SNS COLLEGE OF ENGINEERING, CBE – 107
QUSTION BANK
MATHEMATICS – II
UNIT-1 (ORDINARY DIFFERENTIAL EQUATIONS)
PART-A
1) Define linear differential equations
2) Solve
𝑑2 𝑦
𝑑𝑥 2
𝑑𝑦
−.6 𝑑𝑥 +13y = 0
3) Solve (D2 + 1) y =0 given y(0) =0 , y’(0)=1
4) Solve (4D2-4D +1)y = 4
5) Solve (D2_1)y = x
6) Solve the equation x2 y11-xy1+y= 0
7) Solve (D+2)2y = 𝑒 −2𝑥 sinx
8) Find the particular integral of (D2+4)y = cos2x
9) Find the particular integral of (D2+1)y = sinx
10) Find the particular integral of (D2+1)y = xex
11) Find the particular integral of (D2_4)y = cosh2x
12) Find the particular integral of (D-1)2y = exsinx
13) Find the particular integral of (D2-2D+5)y = ex cos2x
14) Find the particular integral of (D2-2D+1)y = ex ( 3x2-2)
15) Find the particular integral of (D2-4D+4)y = 2x
16) Solve (D2+2D +1) y =π
17) Solve (D2-3D -4) y = e 3x +e-x
18) (D2+1) y = sin2x
19) Transform the equation x2 y11+xy1= x into a linear differential equation with constant
Co-efficient.
𝑑2 𝑦
𝑑𝑦
20) Transform the equation (2x+3)2 𝑑𝑥 2 - 2(2x+3) 𝑑𝑥 - 12y =6x into a linear differential
Equation with constant co-efficients
21) Find the Wronskian of y1, y2 of y11 -2y 1 +y =ex logx
Part-B
1)
2)
3)
4)
5)
Find the particular integral of (D2+4)y = x2 cos2x
Solve (D2+3D+2)y = sin3x cos2x
Solve (D2+5D+6)y = e –7x sinh3x
Solve (D3 - 3D2 + 4D-2)y = sinh2x
(D2+9)y = 11 cos3x
1
6) Solve (D2-6D+13)y =8 e 3x sin4x
7) Solve (D2+4D+4)y = e –2x/ x2
8) Solve (D2-2D+1)y = e x xcosx
9) Solve the equation (D2+a2)y = secax by the method of variation of parameters
10) Solve
11) Solve
𝑑2 𝑦
𝑑𝑥 2
𝑑2 𝑦
𝑑𝑥 2
+ 4y= 4tan2x by the method of variation of parameters
+ y= cosecx by the method of variation of parameters
12) Solve (D2+1)y = xsinx by the method of variation of parameters
13) Solve (D2-2D+1)y = e x logx by the method of variation of parameters
𝑑2 𝑦
𝑑𝑦
14) Solve(1+x2)2 𝑑𝑥 2 +(1+x) 𝑑𝑥 + 𝑦 = 2sin(log(1+x))
15) Solve (x2
𝑑2 𝑦
𝑑𝑥 2
𝑑𝑦
-2x 𝑑𝑥 -4)y = x2+2logx
16) Solve(x2D2+ xD+1)y= logxsin(logx)
17) Solve(x2D2+ xD+4)y=cos(logx)+ xsin(logx)
1
18) Solve (D2+𝑋D)y =
2
12𝑙𝑜𝑔𝑥
𝑥2
2
19) Solve(x D + 3D+1)y= sin(logx)/x2
𝑑2 𝑦
𝑑𝑦
20) Solve((3x+2)2𝑑𝑥 2 +3(3x+2) 𝑑𝑥 -36)y =3x2+4x+1
𝑑2 𝑦
𝑑𝑦
21) Solve(x+1)3𝑑𝑥 2 +3(x+1)2 𝑑𝑥 +(x+1)y =6log(x+1)
𝑑𝑥
𝑑𝑦
22) Solve the system of equations 𝑑𝑡 +y =et ;𝑥 −
𝑑𝑥
23) Solve the system of equations 𝑑𝑡 +y =sint ;
𝑑𝑡
