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Pokhara University
Level: Bachelor; Semester: xxxx; Year: xxxx
Program: BE; Full Marks: 100
Course: Calculus II; Time: 3 hrs.; Pass Marks: 45
MODEL QUESTION
Candidates are required to give their answer in their own words as far as practicable. The figures in the margin indicate full marks.
Attempt all questions.
-
- Evaluate the integral: ∫02 ∫04 cos(x2 + y2) dx dy
- Evaluate the integral: ∫01 ∫0x ∫1x+y e-z dz dy dx
- Find the volume in the first octant bounded by the coordinate planes, the cylinder x2 + y2 = 1 and the plane z + y = 3.
-
- Solve by using power series: y′′−4𝑥𝑦′+(4𝑥2−2)𝑦=0.
-
- Express: 2x2 - 4x +2 as Legendre polynomial.
- Show that: x x x J1(𝜋𝑥) = 0.
- OR
- Find the solution of Bessel’s Equation.
1
Pokhara University
Level: Bachelor Semester: xxxx Year: xxxx
Program: BE Full Marks: 100
Course: Calculus II Time: 3 hrs. Pass Marks: 45
MODEL QUESTION
Candidates are required to give their answer in their own words as far as
practicable. The figures in the margin indicate full marks.
Attempt all questions.
1.
a) Evaluate the integral:
dxdyxy
y
2
0
42
2
)cos(
.
b) Evaluate the integral:
dzdydxe
x yx z
1
0
)1(
0
)(
0
.
c) Find the volume in the first octant bounded by the co-ordinate
planes, the cylinder and the plane z+ y = 3.
5
5
5
2.
a) Solve by using power series: .
b) (i) Express: 2x2 - 4x +2 as Legendre polynomial.
(ii) Show that:
x
x
xJ sin
2
)(
2
1
.
OR
Find the solution of Bessel’s Equation.
0)(''' 222 ynxxyyx
7
4+4
3.
a) (i ) State first shifting theorem of Laplace Transform and find the Laplace
transform of
(ii) Find the inverse Laplace transform of the function
)3(
1
2
ss
s
.
b) Apply Laplace transform to solve the initial value problem
1
4+4
7
2
4.
a) A particle moves along the curve (t3+ 1, t2, 2t+5). Find the component
of the velocity and acceleration at t=1 along
kji 3
.
b) If , then find .
c) Evaluate
where C is the
boundary of the triangle with vertices .
5
5
5
5.
a) Find
,).( dsnF
s
for
,
222 kzjyixF
.20,10);3,sin,cos(
vuvvuvur
b) Evaluate
by using Stoke’s theorem, where
and
OR
State Gauss divergence theorem and use it to evaluate the surface
integral
for
and S is a cube
7
8
6.
a) Find the Fourier series of
for and deduce that
6
........................
16
1
9
1
4
1
12
.
b) Find Fourier half range cosine and sine sires of
7
8
7.
Attempt any two questions:
a) Derive one dimensional traffic flow model using conservation law.
b) Find the breaking time for ut +u ux = 0, u(x,0) =
.
2
x
e
c) Evaluate
3,1,2
2,0,1
2223 ]14926[ dzxzdyyxdxzxy
.
