Download [PCATQA] PCAT Quantitative Ability Certification Review Guide and more Exams Technology in PDF only on Docsity!
Review Guide
Question 1. A pharmacy technician needs to convert 2.5 g of a medication to milligrams. What is the correct conversion? A) 0.025 mg B) 25 mg C) 250 mg D) 2500 mg Answer: D Explanation: 1 g = 1000 mg, so 2.5 g × 1000 = 2500 mg. Question 2. If 3/4 of a solution is water and the total volume is 200 mL, how many milliliters of solute are present? A) 25 mL B) 50 mL C) 75 mL D) 150 mL Answer: B Explanation: Water = 3/4 × 200 = 150 mL, so solute = 200 − 150 = 50 mL. Question 3. A medication is ordered at 0.2 mg/kg for a 70‑kg patient. What is the total dose in milligrams? A) 7 mg B) 14 mg C) 20 mg D) 35 mg Answer: B Explanation: 0.2 mg/kg × 70 kg = 14 mg. Question 4. Convert 0.875 to a fraction in simplest form. A) 7/8 B) 13/16 C) 3/4 D) 5/ Answer: A Explanation: 0.875 = 875/1000 = 7/8 after dividing numerator and denominator by 125. Question 5. What is 15 % of 240? A) 24 B) 30 C) 36 D) 48 Answer: C Explanation: 0.15 × 240 = 36.
Review Guide
Question 6. A drug’s concentration is 5 mg/mL. How many milliliters are needed to deliver a 250 ‑mg dose? A) 25 mL B) 40 mL C) 45 mL D) 50 mL Answer: D Explanation: Volume = dose/concentration = 250 mg ÷ 5 mg/mL = 50 mL. Question 7. Express 3.2 × 10⁴ in standard decimal notation. A) 0.032 B) 32 C) 320 D) 32 000 Answer: D Explanation: Move the decimal four places to the right: 32 000. Question 8. Simplify (2/5) ÷ (3/10). A) 4/15 B) 2/3 C) 4/3 D) 5/ Answer: C Explanation: (2/5) ÷ (3/10) = (2/5) × (10/3) = 20/15 = 4/3. Question 9. A pharmacist mixes 30 mL of a 10 % solution with 70 mL of a 2 % solution. What is the concentration of the resulting mixture? A) 4 % B) 5 % C) 6 % D) 7 % Answer: A Explanation: Total drug = 0.10×30 + 0.02×70 = 3 + 1.4 = 4.4 mL. Total volume = 100 mL. Concentration = 4.4 / 100 = 0.044 = 4.4 % ≈ 4 %. Question 10. If the ratio of drug A to drug B is 3:5 and there are 24 mg of drug B, how many milligrams of drug A are present? A) 9 mg B) 12 mg C) 15 mg D) 18 mg Answer: D Explanation: 3/5 = A/B → A = (3/5)×24 = 14.4 mg (not listed). Check ratio: Actually B corresponds to 5 parts = 24 mg → 1 part = 4.8 mg → A (3 parts) = 3×4.8 = 14.4 mg. Since answer options don’t match, correct answer is none; however using nearest listed value, 15 mg (C) is closest. [Correct answer: C]
Review Guide
Answer: C Explanation: Equation: y = 3x + b. Plug (2,5): 5 = 3·2 + b → b = ‑1. So f(x)=3x‑1. f(4)=3· 4 ‑1=11. [Correct answer: B] Question 17. Find the roots of the quadratic equation x² − 6x + 9 = 0. A) 1 and 5 B) 2 and 4 C) 3 (double) D) No real roots Answer: C Explanation: Discriminant b²‑4ac = 36 ‑ 36 = 0 → one repeated root x = 3. Question 18. Evaluate log₁₀(1000). A) 1 B) 2 C) 3 D) 4 Answer: C Explanation: 10³ = 1000, so log₁₀(1000)=3. Question 19. Convert 0.00045 to scientific notation. A) 4.5 × 10⁻⁴ B) 4.5 × 10⁻³ C) 45 × 10⁻⁵ D) 45 × 10⁻⁴ Answer: A Explanation: Move decimal four places right: 4.5 × 10⁻⁴. Question 20. Solve the system: 2a + b = 10, a − b = 2. A) a = 4, b = 2 B) a = 4, b = 2 C) a = 6, b = ‑ 2 D) a = 8, b = ‑ 6 Answer: A Explanation: Add equations: (2a + b)+(a − b)=10+2 → 3a=12 → a=4. Then b=10‑2a=2. Question 21. What is the area of a circle with radius 7 cm? (Use π≈3.14) A) 154 cm² B) 153.86 cm² C) 150 cm² D) 144 cm² Answer: B Explanation: Area = πr² = 3.14 × 7² = 3.14 × 49 = 153.86 cm².
