Your FREE Praxis Mathematics: Content Knowledge (5165) Practice Test 2026 – 160+ Q&A

Praxis 5165 Practice Questions

1. Which of the following numbers is irrational?
   • the cube root of 1000
   • the square root of 50
   • the ninth root of 512
   • the fifth root of 243

   Correct answer: the square root of 50

   The square root of 50 works out to 5 times the square root of 2, and no ratio of two whole numbers is equal to it, so it is irrational. The cube root of 1000 is exactly 10, the ninth root of 512 is exactly 2, and the fifth root of 243 is exactly 3, so each of those three is a whole number and therefore rational. A radical sign does not by itself make a value irrational; what matters is whether the radicand is an exact power.

2. What is the value of $i^{27}$, where $i = \sqrt{-1}$?
   • val is neg 1
   • val is pos i
   • val is pos 1
   • val is neg i

   Correct answer: val is neg i

   Powers of the imaginary unit repeat in a cycle of four: the first is pos i, the second is neg 1, the third is neg i, and the fourth is pos 1. Dividing 27 by 4 leaves a remainder of 3, which lands on the third step of that cycle, so val is neg i. A remainder of 1 would give val is pos i, a remainder of 2 would give val is neg 1, and a remainder of 0 would give val is pos 1.

3. What is the greatest common divisor of 84 and 126?
   • GCD is 14
   • GCD is 42
   • GCD is 21
   • GCD is 28

   Correct answer: GCD is 42

   84 breaks into the primes 2, 2, 3 and 7, while 126 breaks into 2, 3, 3 and 7. The primes common to both, counted at the lower power each time, are 2, 3 and 7, whose product is 42, so GCD is 42. GCD is 14 keeps only the 2 and the 7, GCD is 21 keeps only the 3 and the 7, and GCD is 28 is ruled out because 28 does not divide 126 evenly at all.

4. Simplify $(3 + 2i)(1 - 4i)$.
   • val is 11 - 10i
   • val is -5 - 10i
   • val is -5 + 14i
   • val is 3 - 18i

   Correct answer: val is 11 - 10i

   Multiplying term by term gives 3, then minus 12i, then plus 2i, then minus 8 times the imaginary unit squared. Since that unit squared equals negative one, the final term turns into plus 8, so the real part is 3 plus 8 and val is 11 - 10i. Treating the unit squared as positive one turns that last term into minus 8 and gives val is -5 - 10i. Multiplying by the conjugate of the second factor, a habit carried over from dividing complex numbers, gives val is -5 + 14i. Replacing the unit squared with the unit itself leaves three imaginary terms to collect and gives val is 3 - 18i.

5. Which expression equals the absolute value (modulus) of the complex number $3 - 4i$?
   • mod is 1
   • mod is 5
   • mod is 7
   • mod is 3

   Correct answer: mod is 5

   The modulus is the square root of the sum of the squares of the two parts: 3 squared is 9, negative 4 squared is 16, and the square root of 25 is 5, so mod is 5. Subtracting one part from the other gives mod is 1, adding the two parts gives mod is 7, and reading off the real part alone gives mod is 3. None of those three is a distance from the origin.

6. A number expressed in scientific notation is $4.5 \times 10^{-3}$. What is this number in standard form?
   • val is 0.00045
   • val is 0.04500
   • val is 0.00450
   • val is 4,500.0

   Correct answer: val is 0.00450

   A negative exponent of three moves the decimal point in 4.5 three places to the left, so val is 0.00450. Moving it only two places gives val is 0.04500, moving it four places gives val is 0.00045, and treating the exponent as positive moves the point three places to the right instead, giving val is 4,500.0. All four are written to five decimal places so that only the placement of the point differs.

7. What is the least common multiple of 12 and 18?
   • LCM is 24
   • LCM is 72
   • LCM is 36
   • LCM is 54

   Correct answer: LCM is 36

   12 breaks into the primes 2, 2 and 3, while 18 breaks into 2, 3 and 3. Taking the highest power of each prime that appears gives 2 times 2 times 3 times 3, so LCM is 36. LCM is 24 is a multiple of 12 but not of 18, LCM is 54 is a multiple of 18 but not of 12, and LCM is 72 is a common multiple of both yet not the smallest one.

8. If $a = 2^{3} \cdot 5$ and $b = 2^{2} \cdot 5^{2}$, what is $a \cdot b$ expressed in prime factorization?
   • 2^6 x 5^2
   • 2^5 x 5^2
   • 2^5 x 5^3
   • 2^6 x 5^3

   Correct answer: 2^5 x 5^3

   Multiplying powers that share a base adds the exponents. On base 2 the exponents 3 and 2 add to 5, and on base 5 the exponents 1 and 2 add to 3, so 2^5 x 5^3 is the product. Both 2^6 x 5^3 and 2^6 x 5^2 add one exponent too many on base 2, treating the first factor as though it held four twos. 2^5 x 5^2 drops the single factor of 5 carried by the first number.

9. Which of the following is equivalent to $8^{2/3}$?
   • val is 2
   • val is 4
   • val is 8
   • val is 6

   Correct answer: val is 4

   A fractional exponent says to take the root named by the bottom number and then raise the result to the top number. The cube root of 8 is 2, and 2 squared is 4, so val is 4. Stopping at the cube root alone gives val is 2. Multiplying that cube root by the denominator 3 gives val is 6. Ignoring the exponent entirely and repeating the base gives val is 8.

10. What is the result of dividing the complex number $1 + i$ by $i$?
    • val is -1 - i
    • val is +1 + i
    • val is +1 - i
    • val is -1 + i

    Correct answer: val is +1 - i

    Multiply top and bottom by the imaginary unit. The bottom becomes that unit squared, which is negative one, and the top becomes the unit plus the unit squared, or negative one plus the unit. Dividing through by negative one reverses both signs, so val is +1 - i. Skipping that final division leaves val is -1 + i. Reversing only the real part gives val is -1 - i, and reversing only the imaginary part gives val is +1 + i.

11. Rationalize the denominator: $\frac{6}{\sqrt{3}}$.
    • sixfold the root of three
    • tripled the root of three
    • halving the root of three
    • doubled the root of three

    Correct answer: doubled the root of three

    Multiplying top and bottom by the root of 3 turns the denominator into 3 and the numerator into six roots of 3; dividing that by 3 leaves doubled the root of three. Stopping before the division gives sixfold the root of three. Dividing the 6 by 2 rather than by 3 gives tripled the root of three. Putting the 2 underneath the radical instead of in front of it gives halving the root of three, which is below 1, whereas the original quotient is above 3.

12. The number 0.4 (repeating, meaning 0.444...) equals which fraction?
    • val is 2/5
    • val is 9/4
    • val is 4/9
    • val is 5/9

    Correct answer: val is 4/9

    Call the repeating decimal x. Then 10 times x is 4.444..., and subtracting x from that removes the tail, leaving 9 times x equal to 4, so val is 4/9. Turning that result upside down gives val is 9/4, which is larger than 1 and cannot equal a decimal below 1. Reading 0.444... as the terminating decimal 0.4 gives val is 2/5. And val is 5/9 is the fraction that produces 0.555..., not 0.444...

13. How many distinct positive divisors does 36 have?
    • 9 whole divisors
    • 4 total divisors
    • 6 exact divisors
    • 12 true divisors

    Correct answer: 9 whole divisors

    36 factors as 2 squared times 3 squared. The count comes from adding one to each exponent and multiplying, so 3 times 3 gives 9 whole divisors. Multiplying the exponents without first adding one gives 4 total divisors. Listing 1, 2, 3, 4, 6 and 9 but stopping before 12, 18 and 36 gives 6 exact divisors. Counting 12 true divisors overshoots, because only nine numbers divide 36 evenly.

14. Which property is illustrated by $a \cdot (b + c) = a \cdot b + a \cdot c$?
    • commutative swap
    • identity element
    • distributive law
    • associative move

    Correct answer: distributive law

    Spreading a single multiplier across a sum, so that it reaches both terms inside the parentheses, is the distributive law. A commutative swap only reorders two terms and never opens a set of parentheses. An associative move only regroups three terms while leaving their order alone. An identity element is a value that leaves another unchanged, such as 0 in addition or 1 in multiplication, and no such value appears here.

15. What is 7 mod 4 (the remainder when 7 is divided by 4)?
    • rem is 0
    • rem is 1
    • rem is 3
    • rem is 2

    Correct answer: rem is 3

    Dividing 7 by 4 gives one whole group of 4 with 3 left over, so rem is 3. A result of rem is 0 would mean 4 divides 7 exactly, which it does not. rem is 1 records the gap between 3 and 4 rather than the leftover itself. rem is 2 is what dividing 6 by 4 would leave, not 7.

16. Express the complex number with magnitude 2 and angle 60 degrees in rectangular form.
    • 1 + 3^0.5 i
    • 3^0.5 + 1 i
    • 2 + 0 i
    • 0 + 2 i

    Correct answer: 1 + 3^0.5 i

    Rectangular form takes the magnitude times the cosine of the angle as the real part and the magnitude times the sine as the imaginary part. With magnitude 2 the cosine of 60 degrees is one half, so the real part is 1, and the sine of 60 degrees is half the root of 3, so the imaginary part is the root of 3, written here as 3^0.5. That gives 1 + 3^0.5 i. Reading the 30 degree row of the table, so that sine and cosine trade places, gives 3^0.5 + 1 i. Never applying the angle at all and leaving the magnitude on the real axis gives 2 + 0 i. Putting the magnitude on the imaginary axis, as a right angle would, gives 0 + 2 i. All four sit at distance 2 from the origin, so the magnitude alone cannot separate them.

17. If $x$ is a positive integer and $3^{x} = 81$, what is $x$?
    • x is 2
    • x is 6
    • x is 8
    • x is 4

    Correct answer: x is 4

    Counting factors of 3 gives 3, then 9, then 27, then 81, which takes four of them, so x is 4. Reading 81 as 9 squared and reporting that exponent gives x is 2, but the base there is 9 rather than 3. Six factors of 3 make 729 and eight make 6561, so x is 6 and x is 8 both overshoot 81 badly.

18. What is the value of log base 2 of 32?
    • log is 4
    • log is 6
    • log is 7
    • log is 5

    Correct answer: log is 5

    The logarithm asks how many factors of 2 multiply together to reach 32, and 2 times 2 times 2 times 2 times 2 arrives in five steps, so log is 5. Four factors of 2 give only 16, so log is 4 falls short. Six factors give 64 and seven give 128, so log is 6 and log is 7 both overshoot.

19. Which set is closed under subtraction?
    • counting terms
    • integer values
    • positive reals
    • odd quantities

    Correct answer: integer values

    Taking one integer away from another always lands on another integer, positive, negative or zero, so integer values make up a set closed under subtraction. Counting terms begin at 1, so 3 minus 7 escapes that set. Positive reals rule out every negative result, so 2 minus 5 escapes as well. Odd quantities fail at once, since 7 minus 3 is 4 and 4 is even.