𝑑𝑦
𝑑𝑡
=t
+x=cost given that x=2,y=0 when t=0
𝑑𝑥
𝑑𝑦
24) Solve the system of equations 𝑑𝑡 +2x+3y =2e2t ; 𝑑𝑡 +3x+2y =0
𝑑𝑥
𝑑𝑦
25) Solve the system of equations 𝑑𝑡 − 𝑦 =t ; 𝑑𝑡 +x =t2
𝑑𝑥
𝑑𝑦
𝑑𝑥
𝑑𝑦
26) Solve the system of equations 𝑑𝑡 − 𝑑𝑡 +2y=cos2t ; 𝑑𝑡 + 𝑑𝑡 -2x =sin2t
27) Solve the simultaneous differential equations
2
𝑑𝑥
𝑑𝑡
+2y=sin2t ;
𝑑𝑦
𝑑𝑡
-2x =cos2t
UNIT II – (VECTOR CALCULUS)
Part – A
1.
2.
3.
4.
Define vector Differential operator(∇)
Define gradient of the scalar function 𝜑.
If f and g are two scalar point function then ∇(fg) = 𝑓∇g + g∇f.
If 𝜑 = log(𝑥 2 + 𝑦 2 + 𝑧 2 ) find ∇𝜑.
5. Prove that ∇𝑓(𝑟) =
𝑓 ′ (𝑟)
𝑟
⃗⃗ .
𝑟⃗ ,𝑟⃗ = 𝑥𝑖⃗ +𝑦𝑗⃗ +𝑧𝑘
⃗⃗
6. Find the directional derivative of 𝜑 = 𝑥𝑦 + 𝑦𝑧 + 𝑧𝑥 in the direction vector 𝑖⃗ + 2𝑗⃗ + 2𝑘
at (1, 2 ,0)
7. Find the directional derivative of 𝜑 = 3𝑥 2 + 2𝑦 − 3𝑧 at (1 , 1 ,1) in the direction of
⃗⃗ .
2𝑖⃗ + 2𝑗⃗ − 𝑘
8. Find the unit vector normal to the surface 𝑥 2 − 𝑦 2 + 𝑧 = 2 at the point (1 , -1 ,2)
9. Find the unit vector normal to the surface 𝑥 2 𝑦 + 2𝑥𝑧 2 = 8 at the point(1 , 0, 2)
⃗⃗⃗⃗ ± ⃗⃗⃗⃗
10. Prove that ∇ × (𝐹
𝐺 )= ∇ × ⃗⃗⃗⃗
𝐹 ± ∇ × ⃗⃗⃗⃗
𝐺.
⃗⃗⃗⃗ ∙ (∇ × 𝐹
⃗⃗⃗⃗).
⃗⃗⃗⃗ × ⃗⃗⃗⃗
⃗⃗⃗⃗)– 𝐹
⃗⃗⃗⃗ ∙ (∇ × 𝐺
11. Prove that ∇ ∙ (𝐹
𝐺)=𝐺
⃗⃗ at
⃗⃗⃗⃗ and ∇ × 𝐹
⃗⃗⃗⃗ of the vector point function 𝐹
⃗⃗⃗⃗ = 𝑥𝑧 3 𝑖⃗ − 2𝑥 2 𝑦 𝑧⃗⃗𝑗 + 2𝑦𝑧 4 𝑘
12. Find ∇ ∙ 𝐹
the point (1 ,-1 ,1).
13. Prove that curl ( grad 𝜑) = 0.
14. Prove that div (grad 𝜑) = ∇2 𝜑.
⃗⃗ is solenoidal and 2𝑥𝑦𝑖⃗⃗ +
15. Show that the vector ⃗⃗⃗⃗
𝐹 = 3𝑦 4 𝑧 2 𝑖⃗ + 4𝑥 3 𝑧 2 ⃗⃗𝑗 − 3𝑥 2 𝑦 2 𝑘
⃗⃗⃗⃗ is irrotational.
(𝑥 2 + 2𝑦𝑧)𝑗⃗⃗ + (𝑦 2 + 1)𝑘
⃗⃗⃗⃗ = (𝑥 + 2𝑦 + 𝑎𝑧) 𝑖⃗⃗ + (𝑏𝑥 − 3𝑦 − 𝑧)𝑗⃗⃗ +
16. Find the value of a , b , c so that the vector 𝐹
⃗⃗⃗⃗ is irrotational.
(4𝑥 + 𝑐𝑦 + 2𝑧)𝑘
17. If ⃗⃗⃗⃗
𝐹 = 𝑥 2 𝑖⃗⃗ + 𝑥𝑦 𝑗⃗⃗ , evaluate ∫ ⃗⃗⃗⃗
𝐹 ∙ ⃗⃗⃗⃗⃗
𝑑𝑟 from (0 ,0) to (1 ,1) along the line y = x.
⃗⃗⃗⃗ and 3𝑖⃗⃗ + 𝑎𝑗⃗⃗ − 2𝑘
⃗⃗⃗⃗ are
18. Find the value of ‘a’ given two vectors 2𝑖⃗⃗ − 3𝑗⃗⃗ + 5𝑘
perpendicular.
⃗⃗ . S is the upper half surface of the sphere 𝑥 2 + 𝑦 2 + 𝑧 2 = 𝑎2 ,then
19. If 𝑟⃗ = 𝑥𝑖⃗ + 𝑦𝑗⃗ + 𝑧𝑘
find ∬𝑆 𝑟⃗⃗⃗ ∙ 𝑛̂ 𝑑𝑠
⃗⃗⃗⃗ = 𝑥 2 𝑖⃗⃗ +
20. If v is the volume of the region enclosed by the cube 0 < x ,y ,z <1 and 𝐹
𝑦 2 𝑗⃗⃗ + 𝑧 2 ⃗⃗⃗⃗
𝑘 , then
𝐹 𝑑𝑉 is
∭ ∇ ∙ ⃗⃗⃗⃗
𝑉
3
Part – B
⃗⃗ prove that (i) ∇𝑟 = 𝑟⃗⃗⃗⃗ , (ii) ∇𝑟 𝑛 = 𝑛𝑟 𝑛−2 ⃗⃗⃗
1. If 𝑟⃗ = 𝑥𝑖⃗ + 𝑦𝑗⃗ + 𝑧𝑘
𝑟 ,where 𝑟 = |𝑟⃗⃗⃗⃗⃗|.
𝑟
