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CSET Mathematics (211/212/213) study materials
CSET Mathematics (211/212/213) study resources

CSET Mathematics (211/212/213) practice questions

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Question 1Number and Quantity (Subtest I)

The set of rational numbers is closed under which of the following operations when the operation is defined?

  1. Taking the square root of a nonnegative rational number
  2. Raising a rational number to a rational power
  3. Taking the cube root of any rational number
  4. Division by a nonzero rational number Correct answer.

Explanation

The quotient of two rationals a/b and c/d with c/d nonzero is (a/b)(d/c) = ad/bc, which is a ratio of integers with nonzero denominator, hence rational. Closure under division requires excluding division by zero, which is why the operation is restricted to a nonzero divisor.

Question 2Number and Quantity (Subtest I)

Which statement correctly describes the relationship between the sets of natural numbers, integers, rationals, and reals?

  1. The irrational numbers are a subset of the rational numbers
  2. Every real number can be written as a ratio of two integers
  3. The integers are a subset of the natural numbers
  4. Every rational number is a real number, and every integer is a rational number Correct answer.

Explanation

Each integer n equals n/1, so the integers sit inside the rationals, and every rational has a decimal expansion, placing it among the reals. This gives the chain of inclusions N ⊆ Z ⊆ Q ⊆ R.

Question 3Number and Quantity (Subtest I)

What is the value of the expression |−7| + |3 − 9| − |−2|?

  1. 7
  2. 3
  3. 15
  4. 11 Correct answer.

Explanation

Evaluate each absolute value: |−7| = 7, |3 − 9| = |−6| = 6, and |−2| = 2. Then 7 + 6 − 2 = 11.

Question 4Number and Quantity (Subtest I)

A number written in scientific notation appears as 4.5 × 10^−3. Which standard decimal representation is equivalent?

  1. 0.045
  2. 4500
  3. 0.00045
  4. 0.0045 Correct answer.

Explanation

A negative exponent of −3 moves the decimal point three places to the left from 4.5, producing 0.0045. Equivalently, 4.5 × 10^−3 = 4.5/1000 = 0.0045.

Question 5Number and Quantity (Subtest I)

Which expression gives the prime factorization of 252?

  1. 2^2 × 3^2 × 7 Correct answer.
  2. 2 × 3^3 × 7
  3. 2^2 × 3 × 7^2
  4. 2^3 × 3 × 7

Explanation

Dividing 252 by 4 yields 63, and 63 = 9 × 7 = 3^2 × 7, so 252 = 2^2 × 3^2 × 7. Checking: 4 × 9 × 7 = 252.

Question 6Number and Quantity (Subtest I)

Simplify the expression (2^5 × 2^−3) / 2^−1, leaving the result as an integer or simple fraction.

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

Explanation

In the numerator, 2^5 × 2^−3 = 2^(5−3) = 2^2. Dividing by 2^−1 subtracts the exponent: 2^2 / 2^−1 = 2^(2−(−1)) = 2^3, which equals 8.

Question 7Number and Quantity (Subtest I)

In the complex numbers, what is the value of i^47, where i^2 = −1?

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

Explanation

Powers of i cycle with period 4: i, −1, −i, 1. Since 47 = 4 × 11 + 3, i^47 = i^3 = −i.

Question 8Number and Quantity (Subtest I)

Which of the following statements about the set of irrational numbers is true?

  1. The product of two irrational numbers is always irrational
  2. The irrational numbers are not closed under addition or multiplication Correct answer.
  3. Every irrational number has a repeating decimal expansion
  4. The sum of two irrational numbers is always irrational

Explanation

Closure fails for both operations: (√2) + (−√2) = 0 and (√2)(√2) = 2 are rational, while √2 itself is irrational. Hence the irrationals are not closed under addition or multiplication.

Question 9Number and Quantity (Subtest I)

A student claims that √50 + √18 can be written as a single radical term. Which simplification supports this claim?

