IB Mathematics: Applications and Interpretation Quiz: Proof And Justification
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Proof And JustificationQuestion 1 of 20

A loan of $25,000 is taken out at an annual interest rate of 6% compounded monthly. The monthly payment is 500. A student calculates the loan balance after the first payment as \(25,000 - $500 = $24,500).

What is the fundamental error in the student's reasoning?

The student did not account for the interest that accrued on the loan during the first month.
The student subtracted the payment from the principal, but they should have added it.
The student used the annual interest rate instead of the monthly interest rate in their calculation.
The student's calculation is correct, but only if the payment is made at the very beginning of the month.
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IB Mathematics: Applications and Interpretation Quiz

IB Mathematics: Applications and Interpretation Quiz: Proof And Justification

Practice Proof And Justification in IB Mathematics: Applications and Interpretation with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Proof And Justification, giving you a quick way to practice the rules, question types, and explanations that matter most for IB Mathematics: Applications and Interpretation.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

A loan of $25,000 is taken out at an annual interest rate of 6% compounded monthly. The monthly payment is 500. A student calculates the loan balance after the first payment as \(25,000 - $500 = $24,500).

What is the fundamental error in the student's reasoning?

  1. The student did not account for the interest that accrued on the loan during the first month. (correct answer)
  2. The student subtracted the payment from the principal, but they should have added it.
  3. The student used the annual interest rate instead of the monthly interest rate in their calculation.
  4. The student's calculation is correct, but only if the payment is made at the very beginning of the month.
Explanation: When working with compound interest loans, you must understand that interest accrues continuously on the outstanding balance, and payments are applied after interest has been calculated for that period. The student's error lies in treating this like a simple subtraction problem. In reality, during the first month, the $25,000 loan accumulates interest before the payment is applied. With 6% annual interest compounded monthly, the monthly rate is 6%/12 = 0.5%. So the balance first grows to $25,000 × 1.005 = $25,125, then the $500 payment is subtracted, leaving $24,625 – not $24,500. Looking at the options: Choice A correctly identifies that the student ignored the interest accrual during the first month. Choice B is wrong because you definitely subtract payments from the balance – adding them would increase the debt. Choice C misses the point; while the student should use the monthly rate (0.5%) in calculations, their fundamental error was ignoring interest altogether, not using the wrong rate. Choice D is incorrect because even if the payment were made at the beginning of the month, interest would still accrue on the remaining balance throughout that month. The key insight for IB Math Applications: compound interest problems always require you to apply interest first, then subtract payments. Never simply subtract payments from the original principal. This sequence matters because interest is earned on the full balance before any payment reduces it.

Question 2

A savings account accrues interest based on a geometric sequence. The balance at the end of year 1 is $1050 and at the end of year 2 is $1102.50. A student claims the initial deposit was $1000.

Which calculation would correctly justify the student's claim?

  1. Find the common difference d=1102.50−1050=52.50d = 1102.50 - 1050 = 52.50 and subtract it from the year 1 balance: 1050−52.50=997.501050 - 52.50 = 997.50.
  2. Find the common ratio r=1102.50/1050=1.05r = 1102.50 / 1050 = 1.05 and divide the year 1 balance by it: 1050/1.05=10001050 / 1.05 = 1000. (correct answer)
  3. Find the average of the two balances, (1050+1102.50)/2=1076.25(1050 + 1102.50)/2 = 1076.25, and assume this was the initial deposit.
  4. Find the common ratio r=1102.50/1050=1.05r = 1102.50 / 1050 = 1.05 and multiply the year 1 balance by it: 1050×1.05=1102.501050 \times 1.05 = 1102.50.
Explanation: The problem states the growth follows a geometric sequence. The initial deposit is the term before year 1 (term 0). To find a previous term in a geometric sequence, we divide by the common ratio. First, calculate the common ratio: r=u2/u1=1102.50/1050=1.05r = u_2 / u_1 = 1102.50 / 1050 = 1.05. Then, to find the initial deposit (let's call it u0u_0), we calculate u0=u1/r=1050/1.05=1000u_0 = u_1 / r = 1050 / 1.05 = 1000. This justifies the student's claim. Choice A incorrectly uses an arithmetic sequence method. Choice C is an irrelevant calculation. Choice D calculates the year 2 balance from the year 1 balance, which is correct but does not find the initial deposit.

