Lke; % 1?kaVk $ 10 feœ ¼vfrfjä½ iw.kkzad % 50 I. (b)laøed (transitive)

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1 SET ¼çk#i i=½&1 SECTION ¼[k.M½ &1 OBJECTIVE QUESTIONS ¼oLrqfu"B ç'u ½ Time : [1 Hrs + 10 Min (Etra)] Full Marks : 50 Lke; % 1?kaVk $ 10 feœ ¼vfrfjä½ iw.kkzad % 50 I ç'u &1 ls 50 rd fueu esa ls fn,, pkj fodyiksa esa ls,d gh mùkj lgh gs] çr;sd ç'u ds lgh mùkj dks mùkj rkfydk esa fpfugr djsaa 50X1= 50 From Question no. 1 to 50 there is one correct answer for each question you have to msark that correct option from the given options. 50X1= lacau/k R = {(1,),(4,),(,4),(,),(,1)} leqpp; A={1,,,4} ij dslk lacau/k gs \ [What type of a relation is R = [{(1,),(4,),(,4),(,),(,1)} on the set A={1,,,4}] (a) LorqY; (refleive) (c)lefer (Symmetric) (b)laøed (transitive) (d) buesa ls dksbz Hkh ugha (None of these) Sin cos-1 1 =? (a) 4π (b) π (c) π (d) bues a ls dksbz Hkh ugha bc 1 1/a ca 1 1/b ab 1 1/c =? (a) 1 abc (b) 0 (c) abc (d) buesa ls dksbz Hkh ugha 4 ;fn (If) A = rks ( then) A =? (a) A (b) A (c) 7A (d) I 1

2 5- ;fn (If) + y 5 + z y = ( then which will be value of y from the following?) Rkks y dk eku fueu esa ls dkssu lk gksk \ (a) (b) 4 (c) 5 (d) buesa ls dksbz Hkh ughaa (None of these) 6 ;fn (if) y = sin(log) rks (then) dy (a) cos (log ) (b) d cos (log ) =? (c) 7- ;fn (If) y = rks (then) d. y d =? sin (log ) (a) + 4 (b) (c) 6 (d) (d) buesa ls dksbz Hkh ughaa 8. f=t;k r ds lkis{k o`r ds {ks=qy dh ifjorzu dh nj] tc r = 14 ls aœehœ gs vksj ¼π= ) 7 ¼The rate of change of area of a circle with respect to its radius r, when r = 14 cm and ¼π= ) 7 is) (a) 44 cm /cm (b) 88 cm /cm (c) 8 cm /cm (d) cm /cm 9. oø y = ds founq (0,0) ij Li'kZ js[kk }kjk -v{k ds /kukred fn'kk ds lkfk cuk;k ;k dks.k gs : [The angle that the tangent to the curve y = at (0,0) makes with the positive direction of the -ais is] (a) 45 (b ) 90 (c) 0 (d) Qyu f () = 4-4, {-1,1} ds fy, jksys ds lk/; esa C dk eku gs : [The value of c in Role s theorem for the function f() = 4-4, {-1,1} is] (a) 4 (b) 1 4 (c) 0 (d) buesa ls dksbz Hkh ughaa (None of these) π 11. sin 7 d π =? (a) 1 (b) -1 (c) 0 (d) buesa ls dksbz Hkh ughaa (None of these) 1. 1 d =.? (a) log e 1/ (b) log e + c (c) log a +c (d) bues a ls dksbz Hkh ughaa (None of these) 1 tan d =.? (a) log cos + c (b ) log sec + c (c) ) log cot + c (d) bues a ls dksbz Hkh ughaa (None of these)

3 14- ijoy; y = 4a vksj mldk latus rectum dk js[kk [k.m ls?ksjk {ks= dk {ks=qy gs [The area bounded by the parabola y = 4a and the line segment of the latus rectum is] (a) 4a. unit. (b ) 8a. unit. 16 a. (c) unit. (d) a. unit. 15. ekuk fd f() = c 1 dt rks lehdj.k f 1 = 0 ds oklrfod ewy gs] [if f() = t 1 dt then real roots of the equation f 1 = 0 are] (a) + 1 (b ) + 1 (c) + 1 (d ) vodyu lehdj.k d. y d.4 + dy d y = cos dk dksfv gsa [The order of the differential equation ( d. y d ).4 + ( dy d. ). + 9 y = cos is] (a) 4 (b ) (c) (d ) buesa ls dksbz ughaa 17. î (ĵ k) dk eku fueufyf[kr esa ls dksu lk gs? [which of the following is the value of Î Ĵ k? ] (a) k (b ) Ô (c) î (d ) ĵ 18 ljy js[kk = y = - z vksj 6 = -y = - 4 z ds chp dk dks.k gsa [The angle between the straight = y = - z and 6 = -y = - 4 z is ] (a) π 6 (b ) π (c) 0 (d ) π nks ikls ds Qsad es a tksm+k ikus dh çf;drk gs [The chance of getting a doublet with dice is] (a) 1 6 (b ) 5 (c) (d ) ;fn a, b vksj c,d nwljs ds yecor bdkbz lfn'k gks rks (a- b) + (b -c ) + (c -a ) =? (a) 6 (b ) (c) 4 (d ) 9 [If a, b and c are unit vectors perpendicular to each other then (a- b) + (b c ) + (c a ) =? a 6 (b ) (c) 4 (d ) 9

