Solved question paper for MATH-2 May-2018 (B-TECH 1st-2nd)

Question paper 1

1. SECTION A
   1. a) Find the general value of log iⁱ.

   Answer:

          log⁡iⁱ = i log⁡i
                 = i(2nπ+ log⁡i)
                 = i [2nπi⁡ + log⁡(cosπ/2 + i sinπ/2) ]
                 = i [2nπi + log⁡ e^iπ/2]             

                = i [2nπi + iπ/2]
                 = i²π [2n+1/2]
                 = - (2n+1/2) π

2. b) Solve the differential equation p = sin (y - x p), where p has its usual meaning.

   Answer:

   P = Sin (y - xp)

         It can be written as

         Sin^-1p = y - xp

         Y= px + sin^-1p which is Clairaut's equation

         So its solution will be given by

         y = ( cx + sin^-1c )

3. c) Test whether the differential equation (5x³ + 12x² + 6y²)dx + 6xydy = 0 is exact or not?
   Give reasons. If not. Find its integrating factor which will make it exact.

   Answer:

   Comparing with Mdx +Ndy=0

         M = 5x³ + 12x² + 6y², N=6xy

        M / y = 12y          N / x = 6y

        Sine M / y    N / x        so it is not exact.

   I.F. is given by M / y - N / x = 12y - 6y / 6xy =1/x = F(x)

   I.F. =     = e^{log x}

4. d) Show that e^z is a periodic function. Find its fundamental period.

   Answer:

   Let Z=x+iy

                e^z = e ^x+iy

            =  e^{x(cos y + i  sin y)}

            =  e^{x[cos(2nπ +y)+i sin(2nπ +y)]}

            =  e^x+i(2nπ+y)

            = e^(x+iy)+2nπi

           = e^z+2nπi

           i.e. e² remains unchanged when Z is increased by an multiple of  2πi

           So e² is periodic function with 2πi

6. e) Test whether the set of vectors {(1,0,0), (1,1,1), (1,2,3)} is linearly independent or
   dependent.

   Answer:

   Consider the equation

         k₁x₁ + k₂x₂ + k₃x₃ = 0

         k₁(1,0,0) + k₂ (1,1,1)+ k₃(1,2,3) = 0                  

         k₁ + k₂ +3k₃ = 0                                              

          k₂ +2k₃ = 0

          k₂ +3k₃ = 0

    

   From matrix we get 

   k₁ + k₂ +3k₃ = 0

   k₂ +2k₃ = 0    

   k₃ = 0                                                                            

   [k_1=0,            k_2=0,           k₃=0]  indepentet

7. f) Examine the convergence / divergence of the series 

   Answer:

   Here U_n =i/n  sin 1/n = 

   

            Taking V_n =  we have

             Lt n→∝         =  Lt n→∝   

   Since ε V_n is convergent, therefore ε U_n is also convergent

8. g)Test the absolute convergence of the series 

   Answer:

   Here U_n(x)                                         [x = can be any integer

                    [p = any real number

   =Mn

   Since   is convergent for p >  1, therefore by M test we can say is also convergent for all real x and p > 1.

9. h) If  is an eigen value or a matrix A then prove that ^- is an eigen value of A^-.

   Answer:

   Let x be a given Eigen vector corresponding to λ , then Ax= λ x

                Multiplying both sides by A^-1, we get

                       A^-1Ax = A^-1 λ  n                         ………….. (1)

          i.e. Ix = λ  A^-1 x or x = λ  (A^-1x)

              = A^-1x = 1/ λx

   This is of same from as (1) shows that 1/ λ  is Eigen value of inverse matrix A^-1.

11. i) For what values of ‘k’ the system of equation 

    x + y + z = 6; x + 2y + 3z = l0; x + 2y + kz =  ,
    has uniue solution.

    Answer:

    it can be written as

                 Ax=b

    A =                 X  =                B  =  

                  Augmented matrix A:B

           

    =  

                                               R₂ -R₁ ,R₃  -R₁

          

                R₁ –R₂ ,R₃  -R₂

         

    If K = 3 then

     Number of unknowns

    Therefore  System has unique solution for K=3

12. j) If x² – 2x cos + 1 = 0, then show that x²ⁿ – 2xⁿ cosn + 1 = 0

    Answer:

    

    By solving it   

    Let  

    

    =

    = 

    = 

    = 

    = 

    = 0 + I x 0

    = 0

    L.H.S = R.H.S.

