Your FREE Praxis Algebra I (5162) Practice Test 2026 – 120+ Q&A

Praxis 5162 Practice Questions

1. Solve for $x$: $3x - 7 = 11$.
   • 6
   • 3
   • 4
   • 9

   Correct answer: 6

   Add 7 to both sides: $3x = 18$. Divide by 3: $x = 6$.

2. Solve for $x$: $5(x - 2) = 3x + 4$.
   • 7
   • 3
   • 5
   • 9

   Correct answer: 7

   Distribute: $5x - 10 = 3x + 4$. Subtract $3x$: $2x - 10 = 4$. Add 10: $2x = 14$, so $x = 7$.

3. Solve the inequality $2(x - 3) < 4x - 6$.
   • $x > 0$
   • $x < 0$
   • $x > 3$
   • $x \geq 3$

   Correct answer: $x > 0$

   Expand: $2x - 6 < 4x - 6$. Subtract $2x$ and add 6: $0 < 2x$, so $x > 0$.

4. Solve for $x$: $\frac{x}{4} + 3 = 9$.
   • 24
   • 18
   • 21
   • 48

   Correct answer: 24

   Subtract 3: $\frac{x}{4} = 6$. Multiply by 4: $x = 24$.

5. A formula gives $d = rt$. Rearranged to solve for $t$, which expression is correct?
   • $t = \frac{d}{r}$
   • $t = dr$
   • $t = \frac{r}{d}$
   • $t = d - r$

   Correct answer: $t = \frac{d}{r}$

   Divide both sides of $d = rt$ by $r$: $t = \frac{d}{r}$.

6. If $x$ and $y$ are inversely proportional and $y = 3$ when $x = 4$, what is $y$ when $x = 12$?
   • 1
   • 2
   • 4
   • 9

   Correct answer: 1

   Inverse variation means $xy = k$. Here $k = 4 \times 3 = 12$. When $x = 12$, $y = \frac{12}{12} = 1$.

7. If $\frac{x}{5} = \frac{y}{3}$ and $x = 10$, what is $y$?
   • 6
   • 2
   • 4
   • 8

   Correct answer: 6

   Substitute $x = 10$: $\frac{10}{5} = \frac{y}{3}$, so $2 = \frac{y}{3}$ and $y = 6$.

8. Solve for $x$: $3^{2x+1} = 81$.
   • $x = \frac{3}{2}$
   • $x = 1$
   • $x = 2$
   • $x = 3$

   Correct answer: $x = \frac{3}{2}$

   Since $81 = 3^{4}$, set $2x + 1 = 4$. Then $2x = 3$, so $x = \frac{3}{2}$.

9. Simplify $\left(2x^{3}\right)^{2} \cdot x^{4}$.
   • $4x^{10}$
   • $2x^{10}$
   • $4x^{9}$
   • $4x^{24}$

   Correct answer: $4x^{10}$

   $\left(2x^{3}\right)^{2} = 4x^{6}$. Multiply by $x^{4}$: $4x^{6+4} = 4x^{10}$.

10. Evaluate $f(-2)$ for $f(x) = 2x^{3} - x^{2} + 4x - 7$.
    • -31
    • -15
    • -17
    • 17

    Correct answer: -31

    $2(-8) - (4) + (-8) - 7 = -16 - 4 - 8 - 7 = -31$.

11. Evaluate $f(-2)$ for $f(x) = 2x^{2} - 5x + 3$.
    • 21
    • -5
    • 7
    • 17

    Correct answer: 21

    $2(4) - 5(-2) + 3 = 8 + 10 + 3 = 21$.

12. Multiply: $(x + 3)(x - 5)$.
    • $x^{2} - 2x - 15$
    • $x^{2} + 2x - 15$
    • $x^{2} - 2x + 15$
    • $x^{2} - 8x - 15$

    Correct answer: $x^{2} - 2x - 15$

    FOIL: $x^{2} - 5x + 3x - 15 = x^{2} - 2x - 15$.

13. Factor completely: $x^{2} - 9$.
    • $(x - 3)(x + 3)$
    • $(x - 3)^{2}$
    • $(x + 3)^{2}$
    • $(x - 9)(x + 1)$

    Correct answer: $(x - 3)(x + 3)$

    This is a difference of squares: $x^{2} - 9 = (x - 3)(x + 3)$.

14. Factor: $2x^{2} + 7x + 3$.
    • $(2x + 1)(x + 3)$
    • $(2x + 3)(x + 1)$
    • $(x + 1)(x + 3)$
    • $(2x - 1)(x - 3)$

    Correct answer: $(2x + 1)(x + 3)$

    $(2x + 1)(x + 3) = 2x^{2} + 6x + x + 3 = 2x^{2} + 7x + 3$.

15. What are the solutions of $x^{2} - 5x + 6 = 0$?
    • $x = 2$ and $x = 3$
    • $x = 1$ and $x = 5$
    • $x = -2$ and $x = -3$
    • $x = 2$ and $x = -3$

    Correct answer: $x = 2$ and $x = 3$

    Factor: $(x - 2)(x - 3) = 0$, so $x = 2$ or $x = 3$.

16. What are the solutions of $x^{2} - 6x + 9 = 0$?
    • $x = 3$ only
    • $x = 3$ and $x = -3$
    • $x = -3$ only
    • no real solution

    Correct answer: $x = 3$ only

    $x^{2} - 6x + 9 = (x - 3)^{2} = 0$, a perfect square with the single repeated root $x = 3$.

17. For what value of $k$ does $x^{2} + kx + 9 = 0$ have exactly one real solution (with $k > 0$)?
    • 6
    • 3
    • 9
    • 12

    Correct answer: 6

    One real solution means the discriminant is zero: $k^{2} - 4(1)(9) = 0$, so $k^{2} = 36$ and $k = 6$ (taking $k > 0$).