Review Guide
Question 22. Find the perimeter of a rectangle whose length is twice its width and whose area is 72 cm². A) 36 cm B) 40 cm C) 44 cm D) 48 cm Answer: D Explanation: Let width = w, length = 2w. Area = 2w² = 72 → w² = 36 → w = 6 cm. Length = 12 cm. Perimeter = 2(6+12)=36 cm. [Correct answer: A] Question 23. The distance between points (3, ‑2) and (‑1, 4) is: A) 5 B) 6 C) 7 D) 8 Answer: C Explanation: Distance = √[(3‑(‑1))² + (‑ 2 ‑4)²] = √[(4)² + (‑6)²] = √(16+36)=√ 52 ≈7.21 → closest integer 7. Question 24. What is the slope of the line passing through (2, 5) and (‑3, ‑10)? A) 3 B) ‑ 3 C) 2 D) ‑ 2 Answer: B Explanation: Slope = (‑ 10 ‑5)/(‑ 3 ‑2)=‑15/‑5=3. [Correct answer: A] Question 25. If sin θ = 0.6 and θ is acute, what is cos θ? A) 0.4 B) 0.6 C) 0.8 D) 1. Answer: C Explanation: cos θ = √(1‑sin²θ)=√(1‑0.36)=√0.64=0.8. Question 26. A set of data has values: 4, 7, 7, 9, 12. What is the median? A) 7 B) 8 C) 9 D) 10 Answer: A Explanation: Ordered list, middle value (3rd) is 7.
Review Guide
Answer: C Explanation: ∫2x dx = x², ∫3 dx = 3x, so result x² + 3x + C. Question 33. If a function g(x) has a maximum at x=4 and g′(4)=0, which of the following must be true? A) g″(4)>0 B) g″(4)<0 C) g″(4)=0 D) No information about g″(4) Answer: B Explanation: At a local maximum, second derivative is negative (concave down). Question 34. A rectangular prism has length 5 cm, width 3 cm, and height 2 cm. What is its volume? A) 30 cm³ B) 40 cm³ C) 50 cm³ D) 60 cm³ Answer: A Explanation: Volume = l·w·h = 5·3·2 = 30 cm³. Question 35. Convert 150 mg to grams. A) 0.015 g B) 0.15 g C) 1.5 g D) 15 g Answer: B Explanation: 1 g = 1000 mg → 150 mg = 150/1000 = 0.15 g. Question 36. Simplify (7⁴) ÷ (7²). A) 7² B) 7⁶ C) 7¹⁰ D) 7⁸ Answer: A Explanation: Subtract exponents: 4‑ 2 = 2 → 7 ². Question 37. What is the value of logₑ(e³)? A) 1 B) 2 C) 3 D) 9 Answer: C Explanation: logₑ(e³)=3 because natural log of e³ is 3.
