NATIONAL BENCHMARK TESTS
PREPARATION COURSE
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
Table of Contents
QUALITATIVE ANALYSIS ......................................................................................................................................5
READING COMPREHENSION QUESTION TYPES AND STRATEGIES ................................................................5
BIG PICTURE QUESTIONS .................................................................................................................................5
LITTLE PICTURE QUESTIONS...........................................................................................................................6
INFERENCE QUESTIONS ...................................................................................................................................6
VOCABULARY-IN-CONTEXT QUESTIONS .........................................................................................................7
QUANTITATIVE LITERACY/ MATHS LITERACY ................................................................................................22
GENERAL ALGEBRA ........................................................................................................................................ 22
DECIMALS AND FRACTIONS...........................................................................................................................23
LCD AND FACTORS ......................................................................................................................................... 24
RATIOS AND PERCENTAGES ..........................................................................................................................25
REMAINDERS AND PROBABILITY.................................................................................................................. 26
SUBJECT OF THE FORMULA, EQUATIONS, SIMULTANEOUS EQUATIONS .................................................... 26
WORD PROBLEMS .......................................................................................................................................... 28
STRANGE SYMBOLS ........................................................................................................................................ 28
MATHEMATICS EXPONENTS ..........................................................................................................................28
LOGARITHMS .................................................................................................................................................. 31
FACTOR AND REMAINDER THEOREM ........................................................................................................... 33
PATTERNS, SEQUENCES AND SERIES ............................................................................................................ 33
FUNCTIONS AND INEQUALITIES.................................................................................................................... 35
STRAIGHT LINES, CO-ORDINATE GEOMETRY AND CIRCLES ........................................................................ 38
EUCLIDEAN GEOMETRY ................................................................................................................................. 39
GENERAL GEOMETRY AND MENSURATION .................................................................................................. 41
TRIGONOMETRY ............................................................................................................................................. 47
CALCULUS .......................................................................................................................................................52
PROBABILITY .................................................................................................................................................. 53
TRICK QUESTIONS AND WORD PROBLEMS .................................................................................................. 53
SIMULTANEOUS EQUATIONS .........................................................................................................................55
SYMBOLS .........................................................................................................................................................55
SPECIAL TRIANGLES....................................................................................................................................... 55
RATIOS ............................................................................................................................................................56
RATES..............................................................................................................................................................56
REMAINDERS .................................................................................................................................................. 56
AVERAGES.......................................................................................................................................................57
PERCENTAGES ................................................................................................................................................ 57
P a g e 3 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
MULTIPLE AND STRANGE FIGURES............................................................................................................... 57
COMBINATIONS .............................................................................................................................................. 58
PRACTICE TEST 2 ........................................................................................................................................... 59
RATIONAL EQUATIONS AND INEQUALITIES................................................................................................. 59
RADICAL EQUATIONS ..................................................................................................................................... 59
MANIPULATION WITH INTEGER AND RATIONAL EXPONENTS ................................................................... 59
ABSOLUTE VALUE ............................................................................................................................................. 59
FUNCTION NOTATION .................................................................................................................................... 59
CONCEPTS OF DOMAIN AND RANGE ............................................................................................................. 60
FUNCTIONS AS MODELS ................................................................................................................................. 60
LINEAR FUNCTONS – ...................................................................................................................................... 60
EQUATIONS AND GRAPHS ..............................................................................................................................60
QUADRATIC FUNCTIONS – EQUATIONS AND GRAPHS ................................................................................. 60
QUALITATIVE BEHAVIOR OF GRAPHS AND FUNCTIONS .............................................................................. 61
ANALYTICAL GEOMETRY ...............................................................................................................................61
PROPERTIES OF TANGENT LINES .................................................................................................................. 61
SEQUENCES INVOLVING EXPONENTIAL GROWTH .......................................................................................61
TRANSFORMATIONS AND THEIR EFFECT ON GRAPHS OF FUNCTIONS ......................................................62
SETS................................................................................................................................................................. 63
DIRECT AND INVERSE VARIATION ................................................................................................................ 63
DATA INTERPRETATION, SCATTER PLOTS, AND MATRICES ....................................................................... 63
GEOMETRIC NOTATION FOR LENGTH, SEGMENTS, LINES, RAYS, AND CONGRUENCE ...............................63
TRIGONOMETRY ............................................................................................................................................. 63
P a g e 4 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
QUALITATIVE ANALYSIS
READING COMPREHENSION QUESTION TYPESAND STRATEGIES
When you read passages in the NBT tests, you’re reading for a specific purpose: to be able to
correctly answer as many questions as possible. Whatever the passage is about and however long it
may be, you can expect the same five basic question categories:
1.
