Your FREE Praxis Mathematics: Content Knowledge (5165) Practice Test 2026 – 160+ Q&A

Praxis 5165 Practice Questions

1. Which of the following numbers is irrational?
   • $\frac{22}{7}$
   • $\sqrt{49}$
   • $\sqrt{50}$
   • 0.272727...

   Correct answer: $\sqrt{50}$

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

2. What is the value of $i^{27}$, where $i = \sqrt{-1}$?
   • -1
   • I
   • -i
   • 1

   Correct answer: -i

   Powers of $i$ cycle with period 4. Since 27 mod 4 = 3 and $i^{3} = -i$, we get $i^{27} = -i$.

3. What is the greatest common divisor of 84 and 126?
   • 6
   • 14
   • 42
   • 21

   Correct answer: 42

   $84 = 2^{2} \cdot 3 \cdot 7$ and $126 = 2 \cdot 3^{2} \cdot 7$, so the GCD is $2 \cdot 3 \cdot 7 = 42$.

4. Simplify $(3 + 2i)(1 - 4i)$.
   • $11 + 10i$
   • $11 - 10i$
   • $3 - 8i$
   • $-5 - 10i$

   Correct answer: $11 - 10i$

   $(3+2i)(1-4i) = 3 -12i +2i -8i^{2} = 3 -10i +8 = 11 -10i$, since $i^{2} = -1$.

5. Which expression equals the absolute value (modulus) of the complex number $3 - 4i$?
   • 7
   • 5
   • 1
   • $\sqrt{7}$

   Correct answer: 5

   The modulus is $\sqrt{3^{2} + (-4)^{2}} = \sqrt{9+16} = \sqrt{25} = 5$.

6. A number expressed in scientific notation is $4.5 \times 10^{-3}$. What is this number in standard form?
   • 0.00045
   • 0.045
   • 4500
   • 0.0045

   Correct answer: 0.0045

   Multiplying 4.5 by $10^{-3}$ moves the decimal three places left, giving 0.0045.

7. What is the least common multiple of 12 and 18?
   • 216
   • 36
   • 6
   • 72

   Correct answer: 36

   $12 = 2^{2} \cdot 3$ and $18 = 2 \cdot 3^{2}$. The LCM takes the highest power of each prime: $2^{2} \cdot 3^{2} = 36$.

8. If $a = 2^{3} \cdot 5$ and $b = 2^{2} \cdot 5^{2}$, what is $a \cdot b$ expressed in prime factorization?
   • $2^{5} \cdot 5^{3}$
   • $2^{5} \cdot 5^{2}$
   • $2^{6} \cdot 5^{3}$
   • $2^{6} \cdot 5^{2}$

   Correct answer: $2^{5} \cdot 5^{3}$

   $a \cdot b = 2^{3+2} \cdot 5^{1+2} = 2^{5} \cdot 5^{3}$.

9. Which of the following is equivalent to $8^{2/3}$?
   • 16
   • 4
   • 6
   • 2

   Correct answer: 4

   $8^{2/3} = (8^{1/3})^{2} = 2^{2} = 4$.

10. What is the result of dividing the complex number $1 + i$ by $i$?
    • $-1 - i$
    • $1 + i$
    • $-1 + i$
    • $1 - i$

    Correct answer: $1 - i$

    $(1+i)/i = (1+i) \cdot (-i)/(i \cdot -i) = (-i - i^{2})/1 = (-i + 1) = 1 - i$.

11. Rationalize the denominator: $\frac{6}{\sqrt{3}}$.
    • $2\sqrt{3}$
    • $\sqrt{18}$
    • $6\sqrt{3}$
    • $\frac{\sqrt{3}}{2}$

    Correct answer: $2\sqrt{3}$

    $\frac{6}{\sqrt{3}} = \frac{6\sqrt{3}}{3} = 2\sqrt{3}$.

12. The number 0.4 (repeating, meaning 0.444...) equals which fraction?
    • $\frac{4}{9}$
    • $\frac{4}{11}$
    • $\frac{2}{5}$
    • $\frac{1}{4}$

    Correct answer: $\frac{4}{9}$

    Let $x = 0.444...$; then $10x = 4.444...$, so $9x = 4$ and $x = \frac{4}{9}$.

13. How many distinct positive divisors does 36 have?
    • 12
    • 6
    • 8
    • 9

    Correct answer: 9

    $36 = 2^{2} \cdot 3^{2}$, so the number of divisors is $(2+1)(2+1) = 9$.

14. Which property is illustrated by $a \cdot (b + c) = a \cdot b + a \cdot c$?
    • Commutative
    • Identity
    • Distributive
    • Associative

    Correct answer: Distributive

    Multiplying a single term across a sum demonstrates the distributive property.

15. What is 7 mod 4 (the remainder when 7 is divided by 4)?
    • 0
    • 3
    • 1
    • 2

    Correct answer: 3

    $7 = 4 \cdot 1 + 3$, so the remainder is 3.

16. Express the complex number with magnitude 2 and angle 60 degrees in rectangular form.
    • $1 + \sqrt{3} \cdot i$
    • $1 + 2i$
    • $\sqrt{3} + i$
    • $2 + 2i$

    Correct answer: $1 + \sqrt{3} \cdot i$

    $x = 2\cos 60 = 2 \cdot (1/2) = 1$; $y = 2\sin 60 = 2 \cdot (\sqrt{3}/2) = \sqrt{3}$. So $z = 1 + \sqrt{3} \cdot i$.

17. If $x$ is a positive integer and $3^{x} = 81$, what is $x$?
    • 3
    • 5
    • 4
    • 27

    Correct answer: 4

    $81 = 3^{4}$, so $x = 4$.

18. What is the value of log base 2 of 32?
    • 6
    • 16
    • 4
    • 5

    Correct answer: 5

    $2^{5} = 32$, so log base 2 of 32 = 5.

