To tackle this question, first identify the two keywords in it “identical” and “roots”. As both functions 𝑓 and 𝑔 are quadratic functions and they are identical, then the coefficient of 𝑥² in 𝑓 is equivalent to the coefficient of 𝑥² in 𝑔, and the coefficient of 𝑥 in 𝑓 is equivalent to the coefficient of 𝑥 in 𝑔, and the constant in 𝑓 is equivalent to the constant in 𝑔.
In other terms, if 𝑓(𝑥) = 𝑎𝑥² + 𝑏𝑥 + 𝑐
and 𝑔(𝑥) = 𝑎'𝑥² + 𝑏'𝑥 + 𝑐' and both functions are identical, then 𝑎 = 𝑎', 𝑏 = 𝑏', and 𝑐 = 𝑐'.
Choice D is correct – In this question, since 𝑓 and 𝑔 are identical, then:
2𝑎 = 𝑎 - 1
Solving for 𝑎: 2𝑎 - 𝑎 = -1
𝑎 = -1
3- 𝑏 = 2𝑏
Solving for 𝑏: -𝑏 - 2𝑏 = -3
-3𝑏 = -3
𝑏 = 1
The next step after finding the values of a and 𝑏 is to replace them in 𝑓(x), and then find the roots of 𝑓. (roots: solutions)
𝑓(𝑥) = 2(-1)𝑥² + (3 - 1)𝑥 + 7
𝑓(𝑥) = -2𝑥² + 2𝑥 + 7
Using the quadratic formula: 𝑥 = -𝑏 ± √𝑏² - 4𝑎𝑐 / 2𝑎 = = -2 ± √2² - 4(-2)(7) / 2(-2)
The two solutions are 2.44 and -1.44

Remember that in a geometric sequence, the ratio between any two consecutive terms is a constant.
Choice B is correct – The three consecutive terms are 3𝑚, 2𝑚 + 5, and 4𝑚 - 5. Taking each two consecutive terms alone, the ratio of each is as follows:
2𝑚 + 5 / 3𝑚 and 4𝑚 - 5 / 2𝑚 + 5 . However, respecting the idea of the common ratio in a geometric sequence, we will have both ratios equal to each other:
2𝑚 + 5 / 3𝑚 = 4𝑚 - 5 / 2𝑚 + 5
Solving for 𝑚:
(2𝑚 + 5)(2𝑚 + 5) = 3𝑚(4𝑚 - 5)
4𝑚² + 20𝑚 + 25 = 12𝑚² - 15𝑚
4𝑚² + 20𝑚 - 12𝑚² + 15𝑚 + 25 = 0
-8𝑚² + 35𝑚 + 25 = 0
Using the quadratic formula: 𝑚 = -𝑏 ± √𝑏² - 4𝑎𝑐 / 2𝑎 = -35 ± √35² - 4(-8)(25) / 2(-8)
The two solutions for 𝑚 are: 5 and -  5 / 8 .
In the question, they stated that 𝑚 has a negative value, which means 𝑚  = 5 is rejected, and 𝑚  = -  5 / 8 is accepted.
To find the common ratio of this sequence, simply choose one of the ratios found before ( 2𝑚 + 5 / 3𝑚 or 4𝑚 - 5 / 2𝑚 + 5) and replace 𝑚 by - 5 / 8 :

2(- 5 / 8 ) + 5 / 3(- 5 / 8 ) = 15 / 4 / - 15 / 8 = -2

This question requires knowledge in the property of perpendicular lines in a coordinate plane: two lines are perpendicular if the product of their slopes is equal to -1.
Choice A is correct – First step to solve this question is by finding the slope of the line that passes through the given points (-4, 5) and (3, 7):
𝑚 = 𝑦₂ - 𝑦₁ / 𝑥₂ - 𝑥₁ = 7 - 5 / 3 - (-4) = 2 / 7
Applying the property of perpendicular lines in a coordinate plane, we will have:
𝑚 ∙ 𝑚' = -1
2 / 7 ∙ 𝑚' = -1
𝑚' = -1 ÷ 2 / 7 then 𝑚' = - 7 / 2

This question is related to composition of functions. If we have (𝑓 ∘ 𝑔)(𝑥), then we need to substitute 𝑔(𝑥) in 𝑓(𝑥).
Choice C is correct – In this question, 𝑔(𝑥) = 𝑥². Therefore, we need to replace 𝑥 in 𝑓(𝑥) by 𝑥².
This will make the function 𝑓 equals to √𝑥² - 1
Since 𝑓 is a square root function, then to find its domain of definition, we need to check the one and only condition regarding square roots: the radicand should always be greater or equal to 0.
Therefore, 𝑥² - 1 ≥ 0
𝑥² ≥ 1
It will result: 𝑥 ≤ -1 or 𝑥 ≥ 1.

