Mathematics xf.kr (311) Assignment - I ewy;kadu i=k & I (Lessons 1-19) ¼ikB 1 ls 19 rd½ Max. Marks: 20 dqy vad % 20

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1 xf.kr (311) Assignment - I ewy;kadu i=k & I (Lessons 1-19) ¼ikB 1 ls 19 rd½ Max. Marks: 0 dqy vad % 0 Note: (i) All questions are compulsory. The marks alloted for each question are given against each. fvii.kh% lhkh iz'uksa ds mùkj nsus vfuok;z gsaa iz'u gsrq fu/kkzfjr vad iz'u ds lkeus fn;s x, gsaa (ii) Write your name, enrolment number, AI name and subject etc. on the top of the first page of the answer sheet. mùkj iqflrdk ds izfke i`"b ij Åij dh vksj viuk uke] vuqøekad] v/;;u dsunz dk uke] fo"k; vkfn Li"V 'kcnksa esa fyf[k,a 1. Answer any two of the following questions: =4 (a) Find the 5 th term in the expansion of (3x + ) 7. (3x + ) 7 ds izlkj dk 5oka in Kkr dhft,a (b) Show that a ab ac ab b bc ac bc c is a perfect square. n'kkzb, fd a ab ac ab b bc ac bc c,d iw.kz oxz gsa 3

2 (c) (d) Show that points A(, 3), B(5, 6) and C(0, 1) are collinear. n'kkzb, fd fcanq A(, 3), B(5, 6) rfkk C(0, 1) lajs[k gsaa Find the centre and radius of the circle: x + y 4x + 6y + 3 = 0 o`ùk x + y 4x + 6y + 3 = 0 dk dsunz rfkk bldh f=kt;k Kkr dhft,a. Answer any two of the following questions: =4 (a) Find the distance between the parallel lines 3x 4y + 7 = 0 and 3x 4y + 5 = 0. lekurj js[kkvksa 3x 4y + 7 = 0 rfkk 3x 4y + 5 = 0 ds chp dh nwjh Kkr dhft,a (b) Which term of A.P.: 5, 11, 17,... is 119? lekurj Js.kh 5, 11, 17,... dk dksu lk in 119 gs\ (c) (d) The third term of a G.P. is 8 and sixth term is 64. Find its 9 th term.,d xq.kksùkj Js.kh dk rhljk in 8 rfkk NBk in 64 gsa bldk 9oka in Kkr dhft,a Prove that: tan 7A tan 4A tan 3A = tan 7A.tan 4A.tan 3A fl) dhft, fd% tan 7A tan 4A tan 3A = tan 7A.tan 4A.tan 3A 3. Prove that: 6 (n)! = n! fl) dhft, fd% (n)! = n! n ( n 1) n ( n 1) Or/vFkok Prove that the equation of the straight line which passes through the point (a cos 3 α, a sin 3 α) and is perpendicular to the straight line x sec α + y cosec α = 0 is x cos α y sin α = a cos α. fl) dhft, fd js[kk x sec α + y cosec α = 0 ij yac js[kk] tks fcanq (a cos 3 α, a sin 3 α) ls xqtjrh gs dk lehdj.k x cos α y sin α = a cos α. gs 4. If the sum of first p terms of an A.P. is q and the sum of first q terms is p, show that the sum of its first (p + q) terms is (p + q). 33

3 ;fn,d lekarj Js.kh ds izfke p inkas dk ;ksx q rfkk izfke q inksa dk ;ksx p gks] rks fn[kkb;s fd blds izfke (p + q) inksa dk ;ksx (p + q) gksxka Or/vFkok Prove that: o o ( 60 A) sin ( 60 + A) sin Asin = fl) dhft, fd o o ( 60 A) sin ( 60 + A) sin Asin = sin3a sin3a 34

4 xf.kr (311) Assignment - II ewy;kadu i=k & II (Lessons 0-31 and Optional Module) ¼ikB 0 ls 31,oa osdfyid ekwm~;wy½ Max. Marks: 0 dqy vad % 5 Note: (i) All questions are compulsory. The marks alloted for each question are given against each. fvii.kh% lhkh iz'uksa ds mùkj nsus vfuok;z gsaa iz'u gsrq fu/kkzfjr vad iz'u ds lkeus fn;s x, gsaa (ii) Write your name, enrolment number, AI name and subject etc. on the top of the first page of the answer sheet. mùkj iqflrdk ds izfke i`"b ij Åij dh vksj viuk uke] vuqøekad] v/;;u dsunz dk uke] fo"k; vkfn Li"V 'kcnksa esa fyf[k,a 1. Answer any two of the following questions: 4 (a) Evaluate: lim x 3 1 x x 6 + x 3 1 x x eku Kkr dhft,% lim x x 3 (b) If y = sin ( x 1) + ;fn = sin ( x 1) + dy, find. dx dy y gs] rks dx Kkr dhft,a (c) A die is rolled three times. If getting 5 or 6 is considered a success, find the 35

