Math Practice Problems

1. A circular pond is centered at the origin with a radius of 3 meters. A gardener installs a decorative ring around it, 4 meters outward from the pond’s edge. What is the equation of the decorative ring in the coordinate plane?

A) x² + y² = 9
B) x² + y² = 16
C) x² + y² = 49
D) (x-4)² + y² = 9

2. A stunt performer jumps off a ramp, and their path follows the parabola y = -1/16 x² + 2x, where x and y are in meters. What is the maximum height reached by the performer?

A) 4 meters
B) 8 meters
C) 16 meters
D) 32 meters

3. An elliptical mirror has its equation given by x²/36 + y²/20 = 1, with the major axis along the x-axis. What are the coordinates of the foci?

A) (±4,0)
B) (±5,0)
C) (±6,0)
D) (0,±4)

4. Two radio towers are positioned at (-5,0) and (5,0). A signal’s time difference between the towers corresponds to a distance difference of 6 units. What is the equation of the hyperbola describing the signal’s possible locations?

A) x²/9 - y²/16 = 1
B) x²/25 - y²/9 = 1
C) y²/9 - x²/16 = 1
D) x²/16 - y²/9 = 1

5. A designer sketches a curve with the equation 4x² - 9y² + 8x + 18y - 5 = 0. What type of conic section is this?

A) Circle
B) Ellipse
C) Parabola
D) Hyperbola

6. A parabolic archway has its vertex at the origin and passes through the point (4,2), opening upward. What is the equation of the parabola?

A) y = 1/8 x²
B) y = 1/4 x²
C) y = 1/2 x²
D) y = 2x²

7. A hyperbolic sculpture has foci at (±10,0) and vertices at (±6,0). What is the length of the transverse axis?

A) 6 units
B) 12 units
C) 20 units
D) 10 units

8. An artist shifts the ellipse x²/16 + y²/9 = 1 left by 2 units and down by 1 unit. What is the equation of the transformed ellipse?

A) (x+2)²/16 + (y+1)²/9 = 1
B) (x-2)²/16 + (y-1)²/9 = 1
C) (x+2)²/9 + (y+1)²/16 = 1
D) x²/16 + y²/9 = 1

9. A student examines the properties of conic sections in the coordinate plane. Which feature is common to both a circle and an ellipse but not a parabola?

A) A single focus
B) A bounded shape
C) Two directrices
D) An axis of symmetry

10. A game designer creates a level where a character’s path is the parabola y = x² - 4x + 4 and a barrier follows the circle (x-2)² + y² = 1. How many times do the path and barrier intersect?

A) 0
B) 2
C) 1
D) 4

1. Liam is building a sandcastle with levels of shells. The bottom level has 20 shells, and each level above it has 5 more shells than the one below. How many shells are on the 8th level?

A) 55
B) 60
C) 65
D) 70

2. A viral video starts with 50 views on day 1 and triples in views each day. How many views does it have on day 5?

A) 4050
B) 1350
C) 450
D) 12150

3. Mia saves $10 in week 1 and increases her savings by $3 each week to buy a skateboard. What’s the total amount she’s saved after 10 weeks?

A) $235
B) $265
C) $295
D) $325

4. A bouncy ball drops from 16 meters and rebounds to ½ its previous height each bounce. What is the total distance traveled after 4 bounces (including the initial drop)?

A) 31 meters
B) 46 meters
C) 63 meters
D) 79 meters

5. Check out this sequence: 2, 6, 18, 54, ... Is it arithmetic, geometric, or neither?

A) Arithmetic
B) Geometric
C) Neither

6. A plant grows leaves recursively: it starts with 4 leaves, and each day it grows 3 fewer leaves than the day before (yes, it’s losing growth power!). What’s the explicit formula for leaves on day n?

A) an = 4 − 3n
B) an = 4 + 3(n − 1)
C) an = 4 − 3(n − 1)
D) an = 4 × 3^(n − 1)

7. A rumor spreads with the explicit formula an = 10 × 4^(n − 1) people hearing it each hour. What’s the recursive formula?

A) a₁ = 10, an = an−1 + 4
B) a₁ = 10, an = 4 × an−1
C) a₁ = 4, an = 10 × an−1
D) a₁ = 10, an = an−1 − 4

8. A staircase has 15 steps. The first step is 7 cm high, and each step after increases by 2 cm in height. What’s the total height of the staircase?

