Q5. Statistics (Mean & Missing Value) Answer: B (89)
The mean is calculated as the sum of all values divided by the quantity: \(\frac{78 + 85 + 92 + x + 88 + 90}{6} = 87 \implies \frac{433 + x}{6} = 87\). Multiply by 6 to isolate variables: \(433 + x = 522 \implies x = 522 - 433 = 89\).
TSI Math Diagnostic Answer Key Page 1 of 4
Section 2: Algebraic Reasoning Solutions (Q6-Q10) Page 2 of 4
Q6. Linear Equations (Rideshare Application) Answer: B (8 miles)
Create a linear equation where \(m\) represents the total miles traveled. The model is: \(\text{Total Fare} = \text{Booking Fee} + (\text{Cost per Mile} \times m)\) \(\implies 21.50 = 3.50 + 2.25m\). Subtract \(3.50\) from both sides: \(18.00 = 2.25m\). Divide by \(2.25\): \(m = \frac{18}{2.25} = 8\) miles.
Q7. Linear Inequalities (Sign Inversion Rule) Answer: A (\(x \ge -1\))
Distribute the coefficient on the left side: \(-3(2x - 5) \le 4x + 25 \implies -6x + 15 \le 4x + 25\). Add \(6x\) to both sides to gather terms: \(15 \le 10x + 25\). Subtract \(25\) from both sides: \(-10 \le 10x\). Divide by \(10\) (retains inequality direction because 10 is positive): \(-1 \le x\), which translates to \(x \ge -1\).
Q8. System of Linear Equations (Elimination Method) Answer: B (\(y = -3\))
Multiply equation (1) by \(3\) to get \(9x - 6y = 36\). Multiply equation (2) by \(2\) to get \(4x + 6y = -10\). Add the modified equations to eliminate \(y\): \((9x + 4x) + (-6y + 6y) = 36 + (-10) \implies 13x = 26 \implies x = 2\). Substitute \(x = 2\) back into (1): \(3(2) - 2y = 12 \implies 6 - 2y = 12 \implies -2y = 6 \implies y = -3\).
Q9. Quadratic Trinomial Equations (Factoring) Answer: A (\(x = -1/2\) and \(x = 3\))
Factor the quadratic expression using grouping: find two numbers that multiply to \(2 \times (-3) = -6\) and add to \(-5\). These are \(-6\) and \(1\): \(2x^2 - 6x + x - 3 = 0 \implies 2x(x - 3) + 1(x - 3) = 0 \implies (2x + 1)(x - 3) = 0\). Set each factor to zero to solve: \(2x + 1 = 0 \implies x = -\frac{1}{2}\) and \(x - 3 = 0 \implies x = 3\).
Q10. Exponential Expressions (Laws of Powers) Answer: A (\(16x^8 / y^{12}\))
Simplify inside the parenthesis first using quotient laws: \(\frac{4x^3 y^{-2}}{x^{-1} y^4} = 4 \cdot x^{3 - (-1)} \cdot y^{-2 - 4} = 4x^4 y^{-6}\). Apply the external squaring exponent to all interior elements: \(\left(4x^4 y^{-6}\right)^2 = 4^2 \cdot (x^4)^2 \cdot (y^{-6})^2 = 16 x^8 y^{-12} = \frac{16x^8}{y^{12}}\).
TSI Math Diagnostic Answer Key Page 2 of 4
Section 3: Geometric & Spatial Solutions (Q11-Q15) Page 3 of 4
Q11. Perimeter & Composite Area (Walkway) Answer: A (96 sq ft)
The inner garden area is \(12 \times 8 = 96\) sq ft. Since the walkway has a uniform width of \(2\) feet on all sides, the outer dimensions are: \(\text{Outer Length} = 12 + 2 + 2 = 16\text{ ft}\) and \(\text{Outer Width} = 8 + 2 + 2 = 12\text{ ft}\). Outer area = \(16 \times 12 = 192\) sq ft. The walkway area is: \(\text{Walkway Area} = 192 - 96 = 96\text{ sq ft}\).
