Reading and Writing + Math · 99 questions · generated from PSAT1.html
Math is the priority. Reading & Writing sits at 73% while Math is at 55% — the bigger opportunity is in Math.
3 Math questions were left blank. The PSAT has no penalty for wrong answers, so every omitted question is a free guess left on the table.
Grid-in (student-produced response) questions are a weak spot: 7 of 11 were missed — worth targeted practice on the free-response computation questions.
| Section | Module | Correct | Score |
|---|---|---|---|
| Reading and Writing | Module 1 | 20 / 27 | 74% |
| Reading and Writing | Module 2 | 20 / 28 | 71% |
| Math | Module 1 | 14 / 22 | 64% |
| Math | Module 2 | 10 / 22 | 45% |
| # | Section | Item | Your ans. | Correct | |
|---|---|---|---|---|---|
| 1 | Reading and Writing | M1 Q1 | D | B | wrong |
| 2 | Reading and Writing | M1 Q3 | B | A | wrong |
| 3 | Reading and Writing | M1 Q7 | D | A | wrong |
| 4 | Reading and Writing | M1 Q11 | A | C | wrong |
| 5 | Reading and Writing | M1 Q12 | A | B | wrong |
| 6 | Reading and Writing | M1 Q18 | B | A | wrong |
| 7 | Reading and Writing | M1 Q21 | B | D | wrong |
| 8 | Reading and Writing | M2 Q1 | A | B | wrong |
| 9 | Reading and Writing | M2 Q3 | D | C | wrong |
| 10 | Reading and Writing | M2 Q12 | C | D | wrong |
| 11 | Reading and Writing | M2 Q16 | C | A | wrong |
| 12 | Reading and Writing | M2 Q19 | C | B | wrong |
| 13 | Reading and Writing | M2 Q21 | B | A | wrong |
| 14 | Reading and Writing | M2 Q22 | D | C | wrong |
| 15 | Reading and Writing | M2 Q26 | B | A | wrong |
| 16 | Math | M1 Q6 | 8.5 | 5 | wrong |
| 17 | Math | M1 Q9 | 1 | 28 | wrong |
| 18 | Math | M1 Q14 | B | D | wrong |
| 19 | Math | M1 Q16 | D | B | wrong |
| 20 | Math | M1 Q17 | A | C | wrong |
| 21 | Math | M1 Q18 | A | C | wrong |
| 22 | Math | M1 Q19 | A | D | wrong |
| 23 | Math | M1 Q22 | 5 | 59/9, 6.555, 6.556 | wrong |
| 24 | Math | M2 Q4 | B | C | wrong |
| 25 | Math | M2 Q5 | B | D | wrong |
| 26 | Math | M2 Q6 | B | C | wrong |
| 27 | Math | M2 Q10 | B | C | wrong |
| 28 | Math | M2 Q11 | 2 | -.9333,-14/15 | wrong |
| 29 | Math | M2 Q13 | 0 | 289 | wrong |
| 30 | Math | M2 Q15 | blank | D | omit |
| 31 | Math | M2 Q16 | blank | 14.5, 29/2 | omit |
| 32 | Math | M2 Q18 | B | C | wrong |
| 33 | Math | M2 Q19 | B | A | wrong |
| 34 | Math | M2 Q20 | blank | 10 | omit |
| 35 | Math | M2 Q22 | C | D | wrong |
Choice B is the best answer because it most logically completes the text’s discussion of Booker T. Whatley. In this context, "hinder" means hold back or obstruct. The text explains that Whatley encouraged farms to allow customers on site to pick their own produce for a fee. He did so despite critics’ concerns that the customers would never pay to do so. This context establishes that the critics’ concerns didn’t hinder Whatley’s efforts to promote the practice.
Choice A is incorrect. The text indicates that critics’ skepticism of the idea that customers would pay to pick their own produce didn’t have some effect on Whatley’s promotion of the practice. The text illustrates this assertion by describing Whatley’s "determined advocacy" for the practice. This context suggests that critics’ concerns didn’t obstruct Whatley’s efforts, not that critics’ concerns didn’t "enhance," or increase or improve, Whatley’s efforts. Choice C is incorrect because in this context, "misrepresent" would mean portray inaccurately, and the text includes no information relevant to the issue of how Whatley’s efforts were portrayed by critics of the practice of charging customers to pick their own produce. Choice D is incorrect. The text indicates that critics’ skepticism of the idea that customers would pay to pick their own produce didn’t have some effect on Whatley’s promotion of the practice. The text illustrates this assertion by describing Whatley’s "determined advocacy" for the practice. This context suggests that critics’ concerns didn’t obstruct Whatley’s efforts, not that critics’ concerns didn’t "aggravate," or irritate or make more severe, Whatley’s efforts.
