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OSAT Advanced Mathematics (211) study materials
OSAT Advanced Mathematics (211) study resources

OSAT Advanced Mathematics (211) practice questions

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Question 1Mathematical Processes and Problem Solving

Let f be a function and a a real number. The statement “If f is differentiable at x = a, then f is continuous at x = a” is logically equivalent to which statement?

  1. If f is not differentiable at x = a, then f is not continuous at x = a.
  2. If f is continuous at x = a, then f is differentiable at x = a.
  3. If f is not continuous at x = a, then f is not differentiable at x = a. Correct answer.
  4. If f is differentiable at x = a, then f is not continuous at x = a.

Explanation

A conditional statement p → q is logically equivalent to its contrapositive, ¬q → ¬p. Here p is “f is differentiable at x = a” and q is “f is continuous at x = a,” so the contrapositive is “If f is not continuous at x = a, then f is not differentiable at x = a.” The converse and inverse are not equivalent to the original statement in general.

Question 2Mathematical Processes and Problem Solving

A car travels 240 miles in 4.5 hours. Which average speed is most reasonable for the trip?

  1. About 53 miles per hour Correct answer.
  2. About 35 miles per hour
  3. About 110 miles per hour
  4. About 75 miles per hour

Explanation

Average speed is distance divided by time, so the calculation is 240 ÷ 4.5 = 53.33… miles per hour. The closest reasonable estimate is about 53 miles per hour. The other values result from incorrect operations or unreasonable rounding.

Question 3Mathematical Processes and Problem Solving

A recipe that makes 12 servings calls for 3 cups of rice. At the same rate, how many cups of rice are needed for 30 servings?

  1. 5 cups
  2. 9 cups
  3. 6 cups
  4. 7.5 cups Correct answer.

Explanation

Set up the proportion 3/12 = x/30, or find the unit rate: 3 ÷ 12 = 0.25 cup per serving. Multiplying 0.25 by 30 gives 7.5 cups. This is a direct proportional relationship between servings and cups of rice.

Question 4Mathematical Processes and Problem Solving

Consider the claim “For every positive integer n, n² + n + 41 is a prime number.” Which value of n provides a counterexample to the claim?

  1. n = 10
  2. n = 40 Correct answer.
  3. n = 20
  4. n = 30

Explanation

For n = 40, n² + n + 41 = 1600 + 40 + 41 = 1681, and 1681 = 41², which is composite. Therefore n = 40 is a counterexample to the claim that the expression is always prime. A single counterexample is enough to disprove a universal statement.

Question 5Mathematical Processes and Problem Solving

A bacterial culture doubles every 6 hours. If the initial population is P₀, which function models the population after t hours?

  1. P(t) = P₀ · (1 + 2)^(6t)
  2. P(t) = P₀ · 2^(t/6) Correct answer.
  3. P(t) = P₀ + 2t + 6
  4. P(t) = P₀ · 6^(t/2)

Explanation

Doubling every 6 hours means the population is multiplied by 2 once for each 6-hour period. The number of such periods in t hours is t/6, so the model is P(t) = P₀ · 2^(t/6). This is exponential growth with base 2 and a doubling time of 6 hours.

Question 6Mathematical Processes and Problem Solving

All multiples of 6 are even. The number 42 is even. Therefore, 42 is a multiple of 6. Which statement best evaluates this argument?

  1. It is valid because the first premise is a true statement.
  2. It is invalid; 4 is even but not a multiple of 6. Correct answer.
  3. It is valid because the conclusion happens to be true in this case.
  4. It is invalid because the number 42 is not actually even.

Explanation

The argument has the form “All A are B; x is B; therefore x is A,” which is invalid reasoning known as affirming the consequent. A counterexample such as 4 shows that an even number need not be a multiple of 6. The truth of the conclusion does not make the argument form valid.

Question 7Mathematical Processes and Problem Solving

In a proof by mathematical induction that 1 + 3 + 5 + … + (2n − 1) = n² for every positive integer n, what is the key requirement of the inductive step?

  1. Assume it holds for all n and derive a contradiction.
  2. Show that the formula holds for several sample values of n.
  3. Assume it holds for n = k; prove it for n = k + 1. Correct answer.
  4. Show that the formula holds when n = 1.