5 probability of getting exactly two successes.,d iklk rhu ckj mnkyk x;ka ;fn 5 vfkok 6 dk vkuk lqyrk ekuk tk,] rks Bhd nks lqyrkvksa ds vkus dh izkf;drk Kkr dhft,a (d) What is the probability that a leap year, selected as random, will contain 53 Sundays? ;kn`fpnd :i ls pqus x, yhi o"kz esa 53 jfookj gksus dh izkf;drk D;k gs\ Option - I. Answer any two of the following questions: 4 r (a) If a = 3 iˆ ˆj + kˆ and, find. r ;fn lfn'k a = 3 iˆ ˆj + kˆ vksj gks rks Kkr dhft,a (b) Find the equation of the line passing through the point (3, 1, 5) and having direction ratios, 4 and 1. ml js[kk dk lehdj.k Kkr dhft, tks fcunq (3, 1, 5) ls gksdj tkrh gs] rfkk ftlds fnd vuqikr, 4 rfkk 1 gsaa (c) Find the equation of the sphere which has its centre at the origin and which passes through the point (3, 6, ). ml xksys dk lehdj.k Kkr dhft,] ftldk dsunz ewy fcanq gs rfkk tks fcanq (3, 6, ) ls gksdj tkrk gsa r (d) The position vectors of vertices A, B and C of ABC are a b r, and c r respectively. Find the position vector of the centroid of ABC. ABC ds 'kh"kks± ds flfkfr lfn'k Øe'k% flfkfr lfn'k Kkr dhft,a OR Option - II rfkk c r gsa f=hkqt ABC ds dsunzd dk Answer any two of the following questions: 4 (a) A man invested ` 60,500 in a 1% stock at ` 10. Find his income if brokerage is ` 1. 36

6 π 4 dx h 3 tan α 7 sin x sin x,d O;fDr 1 izfr'kr LVkd esa 10 ds Hkko ij 60]500 #i, fuos'k djrk gs] ;fn nykyh 1 #i, gks rks mldh vk; Kkr dhft,a (b) The cost function of firm is given by C(x) = 3x + 6x Find the average cost and Marginal cost at x = 10.,d QeZ dk ykxr Qyu C(x) = 3x + 6x + 13 }kjk iznùk gs] rks x = 10 ij vkslr ykxr rfkk lhekar ykxr Kkr dhft,a (c) Find the percent income on 10% debenture of face value ` 10 available in the market for ` #i, vafdr ewy; okys 10 izfr'kr okys fmcsupjkas] tks cktkj esa 150 #i, ij miyc/k gsa] ij izfr'kr vk; Kkr dhft,a (d) If A purchases goods worth ` 0000 from the manufacturer and adds value of ` Calculate the total sale price of the product, if VAT 1.5%. ;fn A, 0000 # ewy; dh olrq, fdlh mriknd ls [kjhnrk gs rfkk ml ij 5000 #i, dk ewy; tksm+ nsrk gs rks mrikn dk dqy foø; ewy; Kkr dhft,] tcfd ns; osv dh nj 1-5 izfr'kr gsa Option - I 3. (a) Show that the maximum volume of a cylinder which can be inscribed in a cone of height h and semivertical α is n'kkzb, fd mapkbz h vksj v)z'kh"kz dks.k α okys 'kadq ds vurxzr cus csyu dk vf/kdre vk;ru π 4 h 3 tan α gsa 7 Or/ vfkok dx Evaluate: sin x sin x eku Kkr dhft,% 37

7 Option I 4. (a) Find the equation of the sphere for which circle given by x + y + z z + = 0 and x + 3y + 4z 8 = 0 is a great circle. (b) Or/ vfkok Find the foot of the perpendicular and the length of perpendicular from the point (1, 1, ) to the plane x y + 4z + 5 = 0. fcunq (1, 1, ) ls lery x y + 4z + 5 = 0 ij [khaps x, yac dk ikn rfkk bldh yeckbz Kkr dhft,a Option-II (a) The marginal revenue function of a commodity is given as MR = 4 3x + 16x. Find the total revenue function, the demand function and average revenue function.,d olrq dk lhekar vk; Qyu MR = 4 3x + 16x }kjk iznùk gsa dqy vk;] ekax Qyu rfkk vkslr vk; Qyu Kkr dhft,a Or/vFkok Using simple average of price relative method, find the price index for 003 and 004 taking 1998 as base year: Commodity A B C D Price (in `) in Price (in `) in Price (in `) in dks vk/kkj o"kz ekudj o"kz 003,oa 004 ds fy, ewy;kuqikrkas ds ljy ek/; dh jhfr ls ewy; lwpdkad Kkr dhft,% olrq A B C D ewy; ¼` esa½ o"kz ewy; ¼` esa½ o"kz ewy; ¼` esa½ o"kz in

