The ACT classifies their math problems into three categories: Pre-Algebra and Elementary Algebra, Intermediate Algebra and Coordinate Geometry, and Plane Geometry and Trigonometry. Topics covered in Intermediate Algebra and Coordinate Geometry, the subject of this post, include:
- Quadratic Equations
- Rational and Radical Expressions
- Absolute Value Equations and Inequalities
- Systems of Equations
- Functions and Their Graphs
- Complex Numbers
- Equations of Lines
- Distance and Midpoint Formulas
- Conic Sections
Below you’ll find a number of problems similar to those that could appear on the ACT Math test. Give them a try and see how you do. The answers appear at the bottom of this post.
1) Given that logba = x and logbc = y, which of the following is equivalent to logba2c ?
(B) x2 + y
(C) 2x + y
(D) x + 2y
2) In the semicircle below, Point N is the point that has the least y-coordinate. Point M has the least x-coordinate and Point P has the greatest x-coordinate. What are the coordinates of Point N?
3) On a map in the standard (x, y) coordinate plane, the town of Abbington and Barnardsville are represented by the points (-2, 3) and (5, -11), respectively. Each unit on the map represents a distance of 4 miles. Which of the following is the closest to the distance, in miles, between the two towns?
4) For the complex numbers 2 + i and 5 – i, what is the value of (2 + i)(5 – i)? Recall that i2 = -1.
(F) 9 + 3i
(G) 9 + 7i
(H) 10 + 3i
(J) 11 + 3i
(K) 11 + 7i
5) If (x – 5) is a factor of 2x2 – 7x + k, what is the value of k ?
6) At a local farm stand, pumpkins that are between 12 and 20 pounds are considered to be “medium” pumpkins. Which of the inequalities below gives the weights, w, of all such medium-sized pumpkins?
(F) | w – 8 | < 20
(G) | w – 4 | < 16
(H) | w – 8 | < 16
(J) | w – 16 | < 4
(K) | w – 16 | < 8
7) In the standard (x, y) coordinate plane, a circle is inscribed in a square with vertices
(-2, -5), (-2, 1), (4, -5) and (4, 1). Which of the following is an equation of the circle?
(A) (x – 1)2 + (y + 2)2 = 3
(B) (x – 1)2 + (y + 2)2 = 6
(C) (x – 1)2 + (y + 2)2 = 9
(D) (x + 1)2 + (y – 2)2 = 3
(E) (x + 1)2 + (y – 2)2 = 9
8) If f(x) = x2 + 3, then f(x + h) = ?
(F) x2 + h2
(G) x2 + h + 3
(H) x2 + h2 + 3
(J) x2 + 2xh + h2
(K) x2 + 2xh + h2 + 3
9) Which of the following is the set of all real numbers x such that x2 > x3
(A) The set containing all real numbers
(B) The set containing all negative real numbers
(C) The set containing all positive real numbers
(D) The set containing all real numbers x such that -1 < x < 1
(E) The set containing all real numbers x such that x < 1
If you have questions about these problems or anything else to do with the ACT, leave a comment below or send me an email at firstname.lastname@example.org .