I85PREP
⌘ K
NYSTCE Math CST study materials
NYSTCE Math CST study resources

NYSTCE Math CST practice questions

One question at a time. Build your NYSTCE Math CST understanding.

First 50 of 200 questions. Each one lists the choices, the correct answer, and an explanation. The checker at the top of this page saves progress in this browser when it is available.

Continue in the practice tool above to load more questions, 50 at a time.

Question 1Mathematical Processes and Applications

A teacher plans a unit on proportional reasoning and wants students to build a mathematical model of a real situation. Which classroom task best exemplifies mathematical modeling as described in the standards for mathematical practice?

  1. Students choose variables, write an equation relating cost to quantity, and revise the model after comparing predictions with data. Correct answer.
  2. Students memorize the cross-multiplication algorithm and complete twenty practice problems.
  3. Students copy definitions of ratio and proportion from the textbook into their notebooks.
  4. Students graph a teacher-provided linear function and identify its slope.

Explanation

Modeling requires cycling through representing a real situation symbolically, computing with the model, interpreting results in context, and revising when predictions diverge from data. Only the option describing variable selection, equation writing, and revision based on data comparison includes that full cycle. The other tasks are procedural or receptive and do not involve building and validating a model.

Question 2Mathematical Processes and Applications

A student claims that because 0.9 repeating has infinitely many 9s, it must be strictly less than 1. Which response is mathematically sound?

  1. The claim is correct because infinity is not a real number.
  2. The claim is incorrect because 0.9 repeating equals 1, as shown by setting x = 0.999... and solving 10x − x = 9. Correct answer.
  3. The claim is correct because no finite truncation equals 1.
  4. The claim is incorrect because 0.9 repeating rounds to 1 under standard rounding rules.

Explanation

A repeating decimal denotes the limit of its truncations, and the algebraic argument with x = 0.999... gives 9x = 9, so x = 1 exactly. The equality is exact, not a rounding convention. Reasoning from finite truncations or from the informality of infinity does not establish the value of the infinite decimal.

Question 3Mathematical Processes and Applications

A student measures the side of a square as 5 cm with a possible error of 0.1 cm. Using differentials, what is the approximate maximum error in the computed area?

  1. 1.0 cm² Correct answer.
  2. 0.25 cm²
  3. 0.1 cm²
  4. 0.5 cm²

Explanation

For A = s², the differential is dA = 2s·ds. Substituting s = 5 and ds = 0.1 gives dA = 2(5)(0.1) = 1.0 cm². This linear approximation estimates how measurement error propagates into the computed area.

Question 4Mathematical Processes and Applications

A researcher reports that a treatment produced a statistically significant improvement at the 0.05 level. Which interpretation is most appropriate?

  1. If there were truly no effect, results at least this extreme would occur about 5% of the time. Correct answer.
  2. The result is practically important because it is statistically significant.
  3. The treatment will improve outcomes for 95% of patients.
  4. There is a 5% chance the treatment is ineffective.

Explanation

A p-value is the probability of observing data at least as extreme as obtained, computed under the assumption that the null hypothesis is true. It is not the probability that the null hypothesis is true, nor a proportion of patients helped. Statistical significance also does not by itself establish practical importance.

Question 5Mathematical Processes and Applications

Which statement correctly distinguishes a parameter from a statistic?

  1. A parameter describes a population, while a statistic is computed from a sample. Correct answer.
  2. A parameter is always larger numerically than the corresponding statistic.
  3. A parameter is computed from sample data, while a statistic describes the population.
  4. A parameter applies only to normal distributions, while a statistic applies to any distribution.

Explanation

Parameters are fixed, usually unknown numerical characteristics of a population, such as μ or p. Statistics are values computed from sample data, such as x̄ or p̂, and are used to estimate parameters. The distinction concerns the source of the data, not relative magnitude or distributional shape.

Question 6Mathematical Processes and Applications

In a geometric sequence, the third term is 12 and the sixth term is 96. What is the common ratio?

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

Explanation

In a geometric sequence, a₆ = a₃·r³, so 96 = 12·r³ and r³ = 8, giving r = 2 for a real common ratio. Dividing the terms directly without accounting for the three-step gap would give the wrong ratio.

Question 7Mathematical Processes and Applications

A student solves a system of two linear equations in two variables and obtains 0 = 5. What does this indicate about the system?

