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

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Stewart Miller
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1 009 (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 from Group A (each of mark). xzqi&a ds lhkh 0 ç'uksa ds mùkj nsaa ¼izR;sd ds vad gsaa½ Group-B consists of 5 questions with internal choice, all five questions are to be answered (each of 4 marks). xzqi&b es a vkarfjd fodyiks a ds lkfk dqy 5 ç'u gsa lhkh ik p iz'uksa ds mùkj nsaa ¼izR;sd ds 4 vad gsaa½ Group-C consists of 5 questions with internal choice, all five questions are to be answered (Each of 8 marks). xzqi&c es a vkarfjd fodyiks a ds lkfk dqy 5 iz'u gsa] lhkh ik p iz'uksa ds mùkj nsaa ¼izR;sd ds 8 vad gsaa½ All parts of question must be answered at one place in sequence, otherwise they may not be evaluated.,d iz'u ds lhkh va'kksa dk mùkj,d gh txg ¼yxkrkj Øe esa½ gksuk pkfg,a vu;fkk] os ugha tk ps tk ldrs gsaa The figures in right hand margin indicate full marks. fueukafdr iz'uksa ds fn;s x;s pkj fodyiksa esa ls lokzf/kd lfvd fodyi pqudj fy[ksa % (i) The sum of (n ) natural numbers is : n ( n+ ) n n + ( ) n ( n ) n (ii) Conjugate of a complex number 5 + i is.... fej jkf'k 5 + i dk Conjugate... gsa 5 + i 5 i ik'oz ds vad iw.kk±d ds lwpd gsaa 5 + i 5 + i

2 4 (iii) Value of cos5º is.... (vi) log 8 =.... cos5º dk eku... gsa (iv) Tan x + cot x =.... sin (sin π /) sin (cosec π /) sec (cos π /) + (v) sin (cosecπ /) radian = x y (vii) Slope of the straight line = is ljy js[kk x 4 y = dk <ky... gsa 4 4 n n n (viii) C + C + C =.... n C n n n+ n n+ (ix) In a triangle ABC, if a = 8, b = 4 and c = 0, 5º 0º 0º º then the area of triangle is sq. units 6 sq. units 6 sq. units 6 sq. units

3 5 6,d f=hkqt ABC esa ;fn a = 8, b = 4 vksj c = 0, sin α + sin β + sin γ = rc f=hkqt dk {ks=qy... gksxka cos α + cos β + cos γ = 06 sq. units 6 sq. units (xii) bytes = sq. units 6 sq. units Kilobyte (x) Harmonic mean between 5 and 5 is.... (xi) 5 vksj 5 la[;kvksa ds chp gjkred ekè;... gsa If a line makes angles α, β and γ with x-axis, y-axis and z-axis respectively then : Megabyte Kilobytes Megabytes. (B) Write down the answer of the following questions : x8=8 fueufyf[kr iz'uksa ds mùkj fy[ksa % (i) The value of the is.... ;fn,d ljy js[kk x-v{k] y-v{k,oa z-v{k ls Øe'k% α, β,oa γ dks.k cukrh gs rks % sin α + sin β + sin γ = cos α + cos β + cos γ = lkjf.kdk dk eku... gsa =

4 8 (ii) 0 th term in the expansion of is.... n + n 50 (vi) Find the angle between two plane : nks ryksa ds chp dk dks.k fudkysa % n 50 + n ds folrkj dk 0oka in... gsa (iii) Find the radius of the circle : (vii) x + y + 4z + 5 = 0 and x + 5y +z + = 0 Convert (00.0) into octal number system. o`r dh f=t;k fudkysa % x + y + 4x 6y = 0 (iv) Find the direction cosines of a line joining the points (4, 5, 6) and (5, 6, ). fcunq (4, 5, 6),oa fcunq (5, 6, ) dks feykus okyh ljy js[kk dh fnd~dkst;k fudkysaa (viii) (00.0) dks octal number system esa cnys aa Convert (FA5C.ED8) 6 into Binary number system. (FA5C.ED8) 6 dks Binary number system es a cnys aa GROUP-B Answer all five questions : 4x5=0 (v) Find the value of ¼eku fudkysa½ % lhkh ik p iz'uksa ds mùkj nsa %. Find the sum of the series : 4 nh gqbz Js.kh ds n inksa dk ;ksx fudkysa % n

