ISEE Upper Level Mathematics Achievement Practice Test: Practice Test 5

Question 1

1. $x^2 + 5x + 6$
2. $x^2 + x - 6$
3. $x^2 + 5x + 7$ (correct answer)
4. $x^2 - x - 6$

Explanation: When you encounter quadratic expressions like these, you're being tested on factoring—specifically whether a quadratic can be written as a product of two binomials with integer coefficients. For an expression $x^2 + bx + c$, you need two integers that multiply to give $c$ and add to give $b$. Let's check each expression systematically. For choice A, $x^2 + 5x + 6$, you need two numbers that multiply to 6 and add to 5. Those numbers are 2 and 3, so this factors as $(x + 2)(x + 3)$. For choice B, $x^2 + x - 6$, you need two numbers that multiply to -6 and add to 1. Those are 3 and -2, giving $(x + 3)(x - 2)$. Choice D, $x^2 - x - 6$, requires two numbers that multiply to -6 and add to -1. Those are -3 and 2, so it factors as $(x - 3)(x + 2)$. However, choice C, $x^2 + 5x + 7$, requires two integers that multiply to 7 and add to 5. Since 7 is prime, its only integer factor pairs are 1×7 and (-1)×(-7). The pair 1 and 7 adds to 8, while -1 and -7 adds to -8. Neither gives us the required sum of 5, making this expression impossible to factor using integer coefficients. The key strategy: when checking if a quadratic factors with integer coefficients, list all factor pairs of the constant term and see if any pair sums to the middle coefficient. If none work, it can't be factored over the integers.

Question 2

1. 144π cubic centimeters
2. 288π cubic centimeters (correct answer)
3. 576π cubic centimeters
4. 2,304π cubic centimeters

Explanation: When you encounter sphere volume problems, remember that you need the radius for the volume formula, even if you're given the diameter. The volume of a sphere is $V = \frac{4}{3}\pi r^3$, where r is the radius. Since the diameter is 12 centimeters, the radius is 6 centimeters (radius = diameter ÷ 2). Substituting into the formula: $V = \frac{4}{3}\pi (6)^3 = \frac{4}{3}\pi (216) = \frac{864\pi}{3} = 288\pi$ cubic centimeters. Let's examine why the other answers are wrong: A) 144π results from using the incorrect formula $V = \pi r^3$ (missing the $\frac{4}{3}$ coefficient) with the correct radius of 6. C) 576π comes from correctly using $\frac{4}{3}\pi r^3$ but mistakenly using the diameter (12) instead of the radius (6) as r: $\frac{4}{3}\pi (12)^3 = \frac{4}{3}\pi (1728) = \frac{6912\pi}{3} = 2304\pi$. Wait, that's answer D. Answer C likely comes from using $4\pi r^3$ (missing the $\frac{1}{3}$) with radius 6. D) 2304π results from the common error of using the diameter instead of radius in the correct volume formula. The correct answer is B) 288π cubic centimeters. Study tip: Always convert diameter to radius immediately when working with spheres. Write "r = d/2" at the start of your work to avoid the most common trap in sphere problems.

Question 3

1. 84
2. 85 (correct answer)
3. 86
4. 87

Explanation: From the stem-and-leaf plot, the scores are: 72, 75, 78, 81, 83, 84, 85, 86, 88, 89, 91, 93, 95. There are 13 scores total. The median is the 7th value (middle value), which is 85.

Question 4

1. The range is 9. (correct answer)
2. The range is 23.
3. The range is 16.
4. The range is 8.

Explanation: This question tests high school mathematics achievement skills: determining range, median, and mode from a data set. The range is the difference between the highest and lowest values, the median is the middle value when data is ordered, and the mode is the most frequently occurring number. In this data set (7, 11, 9, 16, 11, 13, 10, 8, 14, 11), you can find the range by first identifying the highest value (16) and the lowest value (7), then calculating 16 - 7 = 9. Choice A is correct because it accurately reflects the calculation of the range as the difference between the maximum and minimum values. Choice B is incorrect because 23 represents adding the highest and lowest values (16 + 7) rather than subtracting them, a common conceptual error. To help students: Always arrange data from smallest to largest first, clearly identify the maximum and minimum values, then subtract minimum from maximum. Remind students that range measures spread, not sum.

