Princeton Digital SAT 1: Elevate Your Test Prep Experience

In this article you will find the Princeton Digital SAT 1 and its answers and explanations. This will act as your “diagnostic” test. We recommend that you take this test before going any further in order to realistically determine the following:

• Your approximate starting score.
• Which question types you’re ready for and which you might need to practice.
• Which content topics you are familiar with and which you will want to carefully review.

Once you have nailed down your strengths and weaknesses based on this exam, you can focus your test preparation.

The Digital SAT has only two modules in each section, not three like this book. The second module you get in each section will be determined by your performance on the first module in that section.

For both RW and Math, the following test contains a standard first module and two options for the second module, one easier and one harder. You should take the appropriate second module based on your performance in the first module, as detailed below, but you can feel free to use the other module for extra practice later.

Direction for Princeton Digital SAT 1

• Reading and Writing:
  • Take Reading and Writing (RW) Module 1, allowing yourself 32 minutes to complete it.
  • If you get fewer than 15 questions correct, take RW Module 2 – Easier; If you get 15 or more questions correct, take RW Module 2.
  • Princeton Review Digital SAT Premium Prep, 2024
  • Whichever RW Module 2 you take, start it immediately and allow yourself 32 minutes to complete it.

Take a 10-minute break between RW Module 2 and Math Module 1.

• Math:
  • Take Math Module 1, allowing yourself 35 minutes to complete it.
  • If you get fewer than 14 questions correct, take Math Module 2 – Easier; If you get 14 or more questions correct, take Math Module 2 – Harder.
  • Whichever Math Module you take, start it immediately and allow yourself 35 minutes to complete it.

Only after you complete the entire test should you read the explanations for the questions:

Reading and Writing - Module 1

Directions: The questions in this section address a number of important reading and writing skills. Each question includes one or more passages, which may include a table or graph. Read each passage and question carefully, and then choose the best answer to the question based on the passage(s).
All questions in the section are multiple-choice with four answer choices. Each question has a single best answer

Question 1: The Chilean volcano Calabozos is located in __________ area. Therefore, the risk of loss of human life in the event of an eruption is minimal. Which choice completes the text with the most logical and precise word or phrase?
Which choice completes the text with the most logical and precise word or phrase?

A) a hazardous
B) an active
C) a mountainous
D) a remote

Question 2: Contemporaries of American modernist poet H.D. focused only on her important contributions to the Imagist movement in the 1920s, taking __________ view of her work. However, she wrote in a variety of forms and genres, from short, lyrical works to complex, booklength poems.
Which choice completes the text with the most logical and precise word or phrase?

A) an expensive
B) a limited
C) an imaginative
D) a complicated

Question 3: Since the 1950s, scientists have known that rapid eye movement, or REM, occurs when someone is sleeping. Previous studies attempting to determine the meaning of these eye movements have been unsuccessful in part because these studies relied on human subjects recalling the content of their dreams. A recent study by physiologists Yuta Senzai and Massimo Scanziani has avoided this issue by studying dreaming mice instead. Their results suggest that REM is correlated to changes in direction during the dream.
Which choice best describes the function of the second sentence in the overall structure of the text?

A) It names a problem in the approach taken by Senzai and Scanziani.
B) It introduces the difficulty that the study by Senzai and Scanziani was designed to bypass.
C) It presents the findings of studies done prior to the study by Senzai and Scanziani.
D) It clarifies how others studying REM sleep interpret the study by Senzai and Scanziani.

Question 4: Electroreception is the ability of an animal to sense the flow of electricity around it by using specialized organs known as electroreceptors. Most species known to use electroreception are fish, including many sharks, elephant fishes, and eels. However, electroreception is not limited to fish. Monotremes, a group of mammals that includes the platypus and some echidnas, have electroreceptors on or near their mouths to help locate prey. There is also some evidence that bees can detect static electricity on flowers.
Which choice best describes the function of the third sentence in the overall structure of the text?

A) It generalizes the phenomenon discussed beyond fishes.
B) It offers another explanation of electroreception that is different from the explanation of how electroreception is used by fishes.
C) It provides more examples of animals with electroreception.
D) It explains how electroreception evolved in monotremes and bees.

Question 5: Text 1
An animal is said to have a theory of mind when it is able to act according to the mental states of other individuals. Psychologists David Premack and Guy Woodruff studied whether chimpanzees have such a theory of mind. They showed videos of human actors struggling with various problems. The chimpanzees were able to select photographs that showed the best tool to solve each actor’s problem.
Text 2
Biologist Daniel J. Povinelli and psychologists Kurt E. Nelson and Sarah T. Boysen have argued that previous research into whether chimpanzees have a theory of mind have not adequately addressed alternative explanations for the chimpanzees’ behaviors. Specifically, it may be the case that chimpanzees are following learned behaviors in a known environment, rather than applying a theory of mind in a novel situation.
Based on the texts, how would Povinelli, Nelson, and Boysen (Text 2) most likely respond to Premack and Woodruff (Text 1)?

