Problem sets by domain

If 3(2x − 5) + 4 = 7x − 11, what is the value of x?

A) 0   B) 1   C) 2   D) 4

Answer: A. Expand: 6x − 15 + 4 = 7x − 11 → 6x − 11 = 7x − 11 → x = 0.

If 2x + y = 7 and x − y = 2, what is x + y?

A) 3   B) 4   C) 5   D) 6

Answer: B. Add: 3x = 9 → x = 3; then y = 1; x + y = 4.

A tank starts with 240 L and drains at a constant rate. After 5 min it has 195 L. Which equation gives volume V after t minutes?

A) V = 240 + 9t   B) V = 240 − 9t   C) V = 240 − 5t   D) V = 195 − 9t

Answer: B. Rate = (240 − 195)/5 = 9 L/min, decreasing → V = 240 − 9t.

Line ℓ passes through (1, 4) and (3, 10). Line m is parallel and passes through (−2, 0). What is the y-intercept of m?

Answer: 6. Slope = (10 − 4)/(3 − 1) = 3. y = 3(x + 2) = 3x + 6.

The system 2x + 3y = 12, 4x + ky = 30 has no solution. Find k.

Answer: 6. Parallel requires k/3 = 4/2 = 2 → k = 6; intercepts differ → no solution confirmed.

Solve for x: −3(x − 4) ≥ 9.

Answer: x ≤ 1. Distribute: −3x + 12 ≥ 9 → −3x ≥ −3 → x ≤ 1 (flip when dividing by −3).

What is the slope of the line through (−2, 5) and (4, −7)?

Answer: −2. (−7 − 5)/(4 − (−2)) = −12/6 = −2.

If y = 2x + 3 and 3x + y = 18, what is x?

Answer: 3. Substitute: 3x + (2x + 3) = 18 → 5x = 15 → x = 3.

Line k has slope −2/5. Line p is perpendicular to k. What is the slope of p?

Answer: 5/2. Perpendicular slopes are negative reciprocals.

A taxi charges a base fare of $3.50 plus $0.45 per quarter-mile. Sasha pays $12.50. How many miles did she travel?

Answer: 5 miles. 3.50 + 0.45 · (4 · m) = 12.50 → 1.80m = 9 → m = 5.

f(x) = 4x − 7. What is f(3) − f(−2)?

Answer: 20. f(3) = 5, f(−2) = −15, difference = 20. (Or recognize: the change in input is 5, slope is 4, so the change in output is 20.)

How many integer solutions satisfy |2x − 7| < 5?

Answer: 4. −5 < 2x − 7 < 5 → 1 < x < 6 → integers 2, 3, 4, 5.

What is the sum of the solutions to x² − 7x + 10 = 0?

Answer: 7. Sum of roots = −b/a = 7 (roots are 2 and 5).

For f(x) = 2x² − 12x + 7, what is the minimum value of f?

Answer: −11. Vertex at x = 3; f(3) = 18 − 36 + 7 = −11.

Bacteria double every 4 hours. 500 cells at t = 0. Which gives P(t)?

A) P = 500 · 2^(4t)   B) P = 500 · 2^(t/4)   C) P = 500 + 2t   D) P = 500 · 4^t

Answer: B.

Solve (x + 6)/(x − 2) = 4.

Answer: x = 14/3. Cross-multiply: x + 6 = 4x − 8.

For what value of k does x² + kx + 9 = 0 have exactly one real solution?

Answer: k = ±6. Discriminant zero: k² − 36 = 0.

Which is a factor of x³ − 4x?

A) x − 4   B) x + 2   C) x − 1   D) x² + 4

Answer: B. x³ − 4x = x(x − 2)(x + 2).

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

Answer: 19. g(3) = 9; f(9) = 19.

Simplify (2x³)² · x⁻⁴.

Answer: 4x². 4x⁶ · x⁻⁴ = 4x².

Solve √(2x + 1) = x − 1 for x > 1.

Answer: x = 4. Square both sides: 2x + 1 = x² − 2x + 1 → x² − 4x = 0 → x(x − 4) = 0 → x = 4 (reject 0, which doesn't satisfy original).

Rewrite x² − 8x + 13 in vertex form.

Answer: (x − 4)² − 3. Complete the square: (x − 4)² − 16 + 13 = (x − 4)² − 3.

If P(x) = x³ − 6x² + 11x − 6 and P(1) = 0, factor P.

Answer: (x − 1)(x − 2)(x − 3). Divide by (x − 1) to get x² − 5x + 6 = (x − 2)(x − 3).

If f(x) = (x − 3)/2, what is f⁻¹(x)?

Answer: 2x + 3. Swap and solve: y = (x − 3)/2 → x = 2y + 3.

If log₂(8x) = 5, find x.

Answer: 4. 8x = 2⁵ = 32 → x = 4.