18. Rewrite $y = x^{2} - 4x + 1$ in vertex form.
    • $y = (x - 2)^{2} - 3$
    • $y = (x + 2)^{2} + 3$
    • $y = (x - 2)^{2} + 3$
    • $y = (x + 2)^{2} - 3$

    Correct answer: $y = (x - 2)^{2} - 3$

    Complete the square: $x^{2} - 4x = (x - 2)^{2} - 4$. So $y = (x - 2)^{2} - 4 + 1 = (x - 2)^{2} - 3$.

19. What is the product of the roots of $x^{2} - 7x + 12 = 0$?
    • 12
    • -7
    • 7
    • -12

    Correct answer: 12

    For $ax^{2} + bx + c = 0$, the product of the roots is $\frac{c}{a} = \frac{12}{1} = 12$ (the roots are 3 and 4).

20. What is the sum of the roots of $2x^{2} - 8x + 3 = 0$?
    • 4
    • -4
    • 8
    • $\frac{3}{2}$

    Correct answer: 4

    The sum of the roots is $-\frac{b}{a} = -\frac{-8}{2} = 4$.

21. What is the slope of the line $2x - 3y = 6$?
    • $\frac{2}{3}$
    • $-\frac{2}{3}$
    • $\frac{3}{2}$
    • $-\frac{3}{2}$

    Correct answer: $\frac{2}{3}$

    Solve for $y$: $-3y = -2x + 6$, so $y = \frac{2}{3}x - 2$. The slope is $\frac{2}{3}$.

22. What is the equation of the line through $(2, 3)$ and $(4, -1)$?
    • $y = -2x + 7$
    • $y = -2x + 3$
    • $y = 2x - 1$
    • $y = 2x - 5$

    Correct answer: $y = -2x + 7$

    Slope $= \frac{-1 - 3}{4 - 2} = -2$. Using $(2, 3)$: $3 = -2(2) + b$ gives $b = 7$, so $y = -2x + 7$.

23. A line has slope $-\frac{1}{2}$ and passes through $(0, 4)$. What is its $x$-intercept?
    • 8
    • 2
    • 4
    • -8

    Correct answer: 8

    The line is $y = -\frac{1}{2}x + 4$. Set $y = 0$: $0 = -\frac{1}{2}x + 4$, so $x = 8$.

24. Solve the system $2x + y = 7$ and $x - y = 2$. What is $x$?
    • 3
    • 1
    • 2
    • 5

    Correct answer: 3

    Add the equations: $3x = 9$, so $x = 3$ (and $y = 1$).

25. For what value of $k$ does the system $2x + 3y = 6$ and $kx - 5y = 10$ have no solution?
    • $-\frac{10}{3}$
    • $\frac{10}{3}$
    • $\frac{2}{5}$
    • $-\frac{2}{5}$

    Correct answer: $-\frac{10}{3}$

    A system has no solution when the lines are parallel (equal slopes, different intercepts). Slopes are $-\frac{2}{3}$ and $\frac{k}{5}$. Setting $\frac{k}{5} = -\frac{2}{3}$ gives $k = -\frac{10}{3}$.

26. Simplify $\frac{x^{2} - 16}{x^{2} - 5x + 4}$.
    • $\frac{x + 4}{x - 1}$
    • $\frac{x - 4}{x - 1}$
    • $\frac{x + 4}{x + 1}$
    • $\frac{x - 4}{x + 1}$

    Correct answer: $\frac{x + 4}{x - 1}$

    Factor: $\frac{(x - 4)(x + 4)}{(x - 1)(x - 4)}$. Cancel $(x - 4)$: $\frac{x + 4}{x - 1}$.

27. For $g(x) = \frac{2x}{x - 1}$, for what value of $x$ is $g(x)$ undefined?
    • $x = 1$
    • $x = 0$
    • $x = 2$
    • $x = -1$

    Correct answer: $x = 1$

    A rational function is undefined where its denominator is zero: $x - 1 = 0$, so $x = 1$.

28. Add: $\frac{1}{x} + \frac{2}{x^{2}}$.
    • $\frac{x + 2}{x^{2}}$
    • $\frac{3}{x^{2}}$
    • $\frac{3}{x^{3}}$
    • $\frac{x + 2}{x^{3}}$

    Correct answer: $\frac{x + 2}{x^{2}}$

    Common denominator $x^{2}$: $\frac{x}{x^{2}} + \frac{2}{x^{2}} = \frac{x + 2}{x^{2}}$.

29. Solve $|2x - 1| = 7$.
    • $x = 4$ or $x = -3$
    • $x = 4$ or $x = 3$
    • $x = -4$ or $x = 3$
    • $x = 4$ only

    Correct answer: $x = 4$ or $x = -3$

    Either $2x - 1 = 7$ (so $x = 4$) or $2x - 1 = -7$ (so $x = -3$).

30. Solve $|x + 2| < 5$.
    • $-7 < x < 3$
    • $-3 < x < 7$
    • $x < -7$ or $x > 3$
    • $-5 < x < 5$

    Correct answer: $-7 < x < 3$

    $|x + 2| < 5$ means $-5 < x + 2 < 5$. Subtract 2: $-7 < x < 3$.

31. The expression $x^{2} - 6x + 9$ is equivalent to which of the following?
    • $(x - 3)^{2}$
    • $(x + 3)^{2}$
    • $(x - 3)(x + 3)$
    • $(x - 9)(x + 1)$

    Correct answer: $(x - 3)^{2}$

    It is a perfect-square trinomial: $x^{2} - 6x + 9 = (x - 3)^{2}$.