20. Simplify $\sqrt{72} + \sqrt{18}$.
    • 6 x 2^0.5
    • 9 x 2^0.5
    • 9 x 3^0.5
    • 6 x 3^0.5

    Correct answer: 9 x 2^0.5

    The root of 72 reduces to 6 x 2^0.5 and the root of 18 reduces to 3 of that same radical, so like radicals add through their coefficients and 6 plus 3 gives 9 x 2^0.5. Reducing only the first term and stopping there gives 6 x 2^0.5. Reading 72 as 36 times 3 rather than 36 times 2 gives 6 x 3^0.5, and carrying that wrong radical through the addition as well gives 9 x 3^0.5.

21. Solve for $x$: $2x + 7 = 19$.
    • x is 6
    • x is 9
    • x is 5
    • x is 7

    Correct answer: x is 6

    Taking 7 from both sides leaves 2 times x equal to 12, and halving that gives x is 6, which checks out because 12 plus 7 is 19. The value x is 5 takes the 7 away a second time, subtracting it from 12 where the 12 has to be halved. The value x is 7 merely copies the constant already in the equation instead of solving for the unknown. The value x is 9 divides before it subtracts, halving 19 to get nine and a half and then dropping the half instead of removing the 7 first. Substituting 5 gives 17, substituting 7 gives 21 and substituting 9 gives 25, so all three miss the required 19.

22. Factor completely: $x^{2} - 9$.
    • (x+3)(x+3)
    • (x-9)(x+1)
    • (x-3)(x-3)
    • (x-3)(x+3)

    Correct answer: (x-3)(x+3)

    A difference of two squares splits into the sum and the difference of the two roots, so the expression becomes (x-3)(x+3); multiplying that back out cancels the middle terms exactly. (x+3)(x+3) expands with a middle term of plus 6x, and (x-3)(x-3) expands with a middle term of minus 6x, so neither leaves a bare difference. (x-9)(x+1) expands to x squared minus 8x minus 9, wrong in both the middle term and the constant.

23. What are the solutions to $x^{2} - 5x + 6 = 0$?
    • +2 and +3
    • -2 and -3
    • +1 and +6
    • -1 and -6

    Correct answer: +2 and +3

    The quadratic splits into two brackets whose numbers multiply to 6 and add to 5, giving roots that are both positive, so +2 and +3 is correct. The pair -2 and -3 would arise from brackets with plus signs, which produce a middle term of plus 5x instead of minus 5x. The pair +1 and +6 multiplies to 6 but adds to 7. The pair -1 and -6 misses on both counts, needing a middle term of plus 7x.

24. Solve the system: $x + y = 10$ and $x - y = 4$.
    • x=6, y=4
    • x=7, y=3
    • x=8, y=2
    • x=5, y=5

    Correct answer: x=7, y=3

    Adding the two equations cancels y and leaves 2 times x equal to 14, so x is 7, and putting that back into the first equation makes y equal to 3, giving x=7, y=3. The pair x=6, y=4 totals 10 but its difference is only 2. The pair x=8, y=2 has a difference of 6. The pair x=5, y=5 has a difference of 0. Only one pair satisfies both equations together.

25. What is the slope of the line passing through (2, 3) and (6, 11)?
    • m is 1
    • m is 2
    • m is 4
    • m is 3

    Correct answer: m is 2

    Slope is the change in the second coordinate over the change in the first: 11 minus 3 is 8, and 6 minus 2 is 4, and 8 divided by 4 is 2, so m is 2. A value of m is 1 would require the rise and the run to be equal, but they are 8 and 4. A value of m is 4 reports the run on its own instead of the ratio. A value of m is 3 copies the first y coordinate rather than computing any ratio at all.

26. Simplify: $(x^{3} \cdot x^{4}) / x^{2}$.
    • val is x^7
    • val is x^9
    • val is x^5
    • val is x^6

    Correct answer: val is x^5

    Multiplying powers of one base adds the exponents, so 3 plus 4 gives 7; dividing then subtracts, so 7 minus 2 leaves 5 and val is x^5. Stopping after the multiplication and never dividing gives val is x^7. Adding all three exponents, as though the division were another product, gives val is x^9. Multiplying 3 by 4 and then halving gives val is x^6, which applies multiplication rules to the exponents themselves.

27. What is the discriminant of $2x^{2} + 3x - 5 = 0$?
    • val is 19
    • val is 40
    • val is 29
    • val is 49

    Correct answer: val is 49

    The discriminant is the middle coefficient squared minus 4 times the outer two. Here the square is 9, and 4 times 2 times negative 5 is negative 40, so subtracting a negative adds 40 and val is 49. Leaving out the factor of 4 and adding only 10 gives val is 19. Using twice rather than four times the outer product adds 20 and gives val is 29. Reporting the 40 on its own, with the squared term dropped, gives val is 40.

28. Expand $(x + 4)^{2}$.
    • x^2 + 8x + 16
    • x^2 + 4x + 16
    • x^2 + 8x + 32
    • x^2 + 4x + 32

    Correct answer: x^2 + 8x + 16

    Squaring a binomial gives the first term squared, then twice the product of the two terms, then the second term squared: that is x squared, plus 2 times 4 times x, plus 16, so x^2 + 8x + 16. The version x^2 + 4x + 16 leaves the doubling out of the middle term, using 4x where twice 4 times x is needed. The version x^2 + 8x + 32 doubles the constant as well as the middle term, putting twice 16 on the end in place of 4 squared. The version x^2 + 4x + 32 moves the doubling from the middle term onto the constant, so both of those terms come out wrong.

29. Solve for $x$: $3(x - 2) = 2x + 5$.
    • x is -1
    • x is 3
    • x is 7
    • x is 11

    Correct answer: x is 11

    Distributing the 3 gives 3x minus 6, and removing 2x from both sides leaves x minus 6 equal to 5, so x is 11; both sides then come to 27. Distributing as 3x plus 6 instead gives x is -1. Failing to multiply the 2 by 3, so that the left side reads 3x minus 2, gives x is 7. Substituting 3 makes the left side 3 while the right side is 11, so x is 3 fails as well.

30. If $f(x) = 2x^{2} - 3x + 1$, what is $f(2)$?
    • 5 is f of 2
    • 9 is f of 2
    • 7 is f of 2
    • 3 is f of 2

    Correct answer: 3 is f of 2

    Putting 2 in place of x gives 2 times 4, then minus 3 times 2, then plus 1, which is 8 minus 6 plus 1, so 3 is f of 2. Subtracting only 3 instead of 3 times 2, and then dropping the final plus 1, gives 5 is f of 2. Subtracting x rather than 3 times x gives 8 minus 2 plus 1, so 7 is f of 2. Omitting the middle term altogether gives 8 plus 1, so 9 is f of 2.

31. Which expression equals $(x^{2} - 4)/(x - 2)$ for $x$ not equal to 2?
    • x + 2 if x is not 2
    • x - 2 if x is not 2
    • x + 4 if x is not 2
    • x - 4 if x is not 2

    Correct answer: x + 2 if x is not 2

    The numerator is a difference of squares, so it factors into x minus 2 times x plus 2; canceling the shared factor against the denominator leaves x + 2 if x is not 2. Keeping the wrong factor gives x - 2 if x is not 2, which multiplied back by x minus 2 yields a perfect square rather than the original numerator. Carrying the 4 straight through gives x + 4 if x is not 2, and doing that with a reversed sign gives x - 4 if x is not 2.

32. Solve the inequality: $-2x + 3 > 7$.
    • x < -2
    • x > +2
    • x < +2
    • x > -2

    Correct answer: x < -2

    Taking 3 from both sides leaves negative 2x greater than 4, and dividing through by a negative number reverses the direction of the sign, so x < -2. Dividing without reversing gives x > -2. Dropping the negative from the coefficient gives x < +2, and making both slips together gives x > +2. Testing -3 in the original statement gives 9, which is greater than 7, while testing 0 gives 3, which is not.

33. What is the sum of the roots of $x^{2} - 7x + 10 = 0$?
    • x^2 sum is 7
    • x^2 sum is 9
    • x^2 sum is 3
    • x^2 sum is 5

    Correct answer: x^2 sum is 7

    The expression factors into x minus 2 times x minus 5, so the roots are 2 and 5 and x^2 sum is 7, which matches the shortcut of negating the middle coefficient and dividing by the leading one. Mis-factoring into x minus 4 times x minus 5 gives x^2 sum is 9, but that pair multiplies to 20 rather than 10. Reporting the gap between the two roots gives x^2 sum is 3. Reporting the larger root on its own gives x^2 sum is 5.

34. Simplify the rational expression: $(6x^{2} y) / (3x y^{2})$.
    • 2y/x
    • x/2y
    • y/2x
    • 2x/y

    Correct answer: 2x/y

    Dividing the coefficients gives 6 divided by 3, which is 2; dividing x squared by x leaves one x on top; and dividing y by y squared leaves one y underneath. Collecting those three results gives 2x/y. The form 2y/x interchanges the two variables. The form x/2y moves the coefficient 2 into the denominator where nothing put it. The form y/2x inverts the whole quotient.

35. If $2^{x} = 64$, what is $x$?
    • x is 6
    • x is 5
    • x is 8
    • x is 7

    Correct answer: x is 6

    Sixty-four is 2 multiplied by itself six times, so the exponent that produces it is 6 and x is 6. The choice x is 5 gives 32, which is half of what the equation requires. The choice x is 7 gives 128 and the choice x is 8 gives 256, both of which overshoot. Only one whole-number exponent satisfies the equation, so the near neighbors on either side of 6 can all be ruled out by direct substitution.

36. Factor: $x^{2} + 7x + 12$.
    • (x+2)(x+6)
    • (x+2)(x+5)
    • (x+3)(x+4)
    • (x-3)(x-4)

    Correct answer: (x+3)(x+4)

    The factored form needs two numbers whose product is 12 and whose sum is 7. Three and four satisfy both conditions, so the answer is (x+3)(x+4). In (x+2)(x+6) the product is 12 but the sum is 8, giving a middle term of 8x. In (x+2)(x+5) the sum is 7 but the product is 10, giving the wrong constant. In (x-3)(x-4) the constant is right but both signs are negative, producing a middle term of minus 7x.

37. What is the solution to $|x - 3| = 5$?
    • x = 2 or x = -8
    • x = 5 or x = -5
    • x = 8 or x = -8
    • x = 8 or x = -2

    Correct answer: x = 8 or x = -2

    An absolute value equation splits into two branches. Setting the inside equal to 5 gives x = 8, and setting it equal to negative 5 gives x = -2, so the solution set is x = 8 or x = -2. The pair x = 2 or x = -8 subtracts 3 from each branch instead of adding it. The pair x = 5 or x = -5 ignores the shift of 3 altogether. The pair x = 8 or x = -8 finds the first root correctly but then negates it rather than solving the second branch.

