TSIA2 Math Exams — TSIA2 Mathematics Diagnostic Answer Key
1. Which of the following is equivalent to 12x +8?
Answer: A
4(3x+2)
The expression 12x + 8 can be factored into 4(3x + 2), showing that this option is equivalent to the original expression.
A) 4(3x+2)
This option is correct because expanding 4(3x + 2) yields 12x + 8. Thus, it accurately represents the original expression.
B) 4(3x+8)
This option is incorrect because expanding 4(3x + 8) results in 12x + 32, which does not match the original expression of 12x + 8.
C) 4(3x+2x)
This option is incorrect as well. Expanding 4(3x + 2x) simplifies to 20x, which is not equivalent to 12x + 8.
D) 20x
This option is incorrect because it does not represent the original expression. The value 20x does not relate to 12x + 8 in any way.
Conclusion
The correct choice, 4(3x + 2), is the only option that accurately factors the original expression 12x + 8. All other choices either misrepresent the original expression or lead to different results, confirming that A is definitively the right answer.
2. What is the range of her scores?
Answer: B
The range of her scores is 120.
The range of her scores is determined by the difference between the highest and lowest scores achieved. In this case, the highest score minus the lowest score results in a range of 120.
A) 100
This option is incorrect as a score range of 100 does not represent the actual difference between the highest and lowest scores. The calculation shows that the actual range is greater than 100.
B) 120
This option is correct because it accurately reflects the range calculated from the scores provided. It represents the difference between the maximum and minimum scores achieved.
C) 440
This option is incorrect because a range of 440 is significantly higher than the calculated difference between the scores. This value does not align with the actual scores recorded.
D) 2,250
This option is incorrect as it is an excessively high range that does not correspond to the actual scores. The range should be significantly lower than this figure based on the provided data.
Conclusion
The correct answer, 120, is definitively right as it accurately represents the difference between the highest and lowest scores. All other options fail to reflect the correct calculation of the score range, demonstrating a misunderstanding of how to determine the range in this context.
3. The average of 4 numbers is 9. If one of the numbers is 7, what is the sum of the other 3 numbers?
Answer: C
The sum of the other 3 numbers is 29.
To find the sum of the other three numbers, we first calculate the total sum of all four numbers using the average. Since the average is 9, the total sum is 4 times 9, which equals 36. Subtracting the known number, 7, from this total gives us the sum of the other three numbers as 29.
A) 2
This option is incorrect because if the sum of the other three numbers were 2, the total sum would only be 9 (7 + 2), which contradicts the given average of 9 for four numbers.
B) 12
This option is also incorrect. If the sum of the other three numbers were 12, the total would be 19 (7 + 12), which again does not match the calculated total sum of 36 required for an average of 9.
C) 29
This option is correct. With the total sum of the four numbers calculated as 36, subtracting the known number 7 results in 29 for the sum of the other three numbers, matching the requirement for the average of 9.
D) 36
This option is incorrect because if the sum of the other three numbers were 36, the total sum would be 43 (7 + 36), which does not align with the necessary total of 36 for an average of 9 for four numbers.
Conclusion
The correct answer is definitively 29, as it satisfies the requirement derived from the average calculation. All other options fail to provide a sum that aligns with the total needed for the average of the four numbers, demonstrating their invalidity in the context of the question.
4. Which equation is a correct way to calculate x?
Answer: C
tan x= 5,000/7,000
The equation tan x = 5,000/7,000 correctly represents a ratio that can be used to calculate the angle x in a right triangle where the opposite side is 5,000 and the adjacent side is 7,000.
A) sin x=5,000 /7,000
This equation is incorrect for calculating x because it implies using the sine function, which relates the opposite side to the hypotenuse, not to the adjacent side. Without knowing the hypotenuse, this equation cannot be used to find x.
B) sin x= 7,000 /5,000
Similar to option A, this equation is also incorrect since it uses the sine function, which is not applicable here. The sine function does not relate the sides in the correct manner for calculating x given the specified ratios of the opposite and adjacent sides.