19. Which set is closed under subtraction?
    • Whole numbers
    • Positive rationals
    • Integers
    • Natural numbers

    Correct answer: Integers

    The difference of any two integers is always an integer, so integers are closed under subtraction; the others can yield negatives or non-members.

20. Simplify $\sqrt{72} + \sqrt{18}$.
    • $\sqrt{90}$
    • $9\sqrt{2}$
    • $12\sqrt{2}$
    • $6\sqrt{3}$

    Correct answer: $9\sqrt{2}$

    $\sqrt{72} = 6\sqrt{2}$ and $\sqrt{18} = 3\sqrt{2}$, so the sum is $9\sqrt{2}$.

21. Solve for $x$: $2x + 7 = 19$.
    • 5
    • 7
    • 13
    • 6

    Correct answer: 6

    $2x = 12$, so $x = 6$.

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

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

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

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

    Correct answer: 2 and 3

    Factoring gives $(x-2)(x-3) = 0$, so $x = 2$ or $x = 3$.

24. Solve the system: $x + y = 10$ and $x - y = 4$.
    • $x=6, y=4$
    • $x=7, y=3$
    • $x=8, y=2$
    • $x=5, y=5$

    Correct answer: $x=7, y=3$

    Adding the equations gives $2x = 14$, so $x = 7$ and $y = 3$.

25. What is the slope of the line passing through (2, 3) and (6, 11)?
    • 2
    • 3
    • 1
    • 4

    Correct answer: 2

    Slope $= (11-3)/(6-2) = 8/4 = 2$.

26. Simplify: $(x^{3} \cdot x^{4}) / x^{2}$.
    • $x^{24}$
    • $x^{5}$
    • $x^{9}$
    • $x^{6}$

    Correct answer: $x^{5}$

    $x^{3} \cdot x^{4} = x^{7}$, and $x^{7} / x^{2} = x^{5}$.

27. What is the discriminant of $2x^{2} + 3x - 5 = 0$?
    • 9
    • 49
    • -31
    • 29

    Correct answer: 49

    Discriminant $= b^{2} - 4ac = 9 - 4(2)(-5) = 9 + 40 = 49$.

28. Expand $(x + 4)^{2}$.
    • $x^{2} + 8x + 16$
    • $x^{2} + 8x + 8$
    • $x^{2} + 4x + 16$
    • $x^{2} + 16$

    Correct answer: $x^{2} + 8x + 16$

    $(x+4)^{2} = x^{2} + 2 \cdot 4 \cdot x + 16 = x^{2} + 8x + 16$.

29. Solve for $x$: $3(x - 2) = 2x + 5$.
    • -1
    • 1
    • 11
    • 7

    Correct answer: 11

    $3x - 6 = 2x + 5$, so $x = 11$.

30. If $f(x) = 2x^{2} - 3x + 1$, what is $f(2)$?
    • 5
    • 3
    • 9
    • 7

    Correct answer: 3

    $f(2) = 2(4) - 3(2) + 1 = 8 - 6 + 1 = 3$.

31. Which expression equals $(x^{2} - 4)/(x - 2)$ for $x$ not equal to 2?
    • $x^{2} - 2$
    • $x + 4$
    • $x + 2$
    • $x - 2$

    Correct answer: $x + 2$

    Factor the numerator: $(x-2)(x+2)/(x-2) = x + 2$.

32. Solve the inequality: $-2x + 3 > 7$.
    • $x < -2$
    • $x > 2$
    • $x < 2$
    • $x > -2$

    Correct answer: $x < -2$

    $-2x > 4$, and dividing by -2 flips the inequality: $x < -2$.

33. What is the sum of the roots of $x^{2} - 7x + 10 = 0$?
    • -10
    • 7
    • 10
    • -7

    Correct answer: 7

    By Vieta's formulas, the sum of roots $= -b/a = 7$.

34. Simplify the rational expression: $(6x^{2} y) / (3x y^{2})$.
    • $\frac{2y}{x}$
    • $\frac{2x}{y}$
    • $\frac{x}{2y}$
    • $2xy$

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

    $6/3 = 2$, $x^{2}/x = x$, and $y/y^{2} = 1/y$, giving $\frac{2x}{y}$.

35. If $2^{x} = 64$, what is $x$?
    • 8
    • 6
    • 5
    • 7

    Correct answer: 6

    $64 = 2^{6}$, so $x = 6$.

36. Factor: $x^{2} + 7x + 12$.
    • $(x+2)(x+6)$
    • $(x-3)(x-4)$
    • $(x+3)(x+4)$
    • $(x+1)(x+12)$

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

    We need two numbers multiplying to 12 and summing to 7: 3 and 4.

37. What is the solution to $|x - 3| = 5$?
    • $x = -2$ only
    • $x = 2$ or $x = -8$
    • $x = 8$ or $x = -2$
    • $x = 8$ only

    Correct answer: $x = 8$ or $x = -2$

    $x - 3 = 5$ gives $x = 8$; $x - 3 = -5$ gives $x = -2$.

38. The line $y = 3x - 2$ is reflected to be perpendicular through the origin. What is the slope of the perpendicular line?
    • $\frac{1}{3}$
    • 3
    • -3
    • $-\frac{1}{3}$

    Correct answer: $-\frac{1}{3}$

    Perpendicular slopes are negative reciprocals; the negative reciprocal of 3 is $-\frac{1}{3}$.

39. Solve for $x$: $\frac{x}{4} + \frac{x}{6} = 5$.
    • 10
    • 15
    • 6
    • 12

    Correct answer: 12

    Common denominator 12: $\frac{3x}{12} + \frac{2x}{12} = \frac{5x}{12} = 5$, so $x = 12$.

40. What is the vertex form of $y = x^{2} - 6x + 5$?
    • $y = (x-3)^{2} + 4$
    • $y = (x-3)^{2} - 4$
    • $y = (x+3)^{2} - 4$
    • $y = (x-6)^{2} + 5$

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

    Complete the square: $x^{2} - 6x = (x-3)^{2} - 9$, so $y = (x-3)^{2} - 9 + 5 = (x-3)^{2} - 4$.