  1. 68
  2. 2√17
  3. 4√17
  4. 8√2 Correct answer.

Explanation

Simplify each radical first: √50 = 5√2 and √18 = 3√2. Since the radicals match, add the coefficients to obtain 5√2 + 3√2 = 8√2.

Question 10Number and Quantity (Subtest I)

Rationalize the denominator of the expression 6 / (√5 − √2) and simplify fully.

  1. 6(√5 + √2)
  2. 2(√5 − √2)
  3. (√5 + √2) / 2
  4. 2(√5 + √2) Correct answer.

Explanation

Multiply numerator and denominator by the conjugate √5 + √2. The denominator becomes 5 − 2 = 3, giving 6(√5 + √2)/3 = 2(√5 + √2).

Question 11Number and Quantity (Subtest I)

Which expression is equivalent to (3 + 2i)(4 − i) for the complex numbers?

  1. 14 + 5i Correct answer.
  2. 14 − 5i
  3. 10 + 11i
  4. 12 − 2i

Explanation

Expand using the distributive property: 12 − 3i + 8i − 2i^2. Since i^2 = −1, the last term becomes +2, so the result is 12 + 2 + 5i = 14 + 5i.

Question 12Number and Quantity (Subtest I)

The number 0.272727… repeats the block 27 indefinitely. Which fraction in lowest terms represents this number?

  1. 3/11 Correct answer.
  2. 27/100
  3. 11/27
  4. 27/99

Explanation

Let x = 0.272727…, then 100x = 27.2727…, and subtracting gives 99x = 27, so x = 27/99 = 3/11. Reducing by the common factor 9 yields lowest terms.

Question 13Number and Quantity (Subtest I)

Which of the following numbers is irrational?

  1. the cube root of 27
  2. √12 Correct answer.
  3. 0.121212…
  4. √81

Explanation

√12 = 2√3, and since 3 is not a perfect square, √3 is irrational, so √12 is irrational. The other options all equal rational values.

Question 14Number and Quantity (Subtest I)

Solve for x over the real numbers: |2x − 5| = 11.

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

Explanation

An absolute value equation |A| = k with k > 0 splits into A = k or A = −k. So 2x − 5 = 11 gives x = 8, and 2x − 5 = −11 gives x = −3.

Question 15Number and Quantity (Subtest I)

A quantity is given as 3.60 × 10^5 grams. Which expression represents the same quantity in kilograms, using scientific notation?

  1. 3.60 × 10^3 kg
  2. 3.60 × 10^−2 kg
  3. 3.60 × 10^8 kg
  4. 3.60 × 10^2 kg Correct answer.

Explanation

One kilogram equals 1000 grams, so divide by 10^3: (3.60 × 10^5)/10^3 = 3.60 × 10^(5−3) = 3.60 × 10^2 kg. This equals 360 kg, consistent with 360,000 g.

Question 16Number and Quantity (Subtest I)

The greatest common divisor of two positive integers is 12 and their least common multiple is 180. What is the product of the two integers?

  1. 192
  2. 21600
  3. 2160 Correct answer.
  4. 540

Explanation

For any two positive integers, the product equals gcd × lcm. Here 12 × 180 = 2160.

Question 17Number and Quantity (Subtest I)

Which statement correctly compares the cardinalities of the integers and the rational numbers?

  1. The rationals have greater cardinality because they include fractions between integers
  2. Both sets are countably infinite and therefore have the same cardinality Correct answer.
  3. The two sets cannot be compared because both are infinite
  4. The integers have greater cardinality because they are a subset of the rationals

Explanation

A set is countably infinite if its elements can be placed in one-to-one correspondence with the natural numbers. Both the integers and the rationals admit such an enumeration, so they share the same cardinality, even though one is a proper subset of the other.

Question 18Number and Quantity (Subtest I)

The number 5√3 is multiplied by 2√6. Which expression is the simplified product?