38. The line $y = 3x - 2$ is reflected to be perpendicular through the origin. What is the slope of the perpendicular line?
    • m is +1/3
    • m is +3/1
    • m is -1/3
    • m is -3/1

    Correct answer: m is -1/3

    Perpendicular lines have slopes that are negative reciprocals of one another. Inverting 3 gives one third and reversing the sign gives m is -1/3. The value m is +1/3 inverts the slope but leaves the sign alone. The value m is +3/1 simply repeats the original slope, which would give a parallel line. The value m is -3/1 reverses the sign without inverting, which is neither parallel nor perpendicular.

39. Solve for $x$: $\frac{x}{4} + \frac{x}{6} = 5$.
    • x is 10
    • x is 12
    • x is 13
    • x is 11

    Correct answer: x is 12

    Rewriting both fractions over a common denominator of 12 turns the left side into 3x over 12 plus 2x over 12, which is 5x over 12. Setting that equal to 5 gives 5x = 60, so x is 12. The value x is 10 comes from combining the two fractions into x over 2 rather than 5x over 12. The values x is 11 and x is 13 are off-by-one slips that fail when substituted back, since neither makes the two fractions add to exactly 5.

40. What is the vertex form of $y = x^{2} - 6x + 5$?
    • y = (x+3)^2 - 4
    • y = (x-3)^2 + 4
    • y = (x-3)^2 - 4
    • y = (x-6)^2 + 5

    Correct answer: y = (x-3)^2 - 4

    Completing the square on x squared minus 6x gives the square of x minus 3, less 9. Adding back the original constant of 5 leaves y = (x-3)^2 - 4. The form y = (x-3)^2 + 4 adds the 9 instead of subtracting it. The form y = (x+3)^2 - 4 halves the linear coefficient but keeps the wrong sign inside the parentheses. The form y = (x-6)^2 + 5 copies the original coefficients straight across without completing the square at all.

41. Using the quadratic formula, solve $x^{2} - 4x + 1 = 0$.
    • 4 +/- 3^(1/2)
    • 2 +/- 3^(1/2)
    • 4 +/- 5^(1/2)
    • 2 +/- 5^(1/2)

    Correct answer: 2 +/- 3^(1/2)

    The discriminant is 16 minus 4, which is 12, and the square root of 12 is 2 times the square root of 3. Dividing 4 plus or minus that root by 2 gives 2 +/- 3^(1/2), where the exponent one half denotes a square root. The answer 4 +/- 3^(1/2) forgets to divide the whole numerator by 2. The answer 2 +/- 5^(1/2) uses a discriminant of 20 by adding 4 rather than subtracting it. The answer 4 +/- 5^(1/2) makes both of those errors together.

42. What is the value of $x$ in the matrix equation where $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 5 \\ 11 \end{bmatrix}$?
    • x is 1
    • x is 3
    • x is 2
    • x is 4

    Correct answer: x is 1

    The two rows of the product give x plus 2y = 5 and 3x plus 4y = 11. Substituting x = 5 minus 2y into the second row gives 15 minus 2y = 11, so y = 2 and therefore x is 1. The value x is 2 is the solution for y, not for x. The value x is 3 comes from reading the lower-left matrix entry instead of solving the system. The value x is 4 reads the lower-right entry in the same way.

43. The determinant of the matrix $\begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}$ is:
    • det is 11
    • det is +2
    • det is +5
    • det is +8

    Correct answer: det is +5

    For a two-by-two array the determinant is the product along the main diagonal minus the product along the other diagonal. Here 2 times 4 is 8, and 3 times 1 is 3, so 8 minus 3 means det is +5. The value det is +8 is the main-diagonal product on its own, with the subtraction never carried out. The value det is 11 adds the two products instead of subtracting one from the other. The value det is +2 subtracts the off-diagonal product a second time.

44. Simplify: $\frac{2x + 6}{2}$.
    • 2x + 6
    • x + 12
    • 2x + 3
    • x + 3

    Correct answer: x + 3

    Every term of the numerator has to be divided by 2, so 2x becomes x and 6 becomes 3, leaving x + 3. The result 2x + 3 divides only the constant and leaves the coefficient of x untouched. The result 2x + 6 is the original numerator with no division carried out at all. The result x + 12 divides the first term correctly but multiplies the constant by 2 instead of dividing it, inverting the operation on that term.

45. If the roots of $x^{2} + bx + c = 0$ are 4 and -3, what is $c$?
    • c is pos 12
    • c is neg 12
    • c is neg 11
    • c is pos 11

    Correct answer: c is neg 12

    In a quadratic with leading coefficient 1 the constant term equals the product of the roots. Multiplying 4 by negative 3 gives negative twelve, so c is neg 12. The choice c is pos 12 drops the sign of the negative root. The sum of the roots is 1, and folding that sum into the product produces the near misses c is neg 11 and c is pos 11; neither is the product, which is what the constant term records.

46. Solve for $x$: $5^{x+1} = 125$.
    • val is 2
    • val is 4
    • val is 1
    • val is 3

    Correct answer: val is 2

    Since 125 is 5 multiplied by itself three times, the exponent on the left must equal 3. That exponent is x plus 1, so x equals 2 and val is 2. The choice val is 3 treats the exponent as x alone and ignores the added 1. The choice val is 1 produces 5 squared, or 25. The choice val is 4 produces 5 to the fifth power, or 3125. Only one of the four makes both sides equal.

47. What is the next term in the arithmetic sequence 3, 7, 11, 15, ...?
    • nxt is 21
    • nxt is 19
    • nxt is 18
    • nxt is 20

    Correct answer: nxt is 19

    Each listed term rises by 4, so the term after 15 is 15 plus 4 and nxt is 19. The answer nxt is 18 adds the first term 3 rather than the common difference. The answer nxt is 20 uses a difference of 5, which does not hold between any two listed terms. The answer nxt is 21 adds 6, a step that matches no gap in the sequence.

48. What is the 5th term of a geometric sequence with first term 2 and common ratio 3?
    • val is 54
    • val is 216
    • val is 162
    • val is 108

    Correct answer: val is 162

    A geometric sequence multiplies by the ratio one time fewer than the term number, so the fifth term is 2 times 3 raised to the fourth power. Three to the fourth is 81, and twice that means val is 162. The answer val is 54 stops one term early, at the fourth term. The answer val is 108 doubles that fourth term instead of tripling it. The answer val is 216 applies a ratio of 4 at the final step.

49. Simplify: $\log(100) + \log(1000)$, using base 10.
    • sum is 8
    • sum is 5
    • sum is 9
    • sum is 6

    Correct answer: sum is 5

    In base 10 the logarithm of 100 is 2 and the logarithm of 1000 is 3, because those are the exponents that produce each number. Adding them means sum is 5, which also matches the logarithm of their product, 100000. The answer sum is 6 multiplies the two logarithms instead of adding them. The answer sum is 8 counts the larger logarithm twice. The answer sum is 9 squares the larger logarithm.

50. Which is equivalent to $\ln(e^{4})$?
    • ans is 1
    • ans is e
    • 2 as 4/2
    • ans is 4

    Correct answer: ans is 4

    The natural logarithm and the exponential with base e undo one another, so taking the natural logarithm of e raised to a power returns that power itself and ans is 4. The choice ans is 1 is the natural logarithm of e on its own, with the exponent discarded. The choice ans is e returns the base rather than the exponent. The choice 2 as 4/2 halves the exponent, which no logarithm rule permits.

51. Solve for $x$: $\frac{x - 1}{x + 2} = 3$.
    • x is -7/2
    • x is +5/2
    • x is -5/2
    • x is +7/2

    Correct answer: x is -7/2

    Multiplying both sides by the denominator gives x minus 1 equal to 3x plus 6. Collecting terms leaves negative 7 equal to 2x, so x is -7/2. The answer x is +7/2 solves the same equation but drops the negative sign at the last step. The answer x is -5/2 expands 3 times the quantity x plus 2 as 3x plus 4. The answer x is +5/2 combines both of those slips.

52. What is the sum of the first 10 positive integers?
    • sum of 55
    • sum of 45
    • sum of 60
    • sum of 50

    Correct answer: sum of 55

    The running total of the first n counting numbers equals n times the quantity n plus 1, all divided by 2. With n equal to 10 that is 10 times 11 over 2, so sum of 55. The answer sum of 45 is the total for the first nine numbers only. The answer sum of 50 uses n times n over 2 and so leaves out the plus 1. The answer sum of 60 uses n plus 2 in place of n plus 1.

53. The function $f(x) = 3x - 5$ has what inverse?
    • (x-5)/3
    • (5-x)/3
    • (x+3)/5
    • (x+5)/3

    Correct answer: (x+5)/3

    To invert the rule, undo each step in reverse order: add 5 to reverse the subtraction, then divide by 3 to reverse the multiplication. That gives (x+5)/3. The form (x-5)/3 subtracts 5 again rather than undoing the subtraction. The form (5-x)/3 reverses the whole numerator and changes the sign of x. The form (x+3)/5 interchanges the multiplier and the constant.

54. What is the domain of $f(x) = \sqrt{x - 2}$?
    • x >= 2
    • x > 2
    • x > 0
    • x <= 2

    Correct answer: x >= 2

    A square root returns a real number only when the quantity under it is not negative, so x minus 2 must be at least zero and x >= 2. The condition x > 2 wrongly excludes 2 itself, where the quantity under the root is zero and the root is defined. The condition x > 0 puts the requirement on x itself rather than on the quantity under the root. The condition x <= 2 admits values that make that quantity negative.

55. What is the range of $f(x) = x^{2} + 1$?
    • y > 1
    • y > 0
    • y <= 1
    • y >= 1

    Correct answer: y >= 1

    Squaring any real number gives a result that is never negative, so the smallest possible output occurs at x equal to zero and equals 1. Every value from there upward is produced, so y >= 1. The condition y > 1 leaves out the minimum, which the rule does reach. The condition y > 0 reports the squared term on its own, read as strictly positive, before the 1 is added. The condition y <= 1 reverses the direction of the inequality.

56. If $f(x) = x + 2$ and $g(x) = x^{2}$, what is $(f \circ g)(3)$?
    • out is 13
    • out is 11
    • out is 15
    • out is 9

    Correct answer: out is 11

    The inner rule squares 3 to give 9, and the outer rule then adds 2, so out is 11. The answer out is 9 stops after squaring and never applies the outer rule. The answer out is 13 applies the outer rule a second time to 11. The answer out is 15 adds 2 times the input rather than adding 2.

