HEART OF ALGEBRA
This is the slope of the line perpendicular to the line defined by 3x - 4y = 12
What is -4/3?
If a population of bacteria grows exponentially by 12% every 3 hours, this expression represents the population after t hours starting from an initial population of P0.
What is P0 * (1.12)^(t/3)?
For the quadratic equation 2x^2 - kx + 8 = 0 to have exactly one real solution, k must equal one of these two values.
What are 8 and -8?
In a right triangle, if sin(theta) = 5/13, this is the value of cos(90 degrees - theta).
What is 5/13?
In a sequence where each term after the first is created by adding a constant d, if the 3rd term is 12 and the 7th term is 28, this is the first term of the sequence.
What is 4?
For a system of two linear equations in two variables to have infinitely many solutions, these two geometric properties of their graphed lines must be identical.
What are the slope and the y-intercept (or being the exact same line)?
A sample of 200 students was surveyed, and 45% preferred online textbooks. If the margin of error is 4% at a 95% confidence level, this is the plausible range for the actual percentage of all students who prefer online textbooks.
What is 41% to 49%?
If (x - 3) is a factor of the polynomial P(x) = 2x^3 - 5x^2 - ax + 3, according to the Remainder Theorem, this is the value of P(3).
What is 0?
The equation x^2 + y^2 - 6x + 8y = 0 represents a circle in the xy-plane. This is the radius of the circle.
What is 5? (Equation rewrites to (x-3)^2 + (y+4)^2 = 25)
If f(x) = 2x - 5 and g(x) = x^2 + 3, this is the value of the composite function f(g(4)).
What is 33?
If 1/2(4x - 8) = 3x - (x + 4) + k has no solution, this must be the value of k.
What is any number other than 0 (or k not equal to 0) ?
In a set of 11 distinct integers, the median is 25. If the largest number in the set is increased by 30, this is the change in the median of the new set.
What is 0 (or no change)?
Expressed in the form a + bi, this is the simplified result of the complex quotient (3 + 2i) / (1 - i).
What is 1/2 + 5/2 i (or 0.5 + 2.5i)?
An arc of a circle with radius 12 inches subtends a central angle of 75 degrees. This is the exact length of the arc in terms of pi.
What is 5 pi inches?
The distance between the points (-2, 5) and (4, y) in the xy-plane is exactly 10 units. These are the two possible values for y.
What are 13 and -3?
A taxi company charges a flat fee of $3.50 plus $2.25 per mile. If a passenger leaves a 15% tip on the total fare and pays $25.30 in total, this was the distance of the ride in miles. (round to nearest mile)
What is 8 miles?
A car travels 60 miles at an average speed of 30 mph, and then returns along the exact same route traveling at 60 mph. This is the average speed of the car for the entire round trip.
What is 40 mph?
If sqrt(2x + 6) - x = 1, this is the only valid real solution for x after extraneous solutions are eliminated.
What is x = 5? (Note: x = -1 is an extraneous root)
A cone and a cylinder have equal radii and equal volumes. If the height of the cylinder is 6 inches, this is the height of the cone.
What is 18 inches?
An investment account balance is modeled by the function A(t) = 5000(1.04)^(2t), where t is measured in years. This is the effective annual percentage growth rate of the account.
What is 8.16%? (Since 1.04^2 = 1.0816)
In the xy-plane, the solution set to the system of inequalities y >= 2x - 5 and y <= -x + 4 forms a region. This is the maximum possible value of y for any point (x, y) in this solution set.
What is 1? (Occurs at the intersection point (3, 1))
Box plot A has a median of 50 and an interquartile range (IQR) of 20. If every data point in set A is multiplied by 3 and then increased by 5 to create set B, these are the new median and IQR of set B.
What are 155 (median) and 60 (IQR)?
The vertex of the parabola y = -2x^2 + 12x - 10 is located at this point (h, k) in the xy-plane.
What is (3, 8)?
In right triangle ABC with the right angle at C, tan(A) = 3/4. If a circle centered at C with a radius of 6 is tangent to the hypotenuse AB, this is the area of triangle ABC.
What is 37.5 (or 75/2)?
The average (arithmetic mean) of five distinct positive integers is 20. If the median of the five numbers is 18, this is the greatest possible value for the largest integer in the set.
What is 60? (Set values: 1, 2, 18, 19, 60; sum must equal 100)