MATERIAL (2023-24) CLASS : IX MATHEMATICS (ENGLISH MEDIUM) Under the Guidance of Shri Ashok Kumar Secretary (Education) Shri Himanshu Gupta Director (Education) Dr. Rita Sharma Addl. DE (School & Exam.) Coordinators Mr. Sanjay Subhas Kumar Mrs. Ritu Singhal Mr. Raj Kumar Mr. Krishan Kumar DDE (Exam) OSD (Exam) OSD (Exam) OSD (Exam)  DIRECTORATE OF EDUCATION Govt. of NCT, Delhi SUPPORT MATERIAL (2023-24) CLASS : IX MATHEMATICS (ENGLISH MEDIUM) NOT FOR SALE PUBLISHED BY : DELHI BUREAU OF TEXTBOOK  MATHEMATICS (IX) The Syllabus in the subject of Mathematics has undergone changes from time to time in accordance with growth of the subject and emerging needs of the society. The present revised syllabus has been designed in accordance with National Curriculum Framework 2005 and as per guidelines given in the Focus Group of Teaching of Mathematics which is to meet the emerging needs of all categories of students. For motivating the teacher to related the topics to real life problems and other subject areas, greater emphasis has been laid on applications of various concepts The curriculum at secondary stage primarily aims at enhancing the capacity of students to employ Mathematics in solving day-to-day life problem and studying the subject as a separate discipline. IT is expected that students should acquired the ability to solve problem using algebraic methods and apply the knowledge of simple trigonometry to solve problem of height and distances. Carrying out experiments with numbers and forms of geometry, framing hypothesis and verifying these with further observations form inherent part of Mathematics learning at this stage. The proposed curriculum includes the study of number system, algebra, geometry, trigonometry, mensuration, statistics, graphs and coordinate geometry etc. The teaching of Mathematics should be imparted through activities which may involve the use of concrete materials, models, patterns, charts, pictures, posters, games, puzzles and experiments. Objectives The broad objectives of teaching of Mathematics at secondary stage are to help the learners to: consolidate the Mathematical knowledge and skills acquired at the upper primary stage; acquire knowledge and understanding, particularly by way of motivation and visualization, of basic concepts, terms, principles and symbols and underlying processes and skills; develop mastery of basic algebraic skills. develop drawing skills; feel the flow of reason while proving a result or solving a problem: apply the knowledge and skills acquired to solve problems and wherever possible, by more than one method. xiii IX – Mathematics  to develop ability to think, analyze and articulate logically; to develop awareness of the need for national integration, protection of environment, observance of small family norms, removal of social barriers, elimination of gender biases; to develop necessary skills to work with modern technological devices and mathematical software’s. to develop interest in mathematics as a problem-solving tool in various fields for its beautiful structures and patterns, etc. to develop reverence and respect towards great Mathematicians for their contributions to the field of Mathematics; to develop interest in the subject by participating in related competitions; to acquaint students with different aspects of Mathematics used in daily life; to develop an interest in students to study Mathematics as a discipline. xiv IX – Mathematics  TERM-WISE SYLLABUS SESSION: 2023-24 CLASS: IX SUBJECT: MATHEMATICS (CODE: 041) COURSE STRUCTURE Units Unit Name Marks I Number Systems 10 II Algebra 20 III Coordinate Geometry 04 IV Geometry 27 V Mensuration 13 VI Statistics & Probability 06 Total 80 Internal Assessment 20 Grand Total 100 UNIT I: NUMBER SYSTEMS Chapter-1: Real Numbers (18) Periods 1. Review of representation of natural numbers, integers and rational numbers on the number line. Rational numbers as recurring/terminating decimals. Operations on real numbers. 2. Examples of non-recurring/non-terminating decimals. Existence of non-rational numbers (irrational numbers) such as 2, 3 and their representation on the number line. Explaining that every real number is represented by a unique point on the number line and conversely, viz. every point on the number line represents a unique real number. 3. Definition of nth root of a real number. 4. Rationalization (with precise meaning) of real numbers of the type and (and their combinations) where x and y are natural number and a and b are integers. xv IX – Mathematics 5. Recall of laws of exponents with integral powers. Rational exponents with positive real bases (to be done by particular cases, allowing learner to arrive at the general laws.) UNIT II: ALGEBRA Chapter-2: Polynomials (26) Periods Definition of a polynomial in one variable, with examples and counter examples. Coefficients of a polynomial, terms of a polynomial and zero polynomial. Degree of a polynomial. Constant, linear, quadratic and cubic polynomials. Monomials, binomials, trinomials. Factors and multiples. Zeroes of a polynomial. Motivate and State the Remainder Theorem with examples. Statement and proof of the Factor Theorem. Factorization of ax² + bx + c, a ≠ 0 where a, b and c are real numbers, and of cubic polynomials using the Factor Theorem. Recall of algebraic expressions and identities. Verification of identities: (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx (x ± y)3 = x3 ± y3 ± 3xy (x ± y) x3 ± y3 = (x ± y) (x² ± xy + y²) x3 + y3 + z3 – 3xyz = (x + y + z) (x² + y2 + z2 – xy – yz – xz) and their use in factorization of polynomials. Chapter-4: Linear Equations in Two Variables (16) Periods Recall of linear equations in one variable. Introduction to the equation in two variables. Focus on linear equations of the type ax + by + c = 0. Explain that a linear equation in two variables has infinitely many solutions and justify their being written as ordered pairs of real numbers, plotting them, and showing that they lie on a line. UNIT III: COORDINATE GEOMETRY Chapter-3: Coordinate Geometry (7) Periods The Cartesian plane, coordinates of a point, names and terms associated with the coordinate plane, notations. UNIT IV: GEOMETRY Chapter-5: Introduction To Euclid’s Geometry (7) Periods History - Geometry in India and Euclid’s geometry. Euclid’s method of formalizing observed phenomenon into rigorous Mathematics with xvi IX – Mathematics  definitions, common/obvious notions, axioms/postulates and theorems. The five postulates of Euclid. Showing the relationship between axiom and theorem, for example: (Axiom) 1. Given two distinct points, there exists one and only one line through them. (Theorem) 2. (Prove) Two distinct lines cannot have more than one point in common. Chapter-6: Lines and Angles (15) Periods 1. (Motivate) If a ray stands on a line, then the sum of the two adjacent angles so formed is 180° and the converse. 2. (Prove) If two lines intersect, vertically opposite angles are equal. 3. (Motivate) Lines which are parallel to a given line are parallel. Chapter-7: Triangles (22) Periods 1. (Motivate) Two triangles are congruent if any two sides and the included angle of one triangle is equal to any two sides and the included angle of the other triangle (SAS Congruence). 2. (Prove) Two triangles are congruent if any two angles and the included side of one triangle is equal to any two angles and the included side of the other triangle (ASA Congruence). 3. (Motivate) Two triangles are congruent if the three sides of one triangle are equal to three sides of the other triangle (SSS Congruence). 4. (Motivate) Two right triangles are congruent if the hypotenuse and a side of one triangle are equal (respectively) to the hypotenuse and a side of the other triangle. (RHS Congruence). 5. (Prove) The angles opposite to equal sides of a triangle are equal. 6. (Motivate) The sides opposite to equal angles of a triangle are equal. Chapter-8: Quadrilaterals (13) Periods 1. (Prove) The diagonal divides a parallelogram into two congruent triangles. 2. (Motivate) In a parallelogram opposite sides are equal, and conversely. 3. (Motivate) In a parallelogram opposite angles are equal, and conversely. 4. (Motivate) A quadrilateral is a parallelogram if a pair of its opposite sides is parallel and equal. xvii IX – Mathematics 5. (Motivate) In a parallelogram, the diagonals bisect each other and conversely. 