Skoltech Engineering Systems Master program
Mathematics entry test – Training
Calculus, geometry, and algebra.
Academic Year 2022
1. Let ω the differential form: ω = ( x2 + 3y)dx − y3 dy. Check if ω is an exact differential form
or not on R2 .
2. Calculate the integral of ω = y2 dx + x2 dy on the line segment delimited by the points A in
(1, 0) and B in (0, 1).
√
3. Let i the imaginary number such that i2 = −1. Calculate i i.
4. Let un , such that un+1 = − 23 u2n + 52 un + 1 with u0 = 1. Check if the sequence un is periodic,
and if so give the period.
ˆ ∞
2
5. Calculate the integral
e−αx dx with α > 0 being a real number.
0
6. Consider a cube (A,B,C,D,E,F,G,H) of side length `, and such that the surface (ABCD) is
−→
−→
parallel to the surface (EFGH). Let M a point of the segment [AH] such that AM = aAH,
[
with 0 ≤ a ≤ 1 a real number. Calculate the angle BME.
7. Consider a cube (A,B,C,D,E,F,G,H) of side length `, and such that the surface (ABCD) is
−→
−→
parallel to the surface (EFGH). Let M a point of the segment [AH] such that AM = aAH,
−→ −→
with 0 ≤ a ≤ 1 a real number. Calculate the scalar product MB · ME.
8. Let x a discrete random variable that takes the following values 0, 3, 6, 9 with probabilities
1 1 1
1
6 , 3 , 3 and 6 respectively. Calculate the standard deviation.
9. Let X the discrete random variable that follows the probability mass function:
n!
p( X = k) =
p k (1 − p ) n − k ,
k!(n − k )!
where n ∈ N, k = 0, 1, 2, . . . , n is an integer, and p ∈ [0, 1] a real parameter. Calculate the
expectation value.
ˆ ∞
1
10. Calculate the integral
dx, with t > 1 and α > 1 being real numbers.
xα
t
∞
11. Calculate the sum
(−1)n−1
∑ n .
n =1
12. Consider the circles C1 of radius 8 cm and C2 of radius 3 cm. The distance between the centers of the two circles is 13 cm. The circles C1 and C2 have a common tangent. Calculate the
distance between the two points of the tangent that are common with C1 and C2 respectively.
1
13. Find the eigenvalues of the matrix A:
2 0 0
A = 1 2 1
−1 0 1
14. Find the eigenvectors of the 4 × 4 matrix A:
−2
0
0
−
2
A=
24 −12
0
0
0
0
2
0
0
0
0
2
15. Let the vector field F = y2 z~i + y3~j + xz~k expressed in the canonical orthonormal basis of R3 :
(~i;~j;~k). Let S be the surface of a cube such that −1 ≤ x ≤ 1, −1 ≤ y ≤ 1, and 0 ≤ z ≤ 2.
Calculate the flow of the vector field F through the closed surface S:
"
F · ~ndS,
S
where ~n is the unit vector normal to the surface element dS.
2
Skoltech Engineering Systems Master program
Physics entry test – training
Academic Year 2022
1. Calculate the efficiency of a heat engine that during each cycle absorbs 64 kcal of heat and
exhausts 42 kcal, operating during two cycles.
2. A radiant heating lamp has a temperature of 1000 K with emissivity factor e = 0.8. What
should be the surface area to provide 250 W of radiation heat transfer?
3. 1 kg of ice at 0◦ C and 1 kg of boiling water at 100◦ C are mixed. What is the temperature of
the mixture when all ice is liquefied?
4. The circuit shown in Fig. 4 is made of: a dc generator delivering a fixed voltage; four resistors
such that R1 and R3 on the one hand are fixed and known, while on the other hand R is
unknown and needs to be determined experimentally using R2 , which is adjustable; and a
galvanometer. Find the condition using the fact that R2 can be adjusted so that R can be
found; express R as a function of the known parameters.
5. A 5m3 container is filled with 840 kg of granite (density is 2400 kg·m3 ) and the rest of the
volume is air (density is 1.15 kg·m3 ). Find the mass of air present in the container.
6. Given that the power output of an engine is 50 kW and that its thermal efficiency is 24%,
find the engine’s fuel consumption rate knowing that the fuel’s heating value is 44000 kJ/kg.