32. Which expression equals $4x^{2} - 12x$ factored completely?
    • $4x(x - 3)$
    • $4(x^{2} - 3x)$
    • $2x(2x - 6)$
    • $4x(x - 12)$

    Correct answer: $4x(x - 3)$

    The GCF is $4x$: $4x^{2} - 12x = 4x(x - 3)$. The other forms are not fully factored or are incorrect.

33. If $a$ and $b$ are the roots of $x^{2} - 5x + 6 = 0$, what is $a^{2} + b^{2}$?
    • 13
    • 17
    • 21
    • 25

    Correct answer: 13

    $a + b = 5$ and $ab = 6$. Then $a^{2} + b^{2} = (a + b)^{2} - 2ab = 25 - 12 = 13$.

34. What is the standard-form equation of the circle with center $(3, -4)$ and radius 5?
    • $(x - 3)^{2} + (y + 4)^{2} = 25$
    • $(x + 3)^{2} + (y - 4)^{2} = 25$
    • $(x - 3)^{2} + (y - 4)^{2} = 5$
    • $(x + 3)^{2} + (y + 4)^{2} = 5$

    Correct answer: $(x - 3)^{2} + (y + 4)^{2} = 25$

    Standard form is $(x - h)^{2} + (y - k)^{2} = r^{2}$ with center $(h, k) = (3, -4)$ and $r^{2} = 25$.

35. Solve for $x$: $\frac{2x - 1}{3} = 5$.
    • 8
    • 7
    • 9
    • 12

    Correct answer: 8

    Multiply by 3: $2x - 1 = 15$. Add 1: $2x = 16$, so $x = 8$.

36. A rectangle's length is $3$ more than twice its width $w$. Which expression gives its perimeter?
    • $6w + 6$
    • $3w + 3$
    • $4w + 6$
    • $2w + 3$

    Correct answer: $6w + 6$

    Length $= 2w + 3$. Perimeter $= 2(\text{length} + \text{width}) = 2(2w + 3 + w) = 2(3w + 3) = 6w + 6$.

37. If $5x - 2y = 20$, what is the $y$-intercept of the line?
    • -10
    • 10
    • 4
    • -4

    Correct answer: -10

    Set $x = 0$: $-2y = 20$, so $y = -10$. The $y$-intercept is $-10$.

38. Solve the compound inequality $-3 \leq 2x - 1 \leq 5$.
    • $-1 \leq x \leq 3$
    • $-2 \leq x \leq 6$
    • $-1 \leq x \leq 2$
    • $1 \leq x \leq 3$

    Correct answer: $-1 \leq x \leq 3$

    Add 1 throughout: $-2 \leq 2x \leq 6$. Divide by 2: $-1 \leq x \leq 3$.

39. Solve for $x$: $x^{2} = 5x$.
    • $x = 0$ or $x = 5$
    • $x = 5$ only
    • $x = 0$ only
    • $x = \pm 5$

    Correct answer: $x = 0$ or $x = 5$

    Rewrite as $x^{2} - 5x = 0$, factor $x(x - 5) = 0$: $x = 0$ or $x = 5$. Dividing by $x$ would lose the $x = 0$ solution.

40. Using the quadratic formula, what are the solutions of $x^{2} + 2x - 1 = 0$?
    • $x = -1 \pm \sqrt{2}$
    • $x = 1 \pm \sqrt{2}$
    • $x = -1 \pm 2\sqrt{2}$
    • $x = -2 \pm \sqrt{2}$

    Correct answer: $x = -1 \pm \sqrt{2}$

    $x = \frac{-2 \pm \sqrt{4 + 4}}{2} = \frac{-2 \pm 2\sqrt{2}}{2} = -1 \pm \sqrt{2}$.

41. What is the discriminant of $3x^{2} - 4x + 2 = 0$, and what does it indicate?
    • $-8$; two complex (no real) solutions
    • $8$; two real solutions
    • $0$; one real solution
    • $40$; two real solutions

    Correct answer: $-8$; two complex (no real) solutions

    Discriminant $= b^{2} - 4ac = (-4)^{2} - 4(3)(2) = 16 - 24 = -8$. A negative discriminant means no real solutions (two complex roots).

42. Simplify $\frac{6x^{3}y^{2}}{2xy^{5}}$.
    • $\frac{3x^{2}}{y^{3}}$
    • $\frac{3x^{2}}{y^{2}}$
    • $3x^{2}y^{3}$
    • $\frac{3x^{4}}{y^{3}}$

    Correct answer: $\frac{3x^{2}}{y^{3}}$

    Divide coefficients $\frac{6}{2} = 3$, subtract exponents: $x^{3-1} = x^{2}$ and $y^{2-5} = y^{-3}$. Result: $\frac{3x^{2}}{y^{3}}$.

43. If $ab = -24$ with integers $a > b$, which value could $a - b$ equal?
    • 10
    • 6
    • 9
    • 12

    Correct answer: 10

    Try factor pairs with $a > b$: $a = 8, b = -3$ gives $ab = -24$ and $a - b = 11$; $a = 12, b = -2$ gives $a - b = 14$; $a = 2, b = -12$ gives $a - b = 14$; $a = 4, b = -6$ gives $a - b = 10$. Only 10 is attainable among the choices.

44. Solve for $x$: $\sqrt{x + 3} = 4$.
    • 13
    • 1
    • 7
    • 16

    Correct answer: 13

    Square both sides: $x + 3 = 16$, so $x = 13$. Check: $\sqrt{16} = 4$.

45. Which inequality describes the solution to $-2x + 5 > 11$?
    • $x < -3$
    • $x > -3$
    • $x < 3$
    • $x > 3$

    Correct answer: $x < -3$

    Subtract 5: $-2x > 6$. Divide by $-2$ and flip the inequality: $x < -3$.