6. (Motivate) In a triangle, the line segment joining the mid points of any two sides is parallel to the third side and is half of it and (motivate) its converse. Chapter-9: Circles (17) Periods 1. (Prove) Equal chords of a circle subtend equal angles at the centre and (motivate) its converse. 2. (Motivate) The perpendicular from the centre of a circle to a chord bisects the chord and conversely, the line drawn through the centre of a circle to bisect a chord is perpendicular to the chord. 3. (Motivate) Equal chords of a circle (or of congruent circles) are equidistant from the centre (or their respective centres) and conversely. 4. (Prove) The angle subtended by an are at the centre is double the angle subtended by it at any point on the remaining part of the circle. 5. (Motivate) Angles in the same segment of a circle are equal. 6. (Motivate) If a line segment joining two points subtends equal angle at two other points lying on the same side of the line containing the segment, the four points lie on a circle. 7. (Motivate) The sum of either of the pair of the opposite angles of a cyclic quadrilateral is 180° and its converse. UNIT V: MENSURATION Chapter-10: Areas (5) Periods Area of a triangle using Heron’s formula (without proof). Chapter-11: Surface Areas and Volumes (17) Periods Surface areas and volumes of spheres (including hemispheres) and right circular cones. UNIT VI: STATISTICS & PROBABILITY Chapter-12: Statistics (15) Periods Bar graphs, histograms (with varying base lengths) and frequency polygons. Mental Maths Practice Revision from Support Material xviii IX – Mathematics  MATHEMATICS Code (041) QUESTION PAPER DESIGN Class-IX (2023-24) Time: 3 Hrs. Max. Marks: 80 S. Typology of Questions Total %Weight-age No. Marks (approx.) 1. Remembering: Exhibit memory of previously learned material by recalling facts, terms, basic concepts and answers. Understanding: Demonstrate understanding 43 54 of facts and ideas by organizing, comparing, translating, interpreting, giving descriptions and stating main ideas. 2. Applying: Solve problems to new situations by applying acquired knowledge, facts, techniques 19 24 and rules in a different way. 3. Analysing: Examine and break information into parts by identifying motives or causes. Make inferences and find evidence to support generalizations. Evaluating: Present and defend opinions by making judgments about information, validity or ideas, or quality of work based on a set of criteria. 18 22 Creating: Compile information together in a different way by combining elements in a new pattern proposing alternative solutions. Total 80 100 Internal Assessment 20 Marks Pen Paper Test and Multiple Assessment (5+5) 10 Marks Portfolio 05 Marks Lab Practical (Lab activities to be done from the prescribed 05 Marks books) xix IX – Mathematics  List of Group Leader and Subject-Experts For Preparation/Review of Support Material Class-IX (2023-24) Subject : Mathematics 1. Mr. Satyawan Vice Principal SBV, Rouse Avenue, DDU Marg (2127001) 2. Ms. Aakanksha PGT (Mathematics) Core Academic Unit (CAU) 3. Ms. Neha Chaudhary TGT (Mathematics) Core Academic Unit (CAU) 4. Ms. Gagandeep Kaur TGT (Mathematics) GGSS, Majlis Park, Delhi (1309036) 5. Ms. Rinku Gupta TGT (Mathematics) RPSKV Rithala, Delhi (1413026) 6. Mr. Vikas Dongre TGT (Mathematics) SBV, Rouse Avenue, DDU Marg (2127001) 7. Mr. Julfikar Ahmad TGT (Mathematics) Dr. Zakir Hussain Memorial Sr. Sec. School, Jafrabad (1105137) xx IX – Mathematics  CONTENTS Ch. No. Chapters Pages No. 1. Number Systems 1 2. Polynomials 19 3. Co-ordinate Geometry 28 4. Linear Equation in two variables 38 5. Introduction to Euclid’s Geometry 57 6. Lines and Angles 68 7. Triangles 83 8. Quadrilaterals 97 9. Circles 115 10. Heron’s Formula 137 11. Surface Area and Volumes 145 12. Statistics 155 Assertion Reasoning Based Questions 166 Case Study Based Questions 172 Practice Question Paper-I with solution 194 Practice Question Paper-II with solution 206 Practice Question Paper-III with solution 217 xxi IX – Mathematics  CHAPTER-1 NUMBER SYSTEMS MIND MAP 1 IX – Mathematics  CHAPTER-1 NUMBER SYSTEMS KEY POINTS 1, 2, 3,............. are natural numbers which are represented by N. 0, 1, 2, 3,............. are whole numbers which are represented by W.............. –3. –2, –1, 0, 1, 2, 3............. are integers which are represented by Z or I. A number is a rational if p (a) it can be represented in the form of q , where p and q are integers and q ≠ 0. OR (b) its decimal expansion is terminating (e.g. = 0.4) OR (c) its decimal expansion is non-terminating recurring (repeating) (e.g. = 0.1234234..........) A number is irrational number if p (a) it can not be represented in the form of q , where p and q are integers and q ≠ 0. OR (b) its decimal expansion is non-terminating non-recurring (e.g. 0.1010010001...........) All rational and irrational numbers collectively form real numbers. There are infinite rational numbers between any two rational numbers. There is a unique real number corresponding to every point on the number line. Also, corresponding to each real number, there is a unique point on the number line. Rationalisation of a denominator means to change the Irrational denominator to rational form. 2 IX – Mathematics  To rationalise the denominator of , we multiply this by , where a is a natural number and b is an integer. If r is rational and s is irrational then r + s, r – s, r. s are always irrational r r numbers but s may be rational or irrational. For r ≠ 0, r. s and s are always irrational. Law of Exponents: Let a > 0 be a real number and m amd n are rational numbers, then (1) am an = am + n (2) am ÷ an = am – n (3) (am)n = amn (4) am. bm = (ab)m (5) a0 = 1 (6) a–m = For positive real numbers a and b, the following identities hold (1) (2) (3) (4) (5) All natural numbers, whole numbers and integers are rational Prime Numbers: All natural numbers that have exactly two factors (i.e., 1 and itself) are called prime numbers, e.g., 2, 3, 5, 7, 11, 13, 17, 19, 23,... etc. Composite Numbers: Those natural number which have more than two factors are known as composite number. e.g., 4, 6, 8, 9, 10, 12,... 1 is neither prime nor composite. where ‘a’ is positive real number and n is a positive integer where a is positive real number. m and n are co-prime integers and n > 0 3 IX – Mathematics  Types of Numbers 4 IX – Mathematics Very Short Answer Questions ( 1 mark) 1. Which of the following is a rational number? (a) 1 + (b) (c) 0 (d) π 2. Which of the following is irrational? (a) (b) (c) (d) 3. If x = 2 + then (1/x) is equal to (a) 2 + (b) (c) 2 – (d) 4. An irrational number between and is (a) (b) (c) (d) 5. If 52y = 25 then 5–y is equal to (a) (b) (c) (d) Fill in the blanks: 6. = __________ 7. The decimal expansion of the number is __________ and __________ 8. __________ is a whole number but not a natural number. 9. = __________ 5 IX – Mathematics 10. Between two distinct rational number there lie __________ rational numbers. 11. The sum and difference of rational and irrational number is always __________ numbers. 12. Every rational number is a __________ number. 13. Find a rational number between. 14. Express in the form , where p and q are integers and q ≠ 0. 15. Find the value of in the form , where p & q are integres and q ≠ 0. 16. Find the value of x, if 5x – 3. 32x – 8 = 225 17. Find the value of [(4 – 5(4 – 5)4]3 18. Write first five whole numbers in form, where p and q are integers and q ≠ 0. 19. Find two irrational numbers between and. 20. Write two numbers whose decimal expansions are terminating. 21. Find the value of (256)0.16 × (256)0.09 22. Evaluate 23. What can be the maximum number of digits in the repeating block of digits in the decimal expansion of. Short Answer Type-I Questions (2 Marks) 24. Represent following on number line (a) (b) 6 IX – Mathematics 25. Find the value of x, 26. Express the mixed recurring decimal in the form. 27. Simplify 28. Which of the following rational numbers will have a terminating decimal expansion or a non-terminating repeating (recurring) decimal expansion? (a) (b) (c) (b) 29. Classify the numbers as terminating decimal or non-terminating recurring decimal or non-terminating non-recurring decimals. (a) 0.1666 (b) 0.27696 (c) 2.142857142857........ (d) 2.502500250002........ (e) Also classify these numbers as rational and irrational numbers. 