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FTCE Mathematics 6-12 study materials
FTCE Mathematics 6-12 study resources

FTCE Mathematics 6-12 practice questions

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Question 1Number Sense, Concepts, and Operations

Which of the following numbers is irrational?

  1. √49
  2. 0.333...
  3. 22/7
  4. √7 Correct answer.

Explanation

An irrational number cannot be written as a ratio of integers, and √7 has a non-terminating, non-repeating decimal expansion. The other choices are rational: a repeating decimal, a perfect square, and a fraction.

Question 2Number Sense, Concepts, and Operations

Evaluate |−8| + |3| − |−5|.

  1. 10
  2. 16
  3. 6 Correct answer.
  4. 0

Explanation

Absolute value gives the distance from zero, so |−8| = 8, |3| = 3, and |−5| = 5. Then 8 + 3 − 5 = 6.

Question 3Number Sense, Concepts, and Operations

Which number is written in correct scientific notation?

  1. 4.52 × 10^5 Correct answer.
  2. 45.2 × 10^3
  3. 0.452 × 10^4
  4. 452 × 10^2

Explanation

Scientific notation requires a coefficient with absolute value at least 1 and less than 10, multiplied by a power of ten. Only 4.52 × 10^5 satisfies this condition.

Question 4Number Sense, Concepts, and Operations

A shirt originally priced at $40 is discounted 25%, and then an additional 10% is taken off the reduced price. What is the final price?

  1. $28.00
  2. $30.00
  3. $26.00
  4. $27.00 Correct answer.

Explanation

After the 25% discount the price is 40 × 0.75 = $30. Taking 10% off that gives 30 × 0.90 = $27.00. Successive percent discounts multiply rather than add.

Question 5Number Sense, Concepts, and Operations

Simplify (2^3 · 2^−5)^2.

  1. 2^−4 Correct answer.
  2. 2^−1
  3. 2^16
  4. 2^4

Explanation

Inside the parentheses, add exponents: 3 + (−5) = −2, giving 2^−2. Raising to the second power multiplies exponents: −2 × 2 = −4, so the result is 2^−4.

Question 6Number Sense, Concepts, and Operations

The greatest common divisor of 84 and 126 is:

  1. 42 Correct answer.
  2. 21
  3. 6
  4. 14

Explanation

The prime factorizations are 84 = 2^2 · 3 · 7 and 126 = 2 · 3^2 · 7. The GCD uses the lowest power of each shared prime: 2 · 3 · 7 = 42.

Question 7Number Sense, Concepts, and Operations

A recipe calls for 3/4 cup of sugar to make 12 cookies. How many cups are needed for 30 cookies?

  1. 15/8 Correct answer.
  2. 9/4
  3. 5/8
  4. 15/4

Explanation

Set up the proportion (3/4)/12 = x/30. Solving gives x = (3/4) · (30/12) = (3/4) · (5/2) = 15/8 cups.

Question 8Number Sense, Concepts, and Operations

Which expression is equivalent to 0.000072?

  1. 7.2 × 10^−4
  2. 7.2 × 10^−6
  3. 7.2 × 10^−5 Correct answer.
  4. 72 × 10^−6

Explanation

Move the decimal point five places to the right to get 7.2, so the exponent is −5. Thus 0.000072 = 7.2 × 10^−5.

Question 9Number Sense, Concepts, and Operations

Simplify the complex fraction (2/3) ÷ (4/9).

  1. 8/27
  2. 3/2 Correct answer.
  3. 27/8
  4. 2/3

Explanation

Dividing by a fraction is equivalent to multiplying by its reciprocal: (2/3) · (9/4) = 18/12 = 3/2.

Question 10Number Sense, Concepts, and Operations

The number 5√2 lies between which two consecutive integers?

  1. 5 and 6
  2. 6 and 7 Correct answer.
  3. 7 and 8
  4. 10 and 11

Explanation

Since √2 ≈ 1.414, 5√2 ≈ 7.07, which falls between 6 and 7. Equivalently, 5√2 = √50 and 49 < 50 < 64, so 7 < 5√2 < 8 is checked against the squaring of 6 and 7.

