NEPAL ENGINEERING COLLEGE
Changunarayan, Bhaktapur
Mock Test-2021
Level: Bachelor
Programme : BE
Course : Engineering Mathematics II
Semester: Spring
Year: 2021
Full Marks: 70
Pass Marks: 31.50
Time: 2 Hours
Candidates are required to give their answers in their own words as far as practicable.
Attempt ALL questions.
1. When does the shortest distance between two lines becomes zero? Find the points one in each of the
two skew lines
y −4 z−5
y −5 z−7
x −4
x −3
=
=
and
=
=
2
3
4
3
4
5
which are closest each other. Find the equation and length of the shortest distance.
[1+6+3]
OR
If the plane x + y + z + k = 0 cuts the sphere x 2 + y 2 + z 2 − 2x + 2 y + 4z + 9 = 0 in a great circle, find
the value of k. Find the equation of the sphere one of whose great circle is
x 2 + y 2 + z 2 − 2x + 4 y − 6z − 11 = 0 = 2x − 4 y + 6z + 11.
[2+8]
2. Write about Fubini’s theorem for double integral. Sketch the region of integration and evaluate the
R2 R2
integral
sin(x 2 ) d x d y. Why the reversing the order of integration is necessary for some given
0 y
double integration? Justify the answer.
3. How can you check the homogeneity of a function of x, y, z? Is the function u = csc−1
homogeneous? If not how can we use Euler’s theorem to show x
[2+6+2]
p x+ y
a
p
p
3
x+ 3 y
p
∂u
∂u 1
+y
+ tan u = 0. [2+1+7]
∂x
∂y 6
4. Write the necessary and sufficient conditions for f (x, y, z) to have extreme value? A rectangular swimming pool is to be constructed containing 20,000 liter of water. Find the dimensions of the swimming
pool so that cost of plastering of swimming is minimum.
[2+8]
5. Write about initial value problem and using Laplace transform, solve the initial value problem
00
0
0
2 y + 5 y + 2 y = e−2t , y(0) = 1, y (0) = 1
0
6. (a) Reduce the Bernoulli’s equation y + P y = Q y n in to linear form. And hence solve the equation
0
y + x y −1 − y = 0.
[3+7]
(b) Distinguish between the general solution and particular solution of a differential equation? Write
00
0
with examples. Find the general solution of the equation y − 6 y + 13 y = 4e3x . Also find its
0
Particular solution when y(0) = 2 and y (0) = 4.
[3+5+2]
The End