Big Picture
2.
Little Picture
3.
Inference
4.
Vocabulary-in-Context
5.
Function
You can expect slightly more than half of the questions to be Little Picture and inference questions:
fewer (approximately 30 percent – 40 percent) will be about Big Picture issues, Function and
Vocabulary.
BIG PICTURE QUESTIONS
Big Picture questions test how well you understand the passage as a whole. They ask about:
•
The main point or purpose of a passage or individual paragraphs
•
The author’s overall attitude or tone
•
The logic underlying the author’s argument
•
How ideas relate to each other in the passage
If you’re stumped on a Big Picture question, even after reading the passage, do the little Picture
questions first. They can help you fill in the Big Picture. Big Picture questions will usually be at the
end of the question set anyway, so you can often use the question order to help you get a deeper
understanding of the whole passage.
Big Picture questions may be worded as:
•
The passage is primarily concerned with . . .
•
What is the author’s attitude toward . . .
•
What is the main idea of the passage?
•
Why does the author mention . . .
P a g e 5 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
LITTLE PICTURE QUESTIONS
About a third of Critical Reading questions are Little Picture questions that ask about localised bits
of information – usually specific facts or details from the passage. These questions often give you a
line reference – a clue to where in the passage you’ll find your answer. Beware of answer choices
that seem to reasonably answer the question in the stem but that don’t make sense in the context of
the passage or that are true but refer to a different section of the text.
Little Picture questions test:
•
Whether you understand significant information that’s stated in the passage
•
Your ability to locate information within a text
•
Your ability to differentiate between main ideas and specific details
Sometimes the answer to a Little Picture question will be directly in the line or lines that are
referenced. Other times, you might need to read a few sentences before or after the referenced
line(s) to find the correct answer. When in doubt, use the context (surrounding sentences) to
confirm the right choice.
Little Picture questions may be worded as
•
According to the passage . . .
•
In lines 12 – 16, what does the author say about . . .
•
How does the author describe . . .
INFERENCE QUESTIONS
To infer is to draw a conclusion based on reasoning or evidence. For example, if you wake up in the
morning and dark grey clouds cover the sky, you may infer that it will rain later and that you may
want to take an umbrella to school.
Often, writers will use suggestion of inference rather than stating ideas directly. But they will also
leave you plenty of clues so you can figure out just what they are trying to convey. Inference clues
include word choice (diction), tone and other specific details.
For example, say a passage states that a particular idea was perceived as revolutionary. You might
infer from the use of the word perceived that the author believes the idea was not truly
revolutionary but only perceived (or seen) that way.
Thus, inference questions test your ability to use the information in the passage to come to a logical
conclusion. The key to inference questions is to stick to the evidence in the text. Most inference
questions have pretty strong clues, so avoid any answer choices that seem far-fetched. If you can’t
find any evidence in the passage, then it probably isn’t the right answer.
Make sure you read inference questions carefully. Some answer choices may be true, but if they
can’t be inferred from the passage, then they can’t be the correct answer. Inference questions may
be worded as:
P a g e 6 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
•
•
•
It can be inferred from the passage that . . .
The phrase ______ implies that . . .
The author states that ______ . This would indicate which of the following?
VOCABULARY-IN-CONTEXT QUESTIONS
Vocabulary-in-Context questions ask about the usage of a single word. These questions do not test
your ability to define hard words like archipelago and garrulous. Instead, they test your ability to
infer the meaning of a word from context.
Words tested in NBT questions are usually fairly common words with more than one definition.
