21xf.kr (Mathematics)

xf.krMathematics

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Tutor Marked Assignmentdqy vad % 20

Maximum Marks: 20

fVIi.kh% (i) lHkh iz'uksa ds mÙkj nsus vfuok;Z gSaA iz'u gsrq fu/kkZfjr vad iz'u ds lkeus fn;s x, gSaANote: All questions are compulsory. The marks alloted for each question are given against each.

(ii) mRrj 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,AWrite your name, enrolment number, AI name and subject etc. on the top of the first page of theanswer sheet.

1. fuEufyf[kr esa ls fdlh ,d iz'u dk mÙkj yxHkx 40&60 'kCnksa esa nhft;s % 2

Answer any one of the following questions in about 40-60 words :

(a) ;fn 1z gks rks fl) dhft, fd 1, 1

1z zz

,d 'kq) dkYifud la[;k gSA ;fn 1z gks rks

vkidk fu"d"kZ D;k gksxk\

If 1z show that 1, 1

1z zz

is a pure imaginary number. What will you conclude if 1z ?

(b) diM+s ds ,d VqdM+s dk ewY; 35 : gSA ;fn diMs dh yackbZ 4 eh- vf/kd rFk ewY; 1 : izfr eh- degks rks VqdM+s dk ewY; vifjofrZr jgrk gSA VqdM+s dh yackbZ Kkr dhft,AThe cost of a piece of cloth is Rs. 35. If the piece were 4 m longer and each meter costs Rs. 1 less, thecost would remain unchanged. How long is the piece

(c) 45% ek=kk ds 1125 yhVj vEyh; foy;u es fdrus yhVj ikuh feyk;k tk, fd feJ.k es vEy dhek=kk 25% ls vf/kd rFkk 30% ls de gks\How many litres of water will have to be added to 1125 litres of the 45% solution of acid so that theresulting mixture will contain more than 25% but less than 30% acid content.

(d) 15 fcanq ,d o`Ùk ij fLFkr gSaA bu fcanqvksa ls dqy fdrus pØh; prqHkqZt cuk, tk ldrs gSA15 points lie on a circle. How many cyclic quadrilaterals can be drawn using these points.

2. fuEufyf[kr esa ls fdlh ,d iz'u dk mÙkj yxHkx 40&60 'kCnksa esa nhft;s % 2

Answer any one of the following questions in about 40-60 words :

(a) 50 O;fDr;ksa ds ,d lewg es 35 fganh] 25 fganh o vxzsth nksuksa RkFkk izR;sd O;fDr nksuks Hkk"kkvksa es lsde ls de ,d Hkk"kk vo'; cksyrk gSA fdrus O;fDr dsoy vaxzsth cksyrs gSa rFkk fganh ugh cksyrs\fdrus O;fDr vaxzsth cksyrs gSA

22 xf.kr (Mathematics)

In a group of 50 people; 35 speak Hindi, 25 speak both English and Hindi and all people speak at leastone of the two languages. How many people speak only English and not Hindi? How many peoplespeak English?

(b) ,d f=kHkqt dh Hkqtkvksa dk vuqikr a: b: c = 7: 8: 9 gks rks n'kkZb, fd cosA: cosB: cosC = 14: 11: 6gksxkAIf the ratio of three sides of a triangle is a: b: c = 7: 8: 9 then show that cosA: cosB: cosC = 14: 11: 6.

(c) A rFkk B }kjk ,d leL;k dks Lora=k :Ik ls gy djus dh izkf;drk Øe'k% ½ o 1/3 gSA ;fn nksuksaLora=k :Ik ls leL;k dks gy djsa rks (i) leL;k gy gksus (ii) fdlh ,d }kjk leL;k gy djus] dhizkf;drk Kkr dhft,AProbabilities of solving a problem independently by A and B are ½ and 1/3 respectively. If both try tosolve the problem independently find the probability that (i) the problem is solved (ii) exactly one of themsolves the problem

(d) [kehj es cSDVhfj;k dh la[;k 1,00,000 gSA 2 ?kaVs es la[;k 10% c<+ tkrh gSA ;fn cSDVhfj;k dhla[;k dh o`f) nj la[;k ds lekuqikrh gks rks fdrus ?kaVs esa cSDVhfj;k dh la[;k 2,00,000 gkstk,xhAIn yeast the bacteria count is 1,00,000. The number is increased by 10% in 2 hours. In how many hourswill the count reach 2,00,000 if the rate of growth of bacteria is proportional to the numbers present.