This question will test your knowledge in right triangles especially using trigonometric ratios and Pythagorean theorem.
Choice E is correct – Since 3tanβ = 2, then tanβ =  2 / 3
However, tanβ = opposite / adjacent = ℎ / 𝑥
Comparing both fractions, we will have ℎ / 𝑥 = 2 / 3 , then ℎ = 2 and 𝑥 = 3.
Using Pythagorean theorem (𝑐² = 𝑎² + 𝑏²):
𝑡² = ℎ² + 𝑥²
𝑡² = 2² + 3²
𝑡² = 13
𝑡 = √13

To solve this question, you need to combine both equations in one system of equations. Solve for 𝑥 and 𝑦, then answer the question regarding the value of twice the quotient of 𝑥 and 𝑦.
Choice A is correct – The system of equations is:
{ 𝑥 + 4𝑦 = -10 2𝑥 = 35 + 3𝑦
Change the second equation into 2𝑥 - 3𝑦 = 35, then multiply the first equation by 2 to eliminate 𝑥 after subtracting both equations:
{ 2𝑥 + 8𝑦 = -20 2𝑥 - 3𝑦 = 35
11𝑦 = -55
𝑦 = - 55 / 11
𝑦 = = -5
Substitute 𝑦 by -5 in one of the equations to find 𝑥:
𝑥 + 4(-5) = -10
𝑥 = 10
Twice the quotient of 𝑥 and 𝑦 means 2 ( 𝑥 / 𝑦 ).
Substitute 𝑥 and 𝑦 by their values:
2( 10 / -5 ) = 2(-2) = -4

Logic and elementary number theory should be used in this question to solve it.
Choice C is correct – First of all, they stated in the question that the numbers they are looking for are 2-digit numbers, which means we have a range of set starting 10 till 99.
However, the numbers are even numbers and they do not contain a 0 digit. Therefore, if we divide our set into 9 sets (10 till 19, 20 till 29, 30 till 39, 40 till 49, 50 till 59, 60 till 69, 70 till 79, 80 till 89, and 90 till 99), we will have 4 available numbers in each set (example: 12, 14, 16, and 18).
Hence, we have 9 sets in which each has 4 available numbers. Consequently: 4 × 9 = 36

To solve this question, do not try to use the harder method which is by trying to find the dimensions of the shaded polygon, and then dividing it into different shapes. Rather than doing this, find the total area of the rectangle, then subtract the area of the two unshaded triangles.
Choice B is correct – The total area of this rectangle can be found using the formula 𝐴 = 𝐿 × 𝑤, with 𝐿 representing the length of the rectangle and 𝑤 its width.
Therefore, 𝐴 = 14 × 6 = 84𝑐𝑚².
Now, working on the triangle at the top left. Its base (b) is equivalent to (14 - 3) which is 11 𝑐𝑚, while its height (ℎ) is 6 - 1.5 = 4.5 𝑐𝑚.
The area of this triangle: 𝐴₁ = 1 / 2 𝑏ℎ = 1 / 2 (11)(4.5) = 24.75𝑐𝑚².
The triangle at the bottom right has a base (𝑏) equal to 14 - 1.5 = 12.5 𝑐𝑚, and a height (ℎ) equal to 6 𝑐𝑚 (total width of the rectangle).
Its area: 𝐴₂ = 1 / 2 𝑏ℎ = 1 / 2 (12.5)(6) = 37.5𝑐𝑚².
The total area (which we already found equivalent to 84 𝑐𝑚²) is equal to A = 𝐴₁ + 𝐴₂ + 𝐴₃ with 𝐴₃ being the area of the shaded region.
Thus, A₃ = 𝐴 - 𝐴₁ - 𝐴₂ = 84 -24.75 - 37.5 = 21.75 𝑐𝑚².

As the shape of the figure is made out of a cone and a cylinder, then the total volume of the given shape is the sum of the volume of the cone and the volume of the cylinder.
Choice A is correct – The volume of the cone is represented by 𝑉_cone = 1 / 3 π𝑅²ℎ, with 𝑅 the radius of the cone and ℎ its height.
The volume of the cylinder is represented by 𝑉_cylinder = π𝑅²ℎ, with 𝑅 the radius of the cylinder and ℎ its height (= 4 units).
Since the shape is a continuous one, then the radius of the cylinder (3 units) is equivalent to the radius of the cone.
Therefore:
𝑉_Total = 𝑉_cone + 𝑉_cylinder
39π = 1 / 3 π(3)²ℎ + π(3)²(4)
Solving for ℎ:
39π = 3πℎ + 36π
39 = 3ℎ + 36
39 - 36 = 3ℎ
3 = 3ℎ
ℎ = 1

To tackle this question in the correct way, remember that an equation containing absolute value should be separated into two parts (positive and negative components).
Choice D is correct – The equation |𝑥 - 3| = 2𝑥 can be written as:
𝑥 - 3 = 2𝑥 or 𝑥 - 3 = -2𝑥
Solving for 𝑥 in both equations:
𝑥 - 2𝑥 = 3
-𝑥 = 3
𝑥 = -3
or: 𝑥 + 2𝑥 = 3
3𝑥 = 3
𝑥 = 1
However, checking both solutions:
If 𝑥 = -3, then |-3 - 3| = 2(-6). Consequently, |-6| = -6 which is incorrect, therefore 𝑥 cannot be equal to -3.
If 𝑥 = 1, then |1 - 3| = 2(1). Consequently, |-2| = 2 which is correct, and then the only solution for 𝑥 is 1.