A laptop priced at $800 is discounted by 25%, then a 10% sales tax is added. Final price?

Answer: $660. 800 · 0.75 · 1.10 = 660.

20 students have mean 78. Add 4 bonus points to every score. New mean?

Answer: 82.

200 students: 60 play a sport, 80 play an instrument, 30 do both. Of those who play an instrument, what fraction also play a sport?

Answer: 30/80 = 3/8.

A printer prints 90 pages in 6 minutes. At this rate, how many pages in 15 minutes?

Answer: 225. Rate = 15 pages/min × 15 min = 225.

Data set: 4, 7, 7, 9, 10, 12, 80. Which is greater, mean or median, and by how much?

Answer: Mean > median by 9. Median = 9; mean = (4+7+7+9+10+12+80)/7 = 129/7 ≈ 18.4; difference ≈ 9.4 → closest 9. (Outlier 80 pulls mean up.)

A recipe calls for 3 cups flour to 2 cups sugar. To make a batch using 9 cups flour, how much sugar?

Answer: 6 cups.

A bag has 4 red, 3 blue, 5 green marbles. Two are drawn without replacement. Probability both red?

Answer: (4/12) · (3/11) = 12/132 = 1/11.

A poll of 500 randomly selected residents found 320 favor a measure. With a margin of error of ±4 percentage points at 95% confidence, which conclusion is best supported?

A) Exactly 64% favor the measure.   B) Between 60% and 68% favor the measure.   C) Less than half favor the measure.   D) The measure will pass.

Answer: B. 320/500 = 64%; margin ±4 → interval [60%, 68%].

Two data sets have the same mean. Set A: 50, 50, 50, 50. Set B: 30, 50, 50, 70. Which has the larger SD?

Answer: B. More spread from the mean.

A car uses 8 liters of fuel per 100 km. How many liters does it use over 250 km?

Answer: 20 L. 8 · 2.5 = 20.

A scatterplot shows hours studied (x) vs test score (y). A best-fit line is y = 5x + 60. Interpret the slope.

Answer: Each additional hour studied is associated with a 5-point higher predicted score (correlation, not causation).

A stock rises 20% in year 1 and falls 20% in year 2. What is the net change after 2 years?

Answer: Down 4%. 1.20 · 0.80 = 0.96 → −4%.

Right triangle: one leg 9, hypotenuse 15. Other leg?

Answer: 12 (3-4-5 family scaled by 3).

x² + y² − 6x + 4y − 12 = 0. Radius?

Answer: 5. (x − 3)² + (y + 2)² = 25.

Right triangle, right angle at C, AC = 5, BC = 12. Find sin A.

Answer: 12/13. Hypotenuse = 13; opposite to A = 12.

Two similar triangles. The smaller has perimeter 12; the larger has perimeter 30. If the smaller's area is 6, what is the larger's area?

Answer: 37.5. Area scales as the square of the linear ratio: (30/12)² = 6.25; 6 · 6.25 = 37.5.

A circle has radius 6. A central angle measures π/3 radians. Arc length?

Answer: 2π. s = rθ = 6 · π/3 = 2π.

Cylinder, radius 4, height 10. Volume?

Answer: 160π. V = πr²h = π · 16 · 10.

If sin θ = 3/5 for an acute angle θ, find cos(90° − θ).

Answer: 3/5. cos(90° − θ) = sin θ.

In a triangle, two angles are 35° and 85°. What is the third?

Answer: 60°.

Two parallel lines are cut by a transversal. One of the alternate interior angles is 110°. What is the other?

Answer: 110°. Alternate interior angles are equal.

Distance between (1, 2) and (7, 10)?

Answer: 10. √(36 + 64) = √100 = 10.

"Octopuses possess the ability to change the color and texture of their skin within milliseconds. This camouflage capability, controlled by specialized pigment cells called chromatophores, is unmatched by any other invertebrate. Recent studies suggest that even blind octopuses can match background colors, hinting at a light-sensing capability distributed across the skin itself."

Which choice best states the central claim?

A) Octopuses are the most intelligent invertebrates.   B) Octopus camouflage is exceptionally fast, fine-grained, and possibly skin-mediated.   C) Blind octopuses cannot fully camouflage.   D) Chromatophores evolved for predator avoidance.

Answer: B. Paraphrases the full passage: fast change + chromatophores + skin sensing.

"Although solar panel costs have fallen ninety percent since 2010, deployment in many regions has lagged. Permits, grid connections, and labor remain comparatively expensive and slow. ____."

A) Therefore, the price of solar panels is the main barrier to deployment.   B) Therefore, non-panel costs may now dominate the total system cost.   C) Therefore, governments should subsidize panel manufacturing.   D) Therefore, solar deployment will reach grid parity within a decade.