8 Note: fvii.kh% xf.kr (311) Assignment - III ewy;kadu i=k & III (Project Work) ¼ifj;kstuk dk;z½ Max. Marks: 0 dqy vad % 5 (i) Attempt any one project from the list of five projects given below: uhps nh xbz ik p ifj;kstukvksa dh lwph esa ls dksbz,d ifj;kstuk rs;kj dhft,a (ii) Write your name, enrolment number, AI name and subject etc. on the top of the first page of the answer sheet. mùkj iqflrdk ds izfke i`"b ij Åij dh vksj viuk uke] vuqøekad] v/;;u dsunz dk uke] fo"k; vkfn Li"V 'kcnksa esa fyf[k,a Project Work 1 ifj;kstuk dk;z 1 Cricket world cup was held in 011. Collect the data of all such players who scored highest total runs for their country. Find the batsman whose performance was most consistent. lu~ 011 esa fødsv dk fo'o di [ksyk x;ka mu lhkh cyyscktksa }kjk cuk, x, juksa ds vk dm+s,d= dhft,] ftugkasus vius ns'k dk izfrfuf/kro djrs gq;s lokzf/kd ju cuk,a Kkr dhft, fd bu lhkh cyyscktksa esa fdlds izn'kzu esa lokzf/kd fujarjrk Fkh\ Project Work ifj;kstuk dk;z Draw the graphs of sin x, cos x, tan x, cot x, sec x and cosec x. Draw the graphs of sin 1 x, cos 1 x, tan 1 x, cot 1 x, sec 1 x and cosec 1 x. Using these graphs show that graphs of each trigonometric function and its inverse trigonometric function are mirror images of each other. sin x, cos x, tan x, cot x, sec x rfkk cosec x ds vkys[k ¼xzkQ½ cukb,a sin 1 x, cos 1 x, 39

9 tan 1 x, cot 1 x, sec 1 x rfkk cosec 1 x ds vkys[k ¼xzkQ½ cukb,a bu vkys[kksa dh lgk;rk ls iznf'kzr dhft, fd izr;sd f=dks.kferh; Qyu rfkk mldk laxr izfrykse f=dks.kferh; Qyu dk vkys[k,d nwljs dk niz.k izfrfcac gsa Project Work 3 ifj;kstuk dk;z 3 Geometrically interpret Rolle's theorem and Lagrange's Mean Value theorem. Using five different types of functions and intervals verify these theorems. jksys rfkk ykxjkat ds e/;eku izes; dh T;kferh; O;k[;k dhft,a ikap fhkuu izdkj ds Qyuksa o varjkyksa ds fy, bu izes;ksa dks lr;kfir dhft,a Project Work 4 ifj;kstuk dk;z 4 Find 10 examples of appearance of conic sections. Taking 1 example explain the importance of conic section in that example. vius ifjos'k esa nl,sls mnkgj.k <waf<+;s tgka ij gesa 'kadq ifjpnsn fn[kkbz nsrs gsaa buesa ls fdlh,d mnkgj.k esa ml 'kadq ifjpnsn dh mi;ksfxrk dks Li"V dhft,a Project Work 5 ifj;kstuk dk;z 5 How do we solve a system of equations using determinants and matrices. Using different system of equations explain the different conditions when (i) Any system of equations has a unique solution. (ii) Any system of equations has infinity many solutions. (iii) Any system of equations has no solution. lkjf.kd rfkk vko;wg dk iz;ksx djds ge lehdj.k fudk; dks fdl izdkj gy djrs gsa\ fofhkuu lehdj.k fudk;ksa dk iz;ksx djrs gq,s mu lhkh flfkfr;ksa dh O;k[;k dhft, tc (i) fdlh lhedj.k fudk; dk vf}rh; gy gksrk gsa (ii) fdlh lehdj.k fudk; ds vuar gy gksrs gsaa (iii) fdlh lehdj.k fudk; dk dksbz gy ugh gksrka 40

dqy iz uksa dh la[;k % 26 dqy i`"bksa dh la[;k % 11 Total No. of Questions : 26 Total No. of Pages : fo"k; % xf.kr

dqy iz uksa dh la[;k % 26 dqy i`bksa dh la[;k % 11 Total No. of Questions : 26 Total No. of Pages : fok; % xf.kr dqy iz uksa dh la[;k % 6 dqy i`"bksa dh la[;k % Total No. of Questions : 6 Total No. of Pages : fo"k; % xf.kr Subject : MATHEMATICS le; % 03?k.Vs iw.kk±d % 00 Time : 03 Hours Maximum Marks : 00 funsz k%&