Question 5

1. $\frac{1}{2}$ (correct answer)
2. $\frac{1}{4}$
3. $\frac{3}{8}$
4. $\frac{2}{3}$

Explanation: This problem tests your ability to solve equations with fractions and then perform operations with the solution. When you see an equation like this, your goal is to isolate the variable first, then use that value in the second expression. To solve $\frac{3x}{4} = \frac{9}{16}$, multiply both sides by $\frac{4}{3}$ to isolate $x$: $x = \frac{9}{16} \times \frac{4}{3} = \frac{36}{48} = \frac{3}{4}$ Now substitute $x = \frac{3}{4}$ into $x \times \frac{2}{3}$: $\frac{3}{4} \times \frac{2}{3} = \frac{6}{12} = \frac{1}{2}$ This confirms that A) $\frac{1}{2}$ is correct. Let's examine why the other answers are wrong. B) $\frac{1}{4}$ results from incorrectly calculating $x$ as $\frac{3}{8}$ instead of $\frac{3}{4}$, possibly from mishandling the fraction multiplication. C) $\frac{3}{8}$ comes from finding $x$ correctly but then making an error in the final multiplication, perhaps forgetting to reduce $\frac{6}{16}$ properly. D) $\frac{2}{3}$ represents simply stating the fraction from the second expression without actually performing the multiplication with your solved value of $x$. Remember: multi-step fraction problems require careful attention to each calculation. Always double-check your arithmetic when multiplying fractions, and make sure you're using the correct value of your variable in subsequent expressions. Reducing fractions to lowest terms at each step helps prevent errors.

Question 6

1. $-1 \leq x \leq 7$ (correct answer)
2. $-2 \leq x \leq 8$
3. $x \leq -1$ or $x \geq 7$
4. $x \leq -2$ or $x \geq 8$

Explanation: When you encounter an absolute value inequality like $|2x - 6| \leq 8$, you're dealing with a compound inequality that describes all values within a certain distance from a central point. The absolute value inequality $|A| \leq B$ is equivalent to $-B \leq A \leq B$. This means the expression inside the absolute value bars must be between the negative and positive values of the number on the right side. Applying this rule: $|2x - 6| \leq 8$ becomes $-8 \leq 2x - 6 \leq 8$. Now solve both parts of this compound inequality by adding 6 to all three parts: $-8 + 6 \leq 2x \leq 8 + 6$, which gives us $-2 \leq 2x \leq 14$. Finally, divide everything by 2: $-1 \leq x \leq 7$. Choice A gives us exactly this result: $-1 \leq x \leq 7$. Choice B ($-2 \leq x \leq 8$) results from forgetting to divide by 2 in the final step. Choice C ($x \leq -1$ or $x \geq 7$) represents the solution to $|2x - 6| \geq 8$ instead of $|2x - 6| \leq 8$ - this is the common error of confusing "less than or equal to" with "greater than or equal to" in absolute value inequalities. Choice D makes both mistakes: using the wrong inequality direction and forgetting to divide by 2. Remember: $|A| \leq B$ means "between" (compound inequality with "and"), while $|A| \geq B$ means "outside" (compound inequality with "or").