Explanation

A proof by induction first establishes a base case, such as n = 1, and then proves the inductive step. In the inductive step, one assumes the statement is true for n = k and uses that assumption to prove it is true for n = k + 1. Checking several values or deriving a contradiction is not the structure of mathematical induction.

Question 8Mathematical Processes and Problem Solving

A rectangular garden is to be enclosed with 60 feet of fencing on three sides, with the fourth side formed by a barn wall. What dimensions give the largest possible area?

  1. 20 ft by 20 ft
  2. 12 ft by 36 ft
  3. 15 ft by 30 ft Correct answer.
  4. 10 ft by 40 ft

Explanation

Let x be the length of each side perpendicular to the barn and y the side parallel to the barn. The fencing condition is 2x + y = 60, so y = 60 − 2x, and the area is A = x(60 − 2x) = 60x − 2x². This quadratic is maximized at x = 15, giving y = 30 and area 450 square feet, the largest among the choices.

Question 9Mathematical Processes and Problem Solving

How many positive integers less than 1000 contain at least one digit equal to 7?

  1. 271 Correct answer.
  2. 300
  3. 252
  4. 243

Explanation

There are 999 positive integers less than 1000. Count those with no digit 7 by using three-digit strings with leading zeros allowed: 9 choices for each digit gives 9³ = 729 strings, but subtract the string 000 because it does not represent a positive integer, leaving 728. Therefore, 999 − 728 = 271 integers contain at least one 7.

Question 10Mathematical Processes and Problem Solving

What is the negation of the statement “For every real number x, there exists a real number y such that x + y = 0”?

  1. There is a real number x such that x + y ≠ 0 for every real y. Correct answer.
  2. There is a real number x and a real number y such that x + y ≠ 0.
  3. For every real number x, there is a real number y with x + y ≠ 0.
  4. For every real number x and every real number y, x + y ≠ 0.

Explanation

The negation of ∀x ∃y P(x, y) is ∃x ∀y ¬P(x, y). Thus the negation says there is some real number x such that for every real number y, x + y ≠ 0. This asserts that at least one x fails to have any y satisfying the equation.

Question 11Mathematical Processes and Problem Solving

A student observes that 2, 5, 11, 17, and 23 are all prime and concludes that every number of the form 6n − 1 is prime. Which type of reasoning does this conclusion best illustrate?

  1. Inductive reasoning Correct answer.
  2. Proof by contradiction
  3. Contrapositive reasoning
  4. Deductive reasoning

Explanation

Inductive reasoning draws a general conclusion from a collection of specific examples, which is exactly what the student does here. The examples support the conjecture but do not prove it, so the conclusion remains provisional rather than established deductively.

Question 12Mathematical Processes and Problem Solving

What is the contrapositive of the statement “If a quadrilateral is a square, then it has four congruent sides”?

  1. A quadrilateral has four congruent sides if and only if it is a square.
  2. If a quadrilateral has four congruent sides, then it is a square.
  3. If a quadrilateral does not have four congruent sides, then it is not a square. Correct answer.
  4. If a quadrilateral is not a square, then it does not have four congruent sides.

Explanation

For a conditional statement of the form P → Q, the contrapositive is the logically equivalent statement ¬Q → ¬P. Here P is “is a square” and Q is “has four congruent sides,” so the contrapositive negates both parts and reverses their order.

Question 13Mathematical Processes and Problem Solving

Which value of x is a counterexample to the statement “For every real number x, √(x²) = x”?

  1. x = 4
  2. x = 0
  3. x = −5 Correct answer.
  4. x = 1/4

Explanation

A counterexample must make the hypothesis true while making the conclusion false. Substituting x = −5 gives √((−5)²) = √25 = 5, and 5 is not equal to −5, so the statement fails.

Question 14Mathematical Processes and Problem Solving

A drawer contains only red, blue, and green socks. What is the smallest number of socks that must be drawn to guarantee that at least three socks have the same color?

