math set iii

8/12/2019 Math Set III

1/6

ekWMy izu&i= lsV&III

Mathematics xf.kr

Time Allowed : 3 Hours Max. Marks -100

General Instructions :

(i) lHkh izu vfuok;Z gSaA(ii) bl izui= esa29 izu gSatksrhu [k.MksaesafoHkkftr gSa] v] c rFkk lA [k.M ^v* esa10 iz

gSaftuesals izR;sd ,d vad dk gSA [k.M ^c* esa12 izu gSaftuesalsizR;sd pkj vadksadk gS[k.M ^l* esa 7 izu gSaftuesals izR;sd N% vadksdk gSA

(iii) [k.M ^v* esalHkh izuksa dk mkj ,d 'kCn] ,d okD; vFkok izu dh vko;drk dsvuqlfn;s tk ldrsgSaA

(iv) iw.kZ izui= esafodYi ughagSA fQj Hkh pkj vadksaokys 5 izuksaesa,oa N% vadksaokys 4 izuksavarfje fodYi gSaA ,sls lHkh izuksa esa vkidks,d gh fodYi gy djuk gSA

(v) dSydqysV dsiz;ksx dh vuqefr ughagSAGeneral Instruction(i) All question are compulsory.(ii) The question paper consists of 29 questions divided into three sections A, B and C. Section A is

comprised of 10 questions of one mark each, section B is comprised of 12 questions of Four marks

each and section C is comprised of Six marks each.(iii) All questions in section A are to be answered in one word, one sectence or as per the exact

requirement of the question.

(iv) There is no overall choice. However, internal choice has been provided in 5 questions of section Band 4 questions of section C. you have to attempt only one alternates in all such questions.

(v) Use of calculator is not permitted.

[k.M & v Section Aiz'u la[;k 1 ls10 rd izR;sd izu 1 vad dk gSAQuestion number 1 to 10 carry 1 mark each.

(1) leqPp; { }1,2,3S = ij ifjHkkf"kr Qyu :f s s dsfy,] tks fuEu#i esa fn[kk;k x;k gS

{ }(1,3),(3,2),(2,1)f = O;qRe Qyu 1f fy[ksaAWrite the 1f

for the function :f s s defined over a set { }1,2,3S = shown as { (1,3),(3,2),(2,1f =

(2) eku Kkr djsa 1 1sin2

Find the value of1 1

sin 2

(3) 2 2X eokysvkO;wgA dsfy;sdk eku Kkr djsa] tgk 5A = ,oa 20A = gSAWrite the value of for a matrix A of order 2 2X where 5A = and 20A = .

(4) ;fn1 1 3

1 2 1

x

y

=

] rks x y+ dk eku crk;saA

If1 1 3

1 2 1

x

y

=

then find x y+ .


2/6

(5) lkjf.kd2 2 2

2 2 2

x y z

x y z

x y z

dk eku fy[ksaA

Write the value of the determinant2 2 2

2 2 2

x y z

x y z

x y z

.

(6) eku Kkr djsa2

3

2

sin xdx

Evaluate2

3

2

sin xdx

.

(7) vody lehdj.k3 22

2sin 1 0

d y dy dy

dx dx dx

+ + + =

dk dksfV gS

A 3 B 2 C 1 D vifjHkkf"krThe degree of the differential equation

3 22

2sin 1 0

d y dy dy

dx dx dx

+ + + =

is

(A) 3 (B) 2 (C) 1 (D) Not defined

(8) a

vkSj a

ds chp cu jgsdks.k dk eku fy[ksaAWrite the angle between a

and a

.(9) a

vkSj ( )a b

ds chp vfnk xq.kd dseku fy[ksaA

Write the scaler product of a

and ( )a b

.

(10) fn;s x;sljy js[kk 3 4 83 2 6

x y z+ += = dsfy;sfnd~dksT;k dk eku fy[ksaA

Write the direction cosine of the state line3 4 8

3 2 6

x y z+ += = .

[k.M & ^c*Section B

iz'u la[;k 11 ls22 rd izR;sd izu 4 vad dk gSAQuestion number 11 to 22 carry 4 marks each.

(11) izkd`frd la[;kvksadsleqPp; Nij ifjHkkf"kr f}vk/kkjh laf;dsfy,] tgk a b a = ,oab

dsy?kqke lekioR;Z gSa Kkr djsa&i 20 16 ii D;ke fofues; xq.k dks/kkj.k djrk gS \iii D;klkgp;Z fu;e dk ikyu djrk gS \iv dsfy;slRled vo;oFor the binary operation defined on set of natural number N . Where a b = LCM of a and b , fin

(i) 20 16

(ii) Is commutative

(iii) Is associative

(iv) Identify element for


3/6

(12) fl) djsa fd 1 1 15 3 63sin cos tan13 5 16

+ =

Prove that 1 1 15 3 63

sin cos tan13 5 16

+ = .

(13) fn;s x;soxZ vkO;wg2 2 4

1 3 4

1 2 3

A

=

dksl;fer ,oa fo"k; l;fer vkO;wgksads;ksx ds#i

esaiznfkZr djsaA

Express the square matrix

2 2 4

1 3 4

1 2 3

A

=

as sum of symmetric and skened symmetric matrices

OR

;fn3 1

1 2A

=

] rks nkkZb;sfd 2 5 7 0A A I + = ]

If3 1

1 2

A

=

, then show that 2 5 7 0A A I + = .