Question 7

1. $40$ (correct answer)
2. $38$
3. $42$
4. $41$

Explanation: When you encounter a sequence like this, you need to look for the pattern by examining the differences between consecutive terms. Don't assume it's a simple arithmetic sequence where you just add the same number each time. Let's find the differences between consecutive terms: $8 - 5 = 3$, $13 - 8 = 5$, $20 - 13 = 7$, and $29 - 20 = 9$. The differences are $3, 5, 7, 9$—these are consecutive odd numbers! This means each difference increases by $2$. Following this pattern, the next difference should be $9 + 2 = 11$. Therefore, the term after $29$ is $29 + 11 = 40$. Choice A ($40$) is correct because it follows the established pattern of adding consecutive odd numbers. Choice B ($38$) would result from adding $9$ again ($29 + 9 = 38$), which assumes the differences stay constant—but we've established they increase by $2$ each time. Choice C ($42$) might come from adding $13$ ($29 + 13 = 42$), which skips the next difference in our odd number sequence. Choice D ($41$) could result from adding $12$ ($29 + 12 = 41$), which might seem logical if you incorrectly think the differences increase by $3$ each time instead of $2$. For sequence problems, always examine the differences between terms first. If those differences don't form a clear pattern, look at the differences between the differences. Many ISEE sequences involve patterns in these "second differences" rather than simple arithmetic progressions.

Question 8

1. Company X provides more predictable response times due to its lower median value overall
2. Company Y provides more predictable response times due to its smaller interquartile range of values (correct answer)
3. Both companies provide equally predictable service since their median values are very similar
4. Predictability cannot be determined without knowing the minimum and maximum response times for each company

Explanation: When analyzing box plots to determine predictability or consistency in data, you need to focus on measures of spread rather than central tendency. Predictability in response times means less variation - customers can more reliably expect similar service times. The key measure here is the interquartile range (IQR), which captures the spread of the middle 50% of the data. Company X has an IQR of $60 - 30 = 30$ seconds, while Company Y has an IQR of $55 - 40 = 15$ seconds. Company Y's smaller IQR means their response times are more tightly clustered around the median, making their service more predictable. Choice A incorrectly focuses on the median value. While Company X does have a lower median (45 vs 50 seconds), a lower average response time doesn't indicate predictability - it just means faster service on average. Predictability is about consistency, not speed. Choice C makes the error of comparing medians to assess predictability. The medians being similar (45 vs 50 seconds) tells us about typical response times, but nothing about how much variation customers should expect. Choice D suggests you need the minimum and maximum values, but this is incorrect. While the full range could provide additional context, the IQR already gives you the essential information about spread in the central portion of the data, which is more reliable than extreme values. Remember: when questions ask about predictability, consistency, or reliability in data, look for measures of spread like IQR or standard deviation, not measures of central tendency like mean or median.

Question 9

1. $42$ feet below sea level (correct answer)
2. $38$ feet below sea level
3. $48$ feet below sea level
4. $52$ feet below sea level

Explanation: When you encounter problems involving movement above and below a reference point like sea level, think of this as working with positive and negative integers on a number line. Descending means moving in the negative direction (below sea level), while ascending means moving in the positive direction (toward or above sea level). Let's track the submarine's position step by step, starting at sea level (position 0). First, it descends 45 feet, putting it at $-45$ feet. Then it ascends 18 feet: $-45 + 18 = -27$ feet. Next, it descends 27 feet: $-27 - 27 = -54$ feet. Finally, it ascends 12 feet: $-54 + 12 = -42$ feet below sea level. Looking at the wrong answers: Choice B (38 feet) likely comes from incorrectly adding all the movements without considering direction: $45 - 18 - 27 + 12 = 12$, then perhaps subtracting from 50 or making another calculation error. Choice C (48 feet) might result from adding the two descents and subtracting only one ascent: $45 + 27 - 18 = 54$, then making an arithmetic mistake. Choice D (52 feet) could come from adding descents and subtracting ascents but making sign errors: $45 + 27 - 18 - 12 = 42$, then confusing this with 52. The correct answer is A: 42 feet below sea level. Strategy tip: For elevation problems, always establish your reference point (sea level = 0) and consistently use positive values for upward movement and negative values for downward movement. Track your running total after each step to avoid errors.