Q12. Solid Geometry (Cylinder Volume) Answer: C (300\(\pi\) cubic ft)
Determine the radius from the diameter: \(r = \frac{\text{Diameter}}{2} = 5\) feet. The volume of a cylinder is: \(V = \pi r^2 h\). Substitute radius \(r = 5\) and height \(h = 12\): \(V = \pi (5)^2 (12) = \pi (25) (12) = 300\pi\text{ cubic feet}\).
Q13. Pythagorean Theorem (Ladder Wall Lean) Answer: C (12 feet)
Model the leaning ladder using the Pythagorean Theorem (\(a^2 + b^2 = c^2\)). The ladder represents the hypotenuse (\(c = 15\) ft) and the floor distance is a leg (\(a = 9\) ft). \(9^2 + b^2 = 15^2 \implies 81 + b^2 = 225 \implies b^2 = 144 \implies b = \sqrt{144} = 12\text{ feet}\). This corresponds to a standard 3-4-5 right triangle scaled by a factor of 3.
Q14. Cartesian Plane Distance Formula Answer: A (10 units)
Use the coordinate distance formula: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). Substitute points \(A(-2, 5)\) and \(B(4, -3)\): \(d = \sqrt{(4 - (-2))^2 + (-3 - 5)^2} = \sqrt{6^2 + (-8)^2} = \sqrt{36 + 64} = \sqrt{100} = 10\) units.
Q15. Coordinate Geometry (Perpendicular Lines) Answer: A (\(y = \frac{4}{3}x - 5\))
The given line has slope \(m_1 = -\frac{3}{4}\). Perpendicular lines have negative reciprocal slopes: \(m_2 = -\frac{1}{m_1} = \frac{4}{3}\). Using point-slope form with slope \(m = \frac{4}{3}\) and point \((3, -1)\): \(y - (-1) = \frac{4}{3}(x - 3) \implies y + 1 = \frac{4}{3}x - 4 \implies y = \frac{4}{3}x - 5\).
TSI Math Diagnostic Answer Key Page 3 of 4
Section 4: Probabilistic & Statistical Solutions (Q16-Q20) Page 4 of 4
Q16. Categorical Data Analysis (Two-Way Tables) Answer: B (45.0%)
Locate "12th Grade" row (total is \(100\) students). There are \(45\) students in that row who prefer football. \(\text{Percentage} = \frac{\text{12th Graders preferring Football}}{\text{Total 12th Graders}} \times 100\% = \frac{45}{100} \times 100\% = 45.0\%\).
Q17. Statistics (Conceptual Standard Deviation) Answer: C (All 5s)
Standard deviation measures dispersion from the mean. The set \(\{5, 5, 5, 5, 5, 5\}\) has identical data points, meaning there is zero dispersion. The standard deviation is exactly \(0\), the absolute minimum possible.
Q18. IQR & Box Plot Quartiles Answer: B (20)
The interquartile range (IQR) represents the range of the middle \(50\%\) of the data: \(\text{IQR} = Q_3 - Q_1 = 90 - 70 = 20\).
Q19. Combinations Formula (Sub-Committees) Answer: A (56)
Since committee order is irrelevant, calculate using the Combinations formula with \(n = 8\) and \(r = 3\): \(\binom{8}{3} = \frac{8!}{3!(8-3)!} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56\) ways.
Q20. Percent Weighting & Weighted Average Answer: B (87.4)
Multiply each component score by its corresponding decimal weight value: \(\text{Final Grade} = 84(0.60) + 90(0.20) + 95(0.20) = 50.4 + 18.0 + 19.0 = 87.4\).
TSIA2 Domain Diagnostic Mapping
Map the questions missed to target domains below. If a student misses 2 or more questions in a specific domain category, prioritize targeted practice worksheets and remedial study modules.
Quantitative Questions 1 – 5 Ratios, Percent, Rates, Prob
Algebraic Questions 6 – 10 Linear, Inequalities, Quadratic
Geometric Questions 11 – 15 Area, Volume, Distance, Lines
Statistical Questions 16 – 20 Tables, SD, Box Plots, Weights
TSI Math Diagnostic Answer Key Texas Success Initiative Prep
