EST II MATHEMATICS — LEVEL 2 HARD PRACTICE MOCK EXAM — 2026 SPECIFICATIONS 60 Questions | Five Choices | A–E

 

EST II MATHEMATICS — LEVEL 2

HARD PRACTICE MOCK EXAM — 2026 SPECIFICATIONS

60 Questions | Five Choices | A–E

Directions:
Choose the best answer for each question. Each question has five choices: A, B, C, D, and E.

Calculator: A scientific or graphing calculator may be used.

Difficulty: Hard — emphasis on application, reasoning, algebraic manipulation, functions, modeling, logarithms, trigonometry, coordinates, statistics, and probability.

NUMERATION AND OPERATIONS

1. If 12^m/6! is an integer, what is the smallest positive integer value of m?

A) 12

B) 20

C) 60

D) 120

E) 720

Answer: C) 60

2. If z = 3 + 4i, what is |z²|?

A) 5

B) 7

C) 10

D) 25

E) 50

Answer: D) 25

3. If z = 2 − 3i, which expression is equal to z + conjugate(z)?

A) −6i

B) −4

C) 0

D) 2

E) 4

Answer: E) 4

4. The matrix A = [[2, 1], [3, 4]]. What is det(A)?

A) 3

B) 5

C) 7

D) 8

E) 11

Answer: B) 5

5. If A = [[1,2],[0,3]] and B = [[2,1],[4,0]], what is the first entry of AB?

A) 4

B) 6

C) 8

D) 10

E) 12

Answer: B) 10

6. A geometric sequence has first term 5 and common ratio −2. What is its sixth term?

A) −80

B) −160

C) 80

D) 160

E) 320

Answer: B) −160

7. If the sum of the first n terms of an arithmetic sequence is S_n = n(3n + 1)/2, what is the fifth term?

A) 11

B) 13

C) 14

D) 15

E) 16

Answer: C) 14

8. If u = (2, −1, 3) and v = (−1, 4, 2), what is u · v?

A) −4

B) 0

C) 4

D) 8

E) 12

Answer: C) 0

ALGEBRA AND FUNCTIONS

9. If log₂x = 3 and log_xy = 2, what is y?

A) 16

B) 32

C) 64

D) 128

E) 256

Answer: C) 64

10. If log₃(x − 1) + log₃(x + 1) = 2, what is x?

A) √10

B) √11

C) √12

D) 4

E) 5

Answer: B) √10

11. Solve 2^x+1 = 32.

A) 3

B) 4

C) 5

D) 6

E) 7

Answer: B) 4

12. The equation e^0.4t = 5 models exponential growth. Approximately what is t?

A) 2.01

B) 3.02

C) 4.02

D) 4.52

E) 5.02

Answer: C) 4.02

13. If f(x) = x³ − 4x + 7, what is f′(2)?

A) 4

B) 6

C) 8

D) 10

E) 12

Answer: C) 8

14. If f(x) = x² + 3x and g(x) = 2x − 1, what is f(g(2))?

A) 15

B) 18

C) 20

D) 24

E) 30

Answer: D) 24

15. A function has derivative f′(x) = 3x² − 4x − 7. At which value of x can the function have a local maximum?

A) −1

B) 1

C) 7/3

D) 3

E) 4

Answer: A) −1

16. If f(x) = (2x + 3)/(x − 4), which value is excluded from the domain?

A) −4

B) −3/2

C) 0

D) 3

E) 4

Answer: E) 4

17. If f(x) = (x − 2)/(x + 1), what is f⁻¹(x)?

A) (x + 2)/(1 − x)

B) (x + 2)/(x − 1)

C) (x − 2)/(x + 1)

D) (2x − 1)/(x + 2)

E) (x + 1)/(x − 2)

Answer: B) (x + 2)/(1 − x)

18. If f(x) = |x − 3| + 2, what is the minimum value of f(x)?

A) −2

B) 0

C) 2

D) 3

E) 5

Answer: C) 2

19. The equation x³ − 6x² + 11x − 6 = 0 has three roots. What is their product?

A) 1

B) 3

C) 6

D) 11

E) 18

Answer: C) 6

20. If x = 2 is a root of x³ + ax² − 5x + 6 = 0, what is a?

A) −4

B) −3

C) −2

D) 2

E) 3

Answer: B) −3

21. The function f(x) = x⁴ − 5x² + 4 has how many real zeros?

A) 1

B) 2

C) 3

D) 4

E) 5

Answer: D) 4

22. If f(x) = 2x + 1 and g(x) = x² − 3, for which x is f(g(x)) = 11?

A) −3 only

B) 3 only

C) −2 or 2

D) −3 or 3

E) 2 or 3

Answer: D) −3 or 3

23. A population is modeled by P(t) = 1200(1.06)^t. Approximately how long will it take the population to double?

A) 8.2 years

B) 10.5 years

C) 11.9 years

D) 12.7 years

E) 15.2 years

Answer: C) 11.9 years

24. If y = ln(x² + 1), what is dy/dx?

A) 1/(x² + 1)

