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
- Logarithms
- Matrices
- Complex Numbers
- Equations of Lines
- Circles
- 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 log_{b}a = x and log_{b}c = y, which of the following is equivalent to log_{b}a^{2}c ?

(A) x^{2}y

(B) x^{2} + y

(C) 2x + y

(D) x + 2y

(E) xy^{2}

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*?

(F) (-2*a*, *a*)

(G) (-2*a*, -2*a*)

(H) (-1/2*a*, –*a*)

(J) (-*a*, -2*a*)

(K) (-*a*, -3*a*)

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?

(A) 21

(B) 36

(C) 63

(D) 68

(E) 84

4) For the complex numbers 2 + *i* and 5 – *i*, what is the value of (2 + *i*)(5 – *i*)? Recall that *i*^{2} = -1.

(F) 9 + 3*i
*(G) 9 + 7

*i*

(H) 10 + 3

*i*

(J) 11 + 3

*i*

(K) 11 + 7

*i*

5) If (x – 5) is a factor of 2x^{2} – 7x + *k*, what is the value of *k* ?

(A) -15

(B) -10

(C) -5

(D) 5

(E) 12

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) = x^{2} + 3, then *f*(x + h) = ?

(F) x^{2} + h^{2
}(G) x^{2} + h + 3

(H) x^{2} + h^{2} + 3

(J) x^{2} + 2xh + h^{2
}(K) x^{2} + 2xh + h^{2} + 3

9) Which of the following is the set of all real numbers *x* such that x^{2} > x^{3}

(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

10) The matrix multiplication would result in which product matrix?

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 info@cardinalec.com .

Solutions:

1) (C)

2) (J)

3) (C)

4) (J)

5) (A)

6) (J)

7) (C)

8) (K)

9) (E)

10) (J)