UNIVERSITY INTERSCHOLASTIC LEAGUE

Mathematics
Invitational A • 2023

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1. On his birthday, Darius received a $250 gift card from Academy Sports in Groom. He purchased
Pegasus running shoes for $128.00, a Nike running shirt for $24.95, Nike running shorts for $33.85,
and Nike socks for $6.95. If the tax rate is 8.25%, how much is left on his gift card?

(A) $40.15 (B) $40.19 (C) $40.23 (D) $40.27 (E) $40.31

2. Consider line AB with points A(8, 6) and B( −6, − 10) . If the point (1, b) lies on AB , then b = ___.

12 10 8 6
(A) −2 (B) − (C) − (D) − (E) −
7 7 7 7

3. Chay has a jar full of nickels, dimes and quarters. The jar has a total of 264 coins with a value of
$37.20. There are 20 more quarters than nickels in the jar. How many dimes are in the jar?

(A) 83 (B) 88 (C) 93 (D) 98 (E) 103

4. Which of the following values is not in the solution to 3  x + 7  12 ?

(A) −4 (B) −1 (C) 1 (D) 4 (E) 5

5. Samuel is solving the quadratic equation x 2 + 10x − 6 = 0 by completing the square. His second step
is x 2 + 10x + c = 6 + c . The value of c is __________.

(A) 5 (B) 6 (C) 10 (D) 12 (E) 25

6. The perimeter of ∆ABC shown on the right is ____. (nearest tenth) B

(A) 42.6 (B) 42.9 (C) 43.2 (D) 43.5 (E) 43.8

7. The coordinates of the midpoint of BC are (a, b) .

a + b = _______. (nearest tenth)
A
(A) 2.0 (B) 2.2 (C) 2.4 (D) 2.6 (E) 2.8
C

8. The area of ∆ABC is __________. (nearest tenth) Problems 6, 7, 8, 9

(A) 78.8 (B) 79.2 (C) 79.6 (D) 80.0 (E) 80.4

9. Point D (not shown) lies on AC such that BD bisects ABC . DC = _______. (nearest tenth)

(A) 6.6 (B) 6.8 (C) 7.0 (D) 7.2 (E) 7.4

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10. Four times the complement of A is 36° greater than the supplement of A . mA = ________.

(A) 44° (B) 46° (C) 48° (D) 50° (E) 52°

Problems 11-12. The base of a pyramid is a square with each side equal to 14 cm. The height is 10 cm.

11. The volume of the pyramid is ____________. (nearest whole number)

(A) 650 cm3 (B) 653 cm3 (C) 656 cm3 (D) 659 cm3 (E) 662 cm3

12. The total surface area of the pyramid is ________. (nearest whole number)

(A) 522 cm2 (B) 526 cm2 (C) 530 cm2 (D) 534 cm2 (E) 538 cm2

13. The volume of a cone is 768 cm 3 and the height of the cone is 12 cm. What is the diameter of the
circular base? (nearest whole number)

(A) 26.9 cm (B) 27.3 cm (C) 27.7 cm (D) 28.1 cm (E) 28.5 cm

14. The hypotenuse of an isosceles right triangle is 24.0416. The area of the triangle is ________.
(nearest tenth)

(A) 144.5 (B) 145.6 (C) 146.7 (D) 147.8 (E) 148.9

15. Jen can mow 3 large yards in 8 hr. Tom can mow 5 large yards in 16 hr. If they work together,
how long would it take them to mow 12 large yards? (nearest minute)

(A) 16 hr 35 min (B) 16 hr 48 min (C) 17 hr 1 min (D) 17 hr 14 min (E) 17 hr 27 min

5 1 1
16. Find the number that is of the way from −3 to 8 .
12 3 2

37 113 115 13 119

(A) (B) (C) (D) (E)
24 72 72 8 72

f (x + h) − f (x)
17. If f (x) = x 2 , then = __________.
h

2x
(A) 2x (B) 2x + h (C) (D) x2 + h (E) 2x + 2h
h

18. Dad’s age is two more than three times Abe’s age and Dad’s age is eight more than twice Connie’s
age. Connie is six years older than Abe. What is the sum of their ages?

(A) 90 (B) 92 (C) 94 (D) 96 (E) 98

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19. Find the total number of distinct diagonals that can be drawn from the vertices of a regular decagon?