B) 2x/(x² + 1)

C) 2x ln(x)

D) ln(2x)

E) x/(x² + 1)

Answer: B) 2x/(x² + 1)

25. If y = (3x² + 1)⁵, what is dy/dx?

A) 5(3x² + 1)⁴

B) 15x(3x² + 1)⁴

C) 30x(3x² + 1)⁴

D) 15x(3x² + 1)⁵

E) 30x(3x² + 1)⁵

Answer: C) 30x(3x² + 1)⁴

26. The position of an object is s(t) = t³ − 6t² + 9t. At what positive time is its velocity zero?

A) 1

B) 2

C) 3

D) 4

E) 6

Answer: C) 1 and 3

27. If f′(x) > 0 throughout an interval, what must be true about f on that interval?

A) f is decreasing.

B) f is constant.

C) f is increasing.

D) f has a maximum.

E) f has a minimum.

Answer: C) f is increasing.

28. Which equation describes a function with period 2π/3?

A) y = sin(3x)

B) y = sin(x/3)

C) y = 3sin(x)

D) y = cos(x) + 3

E) y = sin(2x)

Answer: A) y = sin(3x)

COORDINATE SYSTEM AND 3-DIMENSIONAL REASONING

29. What is the distance between the points (1, 2, 3) and (4, 6, 15)?

A) √25

B) √34

C) √169

D) 13

E) 15

Answer: D) 13

30. What is the midpoint of (−2, 5, 4) and (6, −1, 10)?

A) (2, 2, 7)

B) (4, 4, 7)

C) (2, 4, 6)

D) (−4, 6, 14)

E) (8, 4, 14)

Answer: A) (2, 2, 7)

31. Which equation represents the sphere centered at (2, −1, 3) with radius 4?

A) (x+2)²+(y−1)²+(z+3)²=16

B) (x−2)²+(y+1)²+(z−3)²=16

C) (x−2)²+(y−1)²+(z−3)²=4

D) (x+2)²+(y+1)²+(z+3)²=4

E) (x−2)²+(y+1)²+(z−3)²=4

Answer: B) (x−2)²+(y+1)²+(z−3)²=16

32. What is the magnitude of the vector v = (−6, 2, 3)?

A) 5

B) 6

C) 7

D) √49

E) √41

Answer: C) 7

33. Which conic has equation x²/25 + y²/9 = 1?

A) Circle

B) Parabola

C) Ellipse

D) Hyperbola

E) Line

Answer: C) Ellipse

34. Which equation represents a hyperbola?

A) x² + y² = 25

B) x²/9 + y²/16 = 1

C) y² = 8x

D) x²/9 − y²/16 = 1

E) y = 3x + 2

Answer: D) x²/9 − y²/16 = 1

35. In polar coordinates, what point is represented by (4, π)?

A) (4, 0)

B) (−4, 0)

C) (0, 4)

D) (0, −4)

E) (−4, −4)

Answer: B) (−4, 0)

36. A point has polar coordinates (6, π/2). What are its rectangular coordinates?

A) (6, 0)

B) (−6, 0)

C) (0, 6)

D) (0, −6)

E) (3, 3)

Answer: C) (0, 6)