Question 3

A city's budget for road maintenance is constrained by the inequality 25000L+40000S≤1,500,00025000L + 40000S \le 1,500,000, where LL is the number of kilometres of local roads paved and SS is the number of kilometres of state roads paved. A contractor proposes a plan to pave 30 km of local roads and 20 km of state roads.

Which statement correctly justifies whether the contractor's plan is feasible under the budget?

  1. The plan is feasible because 30+20=5030 + 20 = 50 km is a reasonable amount of road to pave.
  2. The plan is not feasible because 25000(30)+40000(20)=1,550,00025000(30) + 40000(20) = 1,550,000, which is greater than the budget. (correct answer)
  3. The plan is feasible because 25000(30)+40000(20)=1,550,00025000(30) + 40000(20) = 1,550,000, which is approximately equal to the budget.
  4. The plan is not feasible because state roads cost more than local roads, making the plan inefficient.
Explanation: To check the feasibility, we must substitute the proposed values of L=30L=30 and S=20S=20 into the inequality. The total cost would be 25000(30)+40000(20)=750,000+800,000=1,550,00025000(30) + 40000(20) = 750,000 + 800,000 = 1,550,000. The budget constraint is ≤1,500,000\le 1,500,000. Since 1,550,000>1,500,0001,550,000 > 1,500,000, the plan exceeds the budget and is therefore not feasible. Choice B correctly performs this check and draws the right conclusion. Choice A ignores the costs. Choice C performs the calculation correctly but incorrectly concludes the plan is feasible. Choice D makes a judgement about efficiency, which is not what the inequality is testing.

Question 4

The height of a thrown ball is modelled by the function h(t)=−4.9t2+15t+2h(t) = -4.9t^2 + 15t + 2, where hh is the height in metres and tt is the time in seconds. A student calculates that the ball hits the ground (h=0h=0) at t=3.19t = 3.19 seconds.

Which of the following statements provides the best justification for the validity of the student's solution?

  1. The solution is correct because solving −4.9t2+15t+2=0-4.9t^2 + 15t + 2 = 0 using the quadratic formula gives exactly t=3.19t = 3.19.
  2. The solution is an incorrect approximation because substituting t=3.19t = 3.19 into h(t)h(t) gives a result significantly different from zero.
  3. The solution is a reasonable approximation because substituting t=3.19t = 3.19 into h(t)h(t) gives a result very close to zero. (correct answer)
  4. The solution is incorrect because time, tt, cannot be a non-integer value in this physical model.
Explanation: To check the solution, we substitute t=3.19t = 3.19 into the model: h(3.19)=−4.9(3.19)2+15(3.19)+2≈−49.88+47.85+2=−0.03h(3.19) = -4.9(3.19)^2 + 15(3.19) + 2 \approx -49.88 + 47.85 + 2 = -0.03. This value is very close to 0. Using a calculator's quadratic solver, the exact root is approximately 3.1915... seconds. Therefore, t=3.19t = 3.19 is not an exact solution but is a very reasonable approximation. A is incorrect because it's not the exact value. B is incorrect because the result is not significantly different from zero. D is incorrect as time is a continuous variable and can be non-integer.

Question 5

A student performs a χ2\chi^2 test for independence between a person's preferred music genre and their primary mode of transport. With a significance level of α=0.05\alpha = 0.05, the calculated p-value is 0.041. The student writes, "Since the p-value is less than the significance level, we reject the null hypothesis. This proves that a person's music preference determines their choice of transport."

What is the most significant error in the student's interpretation?