A) They would argue that nonhuman primates other than chimpanzees, such as baboons and gorillas, may also have a theory of mind.
B) They would argue that the chimpanzees would be able to solve the problems themselves without referencing the photographs by struggling with the situation themselves and eventually determining the correct solution.
C) They would encourage Premack and Woodruff to show the same videos and photographs to other nonhuman primates and compare the other nonhuman primates’ reactions to the chimpanzees’ reactions.
D) They would suggest that placing the chimpanzee subjects in novel environments, such as rooms distinct from the chimpanzees’ regular enclosures, may help better ascertain whether chimpanzees have a theory of mind.

Question 6: The following text is from Oscar Wilde’s 1890 novel The Picture of Dorian Gray. Dorian is seeing his portrait, painted by Basil Hallward, for the first time.

Dorian made no answer, but passed listlessly in front of his picture and turned towards it. When he saw it he drew back, and his cheeks flushed for a moment with pleasure. A look of joy came into his eyes, as if he had recognized himself for the first time. He stood there motionless and in wonder, dimly conscious that Hallward was speaking to him, but not catching the meaning of his words. The sense of his own beauty came on him like a revelation. He had never felt it before.

According to the text, what is true about Dorian?

A) Dorian is distracted by the beauty of the painting.
B) Dorian believes that what Hallward is saying is unimportant.
C) Dorian does not recognize his own image.
D) Dorian is prone to embarrassment.

Question 7: The following text is from Frederick Marryat’s 1847 novel The Children of the New Forest.

The old forester lay awake the whole of this night, reflecting how he should act relative to the children; he felt the great responsibility that he had incurred, and was alarmed when he considered what might be the consequences if his days were shortened. What would become of them—living in so sequestered a spot that few knew even of its existence—totally shut out from the world, and left to their own resources?

Based on the text, what is true about the children?

A) They are isolated from people other than the old forester.
B) They are completely unable to take care of themselves.
C) The old forester is resentful of having to take care of them.
D) They attempt to help the old forester with his responsibilities.

Question 8: The following text is Baron George Gordon Byron’s poem “Answer to _______’s Professions of Affection,” written around 1814. The poem is addressed to an unknown person.

In hearts like thine ne’er may I hold a place
Till I renounce all sense, all shame, all grace—
That seat,—like seats, the bane of Freedom’s realm,
But dear to those presiding at the helm—
Is basely purchased, not with gold alone;
Add Conscience, too, this bargain is your own—
‘Tis thine to offer with corrupting art
The rotten borough of the human heart.

What is the main idea of the text?

A) The speaker is expressing disapproval towards the unknown person.
B) The speaker is unimportant to the unknown person.
C) The speaker is thinking of purchasing a seat.
D) The speaker holds a place in the heart of the unknown person.

Question 9:
Sepsis is a life-threatening condition caused by the body’s response to an infection. These infections are typically bacterial but may be fungal, parasitic, or viral. The body’s response to these infections leads to increased inflammation and organ damage. This damage, in turn, results in a weakened immune system, which increases the likelihood of reinfection. In a recent study, a team of doctors and pharmacologists led by Shubham Soni claims that administering ketone esters can reduce inflammation and immune system weakening caused by sepsis.
Which finding from the team led by Soni, if true, would most directly support its claim?

A) Patients with sepsis who were administered ketone esters had fewer signs of inflammation and less organ damage than those administered standard antibiotics.
B) When administered, ketone esters are known to increase blood ketone levels, which in turn are a source of energy for the brain.
C) Both those patients administered ketone esters and those administered standard antibiotics did not have reduced inflammation when treated with medication intended to reduce fever.
D) Those sepsis patients administered ketone esters had reduced inflammation but greater organ damage than those administered standard antibiotics.

Question 10: Horses have been domesticated for thousands of years. Therefore, they show great sensitivity to the emotions of humans. Biologist Océane Liehrmann from the University of Turku, Finland, led a team of researchers in a study of horses to determine the effect of the number of handlers (either only one person or multiple people) on the horses’ responses to a novel object. The researchers determined that horses with only one handler were less reluctant to interact with the novel object than were horses with multiple handlers. For example, 45% of horses with only one handler had no reluctance when interacting with a novel object while __________
Which choice most effectively uses data from the table to complete the example?

A) 13% of horses with only one handler had strong reluctance.
B) 25% of horses with multiple handlers had no reluctance.
C) 26% of horses with multiple handlers had strong reluctance.
D) 42% of horses with only one handler had mild reluctance.

