1- If 12 m / 6! is an integer, what is the smallest possible value of m?
12m
/
6! =
12m
/
6 × 5 × 4 × 3 × 2 × 1 =
1
/
60 m
Since all the alternatives of m are positive, then
1
/
60 m should be a positive integer.
The smallest positive integer is 1.
Then:
1
/
60 m = 1
m = 60
2- If x = 
, which of the following is the value of x2?
3- Let ƒ(x) = -3 + 12 / x + 2 . How many points with integer coordinates lie on the graph of function ƒ ?
Since the points are with integer coordinates then 12 / x + 2 must be an integer and therefore x + 2 must be a positive or a negative divisor of 12. Divisors of 12 are ±1; ±2; ±3; ±4; ±6; ±12, so there are 12 points with integer coordinates.
4- If log2x = 2 and logx15 = y, then what is the value of x + y?
log2x = 2
x = 22
x = 4
logx15 = y
log415 = y
x + y = 4 + log415 = 5.953
5- The radiation level A in a certain laboratory where a technical problem caused an explosion rises exponentially according to the relation A = B e0.5t where t is measured in minutes. Approximately how many minutes, after the measurements started, did the radiation level reach triple its initial value?
The initial radiation level is B (at t = 0).
Radiation level reach triple its initial value means A = 3B
A = B e0.5t
3B = B e0.5t
3 = e0.5t
ln3 = lne0.5t
ln3 = 0.5t lne
t =
ln3
/
0.5
= 2.197 minutes
6- Due to some disease, the level of a particular hormone X in the body of a patient at a certain hospital fluctuates according to the model ƒ(t) = 3 sin (0.2t - 250) + 14, where ƒ(t) represents the level of the hormone in appropriate units and t represents the time in hours since 8:00 a.m. Monday morning (that is t = 0 represents 8:00 a.m. Monday morning). Approximately on which day and at what time would the hormonal level reach its maximum peak for the second time since Monday morning?
Period of the function =
2π
/
0.2
≅ 31.42
This means that it takes 31.42 hours for the hormonal level to goes from one maximum peak (y = 17) to the next maximum peak (y = 17).
Monday 8:00 a.m. + 31.42 hours ≅ Tuesday 3:30 p.m.
Therefore alternative C is the correct one.
OR:
By using a graphical calculator
The second peak will be exactly on point C where t ≅ 32.5 hours
Monday 8:00 a.m. + 32.5 hours ≅ Tuesday 4:30 p.m.
7- The graph given below is that of a function ƒ. Which of the following could be the graph of the function h defined by h(x) = ƒ(x).e-x ?
The graph below represents the function g(x) = e-x.
h(x) = ƒ(x).e-x should have a horizontal asymptote: y = 0. Hence, alternatives A and C are the two only alternatives that could be correct.
In the interval ]0,+∞[, ƒ(x) > 0 because the curve of f is above the x-axis and e-x > 0 because e-x is always positive for every x in ℝ, therefore h(x) > 0.
(i.e. function h must be above the x-axis in the interval ]0,+∞[ )
Then, alternative A is correct.
8- If the parametric equations of a certain line D are x = 4t - 1 and y = 4, which of the following lines is perpendicular to D?
The perpendicular line to D should be perpendicular to y = 4.
From the alternatives, only E is the correct one.
9- Consider the transformation in the plane that takes any point (x, y) into the point
(x + 7, y + 7) in the plane. If the transformation is applied to a parallelogram ABCD and a new parallelogram MNOP is obtained by the transformation, which of the following statements will be true?
I. MNOP has the same area as ABCD.
II. The perimeter of MNOP is 7 units more than the perimeter of ABCD.
III. The measures of the smaller angles in MNOP and ABCD are the same.
The described transformation is a vertical translation and a horizontal translation of equivalent units.
Hence, both parallelograms will have the same area (I), and the two parallelograms are congruent, which means that the angles are congruent (III).