(A) 24 (B) 32 (C) 35 (D) 36 (E) 42

20. Consider ∆ABC with point D on AC such that BD ⊥ AC . If mABC = 90 , AD = 10.8 and
DC = 19.2 , then BD = ________. (nearest tenth)

(A) 14.0 (B) 14.2 (C) 14.4 (D) 14.6 (E) 14.8

x 2 − 25
21. Find the domain of the function f (x) = .
x−6

(A) ( − , − 5]  [5,  ) (B) ( − , − 5)  (5, 6)  (6,  ) (C) ( − , − 5)  (5,  )

(D) ( − ,  ) (E) ( − , − 5]  [5, 6)  (6,  )

22. How many distinguishable permutations can be formed using the letters from the word
Massachusetts?

(A) 64,864,800 (B) 129,729,600 (C) 259,459,200 (D) 518,918,400 (E) 1,037,836,800

23. Consider the circle x 2 + y 2 − 8x + 10y − 8 = 0 . The area of the circle is __________.
(nearest whole number)

(A) 154 (B) 156 (C) 158 (D) 160 (E) 162

 
24. sin ( x ) tan  − x  = ___________.
2 

(A) cos(x) (B) sin(2x) (C) tan(x) (D) − tan(x) (E) − cos(x)

1 4
25. Consider a geometric sequence in which the first term is 22 and the fifth term is 4 . What is the
2 9
seventh term of the sequence?

52 158 160 164

(A) (B) (C) (D) 2 (E)
27 81 81 81

26. Find the eccentricity of the ellipse. 16x 2 + 25y 2 − 128x + 150y + 81 = 0 . (nearest hundredth)

(A) 0.60 (B) 0.67 (C) 0.75 (D) 0.80 (E) 0.83

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27. The pressure of an ideal gas varies directly with the temperature and inversely with the volume.
The values of the initial state were 2.0 atm, 6.0 L, and 300 K. What will the pressure be in the final
state if the volume is reduced to 2.0 L and the temperature is increased to 500 K? (nearest tenth)

(A) 9.8 atm (B) 10.0 atm (C) 10.2 atm (D) 10.4 atm (E) 10.6 atm

28. The value of my RAV4 Prime (plug-in hybrid) depreciates exponentially. I originally paid $55,200
on July 8, 2022. On July 8, 2025, the value had fallen to $51,500. If the value continues to depreciate
exponentially, what is the expected value of my RAV4 Prime on July 8, 2030?

(A) $45,416.16 (B) $45,531.17 (C) $45,646.18 (D) $45,761.19 (E) $45,876.20

29. Consider the baseball diamond at SHS. Home plate and the three bases are located at the vertices of
a square with the length of each side of the square being 90 ft. The pitcher’s mound lies on a straight
line from home plate to second base and it is located 60 ft 6 in from home plate. How far is it from
the pitcher’s mound to first base? (nearest tenth)

(A) 63.7 ft (B) 63.9 ft (C) 64.1 ft (D) 64.3 ft (E) 64.5 ft

30. At the Moulton Fall Festival, cash prizes were awarded for the top twenty places in the peach pie
contest. First place received $500, second place received $475, third place received $450, fourth
placed received $425 and so on. What was the total amount of prize money awarded?

(A) $5200 (B) $5225 (C) $5250 (D) $5275 (E) $5300

31. Six couples plan to attend a concert and sit in the same row. Each row has 12 seats. If the two
members of each couple are to sit together, how many different seating arrangements are possible?

(A) 24,060 (B) 36,060 (C) 42,060 (D) 46,080 (E) 48,040

x = t−5
32. The point ( −7, b) lies of the curve defined by the parametric equations . b = _______.
y = t2

(A) 2 (B) 3 (C) 4 (D) 5 (E) 6

33. Convert the rectangular equation 2x − 3y − 5 = 0 to polar form.

5 5 5
(A) r = (B) r = (C) r =
cos  − sin 3 
2
2cos  − 3sin  2sin  − 3cos 

(D) r = 5sec() (E) r = 5csc()

34. Find the perimeter of the triangle with vertices (4, − 5, 6) , (3, − 2, − 1) and ( −5, 7, − 9) .
(nearest tenth)

(A) 43.4 (B) 43.6 (C) 43.8 (D) 44.0 (E) 44.2

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35. Three of the zeros of f (x) = x4 + bx 3 + cx 2 + dx + 12 = 0 are −2, 3, 1 + 3 . f (2) = _______.