41. Using the quadratic formula, solve $x^{2} - 4x + 1 = 0$.
    • $4 \pm \sqrt{3}$
    • $2 \pm \sqrt{3}$
    • $2 \pm \sqrt{5}$
    • $-2 \pm \sqrt{3}$

    Correct answer: $2 \pm \sqrt{3}$

    $x = \frac{4 \pm \sqrt{16-4}}{2} = \frac{4 \pm \sqrt{12}}{2} = 2 \pm \sqrt{3}$.

42. What is the value of $x$ in the matrix equation where $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 5 \\ 11 \end{bmatrix}$?
    • 2
    • 1
    • 4
    • 3

    Correct answer: 1

    The matrix equation gives the system $x + 2y = 5$ and $3x + 4y = 11$. Solving: from row 1, $x = 5 - 2y$; substitute into row 2 to get $3(5 - 2y) + 4y = 11$, so $15 - 2y = 11$ and $y = 2$. Then $x = 5 - 2(2) = 1$.

43. The determinant of the matrix $\begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}$ is:
    • 11
    • -5
    • 8
    • 5

    Correct answer: 5

    Determinant $= (2)(4) - (3)(1) = 8 - 3 = 5$.

44. Simplify: $\frac{2x + 6}{2}$.
    • $x + 6$
    • $x + 3$
    • $2x + 3$
    • $x + \frac{6}{2}$

    Correct answer: $x + 3$

    Divide each term by 2: $\frac{2x}{2} + \frac{6}{2} = x + 3$.

45. If the roots of $x^{2} + bx + c = 0$ are 4 and -3, what is $c$?
    • 12
    • -1
    • -12
    • 1

    Correct answer: -12

    Product of roots $= c = (4)(-3) = -12$.

46. Solve for $x$: $5^{x+1} = 125$.
    • 3
    • 4
    • 2
    • 1

    Correct answer: 2

    $125 = 5^{3}$, so $x + 1 = 3$ and $x = 2$.

47. What is the next term in the arithmetic sequence 3, 7, 11, 15, ...?
    • 21
    • 19
    • 18
    • 20

    Correct answer: 19

    The common difference is 4, so 15 + 4 = 19.

48. What is the 5th term of a geometric sequence with first term 2 and common ratio 3?
    • 108
    • 162
    • 54
    • 486

    Correct answer: 162

    $a_{n} = 2 \cdot 3^{n-1}$; $a_{5} = 2 \cdot 3^{4} = 2 \cdot 81 = 162$.

49. Simplify: $\log(100) + \log(1000)$, using base 10.
    • 10
    • 8
    • 6
    • 5

    Correct answer: 5

    $\log(100) = 2$ and $\log(1000) = 3$, so the sum is 5.

50. Which is equivalent to $\ln(e^{4})$?
    • 1
    • E
    • 4
    • $4e$

    Correct answer: 4

    ln and e are inverse functions, so $\ln(e^{4}) = 4$.

51. Solve for $x$: $\frac{x - 1}{x + 2} = 3$.
    • 5
    • $-\frac{7}{2}$
    • $-\frac{5}{2}$
    • $\frac{7}{2}$

    Correct answer: $-\frac{7}{2}$

    $x - 1 = 3(x+2) = 3x + 6$, so $-7 = 2x$ and $x = -\frac{7}{2}$.

52. What is the sum of the first 10 positive integers?
    • 100
    • 50
    • 45
    • 55

    Correct answer: 55

    Sum $= \frac{n(n+1)}{2} = \frac{10 \cdot 11}{2} = 55$.

53. The function $f(x) = 3x - 5$ has what inverse?
    • $\frac{x+5}{3}$
    • $3x+5$
    • $\frac{5-x}{3}$
    • $\frac{x-5}{3}$

    Correct answer: $\frac{x+5}{3}$

    Solve $y = 3x - 5$ for $x$: $x = \frac{y+5}{3}$, so $f^{-1}(x) = \frac{x+5}{3}$.

54. What is the domain of $f(x) = \sqrt{x - 2}$?
    • $x \ge 0$
    • $x > 2$
    • All reals
    • $x \ge 2$

    Correct answer: $x \ge 2$

    The radicand must be nonnegative: $x - 2 \ge 0$, so $x \ge 2$.

55. What is the range of $f(x) = x^{2} + 1$?
    • $y \ge 1$
    • $y > 1$
    • All reals
    • $y \ge 0$

    Correct answer: $y \ge 1$

    Since $x^{2} \ge 0$, the minimum of $x^{2} + 1$ is 1, so the range is $y \ge 1$.

56. If $f(x) = x + 2$ and $g(x) = x^{2}$, what is $(f \circ g)(3)$?
    • 13
    • 9
    • 25
    • 11

    Correct answer: 11

    $(f \circ g)(3) = f(g(3)) = f(9) = 9 + 2 = 11$.

57. Which describes the end behavior of $f(x) = -2x^{3}$?
    • Up left, down right
    • Down left, up right
    • Down on both ends
    • Up on both ends

    Correct answer: Up left, down right

    Odd degree with negative leading coefficient: as x goes to negative infinity f goes up, as x goes to positive infinity f goes down.

58. What is the period of $f(x) = \sin(2x)$?
    • $4\pi$
    • $\frac{\pi}{2}$
    • $\pi$
    • $2\pi$

    Correct answer: $\pi$

    Period $= \frac{2\pi}{b} = \frac{2\pi}{2} = \pi$.

59. The graph of $y = (x - 3)^{2} + 2$ has its vertex at:
    • (3, -2)
    • (2, 3)
    • (3, 2)
    • (-3, 2)

    Correct answer: (3, 2)

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

60. What is the amplitude of $f(x) = 4\cos(x)$?
    • $\pi$
    • 1
    • 2
    • 4

    Correct answer: 4

    The amplitude is the absolute value of the coefficient of cosine, which is 4.