5 9 0 Show that ¼lkfcr djsa fd½ % 4 Arithmetic mean > Geometrical mean > Harmonic mean.. Resolve into Partial fraction : 4 vkaf'kd fhkuu esa ifjofrzr djsa % x x + x + x 5 Or, ( x ) ( x+ )( x+ ) 4. Find the square root of (4 i). 4 (4 i) dk oxzewy fudkysaa If, ω, ω be the cubic roots of unity, then show that : 4 ;fn, ω, ω bdkbz dk?kuewy gs rks lkfcr djsa fd % (+ ω) (+ ω ) (+ ω 4 ) (+ ω 8 )... to n factor = x 5. Find the middle terms in the expansion of x 6. 4 x x ds folrkj es a eè; inksa dks Kkr djs 6 aa How many words can be formed with the letters of the word DAUGHTER? So that : 4 (i) The vowels always come together. (ii) The vowels never come together. DAUGHTER 'kcn ds v{kjksa ls dqy fdrus 'kcnksa dks cuk;k tk ldrk gs fd % (i) (ii) lhkh Loj lkfk jgsa] lhkh Loj lkfk ugha jgsaa 6. For a triangle ABC, show that : 4,d f=hkqt ABC ds fy, fn[kk, fd % C ( s a)( s b) sin = ab Prove that ¼lkfcr djsa½ : π sin + sin + sin = 5 65

6 GROUP-C Answer all five questions : 8x5=40 lhkh ik p iz'uksa ds mùkj nsa %. If (4, 6), (8, 9), (6, ) are the vertices of a triangle, find the co-ordinates of the circumcentre Find the equation and area of the circle circumscribing the triangle formed by the lines x + y = 6, x + y = 4 and x + y = 5. 8,d o`r dk lehdj.k,oa {ks=qy Kkr djsa tks rhu ljy js[kkvksas x + y = 6, x + y = 4,oa x + y = 5 ls cus f=hkqt ds 'kh"kz of the triangle. 4 ;fn (4, 6), (8, 9), (6, ) fdlh f=hkqt ds 'kh"kz fcunq gsa rks f=hkqt ds ifjo`r ds dsuæ dk fu;ked fudkysaa Prove that (, ), ( 4, 8) and (6, 6) are collinear. Find the equation of the line on which they lie. 4 fcunqvksa ls xqtjrk gsa 8. Solve the following set of equations by matrix method : 8 fueufyf[kr lehdj.kksa dks esfvªdl fof/k ls gy djsa % x + y + z = ; x y z + = 0; x + y + z = 4 Express 5 6 as a sum of a symmetric 4 lkfcr djsa fd fcunq (, ), ( 4, 8) vksj (6, 6),d jsf[kd gsa ml ljy js[kk dk lehdj.k Kkr djsa ftl ij rhuksa fcunq flfkr gsaa and a skew symmetric matrices. 4 esfvªdl 5 6 dks,d flesfvªd esfvªdl,oa,d 4 LD;w flesfvªdl esfvªdl ds ;ksx ds :i esa n'kkz, A

7 4 If ¼;fn½ A = , B = 9 6, Find the equation of the plane passing through C = 5 0 point (,, ) and parallel to the plane 4x + 5y z = 0 4 Find ¼Kkr djsa½ : 4 (5A + B) (B 4C) 9. Find the symmetrical form of the equation of the line : 4 nh x;h ljy js[kk ds lehdj.k dks flesfvªdy :i esa Kkr djs a % x + y + z + = 0, 4x + y z + = 0, Find the equation of the plane passing through,d ry dk lehdj.k fudkysa tks fcunq (,, ) ls xqtjrk gs,oa ry 4x + 5y z = 0 ds lekukurj gsa x Find the angle between straight line = y 5 z+ = and a straight line passing through points (5, 6, ) and (, 9, 4). 4 ljy js[kk x = y 5 z+ =,oa fcunq (5, 6, ) point (,, 0) and perpendicular to the planes x y z = 5 and x + y z = 5. 4,d ry dk lehdj.k Kkr djs a tks,d fcunq (,, 0) ls xqtjrk gs,oa ry x y z = 5,oa ry x + y z= 5 ij yec gsa rfkk fcunq (, 9, 4) ls xqtjus okyh ljy js[kk ds chp dk dks.k Kkr djs aa 0. Solve the equation ¼lehdj.k dks gy djs a½ % 4 cos θ + sinθ = 0

8 5 6 In a triangle ABC, prove that : 4 fdlh f=hkqt ABC esa] lkfcr djsa fd % cos A = b + c a bc Convert octal number into a decimal number. 4 vkwdvy la[;k dks n'keyo la[;k esa cnysaa If ;fn C C =, find the value of n. 4 n n C C = ] rks n dk eku fudkysaa n n Find the term independent of x in the expansion of x x. 4 x x ds folrkj esa x ls Lora= in Kkr djsaa Convert decimal number into a hexadecimal number n'keyo la[;k dks gsdlkn'keyo la[;k esa cnys aa Explain Boolean Algebra. 4 Boolean Algebra dh O;k[;k djs aa. Find the value of (5F.A) 6 8. using ' complement method. 4 's leiwjd fof/k dk iz;ksx dj (5F.A) 6 8. dk eku fudkysaa

Regional Mathematical Olympiad -2011

Regional Mathematical Olympiad -2011 RMO-011 EXMINTION REER POINT Regional Mathematical Olmpiad -011 Time : 3 Hours December 04, 011 Instructions : alculators (In an form) and protractors are not allowed. fdlh Hkh rjg ds dsydqsvj ;k dks.kekid