  1. The conclusion is wrong because the p-value (0.041) is actually greater than the significance level (0.05).
  2. The null hypothesis should have been accepted because the p-value is very small.
  3. The student should have used a t-test for this type of data, not a χ2\chi^2 test.
  4. The test only suggests that the variables are not independent; it does not provide evidence of causation or determination. (correct answer)
Explanation: When you encounter chi-square tests for independence, remember that these tests can only detect association between variables—they cannot establish causation or directional relationships. The fundamental distinction between correlation and causation is crucial in statistical interpretation. The student's statistical procedure is actually correct: with p-value = 0.041 < α = 0.05, rejecting the null hypothesis is the right decision. However, the fatal error lies in the conclusion. A chi-square test for independence only tells us whether two categorical variables are statistically independent. Rejecting the null hypothesis means we have evidence that the variables are associated, but this says nothing about which variable influences the other, or whether either variable "determines" the other. Let's examine why the other options miss the mark. Option A is mathematically incorrect—0.041 is indeed less than 0.05, so the comparison is accurate. Option B demonstrates a fundamental misunderstanding of hypothesis testing; small p-values lead us to reject the null hypothesis, not accept it. Option C incorrectly suggests using a t-test, but chi-square tests are appropriate for examining relationships between categorical variables like music genre and transport mode. The correct answer is D because it identifies the core statistical misconception: confusing association with causation. The test provides evidence that music preference and transport choice are not independent, but cannot prove that one determines the other. Remember this key principle: statistical tests can reveal relationships between variables, but establishing causation requires additional evidence beyond what any single statistical test can provide.

Question 6

The profit, PP, in thousands of dollars, from selling xx hundred items is modelled by P(x)=−x2+14x−40P(x) = -x^2 + 14x - 40. A student finds that the vertex of this quadratic is at x=7x=7 and concludes that selling 700 items guarantees the company will make a profit.

What additional step is required to fully justify the student's conclusion?

  1. Check that the second derivative is negative to confirm it is a maximum.
  2. Check the y-intercept to ensure the company has a positive starting value.
  3. Calculate the profit at the vertex, P(7)P(7), to ensure it is a positive value. (correct answer)
  4. Find the roots of the equation P(x)=0P(x)=0 to determine the break-even points.
Explanation: Finding the vertex at x=7x=7 only identifies the production level that gives the maximum profit. It does not guarantee that this maximum profit is actually positive (i.e., a profit rather than a minimum loss). To justify the conclusion that the company makes a profit, one must calculate the value of the profit at this point: P(7)=−(7)2+14(7)−40=−49+98−40=9P(7) = -(7)^2 + 14(7) - 40 = -49 + 98 - 40 = 9. Since P(7)=9P(7)=9 (representing $9000), which is positive, the conclusion is justified. Choice C correctly identifies this necessary check. Choice A confirms it's a maximum, but not if it's a profit. Choice B is irrelevant to the profit at x=7x=7. Choice D is an alternative way to analyze profitability but C is the direct check of the specific claim about the vertex.

Question 7

A city's budget for road maintenance is constrained by the inequality 25000L+40000S≤1,500,00025000L + 40000S \le 1,500,000, where LL is the number of kilometres of local roads paved and SS is the number of kilometres of state roads paved. A contractor proposes a plan to pave 30 km of local roads and 20 km of state roads.

Which statement correctly justifies whether the contractor's plan is feasible under the budget?

  1. The plan is feasible because 30+20=5030 + 20 = 50 km is a reasonable amount of road to pave.
  2. The plan is not feasible because 25000(30)+40000(20)=1,550,00025000(30) + 40000(20) = 1,550,000, which is greater than the budget. (correct answer)
  3. The plan is feasible because 25000(30)+40000(20)=1,550,00025000(30) + 40000(20) = 1,550,000, which is approximately equal to the budget.
  4. The plan is not feasible because state roads cost more than local roads, making the plan inefficient.
Explanation: To check the feasibility, we must substitute the proposed values of L=30L=30 and S=20S=20 into the inequality. The total cost would be 25000(30)+40000(20)=750,000+800,000=1,550,00025000(30) + 40000(20) = 750,000 + 800,000 = 1,550,000. The budget constraint is ≤1,500,000\le 1,500,000. Since 1,550,000>1,500,0001,550,000 > 1,500,000, the plan exceeds the budget and is therefore not feasible. Choice B correctly performs this check and draws the right conclusion. Choice A ignores the costs. Choice C performs the calculation correctly but incorrectly concludes the plan is feasible. Choice D makes a judgement about efficiency, which is not what the inequality is testing.

Question 8

A student is solving the equation 2(1.5)x=202(1.5)^x = 20. Their first step is to rewrite the equation as (3)x=20(3)^x = 20.