Question 11: India is the largest democracy in the world, with over 614 million people voting in the 2019 election for the Lok Sabha, the parliament of the federal government. In the early years of Indian independence, from the first election in 1951–52 through the eighth Lok Sabha in 1984, each election resulted in one party winning the majority of seats. However, starting with the 1989 election, the party with the largest number of seats failed to win more than half of the total seats. This trend was eventually broken by the Bharatiya Janata Party, which __________
Which choice most effectively uses data from the graph to illustrate the claim?

A) went from holding the second most seats among the top 3 parties in parliament in 2004 and 2009 to holding a majority of seats in 2014 and 2019.
B) reached its highest percentage of seats the same year that the Indian National Congress had its lowest percentage of seats over the same time period.
C) won a lower percentage of seats in the 2009 election than in the 2004 election.
D) had a lower percentage of seats than the Indian National Congress in 2004 but a higher percentage of seats than the Indian National Congress in 1999.

Question 12: Fatty liver disease (FLD) occurs when excess fat builds up in the liver. While there are often few or no symptoms of FLD, if left untreated, it can lead to cirrhosis or liver cancer. Because FLD is often asymptomatic, doctors and researchers rely on indicators such as steatosis (retention of fat in the liver), fibrosis (scarring), blood glucose (sugar), serum insulin, and insulin resistance to measure and track the development of FLD. A group of researchers led by radiologist Hamid Reza Talari hypothesized that those who take vitamin B12 would experience improvements in fibrosis and insulin resistance when compared to a control group over the same time period.
Which choice best describes data from the table that support the researchers’ hypothesis?

A) Those in the control group had decreases in their steatosis values and fasting blood glucose but had increases in fibrosis values and HOMA-IR.
B) Those in the vitamin B12 group had decreases in fibrosis values and HOMA-IR levels, whereas those in the control group had increases in these same values.
C) Both those in the vitamin B12 group and the control group had decreases in their steatosis values.
D) Those in the control group had a decrease in their fasting blood glucose, but those in the vitamin B12 group had an increase in their fasting blood glucose.

Question 13: The air pollution produced in an area is only one factor in that area’s air quality. Weather patterns, in particular wind and the movement of air masses, can affect the concentration of pollutants such as carbon monoxide. During a smog event that occurred in the northeastern United States in November 1966, levels of carbon monoxide were recorded in Newark, New Jersey, the origin of the smog event, as well as neighboring city New York, NY, and more distant cities such as Philadelphia, PA, and Washington, D.C. The localized nature of weather patterns during this event can be seen by comparing Newark, NJ, and New York, NY, with __________
Which choice most effectively uses data from the table to complete the statement?

A) Washington, D.C., on the 18th and the 19th.
B) Philadelphia, PA, on the 23rd and the 25th
C) Philadelphia, PA, on the 24th and the 26th.
D) Washington, D.C., on the 23rd and the 24th.

Question 14: Neurons respond to stimuli from sensory organs or other neurons. Learning occurs when neurons change how they respond to stimuli based on previous experience, which is a property of memory. Electrical engineers seek to replicate similar processes in their development of computer memory. Recently, research by electrical engineer Mohammad Samizadeh Nikoo has demonstrated that vanadium dioxide (VO₂ ) has a similar memory property to that of neurons, suggesting that __________
Which choice most logically completes the text?

A) VO₂ could be used in the development of computer memory.
B) neurons use VO₂ when forming memories.
C) VO₂ can learn to respond to stimuli from sensory organs.
D) electrical engineers can now use neurons to develop computer memory.

Question 15: Uruguayan-Spanish author Carmen Posadas has written the children’s books Juego de Niños (Child’s Play) and La Cinta Roja (The Red Ribbon). Currently, __________ available in over fifty countries and thirty languages.
Which choice completes the text so that it conforms to the conventions of Standard English?

A) some are
B) this is
C) they are
D) it is

Question 16: During a meeting, a group of twelve young deaf people shared their feelings of isolation and their desire for support. In 1988, the group worked together to form Action Deaf Youth, an __________ provides services and programs for deaf children and youth throughout Northern Ireland.
Which choice completes the text so that it conforms to the conventions of Standard English?

A) organization, that
B) organization
C) organization that
D) organization,

Question 17: In 1986, after a 56-day expedition, Ann Bancroft became the first woman to reach the North Pole. Her experience as a physical education teacher and her leadership of the first all-female team to cross the ice to the South __________ her to create a foundation that supports girls in pursuing their dreams.
Which choice completes the text so that it conforms to the conventions of Standard English?

A) Pole to inspire
B) Pole that inspired
C) Pole, inspiring
D) Pole inspired

Question 18: American artist Simone Leigh creates art in various mediums, including sculptures, video, and __________ the themes and images in her artwork, Leigh has emphasized that Black women are her primary audience and that they would be familiar with the allusions in her work.
Which choice completes the text so that it conforms to the conventions of Standard English?