Answer: B. Only B follows from "non-panel costs remain expensive": those are now the dominant share.

"A researcher hypothesizes that crows learn to recognize individual human faces. Which finding, if true, would most directly support this hypothesis?"

A) Crows in urban areas are more numerous than in rural areas.   B) Crows that previously interacted with a specific masked researcher gave alarm calls when that mask reappeared, but not when a new mask appeared.   C) Crows produce a wide range of vocalizations.   D) Crows can use simple tools.

Answer: B. Direct match: discrimination between a previously-seen mask and a novel one is face recognition behavior.

"A table reports rainfall totals for City X across four years: 2019: 78 cm, 2020: 90 cm, 2021: 85 cm, 2022: 102 cm. Which choice most accurately uses these data to support the claim that rainfall in City X has trended upward over this period?"

A) Rainfall in 2020 exceeded rainfall in 2019.   B) The 2022 value is the highest of the four years.   C) Rainfall increased each year.   D) The four-year mean rainfall is approximately 89 cm.

Answer: B. C is false (2021 dropped from 2020). B is the only choice that supports "trended upward" without overstating.

"Migratory birds rely on a combination of celestial cues, magnetic fields, and landmarks to navigate. When researchers shifted magnetic fields in a controlled environment, even captive-raised birds reoriented in the direction predicted by the new field, suggesting that ____."

A) magnetic orientation can be learned only through prior migration experience   B) magnetic orientation may be present without prior migration experience   C) celestial cues are not used in real migration   D) landmarks are the most important navigational input

Answer: B. Captive-raised birds had no prior migration, yet still responded — so the magnetic compass is at least partly innate.

"In the 19th century, the chemist Marcellin Berthelot synthesized organic compounds from inorganic precursors, challenging the long-held belief that organic molecules required a vital force. His syntheses of methanol and acetylene from carbon, hydrogen, and water marked turning points in chemistry."

According to the passage, what was the significance of Berthelot's syntheses?

A) They produced the first commercial fuels.   B) They disproved the idea that organic molecules require a vital force.   C) They led to the discovery of carbon.   D) They were rejected by his contemporaries.

Answer: B. Direct paraphrase.

"The committee's report was praised for its candid assessment of the program's shortcomings."

As used here, "candid" most nearly means:

A) honest   B) photographic   C) brief   D) hostile

Answer: A. "Candid" = frank/honest; the context "assessment of shortcomings" rewards honesty, not hostility.

"The author's prose is spare, relying on short sentences and concrete nouns."

As used, "spare" most nearly means:

A) extra   B) lean   C) reserved   D) decorative

Answer: B. "Spare" here = stripped-down / minimal, matched by "short sentences, concrete nouns".

(Passage describes the Mariana Trench; an underlined sentence reads: "Despite its depth and darkness, the trench teems with life forms uniquely adapted to high pressure.")

The main purpose of the underlined sentence is to:

A) introduce a counterintuitive finding   B) summarize the geological history of the trench   C) describe specific species   D) argue for further exploration

Answer: A. The "Despite ... teems with life" structure flags a counterintuitive observation.

"Some economists argue that minimum wage increases raise unemployment among low-skill workers. Others find no such effect in their empirical studies. A recent meta-analysis suggests the estimated effect depends heavily on whether neighboring counties are used as controls."

How does the third sentence function?

A) Provides a definitive resolution to the debate.   B) Suggests a methodological reason for the disagreement.   C) Refutes both prior positions.   D) Introduces a third unrelated position.

Answer: B. The third sentence locates the disagreement in methodology (choice of controls).

"Author A argues that universal basic income (UBI) would reduce poverty without significantly affecting labor supply. Author B, citing experimental data from Finland, argues that while UBI did not measurably reduce work, it also did not produce the dramatic well-being gains UBI advocates predict."

Which best describes Author B's stance toward Author A's claim?

A) Agrees with the labor-supply portion but is skeptical of UBI's broader benefits.   B) Rejects the labor-supply portion entirely.   C) Argues UBI worsens poverty.   D) Asserts Finland's experiment failed.

Answer: A.

"The professor's lecture was so dense that students struggled to extract its main argument."

As used, "dense" most nearly means:

A) heavy   B) packed with ideas   C) unclear   D) stubborn

Answer: B. "Dense" here = full of content (hard to extract from), not the colloquial "slow-witted".

"The exhibit drew record crowds in its first week. ____, the museum extended its hours through the end of the month."

A) However   B) For example   C) As a result   D) In contrast

Answer: C.

"Most reviews emphasized the novel's plot. ____, several critics argued the book's real achievement lay in its prose style."

A) Therefore   B) Specifically   C) However   D) Similarly

Answer: C.

"Many crystalline minerals exhibit remarkable structural symmetry. ____, halite forms perfect cubes whose faces meet at right angles."