This question requires knowledge in probability to be able to solve it.
Choice C is correct – Let us define two events:
“𝑀: Ross passed his math assignments”, such that 𝑃(𝑀) = 0.7 as Ross passed 70% of his math assignments, and “𝐶: Ross passed his chemistry assignments”, such that 𝑃(𝐶) = 0.55 as Ross passed 55% of his chemistry assignments.
The complement (the outcomes in which the event does not occur) of event 𝑀 is 𝑀': Ross didn’t pass his math assignments.
The complement of event 𝐶 is 𝐶': Ross didn’t pass his chemistry assignments.
𝑃(𝑀') = 1 - 𝑃(𝑀) = 1 - 0.7 = 0.3
𝑃(𝐶') = 1 - 𝑃(𝐶) = 1 - 0.55 = 0.45
They stated in the question that both subjects’ assignments are independent, then both events are independent, and since we need to get the chance (probability) that on the next attempt of each subject, he will pass the math assignment but not the chemistry, then we are searching for 𝑃(𝑀 and 𝐶') = 𝑃(𝑀) × 𝑃(𝐶') = 0.7 × 0.45 = 0.315

To solve this question, you need first to create equivalent expressions in the equation that have equal bases.
Choice D is correct – In 3125 = 5^{(3𝑥 - 4)}, we have 3125 = 5⁵, which means: 5⁵ = 5^{3𝑥 - 4}.
Since the bases are the same, then their exponents are equivalent:
5 = 3𝑥 - 4
Solving for 𝑥:
5 + 4 = 3𝑥
3𝑥 = 9
𝑥 = 3
Substituting 𝑥 by 3 in 2𝑥³:
2(3)³ = 54

The oblique asymptote of a rational function will exist if the degree of the denominator is one less than the degree of the numerator. To find it out, and if the degree of the denominator is equal to 1, you need to divide the numerator by the denominator using long division, synthetic division, or by factoring if possible.
Choice B is correct – Applying long division in our case:

(2𝑥 + 1) 3 / 2 𝑥 - 11 / 4 / 3𝑥² - 4𝑥 + 1
- 3𝑥² - 3 / 2 𝑥
- - 11 / 2 𝑥 + 1
- 11 / 2 𝑥 - 11 / 4
15 / 4
The result of the division will be the equation of the oblique asymptote (without taking into consideration the remainder).
Therefore: 𝑦= 3 / 2 𝑥 - 11 / 4 equivalent to 𝑦 = 1.5𝑥 - 2.75 in decimal form.

Separate the equation into expressions in which the first contains the variable 𝑥 and the second contains the variable 𝑦, then complete the square in each.
Remember that the equation of a circle is: (𝑥 - ℎ)² + (𝑦 - 𝑘)² = 𝑟²
with (ℎ, 𝑘) the coordinates of the center of the circle, and 𝑟 its radius.
Choice C is correct – Separate/Group the common variables together:
(𝑥² - 6x) + (𝑦² - 4𝑦) = -2
To complete the square in 𝑥² - 6𝑥, we need to add 9. However, the second side of the equation will have this addition as well.
To complete the square of 𝑦² - 4𝑦, we need to add 4. Add it as well to the second side of the equation.
This will result:
(𝑥² - 6𝑥 + 9) + (𝑦² - 4𝑦 + 4) = -2 + 9 + 4
Using perfect squares:
(𝑥 - 3)² + (𝑦 - 2)² = 11
Hence, 𝑟² = 11 so 𝑟 = √11.

The area of a cube is 6𝑎², while the distance between two points in a 3D-coordinate plane is √(𝑥₂ - 𝑥₁)² + (𝑦₂- 𝑦₁)² + (𝑧₂ - 𝑧₁)² .
Choice C is correct – The distance between 𝐴 and 𝐺 is:
√(-3 + 5.84)² + (2 - 2.27)² + (0 - 0)² = 2.8528
Hence: 𝐴 = 6𝑎² = 6(2.8528)² = 48.83 square units.