A) performance. Discussing
B) performance discussing
C) performance and discussing
D) performance, discussing

Question 19: Japanese origamist Akira Yoshizawa is considered the grandmaster of origami, creating more than 50,000 models as well as wet-folding, the most well-known of his invented techniques. __________ dampening the paper before folding, leading to origami models with rounder and more sculpted looks.
Which choice completes the text so that it conforms to the conventions of Standard English?

A) It involves
B) They involve
C) One involves
D) These involve

Question 20: Chinese artist Xu Bing is known for his art installations that showcase his printmaking skills and his creative use of languages and texts. His 1991 installation A Book from the Sky, for example, consists of volumes and scrolls printed with characters he invented, while his 2004 installation The Glassy Surface of a __________ uses the text of Henry David Thoreau’s Walden to create the illusion of a lake.
Which choice completes the text so that it conforms to the conventions of Standard English?

A) Lake:
B) Lake
C) Lake,
D) Lake–

Question 21: Developed along with the swing style of jazz music in the 1920s, swing dance is a group of social dances that once comprised hundreds of styles. Not all of the styles survived beyond that time __________ the dances that are still popular today include Lindy Hop, Balboa, Collegiate Shag, and Charleston.
Which choice completes the text so that it conforms to the conventions of Standard English?

A) period; however,
B) period, however;
C) period, however,
D) period, however

Question 22: Evolutionary biologist Jonathan Calede may have discovered the oldest amphibious beaver species in the world. Calede first compared measurements of the beaver’s ankle to those of almost 350 other rodent species to learn more about how it moved. __________ Calede dated the species to approximately 30 million years ago based on its location between rock and ash layers.
Which choice completes the text with the most logical transition?

A) For example,
B) In conclusion,
C) Next,
D) In fact,

Question 23: Male and female American citizens had starkly different roles during World War II. Men served as soldiers or took part in the workforce to create weapons and other wartime materials. __________ women were responsible for maintaining the home and supporting the men. Some women also ventured into the workforce for the first time, and the famous “We Can Do It” poster featuring “Rosie the Riveter” was created to motivate women to pursue this new role.
Which choice completes the text with the most logical transition?

A) Besides,
B) Instead,
C) Likewise,
D) Meanwhile,

Question 24: While treatment for hearing loss is typically associated with the ears, some patients with damaged ear structures are not able to use traditional cochlear implants. __________ researchers are working to develop hearing aids anchored to patients’ bones in order to combat hearing loss through vibrations in the skull.
Which choice completes the text with the most logical transition?

A) Secondly
B) In addition,
C) Finally,
D) Hence,

Question 25: Korean artist Anicka Yi uses a unique process and materials to generate her art installations. Her materials are often perishable and biological, such as soap and flowers, and are not traditionally used for artwork. __________ Yi spends almost as much time transforming these substances into completely new materials as she does creating the actual art pieces.
Which choice completes the text with the most logical transition?

A) Meanwhile,
B) Instead,
C) In fact,
D) To conclude,

Question 26: While researching a topic, a student has taken the following notes:

• A writing system for expressing numbers is a numeral system.
• Two examples of numeral systems from history are Babylonian cuneiform numerals and Roman numerals.
• The Babylonian cuneiform numeral system is a base-60 system and lacks a zero digit.
• It’s a positional numeral system in which the position of a digit affects its value.
• The Roman numeral system is a base-10 system and lacks a zero digit.
• It’s a non-positional numeral system in which the position of a digit does not affect its value.

The student wants to emphasize a difference between the two numeral systems. Which choice most effectively uses relevant information from the notes to accomplish this goal?

A) Babylonian cuneiform numerals and Roman numerals are two writing systems for expressing numbers.
B) The Roman numeral system is a base-10 nonpositional system that lacks a zero digit.
C) One system for expressing numbers is Babylonian cuneiform; however, another one is the Roman numeral system.
D) The Babylonian cuneiform numeral system is base-60 and positional, while the Roman numeral system is base-10 and non-positional.

Question 27: While researching a topic, a student has taken the following notes:

• Archaeologists studied the burial of an individual at the Newen Antug site in Argentinian Patagonia.
• The individual was buried in a wooden structure over 800 years ago.
• An analysis of the structure revealed that it was carved from a tree with excellent buoyancy.
• The wooden structure was a canoe, suggesting that canoes were used as coffins at that time.

The student wants to present the Newen Antug study and its conclusions. Which choice most effectively uses relevant information from the notes to accomplish this goal?