46. If $f(x) = 3x^{2} - 2x + 1$, what is $f(-1)$?
    • 6
    • 2
    • 4
    • 8

    Correct answer: 6

    $3(1) - 2(-1) + 1 = 3 + 2 + 1 = 6$.

47. If $f(x) = 2x^{2} + 3x - 5$, what is $f(-3)$?
    • 4
    • -20
    • -14
    • 20

    Correct answer: 4

    $2(9) + 3(-3) - 5 = 18 - 9 - 5 = 4$.

48. What is the inverse of $f(x) = 3x - 4$?
    • $f^{-1}(x) = \frac{x + 4}{3}$
    • $f^{-1}(x) = \frac{x - 4}{3}$
    • $f^{-1}(x) = 3x + 4$
    • $f^{-1}(x) = \frac{x}{3} + 4$

    Correct answer: $f^{-1}(x) = \frac{x + 4}{3}$

    Set $y = 3x - 4$, swap and solve: $x = 3y - 4 \Rightarrow y = \frac{x + 4}{3}$.

49. What is the domain of $f(x) = \log_{3}(x + 2)$?
    • $x > -2$
    • $x \geq -2$
    • $x < -2$
    • all real numbers

    Correct answer: $x > -2$

    A logarithm requires a positive argument: $x + 2 > 0$, so $x > -2$.

50. What is the range of $h(x) = \sqrt{x - 2}$?
    • $y \geq 0$
    • $y > 0$
    • $x \geq 2$
    • $x > 2$

    Correct answer: $y \geq 0$

    The principal square root is never negative, so outputs satisfy $y \geq 0$. (The domain is $x \geq 2$, but the range is $y \geq 0$.)

51. What is the domain of $f(x) = \sqrt{9 - x^{2}}$?
    • $-3 \leq x \leq 3$
    • $x \leq -3$ or $x \geq 3$
    • all real numbers
    • $x > 0$

    Correct answer: $-3 \leq x \leq 3$

    Require $9 - x^{2} \geq 0$, i.e. $x^{2} \leq 9$, which gives $-3 \leq x \leq 3$.

52. Which equation defines a function whose graph is symmetric about the $y$-axis?
    • $y = x^{2} - 4$
    • $y = x^{3} - x$
    • $y = x - 2$
    • $y = 2x + 3$

    Correct answer: $y = x^{2} - 4$

    Symmetry about the $y$-axis means an even function, $f(-x) = f(x)$. Only $y = x^{2} - 4$ satisfies this.

53. For $f(x) = |x - 3|$, what is the minimum value of $f(x)$?
    • 0
    • -3
    • 3
    • \text{no minimum}

    Correct answer: 0

    An absolute value is never negative and reaches 0 when $x = 3$, so the minimum value is 0.

54. What is the vertex of $g(x) = -2(x - 1)^{2} + 3$?
    • $(1, 3)$
    • $(1, -3)$
    • $(-1, 3)$
    • $(-1, -3)$

    Correct answer: $(1, 3)$

    In vertex form $a(x - h)^{2} + k$, the vertex is $(h, k) = (1, 3)$.

55. For $h(x) = \frac{1}{x^{2} - 4}$, where are the vertical asymptotes?
    • $x = 2$ and $x = -2$
    • $x = 4$ and $x = -4$
    • $y = 2$ and $y = -2$
    • $x = 0$ only

    Correct answer: $x = 2$ and $x = -2$

    Vertical asymptotes occur where the denominator is zero: $x^{2} - 4 = 0$, so $x = \pm 2$.

56. If $h(x) = 3^{x+1}$, what is $h^{-1}(27)$?
    • 2
    • 3
    • 4
    • 5

    Correct answer: 2

    Solve $3^{x+1} = 27 = 3^{3}$: $x + 1 = 3$, so $x = 2$. Thus $h^{-1}(27) = 2$.

57. What is the end behavior of $f(x) = x^{3} - 2x^{2} + x$?
    • As $x \rightarrow \infty,\ f(x) \rightarrow \infty$; as $x \rightarrow -\infty,\ f(x) \rightarrow -\infty$
    • As $x \rightarrow \infty,\ f(x) \rightarrow -\infty$; as $x \rightarrow -\infty,\ f(x) \rightarrow \infty$
    • As $x \rightarrow \pm\infty,\ f(x) \rightarrow \infty$
    • As $x \rightarrow \pm\infty,\ f(x) \rightarrow -\infty$

    Correct answer: As $x \rightarrow \infty,\ f(x) \rightarrow \infty$; as $x \rightarrow -\infty,\ f(x) \rightarrow -\infty$

    The leading term $x^{3}$ has odd degree and positive coefficient, so the graph rises to the right and falls to the left.

58. Which function has a horizontal asymptote at $y = 1$?
    • $f(x) = \frac{x + 1}{x}$
    • $f(x) = \frac{1}{x} + 2$
    • $f(x) = \frac{x}{x + 1} \cdot 2$
    • $f(x) = x + 1$

    Correct answer: $f(x) = \frac{x + 1}{x}$

    $\frac{x + 1}{x} = 1 + \frac{1}{x} \rightarrow 1$ as $x \rightarrow \pm\infty$. Its horizontal asymptote is $y = 1$.

59. The graph of $g(x) = f(x) + 3$ is the graph of $f(x)$ transformed how?
    • shifted up 3 units
    • shifted down 3 units
    • shifted right 3 units
    • shifted left 3 units

    Correct answer: shifted up 3 units

    Adding a constant to the output value translates the graph vertically: $+3$ shifts it up 3 units.