13. SECTION B
    2. a) Solve the differential equation:
             y – 2px = tan^-1(x p²)
    b). Use method of variation of parameters to find the general solution of the differential
    equation: y" – 2y' + y = e^x log x

    Answer:

    (a)   y = 2px + tan^-1(xp²)       ……… (1)

                              Diff w.r.t  x

       dy/dx = 2(p+xdp/dx) + p² +2xp dp/dn /1+x² p ⁴

    = (p + 2ndp/dn)[1+p/1+x²p4]  =  0

              This given

    P + 2ndp / dn = 0

     Separating variables and integrating

      a constant

    Log x +2 log p = log c

    Log xp²= log c

    xp² =  c or  p =                    ……….(2)

    from (1) and (2) we get y = 2 ( ) x+ tan^-1c

    y = 2  + tan^-1c

    (b)  

    Given equation can be written as

           D² – 2D+11 y = e^x log x

    C.F and A.E are

      A E : (D-1)²  = 0

            D = 1, 1

     Thus C.F y = (C1 + C2n) e^x

    To find P. I here we have

                           y1= e^x, y2 = x eⁿ , X = e^x log n

    therefore  W =   =

    Thus P.I = -y1 

    =     - e⁺ⁿ  -  log n/  

     

    = -    

    
    = -

    =

    =   

    Here C.S  is y  = (C₁+ C₂ n)  + ¼ .

14. 3. a) Find the solution of the differential equation:
                    xy(1 + xy²)dy = dx
    b) Find the particular solution of the differential equation:
                   x²y" + xy' + y = log x sin (log x).
    by using operator method.

    Answer:

    (b) putting x=e^t

                        T=log x

    [D(D-1)+D+1] Y=T sin t

    (D²+1)y  = t sin t

    AE =D²+1

        D=

    C.F =C₁ Cos t + C2sin t

    P.I =   t sint

    =   t [I.P of e^it]                          I.P imaginary part

    = I.P. of e^it  

    =I.P. of e^it   

    =I.P. of e^it  

    =I.P. of  e^it   

    =I.P. of   e^it     

    =I.P. of   e^it    

    =I.P. of     dt

    =I.P. of 

    =I.P. of 

    =I.P. of 

     =

    Hence complete solution is

    [y=C₁ cost +C₂ sin t - ]

16. 4. a) Solve the simultaneous linear differential equation:

     + 2y + sint = 0,  - 2x – cost = 0
    
    b) Find the particular solution of the differential equation:
    y"' – 7y" + 10y' = e^2x sin x

    Answer:

    (a) Given equation are

    Dx + 2y= -sint       ….(i)

    -2x+Dy= cost     ….(ii)

    Eliminating n by multiplying (i) by 2 and (ii) by D and then adding

    4y²+D²y=- 2sin t – sin t

    
    (D²+4)y=-3 sin t

    A.E.=[D=± 2i]               C.F C₁ cos 2t +C₂ sin 2t

    P.I =     -3

    Y = C₁ cos 2t + C₂ sin 2t- cos t       …..(iii)

            dy/dt=-2 sin t + 2C₂ cos 2t- cos t      ….(iv)

    Substituting (iii) in (ii) we get

         2x = Dy - cos t

               = -2 C₁ sin 2t +2C₂ cos 2t- 2cos t

             x = -C₁ sin 2t +C₂ cos 2t- cos t     …..(v)

               When t = 0 , x = 0 , y = 1 (iii) and (v) will give

           C₁ =1 ,C₂ =1    

     Hence

    [x= cos 2t – sin 2t - cos t] 

    [y=cos 2t + sin 2t - sin t]

    (b) Given equation can be written as

                 (D² =7 D²  + 10 D) Y = e²ⁿ  sin x

     A.E =    D³ - 7D² + 10D = 0

    Which will give D=2 , D=5

    C.F = C₁ e^2x + C₂ e⁵ⁿ

    P.I =  

       =

    =     [D^L=-1]

    =

    =

    =

    =e²ⁿ sin x - 1/7 cos n

    C.S = P.I + CF

    Y=C₁ e^2x + C₂e^5x+e^2x  sin n -1/7 cos n

17. 5. An L-C-R circuit, the charge q on a plate of the condensor is given by the
    equation: L   + R +    +   = E sin pt, where  = I The circuit is tuned to
    resonance so that p² =   .If q = i = 0 when t = 0, show that for small values of R/L
    the current in the circuit at time t is given by (Et /2L) sin pt.