To solve this question, you need to think of the basic relations between angles.
Choice E is correct - ∠𝐸𝐷𝐴 and ∠𝐹𝐷𝐵 are vertically opposite angles, which means 𝑚∠𝐸𝐷𝐴 = 𝑚∠𝐹𝐷𝐵 = 2𝑥 + 15.
∠𝐹𝐷𝐵 and ∠𝐵𝐷𝐶 are complementary angles as the figure shows.
Then:
𝑚∠𝐹𝐷𝐵 + 𝑚∠𝐵𝐷𝐶 = 90°
2𝑥 + 15 +𝑚∠𝐵𝐷𝐶 = 90
𝑚∠𝐵𝐷𝐶 = 90 - 2𝑥 -15
𝑚∠𝐵𝐷𝐶 = 75 - 2𝑥

If you have four terms with no 𝐺𝐶𝐹, then try factoring by grouping.
Choice D is correct – Grouping the expression:
(2𝑎² - 3𝑎) + (4𝑏𝑎 - 6𝑏)
Find the greatest common factor 𝐺𝐶𝐹 of each group of terms:
𝑎(2𝑎 - 3) + 2𝑏(2𝑎 - 3)
Then: (2𝑎 - 3)(𝑎 + 2𝑏)

To find the average of a set of numbers, add all numbers and divide the answer by how many numbers there are.
Choice C is correct – First, consider yourself finding the average. However, since Jimmy will equally spend the third and the fourth week on buying clothes, then variable 𝑥 represents the amount spent by Jimmy during each week.
25 + 66 + 𝑥 + 𝑥 / 4 = 40 (Remember that they gave you the average which is equivalent to 40)
Solving for 𝑥:
91 + 2𝑥 = 160
2𝑥 = 69
𝑥 = 34.5

Relate the small base of the trapezoid to its height and its given expression. Do the same thing again for the large base! Remember that the area of a trapezoid is represented by: 𝐴 = 1 / 2 ℎ(𝑏₁ + 𝑏₂), with ℎ the height of the trapezoid, 𝑏₁ its small base, and 𝑏₂ its large base.
Choice B is correct – The height is already given by ℎ = 2𝑥 + 1. Since the small base is half the height, then 𝑏₁ = ℎ / 2 = 2𝑥 + 1 / 2 = 𝑥 + 1 / 2 .
The large base is triple the small one. Hence: 𝑏₂ = 3𝑏₁ = 3(𝑥 + 1 / 2 ) = 3𝑥 + 3 / 2 .
Substituting ℎ, 𝑏₁, and 𝑏₂ by the expressions found:
𝐴 = 1 / 2 (2𝑥 + 1)(𝑥 + 1 / 2 + 3𝑥 + 3 / 2 ) = 1 / 2 (2𝑥 + 1)(4𝑥 + 2)
= 1 / 2 (2)(2𝑥 + 1)(2𝑥 + 1) =(2𝑥 + 1)²

Choice D is correct – In this question, consider another variable 𝑢 such that 𝑢 = 𝑥². We will have 𝑢² = 𝑥⁴.
Substituting 𝑥² by 𝑢 and 𝑥⁴ by 𝑢² in the given equation:

𝑢² - 3𝑢 = -2
𝑢² - 3𝑢 + 2 = 0

The two solutions for 𝑢 are:
𝑢 = 2 or 𝑢 = 1
Therefore:
𝑥² = 2 or 𝑥² = 1
𝑥 = ±√2 or 𝑥 = ±1
The product of all the solutions: √2 ∙ (-√2) ∙ 1 ∙ (-1) = 2

To solve this question, you need to remind yourself of the following:
- A function 𝑓(𝑥) is horizontally shifted if 𝑔(𝑥) = 𝑓(𝑥 ± ℎ).
If ℎ > 0, then shifted ℎ units to the left.
If ℎ < 0, then shifted ℎ units to the right.
- A function 𝑓(𝑥) is vertically shifted if 𝑔(𝑥) = 𝑓(𝑥) ± 𝑘.
If 𝑘 > 0, then shifted 𝑘 units up.
If 𝑘 < 0, then shifted 𝑘 units down.
- If 𝑔(𝑥) = -𝑓(𝑥), then it is reflected about the 𝑥-axis.
- If 𝑔(𝑥) = f(-𝑥), then it is reflected about the y-axis.
- If 𝑔(𝑥) = 𝑐 𝑓(𝑥) with 𝑐 > 1, then it is vertically stretched by a factor of 𝑐.
- If 𝑔(𝑥)) = 𝑐 𝑓(𝑥) with 0 < c < 1, then there is a vertical shrink by a factor of 1 / 𝑐 .
Choice A is correct – The subtraction of 2 means that we have a vertical shift, and since -2  < 0, then the function is vertically shifted 2 units down.
Multiplying 𝑥² by 2 and since 2 > 1, then the function is vertically stretched by a factor of 2.

The mean of a set of numbers is the average of the numbers. The median is the number in the middle of the set.
Choice D is correct – To find the mean of the set of numbers, add them and divide the answer by how many numbers there are:
44.77 + 25.67 + 20.67 + 24.67 + 67.77 + 20.17 / 6
= 33.95
To find the median of the set, arrange all the numbers first from smallest to largest:
20.17; 20.67; 24.67; 25.67; 44.77; 67.77
Then, since we have an even number of numbers (6), then the median will be between 24.67 and 25.67. Consequently, 24.67 + 25.67 / 2 = 25.17
Now adding the mean and the median:
33.95 + 25.17 = 59.12

Choice B is correct – First step is to divide each term in the expression by the value of 𝑎.