61. For what value(s) of $x$ does $f(x) = \frac{x-1}{x^{2}-1}$ have a removable discontinuity?
    • $x = -1$
    • $x = 0$
    • $x = 1$
    • $x = 1$ and $x = -1$

    Correct answer: $x = 1$

    Factor: $\frac{x-1}{(x-1)(x+1)}$; the $(x-1)$ cancels, creating a removable discontinuity (hole) at $x = 1$, while $x = -1$ is a vertical asymptote.

62. Which function is even?
    • $f(x) = 2x - 1$
    • $f(x) = x$
    • $f(x) = x^{3}$
    • $f(x) = x^{2} + 1$

    Correct answer: $f(x) = x^{2} + 1$

    A function is even if $f(-x) = f(x)$. For $x^{2} + 1$, $(-x)^{2} + 1 = x^{2} + 1$, so it is even.

63. What is the horizontal asymptote of $f(x) = \frac{3x^{2} + 1}{x^{2} - 4}$?
    • $y = 0$
    • $y = 1$
    • $y = 3$
    • None

    Correct answer: $y = 3$

    When degrees of numerator and denominator are equal, the asymptote is the ratio of leading coefficients: $\frac{3}{1} = 3$.

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

    Correct answer: 6

    $f(3) = 8$ and $f(1) = 2$, so the difference is 6.

65. The exponential function $f(x) = 100 \cdot (0.5)^{x}$ represents decay. What is $f(2)$?
    • 50
    • 100
    • 25
    • 12.5

    Correct answer: 25

    $f(2) = 100 \cdot (0.5)^{2} = 100 \cdot 0.25 = 25$.

66. What transformation maps $y = x^{2}$ to $y = (x + 4)^{2}$?
    • Shift down 4
    • Shift up 4
    • Shift left 4
    • Shift right 4

    Correct answer: Shift left 4

    Replacing x with x + 4 shifts the graph left by 4 units.

67. What is the y-intercept of $f(x) = 3x^{2} - 2x + 5$?
    • 5
    • -2
    • 0
    • 3

    Correct answer: 5

    The y-intercept occurs at $x = 0$: $f(0) = 5$.

68. A function f is one-to-one if and only if:
    • Every horizontal line crosses it at most once
    • It passes the vertical line test
    • It is continuous
    • It is increasing then decreasing

    Correct answer: Every horizontal line crosses it at most once

    A one-to-one function passes the horizontal line test, meaning each output corresponds to at most one input.

69. What is the value of $\tan(45^\circ)$?
    • 1
    • $\sqrt{2}$
    • $\frac{1}{2}$
    • 0

    Correct answer: 1

    $\tan(45^\circ) = \frac{\sin(45^\circ)}{\cos(45^\circ)} = 1$.

70. The piecewise function $f(x) = x$ for $x < 0$ and $f(x) = x^{2}$ for $x \ge 0$. What is $f(-3) + f(2)$?
    • -1
    • $-3+4=1$
    • 7
    • 1

    Correct answer: 1

    $f(-3) = -3$ and $f(2) = 4$, so the sum is 1.

71. What is the inverse of f(x) = log base 2 of x?
    • $x^{2}$
    • $2^{x}$
    • $\frac{1}{x}$
    • Log base 10 of x

    Correct answer: $2^{x}$

    The inverse of a logarithmic function base 2 is the exponential function $2^{x}$.

72. For $f(x) = x^{2} - 4x$, on what interval is $f$ decreasing?
    • $x > 2$
    • All $x$
    • $x < 0$
    • $x < 2$

    Correct answer: $x < 2$

    The vertex is at $x = 2$; a parabola opening up decreases to the left of its vertex, so $f$ decreases for $x < 2$.

73. What is the value of $\sin(\frac{\pi}{6})$?
    • 1
    • $\frac{\sqrt{3}}{2}$
    • $\frac{1}{2}$
    • $\frac{\sqrt{2}}{2}$

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

    $\sin(\frac{\pi}{6}) = \sin(30^\circ) = \frac{1}{2}$.

74. Which is the correct factorization showing the zeros of $f(x) = x^{2} - x - 6$?
    • $(x-6)(x+1)$, zeros 6 and -1
    • $(x-2)(x-3)$, zeros 2 and 3
    • $(x+3)(x-2)$, zeros -3 and 2
    • $(x-3)(x+2)$, zeros 3 and -2

    Correct answer: $(x-3)(x+2)$, zeros 3 and -2

    $x^{2} - x - 6 = (x-3)(x+2)$, giving zeros at $x = 3$ and $x = -2$.

75. The function $f(x) = \cos(x)$ has what maximum value?
    • 0
    • 1
    • $\pi$
    • 2

    Correct answer: 1

    The cosine function oscillates between -1 and 1, so its maximum value is 1.

76. What is the average rate of change of $f(x) = x^{2}$ on the interval [1, 4]?
    • 3
    • 15
    • 5
    • 7

    Correct answer: 5

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

77. How many real zeros does $f(x) = x^{2} + 4$ have?
    • 4
    • 0
    • 1
    • 2

    Correct answer: 0

    $x^{2} + 4 = 0$ gives $x^{2} = -4$, which has no real solutions, so there are 0 real zeros.

78. What is the limit as $x$ approaches 2 of $\frac{x^{2} - 4}{x - 2}$?
    • 4
    • Undefined
    • 0
    • 2

    Correct answer: 4

    Factor: $\frac{(x-2)(x+2)}{x-2} = x + 2$, so the limit is $2 + 2 = 4$.

79. What is the derivative of $f(x) = 3x^{2} + 2x - 5$?
    • $6x + 2$
    • $6x - 5$
    • $6x^{2} + 2$
    • $3x + 2$

    Correct answer: $6x + 2$

    $\frac{d}{dx}[3x^{2}] = 6x$, $\frac{d}{dx}[2x] = 2$, and the constant differentiates to 0.