3. fuEufyf[kr esa ls fdlh ,d iz'u dk mÙkj yxHkx 40&60 'kCnksa esa nhft;s % 2

Answer any one of the following questions in about 40-60 words :

(a) 95 ij 14% LVkWd vFkok 105 ij 15% LVkWd es ls dkSu lk vf/kd ykHkizn gSAWhich is a better investment 14% stock at 95 or 15% stock at 105

(b) 2005 dks vk/kkj o"kZ ekudj o"kZ 2005, 2006, 2007, 2008 ,oa 2009 ds fy, lwpdkad dh x.kukdhft,ACompute the index number for the years 2005, 2006, 2007, 2008 and 2009 taking 2005 as base year

o"kZ Year 2005 2006 2007 2008 2009

ewY; Price 120 144 168 204 216

(c) xksys 2 2 2 49x y z ij ry 2 3 5 14 0x y z }kjk dkVs x, oÙkkdkj Hkkx dh f=kT;kKkr dhft,AFind the radius of the circular section of the sphere 2 2 2 49x y z by the plane

2 3 5 14 0x y z

(d) ,d ABC ds Å/oZ B rFkk C js[kk 2 1 02 1 4

x y z ij fLFkr gSA ;fn A ds funsZ'kkad 1, 1,2

rFkk js[kk [k.M BC dh yackbZ 5 bdkbZ gks rks ABC dk {ks=kQy Kkr dhft,A

Vertices B and C of a ABC lie along the line2 1 0

2 1 4x y z

. Find the area of the triangle given

that A has Coordinates 1, 1,2 and line segment BC has length 5 units.

23xf.kr (Mathematics)

4. fuEufyf[kr esa ls fdlh ,d dk mÙkj yxHkx 100-150 'kCnksa esa nhft,A 4Answer any one out of the following four questions in about 100 to 150 words

(a) nks izdkj ds QfVZykbtj 1F rFkk 2F miyC/k gSaA 1F esa 10% ukbVªkstu rFkk 6% QkjLQksfjd vEyvkSj 2F es 5%ukbVªkstu rFkk 10% QkLQksfjd vEy gSA feV~Vh dh tkap ls fdlku dks Kkr gksrk gSfd mls Qly ds fy, U;wure 14 fdxzk ukbVªkstu rFkk 14 fdxzk QkLQksfjd vEy dh vko';drkgSA ;fn 1F dk ewY; 6 :0 izfr fdxzk rFkk 2F dk ewY; 5 :0 izfr fdxzk gks rks Kkr dhft, fdfdl izdkj ds QfVZykbtj dks fdl ek=kk es iz;ksx fd;k tk, dh Qly dh vko';drk dksU;wure ewY; esa iwjk fd;k tk ldsA

There are two types of fertilizers 1F and 2F . 1F consists of 10% nitrogen and 6% phosphoric acid and

2F consists of 5% nitrogen and 10% phosphoric acid. After testing the soil conditions, a farmer finds

that he atleast needs 14 kg. of nitrogen and 14 kg of phosphoric acid for the crop. If 1F costs Rs. 6 per

kg. and 2F costs Rs. 5 per kg., determine how much of each type of fertilizer should be used so thatnutrient requirements are met at a minimum cost.

(b) ,d leckgq f=kHkqt dh ,d Hkqtk 24 lseh gSA bldh Hkqtkvksa ds e/; fcnqavks dks feykdj ,d nwljkf=kHkqt cuk;k x;k gSA bl nwljs f=kHkqt dh Hkqtkvksa ds e/; fcUnqvksa dks feykdj ,d vkSj f=kHkqtcuk;k tkrk gSA ;g izfØ;k vlhfer :Ik ls pyrh jgrh gSA bl izfØ;k es cuus okys f=kHkqtksa dsifjekiksa dk ;ksx Kkr dhft,AOne side of an equilateral triangle is 24 cm. The mid points of its sides are joined to form another trianglewhose mid points in turn are joined to form still another triangle. The process continues indefinitely. Findthe sum of the perimeters of all the triangles.

(c) funsZ'kkad T;kfefr dk iz;ksx djds fl) dhft, fd v/kZor es cuk dks.k ledks.k gksrk gSAUsing Coordinate geometry prove that angle in a semi circle is a right angle.

(d) 4 fdxzk I;kt] 3 fdxzk xsagwa o 2 fdxzk pkoy dk ewY; 60 :0 gSA 2 fdxzk I;kt] 4 fdxzk xsagwa o 6 fdxzkpkoy dk ewY; 90:0 gSA 6 fdxzk I;kt 2 fdxzk xsagwa o 3 fdxzk pkoy dk ewY; 70 :0 gSA vkO;wgdk iz;ksx djrs gq, rhuksa oLrqvksa dk izfr fdxzk ewY; Kkr dhft,AThe cost of 4 kg onion, 3 kg wheat and 6 kg rice is Rs 60. The cost of 2 kg onion, 4kg wheat and 6kgrice is Rs. 90. The cost of 6 kg. onion, 2 kg wheat and 3kg rice is Rs. 70. Find the cost of each item perkg. using matrix.