3𝑥² / 3 - 8𝑥 / 3 = 𝑥² - 8 / 3 𝑥

To complete the square of this expression, find the square of half of 𝑏:
( 8 / 3 ÷ 2)² = 16 / 9

The area of a parallelogram is 𝐴 = 𝑏ℎ.
Choice A is correct – The question stated that “the ratio of its height to base is 1:4”, then ℎ / 𝑏 = 1 / 4 , equivalent to 4ℎ = 𝑏
As well, the area of the parallelogram is 16 𝑚². Hence, 𝑏ℎ = 16.
Substituting 𝑏 by 4ℎ:
4ℎ(ℎ) = 16
4ℎ² = 16
ℎ² = 4
ℎ = 2

Use the inequalities in one triangle to solve this question.
Choice E is correct – Consider the dimension of segment 𝐴𝐵 is 𝑥. We will have:
𝑥 + 7 > 11 so 𝑥 > 4
and 𝑥 + 11 > 7 so 𝑥 > -4
and 11 + 7 > 𝑥 so 18 > 𝑥 or 𝑥 < 18
Therefore, 𝑥 is any value between 4 and 18.
All values are rejected except for 4.5.

Use the alternate exterior angles theorem to solve this question.
Choice E is correct – Using alternate exterior angles theorem, we have:
2𝑥 + 5 = 3𝑥 - 15
Solving for 𝑥:
2𝑥 - 3𝑥 = -15 - 5
-𝑥 = -20
𝑥 = 20
The angles with values 2𝑥 + 5 and 5𝑦 + 35 are supplementary angles.
Thus: 2𝑥 + 5 + 5𝑦 + 35 = 180
2(20) + 5 + 5𝑦 + 35 = 180
5𝑦 + 80 = 180
5𝑦 = 100
𝑦 = 20

The inverse function in Mathematics is a function that reverses another one.
Choice D is correct - 𝑓(𝑥) = 𝑥² - 4𝑥 + 4 can be written as:

𝑓(𝑥) = (𝑥 - 2)²

Then 𝑦 = (𝑥 - 2)²
𝑥 = (𝑦 - 2)²
Solving for 𝑦:
±√𝑥 = 𝑦 - 2
𝑦 = 2 ± √𝑥
Thus, 𝑓^-1(𝑥) = 2 ± √𝑥

Think of a way to simplify this expression using laws of exponents.
Choice B is correct - If 𝑓(𝑥) = 343(7^-2𝑥) and 7³ = 343, then:

𝑓(𝑥) = 7³(7^-2𝑥)

Using laws of exponents:

𝑓(𝑥) = 7^3-2𝑥

In an equilateral triangle, all sides are congruent. For that reason, the perimeter is triple one of the sides: 𝑃 = 3𝑠.
Choice C is correct – The perimeter of this equilateral triangle is 3√7 - 6. It means that each side has a dimension equal to 3√7 / 3 - 6 / 3 = √7 - 2
Since in an equilateral triangle, each angle is equal to 60°, then drawing the height of this triangle will divide it into two 30° -60° -90° triangles.
Working on one of the triangles, the height is ℎ and is considered as the large side of this new triangle. The small side (𝑠) of this triangle is equal to the side of the equilateral triangle divided by 2 (The height in an equilateral triangle is at the same time the median of this triangle). It is equivalent to √7 / 2  - 1. And the hypotenuse of this special right triangle is equal to the side of the equilateral triangle.
Hence, in a 30° -60° -90°, the large side is equal to the small side multiplied by √3, so: ℎ = 𝑠√3 in our case, which leads us to:

ℎ = ( √7 / 2 - 1)(√3) = √21 / 2 - √3 = 0.56

The best way to solve this question is by substituting 𝑥 in each equation by the values given in order to check whether they both could be roots of the equation or not.
Checking option A:
if 𝑥 = 2√2, then 0 = (2√2)² + (2√2)( -7 + 4√2 / 2 ) - 7√2
We will have: 0 = 8 + 8 - 7√2 - 7√2
0 = 16 - 14√2 which is incorrect.
Since 2√2 is not a root for option A, then it is useless to check 𝑥 = - 7 / 2 .
Checking option B:
if 𝑥 = 2√2, then 0 = (2√2)² + (2√2)( 7 - 4√2 / 2 ) - 7√2.
We will have: 0 = 8 - 8 + 7√2 - 7√2
0 = 0, then 2√2 is a root for the equation.
if 𝑥 = - 7 / 2 , then 0 = (- 7 / 2 )² + (- 7 / 2 )( 7 - 4√2 / 2 ) - 7√2
We will have: 0 = 49 / 4 + -49 + 28√2 / 4 - 7√2.
0 = 49 / 4 - 49 / 4 + 7√2 -7√2
0 = 0, then - 7 / 2 is a root for the equation.
Choice B is correct.