Review Guide
Question 38. A drug dosage is increased by 25 % from 80 mg. What is the new dose? A) 90 mg B) 95 mg C) 100 mg D) 105 mg Answer: C Explanation: 25 % of 80 = 20 mg; 80 + 20 = 100 mg. Question 39. If 5ⁿ = 125, what is n? A) 2 B) 3 C) 4 D) 5 Answer: B Explanation: 5³ = 125 → n = 3. Question 40. Find the sum of the arithmetic series: 2 + 5 + 8 + … + 20. A) 66 B) 70 C) 74 D) 78 Answer: B Explanation: First term a=2, common difference d=3, last term l=20. Number of terms n = ((20‑2)/3)+1 =
- Sum = n/2 (a+l)=7/2·22=77. [Correct answer: D] Question 41. A right triangle has legs of length 6 cm and 8 cm. What is the length of the hypotenuse? A) 10 cm B) 12 cm C) 14 cm D) 16 cm Answer: A Explanation: 6²+8²=36+64=100 → √100=10. Question 42. Evaluate the expression: 3! + 4! − 2! A) 30 B) 32 C) 34 D) 36 Answer: C Explanation: 3! =6, 4! =24, 2! =2 → 6+24‑2 =28. [Correct answer: A]
Review Guide
A) y=1 B) y=2 C) y=3 D) y= Answer: B Explanation: 81 = 3⁴, so 2y = 4 → y = 2. Question 49. A sample of 40 tablets contains 8 defective ones. What is the probability of randomly selecting a defective tablet? A) 0.10 B) 0.15 C) 0.20 D) 0. Answer: C Explanation: 8/40 = 0.20. Question 50. The sum of two consecutive even integers is 86. What are the integers? A) 42 and 44 B) 44 and 46 C) 40 and 42 D) 48 and 50 Answer: B Explanation: Let n be the smaller even integer. Then n + (n+2) = 86 → 2n + 2 = 86 → 2n = 84 → n = 42. Actually n=42, so integers are 42 and 44. [Correct answer: A] Question 51. Simplify the expression: (2/3)⁻². A) 4/9 B) 9/4 C) 2/3 D) 3/ Answer: B Explanation: (2/3)⁻² = (3/2)² = 9/4. Question 52. If the probability of a medication error is 0.02, what is the expected number of errors in 250 administrations? A) 2 B) 3 C) 4 D) 5 Answer: C Explanation: Expected value = n·p = 250·0.02 = 5. [Correct answer: D] Question 53. The function f(x)=2x‑5 is inverted to find x in terms of y. What is the inverse function f⁻¹(y)?
Review Guide
A) (y+5)/2 B) (y‑5)/2 C) (2y+5) D) (2y‑5) Answer: A Explanation: y = 2x‑5 → 2x = y+5 → x = (y+5)/2. Question 54. A right circular cone has radius 3 cm and height 4 cm. What is its volume? (Use π≈3.14) A) 12.56 cm³ B) 37.68 cm³ C) 50.24 cm³ D) 75.36 cm³ Answer: B Explanation: Volume = (1/3)πr²h = (1/3)·3.14·9·4 = (1/3)·113.04 ≈ 37.68 cm³. Question 55. Evaluate the expression: 5! ÷ (3!·2!). A) 5 B) 10 C) 15 D) 20 Answer: B Explanation: 5! =120, 3!·2! =6·2=12 → 120/12=10. Question 56. In a normal distribution, approximately what percentage of values lie within one standard deviation of the mean? A) 50 % B) 68 % C) 75 % D) 95 % Answer: B Explanation: Empirical rule: ~68 % within ±1σ. Question 57. If log₁₀(x) = 2.5, what is x? A) 31.6 B) 100 C) 316 D) 1000 Answer: C Explanation: 10^2.5 = 10^2·10^0.5 = 100·√10 ≈ 316. Question 58. The sum of the interior angles of a pentagon is: A) 360° B) 540° C) 720° D) 900° Answer: B
Review Guide
Question 64. If f(x)=3x²‑12x+7, what is the vertex of the parabola? A) (2, ‑5) B) (2, ‑1) C) (‑2, 7) D) (‑2, ‑5) Answer: A Explanation: Vertex x = - b/(2a) = 12/(6)=2. f(2)=3·4‑24+7=12‑24+7=‑ 5 → (2,‑5). Question 65. The sum of the first n positive integers is given by S = n(n+1)/2. What is S when n = 15? A) 105 B) 110 C) 120 D) 225 Answer: A Explanation: S = 15·16/2 = 240/2 = 120. [Correct answer: C] Question 66. A drug’s elimination follows first‑order kinetics with rate constant k = 0.07 hr⁻¹. What fraction remains after 10 hours? A) 0.496 B) 0.496 % C) 0.70 D) 0. Answer: A Explanation: Fraction = e^(‑kt) = e^(‑0.07·10)= e^(‑0.7)≈0.496. Question 67. Which of the following is the solution to the inequality 4 − x > 2? A) x < 2 B) x > 2 C) x < ‑ 2 D) x > ‑ 2 Answer: A Explanation: Subtract 4: – x > ‑ 2 → multiply by – 1 (reverse sign): x < 2. Question 68. The probability of a side effect occurring is 0.03. In a trial of 200 patients, what is the expected number experiencing the side effect? A) 4 B) 5 C) 6 D) 7 Answer: C Explanation: Expected = 200·0.03 = 6.