A) 270 cm
B) 285 cm
C) 300 cm
D) 315 cm

9. A discount sale slashes prices: a $100 item drops to 90% of its price each day. What’s the total value of the item over the first 6 days (sum of prices)?

A) $368
B) $469
C) $500
D) $575

10. Option A: Start with 200 trees and add 50 trees each year. Option B: Start with 200 trees and the total number of trees grows by 25% each year. When does Option B first exceed Option A?

A) Year 4
B) Year 3
C) Year 6
D) Year 5

1. A chemist is analyzing the acidity of two solutions. Solution A has a pH of 3, and Solution B has a pH of 5. What is the ratio of the hydrogen ion concentration [H⁺] in Solution A to that in Solution B?

A) 10
B) 100
C) 0.01
D) 2

2. On the Richter scale, each increase of 1 unit represents a tenfold increase in seismic wave amplitude. If Earthquake A has a magnitude of 6 and Earthquake B has a magnitude of 4, what is the ratio of their amplitudes?

A) 10
B) 100
C) 1000
D) 2

3. Sound loudness is measured in decibels (dB) with the formula L = 10 log₁₀(I / I₀). If Sound A is 60 dB and Sound B is 40 dB, how many times more intense is Sound A than Sound B?

A) 10
B) 100
C) 1000
D) 20

4. A physicist is combining variables in an experiment, expressed as log₁₀(a² b³ / c⁴). Given log₁₀(a) = 2, log₁₀(b) = 3, and log₁₀(c) = 1, what is the value of log₁₀(a² b³ / c⁴)?

A) 9
B) 17
C) -1
D) 13

5. Solve for x in log₂(x) + log₂(x - 2) = 3.

A) x = 4 only
B) x = -2 only
C) x = 4 and x = -2
D) No solution

6. Solve for x in 2^x = 3^(x - 1).

A) x = log₁₀(3) / (log₁₀(3) − log₁₀(2))
B) x = log₁₀(2) / (log₁₀(3) − log₁₀(2))
C) x = log₃(2)
D) x = log₂(3)

7. If 60% of a substance remains after 10 years, what is its half-life?

A) 10 years
B) 12 years
C) 14 years
D) 16 years

8. A microbiologist studies a bacterial population doubling every hour. Starting with 1000 bacteria, how many hours until there are 64,000?

A) 4 hours
B) 5 hours
C) 6 hours
D) 7 hours

9. A data scientist transforms a logarithmic model for better visualization. If f(x) = log₁₀(x) is shifted 2 units right and reflected over the x-axis, what is the new function?

A) g(x) = -log₁₀(x + 2)
B) g(x) = log₁₀(-x + 2)
C) g(x) = -log₁₀(x - 2)
D) g(x) = log₁₀(x - 2) - 2

10. A financial analyst calculates doubling time for an investment with continuous compounding. If $1000 grows to $2000 at 5% annual interest compounded continuously, how long does it take?

A) 10 years
B) 12 years
C) 14 years
D) 16 years

1. A taxi company charges a base fee of $3 plus $2 per mile traveled. Due to rising fuel costs, the company increases the base fee by $1 and the per-mile charge by 50%. Which of the following represents the new cost function for a trip of m miles?

A) C(m) = 4 + 3m
B) C(m) = 3 + 3m
C) C(m) = 4 + 2m
D) C(m) = 4.5 + 3m

2. A ball is thrown upwards from ground level, and its height h in meters after t seconds is given by h(t) = −4.9t² + 20t. If the ball is thrown with the same initial speed but from a platform 5 meters above the ground, which of the following represents the new height function?

A) h(t) = −4.9t² + 20t + 5
B) h(t) = −4.9(t − 5)² + 20(t − 5)
C) h(t) = −4.9t² + 25t
D) h(t) = 5(−4.9t² + 20t)

3. A car drives along a straight road, and its distance from a landmark is given by d(t) = |t − 5|, where t is time in hours. If the car’s journey is adjusted so it reaches the landmark 3 hours earlier, which of the following represents the new distance function?

A) d(t) = |t − 2|
B) d(t) = |t − 8|
C) d(t) = |t − 5| − 3
D) d(t) = 3|t − 5|

4. A rectangular garden has a length 3 meters longer than its width. The area A as a function of width w is A(w) = w(w + 3). Due to property boundaries, the width cannot exceed 10 meters. What is the domain of A(w) in this context?