4 1. cos 15 sin75 sin 15 cos 75 =? a 1 (b ) π (c) 0 (d) bues a ls dksbz Hkh ughaa (None of these) - ;fn (If) P(A) =, P(B) = 1 vksj A rfkk B Lora=?kVuk, gs (A and B are mutually 5 independent events) rks (then) P(A B)=? a 5 (b ) 5 (c) 1 5 (d) bues a ls dksbz Hkh ughaa (None of these). ;fn lafø;k ӿ ifjhkkf"kr gs fd (a ӿ b) = a + b rks ( ӿ 4) = 5 gsa [If the operation ӿ is defined as (a ӿ b) = a + b ten ( ӿ 4) = 5 is a 650 (b ) 15 (c) 65 (d) 15 4 ;fn f : R R tgk f () =, rks dslk Qyu gksk? If : R R R such that f () =, then what type of a function is f? ¼ a ½,dSd varjs.kh(one One into) (b ),dsd vkpnknd (One One onto) ¼ c ½ vudsd var{ksih (many One into) (d) vusdsd var{ksih (many One onto) a 5. oklrfod la[;kvksa ds leqpp; es laca/k ^^cm+k gs** fueufyf[kr esa dslk lacu/k gs \ ¼ a ½ dsoy lefer (b ) dsoy LorqY; ¼ c ½ vudsd laøed (d) rqy;rk laca/k [ What type of a relation is greater than in the Set of real numbers?] ¼ a ½ Only Symmetric (b ) Only refleive ¼ c ½ Only transitive (d) Equivalence relation 6. ;fn (If) Sin 1 = π 5, ε ( 1,1) rks (then) cos 1 =? ¼ a ½ 5π Sin (cot 1 ) =? (b ) 7π 10 ¼ c ½ π 10 (d) 9π 10 ¼ a ½ (b ) 1 + ¼ c ½(1 + ) 1/ ¼ d ½(1 + ) / 8. ;fn λ ε R vksj Δ = a b c d [If λ ε R, and Δ = a ¼ a ½ λa λc b d (b ) b c d λa b c d rks λ Δ cjkcj gksk \ then λ Δ is equal to? ¼ c ½ λa λc 4 λb λd ¼ d ½buesa ls dksbz ugha (None of these)

5 9. ;fn 7 vkssj ] lehdj.k [If 7 and are two roots of the equation ¼ a ½ 1 0. ;fn A = [If A = ¼ a ½ ;fn y = log rks dy =0 ds nks ewy gs rks rhljk ewy gksk \ = 0 then the third root is? ] (b ) 9 ¼ c ½ 14 (d ) buesa ls dksbz ugha (None of these) rks fueukafdr esa dksu A ds cjkcj gs? ten wic one of te following is equal to A? d (b ) ¼ c ½ dk eku fueukafdr esa ls dksu gksk \ (d ) [If y = log ten wic one of te following will be value of dy? ] d ¼ a ½ 1 (b ) ¼ c ½ 1 (d ). ;fn y = sin rks d. y d dk eku fueukafdr esa ls dksu gksk \ [If y = sin ten wic one of te following will be value of d y? ] d ¼ a ½ sin4 (b ) 4sin ¼ c ½ 4sin (d ) 4cos. ds ifjorzu dh nj rfkk log. ds ifjorzu dh nj dk vuqikr fueukafdr es a ls dksu gksk \ [which of the following is the ratio of rate of change of and the rate of change of log. ] ¼ a ½ (b ) ¼ c ½ (d ) 4. f = Sin+Cos dk eku egùke ds fueukafdr esa ls fdl eku ds fy, gksk \ [The value of f = Sin +Cos will be maimum for which of the following value of?] ¼ a ½ π (b ) π ¼ c ½ π (d ) π vody lehdj.k {1+. ( dy d ). } / = m d y dk?kkr fueukafdr esa ls dksu gksk \ d [Which one of the following is the degree of the differential equation{1+. ( dy d ). } / = m d y d?] ¼ a ½ (b ) ¼ c ½ 1 (d ) 0. 5

