1- If y varies directly with z and inversely with v, then which of the following could be a correct expression of z as a function of y and v?
y =
Kz
/
v
yv = kz
z =
yv
/
k
The alternative A only contains the product of y and v.
2- If 3n × 9m × 27p = 1, which of the following is the correct expression of n in terms of m and p?
3n × 9m × 27p = 1
3n × (32)m × (33)p = 1
3n × 32m × 33p = 30
3(n+2m+3p) =30
n + 2m + 3p = 0
n = -2m -3p
3- In a certain country, the ratio of male individuals who got infected by the COVID-19 virus to the female individuals who got infected by the same virus is 6 to 4. Moreover, it was observed that only 9% of the males and 5% of the females who got infected by the virus got severe symptoms. If the number of COVID-19 cases in the country is 1200, how many females have severe symptoms?
6 + 4 = 10
1200 ×
4
/
10
= 480
480 ×
5
/
100
= 24
4- For how many strictly positive integer values of x is the following inequality satisfied?
2(3 - x) ≥ -14
2(3 - x) ≥ -14
3 - x ≥ -7
-x ≥ -10
x ≤ 10
Therefore, we have 10 positive integer values for x.
5- The product of three consecutive integers is equal to the middle integer. What is the smallest of these integers?
First integer: x
Second integer: x + 1
Third integer: x + 2
x(x + 1)(x + 2) = x + 1
(x2 + x)(x + 2) = x + 1
x3 + 2x2 + x2 + 2x = x + 1
x3 + 3x2 + x - 1 = 0
The three solutions are:
x = -1 - 
(rejected: not an integer)
x = -1 + (rejected: not an integer)
x = -1 (accepted)
6- Let ƒ be the function defined by ƒ(x) = mx3 + px2 + qx - 1, where m, p, and q are real numbers. If the graph of f passes through the point (1,7), what is the average of m, p, and q?
ƒ(x) = mx3 + px2 + qx - 1
7 = m + p + q - 1
8 = m + p + q
m + p + q
/
3
=
8
/
3
7- Circles C, P, and Q have radii 1, 2, and 3 respectively. If the three circles are tangent to each other, what is the nature of the triangle formed by joining the three centers of the circles?
The circles are tangent to each other, then the sides of the triangle are:
1 + 2 = 3
1 + 3 = 4
2 + 3 = 5
The three dimensions form a set of a Pythagorean triple, then the triangle is a right triangle so the alternatives B and C are the only alternatives that could be correct.
But since the sides of the triangle are of different lengths then the triangle can never be right isosceles, so B is the correct alternative.
8- A circle inscribed in a square has an area of 36π. What is the length of the diagonal of the square?
Area of the circle : πR2 = 36π
R2 = 36
R = 6
Diameter = 2R = 12
Circle is inscribed in a square means diameter of the circle = side of the square = 12
So the hypotenuse of the right isosceles triangle which is the diagonal = side = 12 = 16.97
9- The perimeter of a regular hexagon is 24 and the perimeter of a regular octagon is 5. What is the ratio of the length of one side of the hexagon to the length of one side of the octagon?
Sidehexagon =
24
/
6
= 4
Sideoctagon =
5
/
8
Ratio:
4
/
5
/
8
=
32
/
5
= 6.4
10- A line in the plane passes through the 3 distinct points (a,b), (c,d), and (e,ƒ). Then all of the following give a correct relation between a,b,c,d,e,and ƒ except:
Slope of the line can be one of the following:
d - b
/
c - a
=
b - d
/
a - c
=
ƒ - d
/
e - c
=
d - ƒ
/
c - e
=
ƒ - b
/
e - a
=
b - ƒ
/
a - e
Using the above:
Switching the denominator and numerator in two fractions, will result an equivalent relation – alternative A is correct.
Multiplying a fraction by the reciprocal of a second one should result 1:
d - b
/
c - a
=
d - ƒ
/
c - e
It is equivalent to :
d - b
/
d - ƒ
=
c - a
/
c - e
Applying the multiplication rule:
d - b
/
d - ƒ
×
c - e
/
c - a
= 1
In alternative B, it is written a - c while it should be c - a Thus, alternative B should be selected.
11- C is a circle tangent to the two straight lines of equations y = 2 and y = -4.
Which of the following could be the equation of the circle?
The distance between the two horizontal lines is equal to 6. Therefore, the radius of the circle is 3.
However, y = 2 - 3 = -1 represents the ordinate of the center of the circle.