  1. 30√2 Correct answer.
  2. 7√9
  3. 10√18
  4. 10√3

Explanation

Multiply coefficients and radicands separately: (5 × 2)√(3 × 6) = 10√18. Since √18 = 3√2, the product becomes 10 × 3√2 = 30√2.

Question 19Number and Quantity (Subtest I)

Let z = 3 − 4i. What is the value of z multiplied by its complex conjugate?

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

Explanation

The conjugate of 3 − 4i is 3 + 4i, and z·z̄ = 3^2 + 4^2 = 9 + 16 = 25. This equals the square of the modulus, |z|^2.

Question 20Number and Quantity (Subtest I)

Which value of x makes the expression 1/(x^2 − 9) undefined, and why?

  1. x = 3 and x = −3, because the denominator factors as (x − 3)(x + 3) and equals zero at both Correct answer.
  2. x = 3 only, because the denominator is zero at that point
  3. No value makes it undefined, because x^2 − 9 is always positive
  4. x = 9 only, because the denominator equals zero when x^2 = 9

Explanation

A rational expression is undefined wherever its denominator equals zero. Solving x^2 − 9 = 0 gives x = 3 or x = −3, so both values must be excluded from the domain.

Question 21Number and Quantity (Subtest I)

The number 2^10 × 5^8 is written in scientific notation as a × 10^k with 1 ≤ a < 10. What is the exponent k?

  1. 9
  2. 10
  3. 18
  4. 8 Correct answer.

Explanation

Rewrite 2^10 × 5^8 = 2^2 × (2^8 × 5^8) = 4 × 10^8. Since 4 lies between 1 and 10, the exponent is 8, and the value is 4 × 10^8.

Question 22Number and Quantity (Subtest I)

Which argument establishes that √2 is irrational?

  1. √2 has a non-terminating decimal expansion, and all such decimals are irrational
  2. Assuming √2 = p/q in lowest terms leads to p^2 = 2q^2, forcing p and q both even, contradicting lowest terms Correct answer.
  3. The square of √2 is 2, and 2 is not a perfect square
  4. √2 lies between 1 and 2, and every number between consecutive integers is irrational

Explanation

The standard proof is by contradiction: if √2 = p/q with gcd(p, q) = 1, then p^2 = 2q^2 makes p even, say p = 2m, which forces q even as well. That contradicts gcd(p, q) = 1, so no such fraction exists.

Question 23Number and Quantity (Subtest I)

For the complex number z = −1 + i√3, what is z^3?

  1. 8 Correct answer.
  2. −8
  3. −8i
  4. 8i

Explanation

Write z in polar form: |z| = 2 and the argument is 2π/3, so z = 2(cos 2π/3 + i sin 2π/3). Then z^3 = 8(cos 2π + i sin 2π) = 8.

Question 24Number and Quantity (Subtest I)

A positive integer n has exactly three positive divisors. Which characterization of such integers is correct?

  1. n must be a power of 2
  2. n must be the square of a prime number Correct answer.
  3. n must be the cube of a prime number
  4. n must be a product of two distinct primes

Explanation

If n = p^a q^b …, the number of divisors is (a+1)(b+1)…. A product equal to 3 forces a single prime factor with exponent 2, so n = p^2 for some prime p.

Question 25Number and Quantity (Subtest I)

Which statement about the real numbers is true?

  1. The irrational numbers are countable because they are the complement of a countable set
  2. The set of irrational numbers is uncountable and has the same cardinality as the reals Correct answer.
  3. Every real number can be expressed as a finite or repeating decimal
  4. The real numbers are countable because each has a decimal expansion

Explanation

The reals are uncountable, and removing the countable set of rationals leaves an uncountable set, so the irrationals have the same cardinality as the reals. This follows from Cantor's theorem and the countability of the rationals.

Question 26Number and Quantity (Subtest I)

Which of the following numbers is a rational number?