57. Which describes the end behavior of $f(x) = -2x^{3}$?
    • L up and R up
    • L dn and R up
    • L up and R dn
    • L dn and R dn

    Correct answer: L up and R dn

    Here L up and R dn is shorthand for a graph that rises at the far left and falls at the far right. An odd power with a negative coefficient in front behaves that way: very negative inputs give large positive outputs, and very positive inputs give large negative outputs. The pattern L dn and R up belongs to an odd power with a positive coefficient. The patterns L up and R up and L dn and R dn belong to even powers, where both tails go the same way.

58. What is the period of $f(x) = \sin(2x)$?
    • 2pi
    • pi
    • pi/4
    • pi/2

    Correct answer: pi

    The length of one full cycle of a sine wave is 2 pi divided by the multiplier attached to x. That multiplier is 2, so one cycle spans pi. The answer 2pi is the cycle length before any horizontal compression is applied. The answer pi/2 divides 2 pi by 4 instead of by 2. The answer pi/4 divides by 8 and compresses the wave four times too far.

59. The graph of $y = (x - 3)^{2} + 2$ has its vertex at:
    • at (3, 2)
    • at (3, 0)
    • at (9, 0)
    • at (9, 2)

    Correct answer: at (3, 2)

    When a parabola is written as a squared bracket plus a constant, the turning point sits where the bracket is zero and the height equals that constant. The bracket vanishes at 3 and the constant is 2, so the turning point is at (3, 2). The reading at (3, 0) finds where the bracket is zero but reports a height of zero, forgetting that the plus 2 lifts the whole curve. The reading at (9, 2) treats the 3 in the bracket as already squared, so the shift is taken as 9 to the right. The reading at (9, 0) makes both of those slips at once, squaring the 3 for the shift and leaving the height at zero.

60. What is the amplitude of $f(x) = 4\cos(x)$?
    • amp is 2
    • amp is 4
    • amp is 1
    • 3 as 4-1

    Correct answer: amp is 4

    The height of a cosine wave above its midline equals the size of the number multiplying the wave, ignoring any sign. That number is 4, so amp is 4. The answer amp is 1 is the height of a plain cosine wave with no multiplier at all. The answer amp is 2 halves the multiplier, which would describe a wave of half this height. The answer 3 as 4-1 subtracts 1 from the multiplier and matches no feature of the rule.

61. For what value(s) of $x$ does $f(x) = \frac{x-1}{x^{2}-1}$ have a removable discontinuity?
    • gap at +1
    • gap at -3
    • gap at -1
    • gap at +3

    Correct answer: gap at +1

    The bottom factors into two brackets, one of which matches the top exactly. That shared factor cancels, leaving a single missing point rather than a break that runs off to infinity, so gap at +1. The answer gap at -1 names the input where the surviving bracket is zero; the curve shoots away there, so that break cannot be patched. The answers gap at +3 and gap at -3 name inputs where the bottom equals 8, so the curve is perfectly continuous at both.

62. Which function is even?
    • f(x) = x^3 + x
    • f(x) = -2x + 1
    • f(x) = x^3 - 1
    • f(x) = x^2 + 1

    Correct answer: f(x) = x^2 + 1

    A rule is even when feeding in the opposite of any input returns the same output. Squaring removes the sign, so f(x) = x^2 + 1 gives the same value for an input and its opposite. The rule f(x) = x^3 + x is odd instead: every term flips sign together. The rules f(x) = -2x + 1 and f(x) = x^3 - 1 are neither, because each mixes a term that flips sign with a constant that does not.

63. What is the horizontal asymptote of $f(x) = \frac{3x^{2} + 1}{x^{2} - 4}$?
    • y = 1
    • y = 0
    • y = 3
    • y = 2

    Correct answer: y = 3

    Top and bottom have the same highest power, so far out along either tail the curve levels off at the ratio of the two leading numbers, 3 over 1. That makes the level line y = 3. The line y = 0 applies the rule for a bottom of higher degree than the top. The line y = 1 keeps only the leading number from the bottom. The line y = 2 subtracts the two leading numbers instead of dividing them.

64. If $f(x) = 2^{x}$, what is $f(3) - f(1)$?
    • dif is 6
    • dif is 2
    • dif is 8
    • dif is 4

    Correct answer: dif is 6

    Doubling three times gives 8, and doubling once gives 2; subtracting the second from the first means dif is 6. The answer dif is 8 reports the first value alone and never subtracts. The answer dif is 2 reports the second value alone. The answer dif is 4 divides 8 by 2 rather than subtracting, which mixes up a ratio with a difference.

65. The exponential function $f(x) = 100 \cdot (0.5)^{x}$ represents decay. What is $f(2)$?
    • amt is 50
    • amt is 75
    • amt is 25
    • amt is 0

    Correct answer: amt is 25

    Each step of this decay keeps half of what is left, so after two steps a starting amount of 100 becomes half of 50, and amt is 25. The answer amt is 50 stops after a single step. The answer amt is 75 subtracts 25 from the start rather than halving twice. The answer amt is 0 treats the decay as finished, but halving never reaches zero in a finite number of steps.

66. What transformation maps $y = x^{2}$ to $y = (x + 4)^{2}$?
    • mov R by 4
    • mov U by 4
    • mov D by 4
    • mov L by 4

    Correct answer: mov L by 4

    Adding 4 to x inside the bracket makes the curve reach any given height 4 units sooner, so every point slides 4 units toward the negative side and mov L by 4 is right. The answer mov R by 4 would follow from subtracting 4 inside the bracket instead. The answers mov U by 4 and mov D by 4 describe adding or subtracting 4 outside the bracket, which raises or lowers the curve rather than sliding it sideways.

67. What is the y-intercept of $f(x) = 3x^{2} - 2x + 5$?
    • 5 is the y cut
    • 3 is the y cut
    • 2 is the y cut
    • 0 is the y cut

    Correct answer: 5 is the y cut

    Putting x at zero wipes out both x terms and leaves only the constant, so 5 is the y cut. 3 is the y cut reports the leading coefficient instead of the constant. 2 is the y cut reports the size of the linear coefficient. 0 is the y cut would hold only for a curve passing through the origin, which this one does not.

68. A function f is one-to-one if and only if:
    • Every vertical line meets the sketch at most once
    • Every tangent line meets the outline at most once
    • Every slanted line meets the drawing at most once
    • Every horizontal line meets the plot at most once

    Correct answer: Every horizontal line meets the plot at most once

    One-to-one means no output is reused, and two points at the same height would be two inputs sharing one output, so the test is that every horizontal line meets the plot at most once. Every vertical line meets the sketch at most once is the weaker test that decides whether the relation is a function at all. Every tangent line meets the outline at most once fails on a parabola, which is not one-to-one yet has tangents touching once. Every slanted line meets the drawing at most once fails on the identity function, which is one-to-one yet is met infinitely often by a line of matching slope.

69. What is the value of $\tan(45^\circ)$?
    • 1 to 2 as the tan
    • 1 to 1 as the tan
    • 3 to 1 as the tan
    • 3 to 2 as the tan

    Correct answer: 1 to 1 as the tan

    The tangent is the opposite leg over the adjacent leg, and at forty-five degrees those two legs are equal, so the answer is 1 to 1 as the tan, which is the number 1. 1 to 2 as the tan is the short leg over the hypotenuse in a thirty-sixty triangle, a sine rather than a tangent. 3 to 1 as the tan is the tangent of sixty degrees with its square root sign dropped. 3 to 2 as the tan is the sine of sixty degrees with the same square root sign dropped.

70. The piecewise function $f(x) = x$ for $x < 0$ and $f(x) = x^{2}$ for $x \ge 0$. What is $f(-3) + f(2)$?
    • 4 as the sum
    • 7 as the sum
    • 1 as the sum
    • 5 as the sum

    Correct answer: 1 as the sum

    The input -3 is negative so the first rule returns -3, while 2 is not negative so the second rule squares it to 4, and -3 plus 4 gives 1 as the sum. 4 as the sum counts only the second term and forgets the first. 5 as the sum squares the negative input and then subtracts rather than adds. 7 as the sum drops the minus sign before adding, treating the first term as positive 3.

71. What is the inverse of f(x) = log base 2 of x?
    • 2 to the x
    • x to the 2
    • e to the x
    • x to the e

    Correct answer: 2 to the x

    A logarithm and an exponential sharing a base undo one another, so the inverse is 2 to the x. x to the 2 is squaring, which undoes a square root and not a logarithm. e to the x undoes the natural logarithm, a logarithm of a different base. x to the e swaps base and exponent and inverts nothing.

72. For $f(x) = x^{2} - 4x$, on what interval is $f$ decreasing?
    • x > 2
    • x < 2
    • x < 0
    • x > 4

    Correct answer: x < 2

    The parabola opens upward and its vertex sits at 2, and an upward parabola falls to the left of its vertex, so the function decreases on x < 2. x > 2 names exactly the stretch where that same graph rises. x < 0 covers only part of the falling stretch and abandons everything between 0 and 2. x > 4 lies beyond the second root, well inside the rising part.

73. What is the value of $\sin(\frac{\pi}{6})$?
    • 1/6 as the sin
    • 1/2 as the sin
    • 3/2 as the sin
    • 3/4 as the sin

    Correct answer: 1/2 as the sin

    An angle of pi over 6 is thirty degrees, and the side opposite it is half the hypotenuse, giving 1/2 as the sin. 1/6 as the sin simply copies the 6 out of the angle and describes no ratio of sides. 3/2 as the sin is the cosine of that angle with its square root sign dropped, and no sine can exceed 1 in any case. 3/4 as the sin is the square of that cosine rather than the sine itself.

74. Which is the correct factorization showing the zeros of $f(x) = x^{2} - x - 6$?
    • (x+3)(x-2) at -3 and 2
    • (x-6)(x+1) at 6 and -1
    • (x-3)(x+2) at 3 and -2
    • (x-1)(x+6) at 1 and -6

    Correct answer: (x-3)(x+2) at 3 and -2

    Two numbers whose product is -6 and whose sum is -1 are -3 and 2, so the factoring is (x-3)(x+2) at 3 and -2. (x+3)(x-2) at -3 and 2 expands to x squared plus x minus 6, so its middle term carries the wrong sign. (x-6)(x+1) at 6 and -1 expands to x squared minus 5x minus 6. (x-1)(x+6) at 1 and -6 expands to x squared plus 5x minus 6.

75. The function $f(x) = \cos(x)$ has what maximum value?
    • 1 as the max
    • 0 as the max
    • 2 as the max
    • e as the max

    Correct answer: 1 as the max

    Cosine swings between -1 and 1 and touches the upper end at every even multiple of pi, so 1 as the max is right. 0 as the max is merely a value cosine passes through a quarter turn away from the peak. 2 as the max reports the full width of the band from -1 to 1 rather than its ceiling. e as the max borrows a constant from exponential growth that sits near 2.72, far outside anything cosine can reach.

