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 the equations: 3x = 9 → x = 3, so y = 1; x + y = 4.

A water tank starts with 240 liters and drains at a constant rate. After 5 minutes the tank contains 195 liters. Which equation gives the volume V, in liters, after t minutes?

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

Answer: B. Drain rate = (240 − 195) / 5 = 9 L/min; starts at 240 → V = 240 − 9t.

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

Answer: 6. Slope of ℓ = (10 − 4) / (3 − 1) = 3. Line m: y − 0 = 3(x − (−2)) → y = 3x + 6.

The system 2x + 3y = 12 and 4x + ky = 30 has no solution. What is the value of k?

Answer: 6. No solution → lines are parallel with different intercepts. The second equation must be a non-proportional multiple of the first in x and y but not the constant. Coefficient ratio for x: 4/2 = 2. So k/3 = 2 → k = 6. (Check: 4x + 6y = 30 vs. 4x + 6y = 24 — parallel.)

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

Answer: 7. Sum of roots = −b/a = 7/1 = 7. (Roots are 2 and 5.)

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

Answer: −11. Vertex at x = −b/(2a) = 12/4 = 3; f(3) = 2(9) − 36 + 7 = 18 − 36 + 7 = −11.

A culture of bacteria doubles every 4 hours. If 500 cells are present at time t = 0 (hours), which equation gives the population P at time t?

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

Answer: B. "Doubles every 4 hours" → exponent is t/4.

Solve for x: (x + 6) / (x − 2) = 4.

Answer: x = 14/3. Cross-multiply: x + 6 = 4(x − 2) = 4x − 8 → 14 = 3x → x = 14/3.

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

Answer: $660. 800 × 0.75 = 600; 600 × 1.10 = 660.

A class of 20 students has a mean test score of 78. After the teacher adds 4 bonus points to every student's score, what is the new mean?

Answer: 82. Adding a constant to every value shifts the mean by that constant.

A survey of 200 students records whether they play a sport and whether they play an instrument. 60 play a sport, 80 play an instrument, and 30 do both. What fraction of students who play an instrument also play a sport?

Answer: 30 / 80 = 3/8. "Given that they play an instrument" → divide by 80.

In a right triangle, one leg has length 9 and the hypotenuse has length 15. What is the length of the other leg?

Answer: 12. This is a 3-4-5 triangle scaled by 3: legs 9, 12, hypotenuse 15.

A circle has equation x² + y² − 6x + 4y − 12 = 0. What is the radius?

Answer: 5. Complete the square: (x − 3)² + (y + 2)² = 12 + 9 + 4 = 25; radius = 5.

In right triangle ABC, the right angle is at C, AC = 5, and BC = 12. What is sin A?

Answer: 12/13. Hypotenuse AB = 13. Side opposite to A is BC = 12. sin A = 12/13.

"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. Two independent clauses joined by a coordinating conjunction need a comma before the conjunction. B is also grammatical but the choice between semicolon and "comma + and" usually goes to "comma + and" when both clauses are short and connected.

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

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

Answer: A. Subject is "list" (singular). The prepositional phrase "of approved candidates" doesn't change the subject.

"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. The opening phrase "Walking through the gallery" must modify the subject of the main clause. Only "she" makes sense as the walker.

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

A) —   B) :   C) ;   D) ,

Answer: B. The clause before the punctuation is a complete sentence introducing a list — a colon is the standard choice. (An em dash is also defensible, but on SAT, colon is preferred for list introductions.)

"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. The second sentence is a consequence of the first → cause-effect.

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

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

Answer: C. The second sentence introduces a contrasting view.

Notes:

• The Antikythera mechanism is an ancient Greek analog computer.
• It was discovered in 1901 in a shipwreck off the Greek island of 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. The goal is to introduce — to an audience unfamiliar — so the sentence must explain what it is and what it does. C does both. A, B, D each leave out one half.