60. The graph of $g(x) = f(x - 4)$ is the graph of $f(x)$ transformed how?
    • shifted right 4 units
    • shifted left 4 units
    • shifted up 4 units
    • shifted down 4 units

    Correct answer: shifted right 4 units

    Replacing $x$ with $x - 4$ shifts the graph horizontally to the right by 4 units.

61. A function grows by equal factors over equal intervals. What type of function is it?
    • exponential
    • linear
    • quadratic
    • constant

    Correct answer: exponential

    Growing by equal factors (a common ratio) over equal intervals is the defining property of an exponential function; linear functions grow by equal differences.

62. For $f(x) = 2^{x}$, by what factor does $f$ change when $x$ increases by 1?
    • multiplies by 2
    • adds 2
    • multiplies by 1
    • adds 1

    Correct answer: multiplies by 2

    $f(x + 1) = 2^{x+1} = 2 \cdot 2^{x} = 2 f(x)$, so each unit increase in $x$ multiplies the output by 2.

63. If $f(x) = x^{2}$ and $g(x) = x + 1$, what is $(f \circ g)(2)$?
    • 9
    • 5
    • 7
    • 3

    Correct answer: 9

    $(f \circ g)(2) = f(g(2)) = f(3) = 3^{2} = 9$.

64. If $f(x) = 2x + 1$ and $g(x) = x^{2}$, what is $(g \circ f)(3)$?
    • 49
    • 13
    • 19
    • 37

    Correct answer: 49

    $f(3) = 7$, then $g(7) = 7^{2} = 49$.

65. What is the average rate of change of $f(x) = x^{2}$ on the interval $[1, 4]$?
    • 5
    • 3
    • 8
    • 15

    Correct answer: 5

    Average rate of change $= \frac{f(4) - f(1)}{4 - 1} = \frac{16 - 1}{3} = \frac{15}{3} = 5$.

66. What is the average rate of change of $f(x) = 3x + 2$ on any interval?
    • 3
    • 2
    • 5
    • varies by interval

    Correct answer: 3

    For a linear function the average rate of change equals the slope, which is 3 on every interval.

67. A sequence is defined by $a_{1} = 5$ and $a_{n} = a_{n-1} + 4$. What is $a_{4}$?
    • 17
    • 13
    • 20
    • 21

    Correct answer: 17

    This arithmetic sequence has terms $5, 9, 13, 17, \dots$. The 4th term is $5 + 3(4) = 17$.

68. Which explicit formula matches the arithmetic sequence $5, 9, 13, 17, \dots$?
    • $a_{n} = 4n + 1$
    • $a_{n} = 5n$
    • $a_{n} = 4n + 5$
    • $a_{n} = 5n - 4$

    Correct answer: $a_{n} = 4n + 1$

    The common difference is 4 and $a_{1} = 5$: $a_{n} = 5 + 4(n - 1) = 4n + 1$.

69. A geometric sequence has $a_{1} = 3$ and common ratio 2. What is $a_{4}$?
    • 24
    • 12
    • 18
    • 48

    Correct answer: 24

    $a_{4} = a_{1} r^{3} = 3 \cdot 2^{3} = 3 \cdot 8 = 24$.

70. Which describes $f(x) = -x^{2} + 4$?
    • opens downward, vertex at $(0, 4)$
    • opens upward, vertex at $(0, 4)$
    • opens downward, vertex at $(0, -4)$
    • opens upward, vertex at $(0, -4)$

    Correct answer: opens downward, vertex at $(0, 4)$

    The negative leading coefficient makes the parabola open downward; with no $x$-term the vertex is at $(0, 4)$.

71. Which relation is NOT a function?
    • $\{(1, 2), (1, 3), (2, 4)\}$
    • $\{(1, 2), (2, 3), (3, 4)\}$
    • $y = x^{2}$
    • $y = 3x - 1$

    Correct answer: $\{(1, 2), (1, 3), (2, 4)\}$

    A function assigns exactly one output to each input. The set $\{(1, 2), (1, 3), \dots\}$ pairs the input 1 with two outputs, so it is not a function.

72. For $f(x) = \frac{x^{2} - 4}{x - 2}$, what value is excluded from the domain?
    • $x = 2$
    • $x = -2$
    • $x = 0$
    • $x = 4$

    Correct answer: $x = 2$

    The denominator $x - 2$ is zero at $x = 2$, so $x = 2$ is excluded even though the expression simplifies to $x + 2$.

73. What is the $y$-intercept of $f(x) = 2^{x} - 3$?
    • -2
    • -3
    • 1
    • 0

    Correct answer: -2

    Evaluate at $x = 0$: $2^{0} - 3 = 1 - 3 = -2$.

74. Which function is odd (symmetric about the origin)?
    • $f(x) = x^{3}$
    • $f(x) = x^{2}$
    • $f(x) = |x|$
    • $f(x) = x^{2} + 1$

    Correct answer: $f(x) = x^{3}$

    An odd function satisfies $f(-x) = -f(x)$. For $x^{3}$, $(-x)^{3} = -x^{3}$, so it is odd; the others are even.

75. If $f(x) = x^{2} - 6x + 5$, what are its zeros?
    • $x = 1$ and $x = 5$
    • $x = -1$ and $x = -5$
    • $x = 1$ and $x = 6$
    • $x = 5$ and $x = 6$

    Correct answer: $x = 1$ and $x = 5$

    Factor: $(x - 1)(x - 5) = 0$, so the zeros are $x = 1$ and $x = 5$.

76. A function $f$ is increasing on $(-\infty, 2)$ and decreasing on $(2, \infty)$. What occurs at $x = 2$?
    • a maximum
    • a minimum
    • a vertical asymptote
    • a zero

    Correct answer: a maximum

    Where a function changes from increasing to decreasing, it reaches a local (here, absolute) maximum.