80. What is the integral of $2x \, dx$?
    • $x^{2} + C$
    • $2 + C$
    • $2x^{2} + C$
    • $\frac{x^{2}}{2} + C$

    Correct answer: $x^{2} + C$

    The antiderivative of $2x$ is $x^{2}$, plus a constant of integration $C$.

81. What is the derivative of $f(x) = \sin(x)$?
    • $\cos(x)$
    • $-\sin(x)$
    • $-\cos(x)$
    • $\tan(x)$

    Correct answer: $\cos(x)$

    The derivative of $\sin(x)$ is $\cos(x)$.

82. Evaluate the definite integral from 0 to 2 of $x \, dx$.
    • 1
    • 4
    • 8
    • 2

    Correct answer: 2

    Integral of $x$ is $\frac{x^{2}}{2}$; evaluated from 0 to 2: $\frac{4}{2} - 0 = 2$.

83. What is the slope of the tangent line to $f(x) = x^{2}$ at $x = 3$?
    • 9
    • 3
    • 6
    • 12

    Correct answer: 6

    $f'(x) = 2x$, so $f'(3) = 6$.

84. Using the chain rule, what is the derivative of $f(x) = (2x + 1)^{3}$?
    • $6(2x+1)^{3}$
    • $6(2x+1)^{2}$
    • $3(2x+1)^{2}$
    • $2(2x+1)^{3}$

    Correct answer: $6(2x+1)^{2}$

    $f'(x) = 3(2x+1)^{2} \cdot 2 = 6(2x+1)^{2}$.

85. What is the derivative of $f(x) = e^{x}$?
    • $e^{x-1}$
    • $x \cdot e^{x-1}$
    • $x \cdot e^{x}$
    • $e^{x}$

    Correct answer: $e^{x}$

    The exponential function $e^{x}$ is its own derivative.

86. At a critical point where $f'(x) = 0$ and $f''(x) > 0$, the function has a:
    • Saddle point
    • Local minimum
    • Inflection point
    • Local maximum

    Correct answer: Local minimum

    By the second derivative test, $f''(x) > 0$ indicates concave up, so it is a local minimum.

87. What is the derivative of $f(x) = \ln(x)$?
    • $e^{x}$
    • $\frac{1}{x}$
    • $x$
    • $\frac{\ln(x)}{x}$

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

    The derivative of the natural logarithm $\ln(x)$ is $\frac{1}{x}$.

88. What is the limit as $x$ approaches infinity of $\frac{3x + 1}{x - 2}$?
    • 3
    • 1
    • 0
    • Infinity

    Correct answer: 3

    For equal-degree rational functions, the limit equals the ratio of leading coefficients: $\frac{3}{1} = 3$.

89. Find the integral of $\cos(x) \, dx$.
    • $-\cos(x) + C$
    • $-\sin(x) + C$
    • $\cos(x) + C$
    • $\sin(x) + C$

    Correct answer: $\sin(x) + C$

    The antiderivative of $\cos(x)$ is $\sin(x) + C$.

90. What is the second derivative of $f(x) = x^{4}$?
    • $4x^{2}$
    • $4x^{3}$
    • $24x$
    • $12x^{2}$

    Correct answer: $12x^{2}$

    $f'(x) = 4x^{3}$ and $f''(x) = 12x^{2}$.

91. Using the product rule, what is the derivative of $f(x) = x\sin(x)$?
    • $\sin(x) + x\cos(x)$
    • $\sin(x) - x\cos(x)$
    • $x\cos(x)$
    • $\cos(x)$

    Correct answer: $\sin(x) + x\cos(x)$

    $f'(x) = (1)(\sin x) + (x)(\cos x) = \sin(x) + x\cos(x)$.

92. What is the area under $f(x) = 3$ from $x = 1$ to $x = 5$?
    • 8
    • 12
    • 15
    • 3

    Correct answer: 12

    The area of the rectangle is height times width $= 3 \cdot (5-1) = 12$.

93. What does the Mean Value Theorem guarantee for a function continuous on [a,b] and differentiable on (a,b)?
    • A maximum value
    • A point where f' equals the average rate of change
    • A zero of the function
    • An inflection point

    Correct answer: A point where f' equals the average rate of change

    The MVT guarantees a point $c$ where $f'(c)$ equals $\frac{f(b)-f(a)}{b-a}$, the average rate of change.

94. What is the derivative of $f(x) = \frac{1}{x}$?
    • $\ln(x)$
    • $-x^{-2} + C$
    • $-\frac{1}{x^{2}}$
    • $\frac{1}{x^{2}}$

    Correct answer: $-\frac{1}{x^{2}}$

    Rewrite as $x^{-1}$; the derivative is $-1 \cdot x^{-2} = -\frac{1}{x^{2}}$.

95. Evaluate the integral from 1 to 3 of $2x \, dx$.
    • 10
    • 6
    • 4
    • 8

    Correct answer: 8

    Antiderivative is $x^{2}$; evaluated $3^{2} - 1^{2} = 9 - 1 = 8$.

96. What is the limit as $x$ approaches 0 of $\frac{\sin(x)}{x}$?
    • 0
    • Undefined
    • 1
    • Infinity

    Correct answer: 1

    This is a standard limit: the limit of $\frac{\sin(x)}{x}$ as $x$ approaches 0 equals 1.

97. A particle's position is $s(t) = t^{2} - 4t$. What is its velocity at $t = 3$?
    • 1
    • 3
    • 6
    • 2

    Correct answer: 2

    Velocity $v(t) = s'(t) = 2t - 4$; $v(3) = 6 - 4 = 2$.

98. The function $f(x) = x^{3} - 3x$ has a local maximum at which x-value?
    • $x = 0$
    • $x = -1$
    • $x = 3$
    • $x = 1$

    Correct answer: $x = -1$

    $f'(x) = 3x^{2} - 3 = 0$ gives $x = \pm 1$; $f''(x) = 6x$ is negative at $x = -1$, so there is a local max there.

99. What is the derivative of $f(x) = 5$ (a constant)?
    • $5x$
    • 0
    • 5
    • 1

    Correct answer: 0

    The derivative of any constant is 0.