(A) −8 (B) −4 (C) 0 (D) 4 (E) 8

 4x  
36. Consider the graph of f (x) = 3 − 2cot  +  . The period of the graph is ________.
 3 6

3 4 3
(A) (B) (C) (D) 2 (E) 3
4 3 2

37. Assume that the earth rotates about its axis once every 23 hr 56 min 3.72 sec. Find the linear speed of
a person sitting in a chair on the equator. The radius of the earth is 3960 miles. (nearest tenth)

(A) 1037.4 mph (B) 1038.5 mph (C) 1039.6 mph (D) 1040.7 mph (E) 1041.8 mph

38. The graph of r = 12cos(8) has _________ rose petals.

(A) 6 (B) 8 (C) 12 (D) 16 (E) 24

39. On January 1, 2020, Lily placed $15,000 into account A where it earns 7% annual interest
compounded semiannually. On the same day, she also placed $14,500 into account B that earns 7.5%
annual interest compounded monthly. Of the following choices, which is the earliest day in which the
balance in account B exceeds the balance in account A?

(A) July 10, 2025 (B) Sept 10, 2025 (C) Nov 10, 2025 (D) Jan 10, 2026 (E) March 10, 2026

40. One of the foci of the hyperbola shown on the right has coordinates (a, 0) .
a = ________. (nearest tenth)

(A) 6.5
(B) 6.7
(C) 6.9
(D) 7.1
(E) 7.3

41. Find the area bounded by the graph of the right branch
of the hyperbola and the line x = 10. (nearest tenth)

(A) 20.2
(B) 20.4
(C) 20.6
(D) 20.8
(E) 21.0 Problems 40, 41

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42. A 50-foot ladder is leaning against a building. If the base of the ladder is being pulled away from the
building at a rate of 4 feet per second, then the top of the ladder will move down the wall at a rate of
________ feet per second at the moment the base of the ladder is 14 feet from the base of the building.

7 5 4 3 5
(A) (B) (C) (D) (E)
6 4 3 2 3

43. Consider the graph of f (x) = 3sin(x) + 2cos(2x) . How many values of x exist in the interval (6, 10)
such that there is a horizontal tangent at x?

(A) 1 (B) 2 (C) 3 (D) 4 (E) 5

44. Find the average value of f (x) = 2sin(x) + 5cos(x) on the interval  6, 8 . (nearest tenth)

(A) 4.3 (B) 4.5 (C) 4.7 (D) 4.9 (E) 5.1

dy x
45-46. Let y = f (x) be the solution to the differential equation = with the initial condition
dx 2y
f (4) = 3 .

45. Use Euler’s method to approximate y(5) using two steps of equal size starting at x = 4.
(nearest ten-thousandth)

(A) 3.6648 (B) 3.6668 (C) 3.6688 (D) 3.6708 (E) 3.6728

46. Find the exact value of y(5). (nearest thousandth)

(A) 3.6702 (B) 3.6722 (C) 3.6742 (D) 3.6762 (E) 3.6782

47-48. Consider the region bounded by the graphs of y1 = .5x2 and y 2 = 4 − x .

47. Find the area of the specified region. (nearest tenth)

(A) 18.0 (B) 18.4 (C) 18.8 (D) 19.2 (E) 19.6

48. Find the volume of the solid generated by revolving the specified region about the x-axis.
(nearest whole number)

(A) 350 (B) 353 (C) 356 (D) 359 (E) 362

49. If P3 (x) is the third degree Maclaurin polynomial for f (x) = e x , then f (.5) − P3 (.5) = ______.
(nearest ten-thousandth)

(A) 0.0021 (B) 0.0023 (C) 0.0025 (D) 0.0027 (E) 0.0029

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50. The position of a particle is given by the parametric equations x(t) = 3e4t and y(t) = ln(t 3 + 3t − 2) .
Find the velocity vector when t = 3.