  1. ∛9
  2. π/2
  3. √12/√3 Correct answer.
  4. 0.101101110…, where the runs of 1s grow by one each time

Explanation

A rational number can be written as a quotient of integers. Since √12/√3 = √(12/3) = √4 = 2, this value is the integer 2 and is therefore rational. The other three are irrational: the nonrepeating decimal, π/2, and the cube root of 9 cannot be expressed as a ratio of integers.

Question 27Number and Quantity (Subtest I)

In the complex numbers, what is the value of i^50?

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

Explanation

Powers of i repeat with period 4: i^1 = i, i^2 = −1, i^3 = −i, i^4 = 1. Since 50 = 4·12 + 2, we get i^50 = (i^4)^12 · i^2 = 1 · (−1) = −1.

Question 28Number and Quantity (Subtest I)

What is the complex conjugate of the quotient (4 + 2i)/(1 − i)?

  1. 3 − i Correct answer.
  2. 1 − 3i
  3. 1 + 3i
  4. 3 + i

Explanation

Multiply numerator and denominator by 1 + i: (4 + 2i)(1 + i)/((1 − i)(1 + i)) = (4 + 4i + 2i + 2i^2)/2 = (2 + 6i)/2 = 1 + 3i. The conjugate of 1 + 3i is 1 − 3i.

Question 29Number and Quantity (Subtest I)

Which expression is equivalent to (2 + 3i)^2 − (2 − 3i)^2?

  1. 24i Correct answer.
  2. 10
  3. −10
  4. −24i

Explanation

Use the difference of squares: (a + b)(a − b) with a = 2 + 3i and b = 2 − 3i. Then a + b = 4 and a − b = 6i, so the value is 4 · 6i = 24i. Expanding directly gives the same result.

Question 30Number and Quantity (Subtest I)

A quantity is written as 7.2 × 10^−3 meters. Which expression represents the same quantity in micrometers, given that 1 meter = 10^6 micrometers?

  1. 7.2 × 10^3 micrometers Correct answer.
  2. 7.2 × 10^9 micrometers
  3. 7.2 × 10^−18 micrometers
  4. 7.2 × 10^−9 micrometers

Explanation

Multiply by 10^6 to convert meters to micrometers: 7.2 × 10^−3 · 10^6 = 7.2 × 10^3 micrometers. This equals 7200 micrometers, which is consistent with 7.2 millimeters.

Question 31Number and Quantity (Subtest I)

Two positive integers have greatest common divisor 18 and least common multiple 540. What is the product of the two integers?

  1. 9720 Correct answer.
  2. 558
  3. 1080
  4. 30

Explanation

For any two positive integers, the product of the numbers equals the product of their GCD and LCM. Thus the product is 18 · 540 = 9720. No further information about the individual numbers is needed.

Question 32Number and Quantity (Subtest I)

Which statement about the set of algebraic numbers (roots of nonzero polynomials with integer coefficients) is correct?

  1. It is finite, because each polynomial has only finitely many roots.
  2. It is uncountably infinite, because it contains all real numbers.
  3. It is countably infinite, because the integer-coefficient polynomials can be enumerated. Correct answer.
  4. It is empty, because most real numbers are transcendental.

Explanation

The set of polynomials with integer coefficients is countable, and each such polynomial has finitely many complex roots, so the algebraic numbers form a countable union of finite sets. This union is countably infinite. Since transcendental numbers such as π exist, the algebraic numbers do not include all real numbers.

Question 33Number and Quantity (Subtest I)

Which of the following numbers is an integer?

  1. (√5)^2 − 5.5
  2. √2 + √8
  3. √7 · √7 Correct answer.
  4. 2/3 + 1/3 written as a decimal

Explanation

Since √7 · √7 = 7, the product is the integer 7. The other expressions equal 1, 3√2, and −0.5 respectively, none of which is an integer.