  1. The system has exactly one solution.
  2. The student must have made an arithmetic error, since every linear system has a solution.
  3. The system has infinitely many solutions.
  4. The system is inconsistent and has no solution. Correct answer.

Explanation

Eliminating both variables and reaching a false statement such as 0 = 5 means the equations represent parallel lines with no intersection. Such a system is inconsistent. A true statement like 0 = 0 would indicate dependent equations with infinitely many solutions.

Question 8Mathematical Processes and Applications

A function f is defined on the closed interval [0, 4] and is differentiable on (0, 4). If f(0) = 3 and f(4) = 3, which conclusion follows from Rolle's theorem?

  1. f(x) = 3 for every x in [0, 4].
  2. f has a maximum at x = 2.
  3. f′(x) is constant on (0, 4).
  4. There exists some c in (0, 4) with f′(c) = 0. Correct answer.

Explanation

Rolle's theorem applies because f is continuous on the closed interval, differentiable on the open interval, and takes equal values at the endpoints. It guarantees at least one interior point where the derivative is zero. It does not assert where that point is or that the function is constant.

Question 9Mathematical Processes and Applications

Which reasoning process is deductive rather than inductive?

  1. Measuring several triangles and concluding the angles of any triangle sum to 180°.
  2. Observing that the first ten even numbers are each divisible by 2 and concluding all even numbers are divisible by 2.
  3. Noticing a pattern in a sequence and predicting the next term.
  4. Starting from the definition of an even integer and proving that the square of any even integer is divisible by 4. Correct answer.

Explanation

Deductive reasoning proceeds from established definitions and premises to a conclusion that must follow, as in proving a general claim from the definition of even integers. Inductive reasoning generalizes from observed cases to a probable conclusion. The other options all generalize from specific observations.

Question 10Mathematical Processes and Applications

A student writes a proof that the sum of two odd integers is even as follows: 'Let m = 2a + 1 and n = 2b + 1. Then m + n = 2a + 2b + 2 = 2(a + b + 1), which is even.' Which method of proof is used?

  1. Proof by contradiction
  2. Proof by counterexample
  3. Direct proof Correct answer.
  4. Proof by mathematical induction

Explanation

The argument begins with the definitions of odd integers, performs valid algebraic steps, and reaches the definition of an even integer directly. This is a direct proof. No assumption is negated, no base case or induction step is used, and no counterexample is produced.

Question 11Mathematical Processes and Applications

A population of bacteria doubles every 3 hours. If the initial population is 500, which expression gives the population after t hours?

  1. 500 · 2^(t/3) Correct answer.
  2. 500 · 2^(3t)
  3. 500 · 3^(t/2)
  4. 500 · (1/2)^(t/3)

Explanation

Exponential growth with doubling time d has the form P = P₀ · 2^(t/d). With P₀ = 500 and d = 3, the model is 500 · 2^(t/3). Substituting t = 3 correctly gives 1000.

Question 12Mathematical Processes and Applications

The graph of a function is shifted 2 units left and 3 units down. If the original function is y = f(x), which equation represents the transformed graph?

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

Explanation

A horizontal shift left by h units replaces x with x + h, and a vertical shift down by k units subtracts k from the output. Combining these gives y = f(x + 2) − 3. Signs inside the function act opposite to the direction of the horizontal shift.

Question 13Mathematical Processes and Applications

A teacher wants students to understand why the formula for the area of a circle is πr². Which activity best develops conceptual understanding rather than memorization?

  1. Having students cut a circle into many congruent sectors and rearrange them into a near-parallelogram to see the area relationship. Correct answer.
  2. Having students complete a worksheet of area computations with given radii.
  3. Having students look up the formula in a reference table.
  4. Having students recite the formula daily until it is memorized.

Explanation

Dissecting a circle into sectors and rearranging them approximates a parallelogram whose base is half the circumference and whose height is the radius, yielding πr². This connects the formula to geometric reasoning. Recitation, drill, and lookup build recall or fluency but not understanding of why the formula holds.

Question 14Mathematical Processes and Applications

A student graphs y = x² − 4x + 7 and reports the vertex as (2, 3). Which statement about this result is correct?

  1. The vertex is correct because completing the square gives y = (x − 2)² + 3. Correct answer.
  2. The vertex is correct only if the parabola opens downward.
  3. The vertex is incorrect because the axis of symmetry is x = −2.
  4. The vertex is incorrect because the y-coordinate should be 7.