76. What is the average rate of change of $f(x) = x^{2}$ on the interval [1, 4]?
    • 3 as the avg
    • 9 as the avg
    • 7 as the avg
    • 5 as the avg

    Correct answer: 5 as the avg

    The outputs run from 1 to 16 across a run of 3, and 15 divided by 3 gives 5 as the avg. 3 as the avg reports the run itself, the denominator rather than the quotient. 9 as the avg squares the run instead of dividing by it. 7 as the avg is the step between the last two squares, 16 and 9, which covers one unit rather than the whole interval.

77. How many real zeros does $f(x) = x^{2} + 4$ have?
    • 1 in all
    • 2 in all
    • 4 in all
    • 0 in all

    Correct answer: 0 in all

    A square is never negative, so x squared plus 4 stays at or above 4 and the curve never touches the axis, leaving 0 in all. 1 in all would require a repeated root, which happens only when the graph rests on the axis. 2 in all assumes every quadratic cuts the axis twice, but that needs a positive discriminant and this one is negative. 4 in all just reads the constant term off as though it were a count.

78. What is the limit as $x$ approaches 2 of $\frac{x^{2} - 4}{x - 2}$?
    • 4 as the lim
    • 0 as the lim
    • 2 as the lim
    • 1 as the lim

    Correct answer: 4 as the lim

    The top factors into (x-2)(x+2) and the common factor cancels, leaving x plus 2, which at 2 gives 4 as the lim. 0 as the lim notes only that the numerator vanishes at 2 and ignores that the denominator vanishes with it. 2 as the lim reports the point being approached rather than the height approached there. 1 as the lim borrows the rule for limits at infinity, where the leading coefficients would be compared.

79. What is the derivative of $f(x) = 3x^{2} + 2x - 5$?
    • 6x - 5
    • 6x + 2
    • 3x - 5
    • 3x + 2

    Correct answer: 6x + 2

    Taking the power rule term by term turns the squared term into 6x and the linear term into 2, while the constant contributes nothing, so the derivative is 6x + 2. 6x - 5 drags the constant through instead of dropping it. 3x + 2 lowers the exponent but never multiplies the front coefficient by it. 3x - 5 makes both of those slips together, leaving the front coefficient untouched and carrying the constant through as well.

80. What is the integral of $2x \, dx$?
    • 2 times x squared + C
    • half of x squared + C
    • x squared + C
    • 2 times x + C

    Correct answer: x squared + C

    Raising the power of x by one and dividing by that new power turns 2x into x squared, so the antiderivative is x squared + C. 2 times x squared + C raises the power but skips the division by it. half of x squared + C integrates x while forgetting the factor of 2 in the integrand. 2 times x + C hands back the integrand unchanged instead of integrating it.

81. What is the derivative of $f(x) = \sin(x)$?
    • -sin x
    • cos x
    • -cos x
    • tan x

    Correct answer: cos x

    The sine curve climbs fastest where the cosine peaks and levels off where the cosine is zero, so its derivative is cos x. -sin x is what two rounds of differentiation produce, not one. -cos x is the antiderivative of sine rather than its derivative. tan x is the quotient of sine by cosine and is not produced by differentiating either of them.

82. Evaluate the definite integral from 0 to 2 of $x \, dx$.
    • 2 as the net
    • 4 as 2^2 net
    • 1 as the net
    • 3 as the net

    Correct answer: 2 as the net

    The antiderivative of x is x squared over 2, and at 2 that is 4 halved, giving 2 as the net. 4 as 2^2 net squares the upper limit but forgets to divide by the new power. 1 as the net divides the true area again by the width of the interval, which yields the average height instead. 3 as the net attaches a constant of integration, which a definite integral never carries.

83. What is the slope of the tangent line to $f(x) = x^{2}$ at $x = 3$?
    • m is 27
    • m is 18
    • m is 12
    • m is 6

    Correct answer: m is 6

    The derivative of x squared is 2x, and substituting 3 for x gives a tangent slope of m is 6. m is 12 brings the exponent down a second time, using 4x where the rule gives 2x. m is 18 substitutes into 2x squared instead of into 2x, keeping a square that differentiation has already removed. m is 27 raises the power instead of lowering it and reports 3 cubed.

84. Using the chain rule, what is the derivative of $f(x) = (2x + 1)^{3}$?
    • 3(2x+1) to the 2
    • 6(2x+1) to the 3
    • 2(2x+1) to the 3
    • 6(2x+1) to the 2

    Correct answer: 6(2x+1) to the 2

    The chain rule multiplies the outer derivative, which lowers the cube to a square and brings down the 3, by the inner derivative 2, so the result is 6(2x+1) to the 2. 3(2x+1) to the 2 stops before multiplying by the inner derivative. 6(2x+1) to the 3 multiplies by the inner derivative but never lowers the exponent. 2(2x+1) to the 3 applies only the inner derivative and leaves the outer power untouched.

85. What is the derivative of $f(x) = e^{x}$?
    • e to the x
    • x times e to x
    • e to the x - 1
    • x raised to e

    Correct answer: e to the x

    The exponential with base e is the one function that reproduces itself under differentiation, so the derivative is e to the x. x times e to x brings the exponent down as a factor, which the power rule does only when the variable sits in the base. e to the x - 1 lowers the exponent by one, again borrowing the power rule where it does not apply. x raised to e swaps base and exponent and describes a different function altogether.

86. At a critical point where $f'(x) = 0$ and $f''(x) > 0$, the function has a:
    • Maximum point
    • Flexion point
    • Minimum point
    • Plateau point

    Correct answer: Minimum point

    A vanishing first derivative marks a stationary point and a positive second derivative means the curve is concave up there, which together make it a minimum point. Maximum point would need the second derivative to be negative so that the curve bends downward. Flexion point needs the second derivative to change sign through zero rather than to stay positive. Plateau point describes a stationary place where the curve flattens without turning, which concavity strictly upward rules out.

87. What is the derivative of $f(x) = \ln(x)$?
    • ln x as the der
    • x ln x as the der
    • 1/x as the der
    • 1/ln x as the der

    Correct answer: 1/x as the der

    The natural logarithm grows ever more slowly, and its rate of growth at x is the reciprocal of x, so the derivative is 1/x as the der. ln x as the der assumes the logarithm reproduces itself, a property that belongs to the exponential instead. x ln x as the der multiplies by x where the rule divides by it. 1/ln x as the der takes the reciprocal of the function rather than of its input.

88. What is the limit as $x$ approaches infinity of $\frac{3x + 1}{x - 2}$?
    • 1 as the lim
    • 0 as the lim
    • 3 as the lim
    • 2 as the lim

    Correct answer: 3 as the lim

    Top and bottom both grow like x, so far out the quotient settles at the ratio of the leading coefficients, 3 over 1, giving 3 as the lim. 1 as the lim cancels the x terms but drops the coefficient 3 standing in front. 0 as the lim assumes every rational function flattens to zero, which happens only when the bottom has the higher degree. 2 as the lim lifts a constant out of the denominator, and constants stop mattering once x is large.

89. Find the integral of $\cos(x) \, dx$.
    • sin x + C
    • -cos x + C
    • cos x + C
    • -sin x + C

    Correct answer: sin x + C

    Differentiating sine returns cosine, so reversing the step makes the antiderivative sin x + C. -cos x + C is the antiderivative of sine rather than of cosine. cos x + C hands the integrand straight back and would differentiate to a negative sine. -sin x + C differentiates to a negative cosine, carrying the wrong sign throughout.

90. What is the second derivative of $f(x) = x^{4}$?
    • 16x to the 2
    • 4x to the 3
    • 24x to the 1
    • 12x to the 2

    Correct answer: 12x to the 2

    One round of the power rule gives 4x cubed, and a second round drops the exponent to 2 and multiplies the 4 by 3, leaving 12x to the 2. 16x to the 2 multiplies 4 by itself instead of by the reduced exponent 3. 4x to the 3 stops after a single round of differentiation. 24x to the 1 carries on for a third round and lowers the power one step too far.

91. Using the product rule, what is the derivative of $f(x) = x\sin(x)$?
    • cos x + x sin x
    • sin x - x cos x
    • cos x - x sin x
    • sin x + x cos x

    Correct answer: sin x + x cos x

    The product rule differentiates the first factor against the untouched second, then adds the untouched first against the differentiated second, giving sin x + x cos x. cos x + x sin x pairs each factor with the wrong partner. sin x - x cos x assembles the right two pieces but subtracts them, which the product rule never does. cos x - x sin x is the derivative of x times cosine, a different product.

92. What is the area under $f(x) = 3$ from $x = 1$ to $x = 5$?
    • 8 as the sum
    • 14 as the sum
    • 12 as the sum
    • 10 as the sum

    Correct answer: 12 as the sum

    The region is a rectangle standing 3 high over a base 4 units wide, so its area is 12 as the sum. 14 as the sum measures the whole way round that rectangle, twice 3 plus 4, rather than the space enclosed. 10 as the sum adds the two upright sides and the top, 3 and 3 and 4, and so still measures boundary and not area. 8 as the sum adds the height 3 to the right-hand limit 5 instead of multiplying the height by the width.

93. What does the Mean Value Theorem guarantee for a function continuous on [a,b] and differentiable on (a,b)?
    • A point where f' equals the peak height
    • A point where f' equals the zero amount
    • A point where f' equals the last output
    • A point where f' equals the chord slope

    Correct answer: A point where f' equals the chord slope

    The theorem promises an interior place whose instantaneous rate matches the rate along the line joining the two endpoints, which is a point where f' equals the chord slope. A point where f' equals the peak height confuses this with the Extreme Value Theorem, and a rate is not a height in any case. A point where f' equals the zero amount is Rolle's theorem, which additionally requires the two endpoint heights to agree. A point where f' equals the last output compares a rate with a function value at the right-hand end, which the theorem never claims.

94. What is the derivative of $f(x) = \frac{1}{x}$?
    • 1/x to the 2
    • -1/x to the 2
    • -2/x to the 3
    • -1/x to the 3

    Correct answer: -1/x to the 2

    Rewriting the quotient as x to the minus 1 and applying the power rule multiplies by minus 1 and lowers the exponent, giving -1/x to the 2. 1/x to the 2 has the right size but the wrong sign, and a function that falls everywhere cannot have a positive slope. -2/x to the 3 is the second derivative rather than the first. -1/x to the 3 lowers the exponent by two steps instead of one.

95. Evaluate the integral from 1 to 3 of $2x \, dx$.
    • 9 as the net
    • 8 as the net
    • 6 as the net
    • 7 as the net

    Correct answer: 8 as the net

    The antiderivative is x squared, and 9 less 1 leaves 8 as the net. 9 as the net evaluates at the upper limit and forgets to subtract anything at the lower one. 6 as the net substitutes into the integrand 2x rather than into the antiderivative. 7 as the net subtracts the integrand's value at the lower limit, 2, in place of the antiderivative's value there, 1.