Question 34Number and Quantity (Subtest I)

A number is written in the form a × 10^n, where 1 ≤ a < 10 and n is an integer. Which statement correctly describes what happens to the value of a when the exponent n is decreased by 1 while the overall number remains unchanged?

  1. The value of a is unchanged, since only the exponent shifts.
  2. The value of a is multiplied by 10. Correct answer.
  3. The value of a is divided by 10.
  4. The value of a is squared, since the base 10 is involved.

Explanation

Writing a number as a × 10^n, lowering the exponent by 1 divides the power of ten by 10, so the coefficient must be multiplied by 10 to preserve the same value. For instance, 3.5 × 10^4 equals 35 × 10^3, showing the coefficient grows tenfold as the exponent drops by one.

Question 35Algebra (Subtest I)

If f(x) = 2x + 3 and g(x) = x² − 1, what is (f ∘ g)(x)?

  1. 2x² + 2
  2. x² + 2
  3. 2x² + 1 Correct answer.
  4. 4x² + 12x + 8

Explanation

The composition (f ∘ g)(x) means f(g(x)), so substitute g(x) into f: 2(x² − 1) + 3 = 2x² − 2 + 3 = 2x² + 1.

Question 36Algebra (Subtest I)

What is the domain of the function h(x) = (x + 4)/(x² − 9)?

  1. All real numbers except 3 and −3 Correct answer.
  2. All real numbers
  3. All real numbers except 0
  4. All real numbers except −4

Explanation

A rational function is undefined where its denominator equals zero. Solving x² − 9 = 0 gives x = 3 and x = −3, so those two values must be excluded from the domain.

Question 37Algebra (Subtest I)

Which expression is a factor of the polynomial p(x) = x³ − 2x² − 5x + 6?

  1. x + 1
  2. x − 6
  3. x − 3 Correct answer.
  4. x + 2

Explanation

By the Factor Theorem, x − a is a factor exactly when p(a) = 0. Evaluating p(3) = 27 − 18 − 15 + 6 = 0, so x − 3 divides p(x).

Question 38Algebra (Subtest I)

Solve the inequality x² − x − 6 > 0.

  1. −2 < x < 3
  2. x < −2 or x > 3 Correct answer.
  3. x < −2
  4. x > 3

Explanation

Factor as (x − 3)(x + 2) > 0. The zeros split the number line into intervals; testing shows the product is positive when x < −2 and when x > 3.

Question 39Algebra (Subtest I)

A line passes through the points (2, −1) and (6, 7). What is the slope-intercept equation of this line?

  1. y = 2x − 5 Correct answer.
  2. y = −2x + 3
  3. y = 2x + 3
  4. y = (1/2)x − 2

Explanation

The slope is (7 − (−1))/(6 − 2) = 8/4 = 2. Using y − y₁ = m(x − x₁) with (2, −1): y + 1 = 2(x − 2), which simplifies to y = 2x − 5.

Question 40Algebra (Subtest I)

What is the solution set of the system 3x + 2y = 12 and x − y = 1?

  1. (4, 0)
  2. (14/5, 9/5) Correct answer.
  3. (1, 0)
  4. (2, 3)

Explanation

From x − y = 1 we get x = y + 1. Substituting: 3(y + 1) + 2y = 12, so 5y = 9 and y = 9/5; then x = 14/5. The ordered pair (14/5, 9/5) satisfies both equations.

Question 41Algebra (Subtest I)

The polynomial q(x) = 2x⁴ − 3x³ + 5x − 7 is divided by x + 1. What is the remainder?

  1. −17
  2. −13
  3. 7
  4. −7 Correct answer.

Explanation

By the Remainder Theorem the remainder equals q(−1). Compute 2(1) − 3(−1) + 5(−1) − 7 = 2 + 3 − 5 − 7 = −7.

Question 42Algebra (Subtest I)

Which of the following is the complete factorization of x³ − 27 over the integers?