100. What is the integral of $x^{2} \, dx$?
     • $2x + C$
     • $3x^{3} + C$
     • $x^{3} + C$
     • $\frac{x^{3}}{3} + C$

     Correct answer: $\frac{x^{3}}{3} + C$

     Using the power rule for integration, the antiderivative of $x^{2}$ is $\frac{x^{3}}{3} + C$.

101. Which of the following indicates a function is concave up on an interval?
     • f'(x) > 0
     • f''(x) < 0
     • f''(x) > 0
     • f'(x) < 0

     Correct answer: f''(x) > 0

     A positive second derivative means the function is concave up.

102. What is the area of a circle with radius 7?
     • $14\pi$
     • $21\pi$
     • $7\pi$
     • $49\pi$

     Correct answer: $49\pi$

     Area = $\pi r^{2} = \pi \cdot 49 = 49\pi$.

103. What is the sum of the interior angles of a hexagon?
     • 720 degrees
     • 540 degrees
     • 900 degrees
     • 1080 degrees

     Correct answer: 720 degrees

     Sum = $(n-2) \cdot 180 = (6-2) \cdot 180 = 720$ degrees.

104. In a right triangle with legs 6 and 8, what is the hypotenuse?
     • 12
     • 14
     • 48
     • 10

     Correct answer: 10

     By the Pythagorean theorem, $\sqrt{6^{2} + 8^{2}} = \sqrt{36+64} = \sqrt{100} = 10$.

105. What is the volume of a cube with edge length 4?
     • 48
     • 16
     • 12
     • 64

     Correct answer: 64

     Volume = $s^{3} = 4^{3} = 64$.

106. The measure of each interior angle of a regular pentagon is:
     • 120 degrees
     • 108 degrees
     • 72 degrees
     • 135 degrees

     Correct answer: 108 degrees

     Each interior angle = $\frac{(n-2) \cdot 180}{n} = \frac{540}{5} = 108$ degrees.

107. What is the distance between points $(1, 2)$ and $(4, 6)$?
     • 5
     • 25
     • $\sqrt{7}$
     • 7

     Correct answer: 5

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

108. What is the midpoint of the segment from $(2, 4)$ to $(8, 10)$?
     • $(10, 14)$
     • $(5, 7)$
     • $(4, 5)$
     • $(6, 6)$

     Correct answer: $(5, 7)$

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

109. Two triangles are similar with a scale factor of 3. If the smaller has area 5, what is the larger's area?
     • 9
     • 30
     • 15
     • 45

     Correct answer: 45

     Area scales by the square of the linear scale factor: $5 \cdot 3^{2} = 5 \cdot 9 = 45$.

110. What is the circumference of a circle with diameter 10?
     • $10\pi$
     • $5\pi$
     • $20\pi$
     • $100\pi$

     Correct answer: $10\pi$

     Circumference = $\pi d = 10\pi$.

111. In a 30-60-90 triangle, if the shortest side is 5, what is the hypotenuse?
     • $5\sqrt{3}$
     • $10\sqrt{3}$
     • $5\sqrt{2}$
     • 10

     Correct answer: 10

     In a 30-60-90 triangle, the hypotenuse is twice the shortest side: $2 \cdot 5 = 10$.

112. What is the volume of a cylinder with radius 3 and height 5?
     • $30\pi$
     • $45\pi$
     • $15\pi$
     • $90\pi$

     Correct answer: $45\pi$

     Volume = $\pi r^{2} h = \pi \cdot 9 \cdot 5 = 45\pi$.

113. The equation of a circle centered at $(2, -3)$ with radius 4 is:
     • $(x-2)^{2} + (y-3)^{2} = 16$
     • $(x+2)^{2} + (y-3)^{2} = 16$
     • $(x-2)^{2} + (y+3)^{2} = 4$
     • $(x-2)^{2} + (y+3)^{2} = 16$

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

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

114. What is the surface area of a sphere with radius 3?
     • $27\pi$
     • $12\pi$
     • $9\pi$
     • $36\pi$

     Correct answer: $36\pi$

     Surface area = $4\pi r^{2} = 4\pi \cdot 9 = 36\pi$.

115. In a triangle, two angles measure 50 and 60 degrees. What is the third angle?
     • 70 degrees
     • 90 degrees
     • 60 degrees
     • 80 degrees

     Correct answer: 70 degrees

     Angles in a triangle sum to 180: $180 - 50 - 60 = 70$ degrees.

116. What is the area of a trapezoid with parallel sides 6 and 10 and height 4?
     • 16
     • 40
     • 32
     • 64

     Correct answer: 32

     Area = $\frac{1}{2}(b_{1}+b_{2})(h) = \frac{1}{2}(16)(4) = 32$.

117. A central angle of 90 degrees in a circle of radius 4 subtends an arc of what length?
     • $8\pi$
     • $4\pi$
     • $\pi$
     • $2\pi$

     Correct answer: $2\pi$

     Arc length = $\frac{90}{360} \cdot 2\pi r = \frac{1}{4} \cdot 8\pi = 2\pi$.

118. Which transformation preserves both distance and angle measure?
     • Stretch
     • Dilation
     • Projection
     • Rigid motion (isometry)

     Correct answer: Rigid motion (isometry)

     Rigid motions (translations, rotations, reflections) preserve distances and angles.

119. What is the volume of a cone with radius 3 and height 4?
     • $9\pi$
     • $36\pi$
     • $12\pi$
     • $48\pi$

     Correct answer: $12\pi$

     Volume = $\frac{1}{3}\pi r^{2} h = \frac{1}{3}\pi \cdot 9 \cdot 4 = 12\pi$.

120. Two angles are complementary. If one is 35 degrees, the other is:
     • 145 degrees
     • 125 degrees
     • 55 degrees
     • 65 degrees

     Correct answer: 55 degrees

     Complementary angles sum to 90 degrees: $90 - 35 = 55$ degrees.

