Must For All Maths Talent Search Exams & Olympiads SALIENT FEATURES BMA’S TALENT & l Suitable for curricula of major boards like CBSE / ICSE / State Boards. Fundamental concepts are thoroughly revised l Extensive range of questions that stimulate the interests of l the students while testing their knowledge OLYMPIAD I Application/skill/knowledge/understanding X l oriented questions l Suitable for International/National/Regional Olympiads and Talent exams like NSTSE, Maths Olympiad, RMO, NMO, IMO etc. B EXAMS RESOURCE BOOK M Hundreds of objective questions ` 50 l A Must For All Science Talent Search Exams & Olympiads ’ BTMA’SALENT & s OLYMPIAD T UNIQUE FEATURES EXAMS RESOURCE BOOK a MATHEMATICS SOLUTIONS l CL ASS IX E-Book e n Synopsis: To provide the essence of a chapter in a nutshell t l & l Previous Contest Problems: Problems appeared in various 3.6 SMOLiDll iOovner O Strong Foundation for talent & olympiad exams Brain Mapping Academy scRionepcvieei ssthe oedf 1Ethsdtis iE tsdieiotrniioens ly Better Results Crossword Puzzles: To stimulate the mind to think beyond m l Detailed solutions to regular preparation & to offer fun all problems in Talent pi a l Solutions: Solutions and explanations for maximum no. of & Olympiad Exams d questions Resource Book are E available as this x Questions @ Stimulating Minds: Selected questions to a l e-book on m challenge your higher order thinking www.bmatalent.com s R e s o u r c e B o ` 130/- o k - M A T H E M A www.bmatalent.com T I C YOUR S SOLD Over COACH 3.6 Million India’s FIRST scientifically designed portal for Olympiad preparation copies of this series • Olympiad & Talent Exams preparation packages since the 1st Edition • Analysis Reports • Previous question papers • Free Demo Packages • Free Android Mobile App Brain Mapping Academy Get 15% discount on all packages by using the discount coupon code: KR157N Revised Edition A unique opportunity to take about 50 tests per subject. BMA's T A L E N T & OLYMPIAD EXAMS RESOURCE BOOK CLASS IX J ( Mathematics BRAIN MAPPING ACADEMY Mapping Your Future www.bmatalent.com Published by: Brain Mapping Academy #16-11-16/1/B, First Floor, Farhath Hospital Road, Saleem Nagar, Malakpet. Hyderabad-500 036. t 040-66135169.65165169. E-mail: [email protected] Website: www.bmatalent.com @ BRAIN MAPPING ACADEMY ALL RIGHTS RESERVED No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher. ISBN: 978-93-82058-53-3 Disclaimer Every care has been taken by the compilers and publishers to give correct, complete and updated information. In case there is any omission, printing mistake or any other error which might have crept in inadvertently, neither the compiler I publisher nor any of the distributors take any legal responsibility. In case ofa nyd ispute, all matters are subjected to the exclusive jurisdiction ofth e courts in Hyderabad only. First Edition : 2003 Second Edition : 2008 Revised Edition : 2015 Publisher's Note Sometimes the understanding of fundamental concepts alone does not help the students to crack the competitive exams as most of them are objective in structure. Students need rigorous training to familiarize themselves to the style of the exams they are attempting. The board exams which are of qualifying, but not competitive, nature do not completely address the needs of students in testing them in objective type format. To bridge this gap and to enable the students to face the reality of competitive exams, Brain Mapping Academy, brought out an all-objective questions reference book. A crisp summary of the topics and useful equations were provided at the beginning of each chapter so that the students can memorize the important points. Care has been taken to design thought-provoking questions. These should help students to attain a deeper understanding of principles. The questions have been reviewed to fill the gaps in problem coverage and to build the confidence in the students. They have also been expanded to impart reasoning/logical/analytical skills. This book will cater all the requirements of the students who are approaching national/state level talent search examinations and all Olympiad exams. This book also complements the additional preparation needs of the students for the regular board exams. We took utmost care to make this the best resource book available for the talent I olympiad exams aspirants. We welcome criticism from the students, teacher community and educators, especially concerning any errors and deficiencies which may have remained in this edition and the suggestions for improvement for the next edition. This page is intentionally left blank. For Your Information Test • Assess. Achieve NATIONAL LEVEL SCIENCE TALENT SEARCH EXAMINATION I Aim of this examination The focus on fundamentals is so important that without a firm understanding of them, a child cannot be expected to face the reality of the competitive world once he/she finishes the formal education. Even while opting for higher studies the student has to go through a complete scan of what he/she knows. Exams like IIT-JEE, AIEEE, AIIMS, AFMC, CAT, SAT, GRE, GMAT, etc. are so designed to test the fundamental strength of a student. Hence the need of the hour is building the fundamental base as strong as possible. A successful life emerges out from healthy and sound competition. Competition is the only way for the students to shake lethargy. It's the only way to get introduced for manly worthiness. Firm standards in education and competition are the tonic for a promising and talented future. This exactly is the philosophy behind the Unified Council's NSTSE. I Organisation National Science Talent Search Examination is