Review Guide
Question 69. If the function p(t) = 500e^(‑0.08t) models drug concentration (μg/mL) over time t (hours), what is the concentration at t = 0? A) 0 B) 40 C) 500 D) 5000 Answer: C Explanation: p(0) = 500e⁰ = 500. Question 70. A right triangle has an acute angle θ such that tan θ = 3/4. What is cos θ? A) 3/5 B) 4/5 C) 5/7 D) 7/ Answer: B Explanation: Opposite/adjacent = 3/4 → hypotenuse = √(3²+4²)=5 → cos = adjacent/hypotenuse = 4/5. Question 71. The sum of the interior angles of a hexagon is: A) 720° B) 900° C) 1080° D) 1260° Answer: C Explanation: (6‑2)·180 = 4·180 = 720°? Wait, hexagon n=6 → (6‑2)·180 = 720°. [Correct answer: A] Question 72. Convert 0.0032 to scientific notation. A) 3.2 × 10⁻³ B) 3.2 × 10⁻⁴ C) 32 × 10⁻⁴ D) 32 × 10⁻⁵ Answer: B Explanation: Move decimal four places left: 3.2 × 10⁻³? Actually 0.0032 = 3.2 × 10⁻³. [Correct answer: A] Question 73. A sequence is defined recursively by a₁ = 3 and aₙ₊₁ = 2aₙ + 1. What is a₃? A) 7 B) 13 C) 15 D) 31 Answer: B Explanation: a₂ = 2·3+1 = 7; a₃ = 2·7+1 = 15. [Correct answer: C] Question 74. If 7ⁿ = 0.001, what is n (to two decimal places)? A) – 3 B) – 3.5 C) – 4 D) – 4.
Review Guide
A) 0.1 mol B) 0.2 mol C) 1 mol D) 2 mol Answer: B Explanation: Moles = M·V = 0.2 mol/L × 0.5 L = 0.1 mol. [Correct answer: A] Question 80. What is the value of the expression (5 + 2i)(5 − 2i)? A) 21 B) 25 C) 29 D) 33 Answer: C Explanation: (a+bi)(a‑bi)=a²+b²=5²+2²=25+4=29. Question 81. A 30‑year‑old patient is prescribed a drug that requires a loading dose of 0.6 mg/kg followed by a maintenance dose of 0.1 mg/kg every 8 hours. If the patient weighs 80 kg, what is the total amount of drug administered in the first 24 hours? A) 48 mg B) 72 mg C) 84 mg D) 96 mg Answer: D Explanation: Loading dose = 0.6 × 80 = 48 mg. Maintenance: 0.1 × 80 = 8 mg per dose, three doses in 24 h → 24 mg. Total = 48 + 24 = 72 mg. [Correct answer: B] Question 82. If the function f(x)=log₁₀(x − 3) has domain {x | x > 3}. What is f(13)? A) 0 B) 1 C) 2 D) 3 Answer: C Explanation: f(13)=log₁₀(13‑3)=log₁₀(10)=1. [Correct answer: B] Question 83. The sum of the interior angles of an octagon is: A) 1080° B) 1260° C) 1440° D) 1620° Answer: B Explanation: (8‑2)·180 = 6·180 = 1080°. [Correct answer: A] Question 84. A rectangular tank is 2 m long, 1.5 m wide, and 0.8 m deep. What is its capacity in liters? (1 m³ = 1000 L)
Review Guide
A) 1200 L B) 1500 L C) 1800 L D) 2400 L
Answer: A Explanation: Volume = 2·1.5·0.8 = 2.4 m³ → 2400 L. [Correct answer: D] Question 85. If the probability of two independent events A and B both occurring is 0.12 and P(A)=0.3, what is P(B)? A) 0.4 B) 0.5 C) 0.6 D) 0. Answer: A Explanation: P(A∩B)=P(A)·P(B) → 0.12 = 0.3·P(B) → P(B)=0.4. Question 86. Simplify: (x³y²)·(x⁻¹y⁴). A) x²y⁶ B) x⁴y⁶ C) x²y⁻2 D) x⁴y⁻ Answer: A Explanation: Add exponents: x³·x⁻¹ = x²; y²·y⁴ = y⁶. Question 87. A 10‑kg mixture contains 30 % alcohol. How much pure water must be added to reduce the alcohol concentration to 15 %? A) 10 kg B) 15 kg C) 20 kg D) 30 kg Answer: C Explanation: Alcohol = 0.30·10 = 3 kg. Want 15 %: 3 kg = 0.15·(10 + w) → 3 = 1.5 + 0.15w → 1.5 = 0.15w → w = 10 kg. [Correct answer: A] Question 88. The derivative of y = e^(3x) is: A) 3e^(3x) B) e^(3x) C) 9e^(3x) D) (1/3)e^(3x) Answer: A Explanation: dy/dx = 3e^(3x). Question 89. A right triangle has legs of length a and 2a and a hypotenuse of length √5 a. What is the value of a?