A) All real numbers
B) w > 0
C) 0 < w ≤ 10
D) w ≥ 0

5. A bakery’s cake production cost is $10 plus $2 per ingredient. The number of ingredients n for a cake of size s kilograms is n(s) = 3s. What is the cost C to produce a cake of size s kilograms?

A) C(s) = 10 + 2s
B) C(s) = 10 + 6s
C) C(s) = 30s
D) C(s) = 12 + 2s

6. A machine converts Celsius to Fahrenheit temperatures using F(C) = (9/5)C + 32. What is the inverse function that converts Fahrenheit to Celsius?

A) C(F) = (5/9)(F − 32)
B) C(F) = (9/5)(F − 32)
C) C(F) = (5/9)F + 32
D) C(F) = (9/5)F − 32

7. A delivery service charges $5 for packages up to 2 kg. For packages over 2 kg, the charge is $5 plus $2 per kg for each kg over 2 kg. Which of the following represents the cost C(w) for a package of weight w kg?

A) C(w) = 5 if w ≤ 2, and C(w) = 5 + 2w if w > 2
B) C(w) = 5 if w ≤ 2, and C(w) = 5 + 2(w − 2) if w > 2
C) C(w) = 5 + 2w for all w
D) C(w) = 5 + 2(w − 2) for all w

8. The function f(x) = x³ is transformed by reflecting it over the y-axis, shifting it 4 units to the right, and stretching it vertically by a factor of 2. What is the equation of the resulting function?

A) g(x) = 2(−x + 4)³
B) g(x) = 2(−x)³ + 4
C) g(x) = 2(x − 4)³
D) g(x) = −2(x + 4)³

1. Alice can paint a room in 4 hours, and Bob can do it in 6 hours. They claim working together takes (4+6)/2 = 5 hours. What’s the flaw in their logic?

A) They forgot to invert the rates.
B) They misadded the denominators.
C) They should multiply the rates.
D) The calculation is correct.

2. A student simplifies (x² − 9)/(x² + 3x) to (x − 3)/x. Are they correct (assume domain restrictions are noted)?

A) Yes, because x² cancels out.
B) Yes, after factoring and canceling (x + 3).
C) No, the denominator should be x + 3.
D) No, it simplifies to x − 3.

3. When solving 2x/(x − 3) = 6/(x − 3), a student gets x = 3. What’s the issue?

A) No issue; x = 3 is valid.
B) The solution makes the denominator zero.
C) The student forgot to cross-multiply.
D) The equation has no solutions.

4. To simplify 1 + 2/(x³ − 1/x), what’s the best first step for the fractional term?

A) Multiply the fractional term’s numerator and denominator by x.
B) Add the fractions in the numerator.
C) Subtract the fractions in the denominator.
D) Rewrite as 1/3 + 2 − 1/3.

5. The focal length f of a lens is given by 1/f = 1/dₒ + 1/dᵢ. If dₒ = 10 cm and dᵢ = 15 cm, what is f?

A) 6 cm
B) 12.5 cm
C) 25 cm
D) 150 cm

6. For f(x) = (x² − 4)/(x² − 5x + 6), which x-value causes a removable discontinuity (a hole) in the graph?

A) x = 2
B) x = 3
C) x = 0
D) x = −2

7. Simplify (2 + 3/x) / (5 − 1/x). What is the best first step?

A) Multiply numerator and denominator by x to clear inner denominators.
B) Multiply numerator and denominator by 5x − 1.
C) Divide the numerator by the denominator.
D) Substitute a sample value of x.

8. Solve 3/(x+1) = 2x/5 by cross-multiplying, then state what must be verified before concluding.

A) Verify that x ≠ −1.
B) Verify that x > 0.
C) Verify that the denominators are equal.
D) Nothing; any algebraic solution is valid.

1. A rectangular garden has a length of (3x² + 2x − 1) meters and a width of (x − 4) meters. What is the degree of the polynomial representing its area?

A) 2
B) 3
C) 4
D) 1

2. A student performs synthetic division on 2x³ − 5x + 1 by (x − 3) and gets a remainder of 40. What does this imply?

A) x = 3 is a root of the polynomial.
B) P(3) = 40
C) The polynomial cannot be factored.
D) The student made a mistake.