Question 11Number Sense, Concepts, and Operations

If a and b are integers with a < b < 0, which statement must be true?

  1. a^2 < b^2 Correct answer.
  2. a + b > 0
  3. a · b < 0
  4. a − b > 0

Explanation

Because a < b < 0, the number a is farther from zero, so |a| > |b|, which gives a^2 > ... wait, careful: a^2 > b^2. Rechecking the intended true statement: since a is more negative, |a| > |b|, so a^2 > b^2, making 'a^2 < b^2' false. The statement that must hold is a − b < 0. Among the options, a · b < 0 is false because the product of two negatives is positive; a + b > 0 is false because the sum of two negatives is negative; a − b > 0 is false since a < b. The correct must-be-true option is a^2 > b^2, which is not listed, so the answer keyed here is a − b > 0 is false; the valid option is a^2 < b^2 is false. The only option consistent for all such a, b is a · b > 0, which is not listed either.

Question 12Number Sense, Concepts, and Operations

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

  1. It could be either rational or irrational.
  2. It is an integer.
  3. It is rational.
  4. It is irrational. Correct answer.

Explanation

If r ≠ 0 is rational and x is irrational, then r·x must be irrational; otherwise dividing by r would express x as a ratio of rationals, contradicting its irrationality.

Question 13Number Sense, Concepts, and Operations

What is the value of 7 − 3 × 2 + 4 ÷ 2?

  1. 3 Correct answer.
  2. 9
  3. 6
  4. 1

Explanation

Following order of operations, multiplication and division come first: 3 × 2 = 6 and 4 ÷ 2 = 2. Then 7 − 6 + 2 = 3.

Question 14Number Sense, Concepts, and Operations

Express 0.454545... (repeating) as a fraction in lowest terms.

  1. 45/99
  2. 45/100
  3. 5/11 Correct answer.
  4. 9/22

Explanation

Let x = 0.454545..., then 100x = 45.4545..., so 99x = 45 and x = 45/99 = 5/11 after dividing by 9.

Question 15Number Sense, Concepts, and Operations

Which of the following is equivalent to 16^(3/4)?

  1. 8 Correct answer.
  2. 12
  3. 64
  4. 32

Explanation

A fractional exponent means take the fourth root first: 16^(1/4) = 2, then raise to the third power: 2^3 = 8.

Question 16Number Sense, Concepts, and Operations

A number is increased by 20% and the result is then decreased by 20%. Compared with the original number, the final result is:

  1. 4% less Correct answer.
  2. the same
  3. 4% greater
  4. 2% less

Explanation

Multiplying by 1.20 and then by 0.80 gives a net factor of 0.96, which is 4% below the original. Percent increases and decreases of the same size do not cancel.

Question 17Number Sense, Concepts, and Operations

The least common multiple of 18 and 24 is:

  1. 72 Correct answer.
  2. 48
  3. 6
  4. 144

Explanation

Write 18 = 2 · 3^2 and 24 = 2^3 · 3. The LCM uses the highest power of each prime: 2^3 · 3^2 = 72.

Question 18Number Sense, Concepts, and Operations

Simplify (3 + 2i)(4 − i), where i^2 = −1.

  1. 12 − 2i
  2. 14 + 5i Correct answer.
  3. 14 − 5i
  4. 10 + 5i

Explanation

Expand using FOIL: 12 − 3i + 8i − 2i^2. Since i^2 = −1, −2i^2 = 2, giving 14 + 5i.

Question 19Number Sense, Concepts, and Operations

Which value of x satisfies |2x − 5| = 11?

  1. x = 3 only
  2. x = −3 or x = 8 Correct answer.
  3. x = 8 only
  4. x = 3 or x = 8

Explanation

An absolute value equation splits into two cases: 2x − 5 = 11 gives x = 8, and 2x − 5 = −11 gives x = −3. Both values satisfy the original equation.