77. If $f(x) = 4^{x}$, what is $f\left(\frac{1}{2}\right)$?
    • 2
    • 4
    • 8
    • 16

    Correct answer: 2

    $4^{1/2} = \sqrt{4} = 2$.

78. The amount $A(t) = 500(1.04)^{t}$ models an account. What does 1.04 represent?
    • a 4% growth rate per period
    • a 4% decay rate per period
    • a 40% growth rate per period
    • the initial amount

    Correct answer: a 4% growth rate per period

    In $A = P(1 + r)^{t}$, the base $1.04 = 1 + 0.04$ indicates 4% growth each period; the initial amount is 500.

79. Which best models a quantity that decreases by 15% each year?
    • $A(t) = A_{0}(0.85)^{t}$
    • $A(t) = A_{0}(1.15)^{t}$
    • $A(t) = A_{0}(0.15)^{t}$
    • $A(t) = A_{0} - 0.15t$

    Correct answer: $A(t) = A_{0}(0.85)^{t}$

    A 15% yearly decrease multiplies by $1 - 0.15 = 0.85$ each year, giving $A_{0}(0.85)^{t}$.

80. What is $g(0)$ for the piecewise function $g(x) = x + 1$ if $x \geq 0$, and $g(x) = -x$ if $x < 0$?
    • 1
    • 0
    • -1
    • undefined

    Correct answer: 1

    Since $0 \geq 0$, use the first piece: $g(0) = 0 + 1 = 1$.

81. Which number is irrational?
    • $\sqrt{50}$
    • $\sqrt{49}$
    • $\frac{22}{7}$
    • $0.2727\ldots$

    Correct answer: $\sqrt{50}$

    $\sqrt{50} = 5\sqrt{2}$ cannot be written as a ratio of integers. The others are a perfect-square root, a fraction, and a repeating decimal.

82. The product of a nonzero rational number and an irrational number is always:
    • irrational
    • rational
    • an integer
    • zero

    Correct answer: irrational

    Multiplying a nonzero rational by an irrational always yields an irrational result.

83. Which statement is always true?
    • The sum of two rational numbers is rational.
    • The sum of two irrational numbers is irrational.
    • The product of two irrational numbers is irrational.
    • The sum of a rational and an irrational is rational.

    Correct answer: The sum of two rational numbers is rational.

    Rationals are closed under addition, so a sum of two rationals is rational. The other statements have counterexamples (e.g. $\sqrt{2} + (-\sqrt{2}) = 0$).

84. Simplify $27^{2/3}$.
    • 9
    • 3
    • 18
    • 81

    Correct answer: 9

    $27^{2/3} = \left(27^{1/3}\right)^{2} = 3^{2} = 9$.

85. Simplify $16^{-1/2}$.
    • $\frac{1}{4}$
    • 4
    • -4
    • $\frac{1}{16}$

    Correct answer: $\frac{1}{4}$

    $16^{-1/2} = \frac{1}{16^{1/2}} = \frac{1}{4}$.

86. Write $3.2 \times 10^{4}$ in standard notation.
    • 32,000
    • 3,200
    • 320,000
    • 0.00032

    Correct answer: 32,000

    $3.2 \times 10^{4} = 3.2 \times 10000 = 32{,}000$.

87. What is $(4 \times 10^{3})(2 \times 10^{5})$ in scientific notation?
    • $8 \times 10^{8}$
    • $8 \times 10^{15}$
    • $6 \times 10^{8}$
    • $8 \times 10^{2}$

    Correct answer: $8 \times 10^{8}$

    Multiply coefficients $4 \times 2 = 8$ and add exponents $3 + 5 = 8$: $8 \times 10^{8}$.

88. Simplify $\sqrt{72}$.
    • $6\sqrt{2}$
    • $2\sqrt{18}$
    • $8\sqrt{3}$
    • $36\sqrt{2}$

    Correct answer: $6\sqrt{2}$

    $\sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2}$.

89. A car travels 150 miles in 3 hours. What is its average speed in miles per hour?
    • 50
    • 45
    • 60
    • 75

    Correct answer: 50

    Average speed $= \frac{150 \text{ mi}}{3 \text{ h}} = 50$ miles per hour.

90. Convert 2.5 kilometers to meters.
    • 2,500
    • 250
    • 25,000
    • 0.0025

    Correct answer: 2,500

    There are 1000 meters in a kilometer: $2.5 \times 1000 = 2{,}500$ meters.

91. What is the 20th term of the arithmetic sequence with $a_{1} = 3$ and common difference 4?
    • 79
    • 75
    • 77
    • 81

    Correct answer: 79

    $a_{20} = a_{1} + (20 - 1)d = 3 + 19(4) = 3 + 76 = 79$.

92. What is the 15th term of the arithmetic sequence $5, 9, 13, \dots$?
    • 61
    • 57
    • 53
    • 65

    Correct answer: 61

    Common difference is 4: $a_{15} = 5 + (15 - 1)(4) = 5 + 56 = 61$.

93. What is the sum of the infinite geometric series $1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots$?
    • 2
    • 1
    • 1.5
    • 2.5

    Correct answer: 2

    For $|r| < 1$, the sum is $\frac{a}{1 - r} = \frac{1}{1 - \frac{1}{2}} = 2$.

94. What is the median of $3, 7, 9, 12, 14, 18, 22$?
    • 12
    • 9
    • 11
    • 14

    Correct answer: 12

    With seven sorted values, the median is the 4th value, which is 12.

95. What is the mean of $4, 8, 10, 14, 14$?
    • 10
    • 8
    • 11
    • 12

    Correct answer: 10

    Sum $= 4 + 8 + 10 + 14 + 14 = 50$; divide by 5: $\frac{50}{5} = 10$.