Choice A is the best answer because it most logically completes the text’s discussion of Yu’s novel. In this context, “haphazard” means marked by a lack of plan or order. The text indicates that the quest featured in the novel, which involves the protagonist bouncing across time, is chaotic and causes the reader to often wonder what will happen. This context suggests that the protagonist’s journey seems to be marked by a lack of order.
Choice B is incorrect because the text indicates that the journey featured in Yu’s novel involves a character “ricocheting chaotically,” or bouncing in a disordered way, across time and causes the reader to often wonder what will happen. It wouldn’t make sense to say that a chaotic journey seems “premeditated,” or characterized by forethought and planning. Choice C is incorrect because the text doesn’t give any indication that readers regard the journey in Yu’s novel as “inspirational,” or as causing extraordinarily creative or brilliant thoughts or actions; instead, the text focuses on the idea that the protagonist’s journey is chaotic, or disordered, and doesn’t give readers a clear sense of what will happen. Choice D is incorrect. Rather than suggesting that the journey featured in Yu’s novel is “fruitless,” or has an unsuccessful outcome, the text focuses on the idea that while reading about the protagonist’s chaotic movements across time, readers are often unsure of what will happen—that is, they don’t know whether the protagonist will be successful in finding his father.
Choice A is the best answer because it presents an explanation about a helicopter that is directly supported by the text. The text states that Mars’s atmosphere is much less dense than Earth’s, and as a result, the air on Mars doesn’t provide the resistance required to support the blades of a helicopter built for Earth and to keep the helicopter aloft. In other words, a helicopter built for Earth can’t fly on Mars because of the differences in the two planets’ atmospheres.
Choice B is incorrect because instead of stating that the blades of helicopters built for Earth are too large to work on Mars, the text indicates that the helicopter built to fly on Mars actually has even longer blades than a helicopter built for Earth. Choice C is incorrect because the text never addresses the role of gravity on Mars or on Earth; instead, it focuses on atmospheric conditions. Choice D is incorrect because the text doesn’t indicate that helicopters built for Earth are too small to operate in the conditions on Mars. In fact, the text states that the size of the helicopter built to fly on Mars is the same size as a helicopter built for Earth, even though it has longer blades that rotate faster.
Choice C is the best answer because it uses data from the graph to effectively complete the example of Eludoyin and his colleagues’ findings concerning female farmers in some regions of Ondo State, Nigeria. The graph presents values for the percentage of Ondo State small-scale farmers who are female, by type of crop and region. The graph shows that of the farmers mainly cultivating non-root vegetables, approximately 57% in north Ondo and approximately 54% in south Ondo are female; in other words, most of those farmers are female, which exemplifies the idea that female farmers make up the majority (more than half) of the farmers cultivating specific types of crops in some regions.
Choice A is incorrect because it inaccurately cites data from the graph: the graph shows that in south Ondo, most of the farmers mainly cultivating non-root vegetables are women (approximately 54%), but that only about 35% (less than half) of the farmers mainly cultivating cereals are women. Choice B is incorrect because it inaccurately cites data from the graph: the graph shows that more women in central Ondo mainly cultivate cereals than mainly cultivate root crops (approximately 36% and 20%, respectively). Additionally, it doesn’t effectively complete the example because the graph shows that female farmers don’t make up the majority (more than half) of the farmers for any type of crop in central Ondo. Choice D is incorrect because it doesn’t effectively complete the example; it simply states that a relatively equal proportion of women across the three regions mainly cultivate cereals, which doesn’t address the value for that proportion and thus doesn’t show that a majority (more than half) of the farmers cultivating certain crops are female.
Choice B is the best answer because it most logically completes the text’s discussion of Zelda Fitzgerald’s contributions to literature. The text begins by saying that many scholars view Zelda mainly in terms of her marriage to F. Scott Fitzgerald and “don’t recognize Zelda as a writer in her own right.” The text then mentions a novel and “numerous short stories” that she wrote and that such scholars tend to ignore. Therefore, those scholars who focus on Zelda only as an inspiration for F. Scott’s writings risk misrepresenting the full range of Zelda’s contributions to literature.
Choice A is incorrect. Although the text does mention that Zelda Fitzgerald “likely influenced” her husband’s literary work, its focus is on Zelda’s own writing, not on her husband’s writing or factors that might have influenced it. Choice C is incorrect because the text does not discuss F. Scott and Zelda Fitzgerald’s opinions of each other’s works. Choice D is incorrect. Although the text does suggest that F. Scott Fitzgerald’s works were “likely influenced in part” by his marriage to Zelda, it does not discuss autobiographical interpretations of the works of either F. Scott or Zelda.