121. What is the slope of a line parallel to $y = -2x + 7$?
     • $\frac{1}{2}$
     • -2
     • $-\frac{1}{2}$
     • 2

     Correct answer: -2

     Parallel lines have equal slopes, so the slope is -2.

122. A rectangle has a diagonal of 13 and a width of 5. What is its length?
     • 8
     • 144
     • 10
     • 12

     Correct answer: 12

     By the Pythagorean theorem, length = $\sqrt{13^{2} - 5^{2}} = \sqrt{169-25} = \sqrt{144} = 12$.

123. What is the measure of an exterior angle of a regular octagon?
     • 135 degrees
     • 40 degrees
     • 60 degrees
     • 45 degrees

     Correct answer: 45 degrees

     Exterior angle = $\frac{360}{n} = \frac{360}{8} = 45$ degrees.

124. In similar triangles, corresponding sides are 4 and 6. If another side of the first triangle is 10, the corresponding side of the second is:
     • 15
     • 12
     • 20
     • 8

     Correct answer: 15

     The scale factor is $\frac{6}{4} = 1.5$, so $10 \cdot 1.5 = 15$.

125. What is the law of cosines used to find in a triangle with sides a, b and included angle C?
     • Angle A
     • The perimeter
     • Side c
     • The area

     Correct answer: Side c

     The law of cosines, $c^{2} = a^{2} + b^{2} - 2ab\cos(C)$, finds the side opposite the included angle.

126. A vector has components $(3, 4)$. What is its magnitude?
     • $\sqrt{7}$
     • 7
     • 12
     • 5

     Correct answer: 5

     Magnitude = $\sqrt{3^{2} + 4^{2}} = \sqrt{25} = 5$.

127. What is the probability of rolling a sum of 7 with two fair six-sided dice?
     • $\frac{1}{12}$
     • $\frac{7}{36}$
     • $\frac{5}{36}$
     • $\frac{1}{6}$

     Correct answer: $\frac{1}{6}$

     There are 6 ways to roll a 7 out of 36 outcomes, so the probability is $\frac{6}{36} = \frac{1}{6}$.

128. What is the mean of the data set 4, 8, 6, 10, 2?
     • 5
     • 7
     • 8
     • 6

     Correct answer: 6

     Mean = $\frac{4+8+6+10+2}{5} = \frac{30}{5} = 6$.

129. What is the median of 3, 7, 9, 12, 15?
     • 9.2
     • 9
     • 7
     • 12

     Correct answer: 9

     The middle value of the ordered set of 5 numbers is 9.

130. A bag has 3 red and 5 blue marbles. What is the probability of drawing a red marble?
     • $\frac{3}{8}$
     • $\frac{3}{5}$
     • $\frac{5}{8}$
     • $\frac{1}{3}$

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

     $P(\text{red}) = \frac{3}{3+5} = \frac{3}{8}$.

131. What is the mode of 2, 4, 4, 6, 8, 4, 10?
     • 8
     • 4
     • 6
     • 5.4

     Correct answer: 4

     The value 4 appears most frequently (three times), so it is the mode.

132. If two events are independent with P(A) = 0.5 and P(B) = 0.4, what is P(A and B)?
     • 0.1
     • 0.2
     • 0.9
     • 0.45

     Correct answer: 0.2

     For independent events, $P(A \text{ and } B) = P(A) \cdot P(B) = 0.5 \cdot 0.4 = 0.2$.

133. In how many ways can 4 books be arranged on a shelf?
     • 12
     • 24
     • 16
     • 4

     Correct answer: 24

     The number of arrangements is $4! = 4 \cdot 3 \cdot 2 \cdot 1 = 24$.

134. What is the value of 6 choose 2 (combinations)?
     • 15
     • 12
     • 30
     • 36

     Correct answer: 15

     6 choose 2 = $\frac{6!}{2! \cdot 4!} = \frac{6 \cdot 5}{2} = 15$.

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

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

     There are 3 favorable outcomes (HHT, HTH, THH) out of 8, so the probability is $\frac{3}{8}$.

136. What is the range of the data set 12, 5, 18, 7, 20?
     • 15
     • 13
     • 18
     • 20

     Correct answer: 15

     Range = max - min = 20 - 5 = 15.

137. In a normal distribution, approximately what percent of data falls within one standard deviation of the mean?
     • 95%
     • 68%
     • 99.7%
     • 50%

     Correct answer: 68%

     By the empirical rule, about 68% of data falls within one standard deviation of the mean.

138. A die is rolled. What is the probability of rolling an even number or a 3?
     • $\frac{1}{2}$
     • $\frac{1}{3}$
     • $\frac{5}{6}$
     • $\frac{2}{3}$

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

     Even numbers $\{2,4,6\}$ plus 3 gives 4 favorable outcomes out of 6, so $\frac{4}{6} = \frac{2}{3}$.

139. What is the expected value of rolling a single fair six-sided die?
     • 6
     • 3.5
     • 4
     • 3

     Correct answer: 3.5

     Expected value = $\frac{1+2+3+4+5+6}{6} = \frac{21}{6} = 3.5$.

140. A class has 10 boys and 15 girls. What fraction of the class is boys?
     • $\frac{2}{3}$
     • $\frac{3}{5}$
     • $\frac{1}{2}$
     • $\frac{2}{5}$

     Correct answer: $\frac{2}{5}$

     Boys fraction = $\frac{10}{25} = \frac{2}{5}$.

141. If P(A) = 0.7, what is the probability of the complement of A?
     • 0.07
     • 0.7
     • 1.0
     • 0.3

     Correct answer: 0.3

     $P(\text{not } A) = 1 - P(A) = 1 - 0.7 = 0.3$.

142. How many distinct ways can the letters in the word 'MATH' be arranged?
     • 24
     • 12
     • 4
     • 16

     Correct answer: 24

     All 4 letters are distinct, so there are $4! = 24$ arrangements.

