maths

1. 21xf.kr (Mathematics)
xf.kr
Mathematics
¼311½
f'k{kd vafdr ewY;kadu i=k
Tutor Marked Assignment
dqy vad % 20
MaximumMarks: 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, gSaA
Note: Allquestions are compulsory. The marks allotedfor each question aregivenagainst 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,A
Write your name, enrolment number, AI name and subject etc. on the top of the first page of the
answer sheet.
1. fuEufyf[kr esa ls fdlh ,d iz'u dk mÙkj yxHkx 40&60 'kCnksa esa nhft;s % 2
Answer anyone ofthe following questions in about 40-60 words :
(a) ;fn 1z  gks rks fl) dhft, fd  
1
, 1
1
z
z
z

 

,d 'kq) dkYifud la[;k gSA ;fn 1z  gks rks
vkidk fu"d"kZ D;k gksxk
If 1z  show that  
1
, 1
1
z
z
z

 

is a pure imaginarynumber. What willyou 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- de
gks rks VqdM+s dk ewY; vifjofrZr jgrk gSA VqdM+s dh yackbZ Kkr dhft,A
The cost ofa piece ofclothis Rs. 35. Ifthe piece were 4 mlonger and each meter costs Rs. 1 less, the
cost would remainunchanged. How longis the piece
(c) 45% ek=kk ds 1125 yhVj vEyh; foy;u es fdrus yhVj ikuh feyk;k tk, fd feJ.k es vEy dh
ek=kk 25% ls vf/kd rFkk 30% ls de gks
How many litres ofwater will have to be added to 1125 litres of the 45% solution ofacid so that the
resulting mixturewillcontainmore than25% but less than30% acid content.
(d) 15 fcanq ,d o`Ùk ij fLFkr gSaA bu fcanqvksa ls dqy fdrus pØh; prqHkqZt cuk, tk ldrs gSA
15 points lie on a circle. How manycyclic quadrilaterals can be drawnusing these points.
2. fuEufyf[kr esa ls fdlh ,d iz'u dk mÙkj yxHkx 40&60 'kCnksa esa nhft;s % 2
Answer anyone ofthe 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 ls
de 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

2. 22 xf.kr (Mathematics)
In a group of50 people;35 speak Hindi, 25 speak both English and Hindiand allpeople speak at least
one of the two languages. How many people speak only English and not Hindi? How many people
speak 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: 6
gksxkA
Ifthe ratio ofthree sides ofa 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 nksuksa
Lora=k :Ik ls leL;k dks gy djsa rks (i) leL;k gy gksus (ii) fdlh ,d }kjk leL;k gy djus] dh
izkf;drk Kkr dhft,A
Probabilities ofsolving a problemindependentlybyAand B are ½ and 1/3 respectively. Ifbothtryto
solvethe problemindependentlyfind the probabilitythat (i) theproblemissolved (ii)exactlyoneofthem
solves 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 dh
la[;k dh o`f) nj la[;k ds lekuqikrh gks rks fdrus ?kaVs esa cSDVhfj;k dh la[;k 2,00,000 gks
tk,xhA
In yeast the bacteria count is 1,00,000. The number isincreased by10% in2 hours. Inhow manyhours
willthe count reach 2,00,000 ifthe rate ofgrowth ofbacteria is proportionalto the numbers present.
3. fuEufyf[kr esa ls fdlh ,d iz'u dk mÙkj yxHkx 40&60 'kCnksa esa nhft;s % 2
Answer anyone ofthe following questions in about 40-60 words :
(a) 95 ij 14% LVkWd vFkok 105 ij 15% LVkWd es ls dkSu lk vf/kd ykHkizn gSA
Which 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.kuk
dhft,A
Compute 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, o`Ùkkdkj Hkkx dh f=kT;k
Kkr dhft,A
Find 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 0
2 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 ofa ABC lie along the line
2 1 0
2 1 4
x y z  
  . Find the area ofthe triangle given
thatAhas Coordinates  1, 1,2 and line segment BC has length 5 units.