Choice A is the best answer. The convention being tested here is subject-verb agreement. The singular verb “was” agrees in number with the singular subject “Josephine St. Pierre Ruffin.”
Choice B is incorrect because the plural verb “were” doesn’t agree in number with the singular subject “Josephine St. Pierre Ruffin.” Choice C is incorrect because the plural verb “are” doesn’t agree in number with the singular subject “Josephine St. Pierre Ruffin.” Choice D is incorrect because the plural verb “have been” doesn’t agree in number with the singular subject “Josephine St. Pierre Ruffin.”
Choice D is the best answer. “In fact” logically signals that the information in this sentence—that McFerrin finds the research phase of her work to be just as fascinating as travel—emphasizes and elaborates on the previous sentence’s point that McFerrin regards background research as a rewarding activity.
Choice A is incorrect because “by contrast” illogically signals that the information in this sentence contrasts with the previous sentence’s point about McFerrin’s attitude toward background research. Instead, it emphasizes and elaborates on that point. Choice B is incorrect because “likewise” illogically signals that this sentence merely adds a second, similar point to the previous sentence’s point about McFerrin’s attitude toward background research. Instead, it emphasizes and elaborates on that point. Choice C is incorrect because “besides” illogically signals that this sentence provides a separate point in addition to, or apart from, the previous sentence’s point about McFerrin’s attitude toward background research. Instead, it emphasizes and elaborates on that point.
Choice B is the best answer because it most logically completes the text’s discussion of Ward and colleagues’ findings. As used in this context, “innocuous” means mild or unharmful. The text describes the vibration and warming that Ward and colleagues used to alleviate itching as “harmless applications” and goes on to contrast these applications with another stimulus that actually offers more relief even though it seems to be stronger and “less benign.” This context conveys the idea that vibration and warming were innocuous stimuli.
Choice A is incorrect because the text focuses on a distinction between harmless stimuli and those that seem to be less benign. Nothing in the text suggests that any of the treatments are “deceptive,” or misleading; indeed, even the less effective ones are described as offering some relief. Choice C is incorrect because the text focuses on the amount of relief from itching offered by harmless stimuli and those that seem to be less benign. The text doesn’t suggest that any of these stimuli are “novel,” or original and new; heat, vibration, and electricity aren’t new inventions. Choice D is incorrect because it wouldn’t make sense to describe an application of vibration or warming as “impractical,” or not suitable for use. The text indicates that these harmless applications are useful in that they offer at least some temporary relief.
Choice C is the best answer because it most logically completes the text’s discussion of the legitimacy of the reigns of French monarchs such as Hugh Capet and Henry I. As used in this context, "buttress" means to strengthen or defend. The text indicates that regardless of whether a French monarch’s reign was significant or uneventful, each monarch faced questions about his right to the throne. The text goes on to say that in order to understand the path of a French monarch’s reign, it’s important to understand what contributed to the monarch’s ability to "hold the throne." This context suggests that French monarchs such as Hugh Capet and Henry I had to buttress, or defend, their right to be monarch.
Choice A is incorrect because it wouldn’t make sense in context to discuss factors that enabled a monarch to "disengage," or withdraw his right to the French throne. The text focuses on an examination of people who reigned as French monarchs, not on people who didn’t choose to rule. Choice B is incorrect because it wouldn’t make sense in context to discuss factors that enabled a monarch to "annotate," or add notes to or explain, his right to the French throne. Nothing in the text suggests that the monarchs were writing notes about their right to the throne; instead, faced with questions about the legitimacy of their reign, the monarchs defended their right. Choice D is incorrect. Saying that a monarch who is faced with questions about the legitimacy of his reign was able to "reciprocate" his right to the French throne would mean that he either returned his right to the throne or that he responded in kind to the challenge. Neither of these meanings would make sense in context because the text focuses on people who did reign as French monarchs and defended their right to do so.
Choice D is the best answer because it describes data from the graph that support the researchers’ conclusion that there is a growing interest among CEOs in connecting with more departments in their companies. The graph shows the average number of individuals reporting directly to CEOs during three different time periods: the individuals are divided into managers and department leaders. The average number of department leaders directly reporting to their CEO during the 1991–1995 period was slightly more than three, during the 1996–2001 period it was four, and during the 2001–2008 period it was almost seven. Thus, the average number of department leaders reporting directly to their CEO rose over the three periods studied, which suggests that CEOs were connecting with more departments.
Choice A is incorrect because the average number of managers and department leaders reporting directly to their CEO rose for both categories between the 1991–1995 and 2001–2008 periods; thus, it isn’t true that the average numbers didn’t fluctuate. Choice B is incorrect because the average number of managers reporting directly to their CEO was highest in the 2001–2008 period, not in the 1996–2001 period. Choice C is incorrect. Although it correctly describes a feature of the graph, the observation that more department leaders than managers are reporting to CEOs does not by itself address the question of whether CEOs are connecting with more departments over time—to address that question, one needs to know whether the number of department leaders reporting to CEOs is increasing over time.