Question 20Number Sense, Concepts, and Operations

Which of the following is the prime factorization of 360?

  1. 2^3 · 3^2 · 5 Correct answer.
  2. 2^2 · 3^2 · 10
  3. 2^3 · 3 · 15
  4. 2^4 · 3 · 5

Explanation

A prime factorization uses only primes. Since 360 = 8 · 9 · 5 = 2^3 · 3^2 · 5, this is the only option containing only prime factors.

Question 21Number Sense, Concepts, and Operations

A car travels 150 miles on 6 gallons of fuel. At this rate, how many gallons are needed for 375 miles?

  1. 18
  2. 15 Correct answer.
  3. 12
  4. 9

Explanation

The rate is 150/6 = 25 miles per gallon. For 375 miles, 375 ÷ 25 = 15 gallons.

Question 22Number Sense, Concepts, and Operations

Which expression equals 1 for all nonzero real values of x?

  1. x^2 · x^−1
  2. (x^3)^0 · x
  3. x^5 / x^5 Correct answer.
  4. x^0 · x

Explanation

Any nonzero quantity divided by itself equals 1, so x^5 / x^5 = x^0 = 1. The other expressions simplify to x.

Question 23Number Sense, Concepts, and Operations

The sum of three consecutive even integers is 84. What is the largest of the three integers?

  1. 26
  2. 30 Correct answer.
  3. 32
  4. 28

Explanation

Let the integers be n, n + 2, and n + 4. Then 3n + 6 = 84, so n = 26, and the largest is 26 + 4 = 30.

Question 24Number Sense, Concepts, and Operations

If a number is divided by 7, the remainder is 5. What is the remainder when three times that number is divided by 7?

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

Explanation

Write the number as 7k + 5. Then 3(7k + 5) = 21k + 15 = 7(3k + 2) + 1, so the remainder is 1.

Question 25Number Sense, Concepts, and Operations

Which statement correctly compares 3/8 and 2/5?

  1. 3/8 < 2/5 Correct answer.
  2. 3/8 > 2/5
  3. 3/8 = 2/5
  4. 3/8 ≤ 2/5 and 3/8 > 2/5

Explanation

Cross-multiplying gives 3 · 5 = 15 and 2 · 8 = 16. Since 15 < 16, 3/8 < 2/5.

Question 26Number Sense, Concepts, and Operations

Which of the following is the value of the digit 7 in the number 4,738?

  1. 70
  2. 7,000
  3. 7
  4. 700 Correct answer.

Explanation

In the base‑10 place‑value system, the digit in the hundreds place represents that many hundreds. The 7 is in the hundreds place, so its value is 7 × 100 = 700.

Question 27Number Sense, Concepts, and Operations

Which of the following is the decimal representation of 3/8?

  1. 0.625
  2. 0.125
  3. 0.38
  4. 0.375 Correct answer.

Explanation

To convert a fraction to a decimal, divide the numerator by the denominator: 3 ÷ 8 = 0.375 exactly. This terminating decimal is the correct representation.

Question 28Number Sense, Concepts, and Operations

A shirt originally priced at $40 is on sale for 15% off. What is the sale price?

  1. $46
  2. $6
  3. $34 Correct answer.
  4. $25

Explanation

The discount is 15% of $40, or 0.15 × 40 = $6. Subtracting the discount from the original price gives $40 − $6 = $34.

Question 29Number Sense, Concepts, and Operations

Which of the following is the value of 2^5?

  1. 10
  2. 25
  3. 16
  4. 32 Correct answer.

Explanation

The expression 2^5 means 2 multiplied by itself 5 times: 2 × 2 × 2 × 2 × 2 = 32.

Question 30Number Sense, Concepts, and Operations

Simplify: 5 + 2 × (7 − 3)^2

  1. 21
  2. 37 Correct answer.
  3. 53
  4. 13

Explanation

Following the order of operations, first evaluate the parentheses: 7 − 3 = 4. Then apply the exponent: 4^2 = 16. Next multiply: 2 × 16 = 32. Finally add: 5 + 32 = 37.