(14) fn;s x;sQyu f ds lR;rk dh tkp djsa] tgksin

, 0( )

1, 0

xif x

f x x

x if x


4/6


5/6

Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.OR

i vUrjky Kkr djsa] ftlesaQyu ( ) 2 2 5f x x x= + o/kZeku e esa gSAii eku Kkr djsavodyu dk mi;ksx djrs gq,& 25.3 (i) Find the internal, where function ( ) 2 2 5f x x x= + is in increasing order.

(ii) Using differentiation, find the value (approx) 25.3 .

(25) i[ky; 2x y= ] ljy js[kk 2y x= + ,oax & v{k ls f?kjs{ks= dk {ks=Qy lekdyu fof/k lsKkr djsaAFind the area of the region enclosed by the parabola 2x y= , st line 2y x= + and x - axis.

OR

;ksx dh lhek fof/k vFkkZr izkjafHkd fof/k ls eku Kkr djsa&3

1

xdx

Using limit of sum ie. Ab-initia method evaluate -3

1

xdx

(26) ;fn 3dyx y xdx

+ = ] rks Kkr djsa

i vodyu lehdj.k dh dksfVii LFkkfir #i esa ykdj P ,oaQ dseku

iii lekdyu xq.kkad vFkkZrPdx

e iv O;kid gyv izkpy fLFkjkad dseku ;fn 1x = ,oa 2y = gks]vi fofk"V gy mi;qZDr izkpy fLFkjkad dslanHkZ esa]

If 3dy

x y xdx

+ = , then evaluate

(i) Order of the differential equation.

(ii) P and Q after bringing it into standard from.

(iii) Integrating factor ie.Pdx

e .(iv) General solution.

(v) Value of arbitrary constant if 1x = and 2y = .

(vi) Particular solution.

(27) vfu;ksftr pj x dsfy;sizkf;drk caVu ( )P x fuEu gSa, 0

2 , 1( )

3 , 2

0,

K if x

K if xP x

K if x

otherwise

=

==

=

i Kdk eku Kkr djsas (i) Determine the value of Kii Kkr djsa ( 2)P x < (ii) Find ( 2)P x < iii ( 2)P x (iii) ( 2)P x iv ( 2)P x (iv) ( 2)P x v izkf;drk caVu lkj.kh (v) Probability Distribution Tablevi ek/; (vi) Mean.

(28) (3, 4, 5) ,oa (2, 3,1) dkstksM+usokyh ljy js[kk ds}kjk lery 2 7x y z+ + = dkscs/kusokysfcUnqdk funsZkkad Kkr djsaA


6/6

Find the co-ordinates of the point where the line through (3, 4, 5) and (2, 3,1) crosses the plane

2 7x y z+ + = .

OR

lery ds lehdj.k dks Kkr djsatks (1,1,0), (1,2,1),oa ( 2,2, 1) lsxqtjrsgksaAFind the equation of the plane passing through there points (1,1,0),(1,2,1) and ( 2,2, 1) .

(29) ,d mRiknd la;a= nksmRikn A ,oa B dk mRiknu djrk gSA mRikn A esafuekZ.k Lrj ij 9?kaVs ,oalqlTtkdj.k esa 1 ?kaVk rFkk mRikn B esafuekZ.k Lrj ij 12 ?kaVk ,oalqlTtkdj.k es3 ?kaVsyxrs gSaA fuekZ.k dsfy;svf/kdre 180 ekuo ?kaVs,oa lqlTtkdj.k dsfy;s vf/kdre30 ?kaVs miyC/k gSaA izR;sd mRikn A ij mRiknd dks` 8000 ,oa mRikn B ij ` 12000 feyrsgSaA mRikn A ,oa B fdruh la[;k esa izfr lIrkg cuk;h tk;sa ftllsla;a= dksvf/kdre ykHkgksldsA vf/kd ykHk dh izfr lIrkg jkfk Kkr djsaAA manufacturing company makes two models A and B of product. Each piece of model A requires 9

labour hours for fabricating and 1 labour hour for finishing. Each piece of model B requires 12 labou

hours for fabricating and 3 labour hours for finishing. For fabricating and finishing the maximum

labour hours are available 180 and 30 respectively. The company makes a profit of` 8000 on each

piece of model A and` 12000 on each piece of model B. How many pieces of model A and B shou

be manufactured per week to realise a maximum profit ? What is the maximum profit per week ?

OR

,d vkgkj foKkuh nks izdkj ds HkksT; inkFkksZaX ,oa Y dksbl izdkj feykdj ,d feJ.k cukupkgrk gS ftlesa de ls de 10 bZdkbZ foVkfeu A, 12 bZdkbZ foVkfeu B ,oa8 bZdkbZ foVkfeuC gksA ,d fdyksHkksT; inkFkZ esafoVkfeu dh ek=k bl izdkj gS &

Hkkstu foVkfeu A B CX 1 2 3Y 2 2 1

X dh dher ` 16@fdxzk ,oaY dh dher ` 20@fdxzk gSA feJ.k Hkkstu dh U;wure dher dfu/kkZj.k djsaftlesa okafNr foVkfeu dh ek=k iwjk gks ldsAA dietician wishes to mix together two kind of foods X and Y in such a way that the mixture containat least 10 units of vitamine A, 12 unite of vitamin B and 8 units of vitamin C. The vitamin contents

of one kg food is given below :-

Food Vitamin A B CX 1 2 3Y 2 2 1

One kg of food X costs` 16 and one kg of food Y costs` 20. Find the least cost of the mixturewhich will produce the required diet.

math set iii

Documents