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GACE Mathematics (522) study materials
GACE Mathematics (522) study resources

GACE Mathematics (522) practice questions

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Question 1Algebra and functions

Which of the following is the completely factored form of x^2 - 9x + 18?

  1. (x - 3)(x - 6) Correct answer.
  2. (x + 3)(x + 6)
  3. (x - 2)(x - 9)
  4. (x - 3)(x + 6)

Explanation

To factor, find two numbers that multiply to 18 and add to -9. Those numbers are -3 and -6, so the factored form is (x - 3)(x - 6).

Question 2Algebra and functions

By the Remainder Theorem, what is the remainder when the polynomial p(x) = x^3 - 2x + 5 is divided by (x - 2)?

  1. 9 Correct answer.
  2. 5
  3. 1
  4. 13

Explanation

The Remainder Theorem states that the remainder of p(x) divided by (x - a) equals p(a). Evaluating p(2) = 2^3 - 2(2) + 5 = 8 - 4 + 5 = 9.

Question 3Algebra and functions

What is the coefficient of the x^2 term in the binomial expansion of (x + 3)^4?

  1. 54 Correct answer.
  2. 12
  3. 27
  4. 108

Explanation

By the binomial theorem, the term with x^2 is C(4,2) · x^2 · 3^2 = 6 · x^2 · 9 = 54x^2. The coefficient is therefore 54.

Question 4Algebra and functions

Solve the equation 3(x - 4) = 2x + 5 for x.

  1. 17 Correct answer.
  2. 7
  3. -7
  4. 1

Explanation

Distribute to get 3x - 12 = 2x + 5. Subtract 2x from both sides to get x - 12 = 5, then add 12 to both sides to get x = 17.

Question 5Algebra and functions

Using the quadratic formula, what are the solutions to 2x^2 - 4x - 6 = 0?

  1. x = 3 and x = -1 Correct answer.
  2. x = -3 and x = 1
  3. x = 2 and x = -3
  4. x = 6 and x = -1

Explanation

Divide by 2 to simplify to x^2 - 2x - 3 = 0, which factors as (x - 3)(x + 1) = 0. The solutions are x = 3 and x = -1.

Question 6Algebra and functions

What does the discriminant of the quadratic equation x^2 + 4x + 5 = 0 tell you about its solutions?

  1. There are two complex (non-real) solutions Correct answer.
  2. There are two distinct real solutions
  3. There is exactly one real solution
  4. There are no solutions of any kind

Explanation

The discriminant is b^2 - 4ac = 4^2 - 4(1)(5) = 16 - 20 = -4. A negative discriminant indicates two complex conjugate (non-real) solutions.

Question 7Algebra and functions

Solve the system of equations: 2x + y = 7 and x - y = 2.

  1. x = 3, y = 1 Correct answer.
  2. x = 1, y = 3
  3. x = 2, y = 3
  4. x = 3, y = 2

Explanation

Adding the two equations eliminates y: (2x + y) + (x - y) = 7 + 2 gives 3x = 9, so x = 3. Substituting into x - y = 2 gives 3 - y = 2, so y = 1.

Question 8Algebra and functions

What is the average rate of change of f(x) = x^2 over the interval from x = 1 to x = 4?

  1. 5 Correct answer.
  2. 3
  3. 15
  4. 9

Explanation

Average rate of change is [f(4) - f(1)] / (4 - 1) = (16 - 1) / 3 = 15 / 3 = 5.

Question 9Algebra and functions

A line passes through the points (1, 2) and (4, 11). What is the slope of the line?

  1. 3 Correct answer.
  2. 9
  3. 1/3
  4. -3

Explanation

Slope is the change in y divided by the change in x: (11 - 2) / (4 - 1) = 9 / 3 = 3.

Question 10Algebra and functions

Completing the square on x^2 + 6x = 7 transforms it into which equivalent form?

  1. (x + 3)^2 = 16 Correct answer.
  2. (x + 3)^2 = 7
  3. (x + 6)^2 = 16
  4. (x - 3)^2 = 16

Explanation

Take half of the coefficient of x, which is 6/2 = 3, and square it to get 9. Adding 9 to both sides gives x^2 + 6x + 9 = 7 + 9, which is (x + 3)^2 = 16.

Question 11Algebra and functions

Which expression is the simplified sum of the rational expressions 1/x + 1/(x + 1)?