2. Find the angle of intersection at the point (2 ,-1,2) of the surfaces 𝑥 2 + 𝑦 2 + 𝑧 2 = 9
and 𝑧 = 𝑥 2 + 𝑦 2 − 𝑧 − 3.
3. Find ‘a’ and ‘b’ such that the surfaces 𝑎𝑥 2 − 𝑏𝑦𝑧 = (𝑎 + 2)𝑥 and 4𝑥 2 𝑦 + 𝑧 3 = 4 cut
orthogonally at (1 ,-1,2).
⃗⃗⃗⃗, find the scalar potential 𝜑.
4. If ∇𝜑 = 2𝑥𝑦𝑧𝑖⃗⃗ + 𝑥 2 𝑧𝑗⃗⃗ + 𝑥 2 𝑦𝑘
⃗⃗ and C is the straight line
5. Evaluate ∫𝐶 ⃗⃗⃗⃗
𝐹 ∙ ⃗⃗⃗⃗⃗
𝑑𝑟 where ⃗⃗⃗⃗
𝐹 = 3𝑥 2 𝑖⃗ + (2𝑥𝑧 − 𝑦)⃗⃗𝑗 + 𝑧𝑘
from A(0 ,0 ,0) to B(2 , 1, 3).
⃗⃗ , evaluate ∫ 𝐹
⃗⃗⃗⃗⃗ from the point
⃗⃗⃗⃗ = xz⃗⃗𝑖⃗ + 𝑦𝑧⃗⃗𝑗 − 𝑧 2 𝑘
⃗⃗⃗⃗ ∙ 𝑑𝑟
6. Given the vector field 𝐹
𝐶
(0,0,0) to (1,1,1) where C is the curve (i) x = t , y = 𝑡 2 , z = 𝑡 3 , (ii) the straight path from
(0,0,0) to (1,1,1).
⃗⃗⃗⃗ =
7. Find the total work done in moving a particle in a force field given by 𝐹
⃗⃗⃗⃗ along a circle C in the XY plane
(2𝑥 − 𝑦 + 𝑧) 𝑖⃗⃗ + (𝑥 + 𝑦 − 𝑧)𝑗⃗⃗ + (3𝑥 − 2𝑦 − 5𝑧)𝑘
𝑥 2 + 𝑦 2 = 9, 𝑧 = 0.
8. Find the work done by the force ⃗⃗⃗⃗
𝐹 = (2𝑥𝑦 + 𝑧 3 ) 𝑖⃗⃗ + 𝑥 2 𝑗⃗⃗ + 3𝑥𝑧 2 ⃗⃗⃗⃗
𝑘 when it moves a
particle from (1,-2,1) to (3,1,4) along any path.
9. Evaluate
 F  nˆds
where F  yzi  zx j  xyk. and S is that part of the surface of the
S
sphere x2+y2+z2 = 1 which lies in the first octant.
10. Evaluate
 F  nˆds
where F  18zi  12 j  3 y k as S is the part of the plane 2x + 3y +
S
6z = 12 which is in the first octant.
11. Evaluate
 F  nˆds
where F  ( x  y 2 )i  2 x j  2 yz k where S is the region bounded by
S
2x + y + 2z = 6 in the first octant.
12. If F  (2 x 2  3z)i  2 xy j  4 xk , then evaluate (i)
   F dV
V
(ii)
   FdV ,where V
V
is the region bounded by x = 0 , y = 0 , z = 0 and 2x + 2y + z = 4.
⃗⃗⃗⃗ over the cube
13. Verify the Gauss divergence theorem for ⃗⃗⃗⃗
𝐹 = 4𝑥𝑧 𝑖⃗⃗ − 𝑦 2 𝑗⃗⃗ + 𝑦𝑧𝑘
bounded by x = 0 , x = 1, y = 0, y = 1, z = 0, z = 1.
⃗⃗⃗⃗ taken
⃗⃗⃗⃗ = (𝑥 2 − 𝑦𝑧) 𝑖⃗⃗ + (𝑦 2 − 𝑧𝑥)𝑗⃗⃗ + (𝑧 2 − 𝑥𝑦)𝑘
14. Verify the Divergence theorem for 𝐹
over the rectangular parallelepiped 0 ≤ x ≤ a ,0 ≤ y ≤ b , 0 ≤ z ≤ c .
15. Evaluate
 F  nˆds
where F  4 xzi  y 2 j  yz k. and S is the surface of the cube
S
bounded by x = 0 ,x = 1,
y = 0, y = 1, z = 0, z = 1.
4
16. Use divergence theorem to evaluate ⃗⃗⃗⃗
𝐹 = 4𝑥 𝑖⃗⃗ − 2𝑦 2 𝑗⃗⃗ + 𝑧 2 ⃗⃗⃗⃗
𝑘 and S is the surface
2
2
bounding the region x + y = 4 z = 0 and z = 3.
17. Verify Green’s theorem in a plane for the integral
 ( x  2 y)dx  xdy,taken around the
C
circle x2 + y2 = 1.
18. Verify Green’s theorem for
 ( x
2