96. What is the limit as $x$ approaches 0 of $\frac{\sin(x)}{x}$?
    • 0 as the lim
    • x as the lim
    • 1 as the lim
    • 2 as the lim

    Correct answer: 1 as the lim

    Near zero the sine of a small angle and the angle itself agree to first order, so their quotient settles at 1 as the lim. 0 as the lim reads the vanishing numerator on its own and ignores that the denominator vanishes with it. x as the lim leaves the variable standing where a limit at a point has to be a number. 2 as the lim belongs to the quotient of the sine of twice the angle by the angle, a different member of the same family.

97. A particle's position is $s(t) = t^{2} - 4t$. What is its velocity at $t = 3$?
    • 9 as the vel
    • 2 as the vel
    • 6 as the vel
    • 4 as the vel

    Correct answer: 2 as the vel

    Velocity is the derivative of position, so 2t less 4 evaluated at 3 gives 6 less 4, which is 2 as the vel. 4 as the vel lifts the constant 4 straight out of the position rule without differentiating anything. 6 as the vel differentiates the squared term but never subtracts the 4. 9 as the vel squares the time instead of differentiating at all.

98. The function $f(x) = x^{3} - 3x$ has a local maximum at which x-value?
    • x is -1
    • x is 1
    • x is 3
    • x is -3

    Correct answer: x is -1

    The derivative 3x squared less 3 vanishes at both 1 and -1, and the second derivative 6x is negative only at the left one, so the local maximum sits where x is -1. x is 1 is the other critical point, where the second derivative is positive and the curve turns upward instead. x is 3 lies far to the right of both critical points, on a stretch where the curve only climbs. x is -3 sits equally far to the left, where the curve is also climbing; the 3 in each is the coefficient from the rule, not a solution of anything.

99. What is the derivative of $f(x) = 5$ (a constant)?
    • 0 as the der
    • 5 as the der
    • x as the der
    • 1 as the der

    Correct answer: 0 as the der

    A constant function never rises or falls, so its graph is a flat line whose slope is 0 as the der. 5 as the der copies the constant itself, as though differentiating left the rule untouched. x as the der attaches a variable to the answer, which is the move integration makes rather than differentiation. 1 as the der is the slope of the line y equals x, not of a horizontal line.

100. What is the integral of $x^{2} \, dx$?
     • one third x cubed + C
     • one whole x cubed + C
     • one ninth x cubed + C
     • two times x cubed + C

     Correct answer: one third x cubed + C

     The power rule raises the exponent to three and then divides by that new exponent, which gives one third x cubed + C. One whole x cubed + C raises the exponent but skips the division. One ninth x cubed + C divides by three a second time. Two times x cubed + C multiplies by the original exponent instead of dividing by the new one.

101. Which of the following indicates a function is concave up on an interval?
     • first slope stays positive
     • second rate reads positive
     • second rate goes downwards
     • first slope drops negative

     Correct answer: second rate reads positive

     Concavity is fixed by how the slope itself is changing, so second rate reads positive is the condition for a concave up interval. Second rate goes downwards describes a concave down stretch instead. First slope stays positive only says the function is rising, and first slope drops negative only says it is falling; neither one settles the curvature.

102. What is the area of a circle with radius 7?
     • 14 pi sq
     • 21 pi sq
     • 49 pi sq
     • 28 pi sq

     Correct answer: 49 pi sq

     Area is pi times the radius squared, and 7 squared is 49, so the area is 49 pi sq units. 14 pi sq is the circumference, two pi times the radius. 21 pi sq triples the radius rather than squaring it. 28 pi sq is twice that circumference.

103. What is the sum of the interior angles of a hexagon?
     • 360 deg
     • 540 deg
     • 900 deg
     • 720 deg

     Correct answer: 720 deg

     The interior angles of a polygon with n sides total n minus two, times 180, and six sides give four times 180, or 720 deg. 540 deg is the total for a pentagon. 900 deg is the total for a heptagon. 360 deg is the total for a quadrilateral, and also the exterior angle sum of every polygon.

104. In a right triangle with legs 6 and 8, what is the hypotenuse?
     • 14 un
     • 10 un
     • 20 un
     • 24 un

     Correct answer: 10 un

     Squaring the legs gives 36 and 64, which sum to 100, and the square root of 100 is 10 un. 14 un adds the two legs instead of combining their squares. 24 un is the perimeter of the finished triangle. 20 un doubles the hypotenuse.

105. What is the volume of a cube with edge length 4?
     • 16 cu un
     • 24 cu un
     • 48 cu un
     • 64 cu un

     Correct answer: 64 cu un

     Volume of a cube is the edge cubed, and 4 cubed is 64 cu un. 16 cu un is the area of a single face. 48 cu un is the combined length of the twelve edges. 24 cu un multiplies the edge by the six faces.

106. The measure of each interior angle of a regular pentagon is:
     • 108 deg
     • 120 deg
     • 135 deg
     • 144 deg

     Correct answer: 108 deg

     Each interior angle of a regular polygon is n minus two times 180, divided by n, and for five sides that is 540 over 5, or 108 deg. 120 deg belongs to a regular hexagon. 135 deg belongs to a regular octagon. 144 deg belongs to a regular decagon.

107. What is the distance between points $(1, 2)$ and $(4, 6)$?
     • 3 un
     • 7 un
     • 5 un
     • 9 un

     Correct answer: 5 un

     The horizontal gap is 3 and the vertical gap is 4, so the distance is the square root of 9 plus 16, which is 5 un. 7 un adds the two gaps instead of combining them under a root. 3 un reports the horizontal gap on its own. 9 un squares the horizontal gap and stops there.

108. What is the midpoint of the segment from $(2, 4)$ to $(8, 10)$?
     • (3, 3)
     • (6, 6)
     • (7, 5)
     • (5, 7)

     Correct answer: (5, 7)

     A midpoint averages the two x values and the two y values separately, giving 10 over 2 and 14 over 2, which is (5, 7). (7, 5) reverses those two averages. (3, 3) halves the differences rather than the sums. (6, 6) averages all four coordinates together into one figure.

109. Two triangles are similar with a scale factor of 3. If the smaller has area 5, what is the larger's area?
     • 25 sq un
     • 15 sq un
     • 45 sq un
     • 30 sq un

     Correct answer: 45 sq un

     Area scales by the square of the linear scale factor, so 5 times 3 squared is 5 times 9, or 45 sq un. 15 sq un applies the scale factor once instead of squaring it. 30 sq un doubles that single application. 25 sq un squares the original area rather than the scale factor.

110. What is the circumference of a circle with diameter 10?
     • 10 un
     • 15 pi un
     • 10 pi un
     • 20 pi un

     Correct answer: 10 pi un

     Circumference is pi times the diameter, so a diameter of 10 gives 10 pi un. 10 un drops the factor of pi and hands back the diameter itself. 20 pi un applies two pi times the diameter and so doubles the result. 15 pi un adds the radius to the diameter before multiplying by pi.

111. In a 30-60-90 triangle, if the shortest side is 5, what is the hypotenuse?
     • 10 un
     • 5 un
     • 15 un
     • 20 un

     Correct answer: 10 un

     The sides of a 30-60-90 triangle stand in the ratio one to the square root of three to two, so the hypotenuse is twice the shortest side, which is 10 un. 5 un just repeats the shortest side. 15 un treats the ratio as one to two to three. 20 un multiplies the shortest side by four.

112. What is the volume of a cylinder with radius 3 and height 5?
     • 45 pi cu
     • 15 pi cu
     • 30 pi cu
     • 60 pi cu

     Correct answer: 45 pi cu

     A cylinder holds pi times the radius squared times the height, so pi times 9 times 5 is 45 pi cu units. 15 pi cu leaves the radius unsquared. 30 pi cu is the curved surface area, two pi times radius times height, not a volume. 60 pi cu repeats that curved surface with the diameter standing in for the radius.

113. The equation of a circle centered at $(2, -3)$ with radius 4 is:
     • (x - 2)^2 + (y + 3)^2 = 16
     • (x - 2)^2 + (y - 3)^2 = 16
     • (x + 2)^2 + (y - 3)^2 = 16
     • (x - 2)^2 + (y + 3)^2 = 4

     Correct answer: (x - 2)^2 + (y + 3)^2 = 16

     Standard form subtracts each center coordinate and sets the total equal to the radius squared, so a center at 2 and negative 3 with radius 4 gives (x - 2)^2 + (y + 3)^2 = 16. The version (x - 2)^2 + (y - 3)^2 = 16 keeps the sign of the y coordinate instead of reversing it. The version (x + 2)^2 + (y - 3)^2 = 16 adds each center coordinate where the formula subtracts it, so both brackets carry the wrong sign. The version (x - 2)^2 + (y + 3)^2 = 4 places the radius itself on the right rather than its square.

114. What is the surface area of a sphere with radius 3?
     • 12 pi sq
     • 36 pi sq
     • 18 pi sq
     • 27 pi sq

     Correct answer: 36 pi sq

     Surface area of a sphere is four pi times the radius squared, so four pi times 9 is 36 pi sq units. 12 pi sq leaves the radius unsquared, giving four pi times 3. 18 pi sq covers only half the true surface. 27 pi sq is pi times the radius cubed.

115. In a triangle, two angles measure 50 and 60 degrees. What is the third angle?
     • 70 deg
     • 60 deg
     • 80 deg
     • 90 deg

     Correct answer: 70 deg

     The three angles of a triangle sum to 180, so taking away 50 and 60 leaves 70 deg. 60 deg simply repeats one of the two given angles. 80 deg subtracts 50 twice over. 90 deg assumes the triangle has to be right angled.

116. What is the area of a trapezoid with parallel sides 6 and 10 and height 4?
     • 32 sq un
     • 40 sq un
     • 48 sq un
     • 64 sq un

     Correct answer: 32 sq un

     A trapezoid covers half the sum of the parallel sides times the height, so half of 16 times 4 is 32 sq un. 64 sq un multiplies the full sum of 16 by the height and never halves it. 40 sq un treats the figure as a rectangle on the longer parallel side. 48 sq un averages the parallel sides correctly but then multiplies by the shorter parallel side instead of the height.

117. A central angle of 90 degrees in a circle of radius 4 subtends an arc of what length?
     • 4 pi un
     • 6 pi un
     • 8 pi un
     • 2 pi un

     Correct answer: 2 pi un

     A 90 degree angle covers a quarter of the circle, and the whole circumference is two pi times 4, or 8 pi, so a quarter of that is 2 pi un. 4 pi un takes half the circumference, as though the angle were 180 degrees. 6 pi un takes three quarters, as though the angle were 270 degrees. 8 pi un is the entire circumference.

118. Which transformation preserves both distance and angle measure?
     • scale change
     • rigid motion
     • plane shadow
     • shear effect

     Correct answer: rigid motion

     Only a rigid motion, meaning a translation, a rotation or a reflection, leaves every length and every angle exactly as it was. Scale change resizes the figure, so lengths grow or shrink even though angles hold. Plane shadow flattens the figure onto a line or plane and loses both. Shear effect slides one edge past another, holding area while the angles bend.