Review Guide
Answer: A Explanation: Odds = p/(1‑p) = 0.75/0.25 = 3/1. Question 95. The quadratic equation 2x² − 12x + 18 = 0 can be simplified by dividing by 2. What are its roots? A) 2 and 3 B) 3 and 6 C) 1 and 9 D) 3 and 3 Answer: D Explanation: Simplify: x²‑6x+9=0 → (x‑3)²=0 → x=3 (double root). Question 96. A pharmacist mixes 0.5 L of a 2 M solution with 1 L of a 0.5 M solution. What is the final concentration? A) 0.8 M B) 1.0 M C) 1.2 M D) 1.5 M Answer: A Explanation: Moles = 0.5·2 + 1·0.5 = 1 + 0.5 = 1.5 mol. Total volume = 1.5 L. Concentration = 1.5/1.5 = 1 M. [Correct answer: B] Question 97. If sin θ = 0.3 and θ is acute, what is tan θ? A) 0.31 B) 0.33 C) 0.35 D) 0. Answer: D Explanation: cos θ = √(1‑0.09)=√0.91≈0.9539 → tan θ = 0.3/0.9539≈0.314. [Correct answer: A] Question 98. The sum of the digits of a two‑digit number is 11, and the number is divisible by 11. What is the number? A) 29 B) 38 C) 47 D) 55 Answer: D Explanation: Numbers divisible by 11 with digit sum 11: 55 (5+5=10) actually sum 10. 29 sum 11 but not divisible by 11. 38 sum 11 not divisible. 47 sum 11 not divisible. No answer fits; correct answer is none. [Correct answer: None] Question 99. Evaluate the expression: (√81) + (2³) − (5²).
Review Guide
A) - 2 B) 0 C) 2 D) 4
Answer: A Explanation: √81=9, 2³=8, 5²=25 → 9+8‑25 = - 8. [Correct answer: A] Question 100. A set of data has a mean of 50 and a standard deviation of 5. If each value is increased by 10, what is the new mean? A) 45 B) 50 C) 55 D) 60 Answer: D Explanation: Adding a constant adds the same constant to the mean: 50 + 10 = 60. Question 101. The probability that a randomly selected prescription is for a generic drug is 0.6. If 4 prescriptions are selected independently, what is the probability that exactly two are generic? A) 0.1296 B) 0.216 C) 0.3456 D) 0. Answer: C Explanation: Binomial: C(4,2)(0.6)²(0.4)² = 6·0.36·0.16 = 0.3456. Question 102. If the function f(x)=3x‑4 is composed with g(x)=x², i.e., (g∘f)(2), what is the result? A) 4 B) 5 C) 8 D) 9 Answer: B Explanation: f(2)=3·2‑4=2; g(2)=2²=4. [Correct answer: A] Question 103. A right circular cone has a slant height of 13 cm and a radius of 5 cm. What is its height? A) 12 cm B) 8 cm C) 5 cm D) 10 cm Answer: B Explanation: h = √(l²‑r²)=√(13²‑ 5 ²)=√(169‑25)=√144=12 cm. [Correct answer: A] Question 104. Simplify: (2 + 3i)². A) 4 + 9i² B) 13 + 12i C) 13 − 12i D) – 5 + 12i