30. Classify the following numbers as rational or irrational numbers. (a) (b) (c) (d) (e) π 7 IX – Mathematics 31. Solve (a) Add (b) Multiply (c) Divide by 3 Short Answer Type-II Questions (3 Marks) 32. If , then find the value of 11 (p + q) 33. Simplify 34. If 322x – 5 = 4 × 8x – 5 then find the value of x. 35. Evaluate (a) (b) (c) (d) 36. If 52x – 1 – (25)x – 1 = 2500 then find the value of x? 37. If x = , show that 38. If xyz = 1 then simplify 39. Find the value of x if (a) 252x –3 = 52x + 3 (b) (4)2x – 1 – (16)x – 1 = 384 8 IX – Mathematics 40. Solve 1 1 1 1 1 1 1 1 + + + + + + + 1+ 2 2+ 3 3+ 4 4+ 5 5+ 6 6+ 7 7+ 8 8+ 9 41. Express 0.6 + + 0.47 in the form , where p and q are integers and q ≠ 0. Long Answer type Questions (5 marks) 42. Evaluate 43. Simplify 44. Simplify 45. Show that 46. Show that and , then find the value of a2 + b2 + ab 47. If x = then find (a) (b) (c) (d) (e) (f) 9 IX – Mathematics  (g) (h) (i) 48. If P = find (a) (b) (c) 49. Find the value of 50. If then prove that m – n = 2 51. If x = 2y and. Find the value of y. 52. If a = 2, b= 3 then find the values of the following (a) (ab + ba)–1 (b) (aa + bb)–1 53. If ab + bc + ca = 0, find the value of 10 IX – Mathematics  CHAPTER-1 NUMBER SYSTEM ANSWERS 1. (c) 0 2. (d) 3. (c) 4. (a) 5. (d) 6. 7. Non-terminating and non-repeating 8. 0 9. 3 10. Infinite 11. Irrational 12. Real 13. Hint: or make denominators equal : (other answer are also possible) 14. 15. 16. Hint: Compare powers x=5 17. –1 11 IX – Mathematics 18. 19. Two irrational No. 5.012301234012345........ 5.1378424134876........ (other answers are also possible) 20. (other answers are also possible) 21. 4 22. 23. 6 25. Hint: cubing on both sides = 53 2x + 3 = 125 x = 61 26. 27. 1 28. (a) Terminating decimal (b) Non-terminating but recurring decimal (c) Hint: simplify it first Terminating decimal (d) Non-terminating but recurring decimal 12 IX – Mathematics 29. (a) Terminating decimal/Rational number (b) Terminating decimal/Rational number (c) Non-terminating but repeating/Rational number (d) Non-terminating non-Repeating/Irrational number (e) Non-terminating but Repeating/Rational number. 30. (a) Rational (b) Rational (c) Irrational (d) Rational (e) Irrational 31. (a) (b) (c) 32. Hint: Rationalise the denominator , – 41 33. Hint: 34. Hint: 25(2x – 5) = 22 × 23(x – 5) 210x – 25 = 23x – 15 + 2 10x – 25 = 3x – 13 x= 13 IX – Mathematics 35. (a) Hint : (b) Hint: = (c) 25 (d) 36. Hint: 52x – 1 – 52(x – 1) = 54 × 22 = 54 × 22 x=3 37. Hint: = =4 = ±2 38. Hint: replace y= = −1 −1 1  xz + 1 + x   x + xz + 1  = + 2  +  1 + x + xz  x   x  = = =1 14 IX – Mathematics 39. (a) Hint: 52(2x – 3) = 52x + 3 x= (b) Hint: 22(2x – 1) – 24(x – 1) = 27 × 3 24x – 2 – 24x – 4 = 27 × 3 24x – 2 (1 – 2–2) = 27 × 3 x= 40. Hint: = = = 41. 42. 4 43. 1 44. 1 45. 1 46. Hint: a = 13 − 2 42 b= (a + b) – ab = a2 + b2 + ab 2 a2 + b2 + ab = a2 + b2 + ab = (26)2 – (169 – 168) = 676 – 1 = 675 15 IX – Mathematics 47. (a) 18 (b) (c) 322 (d) (e) Hint: = = 183 – 3 × 18 = 5778 (f) Hint: = = = (g) (h) 4 (i) 135 + 48. (a) 98 2 1  1  1 (b) Hint: P − 2 −40 6  P +  P −  = = P  P  P 2 1  1  (c) Hint: P + 4=  P 2 + 2  − 2= 9602 4 P  P  49. 214 50. Hint: = 16 IX – Mathematics  = 33n – 3m = 3–6 n – m = –2 ∴ m–n = 2 51. Hint: = 360 = 360 32x = 81 x=2 y=1 52. (a) (b) 53. Hint: ab = –(bc + ca); bc = – (ca + ab); ca = –(ab + bc) = = =0 17 IX – Mathematics  CHAPTER-1 NUMBER SYSTEM PRACTICE TEST Time: 1 hr M.M: 20 1. Write one rational number and one irrational number. (1) 2. If p = then find the value of. (1) 5 3. Simplify 4 3 + 3 48 − 12 (2) 2 4. If (5)2x – 1 – (25)x – 1 = 2500 then find the value of x. (2) 5. Find the value of x and y (3) 6. Represent on number line (3) 7. Simplify: (3) 8. Express the following in the form where p and q are integers and q ≠ 0 0.4 + 0.18 + 0.2 (5) 18 IX – Mathematics  CHAPTER-2 POLYNOMIALS MIND MAP 3 19 IX – Mathematics  CHAPTER-2 Polynomials KEY POINTS Definition A polynomial p(x) in one variable x of degree n is an algebraic expression in x of the form p(x) = anxn + an–1xn–1 + an–2xn–2 +.... + a2x2 + a1x + a0, where (i) a0, a1, a2,... an are constants and an ≠ 0 (ii) a0, a1, a2,... an, are respectively the coefficients of x0, x1, x2,................, xn terms of the polynomial. (iii) Each of anxn, an–1xn–1, an–2xn–2,................, a2x2, a1x1 a are called terms of the polynomial. (iv) n is called the degree of the polynomial where n is a non-negative integer. Zeros of Polynomial For a polynomial p(x) if p(a) = 0, where a is a real number we say that ‘a’ is a zero of the polynomial. 1. A polynomial having four or more than four terms does not have particular name. These are simply called polynomials. 2. A polynomial of degree five or more than five does not have any particular name. Such a polynomial is usually called a polynomial of degree five or six or... etc. 3. The degree of zero polynomial is not defined or we can not determine the degree of zero polynomial. Facts about Polynomial: 4. A polynomial of degree ‘n’ can have at most n zeroes. 5. A non-zero constant polynomial has no-zero. 6. Every real number is a zero of the zero polynomial. Very Short Answer type Questions (1 Mark) 1. The coefficient of x2 in the polynomial 4x3 – 7x2 + 2x + 1 is :- (a) 4 (b) 7 (c) –4 (d) –7 20 IX – Mathematics 2. Which of the following is not a polynomial? (a) x + 1 (b) (c) x2 + 1 (d) 3. If x = –1 is a zero of x3 – 2x2 + 3ax + 5, then value of a is :- (a) 2 (b) (c) (d) –5 4. If (x + 2) is a factor of x2 – kx + 14, then find the value of k :- (a) –9 (b) 9 (c) –2 (d) 14 5. When p(x) x3 – 6x2 + 2x – 4 is divided by x – 2 then remainder is :- (a) 16 (b) 24 (c) –16 (d) –24 6. If the side of a square is (x + 2y – z) units, then its area is ___________. 7. The polynomial x2 – a2 has __________ zeroes. 8. A quadratic polynomial can have at most _________ terms. 9. (49)3 – (30)3 + _______ = 3 × 49 × 30 × 19 10. x3 – 64 is a polynomial of degree ________ having ________ terms. 11. Check whether x = 3 is a zero of the polynomial x3 – 3x2 + x – 3 12. If p + q + r = 9, then find the value of (3 – p)3 + (3 – q)3 + (3 – r)3. 13. Find the remainder when x3 + 3x2 + 2x is divided by x. 14. If f(x) = x2 – 3, then find f(1) + f(–1) 15. Find the sum of coefficient of x2 and coefficient of x in the polynomial 3x3 – 4x2 + 5x + 2 21 IX – Mathematics Short Answer Type-I Questions (2 Marks) 16. Check whether q(x) is a multiple of r(x) or not. Where q(x) = 2x3 – 11x2 – 4x + 5, r(x) = 2x + 1. 17. Show that (x – 5) is a factor of x3 – 3x2 – 4x – 30. 18. Evaluate by using suitable identity: (997)2 19. Find the zeroes of the polynomial p(x) = x(x – 2) (x + 3) 20. Find the remainder when 3x2 – 7x – 6 is divided by (x – 3) 21. Factorise : 22. If p(x) = x + 9, then find p(x) + p(–x) 23. Find the product without multiplying directly 106 × 94 24. The factors of 5x2 – 18x + 9 are (ax + b) and (x + b). Find the values of a and b. 25. Find p (1) + p (–1) + p(10) if p(x) = x2 – 3x + 2 2 26. Find (x – y) if 27. Show that –1 is a zero of 3x4 – x3 + 3x – 1. 28. Multiply (x + 1) (x – y) Short Answer Type-II Questins (3 Marks) 29. Factorise: 64a2 + 96ab + 36b2 30. Facrotise: x3 + 6x2 + 11x + 6 31. If x2 + y2 = 49 and x – y = 3, then find the value of x3 – y3. 32. Simplify: (5a – 2b) (25a2 + 10ab + 4ab2) – (2a + 5b) (4a2 – 10ab + 25b2) 33. Find the sum of remainders when x3 – 3x2 + 4x – 4 is divided by (x – 1) and (x + 2). 34. Find the product of 35. Factorise: 36. Simplify: (3x – 4y)3 – (3x + 4y)3 37. Simplify: (x + y + z)2 – (x – y – z)2. 22 IX – Mathematics 38. Factorise: 125x3 + 8y3 + z3 – 30xyz 39. x + 2 is a factor of polynomial ax3 + bx2 + x – 2 and the remainder 4 is obtained on dividing this polynomial by (x – 2). Find the value of a and b. 40. If the polynomials ax3 + 4x2 + 3x – 4 and x3 – 4x + a leave the same remainder when divided by (x – 3), find the value of a. 41. If , find x 42. If (x – 3) and are factors of the polynomial px2 + 3x + r, show that p = r. Long Answer type Questions (5 Marks) 43. A literacy campaign was organised by Class IX girl students under NSS. Students made (x – 5) rows and (3x – 4) columns for the rally. Write the total number of students in the form of a polynomial. 3 3 3 44. (i) Using identity, find the value of (–7) + (5) + (2). (ii) Find dimensions of cuboid whose volume is given be the expression 4x2 + 14x + 6. 45. If a + b + c = 0, find the value of 46. Simplify: 47. Factorize (2a – b – c)3 + (2b – c – a)3 + (2c – a – b)3 48. If the polynomial 4x3 – 16x2 + ax + 7 is exactly divisible by x – 1, then find the value of a. Hence factorise the polynomial. 