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

Explanation

By the pigeonhole principle, the worst case without three matching socks is drawing two socks of each of the three colors, for a total of six socks. The next sock drawn must be a third sock of one of those colors, so seven socks guarantee a triple.

Question 15Mathematical Processes and Problem Solving

One pump can empty a tank in 6 hours, and a second pump can empty the same tank in 4 hours. Working together at these constant rates, how long will the two pumps take to empty the tank?

  1. 2.5 hours
  2. 2.4 hours Correct answer.
  3. 10 hours
  4. 5 hours

Explanation

The first pump removes 1/6 of the tank per hour and the second removes 1/4 per hour, so together they remove 1/6 + 1/4 = 5/12 of the tank per hour. The reciprocal of this combined rate is 12/5 = 2.4 hours.

Question 16Mathematical Processes and Problem Solving

Consider the argument: “If a number is divisible by 9, then it is divisible by 3. The number n is not divisible by 3. Therefore, n is not divisible by 9.” Which statement best evaluates this argument?

  1. It is invalid because the first premise is false.
  2. It is valid by modus ponens.
  3. It is invalid because it affirms the consequent.
  4. It is valid by modus tollens. Correct answer.

Explanation

The argument has the form P → Q, not Q, therefore not P, which is modus tollens and is a valid argument form. Since the premises are true and the form is valid, the conclusion follows necessarily.

Question 17Mathematical Processes and Problem Solving

A taxi charges a base fare of $3.50 plus $1.25 per mile. Which function gives the total cost C(m), in dollars, for a ride of m miles?

  1. C(m) = 3.50(m + 1.25)
  2. C(m) = 1.25m − 3.50
  3. C(m) = 3.50m + 1.25
  4. C(m) = 1.25m + 3.50 Correct answer.

Explanation

The base fare is a fixed charge that does not depend on the number of miles, so it is the y-intercept. The per-mile charge is the constant rate of change, so it multiplies m, giving C(m) = 1.25m + 3.50.

Question 18Mathematical Processes and Problem Solving

A student wants to prove: “If n is an integer and n² is even, then n is even.” Which assumption is the starting point for a proof by contrapositive?

  1. Assume n is odd and prove n² is odd. Correct answer.
  2. Assume n² is odd and prove n is odd.
  3. Assume n² is even and prove n is even.
  4. Assume n is even and prove n² is even.

Explanation

The contrapositive of “if n² is even, then n is even” is “if n is not even, then n² is not even.” For integers, that becomes “if n is odd, then n² is odd,” so the proof begins by assuming n is odd.

Question 19Mathematical Processes and Problem Solving

Which statement is logically equivalent to (P → Q) ∧ (Q → P)?

  1. P or Q
  2. Not P or not Q
  3. P and Q
  4. P if and only if Q Correct answer.

Explanation

The conjunction of a conditional and its converse is the definition of a biconditional statement. Thus (P → Q) ∧ (Q → P) is equivalent to P ↔ Q, which is read as “P if and only if Q.”

Question 20Mathematical Processes and Problem Solving

For real numbers x, which statement correctly describes the relationship between x > 2 and x² > 4?

  1. x > 2 is both necessary and sufficient for x² > 4.
  2. x > 2 is neither necessary nor sufficient for x² > 4.
  3. x > 2 is a sufficient but not necessary condition for x² > 4. Correct answer.
  4. x > 2 is a necessary but not sufficient condition for x² > 4.

Explanation

If x > 2, then squaring both sides of a positive inequality gives x² > 4, so x > 2 is sufficient. It is not necessary because a number such as x = −3 satisfies x² > 4 without satisfying x > 2.

Question 21Number Systems and Quantity

Which of the following real numbers is irrational?

  1. 0.272727...
  2. -7/3
  3. √18 Correct answer.
  4. √16 + 1/2

Explanation

Since √18 = 3√2 and √2 is irrational, √18 is irrational. The repeating decimal 0.272727... equals 27/99, -7/3 is a ratio of integers, and √16 + 1/2 = 9/2, so those three numbers are rational.

Question 22Number Systems and Quantity

What is i^17 + i^18 in standard form a + bi?