6 1 6 e d dk eku fueu esa ls dksu gs \ 0 1 [which of the following is the value of e d (a) e 1 (b) e (c) 1 (d) buesa ls dksbz ugha (None of these) a 7 ;fn (if) f = f rks (then) f() d a 6 0? dk eku fueu esa ls dksu gksk \ a [If f = f then which of the following is the value of f() d a (a) f(a) (b) f(a) (c) 0 (d) buesa ls dksbz ugha (None of these) d =? (a) log - 1 +c (b) 1 - log +k (c) log - 1 +k (d) buesa ls dksbz ugha (None of these) 9. fueufyf[kr esa dksu le?kkrh vodyu lehdj.k ugha gs \ [which of the following is not Homogeneous differential equation?] (a) y d + + y dy = 0 (b)( y)dy + y d = 0 (c) dy d = y y dy (d) = Sin y d 40 a = -5ĵ + k dk b = 4î - 4 ĵ +7k ij ç{ksi dkeku fueu esa ls dksu gksk \ [which of the following will be the value of the projection of a = -5ĵ + k on b = 4î - 4 ĵ +7k]? (a) 7 (b) 47 (c) 7 (d) buesa ls dksbz ugha (None of these) 41. ewy founw ls founw (-,4,5) dh nwjh fueu es a ls dksu gs \ [which of the following is the distance of the point (-,4,5) from the origin?] (a) 5 (b) 5 (c) 10 (d) buesa ls dksbz ugha (None of these) 4. fdlh ljy js[kk ds fnd~ vuqikr 1,, 5 gs rks mldh fnd~dkst;k, fueu esa ls dksu gksk \ [Which of the following will be direction cosines of the line which directions ratio are 1,,5?] (a) 1 5, 5, 5 5 (b) 1 9, 1, 5 9 (c) 5 5, 5, 1 5 (d) buesa ls dksbz ugha (None of these) 4. vlehdj.k a + by c, a + by c, a + by c vksj a a + by c esa ls lar lehdj.k fueu esa ls dksu gs \ [Which of the following is the corresponding equation of the each of in equations a + by c, a + by c, a + by c and a a + by c] (a) a + by = c (b) a + by = 0 (c) b + ay = c (d) buesa ls dksbz ugha (None of these) 44. q.kksùkj O;kojks/k fueu esa ls dksu gs \ [Which of the following is the non-negative constraints?] (a) 0, y 0, (b) 0, y 0, (c) 0, y 0 (d) buesa ls dksbz ugha (None of these)?

7 45. _.kksùkj O;kojks/k 0, y 0 ds dkj.k lqlar {ks= fueu ess a ls fdl ikn esa gksk \ [The feasible region due to non-negative constraints 0, y 0 will lie in which of the following quadrants?] (a) prqfkz (Fourth) (b) çfke (First) (c) r`rh;z (Third) (d) f}rh; (Second) 46. jsf[kd lehdj.k a + by = c, dk vkys[k tks,d ljy js[kk gs] fueu esa ls fdl founq ij v{k ls qtjsh \ a [The graph of linear a + by = c, which is a straight line, will pass through which of the following point on the -ais?] (a) ( c b, 0) (b) (c a, 0) (c) (0, c a ) (d) (0, c b ) 47. ;fn A vksj B dksbz nks?kvuk, gks rkfd P(A) = 0., P(B) = 0.6 rks P(A B)+P A B = \ a [If A and B are any two events such that P(A) = 0., P(B) = 0.6 then P(A B)+P A B = \] (a) 0.4 (b) 0.8 (c) 0.1 (d) ,d iklk dks N% ckj Qsadk tkrk gsa ;fn,d ^^le la[;k Qsduk lqyrk** gks rks N% ckj lqyrk dh çkf;drk gs\ [ A dice is thrown 6 times If getting an even number is a success then the probability of getting 6 success is \] (a) 1 64 (b) (c) 7 64 (d) buesa ls dksbz ugha (None of these) 49. ;fn A vksj B nks ijlij viothz?kvuk, gks rfkk P(A) = 1 rfkk 6 P(B) = 1 rks P(A B) dk eku fueu esa ls D;k gksk \ [ If A and B are two mutually eclusive events P(A) = 1 6 and P(B) = 1 following is the value of P(A B) \] (a) (b) 0 (c) 1 1 then which of the (d) buesa ls dksbz ugha (None of these) 50. ;fn A vksj B nks ijlij viothz?kvuk, gks rfkk P(A) = 1 rfkk 5 P(B) = rks P(A B) dk 5 eku fueu esa ls dksu gksk \ [ If A and B are two mutually eclusive events P(A) = 1 5 and P(B) = 5 following is the value of P(A B) \] then which of the (a) 5 (b) 5 (c) 0 (d) buesa ls dksbz ugha (None of these) 7