Which statement best justifies why this first step is mathematically incorrect?

  1. The student should have divided both sides by 2 first, leading to 1.5x=101.5^x = 10.
  2. The student cannot multiply the base by the coefficient; the order of operations requires exponents to be calculated first. (correct answer)
  3. The student should have taken the logarithm of both sides as the very first step.
  4. The student incorrectly simplified 2×1.52 \times 1.5; the result should be 3.0, not 3.
Explanation: The step is incorrect because it violates the order of operations (PEMDAS/BODMAS). The expression 2(1.5)x2(1.5)^x means the exponentiation (1.5)x(1.5)^x must be performed before the multiplication by 2. The student incorrectly multiplied the coefficient 2 and the base 1.5. Choice B provides the most fundamental reason for the error. Choice A describes the correct first step but doesn't explain why the student's step is wrong. Choice C describes a valid subsequent step, but not why the first step is flawed. Choice D is irrelevant as 3.0 is the same as 3.

Question 9

The line of best fit for a dataset is given by the equation y=−3.2x+15.7y = -3.2x + 15.7. A student states, "Because the point (2,9.3)(2, 9.3) lies perfectly on this line, it must have been one of the original data points from the survey."

What is the most accurate critique of the student's reasoning?

  1. The reasoning is correct, as any point on the line of best fit must be an original data point.
  2. The reasoning is flawed because the correlation must be perfectly r=−1r=-1 for a data point to be on the line.
  3. The reasoning is flawed because a calculation error was made; (2,9.3)(2, 9.3) does not actually lie on the line.
  4. The reasoning is flawed because the line of best fit is a model and does not necessarily pass through any of the original data points. (correct answer)
Explanation: This question tests your understanding of what a line of best fit (regression line) actually represents in statistical analysis. When you encounter problems about lines of best fit, remember that these are mathematical models created to approximate trends in data, not exact representations of every data point. The student's reasoning is fundamentally flawed because a line of best fit is a model that shows the general trend or relationship between variables, but it doesn't necessarily pass through any of the original data points from the dataset. Even though the point (2,9.3)(2, 9.3) does lie on the line y=−3.2x+15.7y = -3.2x + 15.7 (since −3.2(2)+15.7=9.3-3.2(2) + 15.7 = 9.3), this doesn't mean it was an original data point. The line is calculated using methods like least squares regression to minimize overall distance from all points, creating a "best fit" through the scattered data. Looking at the incorrect answers: (A) is wrong because points on the line of best fit are typically not original data points—the line usually passes between actual data points. (B) incorrectly focuses on correlation coefficient requirements that don't determine whether points lie on the line. (C) is mathematically incorrect since (2,9.3)(2, 9.3) does satisfy the equation. The correct answer is (D) because it identifies the core misconception: confusing a mathematical model with the actual data it represents. Study tip: Remember that regression lines are predictive models, not exact fits. On IB exams, distinguish between what the model shows (trends/predictions) versus what the original data points were.

Question 10

The profit, PP, in thousands of dollars, from selling xx hundred items is modelled by P(x)=−x2+14x−40P(x) = -x^2 + 14x - 40. A student finds that the vertex of this quadratic is at x=7x=7 and concludes that selling 700 items guarantees the company will make a profit.

What additional step is required to fully justify the student's conclusion?

  1. Check that the second derivative is negative to confirm it is a maximum.
  2. Check the y-intercept to ensure the company has a positive starting value.
  3. Calculate the profit at the vertex, P(7)P(7), to ensure it is a positive value. (correct answer)
  4. Find the roots of the equation P(x)=0P(x)=0 to determine the break-even points.
Explanation: Finding the vertex at x=7x=7 only identifies the production level that gives the maximum profit. It does not guarantee that this maximum profit is actually positive (i.e., a profit rather than a minimum loss). To justify the conclusion that the company makes a profit, one must calculate the value of the profit at this point: P(7)=−(7)2+14(7)−40=−49+98−40=9P(7) = -(7)^2 + 14(7) - 40 = -49 + 98 - 40 = 9. Since P(7)=9P(7)=9 (representing $9000), which is positive, the conclusion is justified. Choice C correctly identifies this necessary check. Choice A confirms it's a maximum, but not if it's a profit. Choice B is irrelevant to the profit at x=7x=7. Choice D is an alternative way to analyze profitability but C is the direct check of the specific claim about the vertex.