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

Explanation

Powers of i repeat with period 4: i^1 = i, i^2 = -1, i^3 = -i, and i^4 = 1. Since 17 ≡ 1 mod 4 and 18 ≡ 2 mod 4, i^17 = i and i^18 = -1, so the sum is -1 + i.

Question 23Number Systems and Quantity

A quantity is written as 3.20 × 10^-5. In decimal notation, what is this quantity?

  1. 320000
  2. 0.00032
  3. 0.000032 Correct answer.
  4. 0.0000032

Explanation

A factor of 10^-5 moves the decimal point five places to the left. Starting with 3.20 gives 0.0000320, and dropping the trailing zero gives 0.000032.

Question 24Number Systems and Quantity

Let S = {a + b√2 : a, b ∈ Z}. Which statement about closure under the usual operations is correct?

  1. S is closed under addition but not multiplication.
  2. S is closed under neither addition nor multiplication.
  3. S is closed under multiplication but not addition.
  4. S is closed under both addition and multiplication. Correct answer.

Explanation

For elements a + b√2 and c + d√2 with integer coefficients, the sum is (a + c) + (b + d)√2, which remains in S. The product is (ac + 2bd) + (ad + bc)√2, also with integer coefficients, so S is closed under both operations.

Question 25Number Systems and Quantity

Which property of the real numbers states that between any two distinct real numbers there exists a rational number?

  1. Archimedean property
  2. Well-ordering of the natural numbers
  3. Density of the rational numbers Correct answer.
  4. Completeness

Explanation

The density of the rational numbers in the real numbers means that every open interval (x, y) with x < y contains a rational number. Completeness concerns least upper bounds, the Archimedean property concerns arbitrarily large natural numbers, and well-ordering concerns least elements of nonempty subsets of natural numbers.

Question 26Number Systems and Quantity

If z = 3 - 4i, what is |z^2|?

  1. 5
  2. 25 Correct answer.
  3. 7
  4. 49

Explanation

First compute |z| = √(3^2 + (-4)^2) = √25 = 5. For complex numbers, |z^2| = |z|^2, so |z^2| = 25.

Question 27Number Systems and Quantity

Simplify 3/(√5 - √2) by rationalizing the denominator.

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

Explanation

Multiply numerator and denominator by the conjugate √5 + √2. The denominator becomes (√5)^2 - (√2)^2 = 5 - 2 = 3, which cancels the factor 3 in the numerator, leaving √5 + √2.

Question 28Number Systems and Quantity

Which set has the same cardinality as the set of natural numbers?

  1. The set of all subsets of the natural numbers
  2. The set of integers Correct answer.
  3. The set of irrational numbers
  4. The set of real numbers

Explanation

The integers can be listed as 0, 1, -1, 2, -2, ..., so they are countably infinite and have the same cardinality as the natural numbers. The real numbers and the irrational numbers are uncountable, and the power set of the natural numbers is uncountable by Cantor's theorem.

Question 29Number Systems and Quantity

How many distinct complex solutions does z^3 = 1 have, and what is their sum?

  1. One solution, with sum 1
  2. Three solutions, with sum 0 Correct answer.
  3. Three solutions, with sum 1
  4. Two solutions, with sum -1

Explanation

The equation z^3 = 1 has exactly three distinct complex roots, the cube roots of unity. They are the roots of z^3 - 1 = 0, whose z^2 coefficient is 0, so by Vieta's formulas their sum is 0.

Question 30Number Systems and Quantity

The set of all real algebraic numbers is which of the following?

  1. the empty set
  2. an uncountably infinite set
  3. a countably infinite set Correct answer.
  4. a finite set

Explanation

A real algebraic number is a root of a nonzero polynomial with integer coefficients. There are countably many such polynomials, each has finitely many roots, and a countable union of finite sets is countable; the set is infinite because it contains every rational number.

Question 31Number Systems and Quantity

The mass of a particle is recorded as 0.0000067 grams. Written in scientific notation, this mass is

  1. 6.7 × 10^6 g
  2. 67 × 10^-8 g
  3. 6.7 × 10^-5 g
  4. 6.7 × 10^-6 g Correct answer.