8 SECTION ¼[k.M½ &II NON-OBJECTIVE QUESTIONS ¼Sj&oLrqfu"B ç'u ½ Time : [ Hrs + 5 Min (Etra)] Full Marks : 50 Lke; %?kavk $ 5 feœ ¼vfrfjä½ iw.kkzad % 50 I ç'u la[;k &1 ls rd y?kqmùkjh; dksfv ds gsa çr;sd ç'u ds fy, vad fu/kkzfjr gsa fdugha 15 ç'uks a dks gy djuk gsa X15=0 Question no. 1 to carry marks each. These questions are of short answer type. You have to answer any 15 question out of. X15= 0 1. ds fy, gy djsa (Solve for ) tan 1 + tan 1 = π 4 a b a + c. ;fn (If) a b c + d = [Find the value of a, b, c & d) rks (then) a, b, c vksj d dk eku çkir djsaa. ;fn (If) y = e sin + cos s rks n'kkzb;sa fd (then show that) d. y d dy d + y = 0 4. ;fn(if) f()= sin tc (when) 0,f 0 = 1, rks D;k f (), =0 ij larr gs+] D;ks a\ [Is f () continuous at = 0, why?] 5. fl) djsa fd (Prove that) tan tan = π 4 6. ;fn fa B vksj g : B C nks One-One on to Qyu gks rks (gof). 1 = f 1 o g rhu Nk=ksa }kjk,d ç'u dks gy djus dh çkf;drk, 1, 1 ] 1 gs rks ç'u dks gy fd;s tkus 4 dh çkf;drk fudkysaa (The Probabilities of solving a problem by three students are 1, 1 ] 1, find 4 the probability that the problem will be solved) 8. mu ryksa ds lehdj.k çkir dhft, tks ry 6 - y + z + 9 = 0 ds lekukurj gs rfkk ewy founq ls nwjh 5 ij flfkr gs [Find the equation of the planes parallel to the plane 6 - y + z + 9 = 0 and at a distance 5 from the origin] 9. ljy js[kkvksa = y 8 = z 1 1 vksj + [Find the shortest distance between the lines 8 = y+7 = z 6 4 = y 8 = z 1 1 ds chp ds y/kùke nwjh fudkysaa and + = y+7 = z 6 4

9 10. lekdyu djs (Integrate) Sec d 11. eku fudkysa (Evauate) 1. vodyu lehdj.k dy d 1 log d = - 4 y dk gy fudkys vj fn;k gks y =1 tc =0 [Find the particular solution of the different equation dy d = -4 y given that y=1when = dk eku Kkr djsa tcfd (find the value of when) 4 =0 14. vksj y dk eku Kkr djs a tcfd (find the value of and y when) y = 4 rfkk and y 1 = ;fn (If) A = dk gy Kkr dhft, \ 16. (-,) fcanw ij y + 4 =0 ds fy, dy (Find the value of dy d vksj (and) B = d dk eku fudkysa \ for y + 4 =0 at (-,)] 17. ;fn (If) y =e (sin + cos) rks n'kkzb, fd (then show that d y d 18. vody lehdj.k yd - dy = yd dks gy djsa \ Solve the differential equation yd - dy = yd? 9 rks (then find the value of) A-B dy d + y = dk eku Kkr djsa tcfd fueukafdr lfn'k ijlij yec gks (find the value of when the following vectors are perpendicular to one another) î ĵ + 5k,- î + ĵ + k 0. lfn'k î + ĵ + k vksj î ĵ + k ls fu/kkzfjr lekarj prqhkqzt dk {ks=qy Kkr dhft,a (Find the area of of te parllelograme determined by te vectors î + ĵ + k and î ĵ + k) 1.,d f[kykm+h ds gkfk esa 7 iùks gs] bues a 5 iùks yky gs vksj bu ikapksa esa nks ckn'kkg gsa,d iùkk ;n`pn;k [khapk tkrk gsa blds ckn'kkg gksus dh çkf;drk fudkysa tcfd ;g ekywe gs fd og iùkk yky gsa ( A player has 7 cards in hand of which 5 are red and of these five are kings A card is drawn at random. Find the probability that is a king, it is known that it is red).,d iklk ds Qsadus esa ;fn fo"ke la[;k vkrh gks] rks mlds 1 ls vf/kd gksus dh D;k çkf;drk gs \( What is the probability of the occurrence of a number greater than 1 if it is known that only odd number can occur?)

