Std. XI Commerce Mathematics & Statistics - II

Transcription

1 Written s per the revised syllbus prescribed by the Mhrshtr Stte Bord of Secondry nd Higher Secondry Eduction, Pune. Std. XI Commerce Mthemtics & Sttistics - II Slient Fetures Exhustive coverge of entire syllbus. Topic-wise distribution of ll textul questions nd prctice problems t the beginning of every chpter. Covers nswers to ll textul nd miscellneous exercises. Precise theory for every topic. Net, lbelled nd uthentic digrms. Relevnt nd importnt formule wherever required. Printed t: Repro Knowledgecst Ltd., Mumbi Trget Publictions Pvt. Ltd. No prt of this book my be reproduced or trnsmitted in ny form or by ny mens, C.D. ROM/Audio Video Cssettes or electronic, mechnicl including photocopying; recording or by ny informtion storge nd retrievl system without permission in writing from the Publisher. 49_JUP P.O. No. 064

2 Prefce Mthemtics is not just subject tht is restricted to the four wlls of clssroom. Its philosophy nd pplictions re to be looked for in the dily course of our life. The knowledge of mthemtics is essentil for us, to explore nd prctice in vriety of fields like business dministrtion, bnking, stock exchnge nd in science nd engineering. With the sme thought in mind, we present to you Std. XI Commerce: Mthemtics nd Sttistics-II complete nd thorough book with revolutionry fresh pproch towrds content nd thus lying pltform for n in depth understnding of the subject. This book hs been written ccording to the revised syllbus. At the beginning of every chpter, topic wise distribution of ll textul questions including prctice problems hve been provided for simpler understnding of different types of questions. Netly lbelled digrms hve been provided wherever required. We hve provided nswer keys for ll the textul questions nd miscellneous exercises. In ddition to this, we hve included prctice problems bsed upon solved exercises which not only id students in self evlution but lso provide them with plenty of prctice. We ve lso ensured tht ech chpter ends with set of Multiple Choice Questions so s to prepre students for competitive exmintions. We re sure this study mteril will turn out to be powerful resource for students nd fcilitte them in understnding the concepts of Mthemtics in the most simple wy. The journey to crete complete book is strewn with triumphs, filures nd ner misses. If you think we ve nerly missed something or wnt to pplud us for our triumphs, we d love to her from you. Plese write to us on: mil@trgetpublictions.org Yours fithfully Publisher Edition: Second Best of luck to ll the spirnts! Disclimer This book is intended to be study mteril expressing views nd elborting concepts for ese of understnding for students nd purely for their benefits. We mke no representtions s to ccurcy, completeness, correctness, suitbility, or vlidity of ny informtion through this study mteril. And, shll not be held lible or responsible for ny errors, omissions, or differences in this informtion or ny losses, injuries or dmges rising from its use. All informtion is provided on n s it is bsis depending upon the understnding of the uthor nd his/her elbortion of such concepts long with doption nd inspirtion from vrious other texts in reltion to the topics s mentioned in this study mteril. It is the reder s responsibility to verify their own fcts. Through this study mteril we re only explining nd elborting vrious concepts s my be necessry for the students in the present frmework nd context. The views nd opinions expressed in this study mteril re purely s per the understnding of the uthors nd do not necessrily reflect the officil policy or position of ny other gency, orgniztion, employer or compny. Assumptions mde in this nlysis re not reflective of the position of ny other thn the uthors - nd since we re criticlly thinking humn beings with personified opinions, these views re lwys subject to chnge, revision nd rethinking t ny time. Plese do not hold us to them in perpetuity. Reders shll not misconceive this work with ny other work. This work is purely inspired upon the course work s suggested nd prescribed by the Mhrshtr Stte Bord of Secondry nd Higher Secondry Eduction, Pune. Reference of Textbook - Reprint: 07, Print order: N/PB/07-8/5,000 All the fcts nd figures so stted hve been purely dopted from vrious reserch points purely for the purpose of representtion nd explining the students nd reders t lrge s prt of fir deling. By producing nything nd everything in this book the uthor does not intend to clim copyrights on ny such mteril but hs been purely dopted nd used for the purpose of representtion nd for better understnding of the students with pure intention to educte the public t lrge for better Indi. No. Topic Nme Pge No. Logrithms Theory of Attributes 9 Prtition Vlues 58 4 Mesures of Dispersion 5 5 Moments 6 6 Skewness nd Kurtosis 86 7 Permuttions nd Combintions 8 Probbility 6 9 Index Numbers 9 0 Time Series 7

3 0 Logrithms Chpter 0: Logrithms Type of Problems Exercise Q. Nos. Problems bsed on definition of logrithm Lw of Product Q. (i. to iv.). Q. (i. to iv.) Q. (i. to iv.) Q. (i. to iv.) Q. (i. to iv.) (Bsed on Exercise.) Q. (i. to iv.) Miscellneous Q. Q.4, 8 Q.7 (i.). Q.8 (ii.) Q. (i.) Q.6 (i.) (Bsed on Exercise.) Miscellneous Q.,, 4, 7 Q.. Q.5 (i.) Lw of Exponent Lw of Quotient Miscellneous Q. Q.. Q.7 (ii.) Miscellneous Q.9 Problems bsed on Product, Quotient nd Exponent lws. Q.4 (i. to v.) Q.5 (ii to v.) Q.6 (i. to iii.) Q.7 (iii., iv.)