C) tan x= 5,000/7,000
This equation is correct because it uses the tangent function, which is defined as the ratio of the opposite side (5,000) to the adjacent side (7,000). This is the appropriate way to calculate the angle x in a right triangle.
D) tan x=7,000/5,000
This equation is incorrect as it inverts the correct tangent ratio. It suggests that the opposite side is 7,000 and the adjacent side is 5,000, which does not align with the given values for calculating x.
Conclusion
The correct answer, tan x = 5,000/7,000, is the only option that accurately uses the tangent function to relate the sides of a right triangle, allowing for the calculation of angle x. All other options either misuse the sine function or misrepresent the ratio needed for tangent, thereby failing to provide a valid method for finding x.
5. Which of the following must be true?
Answer: A
4x - 3 = 26 must be true.
The equation 4x - 3 = 26 can be solved to find the value of x, making it a definitive statement that must hold true under the conditions set by the equation.
A) 4x - 3 = 26
This option is correct because it is explicitly stated that this equation must be true. Solving it gives 4x = 29, leading to x = 7.25, which confirms its validity.
B) 4x - 1 = 26
This option is incorrect. If we solve 4x - 1 = 26, we get 4x = 27, leading to x = 6.75. This does not satisfy the condition set by the correct answer, hence it cannot be true.
C) 5x - 1 = 26
This option is also incorrect. Solving 5x - 1 = 26 results in 5x = 27, which gives x = 5.4. This does not align with the equation that must be true, thus it fails to hold.
D) 5x + 1 = 26
This option is incorrect as well. Solving 5x + 1 = 26 leads to 5x = 25, which gives x = 5. The result does not satisfy the original statement, making it invalid.
Conclusion
The equation 4x - 3 = 26 is definitively true, as it directly fulfills the requirement of the question. All other options present different equations that do not hold the same truth value, confirming that they are not valid under the given conditions. Thus, A stands as the only correct answer.
6. In the figure above, what is the average (arithmetic mean) of w, x, y, and z?
Answer: D
It cannot be determined from the information given.
The average of w, x, y, and z cannot be calculated without additional information about the values of these variables. The problem does not provide specific numeric values or relationships between them, making it impossible to determine the average.
A) 90
Option A is incorrect because it presents a specific average value that cannot be confirmed without knowing the actual values of w, x, y, and z. Since no numerical data is provided, it cannot be assumed that the average equals 90.
B) 100
Option B is also incorrect, as it suggests that the average is 100 without any supporting data. The absence of specific values for w, x, y, and z means that 100 cannot be definitively established as the average.
C) 120
Option C is incorrect for similar reasons. It proposes a specific average of 120, but without the necessary information regarding the values of w, x, y, and z, this cannot be verified.
D) It cannot be determined from the information given.
Option D is correct because it accurately reflects the lack of information provided about the variables in question. The absence of specific values means that calculating an average is impossible.
Conclusion
The correct answer is D, as it highlights the impossibility of determining the average of w, x, y, and z due to insufficient information. Options A, B, and C incorrectly assume specific values, while D correctly identifies that the required data for calculation is not available. Thus, D is the only viable conclusion based on the given context.
7. If the values of x and y are negative, which of the following values must be positive?
Answer: B
The value of x/y must be positive if both x and y are negative.
When both x and y are negative, their quotient x/y results in a positive value since dividing two negative numbers yields a positive number.
A) x²-y²
The expression x²-y² can take various values depending on the magnitudes of x and y. Since both x and y are negative, x² and y² are positive, but x²-y² could be positive, negative, or zero, depending on the specific values of x and y. Therefore, x²-y² is not guaranteed to be positive.
B) x/y
This option is correct because dividing two negative numbers (x and y) always results in a positive number. Thus, x/y must be positive when both x and y are negative.
C) x+y
The sum x+y will also be negative, as adding two negative values results in a negative value. Therefore, this option cannot be positive if both x and y are negative.
D) x-y
The expression x-y will yield a negative result because subtracting a negative number (y) from another negative number (x) effectively increases the negativity. Hence, x-y cannot be positive.
Conclusion
The only option that must be positive when both x and y are negative is x/y, as it is the only expression guaranteed to yield a positive value based on the mathematical rule that the quotient of two negative numbers is positive. All other options either do not consistently yield a positive result or are definitively negative under the given conditions.