 y 2 )dx  2 xydy , where C is the boundary of the
C
rectangle in the XOY – plane bounded by the lines x = 0,x = a, y = 0 and y = a.
19. Verify Green’s theorem for
 ( xy  y
2

)dx  x 2 dy , where C is the closed curve of the
C
region bounded by y = x and y = x2 .
20. Using Green’s theorem, evaluate
 ( y  sin x)dx  cos xdy,where C is the triangle
C
bounded by
𝑦 = 0, 𝑥 =
𝜋
2
,𝑦 =
2𝑥
𝜋
.
21. By applying Green’s theorem prove that the area bounded by a simple closed curve C is =
1
2
 ( xdy  ydx) and hence find the area of the ellipse.
C
22. Verify Stoke’s theorem for a vector defined by ⃗⃗⃗⃗
𝐹 = (𝑥 2 − 𝑦 2 ) 𝑖⃗ + 2𝑥𝑦⃗⃗𝑗 in the
rectangular region in the XOY plane bounded by the lines x = 0, x = a, y = 0 and y = b.
⃗⃗⃗⃗ ,where S is the
23. Verify Stoke’s theorem for a vector defined by ⃗⃗⃗⃗
𝐹 = y 𝑖⃗ + 𝑧⃗⃗𝑗 + 𝑥𝑘
upper half of the surface of the sphere x2 + y2 + z2 = 1 and C is its boundary.
24. Evaluate the integral
 ( x  y)dx  (2 x  z)dy  ( y  z)dz where C is the boundary of
C
the triangle with vertices (2,0,0), (0,3,0) and (0,0,6) using Stoke’s theorem.
25. Evaluate  ( xydx  xy 2 dy) by the Stoke’s theorem where C is the square in the XY plane
C
with vertices (1,0),
(-1,0), (0,1) and (0,-1).
26. Prove that  r  dr  2 nˆds where S is the surface enclosing a circuit C.
S
5
UNIT 3 - (ANALYTIC FUNCTIONS – COMPLEX VARIABLES)
Part A
𝑥
1. Show that 𝑥 2 +𝑦 2 is harmonic.
2. Is f(z) = z3 analytic?
1
3. Find the invariant point of the transformation w = 𝑧−2𝑖
4.
5.
6.
7.
Show that xy2 cannot be the real part of an analytic function.
Find the image of x2+y2 = 4 under the transformation w=3z.
f(z) = u + iv is such that u and v are harmonic is f(z) analytic always? Justify.
State the Cauchy-Riemann equations in polar coordinates satisfied by an analytic
function.
2𝑧+6
8. Find the invariant points of the transformation w= 𝑧+7 .
9. Find the analytic region of f(z) = (x-y)2+2i (x+y).
10. Find the critical points of the transformation w2= (z-𝛼)(z-𝛽).
11. Define conformal mapping.
12. For what values of a,b and c the function f(z) = x – 2ay + i(bx-cy) is analytic?
13. If u+iv is analytic, show that v-iu is also analytic.
14. Find the image of the circle |z| = 2 under the transformation w = 3z.
15. Define bilinear transformation.
16. Define analytic function of a complex variable.
17. Give an example such that u & v are harmonic but u+iv is not analytic.
18. Find ‘a’ so that u(x,y) = ax2-y2+xy is harmonic.
19. State the orthogonal property of an analytic function.
20. Under the transformation w = iz + I show that the half plane x>0 maps into the half plane
w>1.
21. Find the points in the z plane at which the mapping w = z + 𝑧 −1 , (z≠0) fails to be