10- A parabola in the plane passes through the points (-700,40) and (200,40). Which of the following can be the coordinates of its vertex?
The average of the two abscissas is:
-700 + 200
/
2
= -250
Thus, only alternatives A and B could be correct.
However, A cannot be correct since the ordinate in A is equal to the ordinates of the two points – the three points will not form a parabola.
Alternative B is correct.
11- Given two points A(1, -2, 1) and B(5, 6, zB), find a value of zB if the distance in space between the two points is equal to 9.
Since 2 is an alternative then B is the correct answer.
12- A large container of water has the shape of a cube topped with a pyramid as shown in the figure below (not drawn to scale). The container can be filled entirely with 100 m3 of water. The cube has a side of 3 m. What is the height of the container (from the base plane to point A)?
Volume of the cube: s3 = 33 = 27 m3
Volume of the pyramid:
L w h
/
3
=
3 × 3 × h
/
3
= 3h
27 + 3h = 100
3h = 73
h =
73
/
3
= 24.333 m
The height of the container:
hpyramid + hcube = 27.333 m
13- What is the degree measure of arccos (sin 60°)?
sin 60° =
/
2
arccos (
/
2
) = cos-1 (
/
2
) = 30°
14- For 0 < x < π / 2, if sinx = a / 3, what is secx?
/
3
/
a
sin x =
a
/
3
=
opp
/
hyp
Then we have a triangle of hypotenuse 3, and one of the legs equal to a.
Consider b is the second leg:
c2 = a2 + b2
9 = a2 + b2
b2 = 9 - a2
b = 
Then: cos x =
adj
/
hyp
=
/
3
Thus: sec x =
1
/
cos x
=
1
/
/
3
=
3
/
15- If 1 - 2 sin2 θ = - 1 / 3, what is the value of sin θ / 2 cos 2θ ( 0 ≤ θ ≤ 90°)?
1 - 2 sin2 θ = -
1
/
3
= cos 2θ
As well, 1 - 2 sin2 θ = -
1
/
3
- 2 sin2 θ = -
4
/
3
sin2 θ =
2
/
3
sin θ =
/
3
sin θ
/
2 cos 2θ
=
/
3
/
2 × -
1
/
3
= -
/
2
= -1.22
16- Which of the following data sets has the smallest standard deviation?
The standard deviation of B is the smallest and equals to 0 since all the numbers are the same (the deviation of each data point from the mean is equal to 0).
17- After developing an anti-bacterial agent against a certain bacterium called X, the company wants to test the efficiency of the agent. So, a group of scientists add the agent to a petri dish that initially contains 1,200 bacteria and record the number of bacteria remaining every minute. The results are given in the table below. The scientists use quadratic regression model for the data. According to their regression equation, when will the number of bacteria reach one-third of the actual initial value, that is one-third of 1,200?
| Time (minutes) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| Number of bacteria left | 1,200 | 1,120 | 980 | 740 | 600 | 510 | 210 | 50 |
According to the data given, the equation of the quadratic regression is:
y = - 6.488x2 - 123.036x + 1220.417
With x being the time in minutes and y the number of bacteria.
One-third of 1200 is 400:
400 = - 6.488 x2 - 123.036x + 1220.417
- 6.488x2 - 123.036x + 820.417 = 0
Two solutions:
x = 5.22 (accepted since time is a positive value)
x = -24.19 (rejected since time cannot be a negative value)
5.22 ≅ 5 minutes (5.22 is closer to 5 than to 5.5)
18- A is the set of integers x such that -3 ≤ x ≤ 20. We select one number from this set randomly. What is the probability that the square of this number is also in this set?
We have 3 negative integers with their square also in this set.
Concerning the positive integers, we only have 1, 2, 3 and 4.
The square of 0 is 0, which means it is also in this set.
So the probability is
8
/
24
=
1
/
3
More explanation:
The set is made out of 24 numbers and 8 numbers have a square which is also in this set.