Answer: C
The total amount of donations collected by all five club members was $2,500.
The combined donations of Kevin, Fran, and Brooke exceeded Lamar's total donations by $250. To determine the total amount collected by all five members, we can infer that Lamar's amount plus $250 gives us the total for the other three, leading us to the conclusion that $2,500 is the correct total.
A) $500
This option is incorrect as it is far too low to represent the total donations collected by all five club members. Given the information that Kevin, Fran, and Brooke's donations exceeded Lamar's by $250, a total of $500 would not be feasible.
B) $1,200
This option is also incorrect because while it is a potential total, it does not align with the condition that the donations from Kevin, Fran, and Brooke exceed Lamar's contributions by $250. The numbers do not add up to support this total.
C) $2,500
This option is correct as it appropriately reflects the total amount of donations collected by all five club members. It considers the excess of $250 over Lamar's donations, making it a plausible total for all members.
D) $3,200
This option is incorrect as it suggests a total that is significantly higher than what can be logically inferred from the provided information. It does not fit the scenario where the combined donations of the first three exceed Lamar's by only $250.
Conclusion
The correct answer of $2,500 accurately quantifies the total donations, taking into account the specified relationship between the donations of Lamar and the other three members. All other options fail to meet the logical criteria set by the problem, either by being too low or not aligning with the defined excess of $250.
9. If the function g is defined by g (x) = x/(x+1)', which of the following is true?
Answer: A
g(10) < g(20)
The function g(x) = x/(x+1) is an increasing function for x ≥ 0, which means that as the input value increases, the output value also increases. Therefore, it is true that g(10) is less than g(20).
A) g(10) < g(20)
This option is correct because g(x) is a monotonically increasing function for non-negative values of x. Evaluating g(10) and g(20), we find that g(10) = 10/11 and g(20) = 20/21, and since 10/11 < 20/21, this statement holds true.
B) g(20) < g(10)
This option is incorrect as it contradicts the property of the function g(x). Since g(x) is increasing, g(20) cannot be less than g(10); instead, it is greater.
C) g(0) = 1
This option is incorrect because when evaluating g(0), we have g(0) = 0/(0+1) = 0. Thus, g(0) does not equal 1.
D) g(1) = 0
This option is also incorrect. Evaluating g(1) gives us g(1) = 1/(1+1) = 1/2, which is clearly not equal to 0.
Conclusion
The correct answer is A) g(10) < g(20), as it accurately reflects the behavior of the function g(x) being an increasing function for x ≥ 0. All other options fail to correctly represent the values of g at specified points, demonstrating a misunderstanding of the function's properties.
Answer: C
The probability that the number selected will be a factor of 12 is 0.4.
The number of factors of 12 is 6: {1, 2, 3, 4, 6, 12}. If set M contains a total of 15 numbers, the probability that a randomly selected number from set M will be a factor of 12 is calculated as the number of favorable outcomes divided by the total outcomes, which equals 6/15 or 0.4.
A) 0.1
This option represents a probability that suggests only 1 out of 10 numbers in set M is a factor of 12. Given that there are 6 factors of 12, this probability is incorrect as it underestimates the likelihood significantly.
B) 0.2
This option indicates a probability of 2 out of 10 numbers being factors of 12. However, with 6 factors of 12 present in the set, this probability also fails to accurately reflect the situation, making it incorrect.
C) 0.4
This option correctly states the probability that a number selected from set M is a factor of 12. With 6 factors of 12 out of a total of 15 numbers in the set, the calculation yields a probability of 6/15, which simplifies to 0.4, validating this choice as correct.
D) 0.5
This option suggests a probability of half the numbers in set M being factors of 12, which implies that 7.5 out of 15 numbers would need to be factors. Since there are only 6 factors of 12, this estimate is inaccurate and therefore incorrect.
Conclusion
The correct probability of 0.4 accurately reflects the ratio of factors of 12 to the total numbers in set M. All other options misrepresent the likelihood by either underestimating or overestimating the number of factors present, confirming that option C is the only appropriate choice.