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

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Engineering mathematics-2

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Question paper 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
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 )
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}
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 e²is periodic function with 2πi
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
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
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.
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^-1x 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.
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
j) If x² – 2x cos + 1 = 0, then show that x²ⁿ – 2xⁿcosn + 1 = 0

Answer:

By solving it

Let

=

=

=

=

=

= 0 + I x 0

= 0

L.H.S = R.H.S.
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+x²p⁴

= (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 .
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 =C₁Cos 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 - ]
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
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:
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.
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
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
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

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

Engineering mathematics-2

Previous year question paper with solutions for Engineering mathematics-2 May-2018

These Questions are downloaded from www.brpaper.com You can also download previous years question papers of 10th and 12th (PSEB & CBSE), B-Tech, Diploma, BBA, BCA, MBA, MCA, M-Tech, PGDCA, B-Com, BSc-IT, MSC-IT.

Question paper 1

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