Ming Gong

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Digital SAT Math Test Bank

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Algebra

  1. If the system of linear equations $3x - 5y = 9$ and $kx + 15y = -27$ has infinitely many solutions, where $k$ is a constant, what is the value of $k$?
    • A) -9
    • B) -3
    • C) 3
    • D) 9
  2. A line in the $xy$-plane passes through the points $(2, -5)$ and $(-4, 7)$. If another line that is perpendicular to the first line passes through $(3, 1)$, which of the following is the equation of the second line?
    • A) $y = -2x + 7$
    • B) $y = \frac{1}{2}x - \frac{1}{2}$
    • C) $y = 2x - 5$
    • D) $y = -\frac{1}{2}x + \frac{5}{2}$
  3. The total cost $C$, in dollars, to produce $x$ units of a certain product is given by the function $C(x) = mx + b$, where $m$ and $b$ are constants. If it costs 2,450 to produce 100 units and 4,250 to produce 250 units, what is the cost, in dollars, to produce 400 units?
    • A) $5,650$
    • B) $6,050$
    • C) $6,450$
    • D) $6,850$
  4. For what value of $p$ does the system of inequalities $y > 2x + 5$ and $y < 2x + p$ have no solutions?
    • A) $p \le 5$
    • B) $p > 5$
    • C) $p = 0$
    • D) For all real values of $p$
  5. An absolute value equation is given by $|2x - 7| + 4 = 3$. How many distinct real solutions does this equation have?
    • A) Zero
    • B) One
    • C) Two
    • D) Infinitely many
  6. If $\frac{2}{3}x - \frac{1}{4}y = 5$ and $\frac{1}{2}x + \frac{3}{8}y = 11$, what is the value of $x + y$?
    • A) 12
    • B) 16
    • C) 20
    • D) 24
  7. If a line passes through the origin and has a slope of $\frac{3}{5}$, which of the following points must lie on the line?
    • A) $(5, 3)$
    • B) $(3, 5)$
    • C) $(0, 3)$
    • D) $(5, 0)$
  8. A rental car company charges a flat daily fee plus a fixed charge per mile driven. If a customer is charged 65 for driving 50 miles in one day and $$95$ for driving 110 miles in one day, what is the flat daily fee, in dollars?
    • A) 0.50
    • B) 35.00
    • C) 40.00
    • D) 45.00
  9. Which of the following lines is parallel to the line with the equation $4x - 6y = 15$?
    • A) $y = -\frac{2}{3}x + 4$
    • B) $y = \frac{2}{3}x - 7$
    • C) $y = -\frac{3}{2}x + 1$
    • D) $y = \frac{3}{2}x + 2$
  10. If $3x + 2y = 18$ and $x - y = 1$, what is the value of $2x + 3y$?
    • A) 17
    • B) 19
    • C) 21
    • D) 23
  11. For the function $f(x) = 4x - 7$, if $f(k) = 13$, what is the value of $k$?
    • A) 3
    • B) 4
    • C) 5
    • D) 6
  12. If a system of equations consists of the lines $y = ax + 3$ and $y = 5x - 2$, for which value of $a$ will the system have no solutions?
    • A) -5
    • B) $-\frac{2}{3}$
    • C) 3
    • D) 5
  13. If $5(2x - 3) - 3(x + 4) = 7$, what is the value of $x$?
    • A) 4
    • B) 5
    • C) 6
    • D) 7