Explanation

Completing the square on x² − 4x + 7 yields (x − 2)² + 3, so the vertex is (2, 3) and the parabola opens upward. The axis of symmetry is x = 2, and the y-intercept of 7 is not the vertex value.

Question 15Mathematical Processes and Applications

A survey of 200 students finds that 120 own a bicycle and 80 own a skateboard, with 30 owning both. How many own neither?

  1. 20
  2. 50
  3. 30 Correct answer.
  4. 10

Explanation

By inclusion–exclusion, the number owning at least one item is 120 + 80 − 30 = 170. Subtracting from the 200 surveyed leaves 30 who own neither. Failing to subtract the overlap would incorrectly give 0.

Question 16Mathematical Processes and Applications

A fair six-sided die is rolled twice. What is the probability that the sum of the two rolls is 7?

  1. 1/6 Correct answer.
  2. 1/12
  3. 1/36
  4. 7/36

Explanation

There are 36 equally likely ordered outcomes. The six pairs summing to 7 are (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1), so the probability is 6/36 = 1/6. Counting unordered pairs would undercount the favorable outcomes.

Question 17Mathematical Processes and Applications

A data set of home prices includes one house priced far above all the others. Which pair of measures best describes the center and spread of the typical price?

  1. Median and interquartile range Correct answer.
  2. Mean and standard deviation
  3. Mean and range
  4. Mode and variance

Explanation

The median and interquartile range are resistant to extreme values because they depend on position rather than magnitude. With a strong outlier, the mean, standard deviation, and range are all pulled or inflated. The median and IQR therefore describe the typical price and its spread more reliably.

Question 18Mathematical Processes and Applications

A student uses a calculator to evaluate 2.5 × 4.8 and reports 120. Which reasoning best identifies the error?

  1. The answer is reasonable because both factors are greater than 1.
  2. The answer is correct because multiplication always increases the product.
  3. An estimate of 3 × 5 = 15 shows the answer is off by a factor of about 10, indicating a decimal placement error. Correct answer.
  4. The answer is incorrect because the product of two decimals must be less than 1.

Explanation

Estimation provides a quick check on reasonableness: 2.5 × 4.8 is close to 3 × 5 = 15, so 120 is far too large. The likely error is misplacing the decimal point. Estimation is a core mathematical practice for detecting computational errors.

Question 19Mathematical Processes and Applications

A rectangular garden is to be fenced on three sides using a river as the fourth side. If 60 meters of fencing are available, what dimensions maximize the enclosed area?

  1. 12 m by 24 m
  2. 15 m by 30 m Correct answer.
  3. 10 m by 40 m
  4. 20 m by 20 m

Explanation

With width w perpendicular to the river and length l parallel to it, the fencing constraint is 2w + l = 60, so l = 60 − 2w. The area A = w(60 − 2w) is maximized at w = 15 by the vertex formula, giving l = 30 and area 450 m². This is the largest area among the listed options.

Question 20Mathematical Processes and Applications

A student argues: 'If a number is divisible by 6, then it is divisible by 3. The number 9 is divisible by 3, so it must be divisible by 6.' Which logical error is present?

  1. Denying the antecedent
  2. Circular reasoning
  3. Affirming the consequent Correct answer.
  4. Equivocation

Explanation

The valid inference from 'if P then Q' is Q from P, or not P from not Q. Concluding P (divisible by 6) from Q (divisible by 3) affirms the consequent, which is invalid, as the counterexample 9 shows. The other fallacies involve different structural mistakes.

Question 21Mathematical Processes and Applications

In a right triangle, the legs have lengths 5 and 12. A student claims the hypotenuse is 17 by adding the legs. Which response is mathematically correct?

  1. The claim is incorrect; the hypotenuse is 13 because 5² + 12² = 169 and √169 = 13. Correct answer.
  2. The claim is correct because 5 + 12 = 17.
  3. The claim is incorrect; the hypotenuse is 7 because 12 − 5 = 7.
  4. The claim is incorrect; the hypotenuse is 60 because 5 × 12 = 60.

Explanation

The Pythagorean theorem states that the square of the hypotenuse equals the sum of the squares of the legs. Here 25 + 144 = 169, so the hypotenuse is 13. Adding or multiplying the leg lengths does not follow the theorem.