3. The profit P(x) from selling x items is 4x³ − 2x + 10, while the cost C(x) is x³ + 5x² + 3x. What polynomial represents revenue?

A) 3x³ − 5x² − 2x + 7
B) 5x³ + 5x² − 2x + 13
C) 5x³ + 5x² + x + 10
D) 4x³ − x³ − 2x − 5x² + 7

4. When x⁴ − 3x³ + kx² + 7x − 4 is divided by (x − 2), the remainder is 6. What is k?

A) 1
B) 4
C) 0
D) 3

5. A polynomial P(x) has roots at x = −1 (multiplicity 2) and x = 2 (multiplicity 1). Which describes its graph near x = −1?

A) Linear and passes through the x-axis.
B) Tangent to the x-axis and turns around.
C) A vertical asymptote.
D) A horizontal asymptote.

6. To divide 5x³ + 0x² − 2x + 7 by (x + 1) using synthetic division, what is the correct first row?

A) −1 ∣ 5 0 −2 7
B) 1 ∣ 5 −2 7
C) −1 ∣ 5 −2 7
D) 1 ∣ 5 0 −2 7

7. The volume of a box is x³ − 6x² + 11x − 6. If the height is (x − 1), what could the other dimensions be?

A) (x − 2)(x − 3)
B) (x + 2)(x + 3)
C) (x² − 5x + 6)
D) (x − 1)²

8. A student claims (x² + 4)(x² − 4) = x⁴ − 16. Are they correct?

A) Yes, because difference of squares applies.
B) No, the middle terms cancel out.
C) No, it should be x⁴ + 16.
D) No, it equals x⁴ − 8x² + 16.

9. If P(x) ⋅ (x² − 1) = x⁵ − 2x³ + x, what is P(x)?

A) x³ − 2x
B) x³ − x
C) x³ + x
D) x³ − x²

10. The number of WiFi hotspots in a city is modeled by P(t) = t³ − 9t² + 24t − 20, where t is years. If P(2) = 0, what does this imply?

A) No hotspots existed at t = 0.
B) The number of hotspots will peak at t = 2.
C) There were no hotspots in year 2.
D) The model is linear.

1. A bacteria colony doubles every 3 hours. If you start with 50 bacteria, which expression gives the population after t hours?

A) 50 ⋅ 2^t B) 50 ⋅ 2^(t/3) C) 50 ⋅ 3^(2t) D) 50 + 2^t

2. A student simplifies (2x³)⁴ as 2x¹². What mistake did they make?

A) They added exponents instead of multiplying. B) They forgot to apply the exponent to the coefficient 2. C) They multiplied the exponents incorrectly. D) They confused the base.

3. A substance decays by 20% every 5 years. Which equation models its remaining mass M after t years if it starts at 100g?

A) M = 100 ⋅ 0.2^t B) M = 100 ⋅ 0.8^(t/5) C) M = 100 − 0.2t D) M = 100 ⋅ e^(–0.2t)

4. Which of the following is not equivalent to 8^x ⋅ 4^(–y)?

A) 2^(3x − 2y) B) 8^x / 4^y C) (2^3)^x − (2^2)^y D) 8^x ⋅ 4^(–y)

5. You invest $500 at 4% annual interest, compounded yearly. How does the graph of the investment’s value over time change if interest is compounded monthly instead?

A) The graph becomes linear. B) The graph rises faster but starts at the same point. C) The graph shifts downward. D) The graph becomes steeper only after 10 years.

6. A computer scientist simplifies 2^(n+3) − 2^(n+1). What is the result?

A) 2^3 − 2^1 B) 6 · 2^n C) 2^(n+3) − 2^(n+1) D) 2^(3−n) − 2^(1−n)

7. The equation 9^(x−1) = 27^(2x) is solved as follows:
Step 1: Rewrite bases: (3^2)^(x−1) = (3^3)^(2x).
Step 2: Simplify exponents: 3^(2x−2) = 3^(6x).
Step 3: Set exponents equal: 2x−2 = 6x.
At which step does the error occur?

A) Step 1 B) Step 2 C) Step 3 D) No error

8. A car’s value depreciates exponentially at 10% per year. If it’s worth $20,000 now, which equation gives its value V after t years?