96. What is the mode of $2, 5, 5, 7, 9, 9, 9, 12$?
    • 9
    • 5
    • 7
    • 8.5

    Correct answer: 9

    The mode is the most frequent value; 9 appears three times, more than any other.

97. If a data set has range 20 and minimum value 15, what is the maximum value?
    • 35
    • 20
    • 25
    • 30

    Correct answer: 35

    Range $=$ maximum $-$ minimum, so maximum $= 20 + 15 = 35$.

98. A data set has first quartile 8 and interquartile range 12. What is the third quartile?
    • 20
    • 16
    • 24
    • 28

    Correct answer: 20

    IQR $= Q_{3} - Q_{1}$, so $Q_{3} = Q_{1} + \text{IQR} = 8 + 12 = 20$.

99. What is the coefficient of variation for a data set with mean 20 and standard deviation 5?
    • 25\%
    • 15\%
    • 20\%
    • 30\%

    Correct answer: 25\%

    Coefficient of variation $= \frac{\text{standard deviation}}{\text{mean}} \times 100\% = \frac{5}{20} \times 100\% = 25\%$.

100. For the data set $2, 4, 4, 4, 5, 5, 7, 9$, which measure equals 5?
     • the mean
     • the mode
     • the range
     • the median

     Correct answer: the mean

     Sum $= 40$, mean $= \frac{40}{8} = 5$. The mode is 4, the median is $4.5$, and the range is 7, so only the mean equals 5.

101. Adding 10 to every value in a data set changes which statistic?
     • the mean
     • the range
     • the standard deviation
     • the interquartile range

     Correct answer: the mean

     Shifting every value by a constant increases the mean (a measure of center) by that constant but leaves spread measures (range, standard deviation, IQR) unchanged.

102. What is the probability of drawing a red card or a queen from a standard 52-card deck?
     • $\frac{7}{13}$
     • $\frac{1}{2}$
     • $\frac{9}{26}$
     • $\frac{5}{13}$

     Correct answer: $\frac{7}{13}$

     By inclusion-exclusion: $P = \frac{26 + 4 - 2}{52} = \frac{28}{52} = \frac{7}{13}$ (subtracting the 2 red queens counted twice).

103. A bag has 4 red, 3 blue, and 5 green marbles. What is the probability of drawing a blue or green marble?
     • $\frac{2}{3}$
     • $\frac{1}{3}$
     • $\frac{1}{2}$
     • $\frac{5}{6}$

     Correct answer: $\frac{2}{3}$

     Favorable $= 3 + 5 = 8$ out of $12$: $\frac{8}{12} = \frac{2}{3}$.

104. A box has 6 red, 4 blue, and 5 green balls. What is the probability of selecting a red or green ball?
     • $\frac{11}{15}$
     • $\frac{6}{15}$
     • $\frac{9}{15}$
     • $\frac{1}{3}$

     Correct answer: $\frac{11}{15}$

     Favorable $= 6 + 5 = 11$ out of $15$: $\frac{11}{15}$.

105. A fair die is rolled twice. What is the probability of a 4 on the first roll and a 6 on the second?
     • $\frac{1}{36}$
     • $\frac{1}{18}$
     • $\frac{1}{12}$
     • $\frac{1}{6}$

     Correct answer: $\frac{1}{36}$

     The rolls are independent: $\frac{1}{6} \times \frac{1}{6} = \frac{1}{36}$.

106. If the odds in favor of an event are 4 to 1, what is the probability the event occurs?
     • $\frac{4}{5}$
     • $\frac{1}{4}$
     • $\frac{1}{5}$
     • $\frac{4}{1}$

     Correct answer: $\frac{4}{5}$

     Odds of 4 to 1 means 4 favorable out of $4 + 1 = 5$ total outcomes: $P = \frac{4}{5}$.

107. A coin is flipped 3 times. What is the probability of getting exactly 2 heads?
     • $\frac{3}{8}$
     • $\frac{1}{8}$
     • $\frac{1}{2}$
     • $\frac{2}{3}$

     Correct answer: $\frac{3}{8}$

     There are $2^{3} = 8$ equally likely outcomes; exactly 2 heads occurs in $\binom{3}{2} = 3$ of them, so $P = \frac{3}{8}$.

108. Two cards are drawn without replacement from a standard deck. What is the probability both are aces?
     • $\frac{1}{221}$
     • $\frac{1}{169}$
     • $\frac{1}{13}$
     • $\frac{1}{26}$

     Correct answer: $\frac{1}{221}$

     $\frac{4}{52} \times \frac{3}{51} = \frac{12}{2652} = \frac{1}{221}$.

109. A spinner has 8 equal sections numbered 1–8. What is the probability of landing on a prime number?
     • $\frac{1}{2}$
     • $\frac{3}{8}$
     • $\frac{5}{8}$
     • $\frac{3}{4}$

     Correct answer: $\frac{1}{2}$

     The primes from 1 to 8 are 2, 3, 5, and 7 — that is 4 of the 8 sections, so $\frac{4}{8} = \frac{1}{2}$.

110. What is the equation of the circle with center $(3, -2)$ and radius 4?
     • $(x - 3)^{2} + (y + 2)^{2} = 16$
     • $(x - 3)^{2} + (y + 2)^{2} = 4$
     • $(x + 3)^{2} + (y - 2)^{2} = 16$
     • $(x + 3)^{2} + (y - 2)^{2} = 4$

     Correct answer: $(x - 3)^{2} + (y + 2)^{2} = 16$

     Standard form $(x - h)^{2} + (y - k)^{2} = r^{2}$ with center $(3, -2)$ and $r^{2} = 16$.