  1. (2x + 1) / [x(x + 1)] Correct answer.
  2. 2 / (2x + 1)
  3. (x + 1) / x
  4. 1 / [x(x + 1)]

Explanation

Use the common denominator x(x + 1). The first fraction becomes (x + 1) / [x(x + 1)] and the second becomes x / [x(x + 1)]. Adding the numerators gives (x + 1 + x) / [x(x + 1)] = (2x + 1) / [x(x + 1)].

Question 12Algebra and functions

Solve the radical equation the square root of (x + 6) = x for real x, checking for extraneous solutions.

  1. x = 3 only Correct answer.
  2. x = 3 and x = -2
  3. x = -2 only
  4. There are no real solutions

Explanation

Squaring both sides gives x + 6 = x^2, so x^2 - x - 6 = 0, which factors as (x - 3)(x + 2) = 0. Testing both candidates, x = 3 works but x = -2 is extraneous because the square root cannot equal a negative number, so x = 3 only.

Question 13Algebra and functions

What is the solution to the inequality -2x + 3 < 11?

  1. x > -4 Correct answer.
  2. x < -4
  3. x > 4
  4. x < 4

Explanation

Subtract 3 from both sides to get -2x < 8. Dividing both sides by -2 reverses the inequality sign, giving x > -4.

Question 14Algebra and functions

The formula for the area of a trapezoid is A = (1/2)(b1 + b2)h. Solving for b1 gives which expression?

  1. b1 = (2A / h) - b2 Correct answer.
  2. b1 = 2A - b2 - h
  3. b1 = (2A - b2) / h
  4. b1 = A / h - b2

Explanation

Multiply both sides by 2 to get 2A = (b1 + b2)h, then divide by h to get 2A/h = b1 + b2. Subtracting b2 isolates b1 = (2A / h) - b2.

Question 15Algebra and functions

A system of two linear equations is represented by parallel, non-identical lines. How many solutions does the system have?

  1. No solution Correct answer.
  2. Exactly one solution
  3. Exactly two solutions
  4. Infinitely many solutions

Explanation

Parallel lines never intersect, so there is no point that satisfies both equations simultaneously. The system is inconsistent and has no solution.

Question 16Algebra and functions

Which graph correctly represents the solution set to the linear inequality y > 2x - 1?

  1. A dashed line through (0, -1) with shading above the line Correct answer.
  2. A solid line through (0, -1) with shading above the line
  3. A dashed line through (0, -1) with shading below the line
  4. A solid line through (0, 1) with shading below the line

Explanation

Because the inequality is strict (greater than, not greater than or equal to), the boundary line is dashed. Since y is greater than the expression, the region above the line is shaded, and the y-intercept is -1.

Question 17Algebra and functions

Given the system represented by the matrix equation where the coefficient matrix is [[2, 1], [1, 3]], what is the determinant of this coefficient matrix?

  1. 5 Correct answer.
  2. 7
  3. 6
  4. -5

Explanation

The determinant of a 2x2 matrix [[a, b], [c, d]] is ad - bc. Here that is (2)(3) - (1)(1) = 6 - 1 = 5.

Question 18Algebra and functions

Which expression is equivalent to the difference of squares x^2 - 25?

  1. (x - 5)(x + 5) Correct answer.
  2. (x - 5)^2
  3. (x + 5)^2
  4. (x - 25)(x + 1)

Explanation

A difference of squares a^2 - b^2 factors as (a - b)(a + b). Here a = x and b = 5, so x^2 - 25 factors as (x - 5)(x + 5).

Question 19Algebra and functions

The polynomial p(x) = x^3 - 4x has how many real zeros?

  1. Three: x = 0, x = 2, and x = -2 Correct answer.
  2. One: x = 0 only
  3. Two: x = 2 and x = -2
  4. None

Explanation

Factor out x to get x(x^2 - 4) = x(x - 2)(x + 2). Setting each factor to zero gives x = 0, x = 2, and x = -2, for three real zeros.

Question 20Algebra and functions

A new operation is defined for real numbers by a * b = a + 2b. Is this operation commutative?

  1. No, because a * b generally does not equal b * a Correct answer.
  2. Yes, because addition is commutative
  3. Yes, because multiplication is involved
  4. It is commutative only when a = b

Explanation

Compute both orders: a * b = a + 2b while b * a = b + 2a. These are equal only in special cases (when a = b), so in general the operation is not commutative.