143. A survey reports a sample mean of 50 with standard error 2. A 95% confidence interval is approximately:
     • 50 to 54
     • 45 to 55
     • 46 to 54
     • 48 to 52

     Correct answer: 46 to 54

     Using roughly 2 standard errors: $50 \pm 2 \cdot 2 = 50 \pm 4$, giving 46 to 54.

144. What is the probability of drawing an ace from a standard 52-card deck?
     • $\frac{4}{13}$
     • $\frac{1}{52}$
     • $\frac{1}{13}$
     • $\frac{1}{4}$

     Correct answer: $\frac{1}{13}$

     There are 4 aces out of 52 cards: $\frac{4}{52} = \frac{1}{13}$.

145. Which measure of center is most resistant to outliers?
     • Median
     • Mean
     • Range
     • Standard deviation

     Correct answer: Median

     The median is based on position and is not affected by extreme values, making it resistant to outliers.

146. Two cards are drawn without replacement from a deck. What is the probability both are hearts?
     • $\frac{1}{4} \cdot \frac{1}{4}$
     • $\frac{13}{52} + \frac{12}{51}$
     • $\frac{13}{52} \cdot \frac{12}{51}$
     • $\frac{13}{52} \cdot \frac{13}{52}$

     Correct answer: $\frac{13}{52} \cdot \frac{12}{51}$

     Without replacement, multiply the conditional probabilities: $\frac{13}{52} \cdot \frac{12}{51}$.

147. In a graph theory context, what is the degree of a vertex?
     • The number of edges incident to it
     • The length of the longest path
     • The number of faces
     • The number of vertices

     Correct answer: The number of edges incident to it

     The degree of a vertex is the number of edges connected to it.

148. What is the next term in the recursive sequence $a_{1} = 2$, $a_{n} = 2a_{n-1} + 1$?
     • $a_{2} = 4$
     • $a_{2} = 5$
     • $a_{2} = 3$
     • $a_{2} = 6$

     Correct answer: $a_{2} = 5$

     $a_{2} = 2a_{1} + 1 = 2 \cdot 2 + 1 = 5$.

149. How many subsets does a set with 4 elements have?
     • 16
     • 12
     • 24
     • 8

     Correct answer: 16

     A set with n elements has $2^{n}$ subsets: $2^{4} = 16$.

150. In propositional logic, what is the negation of 'p AND q'?
     • Not p AND q
     • Not p AND not q
     • Not p OR not q
     • P OR q

     Correct answer: Not p OR not q

     By De Morgan's law, the negation of (p AND q) is (not p OR not q).

151. What is the sum $1 + 2 + 4 + 8 + 16$ (a geometric series)?
     • 32
     • 31
     • 30
     • 63

     Correct answer: 31

     This is a geometric series; the sum is $2^{5} - 1 = 32 - 1 = 31$.

152. A binary number 1011 equals what in decimal?
     • 11
     • 9
     • 13
     • 15

     Correct answer: 11

     1011 in binary $= 8 + 0 + 2 + 1 = 11$.

153. In set notation, what is the intersection of $\{1,2,3,4\}$ and $\{3,4,5,6\}$?
     • $\{1,2,5,6\}$
     • $\{1,2,3,4,5,6\}$
     • $\{3, 4\}$
     • Empty set

     Correct answer: $\{3, 4\}$

     The intersection contains elements common to both sets: $\{3, 4\}$.

154. How many edges does a complete graph on 5 vertices (K5) have?
     • 5
     • 15
     • 20
     • 10

     Correct answer: 10

     A complete graph has $\frac{n(n-1)}{2}$ edges: $\frac{5 \cdot 4}{2} = 10$.

155. What is 5! (5 factorial)?
     • 25
     • 60
     • 720
     • 120

     Correct answer: 120

     $5! = 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120$.

156. In a truth table, the statement 'if p then q' is false only when:
     • Both are true
     • P is true and q is false
     • P is false and q is true
     • Both are false

     Correct answer: P is true and q is false

     A conditional p implies q is false only when the hypothesis p is true and the conclusion q is false.

157. What is the union of sets $A = \{1, 2\}$ and $B = \{2, 3, 4\}$?
     • $\{1, 2, 3, 4\}$
     • $\{1, 3, 4\}$
     • $\{2\}$
     • $\{1, 2, 2, 3, 4\}$

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

     The union combines all distinct elements: $\{1, 2, 3, 4\}$.

158. Using mathematical induction to prove a statement, the base case typically establishes the statement for:
     • Even n only
     • N = infinity
     • All n
     • N = 1 (or the smallest value)

     Correct answer: N = 1 (or the smallest value)

     The base case proves the statement true for the smallest value, usually n = 1.

159. How many ways can you choose a president and a vice-president from a group of 5 people (order matters)?
     • 25
     • 20
     • 10
     • 120

     Correct answer: 20

     This is a permutation: $5 \cdot 4 = 20$ ways.

160. The contrapositive of 'if p then q' is:
     • If not p then q
     • If not p then not q
     • If not q then not p
     • If q then p

     Correct answer: If not q then not p

     The contrapositive of p implies q is (not q) implies (not p), and it is logically equivalent.

161. A teacher wants students to discover the relationship between a quadratic's coefficients and its graph. The most appropriate use of the on-screen graphing calculator is to:
     • Find prime factorizations
     • Have students compute factorials
     • Calculate the standard deviation
     • Let students vary parameters and observe how the parabola changes

     Correct answer: Let students vary parameters and observe how the parabola changes

     A dynamic exploration where students adjust coefficients and observe graphical effects supports conceptual discovery, an effective instructional use of the graphing tool.

162. A student claims that squaring always makes a number larger. Which counterexample best refutes this?
     • $(0.5)^{2} = 0.25$
     • $10^{2} = 100$
     • $(-3)^{2} = 9$
     • $5^{2} = 25$

     Correct answer: $(0.5)^{2} = 0.25$

     Squaring 0.5 gives 0.25, which is smaller, demonstrating the claim is false for numbers between 0 and 1.