Question 22Mathematical Processes and Applications

A function f is defined by f(x) = (x² − 9)/(x − 3) for x ≠ 3. Which statement about f at x = 3 is correct?

  1. f has a jump discontinuity at x = 3.
  2. f is continuous at x = 3 with value 3.
  3. f has a removable discontinuity at x = 3, and the limit there is 6. Correct answer.
  4. f has a vertical asymptote at x = 3.

Explanation

Factoring the numerator gives (x − 3)(x + 3)/(x − 3) = x + 3 for x ≠ 3, so the limit as x approaches 3 is 6. Because the function is undefined at 3, the graph has a hole, a removable discontinuity. The limit exists, so there is no asymptote or jump.

Question 23Mathematical Processes and Applications

A teacher analyzes student scores on a unit test and finds the distribution is strongly skewed right. Which measure of center should the teacher report as most representative of a typical score?

  1. The maximum score, because it shows the highest achievement.
  2. The range, because it describes the spread of scores.
  3. The median, because it is less affected by the high scores in the tail. Correct answer.
  4. The mean, because it uses every score.

Explanation

In a right-skewed distribution, extreme high values pull the mean above the median. The median better represents a typical score because it depends on position rather than magnitude. The range and maximum describe spread or an extreme rather than center.

Question 24Mathematical Processes and Applications

A student claims that if a sequence converges, then the sequence must be monotonic. Which counterexample refutes the claim?

  1. aₙ = n², which diverges
  2. aₙ = 1/n, which converges and decreases
  3. aₙ = (−1)ⁿ/n, which converges to 0 but alternates in sign Correct answer.
  4. aₙ = n, which diverges

Explanation

A counterexample must be a convergent sequence that is not monotonic. The sequence (−1)ⁿ/n converges to 0 by the squeeze theorem but alternates between positive and negative values, so it is not monotonic. Divergent sequences and monotonic convergent sequences cannot refute the claim.

Question 25Mathematical Processes and Applications

A teacher wants to assess whether students can critique the reasoning of others, a standard for mathematical practice. Which task most directly targets that practice?

  1. Students copy the teacher's worked example into their notes.
  2. Students read a classmate's written solution, identify a flawed step, and explain why it is invalid. Correct answer.
  3. Students memorize a list of problem-solving strategies.
  4. Students solve ten equations independently and check answers in the back of the book.

Explanation

Critiquing the reasoning of others requires analyzing another person's argument, locating an error, and justifying why the step fails. Only the option describing evaluation of a classmate's solution and explanation of the flaw engages that practice. The other tasks involve independent practice, copying, or memorization.

Question 26Mathematical Processes and Applications

A student is asked to estimate the product 49.7 × 3.1 mentally. Which estimate is most reasonable and shows sound number sense?

  1. 50 × 3 = 150 Correct answer.
  2. 40 × 3 = 120
  3. 50 × 4 = 200
  4. 49 × 3 = 147

Explanation

Rounding 49.7 to 50 and 3.1 to 3 keeps both factors close to their actual values, giving a product near 150. The true product is about 154, so this estimate is accurate and efficient for mental computation.

Question 27Mathematical Processes and Applications

A teacher wants students to recognize when a computational result is unreasonable. A student computes 8.4 ÷ 0.2 and writes 4.2. Which check best exposes the error?

  1. The answer 4.2 is less than 8.4, which is impossible when dividing.
  2. The decimal point should be moved one place in both numbers before dividing.
  3. Dividing 8.4 by 0.2 is the same as multiplying 8.4 by 5, so the answer should be 42. Correct answer.
  4. The quotient of two decimals always has more decimal places than the dividend.

Explanation

Dividing by 0.2 is equivalent to multiplying by 5, since 0.2 = 1/5. Thus 8.4 ÷ 0.2 = 42, and the student's answer is off by a factor of ten.

Question 28Mathematical Processes and Applications

A class collects data on the number of minutes 15 students spend on homework. The teacher wants a display that shows the median and the spread of the middle half of the data. Which display is most appropriate?

  1. A box plot Correct answer.
  2. A line graph
  3. A circle graph
  4. A scatter plot

Explanation

A box plot displays the median, quartiles, and interquartile range, which directly represent the middle half of a distribution. It is the standard choice for summarizing spread and center of a single quantitative variable.