conducted by Unified Council. Unified Council is India's first ISO 9001 certified organisation in the educational testing and assessment. Since its inception, Unified Council has put together the best brains in an endeavour to make the younger generation fundamentally stronger and nourish their brains for a bright and enterprising future. Eligibility: Students of classes 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 & 12 are eligible to participate in this examination. Medium & Syllabus: This exam is conducted in only English medium and is suitable for all the students following CBSE/ICSE/State Board Syllabi. I Examination Pattern There will be a separate question paper for each class. All questions are objective-type multiple choice with no negative marking for wrong answers. Duration: 90 minutes Date : Conducted every year on the last Sunday of January. Test Centres: Spread across the country. DIVISION OF MARKS FOR CLASS I FOR CLASSES VI TO X Mathematics 25 marks Mathematics 25 marks Physics 25 marks General Science 15 Marks Chemistry 20 marks FOR CLASS II Biology 20 marks Mathematics 25 marks General Questions 10 marks General Science 25 Marks FOR CLASS XI & XII(PCM) Mathematics 40 marks FOR CLASS Ill Physics 25 marks Mathematics 40 marks Chemistry 25 marks General Science 35 Marks General Questions 10 marks FOR CLASS XI & XII(PCB) FOR CLASSES IV & V Biology 40 marks Mathematics 45 marks Physics 25 marks General Science 45 Marks Chemistry 25 marks General Questions 10 marks General Questions 10 marks I Infrastructure The Council makes use of ultra-modern equipment such as Optical Mark Recognition (OMR) equipment to evaluate the answer papers to proficiently assess students' performance. The examination procedure is completely computerised. Unique Service from Unified Council: Unique analysis reports like Student's Performance Report for students, General School Re- port & Individual School Report for schools provided. These reports are very much helpful for students & schools to analyse their strengths and weaknesses. General School Report (GSR) analyses the performance of students participating in the exam (subject-wise and class-wise). The report, in graphical format will have Ogive and Histogram Graphs, which are useful to schools that wish to improve their students' performance by benchmarking the areas of weaknesses and building upon them. Individual School Report (ISR) analyses the performance of a particular school when com- pared to the rest of the students participating in this examination (subject-wise, class-wise and question-wise). This report acts as a tool for the schools to improve their students' perfor- mance in the future by benchmarking the areas of weaknesses and building upon them. Awards & Scholarships: Top 100 members in each class will be awarded with Awards & Medals etc. UNIFIED COUNCIL .·tn IIO !J(J()I: 2008 (f.,.an('(!Oqpnlt.uioo ftHJtufJtiOII for SUct'i!'Sl #16-11-16/1 /B, Farhath Hospital Road, Saleem Nagar, Malakpet, Hyderabad-500 036 Phones:040-24557708, 24545862, 66139917 E-mail: [email protected], Website: www.unifiedcouncil.com CONTENTS Mathematics ------~~~··~~------ 1. Number Systems ............................ 9 -16 2. Polynomials ................................... 17- 22 3. Co-ordinate Geometry ................... 24- 28 4. Linear Equations in Two Variables .. 29- 33 5. Introduction to Euclid's Geometry .. 35- 39 6. Lines and Angles ............................ 40-48 7. Triangles ......................................... 49-55 8. Quadrilaterals ............................... 57- 64 9. Area of Parallelograms and Triangles .... 65- 71 10. Circles ............................................ 72- 81 11. Heron's Formula ............................. 83- 88 12. Surface Areas and Volumes ........... 89- 94 13. Statistics ........................................ 96- 102 14. Probability ..................................... 103- 107 Questions@stimulating-minds ........ 109- 110 Model Test Paper ......................... 111 - 112 Explanatory Answers ................... 113 - 134 This page is intentionally left blank. CHAPTER 1 Number Systems + Rational numbers (Q): The numbers of the form ~ where 'p' and 'q' are integers and q ::~; 0 are called rational numbers. a b Note: A number of the form is a fraction. So all fractions are rational numbers. In a b, the fraction 'a' is called the numerator and 'b' is called the denominator. 1 2 7 6 2 2' - 3, 6' u'- 9' ······ e.g., (i) Zero is a rational number. (~ ~) _______N_ o_t_ e_:_O_ _b_y _O _is__u_n_d_ef_in_e_d_._ _____________________________________ (ii) Every integer is a rational number. (iii) A rational number, may or may not be an integer. (iv) To write 'n' distinct rational numbers between any two rational numbers 'a' and 'b', we write a= Pq, and b = Pq2 such that (p2-p,) is a positive integer greater than 'n; (Here a< b); P,, P and q are integers (q ::~= 0). 2 1 2 A set of'n' rational numbers can be written as p, + , p, + , ...... , p, + n q q q i.e., a = £!. < p, + 1 < p, + 2 ...... < p, + n < P2 = b. q q q q q (v) Between two given rational numbers a and b, there are infinitely many rational numbers. + Properties of rational numbers: p . . I b d' ,. . h p pxm (i) If -q IS a rat10na num er an m IS a non-zero mteger, t en - =--. q qxm p p p7m (i i) If -q is a rational number and 'm' is a common divisor of p and q, then - =- - q q7m (iii) Two rational numbers are equivalent only when the product of the numerator of the first rational number and the denominator of the second is equal to the product of the denominator of the first and the numerator of the second. I I 1. Number Systems