INSTITUTE OF AERONAUTICAL ENGINEERING Dundigal, Hyderabad FRESHMAN ENGINEERING

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1 INSTITUTE OF AERONAUTICAL ENGINEERING Dundigl, Hyderbd - FRESHMAN ENGINEERING QUESTION BANK Nme : ENGINEERING MATHEMATICS-I Code : A Clss : I - B. Tech Brnch : Common for ll Brnches Yer : - Fculty : Dr.M.Anit(HOD), Mr.Ch.Kumr Swmy, Mr.J.Suresh Goud, Mrs.L.Indir, Mrs.V.Subblmi, Mrs P.Srilth, Mr Ch.Somshekr. OBJECTIVES To meet the chllenge of ensuring ecellence in engineering eduction, the issue of qulity needs to be ddressed, debted nd tken forwrd in systemtic mnner. Accredittion is the principl mens of qulity ssurnce in higher eduction. The mjor emphsis of ccredittion process is to mesure the outcomes of the progrm tht is being ccredited. In line with this, Fculty of Institute of Aeronuticl Engineering, Hyderbd hs tken led in incorporting philosophy of outcome bsed eduction in the process of problem solving nd creer development. So, ll students of the institute should understnd the depth nd pproch of course to be tught through this question bnk, which will enhnce lerner s lerning process.. Group - A (Short Answer Questions) S. No Question UNIT-I THEORY OF MATRICES i Find the eigen vlues of the mtri i Apply If A is Hermitin mtri Prove tht it is skew- Hermitin mtri Anlyze Stte Cyley- Hmilton Theorem Anlyze i i Prove tht is unitry mtri. i i Anlyze Find the vlue of k such tht rnk of k is. Apply i Find the eigen vlues of the mtri i Apply Apply P g e

2 i Find A if A = i i Define modl mtri. Remember Find the Skew- symmetric prt of the mtri Apply If,, re the eigen vlues of A then find the eigen vlues of dj A Apply UNIT-II DIFFERENTIAL CALCULUS METHODS Define Rolle s Men vlue theorem. Remember Verify Lgrnge s Men Vlue theorem for f() = log in [, e] Verify Lgrnge s Men Vlue theorem for function f() = cos in [, π/]. Verify Cuchy s Men Vlue theorem for f() =, g() = in[, ]. Find first nd second order prtil derivtives of +hy+by nd f f Apply verify. y y When two functions u,v of independent vribles,y re functionl dependent Remember If = u(-v), y = uv prove tht JJ = y y Find the mimum nd minimum vlues of Apply ( u, v) If u e siny, v e cosy then find. (, y) Apply Verify Rolle s Men vlue theorem for f()=(+) (-) in [-,] UNIT-III IMPROPER INTEGRALS, MULTIPLE INTEGRALS AND ITS APPLICATIONS Prove tht ( m, n) ( n, m). Anlyze m n mn Prove tht ( ( ) d ( b) ( m, n), m, n Anlyze Compute ( /), (/), ( /). Apply Write the vlue of (). Remember ydyd sin rdrd. y ( y ) ddy. e logy e log zddydz. P g e

3 find the vlue of ddydz. Apply Write the sphericl polr coordintes Remember UNIT-V DIFFERENTIAL EQUATIONS AND APPLICATIONS (+)dy/d y=e (+) Apply Write the working rule to find orthogonl trjectory in Crtesin form. Remember Form the D.E.by eliminte c in y=+c - Apply (+y+) dy/d = Apply Prove tht the system of prbols y = (+) is self orthogonl. Anlyze Find the O.T. of the fmily of curves Apply Stte Newtons lw of cooling Remember A bcteril culture, growing eponentilly, increses from to grms in the period from.m to.m.. How mny grms will be present t noon. Anlyze Apply - + y = Define S.H.M. nd give its D.E Remember UNIT-V LAPLACE TRANSFORMS AND ITS APPLICATIONS Find Lplce trnsform of Apply Stte nd prove first shifting theorem. Remember Define chnge of scle property of Remember Find Apply If is periodic function with period T then find Apply Find Apply Find Apply Find ) Apply Define inverse L.T. of f(s) Remember Use L.T to solve D.E. when t= Apply. Group - B (Long Answer Questions) S. No Question UNIT-I THEORY OF MATRICES Show tht only rel number for which the system +y+z=, +y+z= y, +y+z= z hs non-zero solution is nd solve them when = Anlyze Epress the mtri s the sum of Hermitin Anlyze mtri nd skew- Hermitin mtri. P g e

