Comprehensive CBSE Question Bank in Mathematics-XI (Term-II)

CBSE II Question Bank in Mathematics CLASS 11 Features Long Answer-I Type Questions Long Answer-II Type Questions Strictly Based on the Latest CBSE Term-wise Syllabus Case Study Based MCQs Chapter-wise Important Results and Formulae Very Short/Short Answer Type Questions



Comprehensive CBSE Question Bank in Mathematics Term–II (FOR CLASS XI) (According to the Latest CBSE Examination Pattern) By A.P. PRABHAKARAN PARMANAND GUPTA Formerly Principal Gokulum Public School, Vatakara B.Sc. (Hons.). M.Sc. (Delhi) M.Phil (KU), Pre. Ph.D. (IIT Delhi) Kerala Associate Professor of Mathematics Former Head of Department of Mathematics Indira Gandhi National College, Ladwa Kurukshetra University Haryana   LAXMI PUBLICATIONS (P) LTD (An ISO 9001:2015 Company) BENGALURU • CHENNAI • GUWAHATI • HYDERABAD • JALANDHAR KOCHI • KOLKATA • LUCKNOW • MUMBAI • RANCHI NEW DELHI

Comprehensive CBSE QUESTION BANK IN MATHEMATICS–XI (TERM–II) Copyright © by Laxmi Publications Pvt., Ltd. All rights reserved including those of translation into other languages. In accordance with the Copyright (Amendment) Act, 2012, no part of this publication may be reproduced, stored in a retrieval system, translated into any other language or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise. Any such act or scanning, uploading, and or electronic sharing of any part of this book without the permission of the publisher constitutes unlawful piracy and theft of the copyright holder’s intellectual property. If you would like to use material from the book (other than for review purposes), prior written permission must be obtained from the publishers. Printed and bound in India Typeset at : Goswami Associates, Delhi New Edition ISBN : 978-93-93738-16-5 Limits of Liability/Disclaimer of Warranty: The publisher and the author make no representation or warranties with respect to the accuracy or completeness of the contents of this work and specifically disclaim all warranties. The advice, strategies, and activities contained herein may not be suitable for every situation. In performing activities adult supervision must be sought. Likewise, common sense and care are essential to the conduct of any and all activities, whether described in this book or otherwise. Neither the publisher nor the author shall be liable or assumes any responsibility for any injuries or damages arising here from. The fact that an organization or Website if referred to in this work as a citation and/or a potential source of further information does not mean that the author or the publisher endorses the information the organization or Website may provide or recommendations it may make. Further, readers must be aware that the Internet Websites listed in this work may have changed or disappeared between when this work was written and when it is read. All trademarks, logos or any other mark such as Vibgyor, USP, Amanda, Golden Bells, Firewall Media, Mercury, Trinity, Laxmi appearing in this work are trademarks and intellectual property owned by or licensed to Laxmi Publications, its subsidiaries or affiliates. Notwithstanding this disclaimer, all other names and marks mentioned in this work are the trade names, trademarks or service marks of their respective owners. & Bengaluru 080-26 75 69 30 & Chennai 044-24 34 47 26 Branches & Guwahati 0361-254 36 69 & Hyderabad 040-27 55 53 83 & Jalandhar 0181-222 12 72 & Kochi 0484-405 13 03 & Kolkata 033-40 04 77 79 & Lucknow 0522-430 36 13 Published in India by & Ranchi 0651-224 24 64 Laxmi Publications (P) Ltd. C—00000/022/01 Printed at : Ajit Printing Press, Delhi. (An ISO 9001:2015 Company) 113, GOLDEN HOUSE, GURUDWARA ROAD, DARYAGANJ, NEW DELHI - 110002, INDIA Telephone : 91-11-4353 2500, 4353 2501 www.laxmipublications.com [email protected]