Question 11

A student is calculating the total amount saved when making monthly contributions that follow a geometric pattern. The contributions are $2 in the first month, $4 in the second month, $8 in the third month, and so on, until the final contribution of $256. The student uses the formula for the sum of a finite geometric series, Sn=u1(rn−1)r−1S_n = \frac{u_1(r^n - 1)}{r-1}, and identifies u1=2u_1 = 2, r=2r = 2, and n=256n = 256.

What is the primary error in the student's justification for their choice of variables?

  1. The variable nn is the number of terms in the series, not the value of the last term. (correct answer)
  2. The first term u1u_1 is incorrect; it should be u1=1u_1 = 1 since the pattern starts from $1.
  3. The first term u1u_1 is incorrect; it should be the last term, 256.
  4. The wrong formula was chosen; an arithmetic series formula should be used.
Explanation: When working with geometric series, you need to carefully distinguish between the parameters in the formula Sn=u1(rn−1)r−1S_n = \frac{u_1(r^n - 1)}{r-1}. Here, u1u_1 is the first term, rr is the common ratio, and crucially, nn represents the number of terms in the series. The student correctly identified u1=2u_1 = 2 (first contribution) and r=2r = 2 (each term doubles), but made a critical error with nn. They set n=256n = 256, confusing the value of the last term with the number of terms. To find the actual number of terms, you need to determine which term equals $256. Since the sequence is $$2, 4, 8, 16, ..., 256$,andeachtermfollows, and each term follows un=2⋅2n−1=2nu_n = 2 \cdot 2^{n-1} = 2^n ,yousolve, you solve 2n=256=282^n = 256 = 2^8 ,giving, giving n=8n = 8 $ terms. Answer A correctly identifies this fundamental error. Answer B is wrong because u_1 = 2 matches the first month's contribution of $2, not $1. Answer C incorrectly suggests using the last term as the first term, which would completely change the series structure. Answer D is incorrect because the contributions clearly form a geometric pattern (each term is double the previous), not an arithmetic one where terms increase by a constant difference. Study tip: Always distinguish between the value of terms and the position of terms in sequences. When using formulas, double-check that each variable represents what you think it does—especially $$n$$, which almost always means "number of terms," not "value of the last term."

Question 12

An investment of $5000 is placed in an account with an annual interest rate of rr, compounded annually. The value of the investment after tt years is given by V(t)=5000(1+r)tV(t) = 5000(1+r)^t. A student claims that for the investment to be worth at least $7000 after 5 years, the interest rate rr must be at least 8%.

How would you justify an evaluation of the student's claim?

  1. The claim is incorrect. An 8% interest rate would result in a value less than $7000.
  2. The claim is correct. An 8% interest rate is the minimum required to reach exactly $7000.
  3. The claim is incorrect. A rate of 7% is sufficient to reach the $7000 target.
  4. The claim is incorrect. An interest rate of just under 7% is sufficient. (correct answer)
Explanation: To check the claim, we first test the 8% rate: V(5) = 5000(1+0.08)^5 \approx \7346.64.Thisisgreaterthan$7000,soan8. This is greater than $7000, so an 8% rate is sufficient, but it might not be the minimum. To find the minimum rate, we solve 7000 = 5000(1+r)^5.Thisgives. This gives 1.4 = (1+r)^5,so, so 1+r = \sqrt[5]{1.4} \approx 1.0696.Thismeans. This means r \approx 0.0696$, or 6.96%. Therefore, a rate of just under 7% is sufficient, and the student's claim that it must be at least 8% is incorrect. D is the most accurate description of the situation.

Question 13

In a survey, the correlation coefficient between hours of weekly exercise (xx) and body mass index (yy) was found to be r=−0.85r = -0.85. A student concludes, "This proves that increasing the hours of weekly exercise causes a person's body mass index to decrease."

What is the primary logical flaw in the student's conclusion?