111. What is the distance between the points $(1, 2)$ and $(4, 6)$?
     • 5
     • 3
     • 4
     • 7

     Correct answer: 5

     $d = \sqrt{(4 - 1)^{2} + (6 - 2)^{2}} = \sqrt{9 + 16} = \sqrt{25} = 5$.

112. What is the midpoint of the segment with endpoints $(2, 4)$ and $(8, 10)$?
     • $(5, 7)$
     • $(6, 6)$
     • $(5, 6)$
     • $(10, 14)$

     Correct answer: $(5, 7)$

     Midpoint $= \left(\frac{2 + 8}{2}, \frac{4 + 10}{2}\right) = (5, 7)$.

113. A shirt regularly priced at $40 is on sale for 25% off. What is the sale price?
     • $30
     • $32
     • $35
     • $28

     Correct answer: $30

     A 25% discount removes $0.25 \times 40 = 10$, so the sale price is $40 - 10 = \$30$.

114. What percent of 80 is 12?
     • 15\%
     • 12\%
     • 18\%
     • 20\%

     Correct answer: 15\%

     $\frac{12}{80} = 0.15 = 15\%$.

115. If 3 pounds of apples cost $6, what is the cost of 7 pounds at the same rate?
     • $14
     • $12
     • $18
     • $21

     Correct answer: $14

     The unit price is $\frac{6}{3} = \$2$ per pound, so 7 pounds cost $7 \times 2 = \$14$.

116. A map uses a scale of 1 inch to 50 miles. How many miles do 3.5 inches represent?
     • 175
     • 150
     • 200
     • 100

     Correct answer: 175

     $3.5 \times 50 = 175$ miles.

117. If $x$ is inversely proportional to $y$ and $y = 2$ when $x = 3$, what is $y$ when $x = 6$?
     • 1
     • 2
     • 3
     • 4

     Correct answer: 1

     Inverse variation: $xy = k = 3 \times 2 = 6$. When $x = 6$, $y = \frac{6}{6} = 1$.

118. Solve for $x$: $4x + 9 = 2x - 5$.
     • -7
     • -2
     • 7
     • 2

     Correct answer: -7

     Subtract $2x$: $2x + 9 = -5$. Subtract 9: $2x = -14$, so $x = -7$.

119. Factor completely: $x^{2} + 5x - 14$.
     • $(x + 7)(x - 2)$
     • $(x - 7)(x + 2)$
     • $(x + 7)(x + 2)$
     • $(x + 14)(x - 1)$

     Correct answer: $(x + 7)(x - 2)$

     Find two numbers with product $-14$ and sum $5$: $7$ and $-2$. So $x^{2} + 5x - 14 = (x + 7)(x - 2)$.

120. Simplify $\frac{2x^{2} - 8}{x + 2}$ for $x \neq -2$.
     • $2x - 4$
     • $2x + 4$
     • $x - 4$
     • $2(x + 2)$

     Correct answer: $2x - 4$

     Factor the numerator: $2x^{2} - 8 = 2(x^{2} - 4) = 2(x - 2)(x + 2)$. Cancel $(x + 2)$: $2(x - 2) = 2x - 4$.

121. What is the axis of symmetry of $f(x) = x^{2} - 4x + 3$?
     • $x = 2$
     • $x = -2$
     • $x = 4$
     • $x = 1$

     Correct answer: $x = 2$

     The axis of symmetry is $x = -\frac{b}{2a} = -\frac{-4}{2(1)} = 2$.

122. Which explicit formula matches the geometric sequence $2, 6, 18, 54, \dots$?
     • $a_{n} = 2 \cdot 3^{n-1}$
     • $a_{n} = 2 \cdot 3^{n}$
     • $a_{n} = 3 \cdot 2^{n-1}$
     • $a_{n} = 2 + 3(n - 1)$

     Correct answer: $a_{n} = 2 \cdot 3^{n-1}$

     The first term is $2$ and the common ratio is $3$: $a_{n} = a_{1} r^{n-1} = 2 \cdot 3^{n-1}$.

123. The graph of $g(x) = \sqrt{x} - 5$ is the graph of $f(x) = \sqrt{x}$ transformed how?
     • shifted down 5 units
     • shifted up 5 units
     • shifted right 5 units
     • shifted left 5 units

     Correct answer: shifted down 5 units

     Subtracting 5 from the output value translates the graph vertically downward by 5 units.

124. Simplify $\frac{6}{\sqrt{3}}$.
     • $2\sqrt{3}$
     • $\frac{2}{\sqrt{3}}$
     • $6\sqrt{3}$
     • $\frac{\sqrt{3}}{2}$

     Correct answer: $2\sqrt{3}$

     Rationalize by multiplying by $\frac{\sqrt{3}}{\sqrt{3}}$: $\frac{6\sqrt{3}}{3} = 2\sqrt{3}$.

125. The mean of 6 numbers is 15. A seventh number, 8, is added to the set. What is the new mean?
     • 14
     • 15
     • 13
     • 16

     Correct answer: 14

     The original sum is $6 \times 15 = 90$. Adding 8 gives $98$; dividing by 7: $\frac{98}{7} = 14$.

126. A jar holds 5 white and 3 black marbles. Two are drawn without replacement. What is the probability that both are white?
     • $\frac{5}{14}$
     • $\frac{25}{64}$
     • $\frac{5}{16}$
     • $\frac{1}{2}$

     Correct answer: $\frac{5}{14}$

     $\frac{5}{8} \times \frac{4}{7} = \frac{20}{56} = \frac{5}{14}$.

127. How many different ways can 2 books be chosen from a set of 5 distinct books?
     • 10
     • 20
     • 25
     • 5

     Correct answer: 10

     Order does not matter, so use combinations: $\binom{5}{2} = \frac{5!}{2!\,3!} = \frac{20}{2} = 10$.