Choice A is the best answer because it most logically completes the text. The text explains that the Cantares Mexicanos contains poems about the Aztec Empire from before the Spanish invasion. Furthermore, it indicates that notes in the collection attest that some of these poems predate the Spanish invasion, while some customs depicted are likely Spanish in origin. The implication is that some poems were composed before the invasion but the references to Spanish customs could have come about only after the invasion, and thus that the collection includes content that predates the invasion and also content from after the invasion.
Choice B is incorrect because the text clearly indicates that the collection is in Nahuatl, not Spanish, so the compilers’ unfamiliarity with Spanish is irrelevant to whether the collection contains material composed after the Spanish invasion. Choice C is incorrect because the text mentions only the Aztec Empire and Spain: there is no information about the relationship of Aztec literature to any traditions other than its own or Spain’s. Choice D is incorrect because the text states that some of the poems make “inarguable references” to common Spanish customs, which conflicts with the idea that these references can reasonably be attributed to mere coincidence.
Choice B is the best answer. The convention being tested is punctuation use between sentences. In this choice, the period after "Bay" is used correctly to mark the boundary between one sentence ("On…Bay") and another sentence that begins with a supplementary phrase ("Having…years"). Here, the supplementary phrase beginning with "having" modifies the subject of the second sentence, "the celebrated ship."
Choice A is incorrect. Without a comma preceding it, the conjunction "and" can’t be used in this way to join sentences. Choice C is incorrect because it results in a comma splice. A comma can’t be used in this way to join two sentences. Choice D is incorrect because it results in a run-on sentence. The sentences ("On…Bay" and "having…years") are fused without punctuation and/or a conjunction.
Choice A is the best answer. The convention being tested is punctuation use between sentences. In this choice, the period after "tombs" is used correctly to mark the boundary between one sentence ("Archaeologist...tombs") and another ("Built...nature").
Choice B is incorrect because it results in a comma splice. A comma can’t be used in this way to mark the boundary between sentences. Choice C is incorrect. Without a comma preceding it, the conjunction "and" can’t be used in this way to join the two sentences. Choice D is incorrect because it results in a run-on sentence. The sentences ("Archaeologist...tombs" and "Built...nature") are fused without punctuation and/or a conjunction.
Choice C is the best answer. The conventions being tested are punctuation use between titles and proper nouns and between verbs and integrated quotations. No punctuation is needed to set off the proper noun "Stina Chyn" from the title that describes Chyn, "critic." Because "Stina Chyn" is essential information identifying the "critic," no punctuation is necessary. Further, no punctuation is needed between the verb "claims" and the following quotation because the quotation is integrated into the structure of the sentence.
Choice A is incorrect because no punctuation is needed before or after the proper noun "Stina Chyn." Setting the critic’s name off with commas suggests that it could be removed without affecting the coherence of the sentence, which isn’t the case. Choice B is incorrect because no punctuation is needed before or after the proper noun "Stina Chyn." Setting the critic’s name off with commas suggests that it could be removed without affecting the coherence of the sentence, which isn’t the case. Additionally, no punctuation is needed between "claims" and the integrated quotation. Choice D is incorrect because no punctuation is needed between the verb "claims" and its subject, "critic Stina Chyn." Additionally, no punctuation is needed between the verb "claims" and the integrated quotation.
Choice A is the best answer. “Nevertheless” logically signals that the information in this sentence—that the spacesuits Suttirat Larlarb designed for the film Sunshine were made in standard sizes in a factory—presents a notable exception to Larlarb’s typical approach of custom-fitting garments to actors, which is described in the previous sentence.
Choice B is incorrect because “thus” illogically signals that the information in this sentence is a result or consequence of the previous information about Larlarb’s typical approach of custom-fitting garments to actors. Instead, it presents a notable exception to Larlarb’s typical approach. Choice C is incorrect because “likewise” illogically signals that the information in this sentence is similar to the previous information about Larlarb’s typical approach of custom-fitting garments to actors. Instead, it presents a notable exception to Larlarb’s typical approach. Choice D is incorrect because “moreover” illogically signals that the information in this sentence merely adds to the previous information about Larlarb’s typical approach of custom-fitting garments to actors. Instead, it presents a notable exception to Larlarb’s typical approach.