49. If where x ≠ 0, y ≠ 0 then find the value of x3 – y3 50. Simplify: 23 IX – Mathematics  CHAPTER-2 POLYNOMIAL Answer 1. (d) – 7 2. (b) 3. (b) 4. (a) – 9 5. (c) – 16 6. x2 + 4y2 + z2 + 4xy – 4yz – 2xz 7. Two 8. Three 9. (–19)3 10. 3, 2 11. Yes 12. p + q + r = 9 (3 – p) + (3 – q) + (3 – r) = 0 ∴ (3 – p)3 + (3 – q)3 + (3 – r)3 = 3(3 – p) (3 – q) (3 – r) 13. 0 14. f(1) + f(–1) = (–2) + (–2) = –4 15. (–4) + (5) = 1 16. Since, ∴ r(x) is not a multiple of q(x). 24 IX – Mathematics 17. Put x = 5 in given polynomial 18. 994009 19. 0, 2, –3 20. –18 21. ( 2x + )( 3 y 4 x 2 − 2 3 xy + 3 y 2 ) 22. 18 23. (100 + 6) (100 – 6) = 9964 24. a = 5, b = –3 25. 8 26. 0 28. x2 – xy + x – y 29. (8a + 6b)2 30. (x + 1) (x + 2) (x + 3) 31. 207 32. 117a3 – 133b3 33. –34 34. 35. 36. – 128y3 – 216x2y 37. 4xy + 4xz 38. (5x + 2y + z) (25x2 + 4y2 + z2 – 10xy – 2yz – 5zx) 25 IX – Mathematics 39. a = 0, b = 1 40. a = –1 41. x = 27, {use, if a + b + c = 0 then a3 + b3 + c3 = 3abc} 43. 3x2 – 19x + 20 44. (i) – 210, (ii) 2, (x + 3), (2x + 1) 45. 3 46. (a + b) (b + c) (c + a) 47. 3 (2a – b – c) (2b – c – a) (2c – a – b) 48. a = 5, (x – 1) (2x + 1) (2x – 7) 49. 0 (155) + 155 × 55 + ( 55) (155) − ( 55) 2 2 3 3 50. = (155) − ( 55) (155 − 55) ( (155) − ( 55) ) 3 3 3 3 1 = = 0.01 100 26 IX – Mathematics  POLYNOMIALS PRACTICE TEST Time: 1 hr. M.M. 20 1. Show that x = 1 is a zero of the polynomial 3x3 – 4x2 + 8x – 7. (1) 2. Find the value of the polynomial 2x + 5 at x = –3. (1) 3. Find the zeroes of the polynomial x2 – 4x + 3. (2) 4. If x + y + z = 6, xy + yz + zx = 11. Find the value of x2 + y2 + z2. (2) 5. If 3x – 4 is a factor of the polynomial p(x) = 2x3 – 11x2 + kx – 20, find the value of k. (3) 6. Factorise: a2 + b2 + 2(ab + bc + ca) (3) 7. Factorise: 2 2a 3 + 8b3 − 27c 3 + 18 2abc (3) 8. Factorise: (5) (i) 4x2 + 20x + 25 (ii) 6x2 + 7x – 3 27 IX – Mathematics  CHAPTER-3 CO-ORDINATE GEOMETRY MIND MAP 28 IX – Mathematics Key Points Co-ordinate Geometry is the branch of Mathematics in which we study the position of any object lying in a plane, called the Cartesian plane. In Cartesian system; there are two mutually perpendicular straight lines xx and yy intersecting at origin O. These mutually perpendicular straight lines, known as x-axis and y-axis, divides the plane into four quadrants. The coordinates of a point is the position of the point in Cartesian plane and are determined by perpendicular distance from x-axis and y-axis. The perpendicular distance of a point from y-axis is called abscissa (x-coordinate) and from x-axis is called ordinate (y-coordinate). Any point in the Cartesian plane is shown by P(a, b) where (a, b) are coordinates of point P. abscissa (x) ordinate (y) Position of point positive (+) positive (+) Quadrant I positive (+) negative (–) Quadrant IV negative (–) negative (–) Quadrant III negative (–) positive (+) Quadrant II The coordinate of a point on x-axis is of the form (x, 0) and on y-axis is of the form (0, y). If x-coordinate of two or more points are same, then the line joining these points is parallel to y-axis. If y-coordinate of two or more points are same, then the line joining these points is parallel to x-axis. NOTE: If a point lie on x-axis or y-axis then it does not lie in any quadrant. The mirror image of a point is just a reflection of this point about one of the axes. Mirror image about x-axis: sign of abscissa remains same but sign of ordinate changes. Mirror image about y-axis: sign of abscissa changes but sign of ordinate remains same. Mirror image about origin: signs of both-abscissa and ordinate changes. 29 IX – Mathematics Very Short Answer Questions (1 mark) 1. The abscissa of a point is the distance of the point from (a) x-axis (b) y-axis (c) origin (d) None of these 2. The y-coordinate of a point is the distance of that point from (a) x-axis (b) y-axis (c) origin (d) None of these 3. If both the coordinates of a point are negative then that point will lie in (a) First quadrant (b) Second quadrant (c) Third quadrant (d) Fourth quadrant 4. If abscissa of a point is zero then that point will lie (a) on x-axis (b) on y-axis (c) at origin (d) in Ist quadrant 5. If x > 0 and y < 0, then the point (x, – y) lies in (a) I quadrant (b) II quadrant (c) III quadrant (d) IV quadrant 6. Point (a, 0) lies (a) on x-axis (b) on y-axis (c) in third quadrant (d) in fourth quadrant 7. The signs of abscissa and ordinate of a point in the second quadrant are respectively. (a) + , + (b) – , – (c) – , + (d) + , – 8. The ordinate of a point is positive in (a) I and IV quadrants (b) I quadrant only (c) I and II quadrants (d) I and III quadrants 9. The point which lies on y-axis at a distance of 10 units in the negative direction of y-axis is (a) (10, 0) (b) (0, 10) (c) (–10, 0) (d) (0, –10) 30 IX – Mathematics 10. The end points of a line lies in I quadrant and III quadrant. The line may pass through (a) origin (b) negative x-axis (c) positive y-axis (d) quadrant II 11. The point whose abscissa and ordinate have different signs will lie in (a) I and II quadrants (b) I and III quadrants (c) II and III quadrants (d) II and IV quadrants 12. Which of the point P(0, 3), Q(1, 0), R(0, –1), S(–5, 0), T(1, 2) do not lie on x-axis? (a) P and R only (b) Q and S only (c) P, R and T (d) Q, S and T 13. If the coordinates of the points are P(–2, 3) and Q(–3, 5) then (abscissa of P) – (abscissa of Q) is (a) –5 (b) 1 (c) –1 (d) –2 14. Point (1, 1), (1, –1), (–1, 1), (–1, –1) (a) lie in I quadrant (b) lie in III quadrant (c) lie in I and III quadrants (d) do not lie in the same quadrant 15. The point of intersection of the coordinates axes is (a) Abscissa (b) Ordinate (c) Quadrant (d) Origin 16. The abscissa and ordinate of the origin are (a) (1, 0) (b) (1, 1) (c) (0, 1) (d) (0, 0) 17. The angle formed between the coordinate axes is (a) Zero angle (b) Right angle (c) Acute angle (d) Obtuse angle 18. The perpendicular distance of the point p(–4, –3) from x-axis is (a) –4 units (b) –3 units (c) 4 units (d) 3 units 31 IX – Mathematics 19. The perpendicular distance of the point p(–7, 2) from y-axis is (a) –7 units (b) 7 units (c) 2 units (d) –2 units 20. The distance of the point p(3, 4) from the origin is (a) 3 units (b) 4 units (c) 7 units (d) 5 units 21. Which of the points A(–5, 0), B(0, –3), C(3, 0), D(0, 4) are closer to the origin (a) A (b) B (c) D (d) Points B and C both 22. The mirror image of the point (0, 3) along y-axis is (a) (0, –3) (b) (0, 3) (c) (3, 0) (d) (–3, 0) 23. The coordinate axes divide the plane into four parts, each part is called _______________. 24. It the coordinates of a point are (–2, 5), then its ordinate is _______________ and its abscissa is _______________. 25. The point (200, –111) lies in the _______________ quadrant. 26. The abscissa of any point on the y-axis is _______________. 27. The ordinate of any point on the x-axis is _______________. 28. The points (0, 0), (0, 4) and (4, 0) form a/an _______________ triangle. 29. If (x, y) represents a point and xy > 0, then the point may lie in _______________ or _______________ quadrant. 30. The points with coordinates (3, –1) and (–1, 3) are at _______________ (same/ different) positions of the coordinate plane. 31. If the ordinate of points is 7 and abscissa is –5, then its coordinates are _______________. 32. The coordinates of a point lying on x-axis having abscissa 5 are _______________. 33. The co-ordinates of point describe the point in the place _______________. 34. The coordinates of a point, which lies on negative x-axis at a distance of 6 units from y-axis, are _______________. 32 IX – Mathematics 35. If the coordinates of the points are P(0, –1) and Q(2, 1) then (abscissa of P) – (abscissa of Q) is _______________. 36. The measure of the angle between coordinate axes is _______________. 37. In which quadrant do the given points lie. (i) (3, –2) (ii) (17, –30) (iii) (–2, 5) (iv) (–50, –20) (v) (10, 100) (vi) (–81, 80) 38. On which axis do the given points lie: (i) (11, 0) (ii) (–11, 0) (iii) (0, – 100) (iv) (0, 14) 39. The abscissa and ordinate of a point A are –3 and –5 respectively then write down the coordinates of A. 40. Do P(7, 0) and Q(0, 7) represent the same point? 41. In which quadrant x coordinate is negative? 42. Name the figure formed when we plot the points (0, 0), (4, 4) and (0, 4) on a graph paper. 