Subject: Mathematics-I Question Bank Section A T T. Find the value of fo which the matix A = T T has ank one. T T i. Is the matix A = i is skew-hemitian matix. i. alculate the invese of the matix = 5 7 4. Is the matix A = is diagonalizable. 5. Find the eigen values of A = n If y= x logx, find 6. ( y n ) 7. If = +, = (, ), Evaluate (,) 8. Find fo the cuve y = x. 9. If y u = f, find + x. Find the symmety and the oigin of the cuve y ( a x) = x. Expand f ( x, y) = x + y + 6x+ about = and =.. Find, if. Find the cente of cuvatue of 4. If x = y, y x find at (, 4) x y= e at the point (, ) 5. Using diffeential calculus, calculate 4.5 appox 6. Find the minimum value of f ( x, y) = x + y + 7. Find the paallel asymptotes of x + y = a xy 8. Find the symmety and pole of the cuve = a sin θ Pape ode: AS 9. What is the asymptote of the cuve y ( a x) = x. If u = x yz 4 y z + xz, find + +. Find the diectional deivative of x yz+ 4 x z at (,, ) in the diection iˆ ˆj kˆ. Detemine the value of fo which F x y yz i xy xz ˆ = ( λ + )ˆ+ ( ) j+ (xyz x y ) kˆ is solenoidal.
. Detemine the value of fo which F xy z i x ˆ = ( λ )ˆ+ ( λ ) j+ ( λ) xz ) kˆ is iotaional. 4. State Geen s theoem fo a plane cuve. 5. State Stoke s fo a closed cuve. 6. State Gauss divegence theoem. 7. Find the value of Γ 8. Evaluate 9. Find the aea between the paabola = and the line =.. Find the value of (, )( +, )( + +, ). Evaluate in tems of gamma function. Section B &. Define the matix. Find the invese of a matix A, by applying elementay tansfomation = 4 4 4. Reduce the matix A to its nomal fom when =. Hence find the ank of A 4 6 7. Find two non-singula matix P and Q such that PAQ is the nomal fom, whee = 4. Find the ank of the matix by educing it into echelon fom, = 6 4 7 5. Test the consistency of the following system of linea equation and hence find the solution. 4 =, + 5 =, + 4 = 8 6. heck the consistency of the following system of linea non-homogeneous equation and find the solution, if exist: 7 + + = 6, + + 5 = 5, + + 4 = 7. Investigate fo what values of and do the systems of equations + + = 6, + + =, + + = have (i) no solution (ii) unique solution (iii) infinite solution. 8. Detemine the value(s) of µ fo which the following set of equations x + z=, x + z=, 4 x + µ z= possess a non-tivial solution and find the solutions fo eal µ 9. Find the value of fo which the vecto,,,, 5, 5, 7 ae linealy dependent. + +. If = 5 4, veify that is a Hemitian matix, whee is a conjugate tanspose of A. +. If =. Is is unitay matix o not. +. Show that the following matix is unitay:. Find the Eigen values and Eigen vectos of the matix: 8 6 = 6 7 4 4 +
4. Veify ayley- Hamilton theoem fo the matix = 4 5. Find the chaacteistic equation of the matix = A and use it to find the matix epesented 5 4 5 4 by A + 5A 6A + A 4A+ 7I. Futhe, expess A 4A 7A+ A A I as a linea polynomial in A. 6. Find the n th deivative of tan. 7. If = cosmlog, Pove that + + + ( + ) =. 8. If =, Pove that = fo n is odd, and =.. 4. 6.. ( ) ; Fo n is even numbe. 9. If = sin, Find at x=.. If x y= 4 then find the n th deivative ( y n) o. If =, show that + + + tan =.. Using Taylo's seies, expand (i) ompute the value of cos to fou decimal place. (ii) sin x in powes of (x π/). Hence find the value of sin 9 o coect to 4 decimal places. Expand in the neighbohood of (, ) up to and inclusive of second degee tems. Hence compute.,.9 appoximately. 4. Expand cos nea the point, 5. Appoximate: [(.8) + (.) ]5 by Taylo s theoem. + x 6. Obtain the seies fo ln( + x) and then find the seies fo ln and hence find ln x 9 7. If = +, pove that = + ( ). Hence, show that =, when = 9 8. If u= log( x + y + z x show that + + u = x y z ( x + y + x y z u u u 9. If + + =, pove + + = x + y + z a + u b + u c + u x x z x y z. A balloon is in the fom of ight cicula cylinde of adius.5m and length 4m and is sumounted by hemispheical ends. If the adius is inceased by.m and length by.5m, find the % change in the volume of balloon.. If =, = (), =, then show that,, ae not independent and find the elation between them.. If u is a homogeneous function of x and y of degee n, pove that + + = du. If u = sin ( x y), x= t, y= 4t, show that =. dt ( t ) 4. If =, 4, 4, pove that + + =. 5. If = sin cos, = sin, = cos, find (,, ). (,,)