The correct answer is 5. It's given that the equation 10x+15y=85 represents the situation, where x is the number of on-site training courses, y is the number of online training courses, and 85 is the total number of hours of training courses the apprentice has enrolled in. Therefore, 10x represents the number of hours the apprentice has enrolled in on-site training courses, and 15y represents the number of hours the apprentice has enrolled in online training courses. Since x is the number of on-site training courses and y is the number of online training courses the apprentice has enrolled in, 10 is the number of hours each on-site course takes and 15 is the number of hours each online course takes. Subtracting these numbers gives 15-10, or 5 more hours each online training course takes than each on-site training course.
The correct answer is 28. The given absolute value equation can be rewritten as two linear equations: 4x-4=112 and -4x-4=112, or 4x-4=-112. Adding 4 to both sides of the equation 4x-4=112 results in 4x=116. Dividing both sides of this equation by 4 results in x=29. Adding 4 to both sides of the equation 4x-4=-112 results in 4x=-108. Dividing both sides of this equation by 4 results in x=-27. Therefore, the two values of x-1 are 29-1, or 28, and -27-1, or -28. Thus, the positive value of x-1 is 28.
Alternate approach: The given equation can be rewritten as 4x-1=112, which is equivalent to 4x-1=112. Dividing both sides of this equation by 4 yields x-1=28. This equation can be rewritten as two linear equations: x-1=28 and -x-1=28, or x-1=-28. Therefore, the positive value of x-1 is 28.
Choice D is correct. Let y represent the number of cells per milliliter x hours after the initial observation. Since the number of cells per milliliter doubles every 3 hours, the relationship between x and y can be represented by an exponential equation of the form y=abxk, where a is the number of cells per milliliter during the initial observation and the number of cells per milliliter increases by a factor of b every k hours. It’s given that there were 300,000 cells per milliliter during the initial observation. Therefore, a=300,000. It’s also given that the number of cells per milliliter doubles, or increases by a factor of 2, every 3 hours. Therefore, b=2 and k=3. Substituting 300,000 for a, 2 for b, and 3 for k in the equation y=abxk yields y=300,0002x3. The number of cells per milliliter there will be 15 hours after the initial observation is the value of y in this equation when x=15. Substituting 15 for x in the equation y=300,0002x3 yields y=300,0002153, or y=300,00025. This is equivalent to y=300,00032, or y=9,600,000. Therefore, 15 hours after the initial observation, there will be 9,600,000 cells per milliliter.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice B is correct. For positive values of a and b, ambm=abm, an=a1n, and ajk=ajk. Therefore, the given expression, x9y97, can be rewritten as xy97. This expression is equivalent to xy917, which can be rewritten as xy9·17, or xy97.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
Choice C is correct. It's given that the population of City A increased by 7% from 2015 to 2016. Therefore, the population of City A in 2016 includes 100% of the population of City A in 2015 plus an additional 7% of the population of City A in 2015. This means that the population of City A in 2016 is 107% of the population in 2015. Thus, the population of City A in 2016 is 107100, or 1.07, times the 2015 population. Therefore, the value of k is 1.07.
Choice A is incorrect. This would be the value of k if the population in 2016 was 7% of the population in 2015.
Choice B is incorrect. This would be the value of k if the population in 2016 was 70% of the population in 2015.
Choice D is incorrect. This would be the value of k if the population increased by 70%, not 7%, from 2015 to 2016.
Choice C is correct. A system of two linear equations in two variables, x and y, has no solution if the graphs of the lines represented by the equations in the xy-plane are distinct and parallel. The graphs of two lines in the xy-plane represented by equations in slope-intercept form, y=mx+b, where m and b are constants, are parallel if their slopes, m, are the same and are distinct if their y-coordinates of the y-intercepts, b, are different. In the equations y=16x+3 and y=16x+19, the values of m are each 16, and the values of b are 3 and 19, respectively. Since the slopes of these lines are the same, and the y-coordinates of the y-intercepts are different, it follows that the system of linear equations in choice C has no solution.
Choice A is incorrect. The lines represented by the equations in this system are a vertical line and a horizontal line. Therefore, this system has a solution, 3,5, rather than no solution.
Choice B is incorrect. The two lines represented by these equations have different slopes and the same y-coordinate of the y-intercept. Therefore, this system has a solution, 0,6, rather than no solution.
Choice D is incorrect. The two lines represented by these equations are a horizontal line and a line with a slope of 5 that have the same y-coordinate of the y-intercept. Therefore, this system has a solution, 0,5, rather than no solution.
Choice D is correct. Since w represents the nth term of the sequence and 9 is the first term of the sequence, the value of w is 9 when the value of n is 1. Since each term after the first is 4 times the preceding term, the value of w is 94 when the value of n is 2. Therefore, the value of w is 944, or 942, when the value of n is 3. More generally, the value of w is 94n-1 for a given value of n. Therefore, the equation w=94n-1 gives w in terms of n.