conformed.
𝑦
22. Prove that tan−1 𝑥 is harmonic.
23. Show that the function f(z) = zz̅ is not analytic at z=0.
24. f(z) = r2 (cos 2𝜃 +i sin 𝑝𝜃) is analytic if the value of p is ….?
a) ½
b) 0
c) 2
d) 1
25. Define Mobius transformation.
1
26. If u ≠ iv = 𝑧; then prove that the families of curves u = c1 and v = c2 ( c1 , c2 being
constants) cut orthogonally.
27. Define Isogonal transformation.
28. Verify whether w = sin 𝑥 cos ℎ𝑦 + 𝑖 cos 𝑥 sin ℎ𝑦 is analytic or not.
6
29. Find the bilinear transformation which maps the points z = -2, 0, 2 into the points w = 0,
i, -i respectively.
𝜕2
𝜕2
30. If f(z) is a regular function of z, prove that (𝜕𝑥 2 + 𝜕𝑦 2) |f(z)|2 = 4 |f ’(z)|2
Part B
1. Find the analytic function whose real part is
sin 2𝑥
cos ℎ2𝑦 − cos 2𝑥
.
2. Find the image of the infinite steps
i)
¼ <y<½
ii)
1
0 < y < ½ under the transformation w = 𝑧
3. Find the bilinear transformation which maps -1, 0, 1 of the z-plane onto -1, -i, 1 of the wplane. Show that under this transformation the upper half of the z-plane maps onto the
interior of the unit circle |w|=1.
2 sin 𝑥 sin ℎ𝑦
4. Find the analytic function f(z) = u + iv, where v = cos 2𝑥+ cos ℎ2𝑦 .
5. Find the bilinear transformation which maps the points z = 0, 1, ∞ into
w=i,-1,-i.
2
2
−2𝑥
6. Prove that x – y + 𝑒
cos 2𝑦 is harmonic and find its harmonic conjugate.
7. If 𝜑 and Ψ are functions of x and y satisfying Laplace equation namely
𝜕2 Ψ 𝜕2 Ψ
𝜕𝑥 2
𝜕𝜑
+𝜕𝑦 2 = 0 and u = 𝜕𝑦 −
𝜕Ψ
𝜕𝑥
;v=
𝜕𝜑
𝜕𝑥
+
𝜕Ψ
𝜕𝑦
𝜕2 𝜑
𝜕𝑥 2
+
𝜕2 𝜑
𝜕𝑦 2
= 0;
. show that u +iv is analytic.
8. Show that the function f(z) = √|𝑥𝑦| is not regular at the origin, through C-R equations
are satisfied at origin.
9. Determine the region D’ of the w-plane into which the triangular region D enclosed by
the lines x=0, y=0, x+y=1 is transformed under the transformation w = 2z.
𝑥
10. Find the analytic function f(z) = u +iv if u + v = 𝑥 2 +𝑦 2 and f(1)=1.
𝑖+𝑧
11. Show that the mapping w = 𝑖−𝑧 , the image of the circle x2+y2 < 1, is the entire half of the
w-plane to the right of the imaginary axis.
12. Show that an analytic function with constant modulus is also constant.
13. Find analytic function f(z)=u(r,𝜃)+iv(r,𝜃) such that v(r,𝜃)=r2cos 2𝜃 −rcos 𝜃+2.
7
UNIT – IV(COMPLEX INTEGRATION)
PART -A
1+𝑖
1. Evaluate ∫0 (𝑥 − 𝑦 + 𝑖𝑥 2 )𝑑𝑧 along the line from z = 0 to 1+i.
2. Evaluate ∫𝑐sin 𝑧 𝑑𝑧 along the line z=0 to z=i.
3. Prove that ∫𝑐(𝑧 − 𝑎)𝑛 𝑑𝑧=0, [n=-1] where c is the circle. |Z-a|=r
2+𝑖
𝑥
4. Evaluate ∫0 (𝑍)2 𝑑𝑧 along the line y= 2
5. State Cauchy’s integral theorem.
𝑑𝑧
6. Evaluate ∫𝑐 2𝑧−3 where c is the circle |Z|=1.
𝑑𝑧
7. Evaluate ∫𝑐 𝑧𝑒 𝑧 where c is the circle |Z|=1.
1