Explanation

To write 0.0000067 in scientific notation, move the decimal point six places to the right to obtain 6.7, so the exponent is -6. Thus the number is 6.7 × 10^-6, and the units remain grams.

Question 32Number Systems and Quantity

What is the product (4 - 3i)(4 + 3i)?

  1. 25 Correct answer.
  2. 16 + 9i
  3. 7
  4. 16 - 9i

Explanation

The product has the form (a - bi)(a + bi) = a^2 + b^2. Here a = 4 and b = 3, so 4^2 + 3^2 = 16 + 9 = 25.

Question 33Number Systems and Quantity

Which expression is equivalent to √50 in simplest radical form?

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

Explanation

Since 50 = 25·2 and √25 = 5, we can write √50 = √25·√2 = 5√2. This is simplest because 2 has no perfect-square factor greater than 1.

Question 34Number Systems and Quantity

Which fraction has a terminating decimal expansion?

  1. 5/12
  2. 2/7
  3. 7/40 Correct answer.
  4. 11/30

Explanation

A fraction in lowest terms has a terminating decimal exactly when its denominator has no prime factors other than 2 and 5. The denominator 40 = 2^3·5 satisfies this, while 12, 7, and 30 contain the prime 3 or 7.

Question 35Number Systems and Quantity

What is (2 + 3i)/(1 - 2i) in standard form a + bi?

  1. 8/5 - 7/5 i
  2. 8/5 + 7/5 i
  3. -4/5 + 7/5 i Correct answer.
  4. -4/5 - 7/5 i

Explanation

Multiply numerator and denominator by the conjugate 1 + 2i. The denominator becomes 1^2 + 2^2 = 5, and the numerator becomes (2 + 3i)(1 + 2i) = -4 + 7i, giving -4/5 + 7/5 i.

Question 36Number Systems and Quantity

After rationalizing the denominator, 1/∛5 is equivalent to which expression?

  1. ∛25
  2. ∛25/5 Correct answer.
  3. 5∛5
  4. ∛5/5

Explanation

To eliminate the cube root, multiply numerator and denominator by ∛(5^2) = ∛25. Then the denominator becomes ∛125 = 5, so the expression is ∛25/5.

Question 37Number Systems and Quantity

The solution set of |2x - 3| ≤ 7 is

  1. x ≥ -2
  2. -5 ≤ x ≤ 2
  3. -2 ≤ x ≤ 5 Correct answer.
  4. x ≤ 5

Explanation

The inequality |2x - 3| ≤ 7 means -7 ≤ 2x - 3 ≤ 7. Adding 3 gives -4 ≤ 2x ≤ 10, and dividing by 2 gives -2 ≤ x ≤ 5.

Question 38Number Systems and Quantity

Which set has cardinality strictly greater than the set of integers?

  1. the set of rational numbers
  2. the set of real numbers Correct answer.
  3. the set of even integers
  4. the set of finite subsets of the integers

Explanation

The integers are countably infinite, and the rationals, even integers, and finite subsets of the integers are also countable. The real numbers are uncountable, so their cardinality is strictly greater than that of the integers.

Question 39Number Systems and Quantity

The set of all finite subsets of the natural numbers is

  1. finite
  2. uncountable but less than the continuum
  3. of cardinality equal to the continuum
  4. countably infinite Correct answer.

Explanation

For each n, the set of n-element subsets of N is countable, and a countable union of countable sets is countable. Including the empty set and all finite sizes still gives a countably infinite collection.

Question 40Number Systems and Quantity

If z = 2(cos(π/12) + i sin(π/12)), then z^6 equals

  1. -64
  2. 64
  3. -64i
  4. 64i Correct answer.

Explanation

De Moivre's theorem gives z^6 = 2^6[cos(6·π/12) + i sin(6·π/12)]. Since 6·π/12 = π/2, this equals 64(0 + i) = 64i.

Question 41Algebra and Functions

What is the domain of the function f(x) = √(2x − 7)?

  1. x ≤ 7/2
  2. x ≥ 7/2 Correct answer.
  3. all real numbers
  4. x ≥ −7/2

Explanation

A square-root function is defined only where its radicand is nonnegative, so the domain requires 2x − 7 ≥ 0. Adding 7 and dividing by 2 gives x ≥ 7/2; at that endpoint the function equals 0, which is permitted.