A) V = 20,000 ⋅ 1.1^t B) V = 20,000 ⋅ 0.9^t C) V = 20,000 − 0.1t D) V = 20,000 ⋅ e^(–0.1t)

9. If 2^a = 3 and 3^b = 8, what is 2^(2ab)?

A) 8 B) 64 C) 12 D) 24

10. The function f(x) = 2^(–x) + 1 is graphed. How does it differ from g(x) = 2^x?

A) f(x) is decreasing and shifted up 1 unit. B) f(x) is increasing and shifted left 1 unit. C) Both functions are identical. D) f(x) grows twice as fast as g(x).

1. Sam has $120 to spend on books ($12 each) and coffee ($3 per cup). They want to buy at least 5 books but no more than 20 coffees. Which system represents the feasible combinations?

A) 12b + 3c ≤ 120, b ≥ 5, c ≤ 20
B) 12b + 3c ≤ 120, b ≤ 5, c ≥ 20
C) 12b + 3c ≥ 120, b ≥ 5, c ≤ 20
D) 12b + 3c = 120, b ≥ 5, c ≤ 20

2. A scientist models the relationship between Fahrenheit (F) and Celsius (C) as linear: F = mC + b. If 0°C = 32°F and 100°C = 212°F, what does the slope (m) represent?

A) The freezing point of water in Fahrenheit.
B) The rate of change of Fahrenheit per degree Celsius.
C) The boiling point of water in Celsius.
D) The y-intercept of the graph.

3. A city mandates that the number of trees (t) in a park must be at least twice the number of benches (b), but no more than 100 total. Which system models this?

A) t ≥ 2b, t + b ≤ 100
B) t ≤ 2b, t + b ≥ 100
C) t > 2b, t + b < 100
D) t = 2b, t + b = 100

4. A company sells gadgets for $15 each. Their costs are $200 fixed + $5 per gadget. How many gadgets must they sell to at least break even?

A) 10
B) 20
C) 25
D) 30

5. The line y = −2x + 5 is graphed. If the slope is doubled and the y-intercept remains the same, how does the new line compare?

A) Steeper and shifts left.
B) Less steep and shifts right.
C) Steeper but same y-intercept.
D) Parallel but higher.

6. A gym charges $30 per month. If you prepay for m months and receive a discount of d dollars off the total (d ≥ 0), the gym caps the total you pay at $400. Which statement follows from 30m − d ≤ 400?

A) The minimum discount needed increases with m.
B) The discount is a fixed $10 per month.
C) The total cost decreases when m > 12.
D) The inequality has no solution.

7. A rectangle’s perimeter is 40 cm. Its length is 2 cm more than twice its width. Which system solves for the dimensions?

A) 2L + 2W = 40, L = 2W + 2
B) L + W = 40, L = 2W
C) 2L + W = 40, L = W + 2
D) LW = 40, L = 2W + 2

8. A phone plan charges $20/month + $0.10/text. Another plan charges $30/month + $0.05/text. After how many texts do the plans cost the same?

A) 50
B) 100
C) 200
D) 300

1. A soccer ball’s height (in meters) is modeled by h(t) = -5t² + 20t. When does the ball hit the ground?

A) After 2 seconds B) After 4 seconds C) When t = 0 D) Never

2. The quadratic f(x) = 2(x - 3)² + 5 is in vertex form. What does the vertex (3, 5) represent?

A) The maximum height of the parabola B) The point where the parabola crosses the x-axis C) The lowest point on the graph, since the parabola opens upward D) The average of the roots

3. The discriminant of 3x² - 6x + c = 0 is zero. What does this imply about c?

A) c = 3 B) c > 3 C) c < 3 D) c could be any real number

4. Why can’t x² + 5x + 7 = 0 be factored into real linear factors?

A) It has one real solution B) The coefficients are prime numbers C) The discriminant is negative D) It’s missing a constant term

5. Why is the quadratic formula guaranteed to work for any quadratic equation?

A) It’s derived from completing the square, a universal method B) It always gives two answers C) It doesn’t require simplifying the equation first D) It’s memorized by all math teachers

6. A drone’s path follows y = -x² + 10x. What does the vertex represent?

A) The drone’s starting position B) The point where the drone runs out of battery C) The maximum height reached D) The time it hits the ground

7. Which quadratic has no real solutions?

A) x² - 4 = 0 B) x² + 4 = 0 C) x² - 4x + 4 = 0 D) x² - 4x - 4 = 0

8. Why is x² + 6x + 10 rewritten as (x + 3)² + 1 useful?

A) It reveals the vertex at (-3, 1) B) It makes the equation easier to factor C) It simplifies the quadratic formula D) It removes the need for a discriminant

9. If a parabola doesn’t intersect the x-axis, which must be true?

A) Its leading coefficient is negative B) Its discriminant is negative C) It has no vertex D) It’s not a quadratic function

10. A farmer wants to enclose a rectangular plot with 100 meters of fencing. Which quadratic models the maximum possible area?