8. Evaluate ∫𝑐𝑒 𝑧 where c is the circle |Z|=1.
9. Define Taylor’s series.
10. Define Laurent’s series
11. Define Singularity.
1
12. Find the residue of f(z)= 𝑧 2 𝑒 𝑧
𝑧+1
13. Find the residue of f(z) = 𝑧 2 (𝑧−2) at each of the poles.
14. State Cauchy’s Residue theorem.
15. State Jordan’s Lemma
PART _ B
1. Evaluate ∫𝑐𝑧 2 𝑑𝑧 where the ends of c are A(1,1) and B(2,4) given that
(i)C is a curve y=𝑥 2
(ii)C is the line y=3x-2
8
1
2. Evaluate , using cauchy’s integral formula 2𝜋𝑖 ∫𝑐
𝑧 2 +5
𝑧−3
𝑑𝑧 on the circles (i) |z|=4 and |z|=1
3. Show that ∫𝑐(𝑍 + 1) 𝑑𝑧 = 0 where C is the boundary of the square whose vertices are at
the point Z=0,Z=1,Z=1+I,Z=i.
(𝑧+4)𝑑𝑧
4. Using Cauchy’s integral formula find the value of ∫𝑐 𝑧 2 +2𝑧+5 where c is the circle |z+1i|=2
𝑒 2𝑧
5. Evaluate ∫𝑐 (𝑧−1)(𝑧−2) dz where c is the circle |z|=3
6. Evaluate ∫𝑐
𝑠𝑖𝑛𝜋 𝑧 2 +cos 𝜋𝑧 2
(𝑧−1)(𝑧−2)
dz where c is the circle |z|=3
3𝑧 2 +𝑧
7. Evaluate ∫𝑐 (𝑧 2 −1) dz where c is the circle |z-1|=1
(𝑧+1)𝑑𝑧
8. Evaluate ∫𝑐 𝑧 2 +2𝑧+4 where c is the circle |Z+1+i|=2
9. Evaluate ∫𝑐
10. Evaluate∫𝑐
𝑠𝑖𝑛6 𝑧 𝑑𝑧
𝜋
(𝑧− )3
where c is the circle |Z|=1
6
𝑡𝑎𝑛𝑧/2 𝑑𝑧
(𝑧−𝑎)2
,-2<a<2 where ‘C is the boundary of the square whose sides are x=
±2 and y=±2.
1+𝑧
11. Evaluate ∫𝑐 (𝑧 3 −2𝑧 2 ) dz where c is the unit circle |z|=1
12. Expand cos z as a Taylors series about the points (i)Z=0 (ii) z= π/4
𝑧 2 −1
13. Expand f(z) = (𝑧+3)(𝑧+2) in a Laurent’s series if (i) |z|>3 (ii) |z|<3
1
14. Expand 𝑧 2 −3𝑧+2 when 1<|z|<2 by Lauren’s Series.
(𝑧−2)(𝑧+2)
15. Obtain the Laurent’s expansion for (𝑧+1)(𝑧+4) which are valid (i) 1< |z|<4 (ii) |z|>4
𝑧
16. If 0<|z-1|<2, then express f(z)= (𝑧−1)(𝑧−3) in a series of positive and negative powers of z1.
𝑧2
17. Find the residue of f(z) = (𝑧−1)2 (𝑧+2) at each of the poles.
9
1
18. Find the residue of f(z) = (𝑧 2 +1)2 about each singularity.
(2𝑧−1)𝑑𝑧
19. Evaluate ∫𝑐 𝑧(𝑧+1)(𝑧−3) where c is the circle |z|=2
(𝑧 2 −2𝑧)𝑑𝑧
20. Evaluate ∫𝑐 (𝑧+1)2 (𝑧 2 +3) where c is the circle |z|=3 using residue theorem
21. Evaluate ∫𝑐
𝑧 sec 𝑧 𝑑𝑧
2𝜋
22. Evaluate ∫0
𝑑𝜃
13+5 𝑠𝑖𝑛𝜃
2𝜋
23. Show that ∫0
2𝜋
24. ∫0
𝑑𝜃
𝑎+𝑏 𝑐𝑜𝑠𝜃
𝑠𝑖𝑛2 𝜃
2𝜋
𝑑𝜃 =
𝑎+𝑏𝑐𝑜𝑠𝜃
2𝜋
25. Evaluate ∫0
𝑏2
∞
∞
29. Evaluate ∫−∞
𝑑𝑥
,0 < 𝑎 < 1
=
𝜋
4
𝑥2
(𝑥 2 +𝑎2 )(𝑥 2 +𝑏 2 )
𝑥 2 −𝑥+2
𝑥 4 +10𝑥 2 +9
∞ cos 𝑎𝑥
(𝑥 2 +1)
, a>b>0
𝑑𝜃
(𝑥 2 +1)2
∞
28. Evaluate ∫−∞
2𝜋
√𝑎2 −𝑏 2
[ 𝑎 − √𝑎2 − 𝑏 2 ] where 0<b< a
𝑑𝜃
5+4 𝑐𝑜𝑠𝜃
27. Prove that ∫0
30. Evaluate ∫0
=
1−2𝑎𝑠𝑖𝑛𝜃+𝑎2
𝜋 1+2 𝑐𝑜𝑠𝜃
26. Evaluate ∫0
where c is the ellipse 4x2+9y2 = 9.
(1−𝑧 2 )
𝑑𝑥 , 𝑎 > 0, 𝑏 > 0
𝑑𝑥 .
𝑑𝑥, 𝑎 > 0
10
UNIT-V (LAPLACE TRANSFORM)
PART-A
1. Define Laplace transforms
2. Find the Laplace transform of
1 cos t
t
k
3. Find the inverse Laplace transform of cot 1  
s