4 Given tht show tht is unitry mtri. Anlyze Find Inverse by elementry row opertions of Apply Find whether the following equtions re consistent if so solve them the equtions using prtil pivoting Guss Elimintion method of Apply Apply Digonlize the mtri A= Prove The Eigen Vlues of Rel symmetric mtri re Rel. Reduce the Qudrtic form form. Reduce the qudrtic form cnonicl form by orthogonl reduction. nd find A Apply -y-yz to the Cnonicl to the Anlyze Anlyze Anlyze Find rnk by reducing to Norml form of mtri Apply the following System of equtions +y+z+w=, +y+z+w=, +y+w= Apply Find nd such tht y z, y z, y z hs (i) no solution (ii) Unique solution (iii) Mny solutions Apply Find the eigen vlues nd eigen vectors of Apply Stte Cyley-Hmilton theorem nd verify the mtri A = Remember P g e

5 Digonlize the mtri A Apply Digonlize the mtri trnsformtion. A by n orthogonl Apply Reduce to sum of squres, the qudrtic form nd find the rnk, inde nd signture. Anlyze Prove tht the eigen vlues of Skew-Hermitin mtri re purely imginry or zeros. Anlyze If A is ny squre mtri then prove tht i) A + A ii) AA, A A re Hermitin iii) A - A re skew Hermitin Anlyze UNIT-II DIFFERENTIAL CALCULUS METHODS if f( ) =, g ( ) = prove tht c of the CMVT is the geometric men of nd b for ny >,b> y z find the minimum vlue of given y z Apply prove using men vlue theorem sin u sin v u v yz (,, ), z, y u v w find u v w if y z (, y, z) using men vlue theorem prove tht the function Apply f ( ) is increses when decresing when Clculte pproimtely by using LMVT Apply P g e find the region in which f( ) =- - is incresing nd the region in which it is decresing using men vlue theorem find three positive numbers whose sum is nd whose product is mimum Apply Apply

6 Find the volume of the gretest rectngulr prllelepiped tht cn be inscribed in the ellipsoid y z b c Apply Using men vlue theorem for b prove tht b b log b log f ( ) ( y ) y Apply cos using LMVT find the mim nd minim of Using men vlue theorem prove tht tn> in Prove tht If Also Show tht If P g e = u y yz z, v y z, w y z re Prove tht functionlly dependent nd find the reltion between them If throughout n intervl [,b] prove using men vlue theorem f() is constnt in tht intervl. Divide into three points such tht the continued product of the first, squre of the second nd cube of the third is mimum using rolle s theorem show tht zero between nd If g( ) hs y z u, y z uv, z uvw show tht u( v), y uv prove tht (, y, z) ( u, v, w) uv JJ UNIT-III IMPROPER INTEGRATION, MULTIPLE INTEGRATION AND APPLICATIONS By trnsforming into polr coordintes the nnulr region between the circles y b with b y ddy y y nd over ( y z) dzdyd where R is the region bounded by the R plnes,, y, y, z, z z yz Find the volume of the tetrhedron bounded by the plne y z nd the coordinte plnes by triple integrtion b c yzddydz Apply

7 (cos ) r cosdrd Find the vlue of ellipse y b yddy tken over the positive qudrnt of the Apply r sin rdzdrd Anlyze ProveTht where p>, q> Show tht ( / ) Anlyze the double integrl y ( ) y dyd Show tht Anlyze Show tht Anlyze r drd ( r ) By chnging the order of integrtion, evlute yddy By chnging the order of integrtion, evlute y dyd Epress sin p cos q d in terms of bet function. P g e

8 ddy y, y,, over the region bounded by hyperbol log log y yz e ddydz by chnging the order of integrtion y dyd y UNIT-IV DIFFERENTIAL EQUATIONS AND APPLICATIONS A bcteril culture, growing eponentilly, in increses from to grms in the period from m to m. How mny grms will be Anlyze present t noon? ( dy yd) d ydy y Apply ydy ( y ) d Apply Find the orthogonl trjectories of the fmily of curves y Apply dy ( ) y e ( ) the D.E d Apply Obtin the orthogonl trjectories of the fmily of curves r( cos ) Apply A prticle is eecuting S.H.M, with mplitude meters time sec.find the time required by the Prticle in pssing between points which re t distnces & meters from the centre of force nd re on the sme side of it. Anlyze ( D D ) y cos( ) e Apply D ( D ) y sin k Apply By using method of vrition of prmeters solve ( D ) y cos ec Apply dy y y d Apply If the ir is mintined t c nd the temperture of the body cools from c to c in minutes, find the temperture of the body fter minutes nd when the temperture will be c Anlyze P g e

9 ( D ) y sin sin e Apply y y ( e ) d e ( ) dy y Apply A copper bll is heted to temperture of c nd time t=, then it is plced in wter which is mintined t c temperture of the bll is reduced to c temperture of the bll is c. If t t=minutes,the find the time t which the Anlyze ( D D D ) y e e Apply dy d Apply sec y tn y cos A body weighing kgs is hung from spring pull of kgs will stretch the spring to cms. The body is pulled down to cms below the sttic equilibrium position nd then relesed. Find the displcement of the body from its equilibrium position t time t seconds, the mimum Velocity nd the period of oscilltion Anlyze ( D D ) y cos the D.E Apply The number N of bcteri in culture grew t rte proportionl to N. The vlue of N ws initilly nd incresed to in one hour. Wht Anlyze ws the vlue of N fter hours? Using L.T P g e UNIT-V LAPLACE TRANSFORMS AND ITS APPLICATIONS t t e e dt t s Find L ( s )( s )( s ) Apply t, if<t<b Find Lf(t) where f(t)= of period b b-t, if b<t<b Apply cos tsin t Find L t Apply cos(t- ), if t> Find Lg() t where g(t)=, if t< Apply Find the.l.t of S S S S S Apply