119. What is the volume of a cone with radius 3 and height 4?
     • 9 pi cu
     • 12 pi cu
     • 15 pi cu
     • 18 pi cu

     Correct answer: 12 pi cu

     A cone holds one third of pi times the radius squared times the height, so one third of pi times 9 times 4 is 12 pi cu units. 9 pi cu is the base area on its own, with no height applied. 18 pi cu halves the matching cylinder instead of taking a third of it. 15 pi cu is the curved surface, pi times the radius times the slant height of 5.

120. Two angles are complementary. If one is 35 degrees, the other is:
     • 35 deg
     • 45 deg
     • 65 deg
     • 55 deg

     Correct answer: 55 deg

     Complementary angles add to 90, so 90 less 35 leaves 55 deg. 35 deg simply repeats the angle already supplied. 45 deg splits 90 into two equal parts whatever the given angle happens to be. 65 deg subtracts from 100 rather than from 90.

121. What is the slope of a line parallel to $y = -2x + 7$?
     • m = +2.0
     • m = -2.0
     • m = +0.5
     • m = -0.5

     Correct answer: m = -2.0

     Parallel lines carry equal slopes, and the given line has slope negative two, so m = -2.0. m = +2.0 drops the minus sign. m = +0.5 is the perpendicular slope, the negative reciprocal rather than the parallel one. m = -0.5 inverts the slope while leaving its sign alone.

122. A rectangle has a diagonal of 13 and a width of 5. What is its length?
     • 8 un
     • 10 un
     • 14 un
     • 12 un

     Correct answer: 12 un

     The diagonal, the width and the length form a right triangle, so the length is the square root of 169 less 25, which is the square root of 144, or 12 un. 8 un subtracts the width straight from the diagonal. 14 un adds the two squares under the root instead of subtracting. 10 un doubles the width.

123. What is the measure of an exterior angle of a regular octagon?
     • 30 deg
     • 45 deg
     • 36 deg
     • 40 deg

     Correct answer: 45 deg

     Exterior angles of any regular polygon total 360, so one of the eight measures 360 over 8, or 45 deg. 40 deg divides 360 by nine sides. 36 deg divides 360 by ten sides. 30 deg divides 360 by twelve sides.

124. In similar triangles, corresponding sides are 4 and 6. If another side of the first triangle is 10, the corresponding side of the second is:
     • 8 un
     • 12 un
     • 15 un
     • 20 un

     Correct answer: 15 un

     The scale factor from the first triangle to the second is 6 over 4, or one and a half, so a side of 10 becomes 15 un. 12 un adds the difference of 2 rather than scaling. 8 un subtracts that same difference. 20 un doubles the side instead of multiplying by one and a half.

125. What is the law of cosines used to find in a triangle with sides a, b and included angle C?
     • the outside angle
     • the circle radius
     • the perimeter sum
     • the opposite side

     Correct answer: the opposite side

     The law of cosines ties two sides and the angle between them to the third side, so what it delivers is the opposite side. The outside angle needs the rearranged form, which requires all three sides to be known already. The circle radius follows from the law of sines. The perimeter sum cannot be reached until that missing side has been found.

126. A vector has components $(3, 4)$. What is its magnitude?
     • mag 1
     • mag 3
     • mag 5
     • mag 7

     Correct answer: mag 5

     Magnitude is the square root of the sum of the squared components, and the square root of 9 plus 16 is 5, so the answer is mag 5. mag 7 adds the components instead of combining their squares. mag 1 subtracts one component from the other. mag 3 reports the first component on its own.

127. What is the probability of rolling a sum of 7 with two fair six-sided dice?
     • p = 1/9
     • p = 1/7
     • p = 1/6
     • p = 1/3

     Correct answer: p = 1/6

     Six of the 36 equally likely pairs give a sum of seven, and 6 over 36 reduces to p = 1/6. p = 1/9 is the chance of a sum of five, where only four pairs work. p = 1/3 doubles the favorable count to twelve pairs. p = 1/7 reads the target sum itself as the denominator.

128. What is the mean of the data set 4, 8, 6, 10, 2?
     • avg 6
     • avg 10
     • avg 18
     • avg 30

     Correct answer: avg 6

     The five values total 30, and 30 divided by 5 gives avg 6. avg 30 reports the total without dividing at all. avg 18 adds only the two largest values. avg 10 picks out the largest single entry in the set.

129. What is the median of 3, 7, 9, 12, 15?
     • med 7
     • med 3
     • med 5
     • med 9

     Correct answer: med 9

     With five values already in order, the middle position holds 9, so the median is med 9. med 7 takes the middle of only the first three values. med 5 averages the two smallest entries. med 3 reports the smallest value in the list.

130. A bag has 3 red and 5 blue marbles. What is the probability of drawing a red marble?
     • p = 3/8
     • p = 3/5
     • p = 5/8
     • p = 5/3

     Correct answer: p = 3/8

     There are 3 red marbles among 3 plus 5, or 8 altogether, so the probability is p = 3/8. p = 5/8 gives the chance of drawing blue instead. p = 3/5 is the ratio of red to blue rather than red to the whole bag. p = 5/3 inverts that same ratio.

131. What is the mode of 2, 4, 4, 6, 8, 4, 10?
     • mod 5
     • mod 6
     • mod 4
     • mod 8

     Correct answer: mod 4

     The value 4 turns up three times while every other value turns up once, so the mode is mod 4. mod 5 rounds the mean of roughly 5.4. mod 6 takes the middle entry of the list as written, without sorting it first. mod 8 is the range, 10 less 2.

132. If two events are independent with P(A) = 0.5 and P(B) = 0.4, what is P(A and B)?
     • p = 0.45
     • p = 0.20
     • p = 0.90
     • p = 0.70

     Correct answer: p = 0.20

     Independent events multiply, so 0.5 times 0.4 gives p = 0.20. p = 0.90 adds the two probabilities instead of multiplying them. p = 0.70 is the chance that at least one of the events happens, 0.5 plus 0.4 less 0.2. p = 0.45 averages the two probabilities.

133. In how many ways can 4 books be arranged on a shelf?
     • 16 in a row
     • 8 in a row
     • 24 in a row
     • 12 in a row

     Correct answer: 24 in a row

     Four different books can be ordered in 4 factorial ways, that is 4 times 3 times 2 times 1, giving 24 in a row. The value 12 in a row stops after 4 times 3 and drops the last two factors. The value 16 in a row squares the number of books instead of multiplying the descending factors. The value 8 in a row merely doubles the number of books.

134. What is the value of 6 choose 2 (combinations)?
     • 12 two of six
     • 15 two of six
     • 8 two of six
     • 10 two of six

     Correct answer: 15 two of six

     Choosing two items from six gives 6 times 5 divided by 2, so the count is 15 two of six. The value 12 two of six multiplies 6 by 2 rather than using the formula. The value 10 two of six is the count for a group of five, one short of the group named. The value 8 two of six adds 6 and 2 instead of combining them.

135. A fair coin is flipped 3 times. What is the probability of getting exactly 2 heads?
     • 1 out of 8
     • 4 out of 8
     • 3 out of 8
     • 2 out of 8

     Correct answer: 3 out of 8

     Three of the eight equally likely sequences show exactly two heads, so the probability is 3 out of 8. The value 1 out of 8 counts a single named sequence such as head, head, tail. The value 2 out of 8 misses one of the three winning orders. The value 4 out of 8 is the chance of a head on one flip, not of two heads in three flips.

136. What is the range of the data set 12, 5, 18, 7, 20?
     • a gap of 20
     • a gap of 25
     • a gap of 18
     • a gap of 15

     Correct answer: a gap of 15

     Range is the largest value less the smallest, 20 minus 5, which is a gap of 15. A gap of 20 reports the maximum itself rather than a difference. A gap of 25 adds the largest and smallest values instead of subtracting them. A gap of 18 subtracts 2, a misreading of the 12 in the list, rather than the true minimum of 5.

137. In a normal distribution, approximately what percent of data falls within one standard deviation of the mean?
     • 68 out of 100
     • 50 out of 100
     • 27 out of 100
     • 16 out of 100

     Correct answer: 68 out of 100

     The empirical rule places about 68 out of 100 observations inside one standard deviation of the mean. The value 50 out of 100 is the share lying below the mean, a half of the curve rather than a band straddling it. The value 27 out of 100 is the ring between one and two standard deviations, the 95 figure less the 68. The value 16 out of 100 is the single tail beyond one standard deviation on one side.

138. A die is rolled. What is the probability of rolling an even number or a 3?
     • 3 out of 6
     • 5 out of 6
     • 2 out of 6
     • 4 out of 6

     Correct answer: 4 out of 6

     The even faces 2, 4 and 6 together with the face 3 make four winning faces, so the chance is 4 out of 6, which is two thirds. The value 3 out of 6 counts the even faces alone and forgets the 3. The value 2 out of 6 counts only the faces divisible by three. The value 5 out of 6 adds 3 out of 6 to 2 out of 6 without noticing that the face 6 has been counted twice.

139. What is the expected value of rolling a single fair six-sided die?
     • 3.5 per die
     • 6.0 per die
     • 4.0 per die
     • 4.5 per die

     Correct answer: 3.5 per die

     Averaging the six faces gives 21 divided by 6, which is 3.5 per die. The value 4.0 per die rounds that average up to a whole face, but an expected value need not be a possible outcome. The value 4.5 per die follows from totaling the faces as 27 rather than 21. The value 6.0 per die names the largest face instead of the average of all six.

140. A class has 10 boys and 15 girls. What fraction of the class is boys?
     • 15 out of 25
     • 15 out of 10
     • 10 out of 25
     • 10 out of 15

     Correct answer: 10 out of 25

     The class holds 25 students and 10 of them are boys, so the boys are 10 out of 25, which reduces to two fifths. The value 15 out of 25 gives the share of girls instead. The value 10 out of 15 compares boys with girls rather than with the whole class. The value 15 out of 10 inverts that same comparison.

141. If P(A) = 0.7, what is the probability of the complement of A?
     • 0.3 for not A
     • 0.1 for not A
     • 0.4 for not A
     • 0.2 for not A

     Correct answer: 0.3 for not A

     A complement takes whatever probability is left over, 1 minus 0.7, so the answer is 0.3 for not A. The value 0.4 for not A comes from a borrowing slip in that subtraction. The value 0.2 for not A would be right only if the given probability were 0.8. The value 0.1 for not A would follow from reading the given probability as 0.9.

142. How many distinct ways can the letters in the word 'MATH' be arranged?
     • 24 in one row
     • 6 in one row
     • 20 in one row
     • 10 in one row

     Correct answer: 24 in one row

     The four letters of the word are all different, so they can be ordered in 4 factorial ways, giving 24 in one row. The value 20 in one row multiplies 4 by 5, using a factor above the number of letters. The value 10 in one row adds the descending factors rather than multiplying them. The value 6 in one row orders only three of the four letters.