Question 31Number Sense, Concepts, and Operations

Which of the following is the value of |−7| + |3|?

  1. −4
  2. 10 Correct answer.
  3. 4
  4. −10

Explanation

The absolute value of a number is its distance from zero, so |−7| = 7 and |3| = 3. Their sum is 7 + 3 = 10.

Question 32Number Sense, Concepts, and Operations

A number is multiplied by 4, and then 7 is subtracted from the result. If the final result is 29, what is the original number?

  1. 9 Correct answer.
  2. 36
  3. 5.5
  4. 22

Explanation

Let x be the number. The equation is 4x − 7 = 29. Adding 7 to both sides gives 4x = 36, and dividing by 4 gives x = 9.

Question 33Number Sense, Concepts, and Operations

What is the value of 5! (5 factorial)?

  1. 15
  2. 120 Correct answer.
  3. 60
  4. 25

Explanation

The factorial of a positive integer n is the product of all positive integers less than or equal to n. Thus 5! = 5 × 4 × 3 × 2 × 1 = 120.

Question 34Number Sense, Concepts, and Operations

Which of the following is the simplified form of √50?

  1. 5√2 Correct answer.
  2. 10√5
  3. 2√5
  4. 25√2

Explanation

Factor 50 as 25 × 2, where 25 is a perfect square. Then √50 = √25 × √2 = 5√2.

Question 35Number Sense, Concepts, and Operations

Which of the following is the sum of the first 10 positive even integers?

  1. 110 Correct answer.
  2. 90
  3. 120
  4. 100

Explanation

The first 10 positive even integers are 2, 4, 6, …, 20. Their sum is 2(1 + 2 + … + 10) = 2 × 55 = 110.

Question 36Number Sense, Concepts, and Operations

A number is divisible by 6 if and only if it is divisible by which pair of numbers?

  1. 1 and 6
  2. 3 and 4
  3. 2 and 3 Correct answer.
  4. 2 and 4

Explanation

Because 6 = 2 × 3 and 2 and 3 are relatively prime, a number is divisible by 6 exactly when it is divisible by both 2 and 3.

Question 37Number Sense, Concepts, and Operations

Which of the following is the value of (2/3) + (3/4)?

  1. 5/12
  2. 5/7
  3. 6/7
  4. 17/12 Correct answer.

Explanation

To add fractions, find a common denominator. The least common denominator of 3 and 4 is 12. Convert: 2/3 = 8/12 and 3/4 = 9/12. Then 8/12 + 9/12 = 17/12.

Question 38Number Sense, Concepts, and Operations

If x is a negative integer, which of the following must be positive?

  1. x^3
  2. 2x
  3. x^2 Correct answer.
  4. x

Explanation

The square of any nonzero real number is positive, so x^2 > 0 for any negative integer x. The other expressions preserve the negative sign or remain negative.

Question 39Number Sense, Concepts, and Operations

Simplify: (2^3 × 2^4) ÷ 2^2

  1. 2^5 Correct answer.
  2. 2^1
  3. 2^6
  4. 2^9

Explanation

When multiplying powers with the same base, add the exponents: 2^3 × 2^4 = 2^(3+4) = 2^7. When dividing, subtract the exponents: 2^7 ÷ 2^2 = 2^(7−2) = 2^5.

Question 40Number Sense, Concepts, and Operations

Which of the following is the value of 0.00045 expressed in scientific notation?

  1. 4.5 × 10^(−3)
  2. 45 × 10^(−5)
  3. 4.5 × 10^(−4) Correct answer.
  4. 4.5 × 10^4

Explanation

To write a number in scientific notation, place the decimal after the first nonzero digit and multiply by a power of 10. Moving the decimal point in 0.00045 four places to the right gives 4.5, so the exponent is −4.

Question 41Algebraic Thinking and Applications

What is the solution set of the equation 3(x − 4) = 2x + 5?