Question 21Algebra and functions

Which relation is NOT a function?

  1. {(1, 2), (1, 3), (2, 4)} Correct answer.
  2. {(1, 2), (2, 2), (3, 2)}
  3. {(0, 1), (1, 2), (2, 3)}
  4. {(-1, 5), (0, 5), (1, 6)}

Explanation

A function assigns exactly one output to each input. The set {(1, 2), (1, 3), (2, 4)} maps the input 1 to two different outputs (2 and 3), so it is not a function.

Question 22Algebra and functions

If f(x) = 2x^2 - 3, what is f(-2)?

  1. 5 Correct answer.
  2. -11
  3. 1
  4. 11

Explanation

Substitute -2 for x: f(-2) = 2(-2)^2 - 3 = 2(4) - 3 = 8 - 3 = 5. Note that squaring -2 yields a positive 4.

Question 23Algebra and functions

What is the domain of the function f(x) = 1 / (x - 4)?

  1. All real numbers except x = 4 Correct answer.
  2. All real numbers
  3. All real numbers except x = 0
  4. All real numbers greater than 4

Explanation

The function is undefined where the denominator equals zero. Setting x - 4 = 0 gives x = 4, so the domain is all real numbers except x = 4.

Question 24Algebra and functions

The graph of g(x) = (x - 3)^2 + 2 is the graph of f(x) = x^2 transformed in which way?

  1. Shifted right 3 units and up 2 units Correct answer.
  2. Shifted left 3 units and up 2 units
  3. Shifted right 3 units and down 2 units
  4. Shifted left 3 units and down 2 units

Explanation

In the form (x - h)^2 + k, the value h shifts horizontally and k shifts vertically. Here h = 3 shifts the graph right 3 units, and k = 2 shifts it up 2 units.

Question 25Algebra and functions

What is the inverse of the function f(x) = 3x + 6?

  1. f inverse (x) = (x - 6) / 3 Correct answer.
  2. f inverse (x) = (x + 6) / 3
  3. f inverse (x) = 3x - 6
  4. f inverse (x) = x / 3 + 6

Explanation

To find the inverse, write y = 3x + 6, swap x and y to get x = 3y + 6, then solve for y: x - 6 = 3y, so y = (x - 6) / 3.

Question 26Algebra and functions

If f(x) = x + 1 and g(x) = x^2, what is the composition (f composed with g)(x), written f(g(x))?

  1. x^2 + 1 Correct answer.
  2. (x + 1)^2
  3. x^2 + 2x + 1
  4. x^3 + x^2

Explanation

The composition f(g(x)) substitutes g(x) into f. Since g(x) = x^2, we get f(x^2) = x^2 + 1.

Question 27Algebra and functions

An exponential function models a population that grows by 8% each year. Which equation represents the population P after t years if the initial population is 500?

  1. P = 500(1.08)^t Correct answer.
  2. P = 500(0.08)^t
  3. P = 500(1.8)^t
  4. P = 500 + 0.08t

Explanation

Exponential growth uses P = P0(1 + r)^t. With r = 0.08 and P0 = 500, the model is P = 500(1.08)^t, where the base 1.08 reflects 100% retained plus 8% growth.

Question 28Algebra and functions

Which statement best distinguishes linear growth from exponential growth?

  1. Linear functions grow by equal differences over equal intervals, while exponential functions grow by equal factors over equal intervals Correct answer.
  2. Linear functions grow by equal factors, while exponential functions grow by equal differences
  3. Both grow by equal differences over equal intervals
  4. Both grow by equal factors over equal intervals

Explanation

A linear function adds a constant amount over each equal interval (equal differences), whereas an exponential function multiplies by a constant factor over each equal interval (equal factors). This is the defining contrast between the two models.

Question 29Algebra and functions

Expressed as a logarithm, the solution to the equation 3 · 2^x = 24 is which value?

  1. x = 3 Correct answer.
  2. x = 8
  3. x = log base 2 of 24
  4. x = 4

Explanation

Divide both sides by 3 to get 2^x = 8. Since 8 = 2^3, the exponent must be x = 3. Equivalently, x = log base 2 of 8 = 3.

Question 30Algebra and functions

On the unit circle, what is the value of cosine of (pi / 3) radians?

  1. 1/2 Correct answer.
  2. square root of 3 / 2
  3. 1
  4. 0

Explanation

The angle pi/3 radians equals 60 degrees. On the unit circle, the cosine of 60 degrees is 1/2.