4. Find L1 cos 1 ( s)
5. Find the Laplace transform of unit step function.
6. State the conditions under which Laplace transform of f(t) exists.
7. State the first shifting theorem on Laplace transforms.
8. State the second shifting theorem on Laplace transforms.
9. Verify initial value theorem for f (t )  1  e-t (sin t  cos t ) .
10. Find the Laplace transform of t cos at
11. Find the Laplace transform of t sin 2t
1


12. Find L1  2

 s  4s  4 

13. Find L e3t sin t cos t

e  as
s
e2 s
15. Find inverse Laplace transform of
s3
1
16. If L f (t ) 
,find Lt f (t ) and Lt f (t )
2
t 0
t 
s( s  a 2 )
14. Find inverse Laplace transform of
17. Verify the finial value theorem for f (t )  3e-t

0,
18. Find the Laplace Transform of f (t )  
cos t - 2 
 
3 
19. Prove that Lsin at  


21. Find Lcos 3t 
22. Find Lsin t cos t 
a2
s2  a2
20. Find L 3e5t  5 cos t
3
2
3
11
2
3
2
t
3
t
 N

23. Find L  a n e-bt cosnt 
 n0

 7 t  12 
24. Find L e .t 


e t ,
25. Find the Laplace Transform of f (t )  
0
0  t 1
t 1
26. Define change of scale property
 sin 2 t 

27. Find L
 t 
1  cos t 
28. Find L



t
29. Find Lt e cosht
-t
t

30. Find the laplace transform of  te t sin tdt 
0




31. Define convolution theorem.
32. Define convolution of two functions.
33. Define initial value theorem.
34. Define finial value theorem

1  s 


35. Find L1 log 2
  s
PART-B
1. Using convolution theorem find the inverse Laplace transform of 1

s
2. Find L 1 
using convolution theorem.
( s 2  1)( s  1)
3. Find
 2
2 2
 (s  a ) 
4. Using Convolution
5. Using Convolution
using
Convolution
theorem


1
L 1  2

theorem,
 4)  1 
 s(s find

1
L 

2
 1)(s Laplace
 1) 
theorem, find the (s
inverse
8.
s2
( s  a )( s 2  b 2 )
2
6. Find the Laplace transform
7.
transform of
2
t,
0ta
with f (t  2a)  f (t )
f (t )  
2a  t
a  t  2a

Find the Laplace transform
t,
0

t

1

and f (t  2)  f (t ) for t  0
f (t )  
1 t  2
0
Find the Laplace transform of square
wave function defined by 1,
f (t )  
with period 2a
0
0ta
a  t  2a
9. Find the Laplace transform of the following triangular wave function given by
t,
f (t )  
2 - t,

0 t
  t  2 and f (t  2 )  f (t )
12
10. Find the Laplace transform of the Half wave rectifier function

sint ,
f (t )  
0,

0t


 2 
with f  t 
  f (t )

2
 

t 


11. Find the Laplace transform of square wave function given by
a

0t
E,
2
f (t )  
a
 E
t  a
2

12. Verify the initial and finial value theorem for the function
f (t )  1  e-t (sin t  cost )
where f (t  a)  f (t )
13. Verify the initial and finial value theorem for the function
f (t )  1 - e-at
14. Solve the differential equation 2
d y
Laplace transform method
dt 2
 y  sin 2t ; y (0)  0 and y'(0)  0
15. Solve the differential equationd 2 y
using Laplace transform method
dt 2
3
by using
by
dy
 2 y  e t ; with y (0)  1 and y'(0)  0
dt
16. using Laplace transform method solve the differential equation
y"3 y'4 y  2et ; with y(0)  1 and y'(0)  1
17. Solve d 2 y
2
4
transform.
dt
using Laplace
dy
dy
 4 y  sin t; if
 0 and y  2 when t  0
dt
dt
18. Solve the differential equation
y"3 y'2 y  4e 2t ; y(0)  3 and y'(0)  5
Laplace transform method.
19. Solve d 2 x
20.
dx
dx
 2 x  2; given x  0
 5 for t  0
dt
Solve dt
the differential
equation dt
2
3
21. Solve d 2 y
22.
23.
dy
 5 y  e t sin t; y(0)  0 , y'(0)  1
dt transform of at
Find the
dt Laplace
e  e bt
t
Evaluate
using Laplace
transform.
2
25.
26.
27.
28.
2

- 2t
2 t dt
 t e scos
 a 2 
2
2 
1
Solve the sequation
 s  b dy
 4 y  5 y dt  e - t when y(0)  0
Find the Laplace transform
of -2t
dt
0
t e cos 3t
Find
cos
at

cos
bt


L

t transform

Find the Laplace
of  4t t
e
t sin 3t dt
24. Find
using Laplace transform.
 