3. 23xf.kr (Mathematics)
4. fuEufyf[kr esa ls fdlh ,d dk mÙkj yxHkx 100-150 'kCnksa esa nhft,A 4
Answer anyone out ofthe followingfour 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 vEy
vkSj 2F es 5%ukbVªkstu rFkk 10% QkLQksfjd vEy gSA feV~Vh dh tkap ls fdlku dks Kkr gksrk gS
fd mls Qly ds fy, U;wure 14 fdxzk ukbVªkstu rFkk 14 fdxzk QkLQksfjd vEy dh vko';drk
gSA ;fn 1F dk ewY; 6 :0 izfr fdxzk rFkk 2F dk ewY; 5 :0 izfr fdxzk gks rks Kkr dhft, fd
fdl izdkj ds QfVZykbtj dks fdl ek=kk es iz;ksx fd;k tk, dh Qly dh vko';drk dks
U;wure ewY; esa iwjk fd;k tk ldsA
There are two typesoffertilizers 1F and 2F . 1F consists of10% nitrogenand 6%phosphoric acid and
2F consists of5% nitrogen and 10% phosphoric acid. After testing the soilconditions, a farmer finds
that he atleast needs 14 kg. ofnitrogenand 14 kg ofphosphoric 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 that
nutrient requirementsare met at aminimumcost.
(b) ,d leckgq f=kHkqt dh ,d Hkqtk 24 lseh gSA bldh Hkqtkvksa ds e/; fcnqavks dks feykdj ,d nwljk
f=kHkqt cuk;k x;k gSA bl nwljs f=kHkqt dh Hkqtkvksa ds e/; fcUnqvksa dks feykdj ,d vkSj f=kHkqt
cuk;k tkrk gSA ;g izfØ;k vlhfer :Ik ls pyrh jgrh gSA bl izfØ;k es cuus okys f=kHkqtksa ds
ifjekiksa dk ;ksx Kkr dhft,A
One sideofanequilateraltriangleis 24 cm. Themid points ofits sides are joined to formanothertriangle
whosemid pointsinturnare joinedto formstillanother triangle. Theprocesscontinuesindefinitely. Find
the sumofthe perimeters ofallthe triangles.
(c) funsZ'kkad T;kfefr dk iz;ksx djds fl) dhft, fd v/kZo`r es cuk dks.k ledks.k gksrk gSA
Using Coordinate geometryprove that angle ina semicircle 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 fdxzk
pkoy dk ewY; 90:0 gSA 6 fdxzk I;kt 2 fdxzk xsagwa o 3 fdxzk pkoy dk ewY; 70 :0 gSA vkO;wg
dk iz;ksx djrs gq, rhuksa oLrqvksa dk izfr fdxzk ewY; Kkr dhft,A
The cost of4 kg onion, 3 kg wheat and 6 kg rice is Rs 60. The cost of2 kg onion, 4kg wheat and 6kg
rice is Rs. 90. The cost of6 kg. onion, 2 kg wheat and 3kg rice is Rs. 70. Findthe cost ofeachitemper
kg. usingmatrix.
5. fuEufyf[kr esa ls fdlh ,d dk mÙkj yxHkx 100-150 'kCnksa esa nhft,A 4
Answer anyone out ofthe followingfour 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 ldsA
Awindowis inthe formofrectangle surmounted bya semi-circle. Thetotalperimeter ofthewindow is
10 m. Find the dimensionsofthe window to admit maximumlight throughthe whole opening.

4. 24 xf.kr (Mathematics)
(b) Lkekdy dk iz;ksx djds o`Ùk 2 2
16x y  ds ml Hkkx dk {ks=kQy Kkr dhft, tks ijoy;
2
6y x ds cfgHkkZx es fLFkr gSA
Using integralfindthe area ofthat part ofcircle 2 2
16x y  whichis exterior to theparabola 2
6y x
(c) ,d QeZ dk ykxr Qyu   2 31
300 10
3
C x x x x   gSA tgka x mRikfnr bdkbZ;ksa dh la[;k gSA
The cost functionofa firmis given by   2 31
300 10
3
C x x x x  
Kkr dhft,
Find
(i) lhekar ykxr Qyu
Marginalcost function
(ii) vkSlr ykxr Qyu
Average cost function
(iii) mRikn] ftl ij lhekar ykxr] vkSlr ykxr ds leku gSA
The out put at whichmarginalcost is equalto average cost.
(d) fcanq  1,6,3 dk js[kk
1 2
1 2 3
x 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 3
x y z 
  . Also find the equation of the line
joining the givenpoint and its image.
6. uhps nh xbZ pkj ifj;kstukvksa dh lwph esa ls dksbZ ,d ifj;kstuk 500 'kCnksa esa rS;kj dhft, 6
Make anyone project out ofthe 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 gSA
Using cone illustrate how different types ofconicsections are generated
(b) eqEcbZ LVkWd ,Dlpst es lwphc) dksbZ 10 daifu;k pqfu,A bu daifu;ks dk (i) 2012-13 es 'ks;j
ewY; es ifjorZu (ii) 2012-13 esa vftZr ykHk (iii) 'ks;j /kkjdkas dks fn;k x;k ykHkka'k] ds vk/kkj ij
v/;;u dhft,A D;k vki bu rhuksa esa dksbZ laca/k ikrs gSaA ;fn gka rks mldh O;k[;k dhft,A
Choose any10 companies listed inBombaystock exchange. Studythe performance ofthese companies
in respect of(i) change intheir share value in2012-13 (ii) profit in2012-13 (iii) dividend givento share
holders. Do you find some relationship between the above mentioned three parameters. If yes then
explainit.

5. 25xf.kr (Mathematics)
(c) lHkh 6 f=kdks.kferh; Qyuksa rFkk izfrykse f=kdks.kferh; Qyuksa ds xzkQ cukb,A ;fn buesa dksbZ
laca/k gS rks mls LFkkfir dhft,A buds izkar o ifjlj es lca/k dh O;k[;k dhft,A
Make a graph ofallsix trigonometric and inverse trigonometric functions. Establish the relationship
betweenthemifany. Also explainthe relationship intheir respectivedomainand range.
(d) lfn'k xq.kuQy dh T;kferh; O;k[;k dhft, A bldk iz;ksx djds rhu fHkUu f=kHkqtksa rFkk rhu
fHkUu lekarj prqHkqZtksa ds {ks=kQy Kkr dhft,A T;kfefr ds iz;ksx ls Hkh bu vkÑfr;ksa ds {ks=kQy
Kkr dhft, A bu nks fof/k;ksa ls Kkr {ks=kQyksa dh rqyuk dhft,A
Interpret the geometrical significance ofvector product. Using this interpretation find area ofthree
different trianglesand three different parallelograms. Also find the area ofthese figuresusing geometry.
Compare the areas calculated bythese two methods.