Question 31Algebra and functions

Which trigonometric identity is the Pythagorean identity?

  1. sin^2(theta) + cos^2(theta) = 1 Correct answer.
  2. sin(theta) + cos(theta) = 1
  3. tan^2(theta) - 1 = sec^2(theta)
  4. sin(theta) / cos(theta) = 1

Explanation

The fundamental Pythagorean identity states that sin^2(theta) + cos^2(theta) = 1 for all angles theta, derived from the unit circle where x^2 + y^2 = 1.

Question 32Algebra and functions

A sinusoidal function is written as y = 3 sin(2x). What are its amplitude and period?

  1. Amplitude 3, period pi Correct answer.
  2. Amplitude 2, period 3
  3. Amplitude 3, period 2pi
  4. Amplitude 6, period pi

Explanation

For y = A sin(Bx), the amplitude is |A| and the period is 2pi / |B|. Here A = 3 gives amplitude 3, and B = 2 gives period 2pi / 2 = pi.

Question 33Algebra and functions

Which graph is equivalent to the graph of y = sin(x), based on the cofunction identity?

  1. y = cos(pi/2 - x) Correct answer.
  2. y = cos(x)
  3. y = cos(x + pi/2)
  4. y = -cos(x)

Explanation

The cofunction identity states that sin(x) = cos(pi/2 - x). This reflects the complementary-angle relationship between sine and cosine, so the graphs are identical.

Question 34Algebra and functions

A function f(x) is defined so that f(-x) = f(x) for all x. This function is classified as which type?

  1. Even Correct answer.
  2. Odd
  3. Neither even nor odd
  4. Both even and odd

Explanation

A function satisfying f(-x) = f(x) is even, which means its graph is symmetric about the y-axis. Examples include f(x) = x^2 and f(x) = cos(x).

Question 35Algebra and functions

What is the y-intercept of the function f(x) = log base 10 of (x) ?

  1. There is no y-intercept Correct answer.
  2. (0, 0)
  3. (0, 1)
  4. (1, 0)

Explanation

The logarithmic function is undefined for x less than or equal to 0, so it never crosses the y-axis (where x = 0). Therefore it has no y-intercept; its x-intercept is at (1, 0).

Question 36Algebra and functions

The function f(x) = |x - 2| is graphed. What are the coordinates of its vertex?

  1. (2, 0) Correct answer.
  2. (-2, 0)
  3. (0, 2)
  4. (0, -2)

Explanation

The vertex of an absolute value function f(x) = |x - h| + k occurs at (h, k). Here h = 2 and k = 0, so the vertex is at (2, 0).

Question 37Algebra and functions

A function is increasing on an interval. What does this mean about its output values as x increases across that interval?

  1. The output values increase as x increases Correct answer.
  2. The output values decrease as x increases
  3. The output values stay constant
  4. The output values alternate up and down

Explanation

A function is increasing on an interval if, for any two inputs a < b in that interval, f(a) < f(b). In other words, as x increases, the output values increase.

Question 38Algebra and functions

For the function f(x) = 2^x to have an inverse that is also a function, which property must f have?

  1. f must be one-to-one (each output comes from exactly one input) Correct answer.
  2. f must be even
  3. f must have a maximum value
  4. f must be a polynomial

Explanation

A function has an inverse that is also a function if and only if it is one-to-one, meaning it passes the horizontal line test. The exponential function 2^x is one-to-one, so its inverse (the logarithm) is a function.

Question 39Algebra and functions

Given a noninvertible function f(x) = x^2, what is the largest domain that produces an invertible function?

  1. x is greater than or equal to 0 Correct answer.
  2. All real numbers
  3. x is between -1 and 1
  4. x is not equal to 0

Explanation

The function x^2 fails the horizontal line test over all reals. Restricting the domain to x greater than or equal to 0 (or alternatively x less than or equal to 0) makes it one-to-one and therefore invertible, with inverse the square root function.

Question 40Algebra and functions

If f(x) = 5x and g(x) = x - 2, what is the function (f - g)(x)?

  1. 4x + 2 Correct answer.
  2. 4x - 2
  3. 5x - 2
  4. 6x - 2

Explanation

Subtract the functions: (f - g)(x) = 5x - (x - 2) = 5x - x + 2 = 4x + 2. Distributing the negative sign over both terms of g is essential.