A) The burial site of an individual over 800 years ago was found at the Newen Antug site in Argentinian Patagonia.
B) Archaeologists studied the burial site of an individual who was buried at the Newen Antug site over 800 years ago.
C) An analysis of a burial site at the Newen Antug site in Argentinian Patagonia provided evidence that canoes were used as coffins over 800 years ago.
D) As part of a study of a burial site at the Newen Antug site in Argentinian Patagonia, a wooden structure buried with an individual was analyzed.

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Reading and Writing - Module 2 - Easier

Directions: The questions in this section address a number of important reading and writing skills. Each question includes one or more passages, which may include a table or graph. Read each passage and question carefully, and then choose the best answer to the question based on the passage(s).
All questions in this section are multiple-choice with four answer choices. Each question has a single best answer.

Reading and Writing - Module 2 - Harder

Directions: The questions in this section address a number of important reading and writing skills. Each question includes one or more passages, which may include a table or graph. Read each passage and question carefully, and then choose the best answer to the question based on the passage(s).
All questions in this section are multiple-choice with four answer choices. Each question has a single best answer.

Math Test - Module 1

DIRECTIONS: The questions in this section address a number of important math skills. Use of a calculator is permitted for all questions.

NOTES: Unless otherwise indicated:

• All variables and expressions represent real numbers.
• Figures provided are drawn to scale.
• All figures lie in a plane.
• The domain of a given function f is the set of all real numbers x for which f(x)is a real number.

The number of degrees of arc in a circle is 360. The number of radians of arc in a circle is 2π. The sum of the measures in degrees of the angles of a triangle is 180.

Question 1:A data set containing only the values 2, 2, 9, 9, 9, 16, 16, 16, 16, 26, 26, and 26 is represented by a frequency table. Which of the following is the correct representation of this data set?

A)
B)
C)
D)

Question 2: The expression x² -x - 56 is equivalent to which of the following?

A) (x - 24) (x + 4)
B) (x - 7) (x + 8)
C) (x - 8) (x + 7)
D) (x - 4) (x + 14)

Question 3: A carpenter hammers 10 nails per minute and installs 7 screws per minute during a project. Which of the following equations represents the scenario if the carpenter hammers nails for x minutes, installs screws for y minutes, and uses a combined total of 200 nails and screws?

A) \(\frac{1}{10}x + \frac{1}{7}y = 200\)
B) \(\frac{1}{10}x + \frac{1}{7}y = 3,420\)
C) 10x + 7y = 200
D) 10x + 7y = 3,420

Question 4: What is the measure of angle F in the triangle DEF, where angle D is 73° and angle E is 35°?

A) 38º
B) 72º
C) 108º
D) 126°

Question 5: The total amount of plastic remaining to be recycled in a facility over x shifts is represented by the graph above. Which of the following represents the y-intercept of the graph?

A) The total amount of plastic remaining at any given time
B) The number of shifts it will take to finish recycling the plastic
C) The amount of plastic that is recycled per shift
D) The initial amount of plastic to be recycled

Question 6: The table below shows the condition and subject type for 200 textbooks at a bookstore

What is the probability that a textbook chosen at random will be a new textbook? (Express your answer as a decimal or fraction, not as a percent.)

A) 12/20
B) 18/20
C) 14/20
D) 16/20

Question 7: A random sample of 5,000 students out of 60,000 undergraduate students at a university were surveyed about a potential change to the registration system. According to the survey results, 75% of the respondents did not support the existing registration system, with a 4% margin of error. Which of the following represents a reasonable total number of students who did not support the existing registration system?

A) 1,250
B) 3,750
C) 13,800
D) 43,800

Question 8: What is the negative solution to the equation \(\frac{32}{a} = a - 4\)

A) -2
B) -4
C) -6
D) 2

Question 9: After a hot air balloon is launched from a plateau 1,000 meters above sea level, it rises at a constant rate of 750 meters per minute. Which of the following best describes the function used to model the balloon’s distance above sea level over time?

A) Increasing linear
B) Increasing exponential
C) Decreasing linear
D) Decreasing exponential

Question 10: What is the x-intercept of the function f(x) = (22)^x -1, when it is graphed in the xy-plane, where y = f(x)?

A) (-1, 0)
B) (0, 0)
C) (21, 0)
D) (22, 0)

Question 11: In parallelogram ABCD shown above, the length of \(\overline{AB}\) is one-third the length of \(\overline{AD}\). The perimeter of the parallelogram is 64. What is the length of \(\overline{AB}\)?

A) 8
B) 16
C) 24
D) 32

Question 12: A triangle with an area of 18 square units has a base of (m + 5) units and a height of m units. What is the value of m?

A) 4
B) 9
C) 13
D) 36

Question 13: The table above shows the exponential growth of a type of yeast over time s, in seconds. There are c total yeast colonies on the count plate. What is the equation that represents this relationship, assuming that no yeast was added or removed after counting began?