30 15 30 15 15
(A) 12e12 , (B) 12e6 , (C) 12e12 , (D) 12e4 , (E) 12e12 ,
39 17 37 17 17

51. The second derivative of a function h(x) is given by h(x) = x 2 (x − 2)3 (x − 8)5 . Of the following values
of x, which of these does the graph of h(x) have a point of inflection? {0, 2, 8}

(A) 2 only (B) 2, 8 only (C) 0, 8 only (D) 0, 2 only (E) 0, 2, 8

52. The graph of the piecewise function f(x) is shown

10
on the right. Find the value of  f (x)dx .
0
(nearest hundredth)

(A) 6.00
(B) 7.75
(C) 9.50
(D) 11.25
(E) 13.00

Problem 52

Use the table below and the information below the table for problems 53 and 54.

1 2 3 4 5 6 7 8 9
Score 166 172 154 205 221 198 144 172 188

Phoenix entered the Irion County Bowling Marathon last week. Each contestant is required to bowl nine
games in six hours. The table above show the results.

53. What is the difference in the mean score and the median score, a positive number?

(A) 2 (B) 4 (C) 6 (D) 8 (E) 10

54. Find the interquartile range of the scores.

(A) 6 (B) 23.75 (C) 41.5 (D) 59.25 (E) 77

55. Alessandra flipped a fair coin 6 times and it came up heads all 6 times. She decided to flip the coin
one more time. What is the probability that the seventh flip will produce a tails?
(nearest ten-thousandth)

(A) 0.0078 (B) 0.1309 (C) 0.2500 (D) 0.3772 (E) 0.5000

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Use the table below and the information below the table for problems 56-58.

M.S. Time 12.25 11.88 11.36 10.75 10.33 9.91

H.S. Score 139 148 153 164 171 178

Christopher decided that he would warm up for his number sense competitions in 2023 by taking
a middle school number sense test 30 minutes before the high school competition started. The table
above shows the time, in minutes, it took him to complete the middle school warm up tests and his
scores at the six high school meets he has attended so far this year.

56. Christopher’s coach plotted the data and observed a strong, negative, linear relationship. His coach
analyzed the data and his statistical software generated a LSRL with a correlation of r = _________.
(nearest ten-thousandth)

(A) -0.9967 (B) -0.9922 (C) -0.9885 (D) -0.9843 (E) -0.9812

57. Use the data in the table above to predict his score at the next competition if he took 9.66 minutes to
complete a middle school warm up test. (nearest whole number)

(A) 180 (B) 182 (C) 184 (D) 186 (E) 188

58. Find the value of the residual for the data point (11.88, 148) . (nearest tenth)

(A) 1.5 (B) 1.7 (C) 1.9 (D) 2.1 (E) 2.3

59-60. A researcher is testing the claim that 62% of Americans believe that the economy is the most
important issue in America today. In a survey involving a simple random sample of 1200 Americans,
782 responded by placing the economy as their top concern. The researcher performed an appropriate
test with H 0 : p = 0.62 and, Ha : p  0.62 . He used an  = 0.05 level as his standard.

59. What is the value of the test statistic? (nearest hundredth)

(A) 2.02 (B) 2.14 (C) 2.26 (D) 2.38 (E) 2.50

60. What was the conclusion of the researcher after studying the test results?

(A) Based on a p-value of 0.06, he failed to reject H 0 .

(B) Based on a p-value of 0.06, he rejected H 0 .
(C) Based on a p-value of 0.02, he failed to reject H 0 .
(D) Based on a p-value of 0.02, he rejected H 0 .
(E) Based on a p-value of 0.01, he failed to reject H 0 .

2023 UIL Mathematics

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University Interscholastic League

MATHEMATICS CONTEST
HS ● Invitational A ● 2023
Answer Key

1. D 21. E 41. A

2. A 22. A 42. A

3. B 23. A 43. C

4. E 24. A 44. A

5. E 25. C 45. D

6. C 26. A 46. C

7. A 27. B 47. A

8. D 28. E 48. E

9. E 29. A 49. E

10. C 30. C 50. E

11. B 31. D 51. B

12. E 32. C 52. A

13. C 33. B 53. D

14. A 34. A 54. C

15. E 35. E 55. E

16. C 36. A 56. A

17. B 37. C 57. B

18. E 38. D 58. D

19. C 39. B 59. C

20. C 40. B 60. D