Question 29Mathematical Processes and Applications

A student claims that the mean of a data set must equal one of the values in the data set. Which counterexample disproves this claim?

  1. The data set {5} has mean 5, which is in the set.
  2. The data set {3, 3, 3} has mean 3, which is in the set.
  3. The data set {2, 4, 9} has mean 5, which is not in the set.
  4. The data set {1, 2, 3, 4} has mean 2.5, which is not in the set. Correct answer.

Explanation

The mean of {1, 2, 3, 4} is 10 ÷ 4 = 2.5, a value not appearing in the data. This single example refutes the claim that the mean must be a member of the set.

Question 30Mathematical Processes and Applications

A teacher wants students to justify each step when solving 3(x − 2) = 15. Which sequence of justifications is mathematically correct?

  1. Associative property, addition property of equality, division property of equality
  2. Distributive property, addition property of equality, division property of equality Correct answer.
  3. Distributive property, multiplication property of equality, subtraction property of equality
  4. Commutative property, subtraction property of equality, multiplication property of equality

Explanation

Expanding 3(x − 2) uses the distributive property, adding 6 to both sides uses the addition property of equality, and dividing both sides by 3 uses the division property of equality. These three steps correctly transform the equation to x = 7.

Question 31Mathematical Processes and Applications

In a proof, a student writes: 'Assume n is an even integer. Then n = 2k for some integer k. Therefore n² = 4k², which is divisible by 4.' Which type of reasoning does this represent?

  1. Deductive reasoning from definitions Correct answer.
  2. Reasoning by counterexample
  3. Inductive reasoning from examples
  4. Statistical inference

Explanation

The argument begins with a definition of even integers and applies valid logical steps to reach a general conclusion. This is deductive reasoning because the conclusion follows necessarily from the premises.

Question 32Mathematical Processes and Applications

A student is asked to find the x-intercepts of y = x² − 5x + 6. The student factors to (x − 2)(x − 3) and concludes the intercepts are x = −2 and x = −3. Which statement identifies the error?

  1. The student should have divided both sides by x before factoring.
  2. The student should have set y equal to 1 instead of 0.
  3. The student set each factor equal to zero correctly but misread the signs; the intercepts are x = 2 and x = 3. Correct answer.
  4. The student should have used the quadratic formula because the expression does not factor.

Explanation

By the zero product property, each factor equals zero, giving x − 2 = 0 and x − 3 = 0, so x = 2 and x = 3. The student reversed the signs when reading the solutions from the factors.

Question 33Mathematical Processes and Applications

A student is asked to determine whether the relation {(1, 2), (2, 4), (1, 5), (3, 6)} is a function. Which conclusion and reason are correct?

  1. It is not a function because the input 1 is paired with two different outputs. Correct answer.
  2. It is a function because every input has at least one output.
  3. It is not a function because the outputs are not in increasing order.
  4. It is a function because each output appears only once.

Explanation

A relation is a function only if each input value corresponds to exactly one output. Here the input 1 maps to both 2 and 5, so the relation fails the definition.

Question 34Mathematical Processes and Applications

A teacher wants students to understand why the sum of the interior angles of any triangle is 180°. Which activity best supports this understanding?

  1. Tearing off the three angles of a paper triangle and arranging them to form a straight line Correct answer.
  2. Drawing a triangle and measuring each angle with a protractor once
  3. Measuring one triangle and concluding the sum is 180° for all triangles
  4. Memorizing the formula (n − 2) × 180° for polygons

Explanation

Arranging the three torn angles along a straight line physically demonstrates that they form a straight angle of 180°. This builds conceptual understanding that the sum is invariant for all triangles, not just the one measured.

Question 35Mathematical Processes and Applications

A student is asked to solve the inequality −2x + 5 > 11. Which solution and reasoning are correct?

  1. x < −3, because dividing by a negative reverses the inequality symbol Correct answer.
  2. x > −3, because the inequality symbol stays the same when dividing
  3. x > 3, because adding 5 to both sides reverses the symbol
  4. x < 3, because the constant term is moved to the right side

Explanation

Subtracting 5 gives −2x > 6, and dividing both sides by −2 reverses the inequality to x < −3. Reversing the symbol when multiplying or dividing by a negative number preserves the truth of the inequality.