  1. The correlation coefficient is not strong enough to draw any conclusion about the relationship.
  2. The conclusion is flawed because correlation does not imply causation. (correct answer)
  3. The conclusion is flawed because body mass index cannot be negative.
  4. The conclusion is backwards; a lower body mass index causes people to exercise more.
Explanation: The student is confusing correlation with causation. A strong correlation (like r=−0.85r = -0.85) indicates a strong statistical association between two variables, but it does not prove that one variable causes the change in the other. There could be other lurking variables (e.g., diet, genetics) that influence both exercise habits and BMI. Choice A is incorrect; r=−0.85r = -0.85 is considered a strong correlation. Choice C is irrelevant to the logical flaw. Choice D suggests reverse causation, which is a possibility, but the primary and most certain logical error is the jump from correlation to causation.

Question 14

A student is solving the equation 2(1.5)x=202(1.5)^x = 20. Their first step is to rewrite the equation as (3)x=20(3)^x = 20.

Which statement best justifies why this first step is mathematically incorrect?

  1. The student should have divided both sides by 2 first, leading to 1.5x=101.5^x = 10.
  2. The student cannot multiply the base by the coefficient; the order of operations requires exponents to be calculated first. (correct answer)
  3. The student should have taken the logarithm of both sides as the very first step.
  4. The student incorrectly simplified 2×1.52 \times 1.5; the result should be 3.0, not 3.
Explanation: The step is incorrect because it violates the order of operations (PEMDAS/BODMAS). The expression 2(1.5)x2(1.5)^x means the exponentiation (1.5)x(1.5)^x must be performed before the multiplication by 2. The student incorrectly multiplied the coefficient 2 and the base 1.5. Choice B provides the most fundamental reason for the error. Choice A describes the correct first step but doesn't explain why the student's step is wrong. Choice C describes a valid subsequent step, but not why the first step is flawed. Choice D is irrelevant as 3.0 is the same as 3.

Question 15

The line of best fit for a dataset is given by the equation y=−3.2x+15.7y = -3.2x + 15.7. A student states, "Because the point (2,9.3)(2, 9.3) lies perfectly on this line, it must have been one of the original data points from the survey."

What is the most accurate critique of the student's reasoning?

  1. The reasoning is correct, as any point on the line of best fit must be an original data point.
  2. The reasoning is flawed because the correlation must be perfectly r=−1r=-1 for a data point to be on the line.
  3. The reasoning is flawed because a calculation error was made; (2,9.3)(2, 9.3) does not actually lie on the line.
  4. The reasoning is flawed because the line of best fit is a model and does not necessarily pass through any of the original data points. (correct answer)
Explanation: This question tests your understanding of what a line of best fit (regression line) actually represents in statistical analysis. When you encounter problems about lines of best fit, remember that these are mathematical models created to approximate trends in data, not exact representations of every data point. The student's reasoning is fundamentally flawed because a line of best fit is a model that shows the general trend or relationship between variables, but it doesn't necessarily pass through any of the original data points from the dataset. Even though the point (2,9.3)(2, 9.3) does lie on the line y=−3.2x+15.7y = -3.2x + 15.7 (since −3.2(2)+15.7=9.3-3.2(2) + 15.7 = 9.3), this doesn't mean it was an original data point. The line is calculated using methods like least squares regression to minimize overall distance from all points, creating a "best fit" through the scattered data. Looking at the incorrect answers: (A) is wrong because points on the line of best fit are typically not original data points—the line usually passes between actual data points. (B) incorrectly focuses on correlation coefficient requirements that don't determine whether points lie on the line. (C) is mathematically incorrect since (2,9.3)(2, 9.3) does satisfy the equation. The correct answer is (D) because it identifies the core misconception: confusing a mathematical model with the actual data it represents. Study tip: Remember that regression lines are predictive models, not exact fits. On IB exams, distinguish between what the model shows (trends/predictions) versus what the original data points were.

Question 16

A substance has a half-life of 20 days. A student is asked to find the time it takes for a 200g sample to decay to 25g. The student reasons: "The sample halves to 100g in 20 days, then to 50g in another 20 days, and finally to 25g in another 20 days. So the total time is 20+20+20=6020+20+20 = 60 days."

Which mathematical principle provides the underlying justification for the student's step-by-step method?