A) However   B) For example   C) Nevertheless   D) In addition

Answer: B. Halite is a specific instance of the general claim.

"The proposed bridge would shorten travel time considerably. ____, environmental groups have raised concerns about its impact on the river ecosystem."

A) Therefore   B) Nonetheless   C) For instance   D) Likewise

Answer: B. Acknowledges a counterpoint.

"The new policy prioritizes prevention over treatment. ____, hospitals will be funded based on community health metrics rather than admissions."

A) However   B) Specifically   C) On the contrary   D) Earlier

Answer: B. The second sentence elaborates / makes the first concrete.

"The data show consistent gains across all three pilot regions. The trend held even after controlling for income. ____, the intervention appears effective regardless of local economic conditions."

A) Nevertheless   B) For example   C) Thus   D) Meanwhile

Answer: C.

Notes:

• The Antikythera mechanism is an ancient Greek analog computer.
• It was discovered in 1901 in a shipwreck off Antikythera.
• It was used to predict astronomical positions and eclipses.
• It dates to approximately the late 2nd century BCE.

Goal: Introduce the Antikythera mechanism to an audience unfamiliar with it.

A) Discovered in 1901, the Antikythera mechanism dates to the late 2nd century BCE.
B) The Antikythera mechanism, discovered in 1901, was used to predict astronomical positions.
C) The Antikythera mechanism is an ancient Greek analog computer used to predict astronomical positions and eclipses.
D) The Antikythera mechanism was discovered off the island of Antikythera in 1901.

Answer: C. Introduces both what it is and what it does.

Notes:

• Quokkas live on Rottnest Island, Australia.
• They weigh 2.5–5 kg.
• Tree kangaroos live in New Guinea rainforests.
• They weigh 7–14 kg.

Goal: Emphasize a key difference between quokkas and tree kangaroos.

A) Quokkas live on Rottnest Island in Australia, while tree kangaroos live in the rainforests of New Guinea.
B) Quokkas weigh 2.5 to 5 kg, considerably less than tree kangaroos, which weigh 7 to 14 kg.
C) Both quokkas and tree kangaroos are marsupials.
D) Tree kangaroos and quokkas inhabit very different ecosystems.

Answer: B. Most quantitatively explicit comparison.

Notes:

• A team studied 240 retired professional musicians.
• Average age was 68.
• Musicians scored higher than non-musicians on tests of auditory working memory.
• No significant difference appeared on visual working memory.

Goal: Summarize the study's primary finding for a general audience.

A) The study's design was robust.   B) Musicians outperformed non-musicians on tests of auditory, but not visual, working memory.   C) Musicians age more slowly than non-musicians.   D) The study examined 240 retired musicians.

Answer: B.

"The committee reviewed the proposal carefully ____ they delivered their verdict the following week."

A) carefully,   B) carefully;   C) carefully and   D) carefully, and

Answer: D. Comma + FANBOYS joins two independent clauses.

"The list of approved candidates ____ posted on the bulletin board."

A) is   B) are   C) were   D) have been

Answer: A. Subject = "list" (singular).

"Walking through the gallery, ____."

A) the paintings impressed her   B) it was an impressive experience   C) she was impressed by the paintings   D) the experience impressed her

Answer: C. Subject of main clause must be the one walking.

"The kit included three essential items ____ a flashlight, a notebook, and a compass."

A) —   B) :   C) ;   D) ,

Answer: B. Complete sentence + introduced list → colon.

"The ____ favorite exhibit closed last week."

A) childrens   B) children's   C) childrens'   D) childrens's

Answer: B. "Children" is already plural; possessive adds 's.

"Each of the volunteers brought ____ own tools to the site."

A) their   B) his or her   C) its   D) there

Answer: B. "Each" is singular on SAT; the safe answer is "his or her". (Spoken English often accepts "their"; SAT does not.)

"By the time the conference started, the keynote speaker ____ already arrived."

A) has   B) had   C) was   D) will have

Answer: B. Past perfect (action completed before another past action).

"The data are surprising, the researchers plan to replicate the study next year."

Which best corrects this sentence?

A) (no change)   B) surprising; the   C) surprising the   D) surprising, the researchers, plan

Answer: B. Two independent clauses → semicolon (or period, or comma+FANBOYS).

"The novel ____ was published in 1924 ____ remains widely read today."

Which choice uses paired commas to set off a non-essential modifier?

A) , which,   B) which   C) , which   D) — which

Answer: A. Non-essential "which" clauses take paired commas on both sides.

"Running across the meadow, the photographer caught the deer in mid-leap."

What is the issue and how to fix?

Answer: "Running across the meadow" attaches to "the photographer", suggesting the photographer was running. Fix: "The photographer caught the deer running across the meadow" or "As the deer ran across the meadow, the photographer caught it in mid-leap."