This is a probability question which is direct to the point, but only after finding how many chocolate bars of type C there are already in the bag.
Choice D is correct – To find the number of chocolate bars of type C in the bag, subtract the number of bars of type A, and the number of bars of type B, from the total number of bars in the bag:
30 - 11 - 13 = 6, then there are 6 chocolate bars of type C in the bag.
The probability of choosing the first bar of chocolate from the bars of type C is 𝑃(𝐶) = number of bars of type 𝐶 / total number of bars = 6 / 30 = 1 / 5

To find the area of the shaded region (grey part), subtract the area of the unshaded region (small white rectangle) from the total area of the whole rectangle.
Choice C is correct – To find the total area of the rectangle, multiply its length by its width:

𝐴 = 𝐿 ∙ 𝑤 = (8𝑥 + 1)(4𝑥 + 3)

Expanding it will result: 𝐴=32𝑥² + 28𝑥 + 3
To find the area of the small white rectangle, we need its length and its width.
To find its width, we need to subtract (𝑥 + 𝑥) = 2𝑥 from the width of the larger rectangle: (4𝑥 + 3) - 2𝑥 = 2𝑥 + 3. While 2𝑥 should be subtracted from the length of the larger rectangle in order to find the length of the small white rectangle: (8𝑥 + 1) - 2𝑥 = 6𝑥 + 1
Now its area is: 𝐴_𝑢𝑛𝑠 = (6𝑥 + 1)(2𝑥 + 3) =12𝑥² + 20𝑥 + 3
Subtracting both areas as mentioned before to find the area of the shaded region:

𝐴_𝑠 = 𝐴 - 𝐴_𝑢𝑛𝑠 = (32𝑥² + 28𝑥 + 3) - (12𝑥² + 20𝑥 + 3)
=32𝑥² + 28𝑥 + 3 - 12𝑥² - 20𝑥 - 3
=20𝑥² + 8𝑥

However, since the area of the shaded region is equal to 204 square units, then: 20𝑥² + 8𝑥 = 204

20𝑥² + 8𝑥 - 204 = 0

As this is a quadratic equation, the two solutions are:
𝑥 = -3.4 which is rejected, or 𝑥 = 3 which is accepted.

Interquartile range and median are parts of statistics. It is important to know that you need to arrange your data from smallest to greatest.
Choice E is correct – The range from smallest to greatest is:

103, 110, 110, 118, 135, 138.8, 140, 142, 143, 187

There are 10 data in this set. To find its median, find the average of the 5^th and 6^th numbers:

135 + 138.8 / 2 = 136.9

The interquartile range is a measure of statistical dispersion. It is equivalent to the difference between the upper quartile (75^th percentile) and the lower quartile (25^th percentile).
This data is to be divided into two as each 5 numbers will be together in a group:
1^st group: 103, 110, 110, 118, 135 – The middle number is 110, which is considered as the lower quartile.
2^nd group: 138.8, 140, 142, 143, 187 – The middle number is 142, which is considered as the upper quartile.
Therefore, the interquartile range is 𝑈𝑄 - 𝐿𝑄 = 142 - 110 = 32.
The sum of the two values found: 136.9 + 32 = 168.9

Use Thales’ property and Pythagorean theorem to find the missing lengths in each triangle in order to find the area of the larger triangle.
Choice C is correct – First, let us find the missing length (base) of the small right triangle, using Pythagorean theorem (𝑐² = 𝑎² + 𝑏²). The hypotenuse is equal to 18, and the small leg is 10.
Therefore, the base 𝑥 is: 18² = 10² + 𝑥²

𝑥² = 224
𝑥 = 4√14

Applying Thales’ property, and considering 𝑦 as the height of the larger right triangle:

𝑦 / 10 = 20 / 4√14
𝑦 = 13.363

The area of the triangle is: 𝐴 = 1 / 2 𝑏ℎ = 1 / 2 (20)(13.363) = 133.63 square units.

Solve the inequality involving absolute value, then compare the values given.
Choice E is correct – The inequality |-8 + 3𝑥| - 2 < 3 will result two inequalities not involving absolute value:
-8 + 3𝑥 - 2 < 3 and -(-8 + 3𝑥) - 2 < 3
3𝑥 < 13 and -3𝑥 < -3
𝑥 < 13 / 3 and 𝑥 > 1
Consequently, any value between 1 and 13 / 3 = 4.33 is correct. Since 19 / 20 = 0.95 < 1 and not included in the range of values that are correct, then 0.95 cannot be a solution.