CONTENTS UNIT I: SETS AND FUNCTIONS 1. Trigonometric Functions ...................................................................................... 1–49 UNIT II: ALGEBRA 2. Linear Inequalities ............................................................................................. 50–73 3. Permutations and Combinations ..................................................................... 74–102 UNIT III: COORDINATE GEOMETRY 4. Conic Sections ................................................................................................. 103–141 5. Introduction to Three-dimensional Geometry .............................................. 142–165 UNIT IV: CALCULUS 6. Derivatives ...................................................................................................... 166–186 UNIT V: STATISTICS AND PROBABILITY 7. Probability ....................................................................................................... 187–213 1

































Linear Inequalities 63 xy 6. Which of the points lie in the half-plane of 2 + 3 – 1  0 : (1, 5), (3, 0), (0, 1), (– 3, – 4). 7. Which of the points lie in the half-plane of x – y – 7  0 : (1, –3), (4, –3), (5, –8), (0, 10). 8. If the point (h, k) lies in the half-plane of ax+ by + c > 0, then show that it cannot lie in the half-plane of ax + by + c < 0. 9. If the point (h, k) lies in the half-plane of ax + by + c  0, then show that it may not lie in the half-plane of ax + by + c  0. Long Answer–I Type Questions 1. Solve graphically: x + y  9, y > x, x  0. 2. Solve: 5  2x x  5. 3 6 3. Solve: 15  3 (x  2)  0. 5 4. Solve: 2x + 10  0 and represent the solution on a number line. 5. Solve: 4  x  3x  12 and represent the solution on a number line. 6. Solve: the inequality: x  5x  2  7x  3 . 2 3 5 7. Solve: the inequality : 37 – (3x + 5)  9x – 8(x – 3). 8. Solve: x  x  x  11. 2 3 9. Solve: 2x  3  8  2  4x . 4 3 10. Solve: | 2x – 1 |  3. 11. Solve the following system of inequations: 3x  7  2(x  6) : 6  x  11  2x. 12. Solve: 1  3x  4   1 ( x  6) . 2  5  3 13. Solve the following system of linear inequalities: 2x + 9 > 5 – 3x; 5x – 7  3x + 11. Long Answer–II Type Questions 1. A manufacturer has 600 litres of a 12% solution of acid. How many litres of the 30% acid solution is to be added to it so that acid content in the resulting mixture will be more than 15% but less than 18%? 2. Solve the following system of inequations graphically: x + 2y  10, x + y  1, x – y  0, x  0, y  0. 3. Solve the following system of inequations graphically: x + 2y  10, x + 2y  1, x – y  0, x  0, y  0.









































84 Mathematics—XI Example 21: If the letters of the word INDIA are arranged as in a dictionary, what is the rank of INDIA? Sol: The letters of the word in the ascending order are A, D, I, I, N No. of words beginning with A= 4!  24  12. 2! 2 No. of words beginning with D = 4!  24  12. 2! 2 No. of words beginning with IA = 3! = 6 No. of words beginning with ID = 3! = 6 No. of words beginning with II = 3! = 6 No. of words beginning with INA = 2! = 2 Next word beginning with IND is INDAI. The word next to it is INDIA.  No. of words before INDIA is 12 + 12 + 6 + 6 + 6 + 2 + 1 = 45. Hence, the rank of INDIA is 46. B. COMBINATIONS Important Results and Formulae 1. Combinations. Combination is the selection of distinct objects taken all or some at a time. 2. The difference between permutation and combination. Permutation is arrangement of objects where order of things is important. Combination is selection or forming groups where order of things is unimportant. The number of permutations exceeds the number of combinations. Notation: The number of combinations of n distinct objects taken r at a time is denoted by nCr or C(n, r). Note: nCr is defined only when n and r are non-negative integers such that 0  r  n. 3. Formula for Calculating nCr . The number of combinations of n distinct objects taken r at a time is given by n Cr = n! r!n  r! . From the above we also get, nCr = n(n 1·12)(·n3\"2)t\"o rtfoarctofarsctors. Note: Second formula is mainly used when we know n and r. 4. Properties of nCrnC1 = n. Property 1: Thus, 5C1 = 5, 20C1 = 20. i.e., the no. of ways selecting n distinct objects taken one at a time = n. Proprety 2: nCn = 1. i.e., the no. of ways selecting n distinct objects taken all at a time = 1.