Choice A is incorrect. This equation describes a sequence for which the first term is 36, rather than 9, and each term after the first is 9, rather than 4, times the preceding term.
Choice B is incorrect. This equation describes a sequence for which the first term is 4, rather than 9, and each term after the first is 9, rather than 4, times the preceding term.
Choice C is incorrect. This equation describes a sequence for which the first term is 36, rather than 9.
The correct answer is 599. When the graph of an equation in the form Ax+By=C, where A, B, and C are constants, is translated down k units in the xy-plane, the resulting graph can be represented by the equation Ax+By+k=C. It’s given that the graph of 9x-10y=19 is translated down 4 units in the xy-plane. Therefore, the resulting graph can be represented by the equation 9x-10y+4=19, or 9x-10y-40=19. Adding 40 to both sides of this equation yields 9x-10y=59. The x-coordinate of the x-intercept of the graph of an equation in the xy-plane is the value of x in the equation when y=0. Substituting 0 for y in the equation 9x-10y=59 yields 9x-100=59, or 9x=59. Dividing both sides of this equation by 9 yields x=599. Therefore, the x-coordinate of the x-intercept of the resulting graph is 599. Note that 59/9, 6.555, and 6.556 are examples of ways to enter a correct answer.
Choice C is correct. It's given from the graph that the points 0,7 and 8,0 lie on the line. For two points on a line, x1,y1 and x2,y2, the slope of the line can be calculated using the slope formula m=y2-y1x2-x1. Substituting 0,7 for x1,y1 and 8,0 for x2,y2 in this formula, the slope of the line can be calculated as m=0-78-0, or m=-78. It's also given that the point d,4 lies on the line. Substituting d,4 for x1,y1, 8,0 for x2,y2, and -78 for m in the slope formula yields -78=0-48-d, or -78=-48-d. Multiplying both sides of this equation by 8-d yields -788-d=-4. Expanding the left-hand side of this equation yields -7+78d=-4. Adding 7 to both sides of this equation yields 78d=3. Multiplying both sides of this equation by 87 yields d=247. Thus, the value of d is 247.
Choice A is incorrect. This is the value of y when x=4.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
Choice D is correct. The equation of a quadratic function can be written in the form fx=ax-h2+k, where a, h, and k are constants. It’s given in the table that when x=-1, the corresponding value of fx is 10. Substituting -1 for x and 10 for fx in the equation fx=ax-h2+k gives 10=a-1-h2+k, which is equivalent to 10=a1+2h+h2+k, or 10=a+2ah+ah2+k. It’s given in the table that when x=0, the corresponding value of fx is 14. Substituting 0 for x and 14 for fx in the equation fx=ax-h2+k gives 14=a0-h2+k, or 14=ah2+k. It’s given in the table that when x=1, the corresponding value of fx is 20. Substituting 1 for x and 20 for fx in the equation fx=ax-h2+k gives 20=a1-h2+k, which is equivalent to 20=a1-2h+h2+k, or 20=a-2ah+ah2+k. Adding 20=a-2ah+ah2+k to the equation 10=a+2ah+ah2+k gives 30=2a+2ah2+2k. Dividing both sides of this equation by 2 gives 15=a+ah2+k. Since 14=ah2+k, substituting 14 for ah2+k into the equation 15=a+ah2+k gives 15=a+14. Subtracting 14 from both sides of this equation gives a=1. Substituting 1 for a in the equations 14=ah2+k and 20=ah2-2ah+a+k gives 14=h2+k and 20=1-2h+h2+k, respectively. Since 14=h2+k, substituting 14 for h2+k in the equation 20=1-2h+h2+k gives 20=1-2h+14, or 20=15-2h. Subtracting 15 from both sides of this equation gives 5=-2h. Dividing both sides of this equation by -2 gives -52=h. Substituting -52 for h into the equation 14=h2+k gives 14=-522+k, or 14=254+k. Subtracting 254 from both sides of this equation gives 314=k. Substituting 1 for a, -52 for h, and 314 for k in the equation fx=ax-h2+k gives fx=x+522+314, which is equivalent to fx=x2+5x+254+314, or fx=x2+5x+14. Therefore, fx=x2+5x+14 defines f.
Choice A is incorrect. If fx=3x2+3x+14, then when x=-1, the corresponding value of fx is 14, not 10.
Choice B is incorrect. If fx=5x2+x+14, then when x=-1, the corresponding value of fx is 18, not 10.
Choice C is incorrect. If fx=9x2-x+14, then when x=-1, the corresponding value of fx is 24, not 10, and when x=1, the corresponding value of fx is 22, not 20.
