1m ago

21 Views

0 Downloads

754.22 KB

23 Pages

Transcription

Written as per the revised syllabus prescribed by the Maharashtra State Boardof Secondary and Higher Secondary Education, Pune.STD. XI Sci.Perfect Mathematics - IFifth Edition: May 2015Salient Features Exhaustive coverage of entire syllabus. Covers answers to all textual and miscellaneous exercises. Precise theory for every topic. Neat, labelled and authentic diagrams. Written in systematic manner. Self evaluative in nature. Practice problems and multiple choice questions for effective preparation.Printed at: Dainik Saamna, Navi MumbaiNo part of this book may be reproduced or transmitted in any form or by any means, C.D. ROM/Audio Video Cassettes or electronic, mechanicalincluding photocopying; recording or by any information storage and retrieval system without permission in writing from the Publisher.TEID : 906

PrefaceIn the case of good books, the point is not how many of them you can get through, but rather how many can getthrough to you.“Std. XI Sci. : PERFECT MATHEMATICS - I” is a complete and thorough guide critically analysed andextensively drafted to boost the students confidence. The book is prepared as per the Maharashtra State board syllabusand provides answers to all textual questions. At the beginning of every chapter, topic – wise distribution of alltextual questions including practice problems has been provided for simpler understanding of different types ofquestions. Neatly labelled diagrams have been provided wherever required.Practice Problems and Multiple Choice Questions help the students to test their range of preparation and theamount of knowledge of each topic. Important theories and formulae are the highlights of this book. The steps arewritten in systematic manner for easy and effective understanding.The journey to create a complete book is strewn with triumphs, failures and near misses. If you think we’venearly missed something or want to applaud us for our triumphs, we’d love to hear from you.Please write to us on : [email protected] of luck to all the aspirants!Yours faithfully,PublisherNo.Topic NamePage No.1Angle and It’s Measurement12Trigonometric Functions223Trigonometric Functions of Compound Angles654Factorization Formulae955Locus1166Straight Line1427Circle and Conics2028Vectors2779Linear Inequations32010Determinants36711Matrices415

Target Publications Pvt. Ltd.01Chapter 01: Angle and it’s MeasurementAngle and it’s MeasurementType of ProblemsCoterminal anglesExercise1.1Practice Problems(Based on Exercise 1.1)1.1Degree measure and radian measurePractice Problems(Based on Exercise 1.1)MiscellaneousPractice Problems(Based on Miscellaneous)Length of an arcArea of a sectorLength of an arc and area of a sector1.2Practice Problems(Based on Exercise 1.2)MiscellaneousPractice Problems(Based on Miscellaneous)1.2Practice Problems(Based on Exercise 1.2)MiscellaneousPractice Problems(Based on Miscellaneous)1.2Practice Problems(Based on Exercise 1.2)MiscellaneousPractice Problems(Based on Miscellaneous)Q. Nos.Q.1 (i. to iv.)Q.1 (i., ii.)Q.2. (i. to vii.)Q.3. (i. to vii.)Q.4. (i., ii.)Q.5, 6, 7Q.8. (i., ii.)Q.9, 10, 11, 12, 13, 14Q.2 (i. to v.)Q.3 (i. to iv.)Q.4 (i. to iii.)Q.5, 6, 7, 8Q.9 (i., ii.)Q.10Q.1. (i., ii.)Q.2. (i., ii.)Q.4. (i. to iii.)Q.3, 5, 13, 14, 15, 16, 17, 18, 20Q.1 (i., ii.)Q.2 (i., ii.)Q.3, 4, 5, 6, 13, 14, 15 (i., ii.), 16, 19Q.1, 2, 3, 4, 5Q.1, 2, 3, 4, 5Q.7, 8, 9, 10, 19Q.8, 9, 10, 11, 17, 18Q.7, 8, 9, 10Q.6, 8, 9, 10Q.6Q.7Q.6Q.7Q.11, 12Q.121