A) A = -l² + 50l B) A = l² + 100l C) A = -2l² + 100l D) A = l² - 50l

1. A fair six-sided die is rolled twice. What is the probability of rolling a 3 both times?

A) 1/6 B) 1/12 C) 1/36 D) 1/18

2. A bag has 3 red marbles and 5 blue marbles. Two marbles are drawn without replacement. What is the probability both marbles are red?

A) 9/64 B) 3/28 C) 3/32 D) 2/7

3. A deck of 52 cards is shuffled. You draw two cards with replacement. What is the probability both cards are aces?

A) 1/169 B) 1/221 C) 4/663 D) 1/52

4. A box has 4 green and 6 yellow marbles. You draw two marbles without replacement. What is the probability of drawing one green and one yellow (in any order)?

A) 24/90 B) 8/15 C) 6/25 D) 12/45

5. A student claims P(A and B) = P(A) × P(B) is always true. When is this statement false?

A) When A and B are independent. B) When A and B are mutually exclusive. C) When A and B are dependent. D) When P(A) = P(B).

6. A coin is flipped 3 times. What is the probability of getting heads all three times?

A) 1/8 B) 1/6 C) 1/2 D) 3/8

7. A jar has 5 red and 7 blue gumballs. Two are drawn without replacement. What is the probability the second gumball is red, given the first was blue?

A) 5/12 B) 5/11 C) 7/12 D) 1/2

8. A bag has 8 white and 2 black socks. You draw two socks without replacement. What is the probability both socks are black?

A) 1/45 B) 1/40 C) 4/45 D) 1/10

9. A spinner has 4 equal sections (red, blue, green, yellow). It’s spun twice. What is the probability of landing on red at least once?

A) 1/16 B) 7/16 C) 1/4 D) 9/16

10. A game requires rolling a die and flipping a coin. What is the probability of rolling an even number and getting heads?

A) 1/4 B) 1/2 C) 1/6 D) 1/12

1. A right triangle ΔABC has ∠B = 90°, AB = 8, and BC = 15. What is the length of AC (hypotenuse)?

A) 17 B) 13 C) 20 D) 23

2. A ladder leans against a wall at a 65° angle to the ground. The ladder’s base is 3 meters from the wall. How tall is the wall? (Use: sin 65° ≈ 0.91, cos 65° ≈ 0.42, tan 65° ≈ 2.14)

A) 1.26 m B) 2.14 m C) 6.42 m D) 7.05 m

3. A right triangle has legs of 7 and 24 units. What is sin(θ), where θ is opposite the 7-unit side?

A) 7/24 B) 7/25 C) 24/25 D) 25/7

4. A kite’s string makes a 50° angle with the ground. The string is 40 meters long. How high is the kite? (Use: sin 50° ≈ 0.77)

A) 30.8 m B) 25.7 m C) 40 m D) 51.2 m

5. A right triangle has hypotenuse 10 and one angle of 30°. What is the side opposite the 30° angle?

A) 5 B) 5√3 C) 10 D) 10√3

6. A 10-meter ramp rises to a height of 2 meters. What is the angle of elevation? (Use: sin⁻¹(0.2) ≈ 11.5°)

A) 11.5° B) 78.5° C) 2° D) 45°

7. A right triangle has legs of lengths 9 and 12. What is the tangent of the smallest angle?

A) 3/4 B) 4/3 C) 10/12 D) 12/9

8. A flagpole casts a 15-meter shadow when the sun is at 45° elevation. How tall is the flagpole?

A) 7.5 m B) 15 m C) 30 m D) 15√2 m

9. A right triangle has ∠A = 20° and adjacent side = 4. What is the hypotenuse? (Use: cos 20° ≈ 0.94)

A) 3.76 B) 4.26 C) 8.51 D) 4.94

10. A ship’s crew measures a lighthouse’s angle of elevation as 10° from 500 meters offshore. How tall is the lighthouse? (Use: tan 10° ≈ 0.18)

A) 90 m B) 500 m C) 2,777 m D) 50 m

1. A quadratic expression is modeled by the area of a rectangle: x² + 7x + 12. The rectangle’s length and width are binomials (x + a) and (x + b). What are the dimensions of the rectangle?