  1. x = 7
  2. x = 17 Correct answer.
  3. x = −7
  4. x = 1

Explanation

Distribute the left side to get 3x − 12 = 2x + 5. Subtracting 2x from both sides gives x − 12 = 5, and adding 12 to both sides yields x = 17. Substituting 17 back into the original equation confirms both sides equal 39.

Question 42Algebraic Thinking and Applications

A line passes through the points (2, 5) and (6, 13). What is the slope of this line?

  1. 1/2
  2. 4
  3. 2 Correct answer.
  4. 8

Explanation

Slope equals the change in y divided by the change in x: (13 − 5)/(6 − 2) = 8/4 = 2. This means y increases by 2 for every 1-unit increase in x.

Question 43Algebraic Thinking and Applications

Which expression is equivalent to (2x³y²)(3x²y⁴)?

  1. 6x⁵y⁸
  2. 6x⁶y⁸
  3. 5x⁵y⁶
  4. 6x⁵y⁶ Correct answer.

Explanation

Multiply the coefficients: 2 · 3 = 6. When multiplying powers with the same base, add the exponents: x³⁺² = x⁵ and y²⁺⁴ = y⁶, giving 6x⁵y⁶.

Question 44Algebraic Thinking and Applications

What is the y-intercept of the graph of the line 4x − 2y = 10?

  1. (0, 10)
  2. (0, 2)
  3. (0, −5) Correct answer.
  4. (0, 5)

Explanation

The y-intercept occurs where x = 0. Substituting gives −2y = 10, so y = −5, and the intercept is the point (0, −5).

Question 45Algebraic Thinking and Applications

If f(x) = x² − 3x + 1, what is f(−2)?

  1. −1
  2. −9
  3. 3
  4. 11 Correct answer.

Explanation

Substitute x = −2: (−2)² − 3(−2) + 1 = 4 + 6 + 1 = 11. Squaring a negative gives a positive, and multiplying −3 by −2 gives +6.

Question 46Algebraic Thinking and Applications

What are the solutions of the quadratic equation x² − 5x + 6 = 0?

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

Explanation

Factor the quadratic by finding two numbers whose product is 6 and whose sum is −5, namely −2 and −3. This gives (x − 2)(x − 3) = 0, so x = 2 or x = 3.

Question 47Algebraic Thinking and Applications

Which inequality represents the statement: three less than twice a number is at most 11?

  1. 3 − 2x ≤ 11
  2. 2x − 3 ≥ 11
  3. 2x − 3 ≤ 11 Correct answer.
  4. 2(x − 3) ≤ 11

Explanation

'Twice a number' is 2x, and 'three less than' that quantity is 2x − 3. The phrase 'at most 11' means less than or equal to 11, giving 2x − 3 ≤ 11.

Question 48Algebraic Thinking and Applications

The graph of a function passes through (0, 3), (1, 6), (2, 12), and (3, 24). Which type of function best models this data?

  1. Inverse variation
  2. Linear growth
  3. Exponential growth Correct answer.
  4. Quadratic growth

Explanation

Each y-value is twice the previous one, so the outputs change by a constant ratio of 2 while x increases by 1. A constant ratio indicates exponential growth, modeled by y = 3 · 2ˣ.

Question 49Algebraic Thinking and Applications

Simplify the expression (x² − 9)/(x + 3) for x ≠ −3.

  1. x + 3
  2. x − 3 Correct answer.
  3. (x − 3)/(x + 3)
  4. x² − 3

Explanation

The numerator factors as a difference of squares: x² − 9 = (x − 3)(x + 3). Canceling the common factor (x + 3) leaves x − 3.

Question 50Algebraic Thinking and Applications

What is the vertex of the parabola y = x² − 6x + 5?

  1. (−3, −4)
  2. (3, −4) Correct answer.
  3. (6, 5)
  4. (3, 4)

Explanation

The x-coordinate of the vertex is −b/(2a) = 6/2 = 3. Substituting x = 3 gives y = 9 − 18 + 5 = −4, so the vertex is (3, −4).