A) c = (1 + 3)^s
B) c = (1 + 5)^s
C) c = 3(1 + 5)^s
D) c = 5(1 + 3)^s

Question 14: The equations 12x = y and 24 + 7 = 2y intersect at how many points when graphed in the xy-plane?

A) 0
B) 1
C) 2
D) 7

Question 15: Several tiles labeled with either an A or a B are placed in a bag, and tiles are worth a different point value depending on the label. The equation 15a + 10b = 100 represents the situation when a of the A tiles and b of the B tiles are drawn from the bag for a total of 100 points. How many points would be earned by drawing one A tile and one B tile from the bag?

A) 32
B) 22
C) 25
D) 35

Question 16: The amount of money remaining in a scholarship fund is reduced by one-fourth every year. The amount of money in the fund is represented by d and the number of years by y. If the fund starts with $10,000, which equation below represents this situation after y years?

A) \(d = \frac{1}{4}(10,000)^y\)
B) \(d = \frac{3}{4}(10,000)^y\)
C) \(d = 10,000 \left(\frac{1}{4}\right)^y\)
D) \(d = 10,000 \left(\frac{3}{4}\right)^y\)

Question 17: What is the diameter, in millimeters (mm), of a cylinder with a volume of 144π mm³ and a height of 4 mm?

A) 6
B) 9
C) 12
D) 36

4x + 2y = 4
19x + 10y = 14
Question 18: When graphed in the xy-plane, the linear equations shown above intersect at (a, b). What is the value of a?

A) -20
B) -10
C) 6
D) 14

Question 19: The longest side of right triangle ABC is opposite angle B. If \(\sin(A) = \frac{9}{41}\), what is the value of sin(C)?

A) 20/21
B) 21/20
C) 41/40
D) 40/41

Question 20: Function g reaches its maximum value when x = a. If g(x) = -6x² -30x - 24, what is the value of a?

A) -5.2
B) 5.2
C) 2.5
D) -2.5

\(f(x) = -\frac{1}{5}x - 3\)
Question 21: The linear function f(x), given above, is perpendicular to g(x) when graphed in the xy-plane. If g(0) = 0, what is the value of g(2)?

A) 6
B) 8
C) 10
D) 12

y = 5kx² + 2x +3
\(\frac{y}{10} = -x\)
Question 22: The system of equations above has exactly one solution. If k is a constant, what is the value of k?

A) -2.4
B) 2.4
C) 3.4
D) -4,3

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Math Module 2 - Easier:

DIRECTIONS: The questions in this section address a number of important math skills. Use of a calculator is permitted for all questions.

NOTES: Unless otherwise indicated:

• All variables and expressions represent real numbers.
• Figures provided are drawn to scale.
• All figures lie in a plane.
• The domain of a given function f is the set of all real numbers x for which f(x)is a real number.

The number of degrees of arc in a circle is 360. The number of radians of arc in a circle is 2π. The sum of the measures in degrees of the angles of a triangle is 180.

Question 1: 33, 34, 38, 41, 43, 44, 47 Which of the following is the median of the data listed above?

A) 38
B) 40
C) 41
D) 42

Question 2: What is the value of the solution to the equation 22 = y – 10?

A) -32
B) 23
C) 22
D) 32

Question 3: A rectangle has a height of 23 inches (in) and a width of 9 in. What is its perimeter, in inches?

A) 32
B) 64
C) 207
D) 1,024

15a – (6a – 2a)
Question 4: Which of the following expressions is equivalent to the one above?

A) 5a
B) 7a
C) 11a
D) 23a

Question 5: Which equation represents the relationship between the numbers a and b if a is half of b?

A) a = 1/2b
B) a = b – 2
C) a = b + 2
D) b = 1/2a

Question 6: For all positive values of y, the expression \(\frac{3}{y+c}\) is equivalent to \(\frac{15}{5y+30}\). What is the value of constant c?

A) 3
B) 6
C) 8
D) 150

Question 7: A total of 200 pets were adopted at an event. If 70% of the adopted pets were dogs, how many of the pets were dogs?

A) 100
B) 120
C) 140
D) 160

Question 8: James must drive 100 miles before he can take his driver’s license test. He knows that when he drives around town running errands, he drives at an average speed of 20 miles per hour. If James maintains this average speed, how many hours must he drive to meet the requirement for his driver’s license test?

A) 5
B) 20
C) 80
D) 100

Question 9: What is the value of 4y – 16 if y – 4 = 11?

A) 42
B) 44
C) 46
D) 48

Question 10: The function g is defined as g(x) = x² - 1. What is the value of g(x) when x = 3?