10 II ç'u la[;k&1 ls 4 rd nh?kz mùkjh; dksfv ds gsa çr;sd ç'u ds fy, 5 vad fu/kkzfjr gsa çr;sd ç'u ds lkfk ^vfkok* dk fodyi fn;k ;k gsaa vkidks ç u ;k ^vfkok* esa ls fdugha,d dk gh mùkj nsuk gsa 4X5=0 Question no. 1 to 4 carry 5 marks each. These questions are of long answer type Each question has an alternative as or. You have to answer each question or its alternative (or)] 4X5=0 1. fl) djsa fd,d o`r ds vurzr o`gùkj {ks=qy okyk vk;r,d oz gsa (Prove that the rectangle of greatest area inscribed in a circle is a square) Or vfkok fl) djsa fd sin θ(1 + cosθ) dk egùke eku θ = π ij gkska [Prove that sin θ 1 + cosθ has maimum value of θ = π ]. nh?ko`r + y 9 4 = 1 dk {ks=qy fudkysa? [Find the area enclosed by the ellipse Or vfkok π fl) djsa fd (Prove that ) log sin d y 4 = 1] π = log cos d = 0. dk eku Kkr djksa tcfd fueukafdr lfn'k ijlij yec gksa π log [Find the value of when the following vectors are perpendicular to one and another] i - j +5k or i +j + k Or vfkok fl) dhft, dh js[kk, = y+1 = z 4 5 and ( + y + z - 9 = 0 = - y + z - 11 ) Lkeryh; gsa ml ry dk lehdj.k Hkh Kkr dhft;s ftles a ;s nksuks js[kk, flfkr gsa (Prove that the lines = y+1 = z and ( +y +z - 9=0= - y + z -11) 4 5 are coplaner, find also te equation of te plane in wic tey lie. ] 10

11 4.,d QuhZpj O;kikjh ek= nks olrq, est vksj dqlhz csprk gsa mlds ikl fuos'k ds fy, 5000 #Œ gs,oa dsoy 60 olrqvksa dks j[kus dk LFkku gsa,d est ij 50 #Œ vksj,d dqlhz ij 50 #Œ dh ykr vkrh gsa og,d est dks 50 #Œ,oa dqlhz dks 15 #Œ ykhk ds lkfk csp ldrk gsa ;g ekurs gq, fd og ftruh olrq, [kjhnrk gs mugsa csp ldrk gs mls viuk /ku fdl çdkj fuosf'kr djuk pkfg, fd mls vf/kdre ykhk gksa [ A furniture dealer deals in only two items tables and chairs. He has Rs. 5000/- to invest and a space to to store at most 60 pieces. A table costs him Rs. 50 and a chair Rs. 50. He can sell a table at a profit of Rs. 50 and a chair at a profit of Rs 15. Assuming that he can sell all the items that he buys how should he invest his money in order that he may maimise his profit) Or vfkok zkq }kjk fueu jsf[kd çkszkfed lel;k dks gy djsaa (Solve the following linear Programming problem graphically) U;wurehdj.k djsa (Minimise) z = y Subject to the constraints: + y 10 +4y 4 0, y 0 11

12 olrqfu"b ç'uksa dk mùkj (Answer of Objective Questions) 1 d 6 c c 7 c b 8 a 4 b 9 b 5 c 0 c 6 b 1 a 7 c b 8 b b 9 c 4 d 10 c 5 a 11 c 6 a 1 b 7 c 1 b 8 a 14 b 9 b 15 b 40 a 16 c 41 b 17 b 4 a 18 b 4 a 19 a 44 c 0 a 45 b 1 c 46 b b 47 b a 48 a 4 b 49 b 5 c 50 a 1

2009 (Odd) xzqi&a ds lhkh 20 ç'uksa ds mùkj nsaa ¼izR;sd ds 1 vad gsaa½

2009 (Odd) xzqi&a ds lhkh 20 ç'uksa ds mùkj nsaa ¼izR;sd ds 1 vad gsaa½ 009 (Odd) Time : Hrs. Full Marks : 80 Pass Marks : 6 SemI-G Engg. Math.-I GROUP-A.(A) Write down the most correct answer for the following question from four given alternatives : = Answer all 0 questions