Question 10

1. $-8$ (correct answer)
2. $-2$
3. $2$
4. $8$

Explanation: When you encounter absolute value expressions with negative numbers, remember that absolute value always gives you the distance from zero, which is always positive or zero. Let's substitute the given values $x = -4$ and $y = -6$ into the expression $|x - y| - |x + y|$. First, calculate what's inside each absolute value:

• $x - y = (-4) - (-6) = -4 + 6 = 2$
• $x + y = (-4) + (-6) = -10$

• $|x - y| = |2| = 2$
• $|x + y| = |-10| = 10$

Question 11

1. 7 (correct answer)
2. 8
3. 9
4. 10

Explanation: From the dot plot, the minimum number of books is 2 and the maximum number of books is 9. The range is 9 - 2 = 7.

Question 12

1. $156$ (correct answer)
2. $132$
3. $144$
4. $182$

Explanation: When you encounter a sequence problem that provides an explicit formula, you're being tested on your ability to substitute values and calculate accurately. The key is to carefully use the given formula rather than trying to find patterns in the listed terms. You're given that the sequence follows the formula $a_n = n(n+1)$, where $n$ represents the position of the term. To find the 12th term, substitute $n = 12$ into this formula: $a_{12} = 12(12+1) = 12 \times 13 = 156$ Let's verify this makes sense by checking a few early terms: $a_1 = 1(2) = 2$, $a_2 = 2(3) = 6$, $a_3 = 3(4) = 12$. These match the given sequence, confirming our formula application is correct. Looking at the wrong answers: Choice B (132) likely results from miscalculating $12 \times 11$ instead of $12 \times 13$—a common error when students forget to add 1 to $n$. Choice C (144) equals $12^2$, suggesting someone might have used $n^2$ instead of $n(n+1)$. Choice D (182) could come from arithmetic errors or misapplying the formula entirely. The correct answer is A (156). Strategy tip: In sequence problems with given formulas, always substitute the position number directly into the formula rather than trying to extend the pattern manually. Double-check by verifying the formula works for the first few given terms, then trust your calculation for the requested term.

Question 13

1. 16 (correct answer)
2. 18
3. 20
4. 22

Explanation: When you encounter range problems, remember that range measures the spread of data by finding the difference between the largest and smallest values. Understanding how changes to these extreme values affect the range is crucial for these types of questions. Let's work through this step by step. Initially, you have a set of positive integers with a range of 24, meaning the difference between the largest and smallest values is 24. If we call the smallest value $s$ and the largest value $L$, then $L - s = 24$. Now, the smallest value increases by 5 (becomes $s + 5$) and the largest value decreases by 3 (becomes $L - 3$). The new range is: $(L - 3) - (s + 5) = L - 3 - s - 5 = L - s - 8 = 24 - 8 = 16$. Looking at the wrong answers: Choice B (18) likely comes from subtracting only 5 instead of 8, perhaps forgetting that both endpoints change. Choice C (20) results from subtracting only the decrease in the maximum value (3) from the original range. Choice D (22) occurs if you subtract only the increase in the minimum value (5) but ignore the decrease in the maximum. The correct answer is A) 16. Study tip: When the range changes due to modifications at both endpoints, always track how each change affects the difference. Increases to the minimum value decrease the range, while decreases to the maximum value also decrease the range. The total change is the sum of both individual effects.

Question 14

1. 214
2. 216
3. 224 (correct answer)
4. 232

Explanation: This question tests ISEE Upper Level Mathematics Achievement: Predicting future terms in a sequence. Predicting sequence terms involves identifying patterns and applying appropriate formulas. Arithmetic sequences increase by a constant difference, while geometric sequences grow by a constant ratio. In this specific sequence, 7, 14, 28, 56, 112, ..., the pattern is a geometric sequence with a common ratio of 2. Choice C is correct because it accurately applies the geometric formula to predict the next term, 112 × 2 = 224, reflecting understanding of sequence growth. Choice B is incorrect because it assumes addition instead of multiplication, which often happens when students misidentify the sequence type. To help students: Practice identifying sequence types and applying corresponding formulas. Encourage checking calculations and understanding the pattern contextually. Use real-world examples to reinforce concepts.