Question 36Mathematical Processes and Applications

A student is asked to find the inverse of f(x) = 3x − 7. The student writes f⁻¹(x) = (x − 7)/3. Which statement identifies the error?

  1. The student's answer is correct because inverses always subtract the constant.
  2. The student should have divided by 7 instead of 3.
  3. The student should have squared both sides before solving.
  4. The student should have added 7 before dividing; the correct inverse is (x + 7)/3. Correct answer.

Explanation

To find an inverse, solve y = 3x − 7 for x: add 7 to get y + 7 = 3x, then divide by 3 to get x = (y + 7)/3. The student subtracted 7 instead of adding it, producing the wrong inverse.

Question 37Mathematical Processes and Applications

A bag contains 4 red and 6 blue marbles. Two marbles are drawn without replacement. What is the probability that both are red?

  1. 2/15 Correct answer.
  2. 1/5
  3. 4/25
  4. 8/45

Explanation

The probability the first marble is red is 4/10, and given that, the probability the second is red is 3/9. Multiplying gives (4/10)(3/9) = 12/90 = 2/15.

Question 38Mathematical Processes and Applications

A city planner models a rectangular park with length 3 meters more than twice its width. If the perimeter is 66 meters, what are the dimensions?

  1. Width 10 m, length 23 m Correct answer.
  2. Width 9 m, length 24 m
  3. Width 11 m, length 22 m
  4. Width 12 m, length 21 m

Explanation

Let w be the width; then the length is 2w + 3. The perimeter equation 2(w + 2w + 3) = 66 simplifies to 6w + 6 = 66, so w = 10 and the length is 23.

Question 39Mathematical Processes and Applications

A teacher asks students to evaluate the claim: 'Every number divisible by 4 is also divisible by 2.' A student responds with the number 12. Which statement best evaluates the student's response?

  1. The response supports the claim but does not prove it, since one example cannot establish a universal statement. Correct answer.
  2. The response proves the claim because 12 is divisible by both 4 and 2.
  3. The response refutes the claim because 12 is even.
  4. The response is irrelevant because divisibility rules apply only to prime numbers.

Explanation

A single confirming example illustrates the claim but cannot establish a universal statement; a proof requires a general argument. Since any multiple of 4 equals 4k = 2(2k), it is always divisible by 2.

Question 40Mathematical Processes and Applications

A data set has a mean of 50 and a standard deviation of 5. A value of 65 is added to the data set. Which statement best describes the position of 65?

  1. It is three standard deviations above the mean. Correct answer.
  2. It is one standard deviation above the mean.
  3. It is fifteen standard deviations above the mean.
  4. It is two standard deviations above the mean.

Explanation

The z-score is (65 − 50)/5 = 3, meaning the value lies three standard deviations above the mean. This is a useful way to compare values across different distributions.

Question 41Mathematical Processes and Applications

A student is asked to describe the transformation that maps y = x² to y = (x + 4)² − 1. Which description is correct?

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

Explanation

Replacing x with x + 4 shifts the graph 4 units left, and subtracting 1 outside the square shifts it 1 unit down. The vertex moves from (0, 0) to (−4, −1).

Question 42Mathematical Processes and Applications

A student is asked to find the limit of (x² − 1)/(x − 1) as x approaches 1. The student substitutes x = 1 and obtains 0/0, then concludes the limit does not exist. Which statement best evaluates this conclusion?

  1. The conclusion is correct because 0/0 is undefined.
  2. The conclusion is correct because the function is undefined at x = 1.
  3. The conclusion is wrong; the limit is 1 because x approaches 1.
  4. The conclusion is wrong; factoring gives (x + 1), so the limit is 2. Correct answer.

Explanation

The form 0/0 is indeterminate, so substitution alone is inconclusive. Factoring the numerator as (x − 1)(x + 1) and canceling x − 1 gives x + 1, whose limit as x approaches 1 is 2.

Question 43Mathematical Processes and Applications

A teacher wants to assess whether students can construct viable arguments and critique the reasoning of others. Which task best targets this practice?

  1. Students watch a video of a proof and copy the steps into their notebooks
  2. Students memorize the statement of the Pythagorean theorem and recite it
  3. Students exchange written proofs of a number property and identify any gaps in each other's reasoning Correct answer.
  4. Students complete a worksheet of twenty computation problems and check answers with a calculator

Explanation

Exchanging proofs and identifying gaps requires students both to construct arguments and to evaluate the reasoning of peers. This directly addresses the practice of critiquing the reasoning of others.