  1. The property of exponential functions that for a fixed time interval, the value is multiplied by a constant factor. (correct answer)
  2. The principle of linear decay, where a constant amount is lost in each time period.
  3. The use of an arithmetic sequence to model the remaining mass at fixed time intervals.
  4. The property that the decay time is directly proportional to the amount of mass lost.
Explanation: When you encounter radioactive decay problems, you're dealing with exponential decay, where the rate of change is proportional to the current amount. This creates a specific pattern that's crucial to recognize. The student's method works because of a fundamental property of exponential functions: over equal time intervals, the quantity gets multiplied by the same constant factor. In this half-life problem, every 20 days the mass gets multiplied by 0.5 (or divided by 2). So 200g → 100g → 50g → 25g, with each step taking exactly 20 days. This multiplicative pattern is the hallmark of exponential decay. Option A correctly identifies this multiplicative property of exponential functions. The "constant factor" here is 0.5, applied every 20-day interval. Option B is wrong because radioactive decay isn't linear. In linear decay, you'd subtract the same amount each period (like 200g → 150g → 100g → 50g), but that's not what happens here. The amount lost gets smaller each period (first 100g, then 50g, then 25g). Option C incorrectly suggests an arithmetic sequence, where you add the same value each time. The masses 200, 100, 50, 25 don't form an arithmetic sequence since the differences aren't constant. Option D is incorrect because decay time isn't directly proportional to mass lost. Notice that losing the first 100g takes 20 days, but losing the next 25g also takes 20 days. Remember: Half-life problems always involve exponential decay with a constant multiplicative factor over fixed time periods. When you see step-by-step halving, think exponential properties, not linear relationships.

Question 17

A student is calculating the total amount saved when making monthly contributions that follow a geometric pattern. The contributions are $2 in the first month, $4 in the second month, $8 in the third month, and so on, until the final contribution of $256. The student uses the formula for the sum of a finite geometric series, Sn=u1(rn−1)r−1S_n = \frac{u_1(r^n - 1)}{r-1}, and identifies u1=2u_1 = 2, r=2r = 2, and n=256n = 256.

What is the primary error in the student's justification for their choice of variables?

  1. The variable nn is the number of terms in the series, not the value of the last term. (correct answer)
  2. The first term u1u_1 is incorrect; it should be u1=1u_1 = 1 since the pattern starts from $1.
  3. The first term u1u_1 is incorrect; it should be the last term, 256.
  4. The wrong formula was chosen; an arithmetic series formula should be used.
Explanation: When working with geometric series, you need to carefully distinguish between the parameters in the formula Sn=u1(rn−1)r−1S_n = \frac{u_1(r^n - 1)}{r-1}. Here, u1u_1 is the first term, rr is the common ratio, and crucially, nn represents the number of terms in the series. The student correctly identified u1=2u_1 = 2 (first contribution) and r=2r = 2 (each term doubles), but made a critical error with nn. They set n=256n = 256, confusing the value of the last term with the number of terms. To find the actual number of terms, you need to determine which term equals $256. Since the sequence is $$2, 4, 8, 16, ..., 256$,andeachtermfollows, and each term follows un=2⋅2n−1=2nu_n = 2 \cdot 2^{n-1} = 2^n ,yousolve, you solve 2n=256=282^n = 256 = 2^8 ,giving, giving n=8n = 8 $ terms. Answer A correctly identifies this fundamental error. Answer B is wrong because u_1 = 2 matches the first month's contribution of $2, not $1. Answer C incorrectly suggests using the last term as the first term, which would completely change the series structure. Answer D is incorrect because the contributions clearly form a geometric pattern (each term is double the previous), not an arithmetic one where terms increase by a constant difference. Study tip: Always distinguish between the value of terms and the position of terms in sequences. When using formulas, double-check that each variable represents what you think it does—especially $$n$$, which almost always means "number of terms," not "value of the last term."

Question 18

A substance has a half-life of 20 days. A student is asked to find the time it takes for a 200g sample to decay to 25g. The student reasons: "The sample halves to 100g in 20 days, then to 50g in another 20 days, and finally to 25g in another 20 days. So the total time is 20+20+20=6020+20+20 = 60 days."

Which mathematical principle provides the underlying justification for the student's step-by-step method?