Remember that sec⁡α = 1 / cos⁡α , then cos⁡α =  1 / sec⁡α .
Choice A is correct – Since sec⁡𝑅 = 1.887, then cos⁡𝑅 = 1 / 1.887 = 0.5299.
To find 𝑚∠𝑅:
𝑚∠𝑅 =cos^-1⁡0.5299 = 57.99 ≅ 58°.
In a triangle, the sum of the measures of the interior angles is 180°:

You can use synthetic division to find the remainder of the division of the polynomial by 𝑥 - 1 when a exists.
Choice D is correct – The division of the polynomial by 𝑥 - 1 using synthetic division is:

Remember that the circumference of a circle is equal to 2π𝑅 with 𝑅 being the radius of the circle. Its area is equal to π𝑅².
Choice B – First, using the circumference formula, find the radius R of the small circle:

2π𝑅 = 5π

Then: 𝑅 = 5 / 2 .
The area of this small circle will be: 𝐴 = π𝑅²=  π( 5 / 2 )² = 25π / 4 .
Adding the area of the small circle to the area of the shaded region will result the area 𝐴' of the larger circle:
𝐴' = 42.75π + 6.25π = 49π.
Hence: 49π = π𝑟²
𝑟² = 49
𝑟 = 7 units

Find the center of the circle using the equation of the circle, and remember that two lines are parallel if their slopes are equal.
Choice A is correct – The equation (𝑥 - 3√2)² + (𝑦² - 8𝑦 + 16) = 25 shows that the abscissa of the center of the circle is 3√2. Factoring 𝑦² - 8𝑦 +16 will result: (𝑦 - 4)², and then the ordinate of the center of the circle is 4.
Line √2𝑥 - 3𝑦 = 1 can be written as: 3𝑦 = √2𝑥 - 1

𝑦 = √2 / 3 𝑥 - 1 / 3

The line parallel to it will have the same slope = √2 / 3 .
Hence: 𝑦 = √2 / 3 𝑥 + 𝑏
It is passing through point (3√2; 4), then:

4 = √2 / 3 (3√2) + 𝑏

Solving for 𝑏:

𝑏 = 2

The equation of the line is: 𝑦 = √2 / 3 𝑥 + 2

Choice B is correct – Consider 𝑚∠𝐴𝐵𝐶 = 𝑦. The sum of the measures of the interior angles in a triangle is 180°. Then:

𝑥 + 50 + 3𝑥 - 5 + 𝑦 = 180
4𝑥 + 𝑦 = 135
𝑦 = -4𝑥 + 135

Given that 𝑚∠𝐴𝐵𝐷 = 2 / 3 𝑚∠𝐶𝐵𝐸, then: 2𝑥 + 15 =  2 / 3 𝑚∠𝐶𝐵𝐸

𝑚∠𝐶𝐵𝐸 = 3𝑥 + 22.5

From the figure, we have ∠𝐴𝐵𝐷, ∠𝐴𝐵𝐶 and ∠𝐶𝐵𝐸 are supplementary angles.
Hence:

2𝑥 + 15 + (-4𝑥 + 135) + 3𝑥 + 22.5 = 180
𝑥 + 172.5 = 180
𝑥 = 7.5

Choice E is correct – The expression 3√135𝑥⁵𝑦⁷ . √180𝑥³𝑦 can be simplified as:

3𝑥𝑦²3√5𝑥²𝑦 . 6𝑥√5𝑥𝑦

= 18𝑥²𝑦² (5 ^{1 / 3} 𝑥 ^{2 / 3} 𝑦 ^{1 / 3} ) . (5 ^{1 / 2} 𝑥 ^{1 / 2} 𝑦 ^{1 / 2} )

= 18𝑥²𝑦² (5 ^{5 / 6} 𝑥 ^{7 / 6} 𝑦 ^{5 / 6} )
= 18𝑥²𝑦² 6√5⁵𝑥⁷𝑦⁵
= 18𝑥²𝑦² . 𝑥 6√5⁵𝑥𝑦⁵
= 18𝑥³𝑦² . 6√5⁵𝑦⁵𝑥

In a 3D-coordinate system, the distance between two points is:

√ (𝑥₂ - 𝑥₁)² +(𝑦₂ - 𝑦₁)²+(𝑧₂ - 𝑧₁)²

Choice D is correct – The distance between 𝐴 and 𝐵 is

𝐴𝐵 = √ (2 - 3𝑚)² + (3 + 2𝑚 - 1)² + (-7 - 6)²
𝐴𝐵 = √ (2 - 3𝑚)² + (2 + 2𝑚)² + 169
𝐴𝐵 = √ 4 - 12𝑚 + 9𝑚² + 4 + 8𝑚 + 4𝑚² + 169
𝐴𝐵 = √ 13𝑚² - 4𝑚 + 177

Since 𝐴𝐵² = 221, then 221 = 13𝑚² - 4𝑚 + 177

13𝑚² - 4𝑚 - 44 = 0

The solutions of this quadratic equation are: 2 or -1.69 ≅ -1.7
Then choice D is to be chosen.