Advanced Math

  1. The function $f$ is defined by $f(x) = 3x^2 - kx + 12$. If the graph of $y = f(x)$ in the $xy$-plane is tangent to the $x$-axis, and $k > 0$, what is the value of $k$?
    • A) $6$
    • B) $12$
    • C) $12\sqrt{2}$
    • D) $144$
  2. If $x > 0$ and $x^2 - 5x - 6 = 0$, what is the value of $x + 3$?
    • A) 4
    • B) 5
    • C) 8
    • D) 9
  3. The graph of the function $g(x) = a^x + b$, where $a$ and $b$ are constants, passes through the points $(0, 4)$ and $(2, 12)$. If $a > 0$, what is the value of $g(3)$?
    • A) 19
    • B) 24
    • C) 27
    • D) 30
  4. Which of the following expression is equivalent to $\frac{2x^2 - 5x - 3}{x - 3}$ for all $x \neq 3$?
    • A) $2x - 1$
    • B) $2x + 1$
    • C) $2x - 3$
    • D) $2x + 3$
  5. If $\sqrt{2x + 7} - x = 2$, what is the solution set containing all real values of $x$?
    • A) ${-1}$
    • B) ${3}$
    • C) ${-1, 3}$
    • D) ${}$
  6. The function $f(x) = 2(x - 4)^2 + 3$ defines a parabola in the $xy$-plane. If the function $g(x)$ is defined by $g(x) = f(x + 2) - 5$, what are the coordinates of the vertex of the parabola defined by $g(x)$?
    • A) $(6, -2)$
    • B) $(2, -2)$
    • C) $(2, 8)$
    • D) $(6, 8)$
  7. If $x^2 + y^2 = 25$ and $xy = 12$, what is the positive value of $x + y$?
    • A) 5
    • B) 7
    • C) 37
    • D) 49
  8. The expression $\frac{x^{-2}y^3}{x^4y^{-1}}$ is equivalent to which of the following expressions for all non-zero values of $x$ and $y$?
    • A) $\frac{y^2}{x^2}$
    • B) $\frac{y^4}{x^6}$
    • C) $\frac{y^2}{x^6}$
    • D) $x^2y^4$
  9. If $p(x)$ is a polynomial function where $p(3) = 0$, which of the following expressions must be a factor of $p(x)$?
    • A) $x + 3$
    • B) $x - 3$
    • C) $x^2 - 9$
    • D) $3x$
  10. What is the sum of all solutions to the quadratic equation $2x^2 - 8x + 3 = 0$?
    • A) -4
    • B) 2
    • C) 4
    • D) 8
  11. If $3^{2x-1} = 27^{x+2}$, what is the value of $x$?
    • A) -7
    • B) -5
    • C) -3
    • D) 5
  12. The population of a bacteria culture doubles every 4 hours. If the initial population is 500, which of the following functions models the population $P(t)$ after $t$ hours?
    • A) $P(t) = 500(2)^{4t}$
    • B) $P(t) = 500(2)^{\frac{t}{4}}$
    • C) $P(t) = 500(4)^{2t}$
    • D) $P(t) = 500(4)^{\frac{t}{2}}$
  13. If $\frac{1}{x} + \frac{1}{y} = \frac{1}{3}$ and $x + y = 12$, what is the value of the product $xy$?
    • A) 4
    • B) 9
    • C) 36
    • D) 48
  14. A quadratic function can be written in vertex form as $f(x) = -(x - h)^2 + 4$. If the function has an $x$-intercept at $(1, 0)$ and $h > 0$, what is the value of $h$?
    • A) 1
    • B) 2
    • C) 3
    • D) 4
  15. If $i = \sqrt{-1}$, what is the value of the complex expression $(4 - 3i)(2 + i)$?
    • A) $5 - 2i$
    • B) $11 - 2i$
    • C) $11 + 2i$
    • D) $5 + 2i$