A) 4
B) 5
C) 7
D) 8

Question 11: The production cost p(x), in dollars, to produce x units of an item when materials cost $2 per item is given by p(x) = 2x + 150. What is the total cost to produce 2,000 units of this item?

A) $1,850
B) $2,300
C) $3,850
D) $4,150

Question 12: The function f is given as \(f(x) = \frac{2}{3}x\). When x = 6, what is the value of f (x)?

A) 2
B) 4
C) 6
D) 9

Question 13: In the figure above, what is the value of d if line a is parallel to line b?

A) 118
B) 120
C) 116
D) 100

3x – 4y = 17
Question 14: In the xy-plane, the graph of a line with an x-intercept of (c, 0) and a y-intercept of (0, k), where c and k are constants, can be represented by the equation above. What is the value of c/k?

A) -4/3
B) -3/4
C) 3/4
D) 4/3

Question 15: A postal machine processes mail at a constant rate of 21 pieces of mail per minute. At this rate, how many pieces of mail would the machine process in 7 minutes?

A) 3
B) 14
C) 28
D) 147

Question 16: Stella had 211 invitations to send for an event. She has already sent 43 invitations and will send them all if she sends 24 each day for the next d days. Which of the following equations represents this situation?

A) 24d – 43 = 211
B) 24d + 43 = 211
C) 43d – 24 = 211
D) 43d + 24 = 211

Question 17: When the linear function y = f (x) is graphed in the xy-plane, the graph contains the corresponding values of x and f (x) shown in the table above. Which of the following could represent the function?

A) f(x) = 3x + 12
B) f(x) = 3x + 15
C) f(x) = 15x + 12
D) f(x) = 15x + 15

Question 18: The height of a rocket launched from a rooftop can be modeled by the equation h = -16s² + 64s + 21, where h is the height of the rocket above the ground, in feet, and s is the number of seconds since the rocket was launched. Which of the following represents the height, in feet, of the rooftop from which the rocket was launched?

A) 0
B) 16
C) 21
D) 64

Question 19: Function f is defined by f(x) = x³ + 1. Which of the following tables gives three values of x and their corresponding values of y?

A)

B)
C)
D)

Question 20: If h(–1) = 3 and h(0) = 5 in linear function h, which of the following is the equation of function h?

A) h(x) = 2x + 5
B) h(x) = 2x + 3
C) h(x) = 2x
D) h(x) = 3x + 5

Question 21: Which of the following equations correctly expresses r in terms of p and s if the relationship between the numbers p, r, and s can be expressed as p = 13r – 6s?

A) \(r = \frac{-6s - p}{13}\)
B) r = 13p + 6s
C) \(r = \frac{1}{13}p + 6s\)
D) \(r = \frac{p+6s}{13}\)

Question 22: Right triangle ABC has sides of the following lengths: AB = 165, BC = 280, and AC = 325. Another triangle, LMN, is similar to ABC such that A corresponds to L and B corresponds to M. What is the value of cos(L)?

A) 33/65
B) 33/56
C) 56/65
D) 65/33

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Math Module 2 - Harder:

DIRECTIONS: The questions in this section address a number of important math skills. Use of a calculator is permitted for all questions.

NOTES: Unless otherwise indicated:

• All variables and expressions represent real numbers.
• Figures provided are drawn to scale.
• All figures lie in a plane.
• The domain of a given function f is the set of all real numbers x for which f(x)is a real number.

The number of degrees of arc in a circle is 360. The number of radians of arc in a circle is 2π. The sum of the measures in degrees of the angles of a triangle is 180.

Question 1: Which of the following is equivalent to 3a³ - 5a³ + 6a?

A) -2a³ - 6a
B) 3a³ - a
C) 4a
D) -15a⁹ + 6a

Question 2: In a shipment of 45,000,000 shirts, 4,950,000 are white. What percentage of the shirts are white shirts?

A) 11%
B) 22%
C) 78%
D) 89%

Question 3: If 3(x – 8) – 16 = 8(x + 10) + x, what is the value of 6x?

A) 120
B) 140
C) -110
D) -120

8(a – 3) – 17 = 9(a – 3)
Question 4: In the equation above, what is the value of a – 3?

A) -20
B) -17
C) -14
D) 3

Question 5: A school classroom with a total of 4,200 floor tiles is divided into a 30 square-foot lab area and an 80 squarefoot seating area. The number of tiles on the entire classroom floor can be represented by the equation 30a + 80b = 4,200. In this context, which of the following does b represent?

A) The average number of tiles per square foot in the lab area
B) The total number of tiles in the lab area
C) The average number of tiles per square foot in the seating area
D) The total number of tiles in the seating area

Question 6: A triangle has a base that is 65% of its height. If the base were decreased by 13 inches, how would the height need to change to keep the same proportions?

