Engineering mathematics-2
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Question paper 1
SECTION A
1. a) Find the general value of log ii.Answer:
logii = i logi
= i(2nπ+ logi)
= i [2nπi + log(cosπ/2 + i sinπ/2) ]
= i [2nπi + log eiπ/2]= i [2nπi + iπ/2]
= i2π [2n+1/2]
= - (2n+1/2) πb) 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 = 6ySine
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. =
= elog xd) Show that ez is a periodic function. Find its fundamental period.
Answer:
Let Z=x+iy
ez = e x+iy
= ex(cos y + i sin y)
= ex[cos(2nπ +y)+i sin(2nπ +y)]
= ex+i(2nπ+y)
= e(x+iy)+2nπi
= ez+2nπi
i.e. e2 remains unchanged when Z is increased by an multiple of 2πi
So e2 is 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
[k1=0, k2=0, k3=0] indepentet
f) Examine the convergence / divergence of the series
Answer:
Here Un =i/n sin 1/n =
Taking Vn =
we haveLt n→∝
= Lt n→∝ 
Since ε Vn is convergent, therefore ε Un is also convergent
g)Test the absolute convergence of the series
Answer:
Here Un(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-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.
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 unknownsTherefore System has unique solution for K=3
j) If x2 – 2x cos
+ 1 = 0, then show that x2n – 2xn cosn + 1 = 0Answer:
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 = ex log xAnswer:
(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 constantLog 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-1cy = 2
+ tan-1c(b)
Given equation can be written as
D2 – 2D+11 y = ex log x
C.F and A.E are
A E : (D-1)2 = 0
D = 1, 1
Thus C.F y = (C1 + C2n) ex
To find P. I here we have
y1= ex, y2 = x en , X = ex 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=et
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 eit] I.P imaginary part = I.P. of eit
=I.P. of eit
=I.P. of eit
=I.P. of
eit 
=I.P. of
eit 
=I.P. of
eit 
=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' = e2x sin xAnswer:
(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 tA.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 e2x + C2 e5n
P.I =
=
=
[DL=-1]=
=
=
=e2n sin x - 1/7 cos n
C.S = P.I + CF
Y=C1 e2x + C2e5x+e2x 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 Un+1-UN =
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 = R3/16R2 - 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
C2 - 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
=
=xThus
is convergent for X< 1, divergent forX> 1 and fails at x=1When 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 convergent9. 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 = 0Answer:
(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