143. A survey reports a sample mean of 50 with standard error 2. A 95% confidence interval is approximately:
     • 48 up to 52
     • 46 up to 54
     • 42 up to 58
     • 45 up to 55

     Correct answer: 46 up to 54

     Roughly two standard errors on each side of the mean gives 50 plus or minus 4, so the interval runs 46 up to 54. The span 48 up to 52 stretches only one standard error each way and covers about two thirds of the sampling distribution. The span 42 up to 58 doubles the margin to four standard errors. The span 45 up to 55 uses a multiplier of 2.5, which is wider than the 95 percent level calls for.

144. What is the probability of drawing an ace from a standard 52-card deck?
     • 13 out of 52
     • 4 out of 52
     • 16 out of 52
     • 12 out of 52

     Correct answer: 4 out of 52

     A deck holds four aces among 52 cards, so the chance is 4 out of 52, which reduces to one in thirteen. The value 12 out of 52 counts the jacks, queens and kings instead of the aces. The value 13 out of 52 is the size of one suit, not the number of aces. The value 16 out of 52 multiplies the four aces by the four suits and so counts each ace four times over.

145. Which measure of center is most resistant to outliers?
     • the middle value
     • the averaged sum
     • the outer spread
     • the scatter size

     Correct answer: the middle value

     A median is reported as the middle value of an ordered list, so one extreme observation shifts it by at most one place. The averaged sum moves directly with every extreme observation added to the data. The outer spread is fixed by the two most extreme observations, so it is entirely at their mercy. The scatter size squares each deviation from center, which magnifies distant points rather than resisting them.

146. Two cards are drawn without replacement from a deck. What is the probability both are hearts?
     • 13 of 52 x 13 of 52
     • 13 of 52 + 12 of 51
     • 13 of 52 x 12 of 51
     • 13 of 52 x 12 of 52

     Correct answer: 13 of 52 x 12 of 51

     After a heart is removed, 12 hearts remain among 51 cards, so the chance of two hearts is 13 of 52 x 12 of 51. The expression 13 of 52 x 13 of 52 describes drawing with the first card put back. The expression 13 of 52 x 12 of 52 reduces the hearts but leaves the deck at full size, which no longer matches a second draw. The expression 13 of 52 + 12 of 51 adds the two chances, which is the rule for either-or events and not for both happening.

147. In a graph theory context, what is the degree of a vertex?
     • how many edges are tied to it
     • how many faces are tied to it
     • how many paths are tied to it
     • how many sides are tied to it

     Correct answer: how many edges are tied to it

     Degree counts the edge ends meeting at a vertex, so the measure is how many edges are tied to it. Counting how many faces are tied to it describes a region count that exists only for a planar drawing, not a property of the vertex. Counting how many paths are tied to it measures routes through the graph, a quantity that can be large even at a vertex of degree one. Counting how many sides are tied to it borrows a polygon idea that has no meaning for a vertex in a graph.

148. What is the next term in the recursive sequence $a_{1} = 2$, $a_{n} = 2a_{n-1} + 1$?
     • a2 is 4
     • a2 is 3
     • a2 is 6
     • a2 is 5

     Correct answer: a2 is 5

     Substituting the first term into the rule gives 2 times 2 plus 1, so a2 is 5. The result a2 is 4 doubles the first term but leaves off the added 1. The result a2 is 3 adds 1 to the first term without doubling it. The result a2 is 6 adds 1 first and doubles afterwards, reversing the order the rule sets out.

149. How many subsets does a set with 4 elements have?
     • 24
     • 8
     • 12
     • 16

     Correct answer: 16

     Every element is either in or out of a given subset, so the tally is 2 raised to the fourth power, which is 16. The reading 8 doubles the element count instead of raising 2 to it. The reading 12 multiplies 4 by 3. The reading 24 counts the orderings of the four elements, which is a different quantity from the number of subsets.

150. In propositional logic, what is the negation of 'p AND q'?
     • p is T or q is T
     • p is T or q is F
     • p is F or q is T
     • p is F or q is F

     Correct answer: p is F or q is F

     De Morgan's law denies each part and turns the connective into a disjunction, so the negation reads p is F or q is F. The form p is T or q is T is the plain disjunction of the two parts with nothing denied at all, and it is true in cases where the original conjunction is also true. The form p is T or q is F denies only the second part and leaves the first standing. The form p is F or q is T denies only the first part and leaves the second standing.

151. What is the sum $1 + 2 + 4 + 8 + 16$ (a geometric series)?
     • a sum of 30
     • a sum of 31
     • a sum of 27
     • a sum of 29

     Correct answer: a sum of 31

     The doubling terms 1, 2, 4, 8 and 16 total 2 raised to the fifth power less 1, which is a sum of 31. A sum of 30 leaves the leading 1 out of the addition. A sum of 29 leaves the 2 out. A sum of 27 leaves the 4 out.

152. A binary number 1011 equals what in decimal?
     • 13 in dec
     • 19 in dec
     • 15 in dec
     • 11 in dec

     Correct answer: 11 in dec

     The four bit columns stand for 8, 4, 2 and 1, and the set bits give 8 plus 2 plus 1, so the value is 11 in dec. The value 13 in dec puts the empty column at the twos place instead of the fours. The value 15 in dec treats every column as filled. The value 19 in dec starts the columns at 16 rather than at 8.

153. In set notation, what is the intersection of $\{1,2,3,4\}$ and $\{3,4,5,6\}$?
     • a set of 1, 2
     • a set of 3, 4
     • a set of 5, 6
     • a set of 1, 6

     Correct answer: a set of 3, 4

     An intersection keeps only the members that both collections hold, and those are 3 and 4, so the answer is a set of 3, 4. The listing a set of 1, 2 gives the members the first collection holds alone. The listing a set of 5, 6 gives the members the second holds alone. The listing a set of 1, 6 takes the smallest and largest of the six values, which describes a spread rather than an overlap.

154. How many edges does a complete graph on 5 vertices (K5) have?
     • 10
     • 15
     • 5
     • 4

     Correct answer: 10

     Each of the 5 corners joins the other 4, and that tally counts every edge from both ends, so halving it gives 10. The reading 15 uses n times n plus 1 over 2, which lets every corner join itself as well as the others. The reading 5 reports the number of corners instead of the number of edges. The reading 4 gives the degree of a single corner rather than a total.

155. What is 5! (5 factorial)?
     • 60
     • 120
     • 720
     • 100

     Correct answer: 120

     Multiplying 5 by 4 by 3 by 2 by 1 gives 120. The reading 60 stops after 5 times 4 times 3 and drops the last two factors. The reading 100 repeats the 5 as a factor and then multiplies by 4, so it uses 5 times 5 times 4 rather than the descending chain. The reading 720 is the factorial of 6, one factor too many.

156. In a truth table, the statement 'if p then q' is false only when:
     • p is T and q is F
     • p is T and q is T
     • p is F and q is T
     • p is F and q is F

     Correct answer: p is T and q is F

     A conditional breaks down only where its hypothesis holds and its conclusion fails, which is the row p is T and q is F. The row p is T and q is T meets the promise the conditional makes. The row p is F and q is T leaves the promise untested, so the conditional counts as true there. The row p is F and q is F is true for the same reason, because a false hypothesis demands nothing of the conclusion.

157. What is the union of sets $A = \{1, 2\}$ and $B = \{2, 3, 4\}$?
     • a set of 1, 2, 2, 3, 4
     • a set of 1, 2, 3 and 4
     • a set of 2 and no more
     • a set of 2, 3 and 4

     Correct answer: a set of 1, 2, 3 and 4

     A union gathers every member appearing in either collection, each written once, giving a set of 1, 2, 3 and 4. The listing a set of 1, 2, 2, 3, 4 writes the shared 2 twice, which set notation never does. The listing a set of 2, 3 and 4 returns the second collection on its own and loses the 1 that only the first holds. The listing a set of 2 and no more gives the overlap instead of the union.

158. Using mathematical induction to prove a statement, the base case typically establishes the statement for:
     • all n, in a huge case
     • n = 1, the small case
     • n = 2, the even case
     • n = top, the end case

     Correct answer: n = 1, the small case

     Induction starts by verifying the claim at the lowest value it is asserted for, usually n = 1, the small case. Verifying all n, in a huge case would finish the proof on its own and leave the inductive step with nothing to do. Verifying n = 2, the even case covers half the values and skips the odd ones. Verifying n = top, the end case works from the largest value down, which reverses the direction the inductive step runs.

159. How many ways can you choose a president and a vice-president from a group of 5 people (order matters)?
     • 10
     • 9
     • 15
     • 20

     Correct answer: 20

     Filling two ranked posts from five people leaves 5 choices then 4, so the tally is 20. The reading 10 counts unordered pairs and so treats the two posts as interchangeable. The reading 9 adds 5 and 4 rather than multiplying them. The reading 15 multiplies 5 by 3, dropping one of the people still available for the second post.

160. The contrapositive of 'if p then q' is:
     • if not p, not q
     • if yes q, yes p
     • if not q, not p
     • if yes p, not q

     Correct answer: if not q, not p

     A contrapositive swaps the two parts and denies each of them, so it reads if not q, not p, and it holds exactly when the original does. The form if not p, not q is the inverse, which denies without swapping and can fail in a case where the original holds. The form if yes q, yes p is the converse, which swaps without denying and makes a claim the original never supports. The form if yes p, not q contradicts the original outright instead of restating it.

161. A teacher wants students to discover the relationship between a quadratic's coefficients and its graph. The most appropriate use of the on-screen graphing calculator is to:
     • let them break each sum and print the prime parts
     • let them work out how big each factorial gets
     • let them shift each term and watch the curve move
     • let them find the spread within a test score set

     Correct answer: let them shift each term and watch the curve move

     Discovery comes from changing a coefficient and seeing the parabola respond, so the strongest use is let them shift each term and watch the curve move. To let them break each sum and print the prime parts is arithmetic drill that never puts a graph on the screen. To let them work out how big each factorial gets belongs to counting problems and shows nothing about a quadratic. To let them find the spread within a test score set is a statistics task with no parabola in it.

162. A student claims that squaring always makes a number larger. Which counterexample best refutes this?
     • 1.5 x 1.5 is 2.25
     • 10 x 10 is 100
     • 0.5 x 0.5 is 0.25
     • -3 x -3 is 9

     Correct answer: 0.5 x 0.5 is 0.25

     Because 0.5 x 0.5 is 0.25 lands below the number that was squared, it is the case the claim forbids and so refutes it. The case 1.5 x 1.5 is 2.25 grows, so it agrees with the claim rather than breaking it. The case 10 x 10 is 100 grows as well. The case -3 x -3 is 9 also ends up larger than the number squared, so the change of sign does not make it a counterexample.