A) (x + 3)(x + 4) B) (x + 2)(x + 6) C) (x + 1)(x + 12) D) (x + 7)(x + 1)

2. A student claims that x² – 9 can be rewritten as (x – 3)². What is the flaw in their reasoning?

A) They forgot to include the middle term. B) They confused difference of squares with a perfect square. C) They incorrectly squared the binomial. D) They ignored the negative sign.

3. The expression 6x² + 11x – 10 represents the area of a trapezoid. Which factored form correctly represents its dimensions?

A) (2x + 5)(3x – 2) B) (6x – 5)(x + 2) C) (3x + 5)(2x – 2) D) (6x + 2)(x – 5)

4. A quadratic’s graph has roots at x = –4 and x = 5. Which expression is not a possible form of this quadratic?

A) (x + 4)(x – 5) B) x² – x – 20 C) 2(x + 4)(x – 5) D) x² + x – 20

5. A physics problem simplifies to the equation 3x² – 10x – 8 = 0. Which factorization solves for x?

A) (3x + 2)(x – 4) B) (3x – 2)(x + 4) C) (x – 2)(3x + 4) D) (x + 2)(3x – 4)

6. A quadratic is written as x² + (a + b)x + ab. What is its factored form?

A) (x + a)(x + b) B) (x + a)(x – b) C) (x + a + b)(x + 1) D) (x + a)(x + a)

7. A student factors 4x² – 25 as (2x – 5)(2x + 5). What principle does this demonstrate?

A) Perfect square trinomial B) Difference of squares C) Sum of cubes D) Grouping

8. The expression x² + 6x + c is a perfect square trinomial. What is the value of c?

A) 3 B) 6 C) 9 D) 12

9. A quadratic ax² + bx + c has roots at x = 2/3 and x = –5. Which quadratic is not possible?

A) 3x² + 13x – 10 B) 6x² + 26x – 20 C) x² + (13/3)x – 10/3 D) x² – (13/3)x + 10/3

10. A quadratic x² + bx + c is factorable. If b is odd, what must be true about c?

A) c must be odd. B) c must be even. C) c must be prime. D) c must be negative.

1. A soccer ball is kicked upward. Its height (h in meters) at time (t in seconds) is modeled by: h = −5t² + 20t. When does the ball hit the ground?

A) 2 seconds B) 4 seconds C) 5 seconds D) 10 seconds

2. The area of a rectangular garden is 24 m², and its length is 2 meters longer than its width. What is the width of the garden?

A) 3 meters B) 4 meters C) 6 meters D) 8 meters

3. A quadratic equation 2x² − 8x + c = 0 has exactly one real solution. What is the value of c?

A) 2 B) 4 C) 8 D) 16

4. The graph of y = ax² + bx + c opens downward and has its vertex at (3, 5). Which statement is true?

A) The equation has no real roots. B) The maximum value of y is 5. C) The axis of symmetry is x = −3. D) The y-intercept is (0, 5).

5. A company’s profit (P in dollars) is modeled by P = −2x² + 100x − 800, where x is units sold. How many units must be sold to break even (P = 0)?

A) 10 or 40 B) 20 or 30 C) 25 only D) 50 only

6. The quadratic equation x² + bx + 9 = 0 has two distinct real roots. Which interval must b fall into?

A) b < −6 or b > 6 B) −6 < b < 6 C) b > 0 only D) b < 0 only

7. A student solves 3x² − 12x + 9 = 0 by factoring as (3x − 3)(x − 3) = 0. What error did they make?

A) Forgot to divide by 3 first. B) No error—the factoring is correct. C) Misapplied the quadratic formula. D) Incorrectly factored the middle term.

8. A ball is thrown upward from a cliff. Its height (h in meters) is given by: h = −4.9t² + 29.4t + 50. What is the ball’s maximum height?

A) 50 meters B) 79.4 meters C) 94.1 meters D) 144.1 meters