  1. (x − 3)³
  2. (x − 3)(x² + 3x + 9) Correct answer.
  3. (x + 3)(x² − 3x + 9)
  4. (x − 3)(x² − 3x + 9)

Explanation

This is a difference of cubes: a³ − b³ = (a − b)(a² + ab + b²). With a = x and b = 3, the factorization is (x − 3)(x² + 3x + 9).

Question 43Algebra (Subtest I)

If f(x) = x² and g(x) = √(x − 4), what is the domain of (f ∘ g)(x)?

  1. All real numbers
  2. x ≥ 4 Correct answer.
  3. x ≥ 0
  4. x ≥ 2

Explanation

(f ∘ g)(x) = f(g(x)) = (√(x − 4))² = x − 4. The expression under the square root must be nonnegative, so x − 4 ≥ 0, giving x ≥ 4.

Question 44Algebra (Subtest I)

What are all the zeros of the polynomial p(x) = x³ − 4x² + x + 6?

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

Explanation

Test rational candidates: p(−1) = −1 − 4 − 1 + 6 = 0, so x + 1 is a factor. Dividing gives x² − 5x + 6 = (x − 2)(x − 3), so the zeros are −1, 2, and 3.

Question 45Algebra (Subtest I)

What is the inverse of the function f(x) = (3x − 6)/5?

  1. f⁻¹(x) = 5/(3x − 6)
  2. f⁻¹(x) = (5x − 6)/3
  3. f⁻¹(x) = (5x + 6)/3 Correct answer.
  4. f⁻¹(x) = (3x + 6)/5

Explanation

Set y = (3x − 6)/5 and solve for x: 5y = 3x − 6, so x = (5y + 6)/3. Swapping the variable name gives f⁻¹(x) = (5x + 6)/3.

Question 46Algebra (Subtest I)

Which statement is true about the graph of f(x) = −2(x + 3)² + 4?

  1. The vertex is (3, 4) and the parabola opens downward.
  2. The vertex is (−3, 4) and the parabola opens downward. Correct answer.
  3. The vertex is (3, −4) and the parabola opens upward.
  4. The vertex is (−3, 4) and the parabola opens upward.

Explanation

Vertex form f(x) = a(x − h)² + k has vertex (h, k); here h = −3 and k = 4. The negative leading coefficient a = −2 means the parabola opens downward.

Question 47Algebra (Subtest I)

Using the quadratic formula, what are the solutions of 2x² + 4x − 1 = 0?

  1. x = (−4 ± √6)/4
  2. x = (2 ± √6)/2
  3. x = (−2 ± √6)/2 Correct answer.
  4. x = (−2 ± √6)

Explanation

Here a = 2, b = 4, c = −1. The discriminant is b² − 4ac = 16 + 8 = 24, so x = (−4 ± √24)/4 = (−4 ± 2√6)/4 = (−2 ± √6)/2.

Question 48Algebra (Subtest I)

What is the sum of the roots of the equation 3x² − 9x + 4 = 0?

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

Explanation

By Vieta's formulas, the sum of the roots of ax² + bx + c = 0 is −b/a. Here −(−9)/3 = 3.

Question 49Algebra (Subtest I)

Which expression is equivalent to (x² − 9)/(x² + 5x + 6) for values where it is defined?

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

Explanation

Factor numerator as (x − 3)(x + 3) and denominator as (x + 2)(x + 3). Canceling the common factor x + 3 leaves (x − 3)/(x + 2), valid for x ≠ −3.

Question 50Algebra (Subtest I)

Solve the equation 1/(x − 2) + 1/(x + 2) = 4/(x² − 4) for x, considering any extraneous solutions.

  1. No real solution
  2. x = 0 Correct answer.
  3. x = 2
  4. x = −2

Explanation

Multiply both sides by (x − 2)(x + 2): (x + 2) + (x − 2) = 4, so 2x = 4 and x = 2. But x = 2 makes the denominators zero, so it is extraneous; thus no real solution remains.