Heart of Algebra and Linear Systems
If 3x + 2y = 12 and x - y = 4, what is the value of x + y?
Answer: What is 4?
If x^2 - kx + 36 = (x - a)^2 for a positive constant a, what is the value of k?
Answer: What is 12?
stock price increases by 20% on Monday and then decreases by 20% on Tuesday. What is the net percentage change from its original price?
Answer: What is a 4% decrease (or -4%)?
In a right triangle, sin(theta) = 5/13 for an acute angle theta. What is the value of tan(theta)?
Answer: What is 5/12?
For i = sqrt(-1), what is the simplified value of i^45?
Answer: What is i?
A line passes through (2, k) and (6, 3k). If this line is perpendicular to 2x + 3y = 7, what is the value of k? Answer: What is 3?
Answer: What is 3?
The vertex of the parabola y = 2x^2 - 12x + c lies on the horizontal line y = 5. What is the value of c?
Answer: What is 23?
The mean of five numbers is 18. If one number is removed, the mean of the remaining four numbers becomes 20. What was the removed number?
Answer: What is 10?
What is the radius of the circle defined by the equation x^2 + y^2 - 6x + 8y = 0?
Answer: What is 5?
If g(x) = 3 * 2^(x-1), for what value of x does g(x) = 96?
Answer: What is 6?
For what value of c does the system of equations 2x + 2y = 8 and 3x + cy = 12 have infinitely many solutions?
Answer: What is 3?
Assuming x is not equal to -3, for what value of x does (x^2 - 9)/(x + 3) = 2x - 7?
Answer: What is 4?
An investment account balance grows according to V(t) = 500(1.04)^(2t), where t is time in years. What is the annual percentage growth rate of the account?
Answer: What is 8.16%?
An arc of a circle with radius 12 has an arc length of 4*pi. What is the measure of its central angle in degrees?
Answer: What is 60 degrees?
What is the product of the complex conjugate pair (3 + 4i) and (3 - 4i)?
Answer: What is 25?
The line y = mx + b has a negative slope, a positive y-intercept, and an x-intercept at (6, 0). If m * b = -18, what is the slope m?
Answer: What is -sqrt(3)?
The quadratic function g(x) = a(x - 3)(x + 5) reaches its maximum value at y = 32. What is the value of a?
Answer: What is -2?
In a group of 80 students, 45 take Spanish, 35 take French, and 15 take both. What is the probability that a randomly chosen student takes neither subject?
Answer: What is 3/16 (or 0.1875)?
: Right triangle ABC is similar to right triangle DEF, where angle A corresponds to angle D. If cos(A) = 3/5, what is sin(F)?
Answer: What is 3/5?
If 2^(x+1) * 8^x = 32, what is the value of x?
Answer: What is 1?
The function f(x) = |2x - 8| + k has a minimum value of -5. What is the x-intercept with the greater value?
Answer: What is 6.5 (or 13/2)?
If x > 0 and x^4 - 13x^2 + 36 = 0, what is the sum of all possible positive values of x?
Answer: What is 5?
How many liters of pure acid must be added to 10 liters of a 20% acid solution to produce a new solution that is 50% acid?
Answer: What is 6 liters?
A regular hexagon is inscribed in a circle with radius 6. What is the exact area of the region inside the circle but outside the hexagon?
Answer: What is 36pi - 54sqrt(3)
When written in standard form a + bi, what is the value of b for the complex fraction (2 + i)/(1 - 2i)?
Answer: What is 1?