7. Consider a car moving along a road at varying speed between two particular points A and
B: the car accelerates and then decelerates. The mean speed measured between points A
and B is 60 km·h−1 . Is there at least one moment between the points A and B when the car
reached exactly the speed of 60 km·h1 ? Prove your answer mathematically.
8. A water filtration device can filter out a fraction 1/n of impurity in the first pass of water
through the device. At the next pass a fraction 1/n of the amount removed during the
previous stage. Find the value of np such that the water can be made pure. Knowing this
value, explain what happens if n = np .
1
9. Two identical balls were charged in such a way that the charge of one of them turned out to
be n times greater in modulus than the other. The balls were brought into contact and then
the distance between them was increased twice as far as before. How many times has the
strength of their Coulomb interaction changed if their charges before contact were opposite?
10. What is the induction of the magnetic field in which a force of 50 mN acts on a conductor
with an active part of 5 cm? The current in the conductor is 25 A. The conductor is located
perpendicular to the magnetic induction.
11. Two bodies of equal mass are on a smooth horizontal surface and are connected by an
inextensible rope of negligible mass. A force F1 = 7 N is applied to the first body, and a
force F2 = 3 N to the second one. The forces are directed along one straight line in opposite
directions. Determine the tension of the rope.
12. A homogeneous cylinder, with radius R, mass m and moment of inertia I around its axis,
rolls without slipping on a horizontal tray. A translational motion is imposed on the tray,
perpendicular to the axis of the cylinder, with a given time law a(t). As a generalized
coordinate, one can choose the angle θ that specifies an arbitrary point of the cylinder along
the horizontal direction. Show that the position x of the center of the cylinder in the Galilean
frame is linked to θ by a holonomic constraint which is to be determined.
2
Matlab
Preparation test
P1. What is the output of this code?
A) (1,8)
B) (2,8)
C) (2,4)
D) (4,1)
E) (4,8)
F) (8,1)
G) (8,4)
H) (4,2)
E) 4
F) 2.8
G) 3.5
H) 9
G) 0.26
H) 9
P2. What is the output of this code?
A) 0
B) 1
C) 5
D) 3
P3. What is the output of this code?
A) 0.50
B) 1.50
C) 0.255
D) 2.4
E) 7
F) 2.8
P4. What is the output of this code?
A) 258
B) 326
C) 1357
D) 3
E) 36
F) 6
G) 1
H) 16
P5. What is the output of this code?
A) 579
B) 152
C) 115
P6. What is the output of this code?
D) 256
E) 912
F) 121
G) 300
H) 84
A.
B.
C.
D.
Correct Answers:
P1
C
P2
E
P3
G
P4
A
P5
B
P6
A
Programming problems: Python
SAMPLE TEST
General note:
The following code is executed as the first cell of the notebook in which all other problems’
codes are run.
import numpy as np
from scipy.integrate import solve_ivp
import scipy.stats as st
from scipy import signal
import matplotlib.pyplot as plt
T1. What is the output of this code?
A = np.array([
[1, 2],
[3, 4]
])
b = np.array([5, 6])
c = np.array([0, 1])
print(np.dot(A.T + c, b))
A. [23 45]
B. [17 50]
C. [34 34]
D. [28 39]
E. [29 40]
F. [22 44]
T2. What is the output of this code?
A = np.arange(9).reshape(3, -1)
B = np.eye(3)
print(np.trace(A - 2 * B))
A. -10
B. -6
C. -1
D. 0
E. 1
F. 6
G. 10
T3. What is the output of this code?
I = np.array([
[1,
[1,
[1,
[1,
[0,
[0,
[0,
[0,
1,
1,
1,
1,
0,
0,
0,
0,
1,
1,
1,
1,
0,
0,
0,
0,
1,
1,
1,
1,
0,
0,
0,
0,
0,
0,
0,
0,
1,
1,
1,
1,
0,
0,
0,
0,
1,
1,
1,
1,
0,
0,
0,
0,
1,
1,
1,
1,
0],
0],
0],
0],
1],
1],
1],
1]
])
S_x = np.outer(
np.array([1, 2, 1]).T,
np.array([-1, 0, 1])
)
print(signal.convolve2d(I, S_x, boundary='symm'))
A.