43. In which quadrant, does the point A(x, y) with values x > 0 and y > 0 exists? 44. Write the coordinates of the fourth vertex of a square when three of its vertices are given by (1, 2) (5, 2) (5, –2). 45. If abscissa of any point is positive & ordinate is negative then in which quadrant do the point lie? 46. Write the coordinates of point whose perpendicular distance from x-axis is 5 units & perpendicular distance from y-axis is 3 units & it lies in II quadrant. 47. In which quadrant will a point lie if its both the coordinates are positive? 48. Write the coordinates of the point at which two coordinate axes meet. 49. Write the coordinates of the point which lies at a distance of x-units from x-axis and y units from y-axis. 50. Find the coordinates of the point which lies on x-axis at a distance of 5 units from y-axis. 51. Find the coordinates of the point which lies on y-axis at a distance of 9 units from x-axis in the negative direction. 33 IX – Mathematics 52. In which quadrant of a Cartesian plane the ordinate of a point will be positive and abscissa will be negative? 53. On which axis the point A(–3, 0) lies? 54. Which axis is parallel to the line joining the points (2, 4) and (2, –5)? 55. Find the image of the point (2, 3) about x-axis. 56. Find the mirror image of the point (–5, 6) about y-axis. 57. In which quadrant the mirror image of (–1, –4) lie about y-axis? 58. A point is in II quadrant. In which quadrant will its mirror image lie along x-axis? Short answer type-I questions (2 marks) 59. Find the co-ordinates of two points on x-axis and two points on y-axis which are at equal distance from the origin. 60. Name the quadrant in which the graph of point A(x, y) lies when (i) x > 0 and y > 0 (ii) x < 0 and y < 0 61. Find the coordinates of the vertices of a rectangular figure placed in III quadrant in the Cartesian plane with length p unit on x-axis and breadth q units on y-axis. 62. Write the coordinates of any two points on the line segment joining the points A (4, –1) and B(4, 5). Short answer type-II questions (3 marks) 63. If we plot the points P(5, 0), Q(5, 5), R(–5, 5) and S(–5, 0), which figure will we get? Name the axis of symmetry of this figure? 64. Find the coordinates of a point which is equidistant from the two points (–4, 0) and (4, 0). How many of such points are possible satisfying this condition? 65. A rectangular field is of length 10 units & breadth 8 units. One of its vertex lie on the origin. The longer side is along x-axis and one of its vertices lie in first quadrant. Find all the vertices. 66. Name the figure obtained by joining the points B(5, 3), E(5,1), S(0, 1) and T(0, 3). Also find the area of the figure. 67. Plot the point P(–5, 4) and from it draw PM and PN as perpendicular to x-axis and y-axis respectively. Write the coordinates of the points M and N. 34 IX – Mathematics  CHAPTER-3 CO-ORDINATE GEOMETRY Hints and Solutions 1. (b) y-axis 26. 0 2. (a) x-axis 27. 0 3. (c) Third quadrant 28. isosceles 4. (b) y-axis 29. I or III 5. (d) IV quadrant 30. different 6. (a) on x-axis 31. (–5, 7) 7. (c) – , + 32. (5, 0) 8. (c) I and II quadrants 33. uniquely 9. (d) (0, –10) 34. (–6, 0) 10. (a) origin 35. –2 11. (d) II and IV quadrants 36. 90° 12. (c) P, R and T 37. (i) & (ii) IV quadrant 13. (b) 1 (iii) & (vi) II quadrant 14. (d) do not lie in same quadrant (iv) III quadrant 15. (d) origin (v) I quadrant 16. (d) (0, 0) 38. (i) & (ii) x-axis 17. (b) Right angle (iii) & (iv) y-axis 18. (d) 3 units 39. (–3, –5) 19. (b) 7 units 40. No because abscissa and ordinates 20. (d) 5 units are different for both the points. 21. (d) points B and C both 41. II and III 22. (b) (0, 3) 42. Triangle 23. quadrant 43. I quadrant 24. 5, –2 44. (1, –2) 25. IV quadrant 45. IV quadrant 35 IX – Mathematics 46. (–3, 5) 47. I quadrant 48. (0, 0) 49. (y, x) 50. (5, 0) 51. (0, –9) 52. II quadrant 53. x-axis 54. y-axis 55. (2, –3) 56. (5, 6) 57. IV quadrant 58. III quadrant 59. (± 9, 0), (0, ± a) where a is any real number 60. (i) I quadrant (ii) III quadrant 61. (0, 0), (–p, 0), (–p, –q), (0 –q) 62. Any two point with abscissa = 4 and ordinate lying between –1 and 5. 63. Rectangle, y-axis 64. Any point on y-axis, infinite 65. (0, 0), (10, 0), (10, 8), (0, 8) 66. Figure : Rectangle Area : 10 sq. units. 67. M (–5, 0) N (0, 4) 36 IX – Mathematics  PRACTICE TEST COORDINATE GEOMETRY Time: 1 hr. M.M.: 20 1. In which quadrant, the point (x, y) will lie, where x is positive and y is negative number? (1) 2. Write the coordinate of a point at a distance of 5 units from x-axis lying in II quadrant. (1) 3. Find the value of x and y if: (2) (a) (x – 4, 7) = (4, 7) (b) (1, 2y – 3) = (1, 7) 4. What is the distance of a point (7, –6) from x-axis and y-axis? (2) 5. In which quadrant, do the following points lie? (3) (i) (4, –2) (ii) (–3, 7) (iii) (–1, –2) 6. Write the mirror image of following points along x-axis. (3) (–3, 5) , (2, 0) , (–4, –7) 7. Consider the points O(0, 0), A(4, 0) and B(4, 6). Find the length of OA and AB. Find the coordinates of the fourth point C such that OABC forms a rectangle. (3) 8. The base AB of two equilateral triangles ABC and ABD with side 2a, lies along the x-axis such that the mid point of AB is at the origin. Find the coordinates of two vertices C and D of the triangles. Which type of Quadrilateral in ABCD? (5) 37 IX – Mathematics  CHAPTER-4 LINEAR EQUATIONS IN TWO VARIABLES MIND MAP O 38 IX – Mathematics Key points Linear equation in one variable: An equation which can be written in the form ax + b = 0, where a, b are real numbers and a ≠ 0 is called a linear equation in one variable. Linear equation in two variables: An equation which can be written in the form ax + by + c = 0, where a, b and c are real numbers and a, b ≠ 0, is called a linear equation in two variables. Linear equation in one variable has a unique solutions. ax + b = 0 ⇒ Linear equation in two variables has infinitely many solutions. The graph of every linear equation in two variables is a straight line. Every point on the line satisfies the equation of the line. Every solution of the equation is a point on the line. Thus, a linear equation in two variables is represented geometrically by a line whose points make up the collection of solutions of the equation. Graph The pair of values of x and y which satisfies the given equation is called solution of the linear equation in two variables. Example: x + y = 4 Solutions of equation x + y = 4 are (0, 4) (1, 3) (2, 2) (4, 0) and many more. Very Short Answer Questions (1 Mark) 1. Which of the following is not a linear equation? (a) 3x + 3 = 5x + 2 (b) x2 + 5 = 3x – 5 7 (c) x − 5 = 4x − 3 (d) (x + 2)2 = x2 – 8 3 39 IX – Mathematics 2. Which of the following is not a linear equation in two variables? (a) 2x + 3y = 5 (b) 3t + 2s = 6 (c) ax2 + by = c (d) ax + by = c 3. A linear equation in two variables has maximum (a) Only one solution (b) Two solutions (c) Infinite solutions (d) None of these 4. The graph of ax + by + c = 0 is (a) a straight line parallel to x-axis (b) a straight line parallel to y-axis (c) a general straight line (d) None of these 5. If x = 1, y = 1 is a solution of equation 9ax + 12ay = 63, then the value of a is (a) 3 (b) 0 (c) –3 (d) 4 6. The equation of x-axis is (a) x = k (b) x = 0 (c) y = k (d) y = 0 7. Any point on the line y = x is of the form (a) (a, 0) (b) (0, a) (c) (a, a) (d) (a, –a) 8. The equation x = 0 represents – (a) x-axis (b) y-axis (c) a line parallel to x-axis (d) a line parallel to y-axis 9. Which of the linear equation has solution as x = 2, y = 3? (a) 2x + y = 8 (b) x + 2y = 8 (c) x + y = 8 (d) –x + y = 8 10. The graph of 2x + 3y = 6 is a line which meets the y-axis at the point. (a) (2, 0) (b) (3, 0) (c) (0, 2) (d) (0, 3) 40 IX – Mathematics 11. At what point, the graph of 3x + 2y = 9, cuts the y-axis? 12. Let y varies directly as x. If y = 15 when x = 5, then write a linear equation. 13. Write the point of intersection of the lines x = 2 and y = –3 14. What is the distance of the point (3, –7) from x-axis? 15. What is the distance of the point (–5, –4) from y-axis? 16. Express the linear equation 2 x − 4 = 5 y in the form of ax + by + c = 0 and thus indicate the values of a, b and c. 17. Express x in terms of y for the equation 3x + 4y = 7. 18. Express y in the terms of x. 3y + 5x = 9 19. Point (9, 0) lie on which axis? 20. Find a solution of x + y = 5 which lies on y-axis. 21. Express the equation 5y = 9 as linear equation in two variables. 