Question 15

1. Winning by matching all 3 numbers is more likely.
2. Matching the first drawn number is more likely. (correct answer)
3. They are equally likely.
4. Neither event has a probability that can be found.

Explanation: This question tests upper-level ISEE mathematics skills: calculating probability of single and compound events. Probability is the measure of the likelihood that an event will occur, calculated by dividing the number of favorable outcomes by the total number of possible outcomes. In this scenario, a player chooses 3 numbers from 1 to 10, and 3 are drawn, comparing probability of matching the first versus all 3. Choice B is correct because matching the first is 3/10, far higher than 1/120 for all 3. Choice A is incorrect due to reversing the comparison, demonstrating a common error where students underestimate compound event rarity. To help students: Teach them to carefully analyze whether events are independent or dependent, and practice calculating probabilities using real-life scenarios. Encourage the use of probability trees or diagrams to visualize complex problems and identify potential errors.

Question 16

1. 45
2. 46
3. 47
4. 48 (correct answer)

Explanation: This question tests ISEE Upper Level Mathematics Achievement: Predicting future terms in a sequence. Predicting sequence terms involves identifying patterns and applying appropriate formulas. Arithmetic sequences increase by a constant difference, while geometric sequences grow by a constant ratio. In this specific sequence, 3, 12, 21, 30, 39, ..., the pattern is an arithmetic sequence with a common difference of 9. Choice D is correct because it accurately applies the arithmetic formula to predict the next term, 39 + 9 = 48, reflecting understanding of sequence growth. Choice A is incorrect because it assumes a smaller difference, which often happens when students average incorrectly. To help students: Practice identifying sequence types and applying corresponding formulas. Encourage checking calculations and understanding the pattern contextually. Use real-world examples to reinforce concepts.

Question 17

1. 0 (correct answer)
2. 1
3. 2
4. 3

Explanation: When you see a complex fraction with multiple operations, you must follow the order of operations (PEMDAS) carefully in both the numerator and denominator before dividing. Let's work through this step by step. In the numerator $24 - 3 \times 2^3$, you first handle the exponent: $2^3 = 8$. Then multiply: $3 \times 8 = 24$. Finally subtract: $24 - 24 = 0$. So the numerator equals 0. In the denominator $4 + 2 \times 3$, you multiply first: $2 \times 3 = 6$. Then add: $4 + 6 = 10$. So the denominator equals 10. Therefore: $\frac{24 - 3 \times 2^3}{4 + 2 \times 3} = \frac{0}{10} = 0$ Choice A (0) is correct because any fraction with 0 in the numerator equals 0, regardless of the denominator value. Choice B (1) likely comes from incorrectly getting the same value for both numerator and denominator, perhaps by making order of operations errors that coincidentally produce equal results. Choice C (2) might result from calculating $24 \div 12$ if you incorrectly simplified the original expression or made computational errors. Choice D (3) could come from various order of operations mistakes, such as working left to right instead of following PEMDAS. Strategy tip: Always work out the numerator and denominator completely and separately before dividing. Double-check your exponents and multiplication before moving to addition and subtraction. Remember that any fraction with 0 on top equals 0.

Question 18

1. 9th graders at both schools check out approximately the same number of books per student on average
2. School B's 9th graders must check out significantly more books to compensate for lower grade performance (correct answer)
3. School A's 9th graders are more consistent readers compared to School B's 9th grade students
4. The data suggests that 9th grade represents a critical transition year for developing reading habits

Explanation: Since both schools have the same total number of books checked out across all grades, but School B shows decreases in 6th-8th grades while School A shows steady increases, School B must compensate for this deficit somewhere. The graph shows School B has an increase in 9th grade, so to balance out the lower numbers in earlier grades and match School A's total, School B's 9th graders must check out significantly more books than would be expected from a simple trend.