Target Publications Pvt. Ltd.Std. XI Sci.: Perfect Maths - ISyllabus:Directed angles, zero angle, straight angle,coterminal angles, standard angles, angle in aquadrant and quadrantal angles.Systems of measurement of angles:Sexagesimal system (degree measure), Circularsystem (radian measure), Relation between degreemeasure and radian measure, length of an arc of acircle and area of sector of a circle.Positive angle:If a ray rotates about the vertex (the point) O frominitial position OX in anticlockwise direction, thenthe angle described by the ray is positive angle.PTerminal ray ve angleOIntroductionIn school geometry we have studied the definition ofangle and trigonometric ratios of some acute angles.In this chapter we will extend the concept fordifferent angles.Initial rayXIn the above figure, XOP is obtained by therotation of a ray in anticlockwise direction denotedby arrow. Hence XOP is positive i.e., XOP.You are familiar with the definition of angle as the“union of two non-collinear rays having commonend point”. But according to this definition measureof angle is always positive and it lies between 0 to180 . In order to study the concept of angle inbroader manner, we will extend it for magnitude andsign.The measurement of angle and sides of a triangleand the inter-relation between them was first studiedby Greek astronomers Hipparchus and Ptolemy gative angle:Directed anglesIn the above figure, XOP is obtained by therotation of a ray in clockwise direction denoted byarrow. Hence XOP is negative angle i.e., XOP.Suppose OX is the initial position of a ray. This rayrotates about O from initial position OX and takes afinite position along ray OP. In such a case we saythat rotating ray OX describes a directed angle XOP.PTerminal rayOVertexInitial rayXIn the above figure, the point O is called the vertex.The ray OX is called the initial ray and ray OP iscalled the terminal ray of an angle XOP. The pairof rays are also called the arms of angle XOP.In general, an angle can be defined as the orderedpair of initial and terminal rays or arms rotating frominitial position to terminal position.The directed angle includes two thingsi.Amount of rotation (magnitude of angle).ii.Direction of rotation (sign of the angle).2If a ray rotates about the vertex (the point) O, frominitial position OX in clockwise direction, then theangle described by the ray is negative angle.Initial rayOX ve angleTerminal rayPAngle of any magnitude:i.Suppose a ray starts from the initial positionOX in anticlockwise sense andmakescomplete rotation (revolution) about O andtakes the final position along OX as shown inthe figure (i), then the angle described by theray is 360 .OFig. (i)XIn figure (ii) initial ray rotates about O inanticlockwise sense and completes tworotations (revolutions). Hence, the angledescribed by the ray is 2 360 720 .OFig. (ii)X

Target Publications Pvt. Ltd.Chapter 01: Angle and it’s MeasurementIn figure (iii) initial ray rotates about O inclockwise sense and completes two rotations(revolutions). Hence, the angle described bythe ray is 2 360 720 Oii.Straight angle:In figure, OX is the initial position and OP is the finalposition of rotating ray. The rays OX and OP lie alongthe same line but in opposite direction. In this case XOP is called a straight angle and m XOP 180 .XFig. (iii)Suppose a ray starting from the initial positionOX makes one complete rotation inanticlockwise sense and takes the position OPas shown in figure, then the angle described bythe revolving ray is 360 XOP.POPCoterminal angles:Two angles with different measures but having thesame positions of initial and terminal ray are calledas coterminal angles.P O410 XIf XOP , then the traced angle is360 .If the rotating ray completes two rotations, thenthe angle described is 2 360 720 and so on.iii.Suppose the initial ray makes one completerotation about O in clockwise sense and attainsits terminal position OP, then the describedangle is (360 XOP).OX O50 –310 XIn figure, the directed angles having measures 50 ,410 , –310 have the same initial arm, ray OX andthe same terminal arm, ray OP. Hence, these anglesare coterminal angles.Note:If two directed angles are co-terminal angles, thenthe difference between measures of these twodirected angles is an integral multiple of 360 .Standard angle:PIf XOP , then the traced angle is (360 ).If final position OP is obtained after 2,3,4, .complete rotations in clockwise sense, thenangle described are (2 360 ), (3 360 ), (4 360 ), .Types of anglesZero angle:If the initial ray and the terminal ray lie along sameline and same direction i.e., they coincide, the angleso obtained is of measure zero and is called zeroangle.OXPXAn angle which has vertex at origin and initial armalong positive X axis is called standard angle.YQPX OXRY In figure XOP, XOQ, XOR with vertex O andinitial ray along positive X axis are called standardangles or angles in standard position.Angle in a Quadrant:An angle is said to be in a particular quadrant, if theterminal ray of the angle in standard position lies inthat quadrant.3

Target Publications Pvt. Ltd.Std. XI Sci.: Perfect Maths - IYQX Divide 1 into 60 equal parts. Each part iscalled as a one minute ( 1 ).POi.e.,X1 60 Divide 1 into 60 equal parts. Each part iscalled as a one second (1 )Ri.e., 1 60 Y In figure XOP, XOQ and XOR lie in first,second and third quadrants respectively.Note:The sexagesimal system is extensively used inengineering, astronomy, navigation and surveying.Quadrantal Angles:ii.If the terminal arm of an angle in standard positionlie along any one of the co-ordinate axes, then it iscalled as quadrantal angle.YPX QOXRY In figure XOP, XOQ, and XOR are quadrantalangles.Note:The quadrantal angles are integral multiples of 90 2i.e., n , where n N.Systems of measurement of anglesThere are two systems of measurement of an angle:i.Sexagesimal system (Degree measure)ii.Circular system (Radian measure)i.Sexagesimal system (Degree Measure):In this system, the unit of measurement of anangle is a degree.Suppose a ray OP starts rotating in theanticlockwise sense about O and attains theoriginal position for the first time, then theamount of rotation caused is called1 revolution.Divide 1 revolution into360 equal parts. EachPpart is called as a oneOdegree(1 ).i.e.,41 revolution 360 1 revolutionCircular system (Radian measure):In this system, the unit of measurement of anangle is radian.Angle subtended at the centre of a circle by anarc whose length is equal to the radius iscalled as one radian denoted by 1c.Draw any circle with centre O and radius r.Take the points P and Q on the circle suchthat the length of arc PQ is equal to radius ofthe circle. Join OP and OQ.Qrrc1PrOThen by the definition, the measure of POQis 1 radian (1c).Notes:i.This system of measuring an angle is used inall the higher branches of mathematics.ii.The radian is a constant angle, therefore radiandoes not depend on the circle i.e., it does notdepend on the radius of the circle as shownbelow.CQcO1r1r2r1PB1cr2AIn figure we draw two circles of different radii r1 andr2 and centres O and B respectively. Then the angleat the centre of both circles is equal to 1c.i.e., POQ 1c ABC.