A) It must increase by 13 inches.
B) It must increase by 20 inches.
C) It must decrease by 13 inches.
D) It must decrease by 20 inches.

Question 7: If \(\frac{a}{3} = 10 - 7b\) and a ≠ 0, which of the following correctly expresses b in terms of a?

A) \(b = \frac{a - 21}{30}\)
B) \(b = \frac{30 - a}{21}\)
C) \(b = 10 + \frac{a}{3}\)
D) \(b = 10 + \frac{3}{a}\)

Question 8: For all positive values of y, the expression \(\frac{3}{y+c}\) is equivalent to \(\frac{15}{5y+30}\). What is the value of constant c?

A) 3
B) 6
C) 8
D) 150

Question 9: In the xy-plane, the equation (x - 7)² + (y + 7)² = 64 defines circle O, and the equation (x - 7)² + (y + 7)² = c defines circle P. If the two circles have the same center, and the radius of circle P is three less than the radius of circle O, what is the value of constant c?

A) -52
B) 52
C) 25
D) -25

Question 10: A school has received a donation of $20,000 for the purchase of new laptops. If each laptop costs $149, no tax is charged, and the laptop manufacturer offers a 7.5% discount on orders of at least 100 laptops, what is the maximum number of laptops the school can purchase with the donation?

A) 124
B) 134
C) 145
D) 146

3x² - y - 26 = 0
y = –3x + 10
Question 11: The point (a, b) is an intersection of the system of equations above when graphed in the xy-plane. What is a possible value of a?

A) -4
B) 6
C) 20
D) 26

Question 12: How many values for y satisfy the equation –6(4y + 2) = 3(4 – 8y)?

A) Zero
B) Exactly one
C) Exactly two
D) Infinitely many

Question 13: A parabola represents the graph of the function f in the xy-plane, where y = f (x). If the vertex of the parabola is (5, –4) and one of the x-intercepts is (–1.5, 0), what is the other x-intercept?

A) (–6.5, 0)
B) (1.5, 0)
C) (3.5, 0)
D) (11.5, 0)

Question 14: Which equation defines function g, if the graph of y = g(x) – 10 is shown above?

A) \(y = \frac{3}{5}x - 15\)
B) \(y = \frac{3}{5}x - 5\)
C) \(y = \frac{3}{5}x + 5\)
D) \(y = \frac{3}{5}x + 10\)

Question 15: If c is a constant in the equation 10x² + c = -5x, and the equation has no real solutions, what is the value of c?

A) -20
B) -5
C) 0
D) 1

3x – 4y = 17
Question 16: In the xy-plane, the graph of a line with an x-intercept of (c, 0) and a y-intercept of (0, k), where c and k are constants, can be represented by the equation above. What is the value of c/k?

A) -4/3
B) -3/4
C) 3/4
D) 4/3

–7 + 2f = cg
21g + 21 = 6f – 15g
Question 17: If c is a constant, and the system of equations shown above has infinitely many solutions, what is the value of c?

A) 10
B) 12
C) 14
D) 16

Question 18: Triangle A has angles measuring 30º, 60º, and 90º. What is the perimeter, in centimeters, of this triangle if the smallest side has a length of 15 centimeters?

A) \(15\sqrt{3}\)
B) 15 + \(15\sqrt{3}\)
C) 45 + \(15\sqrt{3}\)
D) \(45\sqrt{3}\)

Question 19: Four values of x and their corresponding values of g(x) are shown in the table above for the linear function g. The equation g(x) = cx + d defines function g, and c and d are constants. What is the value of c + d?

A) -23
B) 69
C) 92
D) 115

114, 109, 106, 111
Question 20: A data set consists of 5 positive integers greater than 101. What is the value of the smallest integer in the data set if the mean of the entire data set is an integer that is less than the mean of the four integers from the data set shown above?

A) 105
B) 108
C) 110
D) 113

Question 21: A teacher awards points to a class based on completed assignments. He gives 5 points per assignment for the first 50 completed assignments and 3 points for each additional completed assignment beyond 50. When a ≥ 50, which function g gives the total number of points earned by the class for a completed assignments?

A) g(a) = 3a + 5
B) g(a) = 3a + 100
C) g(a) = 3a + 250
D) g(a) = 8a - 150

Question 22: In triangles ABC and XYZ, AB = 22, XY = 11, and angles A and X both measure 77°. Which of the following pieces of information, if any, would be enough to prove that the two triangles are similar to each other?
I. Angle B measures 40°
II. Angle Y measures 50°
III. Angle Z measures 63°

A) No additional information is necessary.
B) Angle measures alone do not provide enough information.
C) I and II together provide enough information.
D) I and III together provide enough information.

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Answer Key Digital SAT 1

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Princeton Digital SAT 1 - Reading and writing explanations