The only possible equation is E since it is equivalent to:
(x - h)2 + (y - k)2 = r2
with (h,k) the coordinates of the center of the circle and r is its radius.
12- The volume of a certain pyramid of rectangular base is 12. We alter its dimensions in such a way that the volume does not change. If the height has been doubled, which of the following could be possible alterations to the length L and the width W of the base of the pyramid?
Volume of the pyramid is :V =
L w h
/
3
if
Lwh
/
3
= 12, then Lwh = 36
After the alternations:
h' = 2h
Checking the alternatives:
A: L' = 12L and w' =
w
/
6
Then L' w' h' =12L ×
w
/
6
× 2h = 4Lwh
Incorrect
B: L' = L and w' =
w
/
0.5
= 2w
Then L' w' h' = L × 2w × 2h = 4Lwh
Incorrect
C: L' = 9L and w' =
w
/
18
Then L' w' h' = 9L ×
w
/
18
× 2h = Lwh
Correct
13- A plane intersects a sphere of radius 4 cm in a circle. Which of the following cannot be the area of this circle?
The greatest intersection will form a circle in the middle of the sphere, and then the greatest dimension of the radius of the circle is 4 cm.
Therefore, the greatest area of the circle is A = πR2 = π42 = 16π.
Hence, alternative E cannot be the area of this circle.
14- Consider all spheres of nonzero volume having the point A in space as their center. How many of these spheres can have a surface area whose value is numerically equal to the volume of the sphere?
Surface area of a sphere: 4πr2
Volume of a sphere:
4
/
3
πr3
Numerically, if both are equal:
4πr2 =
4
/
3
πr3
3r2 = r3
3 = r or r = 0 (rejected)
So only when the radius is equal to 3, the numerical value of the volume is equal to the numerical value of the surface area.
Therefore, only in one sphere this will be true. (alternative B).
15- In the figure below, B is a point on the circle of center A and diameter DC such that BC = 20 and BD = 15. What is the perimeter of triangle BDC?
c2 = a2 + b2
DC2 = BC2 + BD2
DC2 = 202 + 152
DC2 = 625
DC = 25
Perimeter: BC + BD + DC = 60
OR:
tan CDB =
CB
/
BD
=
20
/
15
CDB ≅ 53.13
cos CDB̂ =
DB
/
CD
3
/
5
=
15
/
CD
CD = 25
Perimeter: BC + BD + DC = 60
16- Celine and Lia are standing on a horizontal track at points A and D respectively. Celine is 1.5 m tall and Lia is 1.2nbsp;m tall (that is AC = 1.5 m and DL = 1.2 m). Celine’s shadow occupies the length AB and Lia’s shadow occupies the length BD on the ground so the two shadows meet at their heads. The angles between the ground and the ray from the top of the head of each individual to the touching point of the shadows is shown. How far apart are Celine and Lia standing?
Celine:
tan 40° =
AC
/
AB
AB =
AC
/
tan 40°
=
1.5
/
tan 40°
= 1.788 m
Lia:
tan 30° =
DL
/
DB
DB =
DL
/
tan 30°
DB =
1.2
/
tan 30°
= 2.078 m
1.788 + 2.078 = 3.866 m
17- The triangle shown in the below figure (not drawn to scale) is right angled at G with GF = 4 and
∠GFH = 60°. I is the midpoint of segment FH and GJ is the altitude drawn from G to FH. What is the area of triangle GJI?
As triangle GFH is a 30° - 60° - 90° triangle, and GF = 4, then:
GH = 4
(side facing 60°)
and FH = 8 (hypotenuse)
Thus, FI = IH = 4 (since I is the midpoint of FH)
As well, GFJ is a 30° - 60° - 90° triangle, and GF = 4, then:
FJ = 2
GJ = 2
JI = FI - FJ = 4 - 2 = 2
Area of triangle GJI =
1
/
2
bh =
1
/
2
(2)(2)
= 2 = 3.46
18- Mr. Hernandez has 8 Christmas greeting cards and he wants to send them to 5 of his friends. How many ways can he send the greeting cards?
8P 5 = 6720
(Another way to write it:
8
/
5
P = 6720)
19- In a certain company of 400 employees, 70% wear face masks at all times. Out of those who do not wear face masks, 15% got the disease. Out of those who do wear a mask, only 5% got the disease. If an employee is chosen at random from this company, what is the probability that he or she got the disease?
0.3 × 0.15 + 0.7 × 0.05 = 0.08