Target Publications Pvt. Ltd.Chapter 01: Angle and it’s MeasurementTheorem:A radian is a constant angle.ORAngle subtended at the centre of a circle by an arcwhose length is equal to the radius of the circle isalways constant.Proof:Let O be the centre and r be the radius of the circle.Take points P, Q and R on the circle such that arcPQ r and POR 90 .By definition of radian, POQ 1c1arc PR circumference of the circle4R1 rQ 2 r 42rBy proportionality theoremc1 POQ arcPQPOr POR arcPR POQ 1c i.e., Exercise 1.11.Determine which of the following pairs ofangles are coterminal:i.210 , 150 ii.330 , 60 iii. 405 , 675 iv. 1230 , 930 Solution:i.210 ( 150 ) 210 150 360 1(360 )which is a multiple of 360 .Hence, the given angles are coterminal.ii.330 ( 60 ) 330 60 390 which is not a multiple of 360 .Hence, the given angles are not coterminal.iii.405 ( 675 ) 405 675 1080 3(360 )which is a multiple of 360 .Hence, the given angles are coterminal.iv.1230 ( 930 ) 1230 930 2160 6(360 )which is a multiple of 360 .Hence, the given angles are coterminal.2.Express the following angles in degrees:arcPQ PORarcPRr 1 right angle r 2 2 (1 right angle) .(i)1c constantR.H.S. of equation (i) is constant and hence aradian is a constant angle.Relation between degree measure and radianmeasure:22 90 180 i.1c (1. right angle) ii. c1 0.01745 (approx.) 180 v. 5 7 vii. 7 24 180 y x cIn general x and y 180 7 12 iv. 1 3 vi. 2 9 cc 180 630 3 1c 57 22 / 7 11 11 57.3 (approx). i. 5 5 180 75 12 12 ii. 7 7 180 105 12 12 iii. 180 1440 8 8 iv. 1 1 180 60 3 3 cc cccc cv.8c 180 180 1 57 17 48 3.142 cii.Solution: civ.ciii. c 180 ciii. 5 12 i. 5

Target Publications Pvt. Ltd.cStd. XI Sci.: Perfect Maths - I v. 5 5 180 (128.57) approx 7 7vi. 2 2 180 40 9 9vii. 7 7 180 ( 52.5) 24 24 c c3.Express the following angles in radians:i.120 ii.225 iv. 600 iii. 945 v.vii.Solution: 2 120 120 180 3 cii. 5 225 225 180 4 9 xc 405 180 4 9 x 4c Also, y 12 180 y 15 12 y 15 If cciv. 10 600 600 180 3 c 1 1 5 5 180 900 vi. 3 108 108 180 5 vii. 4 144 144 180 5 c4.ccv.c5 c9 .(given) 21 945 945 180 4 c5 cand c 900 , find and .9Solution:iii. (given)c cc 5 180 9 –100 100Also, c 900 ccExpress the following angles in degrees,minutes and seconds form:ii.(200.6) i.(321.9) Solution:i.(321.9) 321 0.9 321 (0.9 60) 321 54 321 54 900 180 c 5 c 5 cc2 cand m B 45 .3Find m C in both the systems.Solution:In ABC,7.In ABC, m A 2 c 2 180 120 3 3and m B 45 But, m A m B m C 180 .(sum of measures of angles of a triangle is 180 ) 120 45 m C 180 m C 180 165 15 m A cii.(200.6) 200 (0.6) 200 (0.6 60) 200 36 200 36 6 c, find x and y.12Solution:xc 405 6.ci. 108 vi.If xc 405 and y c 1 5 144 5. 15 180 12 m C 15 12 cc

Target Publications Pvt. Ltd.Chapter 01: Angle and it’s Measurement8.If the radian measures of two angles of atriangle are as given below. Find the radianmeasure and the degree measure of thethird angle:5 5 3 4 ,,i.ii.9 185 15Solution:i.The measures of two angles of a triangle arecc5 5 ,9 18 5 180 5 180 , 18 9i.e., 100 , 50 Let the measure of third angle of the trianglebe x .100 50 x 180 . (sum of measures of angles of a triangle is180 )150 x

Target Publications Pvt. Ltd. Std. XI Sci.: Perfect Maths - I 4 In figure XOP, XOQ and XOR lie in first, second and third quadrants respectively. Quadrantal Angles: If the terminal arm of an angle in standard position