TRIGONOMETRY

37. If sin θ = 3/5 and θ is in the first quadrant, what is cos θ?

A) 2/5

B) 3/5

C) 4/5

D) 5/3

E) 5/4

Answer: C) 4/5

38. If tan θ = 3/4 and θ is in the first quadrant, what is sec θ?

A) 3/5

B) 4/5

C) 5/4

D) 5/3

E) 4/3

Answer: C) 5/4

39. What is the value of sin(2θ) if sin θ = 3/5 and cos θ = 4/5?

A) 7/25

B) 12/25

C) 16/25

D) 24/25

E) 1

Answer: D) 24/25

40. What is the radian measure of 150°?

A) 2π/3

B) 3π/4

C) 5π/6

D) 7π/6

E) 4π/3

Answer: C) 5π/6

41. What is arccos(sin 60°), in degrees?

A) 15°

B) 30°

C) 45°

D) 60°

E) 90°

Answer: B) 30°

42. If 1 − 2sin²θ = −1/3 and 0° ≤ θ ≤ 90°, what is cos(2θ)?

A) −1

B) −2/3

C) −1/3

D) 1/3

E) 2/3

Answer: C) −1/3

43. Which identity is always true?

A) sin²x − cos²x = 1

B) 1 + tan²x = sec²x

C) 1 + cot²x = sin²x

D) tan x = cos x/sin x

E) sec x = sin x

Answer: B) 1 + tan²x = sec²x

44. Solve 2sin x = √3 for 0 ≤ x ≤ 2π.

A) π/6, 5π/6

B) π/3, 2π/3

C) π/4, 3π/4

D) π/3, 5π/3

E) 2π/3, 4π/3

Answer: B) π/3, 2π/3

45. Using the law of cosines, if a = 5, b = 7, and the included angle is 60°, what is c²?

A) 39

B) 49

C) 59

D) 74

E) 84

Answer: C) 39

46. If sin x = 5/13 and x is in the second quadrant, what is cos x?

A) −12/13

B) −5/13

C) 5/13

D) 12/13

E) 13/12

Answer: A) −12/13

DATA ANALYSIS, STATISTICS AND PROBABILITY

47. Which data set has the smallest standard deviation?

A) 1, 2, 3, 4, 5

B) 9, 9, 9, 9, 9

C) 0, 2, 4, 6, 8

D) 3, 3, 3, 3, 7

E) −5, 0, 5, 10, 15

Answer: B) 9, 9, 9, 9, 9

48. A data set has mean 20 and standard deviation 4. If every value is multiplied by 3, what are the new mean and standard deviation?

A) Mean 23, SD 7

B) Mean 60, SD 4

C) Mean 60, SD 12

D) Mean 20, SD 12

E) Mean 60, SD 16

Answer: C) Mean 60, SD 12

49. A least-squares regression equation is y = 2.4x + 7. If x = 15, what is the predicted value of y?

A) 36

B) 39

C) 43

D) 45

E) 49

Answer: C) 43

50. If the correlation coefficient between x and y is r = 0.94, which statement is most appropriate?

A) There is a strong negative linear relationship.

B) There is a weak positive relationship.

C) There is a strong positive linear relationship.

D) x causes y to increase.

E) There is no relationship.

Answer: C) There is a strong positive linear relationship.

51. How many different arrangements can be made using 6 different books on a shelf?

A) 36

B) 120

C) 360

D) 720

E) 1440

Answer: D) 720

52. How many different groups of 4 students can be selected from 10 students?

A) 40

B) 120

C) 180

D) 210

E) 5040

Answer: D) 210

53. A fair coin is tossed 5 times. What is the probability of getting exactly 2 heads?

A) 1/8

B) 5/16

C) 3/8

D) 1/2

E) 5/8

Answer: B) 5/16

54. A bag contains 4 red, 5 blue, and 3 green balls. Two balls are selected without replacement. What is the probability that both are blue?

A) 5/33

B) 5/22

C) 10/33

D) 1/3

E) 25/144

Answer: B) 5/33

55. A regression model is y = 100(0.85)^x. What does 0.85 represent?

A) A 15% increase per unit of x

B) A 15% decrease per unit of x

C) An 85% increase per unit of x

D) The initial value of y

E) The slope of a linear function

Answer: B) A 15% decrease per unit of x

HARD MIXED EST-STYLE REASONING

56. If x + 1/x = 3, what is x³ + 1/x³?

A) 9

B) 12

C) 15

D) 18

E) 21

Answer: C) 18

57. A function is defined recursively by a₁ = 2 and aₙ₊₁ = 3aₙ − 1. What is a₄?

A) 41

B) 42

C) 43

D) 44

E) 45

Answer: C) 43

58. If f(x) = x² − 4x + 7, what is the minimum value of f(x)?

A) 1

B) 2

C) 3

D) 4

E) 7

Answer: C) 3

59. A quantity follows the model A(t) = 500e^0.08t. Approximately how long will it take for A(t) to reach 1,000?

A) 6.66

B) 8.66

C) 10.66

D) 12.66

E) 14.66

Answer: C) 8.66

60. If f(x) = x³ − 3x² − 9x + 5, at which value of x is f′(x) = 0 and the function changes from increasing to decreasing?

A) −1

B) 0

C) 1

D) 2

E) 3

Answer: D) 3