5. fuEufyf[kr esa ls fdlh ,d dk mÙkj yxHkx 100-150 'kCnksa esa nhft,A 4Answer any one out of the following four questions in about 100 to 150 words(a) ,d f[kM+dh vk;rkdkj gS ftlds mij ,d v/kZo`Ùk gSA f[kM+dh dk ifjeki 10 ehVj gSA f[kM+dh

dh foek,a Kkr dhft, rkfd vf/kdre laHko izdk'k dh ek=kk vanj vk ldsAA window is in the form of rectangle surmounted by a semi-circle. The total perimeter of the window is10 m. Find the dimensions of the window to admit maximum light through the whole opening.

24 xf.kr (Mathematics)

(b) Lkekdy dk iz;ksx djds oÙk 2 2 16x y ds ml Hkkx dk {ks=kQy Kkr dhft, tks ijoy;2 6y x ds cfgHkkZx es fLFkr gSA

Using integral find the area of that part of circle 2 2 16x y which is exterior to the parabola 2 6y x

(c) ,d QeZ dk ykxr Qyu 2 31300 103

C x x x x gSA tgka x mRikfnr bdkbZ;ksa dh la[;k gSA

The cost function of a firm is given by 2 31300 103

C x x x x

Kkr dhft,Find

(i) lhekar ykxr QyuMarginal cost function

(ii) vkSlr ykxr QyuAverage cost function

(iii) mRikn] ftl ij lhekar ykxr] vkSlr ykxr ds leku gSAThe out put at which marginal cost is equal to average cost.

(d) fcanq 1, 6,3 dk js[kk 1 21 2 3x y z s esa izfrfcEc Kkr dhft,A fcanq rFkk blds izfrfcEc dks

feykus okyh js[kk dk lehdj.k Kkr dhft,A

Find the image of the point 1, 6,3 in the line 1 2

1 2 3x y z . Also find the equation of the line

joining the given point and its image.

6. uhps nh xbZ pkj ifj;kstukvksa dh lwph esa ls dksbZ ,d ifj;kstuk 500 'kCnksa esa rS;kj dhft, 6Make any one project out of the four given below in 500 words.

(a) 'kadq dk iz;ksx djds n'kkZb, fd fdl izdkj fofHkUu izdkj ds 'kadq ifjPNsnksa dh mRifÙk gksrh gSAUsing cone illustrate how different types of conic sections are generated

(b) eqEcbZ LVkWd ,Dlpst es lwphc) dksbZ 10 daifu;k pqfu,A bu daifu;ks dk (i) 2012-13 es 'ks;jewY; es ifjorZu (ii) 2012-13 esa vftZr ykHk (iii) 'ks;j /kkjdkas dks fn;k x;k ykHkka'k] ds vk/kkj ijv/;;u dhft,A D;k vki bu rhuksa esa dksbZ laca/k ikrs gSaA ;fn gka rks mldh O;k[;k dhft,AChoose any 10 companies listed in Bombay stock exchange. Study the performance of these companiesin respect of (i) change in their share value in 2012-13 (ii) profit in 2012-13 (iii) dividend given to shareholders. Do you find some relationship between the above mentioned three parameters. If yes thenexplain it.

25xf.kr (Mathematics)

(c) lHkh 6 f=kdks.kferh; Qyuksa rFkk izfrykse f=kdks.kferh; Qyuksa ds xzkQ cukb,A ;fn buesa dksbZlaca/k gS rks mls LFkkfir dhft,A buds izkar o ifjlj es lca/k dh O;k[;k dhft,AMake a graph of all six trigonometric and inverse trigonometric functions. Establish the relationshipbetween them if any. Also explain the relationship in their respective domain and range.

(d) lfn'k xq.kuQy dh T;kferh; O;k[;k dhft, A bldk iz;ksx djds rhu fHkUu f=kHkqtksa rFkk rhufHkUu lekarj prqHkqZtksa ds {ks=kQy Kkr dhft,A T;kfefr ds iz;ksx ls Hkh bu vkÑfr;ksa ds {ks=kQyKkr dhft, A bu nks fof/k;ksa ls Kkr {ks=kQyksa dh rqyuk dhft,AInterpret the geometrical significance of vector product. Using this interpretation find area of threedifferent triangles and three different parallelograms. Also find the area of these figures using geometry.Compare the areas calculated by these two methods.