  1. The property of exponential functions that for a fixed time interval, the value is multiplied by a constant factor. (correct answer)
  2. The principle of linear decay, where a constant amount is lost in each time period.
  3. The use of an arithmetic sequence to model the remaining mass at fixed time intervals.
  4. The property that the decay time is directly proportional to the amount of mass lost.
Explanation: When you encounter radioactive decay problems, you're dealing with exponential decay, where the rate of change is proportional to the current amount. This creates a specific pattern that's crucial to recognize. The student's method works because of a fundamental property of exponential functions: over equal time intervals, the quantity gets multiplied by the same constant factor. In this half-life problem, every 20 days the mass gets multiplied by 0.5 (or divided by 2). So 200g → 100g → 50g → 25g, with each step taking exactly 20 days. This multiplicative pattern is the hallmark of exponential decay. Option A correctly identifies this multiplicative property of exponential functions. The "constant factor" here is 0.5, applied every 20-day interval. Option B is wrong because radioactive decay isn't linear. In linear decay, you'd subtract the same amount each period (like 200g → 150g → 100g → 50g), but that's not what happens here. The amount lost gets smaller each period (first 100g, then 50g, then 25g). Option C incorrectly suggests an arithmetic sequence, where you add the same value each time. The masses 200, 100, 50, 25 don't form an arithmetic sequence since the differences aren't constant. Option D is incorrect because decay time isn't directly proportional to mass lost. Notice that losing the first 100g takes 20 days, but losing the next 25g also takes 20 days. Remember: Half-life problems always involve exponential decay with a constant multiplicative factor over fixed time periods. When you see step-by-step halving, think exponential properties, not linear relationships.

Question 19

The height of a thrown ball is modelled by the function h(t)=−4.9t2+15t+2h(t) = -4.9t^2 + 15t + 2, where hh is the height in metres and tt is the time in seconds. A student calculates that the ball hits the ground (h=0h=0) at t=3.19t = 3.19 seconds.

Which of the following statements provides the best justification for the validity of the student's solution?

  1. The solution is correct because solving −4.9t2+15t+2=0-4.9t^2 + 15t + 2 = 0 using the quadratic formula gives exactly t=3.19t = 3.19.
  2. The solution is an incorrect approximation because substituting t=3.19t = 3.19 into h(t)h(t) gives a result significantly different from zero.
  3. The solution is a reasonable approximation because substituting t=3.19t = 3.19 into h(t)h(t) gives a result very close to zero. (correct answer)
  4. The solution is incorrect because time, tt, cannot be a non-integer value in this physical model.
Explanation: To check the solution, we substitute t=3.19t = 3.19 into the model: h(3.19)=−4.9(3.19)2+15(3.19)+2≈−49.88+47.85+2=−0.03h(3.19) = -4.9(3.19)^2 + 15(3.19) + 2 \approx -49.88 + 47.85 + 2 = -0.03. This value is very close to 0. Using a calculator's quadratic solver, the exact root is approximately 3.1915... seconds. Therefore, t=3.19t = 3.19 is not an exact solution but is a very reasonable approximation. A is incorrect because it's not the exact value. B is incorrect because the result is not significantly different from zero. D is incorrect as time is a continuous variable and can be non-integer.

Question 20

In a survey, the correlation coefficient between hours of weekly exercise (xx) and body mass index (yy) was found to be r=−0.85r = -0.85. A student concludes, "This proves that increasing the hours of weekly exercise causes a person's body mass index to decrease."

What is the primary logical flaw in the student's conclusion?

  1. The correlation coefficient is not strong enough to draw any conclusion about the relationship.
  2. The conclusion is flawed because correlation does not imply causation. (correct answer)
  3. The conclusion is flawed because body mass index cannot be negative.
  4. The conclusion is backwards; a lower body mass index causes people to exercise more.
Explanation: The student is confusing correlation with causation. A strong correlation (like r=−0.85r = -0.85) indicates a strong statistical association between two variables, but it does not prove that one variable causes the change in the other. There could be other lurking variables (e.g., diet, genetics) that influence both exercise habits and BMI. Choice A is incorrect; r=−0.85r = -0.85 is considered a strong correlation. Choice C is irrelevant to the logical flaw. Choice D suggests reverse causation, which is a possibility, but the primary and most certain logical error is the jump from correlation to causation.