22. Write the linear equation which is parallel to x-axis and is at a distance of 2 units from the origin in upward direction. 23. Check whether (1, –2) is a solution of 2x –y = 6. 24. Check whether x = 2 and y = 2 is a solution of 2x + y = 6. 25. How many solutions are there for equation y = 5x + 2. 26. Find the value of K, if x = –1 and y = 1 is a solution of equation Kx – 2y = 0 27. If the graphs of equation 2x + Ky = 10K intersects x-axis at point (5, 0), find the value of K. 28. The graph of the linear equation 4x = 6 is parallel to which axis? 29. At what point the graph of 2x – y = 6, cuts x-axis? 30. On which side of y-axis, x + 3 = 0 lies? 31. On which side of x-axis, 2y – 1 = 0 lies? 41 IX – Mathematics Fill in the blanks: 32. (a) The equation of a line parallel to x-axis is ____________ = a, where a is any non-zero real number. (b) The equation of a line parallel to y-axis is ____________ = a, where a is any non-zero real number. 33. The graph of every linear equation in two variables is a ____________. 34. An equation of the form ax + b = 0, where a, b are real numbers and a ≠ 0, in the variable x, geometrically represents ____________. 35. The coefficient of x in the linear equation 2 (x + y) – x = 7 is ____________ 36. State whether the following statements are true or false :- (a) The linear equation 7x + 9y = 8 has a unique solution (b) All the points (2, 0), (–3, 0), (4, 2) lie on the x-axis (c) The line parallel to y-axis at a distance of 5 units to the left of y-axis is given by the equation x = –5. (d) The graph of every linear equation in two variables need not be a line. (e) The graph of the linear equation x + 2y = 5 passes through the point (0, 5) Short Answer Type-I Questions (2 marks) 37. Find any two solutions of equation 2x + y = x + 5 38. Find the value of P if x = 2, y = 3 is a solution of equation 5x + 3 Py = 4a 39. If the points A(3, 5) and B(1, 4) lies on the graph of line ax + by = 7, find the value of a. 40. Write the coordinates of the point where the graph of the equation 5x – 2y = 10 intersect both the axes. 41. Write the equations of two lines passing through (3, 10). 42. The cost of coloured paper is 7 more than 1/3 of the cost of white paper. Write this statements in linear equation in two variables. 43. Draw the graph of equation x + y = 5. 42 IX – Mathematics 44. The graph of linear equation 2x – y = 6 will pass through which quadrants(s). 45. How many solution of the equations 3x – 2 = x – 3 are there on the (i) Number line (ii) Cartesian plane... 46. Find the points where the graph of x + y = 4 meets line which is (i) parallel to x-axis at 3 units from origin in positive direction of y-axis. (ii) parallel to y-axis at 2 units on left of origin. Short Answer Type-II Questions (3 marks) 47. If total number of legs in a herd of goats and hens is 40. Represent this situation in the form of a linear equation in two variables. 48. Find the value of a and b, if the line 6ax + by = 24 passes through, (2, 0) and (1, 2) 49. Determine the point on the graph of the linear equation 2x + 5y = 19 whose ordinate is 1½ times its abscissa. 50. Find the points where the graph of the following equation cuts the x-axis and y-axis 2x = 1– 5y. 51. Write the equation of the line parallel to x-axis at a distance of 4 units above the origin. 52. If the points A(4, 6) and B(1, 3) lie on the graph of ax + by = 8 then find the value of a and b. 53. Find the value of ‘a’ if (1, –1) is the solution of the equation 2x + ay = 5. Find two more solutions of the equation. 54. Find two solutions of the equation 4x + 5y = 28. Check whether (–2, 10) is solution of the given equation. 55. Write the equation of line passing through (3, –3) and (6, –6). 56. If x = 3k – 2, y = 2k is a solution of equation 4x – 7y + 12 = 0, then find the value of K. 57. If (m –2, 2m + 1) lies on equation 2x + 3y – 10 = 0, find m. 43 IX – Mathematics 58. F = (9/5)C + 32, where F is temperature Fahrenheit and C is temperature in Celsius. (i) If the temperature is 35°C, what is the temperature in Fahrenheit? (ii) If the temperature is 30°C, what is the temperature in Fahrenheit? 59. Draw the graph of the linear equation 2x + 3y = 6. Find out the coordinates of the points where the line intersets x-axis and y-axis. 60. Draw the graph for the linear equation 3x + 4y = 12. If x = 8, find the value of y with the help of graph. 61. Draw the graph of y = x and 2y = –5x on the same graph. 62. Give the geometrical representation of 5x + 7 = 0 as equation. (i) in one variable (ii) in two variables 63. Draw the graph of the linear equations 2y – x = 7. With the help of graph check whether x = 3 and y = 2 is the solution of the equation? 64. Draw the graph of linear equation 3x – y = 4. From the graph find the value of p and q if the graph passes through (p, –4) and (3, q) 65. Draw the graph of equations 2x + 3y = –5 and x + y = –1 on the same graph. Find the co-ordinate of the point of intersection of two lines. 66. Show that the points A(1, –1), B(2, 6) and C(0, –8) lie on the graph of the linear equation 7x – y = 8. Long answer type questions (5 Marks) 67. Write 3y = 8x in the form of ax + by + c = 0. Write x in terms of y. Find any two solutions of the equation. How many solutions you can find out? 68. Rohan and Ramita of Class IX decided to collect \25 for class cleanliness. Write it in linear equation in two variables. Also draw the graph. 44 IX – Mathematics 69. Sarika distributes chocolates on the occasion of children’s Day. She gives 5 chocolates to each child and 20 chocolates to adults. If number of children is represented by ‘x’ and total distributed chocolates as ‘y’. (i) Write it in the form of linear equation in two variables. (ii) If she distributed 145 chocolates in total, find number of children? 70. Priyanka and Arti decided to donate \1600 for the Army widows. Assuming Priyanka’s share as ‘x’ and Arti’s share as ‘y’. (a) Form a linear equation in two variables. (b) If Priyanka donates thrice the amount donated by Arti, then find out the amount donated by both. 71. Riya participates in Diwali Mela with her friends for the charity to centre of handicapped children. They donate \3600 to the centre from the amount earned in Mela. If each girl donates \150 and each boy donates \200. (a) Form the linear equation in two variables. (b) If number of girls are 8, find number of boys. 72. Aftab is driving a car with uniform speed of 60 km/hr. Assuming total distance to be y km and time taken as x hours, form a linear equation. Draw the graph. From the graph read the following: (i) distance travelled in 90 minutes. (ii) Time taken to cover a distance of 150 km. 73. The parking charges of a car in a private parking is \20 for the first hour and \10 for subsequent hours. Taking total parking charges to be y and total parking time as x hours form a linear equation. Write it in standard form and indicate the values of a, b and c. Draw the graph also. 74. We know that C = 2πr, taking π = 22/7, circumference as y units, radius as x units, form a linear equation. Draw the graph. Check whether the graph passes through (0, 0). From the graph read the circumference when radius is 2.8 units. 45 IX – Mathematics  CHAPTER-4 LINEAR EQUATIONS IN TWO VARIABLES Answers 1. (b) x2 + 5 = 3x – 5 2. (c) ax2 + by = c 3. (c) Infinite solutions 4. (c) a general straight line 5. (a) 3 6. (d) y = 0 7. (c) (a, a) 8. (b) y-axis 9. (b) x + 2y = 8 10. (c) (0, 2) 11. (0, 4.5) 12. y = 3x 13. (2, –3) 14. 7 units 15. 5 units 16. √2x – 5y – 4 = 0 a = √2, b = –5, c = –4 17. 9 − 5x 18. y = 3 19. x-axis 20 (0, 5) 21. 0x + 5y = 9 46 IX – Mathematics 22. y = 2 23. No 24. Yes 25. Infinitely many solutions 26. K(–1) –2 (1) = 0 k = –2 27. 2(5) + k(0) = 10k k=1 28. Parallel to y-axis 29. (3, 0) 30. On left side 31. On right side 32. (a) y (b) x 33. Straight line 34. a point on number line 35. 1 36. (a) F (b) F (c) T (d) F (e) F 37. (1, 4) (0, 5) (or any other possible solutions) 38. As x = 2, y = 3 is a solution 5(2) + 3p(3) = 4a 10 + 9p = 4a p= 39. 3a + 5b = 7; a + 4b = 6 3(7 – 4b) + 5b = 7 b = 2, a = –1 47 IX – Mathematics 40. Graph of 5x – 2y = 10 will intersect x-axis when y = 0 ie 5x – 2(0) = 10 ⇒ 5x = 10 x = 2 i.e pt. (2, 0) Similarly for y-axis put x = 0 i.e. 5 (0) – 2y = 10 y = –5 i.e pt (0, –5) 41. 