[[0
[ 0
[ 0
[ 0
[ 0
[ 0
[ 0
[ 0
[ 0
[ 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
2
2
2
1
-1
-2
-2
-2
-2
2
2
2
2
1
-1
-2
-2
-2
-2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0]
0]
0]
0]
0]
0]
0]
0]
0]
0]]
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
4
4
4
4
2
-2
-4
-4
-4
-4
4
4
4
4
2
-2
-4
-4
-4
-4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0]
0]
0]
0]
0]
0]
0]
0]
0]
0]]
B.
[[0
[ 0
[ 0
[ 0
[ 0
[ 0
[ 0
[ 0
[ 0
[ 0
C.
[[0
[ 0
[ 0
[ 0
[ 2
[ 2
[ 0
[ 0
[ 0
[ 0
0
0
0
0
2
2
0
0
0
0
0
0
0
0
2
2
0
0
0
0
0
0
0
0
2
2
0
0
0
0
0 0 0 0 0 0]
0 0 0 0 0 0]
0 0 0 0 0 0]
0 0 0 0 0 0]
1 -1 -2 -2 -2 -2]
1 -1 -2 -2 -2 -2]
0 0 0 0 0 0]
0 0 0 0 0 0]
0 0 0 0 0 0]
0 0 0 0 0 0]]
0
0
0
0
4
4
0
0
0
0
0
0
0
0
4
4
0
0
0
0
0
0
0
0
4
4
0
0
0
0
0 0 0 0 0 0]
0 0 0 0 0 0]
0 0 0 0 0 0]
0 0 0 0 0 0]
2 -2 -4 -4 -4 -4]
2 -2 -4 -4 -4 -4]
0 0 0 0 0 0]
0 0 0 0 0 0]
0 0 0 0 0 0]
0 0 0 0 0 0]]
D.
[[0
[ 0
[ 0
[ 0
[ 4
[ 4
[ 0
[ 0
[ 0
[ 0
T4. What is the output of this code?
from sympy import Symbol, Limit, sin
x = Symbol('x')
ans = Limit(np.power((1 + (1 / x)), x), x, np.inf).doit()
print(round(ans, 2))
A. -7.39
B. -3.14
C. -2.72
D. -1.0
E. 0
F. 1.0
G. 2.72
H. 3.14
I. 7.39
T5. What is the output of this code?
def is_prime(num):
for n in range(2, int(num ** 1/2) + 1):
if num % n == 0:
return False
return True
def sum_digits(num):
return (num % 10) + (num // 10)
a, b = [2**i for i in range(4, 6)]
c, d = [10*i for i in range(2, 4)]
a1 = np.arange(a, b)
a2 = np.arange(c, d)
s1 = set(filter(is_prime, a1))
s2 = set([i for i in a2 if sum_digits(i) < 2**3])
print(len(s1.union(s2)))
A. 0
B. 7
C. 9
D. 10
E. 11
F. 12
G. 13
H. 42
T6. What is the output of this code?
import pandas as pd
from sklearn.neighbors import KernelDensity
from scipy.integrate import quad
def pdf2cdf(X):
'''
Calculates cumulative distribution function (CDF) using
integration
'''
kde = KernelDensity(kernel='gaussian', bandwidth=0.75)
.fit(X[:, np.newaxis])
gmm_pdf = lambda x: np.exp(kde.score(np.array([x]).reshape(-1, 1)))
x_cdf = np.arange(-5, 20, 0.1)
y_cdf = np.array([tup[0]
for tup in [quad(gmm_pdf, a, b)
for a, b in [(a, b)
for a, b in zip(x_cdf, x_cdf[1:len(x_cdf)])]]] + [0]
).cumsum()
return x_cdf, y_cdf
N = 10000
# a guassian-mixture PDF
X = np.concatenate((
np.random.normal(10, 2, 10*N),
np.random.normal(0, 1, 2 *N)
))
s = pd.Series(X)
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(20, 6))
s.plot(kind='kde', bw_method='scott', ax=ax1)
ax1.set_title('Mixed gaussian PDF')
x, y_cdf = pdf2cdf(X)
ax2.plot(x, y_cdf, color='b')
ax2.set_title('CDF from integration over gaussian-mixture PDF')
A.
B.
C.
D.
E.
F.
Answers:
1. E
2. F
3. B
4. G
5. D
6. B