10 '' ' t ' the D.E using L.T, y y y t e, y( o) y () Apply d y dy ' the D.E using L.T, ysin t, y() y() dt dt Apply s Find the.l.t of log( ) s Apply Use Lplce trnsforms, solve ' ( D ) t cos t, given() () Apply S S S S S s e Find the Inverse L.T of Apply Find L Apply s s ' Using L.T solve ( D D ) y sin, y() y () Apply Find the Lplce trnsform of t te sint Apply Find the L.T of periodic function f() t with period T where ET T -E, if t f(t)= T Apply ET T E-, if t T T - S Find L S S S Apply d d t the D.E using L.T e with dt dt d (), t t= dt Apply L t sint cost Apply Find Using L.T solve ( D D D ) y e sin, y, Dy, D y t Find Le t sinht Apply using chnge of scle property Apply. Group - III (Anlyticl Questions) S. No Questions If A is n rowed mtri ij integer then find the vlue of det A A where UNIT-I THEORY OF MATRICES i, ij. denotes gretest j Progrm Apply P g e

11 s Progrm If =digonl( - )nd b=digonl ( -)then find +b Apply i If = then find A n i Apply Find the vlue of k for which mtri A= k k Is invertible. Apply Wht is rnk of mtri Remember If,, re eigen vlue of A then find eigen vlue of Adj A. Apply If vector & re two orthogonl vectors of mtri then find third If A is nn mtri, rnk is k nd norml form is of null mtri below side of If A= I k then epress I k then find order Apply Apply A in terms of A. Apply Find the rnk of qudrtic form whose eigen vlues re,, Apply UNIT-II DIFFERENTIAL CALCULUS METHODS When the Jcobin Trnsformtion is used? Remember Find the functionl reltionship between u= + y +z, v= y +y z + z, w = +y +z Apply Wht re criticl points? Remember y Write the reltionship betweenu, v tn tn y. y Remember Find the sttionry vlues of y (--y). Apply Wht re sddle points? Remember Wht is condition for f(, y)to hve mimum nd minimum vlues t (, b)? Remember Wht is the demerit of Lgrnge s method of undetermined multipliers? Remember If f(, y) = y+(-y) then find sttionry points. u Apply If u= y then. Apply UNIT-III IMPROPER INTEGRALS, MULTIPLE INTEGRALS AND ITS APPLICATIONS Write the reltionship between bet nd gmm functions. Remember P g e

12 s Progrm Wht is the vlue of sin cos d using -function Remember Wht is the vlue of (p+,q) + (p,q+). Remember Find (m, m) Apply An equivlent iterted integrl with order of integrtion reversed for e Anlyze dyd is How to find the re of bounded region. Remember How to find the volume of closed surfce. Remember Wht is difference between proper nd improper integrls Remember Convert Wht is the re of ( y ) dyd to polr co-ordintes. Anlyze circles r sin, r sin. r drd over the region included between the UNIT-IV DIFFERENTIAL EQUATIONS d y dy Find the order nd degree of y dy d A sphericl rin drop evportes t rte proportionl to its surfce re t ny instnt t. The differentil eqution giving the rte of chnge of the rdius (r) of the rin drop is. dy Anlyze Apply Anlyze If y, y() then y() is d Apply When the differentil eqution is sid to be homogeneous? Remember Mention two pplictions of higher order differentil equtions. Remember Wht is generl solution of higher order differentil equtions? Remember d dr wht is orthogonl form of the function f(, y, ), f( r,, ) d Remember dy wht is generl solution of liner differentil eqution Remember when the Bernoull s differentil eqution becomes liner differentil eqution Remember Give the complementry function for (D +D+)y= Anlyze UNIT-V LAPLACE TRANSFORMS Give emple where the Lplce Trnsforms technique is used Remember Wht re the conditions tht the functions hs to stisfy to Apply lplce Remember trnsform Where the convolution theorem is in lplce trnsforms Remember Wht re periodic functions Remember Find L s Apply P g e

13 s Progrm Wht is Lplce Trnsform of unit impulse function Remember ' If f()= find L f ( t) Apply Find the vlue of L t Apply When L is possible n Remember s If y + y=sin t with y=y = then find L f(t) Apply Prepred by : Dr M. Anith, Mr Ch. Kumr Swmy, Ms L. Indir, Ms V. Subb Lkshmi, Mr J. Suresh Goud, Ms P. Sri Lth nd Mr Ch. Somshekr. Dte: August, P g e