3x – y + 1 = 0, x + y = 13 (or any other possible equation) 42. Let the cost of coloured paper be ` x Let the cost of white paper be ` y, then A/Q x = 1/3 y + 7 or 3x = y + 21 43. x + y = 5 x 0 5 1 y 5 0 4 44. I, IV, III 48 IX – Mathematics 45. (i) 3x – 2 = x – 3 ⇒ x= On number line only one solution i.e., (ii) On Cartesian plane infinitely many solutions i.e., 1. x + 0. y = x y –1 0 1 (A line parallel to y-axis) 49 IX – Mathematics 46. (i) (1, 3) (ii) (–2, 6) 47. Let number of goats = x Number of hens = y 4x + 2y = 40 or 2x + y = 20 50 IX – Mathematics 48. 6a(2) + b(0) = 24 12a = 24 a=2 6(2)(1) + b(2) = 24 12 + 2b = 24 2b = 12 b=6 49. Let required pt. be (x′, y′) A/Q, y′ = ------- (1) (x′, y′) lies on graph of 2x + 5y = 19 2x′ + 5y′ = 19 ------- (2) from (1) and (2) = 19 4x′ + 15x′ = 38 ⇒ x′ = 2 y′ = ×2=3 point will be (x′, y′) ie (2, 3) 50. cuts x-axis at , cuts y-axis at 51. y = 4 52. 4a + 6b = 8 or 2a + 3b = 4 a + 3b = 8 After solving → a = –4 and b = 4 53. 2(1) + a(–1) = 5 –a = 3 a = –3 2x – 3y = 5, two solutions are (7, 3) and (10, 5) or any two solutions possible. 51 IX – Mathematics 54. (2, 4); (7, 0) [or any other possible solution] (–2, 10) 4x + 5y = 28 L.H.S R.H.S. 4(–2) + 5(10) 28 = –8 + 50 = 42 L.H.S. ≠ R.H.S ⇒ (–2, 10) is not a solution of equation 4x + 5y = 28 55. x + y = 0 56. 4[3k – 2] –7[2k] + 12 = 0 12k – 8 – 14k + 12 = 0 k=2 57. 2 [m – 2] + 3[2m + 1] – 10 = 0 2m – 4 + 6m + 3 – 10 = 0 m= 58. (i) F = C + 32 when C = 35° F= (35) + 32 F = 95° F (ii) F = (30) + 32 = 9 × 6 + 32 = 86° F 52 IX – Mathematics 59. 2x + 3y = 6 x-axis co-ordinates (3, 0) ; y-axis co-ordinates (0, 2) 60. y = –3 62. x = or x = –1.4 (i) (ii) 53 IX – Mathematics 63. No 64. Graph of 3x – y = 4 passes through (p, –4) ⇒ (p, –4) lies on line of graph of 3x – y = 4, when y = –4 , x = 0 ⇒ p=0 Similarly (3, q) lies on this line when x = 3, y = 5 ⇒ q=5 65. 2x + 3y = –5 ⇒x=.... (1) x –2.5 –4 –1 y 0 1 –1 x + y = –1 x 0 –1 1 y –1 0 –2 Point of intersection is (2, –3) 54 IX – Mathematics 67. 8x – 3y + 0 = 0; x = (0, 0) (3, 8) Infinitely many solutions. 68. x + y = 25 [where x-Rohan’s collection and y-Ramita’s collection 69. (i) 5x + 20 = y (ii) 25 70. (a) x + y = 1600 (b) Priyanka = s1200 [ x = 3y] Arti = s400 71. (a) 150x + 200y = 3600 (b) Number of boys = 12 72. Using speed = y = 60x (i) 90 km (ii) 2 hours 30 min. 73. 20 + 10 (x – 1) = y 20 + 10x – 10 = y 10x – y + 10 = 0 a = 10, b = –1, c =10 74. y = 2πx yes when r = 2.8 units c = 17.6 units 55 IX – Mathematics  Chapter - 4 Linear Equations in Two Variables Practice Test Time: 1 hr. M.M.: 20 1. The graph of linear equation 2y = 5 is parallel to which axis? (1) 2. Write the linear equation of the graph which is parallel to y-axis and is at a distance 3 units on left from the origin (1) 3. Find the value of a and b if the line 5bx – 3ay = 30 passes through (–1, 0) and (0, –3) (2) 4. Write two linear equations passing through the points (2, –3) (2) 5. Write the linear equations x + = 4 in the form of ax + by + c = 0 and hence write the values of a, b and c. Write also x in terms of y (3) 6. Find the solutions of linear equation 2x + y = 4 which represents a point on/ which (3) (i) x-axis (ii) y-axis (iii) is at 3 unit perpendicular distance above x-axis 7. Give the geometrical representation of 2x + 5 = 0 as a linear equation in (3) (a) one variable (b) two variables 8. A taxi charges `15 for first kilometer and `8 each for every subsequent kilometer. For a distance of x km, an amount of `y is paid. Write the linear equation representing the above information and draw the graph. (5) 56 IX – Mathematics  Chapter-5 INTRODUCTION TO EUCLID’S GEOMETRY Key points Introduction: Euclidean geometry, which is taught today is named after Euclid – he is known as “the father of geometry”. Euclid also studied and contributed in other areas of mathematics, including number theory and astronomy. Axiom or Postulates: Axiom or Postulates are the assumptions which are obvious universal truths. They are not proved. Theorems: Theorems are statements which are proved, using definitions, axioms, previously proved statements and deductive reasoning. Some of Euclids Axioms 1. Things which are equal to the same thing are equal to one another. 2. If equals are added to equals, the whole are equal. 3. If equals are subtracted from equals, the remainders are equal. 4. Things which coincide with one another are equal to one another. 5. The whole is greater than the part. 6. Things which are double of the same things are equal to one another 7. Things which are halves of the same things are equal to one another. 57 IX – Mathematics Euclid’s Postulates and Definitions Postulates 1: A straight line may be drawn from any one point to any other points. Postulate 2: A terminated line can be produced indefinitely. Postulate 3: A circle can be drawn with any centre and any radius. Postulate 4: All right angles are equal to one-another. Postulate 5: If a straight line falling on two straight line makes the interior angles on the same side of it taken together less than two right angles, then the two straight lines, if produces indefinitely, meet on that side on which the sum of angles is less than two right angles. Definitions 1. A Point is that which has no part. 2. A line is breadthless length. 3. The ends of a line are points 4. A straight line is a line which lies evenly with the points on it self. 5. A surface is that which has length and breadth only. 6. The edges of a surface are lines. 7. A plane surface is a surface which lies evenly with the straight lines on it self. Very Short Answer type Questions (1 Marks) 1. Through two points: (a) A unique line can be drawn (b) No line can be drawn (c) Two lines can be drawn (d) More than two lines can be drawn 2. Euclid arranged all known work in the field of mathematics in his treatise called: (a) Elements (b) Axioms (c) Theorems (d) Postulates 3. Things which are double of the same things are: (a) Halves of the same thing (b) Double of the same thing (c) Equals (d) Four times of the same thing 58 IX – Mathematics 4. A mathematical statement whose truth has been logically established is called: (a) An Axiom (b) A postulate (c) A Theorem (d) None of the above 5. Two lines having a common point are called: (a) parallel lines (b) intersecting lines (c) coincident (d) None of the above 6. A proof is required for __________ (Postulate, Axioms, Theorem) 7. The number of line segments determined by three collinear points is __________ (Two, three, only one) 8. Euclid stated that if Equals are subtracted from equal then the remainders are equal in the form of __________ (an axiom, a definition, a postulate) 9. A point has __________ dimensions. 10. There are __________ number of Euclid’s postulates. 11. Write the number of dimensions, that a surface contain. 12. In given figure AB = CD then AC and BD are equal or not? A B C D 13. How many lines can pass through a single point? 14. Write Euclid’s fifth postulate. 15. If a + b = 15 and a + b + c = 15 + c which axiom of Euclid does the statement illustrate? Short Answer type-I Questions (2 Marks) 16. If x + y = 10 and x = z then show that z + y = 10 17. In given figure AX = AY, AB = AC show that BX = CY 59 IX – Mathematics 18. In the given figure ∠ABC= ∠ACB, ∠3 = ∠4 show that ∠1 = ∠2 A D 4 3 1 2 B C 19. In the given figure if AD = CB then prove that AC = BD A C D B 20. Solve the equation x – 10 = 15, state which axiom do you use here. 21. In the given figure if AM = AB, AN = AC and AM = AN then show that AB = AC A M N B C 22. In the given figure AC = DC, CB = CE then show that AB = DE A E C B D 23. In figure, A and B are centres of the two intersecting circles, which intersect at C. Prove that AB = AC = BC 60 IX – Mathematics  C A B 24. Prove that every line segment has one and only one mid point. 25. Kartik and Himank have the same weight. If they each gain weight by 3 kg how will their new weight be compared? State Euclid’s axiom used. Short Answer type-II Questions (3 Marks) 26. In the given figure ∠1 = ∠2 and ∠2 = ∠3 then show that ∠1 = ∠3 B 1 3 A C 2 4 D 27. In the given figure AB = BC, M is the mid point of AB and N is the mid-point of BC. Show that AM = NC B A M N C P T 28. In the given figure PR = RS and RQ = RT. Show that PQ = ST and write the Euclid’s axiom to supports R this. S Q 61 IX – Mathematics 29. An equilateral triangle is a polygon made up of three line segments out of which two line segments are equal to the third one and all the angles are 60° each. Can you justify that all the sides and all the angles are equal in equilateral triangle? 