Choice C is correct. It's given that fx=x+155 and fa=10, where a is a constant. Therefore, for the given function f, when x=a, fx=10. Substituting a for x and 10 for fx in the given function f yields 10=a+155. Multiplying both sides of this equation by 5 yields 50=a+15. Subtracting 15 from both sides of this equation yields 35=a. Therefore, the value of a is 35.
Choice A is incorrect. This is the value of a if fa=4.
Choice B is incorrect. This is the value of a if fa=5.
Choice D is incorrect. This is the value of a if fa=16.
Choice C is correct. It's given that the function P models the population of the city t years after 2005. Since there are 12 months in a year, 18 months is equivalent to 1812 years. Therefore, the expression 1812x can represent the number of years in x 18-month periods. Substituting 1812x for t in the given equation yields P1812x=2901.04461812x, which is equivalent to P1812x=2901.04x. Therefore, for each 18-month period, the predicted population of the city is 1.04 times, or 104% of, the previous population. This means that the population is predicted to increase by 4% every 18 months.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect. Each year, the predicted population of the city is 1.04 times the previous year's predicted population, which is not the same as an increase of 1.04%.
Choice D is incorrect and may result from conceptual or calculation errors.
The correct answer is -1415. A linear equation in the form ax+b=cx+d has no solution only when the coefficients of x on each side of the equation are equal and the constant terms are not equal. Dividing both sides of the given equation by 2 yields kx-n=-2830x-3638, or kx-n=-1415x-1819. Since it’s given that the equation has no solution, the coefficient of x on both sides of this equation must be equal, and the constant terms on both sides of this equation must not be equal. Since 1819<1, and it's given that n>1, the second condition is true. Thus, k must be equal to -1415. Note that -14/15, -.9333, and -0.933 are examples of ways to enter a correct answer.
The correct answer is 289. A quadratic equation of the form ax2+bx+c=0, where a, b, and c are constants, has no real solutions when the value of the discriminant, b2-4ac, is less than 0. In the given equation, x2-34x+c=0, a=1 and b=-34. Therefore, the discriminant of the given equation can be expressed as -342-41c, or 1,156-4c. It follows that the given equation has no real solutions when 1,156-4c<0. Adding 4c to both sides of this inequality yields 1,156<4c. Dividing both sides of this inequality by 4 yields 289<c, or c>289. It's given that the equation x2-34x+c=0 has no real solutions when c>n. Therefore, the least possible value of n is 289.
Choice D is correct. It's given that 4x2+bx-45 can be rewritten as hx+kx+j. The expression hx+kx+j can be rewritten as hx2+jhx+kx+kj, or hx2+jh+kx+kj. Therefore, hx2+jh+kx+kj is equivalent to 4x2+bx-45. It follows that kj=-45. Dividing each side of this equation by k yields j=-45k. Since j is an integer, -45k must be an integer. Therefore, 45k must also be an integer.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
The correct answer is 292. According to the first equation in the given system, the value of y is -1.5. Substituting -1.5 for y in the second equation in the given system yields -1.5=x2+8x+a. Adding 1.5 to both sides of this equation yields 0=x2+8x+a+1.5. If the given system has exactly one distinct real solution, it follows that 0=x2+8x+a+1.5 has exactly one distinct real solution. A quadratic equation in the form 0=px2+qx+r, where p, q, and r are constants, has exactly one distinct real solution if and only if the discriminant, q2-4pr, is equal to 0. The equation 0=x2+8x+a+1.5 is in this form, where p=1, q=8, and r=a+1.5. Therefore, the discriminant of the equation 0=x2+8x+a+1.5 is 82-41a+1.5, or 58-4a. Setting the discriminant equal to 0 to solve for a yields 58-4a=0. Adding 4a to both sides of this equation yields 58=4a. Dividing both sides of this equation by 4 yields 584=a, or 292=a. Therefore, if the given system of equations has exactly one distinct real solution, the value of a is 292. Note that 29/2 and 14.5 are examples of ways to enter a correct answer.
Choice C is correct. It's given that fx=ax+b and that the graph shown is a partial graph of y=fx. Substituting y for fx in the equation fx=ax+b yields y=ax+b. The graph passes through the point -7,-2. Substituting -7 for x and -2 for y in the equation y=ax+b yields -2=a-7+b. Multiplying each side of this equation by -7+b yields -2-7+b=a, or 14-2b=a. The graph also passes through the point -5,-6. Substituting -5 for x and -6 for y in the equation y=ax+b yields -6=a-5+b. Multiplying each side of this equation by -5+b yields -6-5+b=a, or 30-6b=a. Substituting 14-2b for a in this equation yields 30-6b=14-2b. Adding 6b to each side of this equation yields 30=14+4b. Subtracting 14 from each side of this equation yields 16=4b. Dividing each side of this equation by 4 yields 4=b. Substituting 4 for b in the equation 14-2b=a yields 14-24=a, or 6=a. Substituting 6 for a and 4 for b in the equation fx=ax+b yields fx=6x+4. It's given that gx=fx+4. Substituting x+4 for x in the equation fx=6x+4 yields fx+4=6x+4+4, which is equivalent to fx+4=6x+8. It follows that gx=6x+8.