Question 42Algebra and Functions

Given f(x) = 3x − 2 and g(x) = x² + 1, what is the value of f(g(2))?

  1. 15
  2. 26
  3. 13 Correct answer.
  4. 17

Explanation

Composition works from the inside out, so first evaluate g(2) = 2² + 1 = 5. Then apply f to that result: f(5) = 3(5) − 2 = 13.

Question 43Algebra and Functions

What is the value of log₃ 72 − log₃ 8?

  1. 9
  2. 64
  3. 80
  4. 2 Correct answer.

Explanation

The quotient rule for logarithms gives log₃ 72 − log₃ 8 = log₃(72/8) = log₃ 9. Since 3² = 9, the value of the expression is 2.

Question 44Algebra and Functions

If f(x) = (2x − 5)/3, what is f⁻¹(x)?

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

Explanation

Write y = (2x − 5)/3 and solve for x: multiplying by 3 gives 3y = 2x − 5, so 2x = 3y + 5 and x = (3y + 5)/2. Replacing y with x gives f⁻¹(x) = (3x + 5)/2.

Question 45Algebra and Functions

The graph of f(x) = (2x² + 3)/(x² − x − 6) has which horizontal asymptote?

  1. There is no horizontal asymptote
  2. y = 0
  3. y = −1/2
  4. y = 2 Correct answer.

Explanation

The numerator and denominator both have degree 2, so the horizontal asymptote is the ratio of the leading coefficients, 2/1 = 2. The zeros of the denominator, x = 3 and x = −2, produce vertical asymptotes rather than horizontal ones.

Question 46Algebra and Functions

A savings account is opened with $2000 at an annual interest rate of 4.5%, compounded quarterly. Which expression gives the account balance after 6 years, assuming no additional deposits or withdrawals?

  1. 2000(1 + 0.045/4)^6
  2. 2000(1 + 0.045/12)^24
  3. 2000(1 + 0.045/4)^24 Correct answer.
  4. 2000(1 + 0.045)^6

Explanation

Quarterly compounding means 4 compounding periods each year, so the periodic rate is 0.045/4 and the number of periods is 4 · 6 = 24. Substituting into the compound-interest model A = P(1 + r/n)^(nt) gives 2000(1 + 0.045/4)^24.

Question 47Algebra and Functions

An arithmetic sequence has first term 7 and common difference 3. What is the sum of its first 20 terms?

  1. 1420
  2. 720
  3. 355
  4. 710 Correct answer.

Explanation

For an arithmetic series, Sₙ = (n/2)[2a₁ + (n − 1)d]. Substituting n = 20, a₁ = 7, and d = 3 gives S₂₀ = 10[14 + 19(3)] = 10[14 + 57] = 710.

Question 48Algebra and Functions

When the polynomial p(x) = 2x³ − 5x² + 4x − 7 is divided by x − 2, what is the remainder?

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

Explanation

By the remainder theorem, dividing p(x) by x − 2 leaves a remainder equal to p(2). Evaluating gives p(2) = 2(8) − 5(4) + 4(2) − 7 = 16 − 20 + 8 − 7 = −3.

Question 49Algebra and Functions

What is the solution set of the equation 4^(x + 1) = 8^(2x − 3)?

  1. x = 1/2
  2. x = 11/4 Correct answer.
  3. x = 5/2
  4. x = 7/4

Explanation

Rewrite both sides with base 2: 4^(x + 1) = 2^(2x + 2) and 8^(2x − 3) = 2^(3(2x − 3)) = 2^(6x − 9). Equating exponents gives 2x + 2 = 6x − 9, so 4x = 11 and x = 11/4.

Question 50Algebra and Functions

What is the coefficient of x⁵ in the expansion of (2x − 3)⁷?

  1. 6048 Correct answer.
  2. 12096
  3. −20412
  4. 2016

Explanation

By the binomial theorem, the x⁵ term is C(7,5)(2x)⁵(−3)². That equals 21 · 32 · 9 = 6048, and the sign is positive because (−3)² = 9.