Question 19

1. 3 (correct answer)
2. 4
3. 5
4. 6

Explanation: The mode is the value that appears most frequently. From the frequency table, the number 3 appears 8 times, which is more than any other number. Therefore, 3 is the mode.

Question 20

1. $-52$
2. $52$
3. $-28$ (correct answer)
4. $28$

Explanation: This question tests integer operation skills for ISEE Upper Level, including addition, subtraction, multiplication, and division of integers. Integer operations are fundamental in mathematics and involve combining numbers using these operations, respecting the rules of arithmetic. In this specific scenario, integers are used to track warehouse inventory changes with hourly shipments, requiring precise calculation to determine the net change after shipping and receiving. The correct answer, choice C ($-28$), accurately applies integer operations: first calculating $-8 × 5 = -40$ for the total shipped, then adding received boxes: $-40 + 12 = -28$, demonstrating understanding of how shipments decrease inventory. A common error, as seen in choice D ($28$), might involve forgetting the negative sign or misinterpreting the problem, which results from not recognizing that shipping out means subtraction. To improve, students should practice integer operations in logistics contexts, ensuring they understand that items shipped out are negative changes. Teachers can use warehouse diagrams or flow charts to help students visualize inventory movements.

Question 21

1. The median is 79. (correct answer)
2. The median is 82.
3. The median is 71.
4. The median is 77.

Explanation: This question tests high school mathematics achievement skills: determining range, median, and mode from a data set. The range is the difference between the highest and lowest values, the median is the middle value when data is ordered, and the mode is the most frequently occurring number. In this data set (64, 71, 85, 90, 77, 82, 68, 75, 88, 92, 71, 79, 83, 71, 86), first order the 15 values: 64, 68, 71, 71, 71, 75, 77, 79, 82, 83, 85, 86, 88, 90, 92, then find the middle (8th) value: 79. Choice A is correct because with 15 values, the median is the 8th value in the ordered list, which is 79. Choice D is incorrect because 77 is the 7th value, not the 8th - this error occurs when students miscount the position or start counting from zero instead of one. To help students: For odd-numbered data sets, the median position is (n+1)/2 = (15+1)/2 = 8. Always verify by counting from both ends to ensure you've found the true middle.

Question 22

1. City A likely has more extreme outliers due to its lower median temperature value
2. City B likely has more extreme outliers since its larger IQR indicates greater overall variation
3. City A likely has more extreme outliers since its smaller IQR with the same range suggests more data in the tails (correct answer)
4. Both cities have identical outlier patterns since they share the same total temperature range

Explanation: When analyzing box-and-whisker plots, understanding the relationship between the interquartile range (IQR) and the overall range helps you identify where data concentrations occur. The IQR captures the middle 50% of data, while the range shows the spread from minimum to maximum values. City A has a smaller IQR (8°F) compared to City B (12°F), but both cities share the same total range. This means City A's middle 50% of temperatures are more tightly clustered around the median, leaving more "room" in the tails of the distribution for extreme values. Since the same total range must be distributed across the data, City A likely has more extreme outliers extending toward the minimum and maximum temperatures. Choice A incorrectly focuses on the median value itself. A lower median doesn't inherently create more outliers—it's the distribution pattern that matters. Choice B makes the common error of assuming larger IQR means more outliers, but larger IQR actually indicates the middle data is more spread out, potentially leaving less extreme variation in the tails. Choice D ignores the crucial relationship between IQR and range—having the same range doesn't guarantee identical outlier patterns when the central tendencies differ. Study tip: Remember that outliers appear in the tails of distributions. When comparing datasets with equal ranges, the one with the smaller IQR (tighter middle clustering) typically has more extreme values pushed toward the minimum and maximum, creating more potential outliers.