y"9 y  cos 2t ; y (0)  1 and y   -1
2
transform method.
1 
L1 0 ln 


0
13
using
using Laplace
using Laplace transform.
SNS COLLEGE OF ENGINEERING, CBE – 107
INTERNAL ASSESSMENT – I
(COMMON FOR ALL BRANCHES)
MATHEMATICS – II
PART – A
1. Define linear differential equations
2. Solve (4D2-4D +1)y = 4
3. Solve (D2-3D -4) y = e 3x +e-x
4. Transform the equation (2x+3)2
𝑑2 𝑦
𝑑𝑥 2
𝑑𝑦
- 2(2x+3) 𝑑𝑥 - 12y =6x into a linear differential
Equation with constant co-efficient
5. Find the particular integral of (D2_4)y = cosh2x
6. Define gradient of the scalar function𝜑.
14
7. Find the value of a , b , c so that the vector ⃗⃗⃗⃗
𝐹 = (𝑥 + 2𝑦 + 𝑎𝑧) 𝑖⃗⃗ + (𝑏𝑥 − 3𝑦 − 𝑧)𝑗⃗⃗ +
⃗⃗⃗⃗ is irrotational.
(4𝑥 + 𝑐𝑦 + 2𝑧)𝑘
8. If 𝜑 = log (𝑥 2 + 𝑦 2 + 𝑧 2 ) find∇𝜑.
9. Find the unit vector normal to the surface 𝑥 2 − 𝑦 2 + 𝑧 = 2 at the point (1 , -1 ,2)
10. Prove that curl (grad𝜑) = 0.
PART – B
𝑑𝑥
11. (a)(i) Solve the system of equations
𝑑𝑡
𝑑𝑦
+2x+3y =2e2t ; 𝑑𝑡 +3x+2y =0
(ii) (D2+9)y = 11 cos3x
(10)
(6)
(OR)
𝑑2 𝑦
𝑑𝑦
(b)(i) Solve ((3x+2)2𝑑𝑥 2 +3(3x+2) 𝑑𝑥 -36)y =3x2+4x+1
(10)
(ii) Solve (D2-3D -4) y = e 3x +e-x
12. (a)(i)Solve (D2+3D+2)y = sin3x cos2x
(ii)Solve(x2D2+ xD+1) y= logx sin (logx)
(OR)
2
(b)(i) Solve (4D -4D +1) y = 4
(ii) Solve (D2+1) y = sin2x
13. (a) Solve the simultaneous differential equations
(6)
(8)
(8)
𝑑𝑥
𝑑𝑡
𝑑𝑦
+2y=sin2t ; 𝑑𝑡 -2x =cos2t
(16)
(OR)
(b) Verify Green’s theorem for
 ( x
2

 y 2 )dx  2 xydy , where C is the boundary of
C
the rectangle in the XOY – plane bounded by the lines x = 0, x = a, y = 0 and y = a.
⃗⃗ prove that (i) ∇𝑟 = 𝑟⃗⃗⃗⃗ , (ii) ∇𝑟 𝑛 = 𝑛𝑟 𝑛−2 ⃗⃗⃗
14. (a)(i) If 𝑟⃗ = 𝑥𝑖⃗ + 𝑦𝑗⃗ + 𝑧𝑘
𝑟 ,where 𝑟 = |𝑟⃗⃗⃗⃗⃗|.
𝑟
⃗⃗⃗⃗ × ⃗⃗⃗⃗
⃗⃗⃗⃗)– 𝐹
⃗⃗⃗⃗ ∙ (∇ × ⃗⃗⃗⃗
(ii) Prove that ∇ ∙ (𝐹
𝐺 ) = ⃗⃗⃗⃗
𝐺 ∙ (∇ × 𝐹
𝐺)
(OR)
⃗⃗⃗⃗
⃗⃗⃗⃗ = (𝑥 2 − 𝑦𝑧) 𝑖⃗⃗ + (𝑦 2 − 𝑧𝑥)𝑗⃗⃗ + (𝑧 2 − 𝑥𝑦)𝑘
(b) Verify the Divergence theorem for 𝐹
taken over the rectangular parallelepiped 0 ≤ x ≤ a ,0 ≤ y ≤ b , 0 ≤ z ≤ c .
⃗⃗⃗⃗ = (𝑥 2 − 𝑦 2 ) 𝑖⃗ + 2𝑥𝑦⃗⃗𝑗 in the
15. (a) Verify Stoke’s theorem for a vector defined by 𝐹
rectangular region in the XOY plane bounded by the lines x = 0, x = a, y = 0 and y = b.
(OR)
⃗⃗⃗⃗ , where S is
(b) Verify Stoke’s theorem for a vector defined by ⃗⃗⃗⃗
𝐹 = y 𝑖⃗ + 𝑧⃗⃗𝑗 + 𝑥𝑘
the upper half of the surface of the sphere x2 + y2 + z2 = 1 and C is its boundary.
15
16