Question 44Mathematical Processes and Applications

A researcher wants to estimate the mean height of students at a large university. Which sampling method is most likely to produce an unbiased estimate?

  1. Selecting 100 students who volunteer to be measured
  2. Selecting the first 100 students who arrive on campus one morning
  3. Selecting 100 students from the basketball team
  4. Selecting 100 students uniformly at random from the full enrollment list Correct answer.

Explanation

A simple random sample gives every student an equal chance of selection, so the sample mean is an unbiased estimator of the population mean. Other methods introduce selection bias related to height or convenience.

Question 45Mathematical Processes and Applications

A student claims that if a function is continuous at x = a, then it must be differentiable at x = a. Which counterexample refutes the claim?

  1. f(x) = |x| at x = 0 Correct answer.
  2. f(x) = 2x + 1 at x = 0
  3. f(x) = x³ at x = 0
  4. f(x) = x² at x = 0

Explanation

The absolute value function is continuous at x = 0 but has a sharp corner there, so its derivative does not exist at that point. This shows continuity does not guarantee differentiability.

Question 46Mathematical Processes and Applications

A student is asked to find the maximum value of f(x) = −x² + 6x − 5 on the interval [0, 5]. Which reasoning correctly identifies the maximum?

  1. The maximum occurs at x = 0 because it is the left endpoint of the interval.
  2. The vertex at x = 3 gives f(3) = 4, which exceeds the endpoint values f(0) = −5 and f(5) = 0. Correct answer.
  3. The function has no maximum because it opens downward.
  4. The maximum occurs at x = 5 because it is the right endpoint of the interval.

Explanation

Completing the square gives f(x) = −(x − 3)² + 4, so the vertex is at x = 3 with value 4. Comparing with the endpoint values f(0) = −5 and f(5) = 0 confirms that 4 is the maximum on the interval.

Question 47Mathematical Processes and Applications

A teacher analyzes a scatter plot showing a strong positive linear association between hours studied and test score, with r = 0.85. A student concludes that studying more hours causes higher scores. Which statement best evaluates this conclusion?

  1. The conclusion is valid because r is close to 1 and the association is strong.
  2. The conclusion is invalid because a positive correlation cannot indicate any relationship.
  3. The conclusion is valid because scatter plots always reveal cause and effect.
  4. Correlation does not establish causation, though the strong association is consistent with a causal relationship that requires further study. Correct answer.

Explanation

A high correlation coefficient indicates a strong linear association but does not by itself demonstrate that one variable causes the other. Confirming causation requires controlled experiments or additional evidence.

Question 48Mathematical Processes and Applications

A student is asked to prove that the square of any odd integer is odd. Which proof strategy is most direct and valid?

  1. Test n = 1, 3, and 5 and observe that each square is odd.
  2. Use the fact that odd numbers are not divisible by 4.
  3. Assume n² is even and derive a contradiction.
  4. Let n = 2k + 1, then n² = 4k² + 4k + 1 = 2(2k² + 2k) + 1, which is odd. Correct answer.

Explanation

Representing an arbitrary odd integer as 2k + 1 and algebraically showing n² has the form 2m + 1 proves the result for all odd integers. This direct proof establishes the general claim rather than illustrating it.

Question 49Mathematical Processes and Applications

A student computes the derivative of f(x) = x² sin x and writes f′(x) = 2x cos x. Which statement identifies the error?

  1. The product rule requires both terms, giving f′(x) = 2x sin x + x² cos x. Correct answer.
  2. The student should have used the quotient rule instead of the product rule.
  3. The derivative of sin x is cos x, so the student's result is correct.
  4. The student should have differentiated only the sine factor.

Explanation

The product rule states that (uv)′ = u′v + uv′, so the derivative is 2x sin x + x² cos x. The student omitted the term in which the power function is left unchanged and only the sine factor is differentiated.

Question 50Number and Quantity

Which of the following numbers is irrational?

  1. √49
  2. √18 Correct answer.
  3. −6
  4. 0.75

Explanation

An irrational number cannot be written as a ratio of two integers. The square root of 18 equals about 4.243, and 18 is not a perfect square, so its root is non-terminating and non-repeating. The other three values are all rational.