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Question Bank of B.TECH(CSE) Second semester
(1) State Gauss Divergence
Theorem.
(2) Evaluate .(x?+y')dxdy
(3) Change the
order of integration
y²
(4) Find volume bounded
by the cylinder x+y'=4 and
planes y+z=4 and z=0.
(5) Verify green theorem for
S. [(xy + y')dx + x*dy]
where c is bounded by the y=x and y=x*
Evaluate S. F.ds where F=4xi - 2y'j + z²k and S is the surface bounded
by region
x+y'=4,2=0 & z=3.
0 (7) Evaluate
(1)- S logzdz dx dy
(2)
(8) Change of the order
S(+ j? + z)dzdxdy
of integration
dxdy
A
(9) If F=3xyi -yj so evaluate F.dR where C
is thecurve y=2x* in the xy plane from
(0,0) to (1,2).
(10) Showthat area enclosed by parabola y' =4ax & x=4ay is a.
(11) Verify green theorem for
J. [(3x- y²)dx+ (4x – 6xy)dy] where cis bounded by the x=0,y=0 & x+y=1.
(12) Change of the order of integration
odydx
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UNIT-II
(1) (a) find General form of y=sin(y-xp)
(b)Write Bernoulli's and Laurent's equation
(c) State Necessary condition for a differential
equation Mdx+Ndy=0 to be exact.
(2) Solve (x 'e - 2mxy ) dx + 2mx "ydy =
0
(3) Solve y-2px=tan(xp )
(4) Solve y =x (p+/1 +
p² )
(5)Solve (xy '+y)dx+2(x 'y '+x+y )dy=0
(6) Solve dy dx
dx dy
6 () Solve (x y
-ez)dx-x*ydy=0
(8) Solve p=tan(x
-)1+p?
(9) Solve p +2pycotx -y=0
(10)Solve (y loy )dx+(x-logy)
dy=0
- y
(11) Solve x dy dx +
xdy-ydx.
=0
*2+y?
(12) Solve [x - +
(13) Solve y+ p x=x'p
[y+y =0
(14) Solve (p x-y) (py +x)=0
UNIT-I|
(1) Explain Method of Variation
of paramter.
(2) Define singular point
(5) Define cauchy & legendre's polynomial.
(4) Solve +y=xe "sinx
+son toit wo2
dx2 -2dx
(5) Solve by method of Variation of parameter (1) y"-6y'+9y e3x
(2))y"-2y'+y=e "logx
(6) Solve the series of equation dy +xy
dx2 dx
=0
(7) (a) Define generation function
of pa(x)
(2) Prove that (a)
j(x) =in(x) - jne(x)
(b) jnei(x) jn(x) - jn-'(*)
(8) express f(x) =x *+3x* -x+5x-2=0
in terms of legendre's polynomial.
G (9) Solve the series of equation (a) x
dx3
+2x d²y +2y =10(x+) (b)
9x(1-x)-12 +4y =0
dx2 dx?
(10) Solve d²y -4dx +4y=8x e 2"sin2x dx
dx2
(9 (11)y "-2y'+2yze "tanx
6 (12) prove that [pm(x).pa(x)dx = 2
if m=n
2n+1
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QUESTION BANK OF UNIT- 4
r (1) Find real& imaginary part of sin z
(2) (a) Define Analytic Function (b)
write Cauchy -Riemann equation.
C (3) If tan( +i0 =tan
) sec a
a+i then prove e = that tcot & 20= (nt;)r + a
2
(4) Determine the analytic function of f(z) =u+iv if U-vi= cosx+sinx-e-Y
&
2(cosx-coshy) f()=0
6 (5) If f(z) is analytic function then prove that( +
)f(2)! =4!f'(z)!
(6)1f
il'=A+iB then prove that
tan" = & A+ B
2 A
(7) Prove that cosh'x= sinhx=1
(8) Prove that tan(x + iy) = sin (u+iv)
Prove that sin2x tanu
sin2y tanv
(9)Prove that tan'z=
log)
(10) Solve (x- ,y ldx +(y+ildy =0
(11)Solvey+ px =x'p
(12) Solve (px -yl(py + x)=a' p
(13) Find Analyticfunction of u-v = X-y
x²+4xy+y?
(14) if u=log tan("
+) prove that (a) tan h ; tan (b) e =-ilogtan (
(15) if sin(a + iß) =x+iy then prove that
sin²a
y² =1
cos²a
(16) if w=logz then find dw
dz
(17) if f(z) is an analytic function of contant modulus then f(z) is constant.
(18)find analytic function of f(z) = utiv if 2u+v= e(cosy-siny)
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QUESTION BANK OF UNIT-6
(a) Write the Cauchy integral formula.
{b) Evaluate sin'z c
.dz,where is the circle |z =1.
(c) Find Laurent expansion of f(2) = 7z-2
in the region of 1<|z+1|<3
(2+1)z(z-2)
(d) Prove that (2r_sin²o d = la-vaz-b²),0<b<a
a+bcos0 b2
(e) Find nature and location of 2-sinz
() Evaluate . ez
(z2+12)
z2
dz,where c if=s |z]=4
(g) Apply calculus of residue 0
de
17-8cos
;dz =
15
(h) Apply calculus of residue cos20de 2ra?
0 1-2acos0 +a2 1-a2
(0) Apply calculus of residue de 2rra2
1-2pcose +p2 1-p?
i) Find the sum of residue of f(z) = sinz at its poles |z|=2
COSZ
(k) Evaluate J sinnz²+cosnz2 dz, where c is |z|=3
(z-1)2(z-2)
() EvaluateJ, tanz dz, where c if]z=2