Question 23

1. 0.441
2. 0.784 (correct answer)
3. 0.832
4. 0.973

Explanation: When you encounter probability questions asking for "at least" a certain number of successes, you're dealing with binomial probability. The key insight is that "at least two rounds" means exactly two rounds OR exactly three rounds. To find the probability of winning exactly k rounds out of 3, use the binomial formula: $P(X = k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k}$, where n = 3 rounds and p = 0.7. For exactly 2 wins: $P(X = 2) = \binom{3}{2} \cdot (0.7)^2 \cdot (0.3)^1 = 3 \cdot 0.49 \cdot 0.3 = 0.441$ For exactly 3 wins: $P(X = 3) = \binom{3}{3} \cdot (0.7)^3 \cdot (0.3)^0 = 1 \cdot 0.343 \cdot 1 = 0.343$ Therefore, P(at least 2 wins) = 0.441 + 0.343 = 0.784. Choice A (0.441) represents only the probability of winning exactly two rounds—this is a common trap where students forget to include the "exactly three" case. Choice C (0.832) likely comes from calculation errors in the binomial coefficients or powers. Choice D (0.973) is far too high and might result from misunderstanding what "at least two" means or incorrectly calculating 1 minus the probability of winning zero or one round. Remember: when you see "at least" in probability questions, always break it down into separate "exactly" cases and add them together. This systematic approach prevents missing scenarios.

Question 24

1. $k = 0$
2. $k = 2$ (correct answer)
3. $k = 4$
4. $k = 8$

Explanation: When you encounter a linear equation with a parameter that asks "for what value does the equation have no solution," you're dealing with a situation where the equation becomes inconsistent—meaning the variables cancel out but leave behind a false statement. Let's expand and simplify the equation $2(x + 3) = k(x - 1) + 14$. First, distribute on both sides: $2x + 6 = kx - k + 14$. Now collect all terms with $x$ on one side and constants on the other: $2x - kx = -k + 14 - 6$, which gives us $(2 - k)x = -k + 8$. For this equation to have no solution, the coefficient of $x$ must equal zero while the right side is non-zero. This means $2 - k = 0$ (so $k = 2$) and $-k + 8 \neq 0$. When $k = 2$, we get $0 \cdot x = -2 + 8 = 6$, or $0 = 6$, which is impossible. Looking at the wrong answers: Choice A ($k = 0$) gives us $2x = 8$, so $x = 4$—this has a solution. Choice C ($k = 4$) results in $-2x = 4$, so $x = -2$—also a solution. Choice D ($k = 8$) yields $-6x = 0$, so $x = 0$—again, a solution exists. Remember this pattern: a linear equation has no solution when the coefficient of the variable becomes zero but the constant term remains non-zero. Always check that both conditions are met when working with parametric equations.

Question 25

1. $\frac{3}{8}$ (correct answer)
2. $\frac{15}{40}$
3. $\frac{1}{4}$
4. $\frac{15}{60}$

Explanation: When you see "given that" in a probability question, you're dealing with conditional probability. This means you're finding the probability of one event happening when you already know another event has occurred. The question asks for P(plays sports | plays music), which reads as "the probability a student plays sports given that the student plays music." To find this, you use the conditional probability formula: P(A|B) = P(A and B) ÷ P(B). Here, you need P(sports and music) ÷ P(music) = 15% ÷ 40% = $\frac{15}{40} = \frac{3}{8}$. Think of it this way: among the 40% of students who play music, 15% play both sports and music. So $\frac{15}{40}$ of music students also play sports. Looking at the wrong answers: Choice B shows $\frac{15}{40}$, which is the correct calculation before simplifying to lowest terms. While mathematically equivalent to $\frac{3}{8}$, choice A is in simplest form. Choice C gives $\frac{1}{4}$, which you might get by incorrectly using $\frac{15}{60}$ (the percentage who play both divided by those who play sports). Choice D shows $\frac{15}{60}$ exactly, representing the common error of finding P(music | sports) instead of P(sports | music). The key strategy is to carefully identify what's given and what you're looking for. "Given that" tells you the denominator in conditional probability - it's your restricted sample space. Always double-check that you have the conditional relationship in the right direction.