Choice A is incorrect. This could define function g if gx=fx-4.
Choice B is incorrect. This could define function g if gx=fx.
Choice D is incorrect. This could define function g if gx=fx·x+4.
Choice A is correct. The left-hand side of the given equation is the expression 57x2+57b+ax+ab. Applying the distributive property to this expression yields 57x2+57bx+ax+ab. Since the first two terms of this expression have a common factor of 57x and the last two terms of this expression have a common factor of a, this expression can be rewritten as 57xx+b+ax+b. Since the two terms of this expression have a common factor of x+b, it can be rewritten as x+b57x+a. Therefore, the given equation can be rewritten as x+b57x+a=0. By the zero product property, it follows that x+b=0 or 57x+a=0. Subtracting b from both sides of the equation x+b=0 yields x=-b. Subtracting a from both sides of the equation 57x+a=0 yields 57x=-a. Dividing both sides of this equation by 57 yields x=-a57. Therefore, the solutions to the given equation are -b and -a57. It follows that the product of the solutions of the given equation is -b-a57, or ab57. It’s given that the product of the solutions of the given equation is kab. It follows that ab57=kab, which can also be written as ab157=abk. It’s given that a and b are positive constants. Therefore, dividing both sides of the equation ab157=abk by ab yields 157=k. Thus, the value of k is 157.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
The correct answer is 10. It's given that the graph of x2+x+y2+y=1992 in the xy-plane is a circle. The equation of a circle in the xy-plane can be written in the form x-h2+y-k2=r2, where the coordinates of the center of the circle are h,k and the length of the radius of the circle is r. The term x-h2 in this equation can be obtained by adding the square of half the coefficient of x to both sides of the given equation to complete the square. The coefficient of x is 1. Half the coefficient of x is 12. The square of half the coefficient of x is 14. Adding 14 to each side of x2+x+y2+y=1992 yields x2+x+14+y2+y=1992+14, or x+122+y2+y=1992+14. Similarly, the term y-k2 can be obtained by adding the square of half the coefficient of y to both sides of this equation, which yields x+122+y2+y+14=1992+14+14, or x+122+y+122=1992+14+14. This equation is equivalent to x+122+y+122=100, or x+122+y+122=102. Therefore, the length of the circle's radius is 10.
Choice D is correct. The equation of a parabola in the xy-plane can be written in the form y=ax-h2+k, where a is a constant and h,k is the vertex of the parabola. If a is positive, the parabola will open upward, and if a is negative, the parabola will open downward. It’s given that the parabola has vertex 9,-14. Substituting 9 for h and -14 for k in the equation y=ax-h2+k gives y=ax-92-14, which can be rewritten as y=ax-9x-9-14, or y=ax2-18x+81-14. Distributing the factor of a on the right-hand side of this equation yields y=ax2-18ax+81a-14. Therefore, the equation of the parabola, y=ax2-18ax+81a-14, can be written in the form y=ax2+bx+c, where a=a, b=-18a, and c=81a-14. Substituting -18a for b and 81a-14 for c in the expression a+b+c yields a+-18a+81a-14, or 64a-14. Since the vertex of the parabola, 9,-14, is below the x-axis, and it’s given that the parabola intersects the x-axis at two points, the parabola must open upward. Therefore, the constant a must have a positive value. Setting the expression 64a-14 equal to the value in choice D yields 64a-14=-12. Adding 14 to both sides of this equation yields 64a=2. Dividing both sides of this equation by 64 yields a=264, which is a positive value. Therefore, if the equation of the parabola is written in the form y=ax2+bx+c, where a, b, and c are constants, the value of a+b+c could be -12.
Choice A is incorrect. If the equation of a parabola with a vertex at 9,-14 is written in the form y=ax2+bx+c, where a, b, and c are constants and a+b+c=-23, then the value of a will be negative, which means the parabola will open downward, not upward, and will intersect the x-axis at zero points, not two points.
Choice B is incorrect. If the equation of a parabola with a vertex at 9,-14 is written in the form y=ax2+bx+c, where a, b, and c are constants and a+b+c=-19, then the value of a will be negative, which means the parabola will open downward, not upward, and will intersect the x-axis at zero points, not two points.
Choice C is incorrect. If the equation of a parabola with a vertex at 9,-14 is written in the form y=ax2+bx+c, where a, b, and c are constants and a+b+c=-14, then the value of a will be 0, which is inconsistent with the equation of a parabola.