Question 41Number and quantity

Which of the following expressions is equivalent to (x^(1/2) · x^(1/3)) for x > 0?

  1. x^(5/6) Correct answer.
  2. x^(1/6)
  3. x^(2/5)
  4. x^(1/5)

Explanation

When multiplying powers with the same base, you add the exponents. Adding 1/2 + 1/3 gives a common denominator of 6, so 3/6 + 2/6 = 5/6, making the product x^(5/6).

Question 42Number and quantity

Which statement about the sum of a nonzero rational number and an irrational number is always true?

  1. The sum is always irrational Correct answer.
  2. The sum is always rational
  3. The sum is always an integer
  4. The sum can be either rational or irrational

Explanation

If a rational number r plus an irrational number i equaled a rational number q, then i = q - r would be a difference of two rationals, which is rational. This contradicts i being irrational, so the sum must always be irrational.

Question 43Number and quantity

A vector has an initial point at (2, -1) and a terminal point at (5, 3). What are the components of this vector?

  1. (3, 4) Correct answer.
  2. (7, 2)
  3. (-3, -4)
  4. (3, -4)

Explanation

The components of a vector are found by subtracting the coordinates of the initial point from those of the terminal point. This gives (5 - 2, 3 - (-1)) = (3, 4).

Question 44Number and quantity

If A is a 2x3 matrix and B is a 3x4 matrix, what are the dimensions of the product AB?

  1. 2x4 Correct answer.
  2. 3x3
  3. 2x3
  4. The product is undefined

Explanation

Matrix multiplication is defined when the number of columns of the first matrix equals the number of rows of the second. Here A has 3 columns and B has 3 rows, so AB is defined and has dimensions equal to the rows of A by the columns of B, which is 2x4.

Question 45Number and quantity

A recipe that serves 4 people requires 3 cups of flour. How many cups of flour are needed to serve 10 people, using the same proportion?

  1. 7.5 Correct answer.
  2. 6
  3. 8
  4. 12

Explanation

Set up the proportion 3/4 = x/10. Cross-multiplying gives 4x = 30, so x = 7.5 cups of flour are needed to serve 10 people.

Question 46Number and quantity

The distance from Earth to the Sun is approximately 1.5 × 10^8 km, and the diameter of a hydrogen atom is approximately 1.0 × 10^(-7) mm. By what order of magnitude (power of 10) does the larger quantity exceed the smaller, treating both in millimeters?

  1. About 10^21 Correct answer.
  2. About 10^15
  3. About 10^1
  4. About 10^(-15)

Explanation

Convert 1.5 × 10^8 km to mm: multiply by 10^6 to get 1.5 × 10^14 mm. Dividing by 1.0 × 10^(-7) mm gives 1.5 × 10^21, so the larger quantity exceeds the smaller by about 10^21.

Question 47Number and quantity

Which of the following is a prime number?

  1. 29 Correct answer.
  2. 27
  3. 33
  4. 51

Explanation

A prime number has exactly two distinct positive factors: 1 and itself. The number 29 has no divisors other than 1 and 29, so it is prime.

Question 48Number and quantity

What is the product (3 + 2i)(1 - 4i), where i is the imaginary unit?

  1. 11 - 10i Correct answer.
  2. 3 - 8i
  3. -5 - 10i
  4. 11 + 10i

Explanation

Using the distributive property: 3(1) + 3(-4i) + 2i(1) + 2i(-4i) = 3 - 12i + 2i - 8i^2. Since i^2 = -1, the term -8i^2 = +8, giving 3 + 8 + (-12i + 2i) = 11 - 10i.

Question 49Number and quantity

What is the modulus (absolute value) of the complex number 3 - 4i?

  1. 5 Correct answer.
  2. 7
  3. 1
  4. 25

Explanation

The modulus of a complex number a + bi is the square root of (a^2 + b^2). Here that is the square root of (3^2 + (-4)^2) = square root of (9 + 16) = square root of 25 = 5.

Question 50Number and quantity

A measured length is reported as 12.0 cm with a possible absolute error of 0.2 cm. What is the relative error, expressed as a percent?

  1. About 1.7% Correct answer.
  2. About 2.0%
  3. About 12%
  4. About 0.2%

Explanation

Relative error is the absolute error divided by the measured value: 0.2 / 12.0 = 0.0167, which is about 1.7% when converted to a percent.