4 Std. XI : Commerce (Mths II) Q.8 (i., iii.) Q.9 (i. to iii.) Q. (ii., iii.) Q. to Q.4 (Bsed on Exercise.) Miscellneous Q.4, 5 Q.6 (ii.) Q.7, 8, 0, Q.5 to Q.8 Q.0,, 4, 6, 9, 0 Q., 5, 6, 7,, 5. Q.0 (i. to iv.) To solve problems without using log tble Q.9 (i., ii.) (Bsed on Exercise.) Miscellneous Q. Q.9. Q. to Q. Chnge of Bse lw Q. to Q.7 (Bsed on Exercise.) Miscellneous Q.5, 8 Q.0,,, 4. Q. to Q. To solve problems by using log tble Q. to Q.7 (Bsed on Exercise.) Miscellneous Q. to Q.4 Q.6

5 Chpter 0: Logrithms Syllbus:. Definition. Lws of Logrithms. Chnge of bse lw.4 Numericl problems Introduction In mthemtics, logrithm of number to given bse is the power of exponent to which the bse must be rised in order to produce the number. For exmple, the logrithm of to the bse is 5 becuse 5 is how mny s one must multiply to get. Thus =. In the lnguge of exponent, 5 = so log = 5.. Definition If x = b, then x = log b ( > 0, ), (b > 0) where is clled the bse of the logrithm. The two sttements x = b nd x = log b re equivlent. The sttement x = b is sid to be in the exponentil form nd the sttement x = log b is sid to be in the logrithmic form. We cn convert n exponentil form into the logrithmic form. Exmple: Exponentil form Logrithmic form 4 = 8 log 8= 4 5 = log = 5 = 9 log 9 = = log = Remrks. We hve m = x if nd only if x = log m. Negtive numbers nd zero hve no logrithms.. i. log = 0, > 0, Let log = x x = = 0 x = 0 log = 0 i.e., logrithm of to ny bse is 0 ii. log =, > 0, Proof : Let log = x x = = x = log = i.e., logrithm of number to the sme bse is. log x iii. x, > 0, Let log x = y y = x log x x iv. If log m = log n, then m = n. v. If > nd m > n, then log m > log n nd conversely.. Lws of Logrithms. Lw of Product: log (xy) = log x + log y, (, x, y > 0, ) Let log x = m nd log y = n x = m nd y = n m. n = xy m + n = xy log (xy) = m + n log (xy) = log x + log y Thus logrithm of the product of two numbers is equl to the sum of their logrithms with reference to the sme bse. Corollry: log (xyz ) = log x + log y + log z +... Lw of Quotient: x log y = log x log y, (, x, y > 0, ) Let log x = m nd log y = n m = x nd n = y m = x n y m n = x y

6 Std. XI : Commerce (Mths II) x log = m n y x log y = log x log y Corollry: i. log x = log log x = 0 log x = log x xy pq ii. log log x log y log p log q. Lw of Exponent: log x y = ylog x, (x > 0, > 0, ) Let log x = m x = m Now, x y m y = x y = my log x y = my log x y = ylog x Corollry: i. log m x = m log x p q x y ii. log r s =plo x+qlog y rlog z slog w zw Exercise.. Write the following in logrithmic form: i. 8 = 5 ii. /5 = 8 iii. 7 = 49 iv. 0 = 0.0 i. 8 = 5 = log 8 5.[By definition of logrithm] ii. 5 = 8 5 = log 8.[By definition of logrithm] iii. 7 = 49 = log 7.[By definition of logrithm] 49 iv. 0 = 0.0 = log 0 (0.0).[By definition of logrithm]. Express the following in exponentil form: i. log = 4 ii. log /6 8 = 4 iii. log = iv. log 4 =. i. log = = 656.[By definition of logrithm] ii. log/6 4 6 = 8 8 = 4.[By definition of logrithm] iii. log = (0.5) = 0.5.[By definition of logrithm] iv. log 4 = = 4.[By definition of logrithm]. Find the vlues of: i. log / 8 ii. log iii. log 5 5 iv. log 7 7. i. Let x = log/8 x = 8.[By definition of logrithm] ( ) x = x = x = x = log/8 = 4

7 Chpter 0: Logrithms ii. Let x = log 5 (0.008) 5 x = [By definition of logrithm] 5 x 8 = x = 0 5 x = 5 5 x = 5 x = log 5 (0.008) = iii. Let x = log x = 5.[By definition of logrithm] 5 x = 5 5 x = 5 log 5 5 = 5 iv. Let x = log x = 7.[By definition of logrithm] 7 x = x = 7 log 7 7 = 4. Simplify the following s single logrithm: i. log log 0 4 ii. log 7 log 4 iii. log log 5 7 log 5 iv. log 0 + log 0 log 0 5 v. log log 6 + log. i. log log 0 4 = log log 0 4.[By exponent lw] = log log 0 6 = log 0 (5 6).[By product lw] = log 0 80 ii log 7 log 4 = log 7 log 4.[By exponent lw] = log 49 log 4 = log 49.[By quotient lw] 4 = log 7 iii. log log 5 7 log 5 = log (log 5 6 log 5 ) = log log 6 5.[By quotient lw] = log log 5 = log5 7 + log5.[by exponent lw] = log log 5 = log 5 49.[By product lw] iv. log 0 + log 0 log 0 5 = log 0 ( ) log 0 5.[By product nd exponent lw] = log 0 6 log = log 0.[By quotient lw] 5 v. log log 6 + log = log log6 + log.[by exponent lw] = log 9 log 6 + log = log 9 log 4 + log = log 9 + log.[by quotient lw] 4 = log 9.[By product lw] 4 = log 7 5. Evlute: i. 4 5 log 5 65 ii. log log 5 0 log 0 7 iii. 5 log log log iv. 6 log log log log v. log log log