6. If =, = + +, = + +, find the Jacobian,,,, 7. If + + =, + =, =, show that (,,) (,,) =. 8. Find the exteme value of the function +. 9. Show that the ectangula solid of maximum volume that can be inscibed in a given sphee is a cube. 4. Find the maximum and minimum distance of the point (, 4, ) fom the sphee + + =. 4. Find the dimension of a ectangula box of maximum capacity whose suface aea is given by when (i) box is open of the top (ii) box is closed. 4. Find all the asymptotes of the following cuves: (i) x y x y xy + x+ = (ii) x + x y 4y x+ = 4. Show that the asymptotes of the cuve = + fom a squae of sid. 44. Show that fo the paabola y = 4ax, ρ vaies as (SP), whee ρ is the adius of cuvatue at any point P of the paabola and S is the focus of the paabola. 45. Find the adius of uvatue of the cuve = at the oigin. 46. Define gadient of a scala field and give its physical intepetation. 47. Discuss the physical significance of divegence and cul of a vecto field. 48. If = xiˆ + yj ˆ+ zkˆ and =, show that n n (i) gad = ; (ii) = n whee a n is a constant vecto (iii) div ( ) = ( n+ ) 49. Find the diectional deivative of the function f ( x, = xy + yz at the point (,,) in the diection of the vecto iˆ + ˆj+ kˆ 5. Pove: (i) ( A B) = B ( A) A ( B) (ii) ( A B) = ( B) A ( A) B + ( B. ) A ( A ) B 5. Evaluate cuve = 4. 5. Evaluate line., whee A is the domain bounded by -axis, odinate = and the ove the aea of the cadioids = ( + co ) about the initial 5. Detemine the aea by the cuves =, 4 = and = 4. n dzdydx 54. Evaluate, x y z x x y by changing it into spheical pola co-odinates. 55. Evaluat, by changing to pola coodinates. 56. Find by double integation, the aea lying between the paabola = 4 and the line =. 57. hange the ode of the integation, hence evaluate the same. 58. hange the ode of the integation,. 59. By changing the ode of the integation of sin, show that 6. Pove that Γ =. =
6. Pove that, =, >, >. 6. Pove that, = +, +, +, whee, >. 6. Show that = 64. Evaluate the following integals: e logye x (a) log z dzdxdy (b) 4 6 65. Pove that x log dx =. x 65 66. Pove that = + 67. Evaluate F S log x x+ log y x+ z e dzdydx nˆ ds, whee F= 8 z iˆ ˆj+ y kˆ x + 6z= in the fist octant. 68. If F= (x ˆ i xyj ˆ 4xkˆ, then evaluate V, whee D is the domain, and and S is the suface of the plane F dv, whee V is bounded by the planes x =, y=, z=, and x+ z= 4. 69. If A = xz iˆ x ˆ j+ y kˆ, evaluate AdV, whee V is the egion bounded by the sufaces x =, y=, x=, y= 6, z= x, and z= 4. V 7. Apply Geen s theoem to evaluate (x y ) dx+ ( x + y ) dy, whee is the bounday of the aea enclosed by the x -axis and the uppe half of the cicle x + y = a. 7. Veify Geen s in the plane fo ( x y dx+ x dy, whee is the closed cuve of the egion ) bounded by y = x andy= x. 7. State Stoke s theoem and veify it fo the vecto field F( x, = ( x + y ) iˆ xy ˆj taken ound the ectangle bounded by the lines x = ± a, y =, and y = b. 7. Evaluate the suface integal S culf. nˆ ds by tansfoming it into a line integal, S being that pat of the suface of the paaboloid z= x y fo which z and F ( x, = yiˆ+ zj ˆ+ xkˆ. 74. Using stoke s theoem, evaluate the line integal ( yz dx+ zx d xy d whee is the cuve of intesection of the sufaces x + y =, and z= y. Note: Do not confine youself only upto these question. Such types of questions ae usually asked in Univesity examination. Kindly ty moe questions of such types. Best of Luck