30. Ram and Shyam are two students of class IX. They given equal donation to a blind school in the month of March. In April each student double their donation. (a) compare their donation in April. (b) which mathematical concept have been covered in this question? 62 IX – Mathematics  CHAPTER-5 INTRODUCTION TO EUCLID’S GEOMETRY Answers 1. (a) A unique line can be drawn 2. (a) Elements 3. (c) Equals 4. (a) An axiom 5. (b) Intersecting lines 6. Theorem 7. only one 8. An axiom 9. Zero 10. Five 11. Two 12. Equal 13. Infinite 14. If a straight line falling on two straight lines makes the interior angles on the same side of it taken together less than two right angles, then the two straight lines if produced indefinitely, meet on that side on which the sum of angles is less than two right angles. 15. Second axiom 16. Given x + y = 10 --- (1) and x = z --- (2) on subtracting y from both sides, of eq (1) n x + y – y = 10 – y [by axiom 3] z = 10 – y [from eq 2] on adding y both sides, we get z + y = 10 – y + y [by axiom 2] z + y = 10 63 IX – Mathematics 17. AB = AC --- (1) AX = AY, AY --- (2) According to Euclid’s axiom (3), if equals are subtracted from equals then remainders are also equal Subtracting equation (2) from equation (1) AB – AX = AC – AY BX = CY (Hence proved) 18. ∠ABC = ∠ACB --- (1) ∠4 = ∠3 --- (2) eqn (1) – eqn (2) ∠ABC – ∠4 = ∠ACB – ∠3 [using axiom 3] ∠1 = ∠2 19. AD = CB AC + CD = CD + DB on subtracting CD from both sides AC + CD – CD = CD + DB – CD (using axiom 3) AC = DB 20. x – 10 = 15 Adding 10 both sides x – 10 + 10 = 15 + 10 [by axiom 2] x = 25 21. Given; AM = AB --- (i) AN = AC --- (ii) AM = AN --- (iii) from eqns (i), (ii) & (iii), we get AB = AC AB = AC [by axiom 7] 64 IX – Mathematics 22. AC = DC --- (1) CB = CE --- (2) By Euclid’s axiom 2 If two equals are aded to equals, then the wholes are equal. Adding eqn (1) and eqn (2) AC + CB = DC + CE AB = DE 23. AB = AC --- (1) [Radius of the same circle] BC = AB --- (2) [Radius of the same circle] from eqn (1) and eqn (2) AB =AC = BC[ by axiom 1] 24. We have C as the mid point of the line segment AB, so AC = BC Let there are two mid-point C & C of AB A B C A B C C Then, AC= AB AC = AB ⇒ AC = AC [by axiom 1] which is possible only when C coincider C , so point C lies on C. 25. Kartik’s weight = Himank’s weight Kartik’s weight + 3 kg = Himank’s weight + 3kg [by axiom 2] Their new weight will be equals By Euclid’s second axiom. If equals are added to equals then wholes are equal. 26. ∠1 = ∠2 --- (1) ∠2 = ∠3 --- (2) from equation (1) and (2) ∠1 = ∠3 [By axiom 1] 65 IX – Mathematics 27. AB = BC AM + BM = BN + CN 2AM = 2CN [M & N are mid-point of AB & BC respectively] AM = CN [By Euclid’s axiom 6] 28. PR = RS --- (1) RQ = RT --- (2) Adding equation (1) and (2) PR + RQ = RS + RT PQ = ST [By axiom 2] 29. a b c a = b and b = c ⇒ a=b=c [By axiom 1] All sides of triangle are equal since all the angles are of 60° in an equilateral triangle so they must be equal to one another. 30. Ram’s donation in March = Shyam’s donation in March --- (1) Ram’s donation in April = 2 × Ram’s donation in March --- (2) Shyam’s donation in April = 2 × Shyam’s donation in March --- (3) Using equation (1), (2) & (3) ⇒ Ram’s donation in April = Shyam’s donation in April [using axiom 6] 66 IX – Mathematics  Practice Test Introduction to Euclid’s Geometry Time: 1 hr. M.M. 20 1. How many line segments can be determined by three collinear points. (1) 2. How many lines can pass through a given point? (1) 3. State Euclid’s first postulate. (2) 4. Solve the equation x + 3 = 10 and state the Euclid’s axiom used (2) 5. If a point C lies between two points A and B such that AC = BC then prove that AC = AB. Explain by drawing the figure. (3) 6. It is known that x + y = 10, then x + y + z = 10 + z. State the Euclid’s axiom that illustrates the statement. (3) 7. State Euclid’s fifth postulate, explain it and compare it with version of parallel lines (3) 8. In the figure PQ = RS, A and B are points on PQ and RS such that AP = PQ and RB = RS show that AP = RB. State which axiom you use here. Also give two more axioms other than the axiom used in the above situation. (5) Q R B A P S 67 IX – Mathematics  CHAPTER-6 LINES AND ANGLES MIND MAP 68 IX – Mathematics Key points Line is a collection of points which has only length, neither breadth nor thickness. Line Segment: A part or portion of a line with two end points. Ray: A part of a line with one end point. Collinear points: Three or more points lying on the same line. Non-Collinear Points: Three or more points which do not lie on same line. Angle: An angle is formed when two rays originate from the same end point. The rays making angle are called the arms and the end point is the vertex. Acute Angle: An angle measure between 0° and 90° Right angle: Angle exactly equal to 90° Obtuse angle: An angle greater than 90° but less than 180° Straight angle: An angle exactly equal to 180° Reflex angle: An angle greater than 180° but less than 90° Complimentary angles: A pair of angles whose sum is 90° Supplementary angle: A pair of angles whose sum is 180° Complete angle: An angle whose measure is 360° Adjacent angles: Two angles are adjacent if (i) they have a common vertex, (ii) a common arm, (iii) their non common arms are on opposite side of common arm. Linear pair of angle: A pair of adjacent angles whose sum is 180° ∠AOB and ∠COB are forming linear pair. Vertically opposite angles: Angles formed by two intersecting lines on opposite side of the point of intersection. 69 IX – Mathematics  Intersecting lines: Two lines are said to be intersecting when the perpendicular distance between the two lines is not same every where. They intersect at some point. Non Intersecting lines: Two lines are said to be non-intersecting lines when the perpendicular distance between them is same every where. They do not intersect. If these lines are in the same plane these are known as Parallel lines. Transversal line: In the given figure l || m and t is transversal then (a) Vertically opposite angle (b) Corresponding angle (c) Alternate Interior angle (d) Alternate Exterior angle (e) Angles on the same sides of a transversal are supplementary. ∠3, ∠6 and ∠4, ∠5 are called co-interior angles or allied angles or consecutive interior angles. Sum of all interior angles of a triangle is 180°. Two lines which are parallel to the third line are also parallel to each other. 70 IX – Mathematics Very Short Answer Questions ( 1 mark) 1. If an angle is equal to its complement, then the angle is (a) 90° (b) 0° (c) 48° (d) 45° 2. In the given fig. for what value of x + y, ABC will be a straight line? (a) 90° (b) 180° (c) 360° (d) 270° A D y x C B 3. In fig. ∠AOC and ∠BOC form a linear pair. Determine the value of x (a) 30° (b) 150° (c) 15° (d) 75° C 5x x A O B 4. The reflex angle of 110° is (a) 70° (b) 90° (c) 250° (d) 190° 5. One of the angles of a pair of supplementary angle is 10° more than its supplement, the angles are: (a) 90°, 90° (b) 86°, 94° (c) 85°, 95° (d) 42.5°, 47.5° 6. If three or more points does not lie on the same straight line, the points are called (a) Concurrent points (b) Collinear points (c) Non-collinear points (d) Adjacent point 71 IX – Mathematics 7. If angles x and y form a linear pair and x – 2y = 30°, then the value of y is (a) 50° (b) 110° (c) 210° (d) 60° 8. In the figure, AB is a straight line, then the value of (a + b) is (a) 0° (b) 90° (c) 180° (d) 60° 9. If ∠AOC = 50° then the value of ∠BOD is ________ (a) 50° (b) 40° (c) 130° (d) 25° A C 50° O D B 10. If two parallel lines are intersected by a transversal, then the interior angles on the same side of transversal are (a) equal (b) Adjacent (c) supplementary (d) complementary 11. In figure, l || m value of x is _________ (a) 70° (b) 35° (c) 210° (d) 110° n l x 70° m 72 IX – Mathematics 12. Three parallel lines intersect at __________ points (a) one (b) two (c) three (d) zero 13. If one angle of a linear pair is acute, then the other angle will be (a) right angle (b) obtuse angle (c) acute angle (d) straight angle 14. In the given figure, find the value of y (a) 18° (b) 9° (c) 30° (d) 36° 5y 2y 3y 15. A ray has only __________ end point. 16. A line segment has a __________ length. 17. If two lines are non-intersecting, then they will be __________. 18. An angle whose measure is more than 0° but less than 90°, is called an __________ angle. 19. A straight angle has __________ right