Problem Solving and Data Analysis

  1. A sample of 200 items from a factory production line was tested, and 4 items were found to be defective. If the factory produces 15,000 items daily, what is the best estimate for the total number of defective items produced per day?
    • A) 300
    • B) 400
    • C) 600
    • D) 1,200
  2. The median of a set of 7 distinct integers is 14. If the 3 largest numbers are each increased by 5, what happens to the median of the new set?
    • A) It decreases by 5.
    • B) It remains 14.
    • C) It increases by 5.
    • D) It cannot be determined.
  3. A box contains 6 red marbles, 4 blue marbles, and 10 green marbles. If two marbles are drawn at random without replacement, what is the probability that the first marble is blue and the second marble is green?
    • A) $\frac{2}{19}$
    • B) $\frac{1}{10}$
    • C) $\frac{4}{19}$
    • D) $\frac{1}{5}$
  4. The price of a stock decreased by $20\%$ on Monday and then increased by $30\%$ on Tuesday. What was the net percentage change in the stock price over these two days?
    • A) $4\%$ decrease
    • B) $4\%$ increase
    • C) $10\%$ increase
    • D) $14\%$ increase
  5. A researcher conducted a survey to estimate the proportion of city residents who support a new park proposal. Based on a random sample of 400 residents, the support rate was found to be $64\%$, with a margin of error of $4\%$. Which of the following statements is best supported by this finding?
    • A) Exactly $64\%$ of all city residents support the proposal.
    • B) The true proportion of city residents who support the proposal is likely between $60\%$ and $68\%$.
    • C) If another 400 residents are surveyed, the support rate must be $64\%$.
    • D) The sample size is too small to make any valid conclusions.
  6. The mean score of a class of 20 students on an exam is 82. If 5 more students take the exam later and their average score is 92, what is the new mean score for the entire class of 25 students?
    • A) 84
    • B) 85
    • C) 86
    • D) 87
  7. A high school track coach records the 100-meter sprint times of 12 athletes. The times, in seconds, are: 11.2, 11.5, 11.8, 11.9, 12.1, 12.2, 12.2, 12.5, 12.8, 13.1, 13.4, 15.2. Which of the following statistical measures will change the most if the outlier of 15.2 seconds is removed from the dataset?
    • A) Median
    • B) Mode
    • C) Range
    • D) Interquartile Range
  8. A particular medicine has a half-life of 6 hours in the human body. If a patient takes a dose of 400 milligrams, how many milligrams of the medicine will remain in their body after 24 hours?
    • A) 12.5 mg
    • B) 25 mg
    • C) 50 mg
    • D) 100 mg
  9. A machine can fill 240 bottles of water in 15 minutes. At this exact rate, how many bottles can the machine fill in 2 hours?
    • A) 960
    • B) 1,440
    • C) 1,920
    • D) 2,880
  10. A survey of 120 students found that $60\%$ play sports, $40\%$ play a musical instrument, and $15\%$ do both. How many students surveyed do not participate in sports or music?
    • A) 12
    • B) 18
    • C) 24
    • D) 36
  11. The scatterplot shows a strong negative linear relationship between variables $x$ and $y$. If the line of best fit is given by $\hat{y} = -2.5x + 85$, what is the predicted value of $y$ when $x = 10$?
    • A) 55
    • B) 60
    • C) 65
    • D) 70
  12. In a heavy data center, 3 computer networks handle traffic. Network A handles $40\%$ of traffic with an error rate of $1\%$. Network B handles $35\%$ with an error rate of $2\%$. Network C handles the remaining $25\%$ with an error rate of $4\%$. What is the overall error probability for data sent through the center?
    • A) $1.85\%$
    • B) $2.10\%$
    • C) $2.33\%$
    • D) $2.50\%$

Geometry and Trigonometry

  1. In a right triangle, the tangent of one acute angle $\theta$ is $\frac{5}{12}$. What is the value of $\sin(\theta)$?
    • A) $\frac{5}{13}$
    • B) $\frac{12}{13}$
    • C) $\frac{13}{12}$
    • D) $\frac{13}{5}$
  2. A circle in the $xy$-plane has its center at $(3, -2)$ and a radius of 5. Which of the following is the standard equation of this circle?
    • A) $(x - 3)^2 + (y + 2)^2 = 5$
    • B) $(x + 3)^2 + (y - 2)^2 = 25$
    • C) $(x - 3)^2 + (y + 2)^2 = 25$
    • D) $(x + 3)^2 + (y - 2)^2 = 5$
  3. An arc of a circle has a measure of $60^\circ$. If the radius of the circle is 9 centimeters, what is the length of the arc, in centimeters?
    • A) $\frac{3\pi}{2}$
    • B) $3\pi$
    • C) $6\pi$
    • D) $18\pi$
  4. In $\triangle ABC$, the measure of $\angle B$ is $90^\circ$, $AB = 6$, and $BC = 8$. If $\triangle ABC$ is dilated by a scale factor of 3 to form $\triangle DEF$, what is the value of $\cos(\angle E)$ where angle E corresponds to angle A?
    • A) $\frac{3}{5}$
    • B) $\frac{4}{5}$
    • C) $\frac{6}{5}$
    • D) It cannot be determined.
  5. A cylinder has a base radius of 4 inches and a height of 10 inches. What is the total surface area, in square inches, of the cylinder (including both bases)?
    • A) $80\pi$
    • B) $96\pi$
    • C) $112\pi$
    • D) $160\pi$
  6. If $\sin(x^\circ) = \cos(24^\circ)$, where $0 < x < 90$, what is the value of $x$?
    • A) 24
    • B) 48
    • C) 66
    • D) 76
  7. A right triangle has a hypotenuse of length 20 and one leg of length 12. What is the area of the triangle?
    • A) 96
    • B) 120
    • C) 192
    • D) 240
  8. The equation of a circle is given by $x^2 + y^2 - 6x + 8y = 0$. What is the radius of this circle?
    • A) 3
    • B) 4
    • C) 5
    • D) 25
  9. A regular hexagon is inscribed inside a circle with a radius of 6 cm. What is the perimeter of the hexagon, in centimeters?
    • A) 18
    • B) 36
    • C) $36\sqrt{3}$
    • D) $72$
  10. If a line segment $AB$ has endpoints $A(-2, 3)$ and $B(6, 9)$, what is the length of segment $AB$?
    • A) 8
    • B) 10
    • C) 12
    • D) 14

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