𝑓(𝑥) is an exponential function such that its general form is 𝑦 = 𝑎 × 𝑏^{𝑥 - 𝑐} + 𝑑.
Choice B is correct – In this function 𝑓(𝑥) = 2 ∙ 3^𝑥 + 2, we have:

𝑎 = 2
𝑏 = 3
𝑐 = 0
𝑑 = 2

Since 𝑎 = 2 > 0 and 𝑏 = 3 > 1, then the graph of this function is increasing.
Since 𝑑 = 2 then 𝑦 = 2 is the equation of the horizontal asymptote of this function.

To find the coordinates of the vertex of the graph of a quadratic function, find the equation of the axis of symmetry (it will represent the abscissa of the vertex), then replace 𝑥 by its value in the equation to find the ordinate of the vertex.
Choice C is correct – The equation of the axis of symmetry of 𝑓, is:

𝑥 = - 𝑏 / 2𝑎 = -(-6) / 2(3) = 1

Then: 𝑦 = 3(1)² - 6(1) + 1 = -2.
The vertex has the coordinates (1,-2). The product of the abscissa and the ordinate is:

1 × -2 = -2

Choice E is correct – Using alternate exterior angles theorem:

3𝑥 - 5 = 4𝑦 + 14

The angle with measure (4𝑦 + 14) is vertically opposite angle with the angle that is considered with the angle of measure 92.1° as interior angles on the same side of the transversal (these two angles are supplementary).

Then 4𝑦 + 14 = 180 - 92.1

4𝑦 = 73.9
𝑦 = 18.475

Therefore: 3𝑥 - 5 = 4(18.475) + 14

3𝑥 - 5 = 87.9
3𝑥 = 92.9
𝑥 = 30.96

We will have: 2𝑥 / 𝑦 = 2(30.96) / 18.475 = 3.35

Choice A is correct – Compare the real parts alone, then the imaginary parts.
We will have for the real parts: 21𝑥 + 8𝑦 = 2𝑦 - 1

21𝑥 + 6𝑦 = -1

For the imaginary parts: -1 = 7𝑥

𝑥 = - 1 / 7

Substituting 𝑥 by - 1 / 7 in 21𝑥 + 6𝑦 = -1:

21(- 1 / 7 ) + 6𝑦 = -1
-3 + 6𝑦 = -1
𝑦 = 1 / 3

Analyze each expression alone.
Choice A is: |2𝑥 - 3(𝑥 - 5)| - 1.
The absolute value is always positive, then |2𝑥 - 3(𝑥 - 5)| is positive. However, it can be between 0 and 1, and then we will have in that case, after subtracting 1, a negative value. Choice A is incorrect.
Choice B is: 𝑥² - 𝑥 - 6 / 𝑥⁴ . The denominator has an even power, and then it will be always positive. However, the quadratic expression in the numerator might be positive or negative (take 𝑥 = -1 as an example), and then the expression can be positive or negative as well. Choice B is incorrect.
Choice C is: (-2|𝑥 - 3|)² - 3. The parenthesis (-2|𝑥 - 3|)² is always positive as it is to the power of an even number. However, subtracting the answer by 3 might result a negative answer. Choice C is incorrect.
Choice D is: 𝑥² + 3 / (𝑥 - 7)² . The denominator is to the power of an even number so it is always positive. The numerator has the expression 𝑥² + 3. Nonetheless, 𝑥² is always positive, and adding 3 will make the answer to remain positive. Therefore, the answer of this expression is always positive and never negative. Choice D is correct.
It is useless to check choice E, but to explain it: (𝑥 - 1)³ is to the power of an odd number, and then it could be positive or negative according to the value you will have inside the parenthesis.

This question is to be solved using a system of equations.
Choice D is correct – Define two variables:
Let 𝑥 be the price of one pencil, and 𝑦 the price of one pen.
We will have: 7𝑥 + 11𝑦 = 11.2 and 5𝑦 = 7𝑥.
Solving for 𝑥 in the second equation:
𝑥 = 5 / 7 𝑦.
Substituting 𝑥 by 5 / 7 𝑦 in the first equation:

7( 5 / 7 𝑦 )+ 11𝑦 = 11.2
5𝑦 + 11𝑦 = 11.2
16𝑦 = 11.2
𝑦 = 0.7

The price of a pen is $0.7.
Hence, the price of 3 pens is: 3(0.7) = $2.1

Tackle this question by understanding first its main concept; having at least 3 women in the chosen group means we could have the following possibilities:
The group might have 3 women and 4 men, or 4 women and 3 men. It cannot have 5 women since the maximum number of women to choose from is 4.
Choice B is correct – In this question, we need to apply the concept of combinations which determines the number of possible arrangements taking into consideration that the order of the selection does not matter.
Therefore, for the first group (3 women chosen from 4, and 4 men chosen from 5), we have: 𝐶³₄ . 𝐶⁴₅ = 4 × 5 = 20.
For the second group (4 women chosen from 4, and 3 men chosen from 5), we have: 𝐶⁴₄ . 𝐶³₅ = 1 × 10 = 10
Adding both answers will result: 20 + 10 = 30 ways.