    Answer:

18. 6. a) Find the eigen values and the corresponding eigen vectors of the matrix:
    

    (b) Test the conditional convergence of the series. 
     

    Answer:

    (a) Characteristic equation is

        (A-λ I)=0

    

      (3-λ )(2-λ )(5-λ )=0

     Eigen values are

                 2, 3, 5

    If x, y, z be the components of an Eigen vector corresponding to the Eigen value λ

    [A-λ I]X  = 

    Putting λ=2 , we have x+y+4z =0, 6z=0, 3z=0

                                 .i. e         x + y = 0 and z = 0

    x/1 = y/-1 = z/0 = k1 say

    Eigen vector corresponding to λ =2 is K₁(1,-1,0)

    Putting λ =3, we have y+4z = 0, -y+6z = 0, 2z = 0 i.e y = 0,z = 0

    

    Eigen vector corresponding to

    Eigen vector corresponding to

    (b)

    The given series is an alternating series of from  U₁-U₂+U₃-U₄+……..

    For all n.

    Here U_n+1-U_N =

    Thus above terms are of decreasing order

    Also  

    Hence be Leibnitz test the given series in convergent

    Or

    To discuss absolute  convergent considers the corresponding series with positive  terms, that is    

    Choose Un =1/n

    We have

    

     

    

    Hence by comparison test, it is convergent.

19. 7. a) Test the consistency of the system of equations:
    x + 2y – z =1; 3x – 2y + 2z = 2; 7x – 2y + 3z = 5,
    and if consistent then solve it completely.

    b) Reduce the matrix  to normal form and hence find its rank.

    Answer:

    (a) [A:B]

    Apply R₃ = R₃/16

                                           R₂  - 3R₁

                                           R₃  -  7R₁

    R₃ = R₂ - 8R₃

    

    Since ρ  (A)  = ρ  (A : B) = no. of unknowns so it is consistent

    X + y + -z = 1

    -8y + 5z = -1

    28/4 z = -2

    Z = -8/28

    Y = 2/56

    X = 42/56

    
      (b) A = /AI

    

                                                                                  C_{2 -} C₁,C₃ - 2C₁,

          .

                                                                                    R₂ - R₁

    

                                                                                    C3 - C2     

    

                                                                                    R₃ - R₂

    

                                                                                         Which is normal form of  

    P =

                             Rank ρ (A) = 2

21. 8. a) Test for what values of ‘x’ for which the series

     +   ² +   ³ + ⁴ + ……..∞

    Converges / diverges.

    b) Examine the convergence / diverge of the series:
    

    Answer:

    (a)  

         Here Un xⁿ/(2n-1).2n and Un+1=xⁿ+1/(2n+1)(2n+2)

    Consider

    

             = =x

     Thus  is convergent for X< 1, divergent forX> 1 and fails at x=1

     When x+1

                          Consider        

    

    

    By Raabe’s test series is convergent for x≥1  given series is convergent for  x≤1  and divergent at x >1 .

    (b) we have Un  

     

    Taking Vn =   we find that

     

    

      is convergent , therefore  is also convergent

22. 9. a) If a, b, g are the roots of the equation x³ + px² + qx + p = 0, then prove that tan^-1 a + tan^-1 b + tan-1 g = np radians except in one particular case. Mention this case.

    b) If sin^-1 (u + iv) = a + ib, then prove that sin² a and cosh² b are the roots of the equation
    x² - x(1+ u² + v²) + u² = 0

    Answer:

    (a)  let

    given equation x³+px²+qn+p=0

    It’s roots are α,β,γ.

                    S₁ =

    S₂ =

    S₃ =

    

        Unless q = 1. in which case the fraction takes

                    Indeterminate from 0/0

    

    

    

    Hence

      Radiation except when q=1

     

    (b)

    

                 = 

     = 

    Comparing real and imaginary parts

    

    

    

    

    

    =

    Equation whose roots are

    

    

    Hence  are roots of