But that’s the trick! Many of the answer choices will be definitions of the tested word, but only one
will work in context. Vocabulary-in-Context questions almost always have a line reference, and you
should always use it!
Sometimes one of the answer choices will jump out at you. It will be the most common meaning of
the word in question – but it’s RARELY right. You can think of this as the obvious choice. Say
curious is the word being tested. The obvious choice is inquisitive. But curious also means odd, and
that’s more likely to be the answer.
Using context to find the answer will help keep you from falling for this kind of trap. But you can
also use these obvious choices to your advantage. If you get stuck on a Vocabulary-in-Context
question, you can eliminate the obvious choice and guess from the remaining answers.
Here’s our strategy for Vocabulary-in-Context questions:
•
Once you find the tested word in the passage, treat the question like aSentence Completion
question.
•
Pretend the word is a blank in the sentence.
•
Read a line or two around the imaginary blank if you need to.
•
Then predict a word for that blank.
•
Check the answer choices for a word that comes close to your prediction.
Vocabulary-in-Context questions may be worded as:
•
As used in line 8, ______ most nearly means . . .
•
Which of the following is the best description of this word’s meaning in the context of the
passage?
•
The term ______ most likely refers to . . .
P a g e 7 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
FUNCTION QUESTIONS
These Why? questions are a little tricky because they require you to take an extra step Beyond the
What? of the passage. To answer these questions affectively, you must put yourself in the author’s
place.
A Function question will ask you:
•
Why include this detail?
•
Why include this word?
•
Why include this sentence?
•
Why include this quote?
•
Why include this paragraph?
Your job is to look back in the passage using the line references or other hints the question stem
gives you. There you will discover clues to the answer. Most often, you have to read around any
lines they give you to find the answer.
Once you have seriously skimmed the passage, here’s how to attack the
questions:
Step 1. Read the question stem.
Step 2. Locate the material you need.
Step 3. Predict the answer.
Step 4. Select the best answer choice.
Question 1 refers to the following passage.
Recently, at my grandmother’s eightieth birthday party,
my family looked at old photographs. In one of them I saw
a scared little boy holding tightly to his mother’s skirt, and
I scarcely recognized myself. My foremost memory of that
(5)
time is simply being cold – the mild Vietnamese winters that
I had known couldn’t prepare me for the bitter winds of the
American Midwest. The cold seemed emblematic of everything I hated about my new country – we had no friends, no
extended family, and we all lived together in a two-room apart(10)
ment. My mother, ever shrewd, remarked that selling heat in
such a cold place would surely bring fortune, and she was right.
My parents now own a successful heating supply company.
1.
The author’s attitude towards the “scared little boy”
mentioned in line 3, indicates that the author
(A) is unsure that the photograph is actually of his
family
(B) believes that the boy is likely overly dependent
on his mother
P a g e 8 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
(C)
(D)
(E)
feels he has changed considerably since
childhood
regards his mother’s strategy to sell heating
supplies as clever
regrets his family’s move to the United States
Let’s take a look at your answer choices.
(A) is too literal – the author is speaking figuratively when he says he scarcely
recognizes the boy.
(B) goes too far in making an inference. Although the boy is sticking close to his
mother in the picture, there’s no evidence that the author thinks this is a bad
thing.
(C) is a great match for your prediction and is the correct answer.
(D) might be a true statement, but it comes later in the passage. It has nothing to
do with the author’s attitude toward the boy in the picture.
(E) like (B), is too great a leap and can’t safely be inferred from the information
in the passage.
PRACTICE TEST
Question 1 – 2 refer to the following passage
Recently, at my grandmother’s eightieth birthday party,
my family looked at old photographs. In one of them I saw
a scared little boy holding tightly to his mother’s skirt, and
I scarcely recognized myself. My foremost memory of that
(5)
time is simply being cold – the mild Vietnamese winters
that I had known couldn’t prepare me for the bitter winds
of theAmerican Midwest. The cold seemed emblematic of
everything I hated about my new country – we had no
friends, no extended family, and we all lived together in a
(10)
two-room apartment. My mother, ever shrewd, remarked
that selling heat insuch a cold place would surely bring
fortune, and she was right.My parents now own a successful heating supply company.
1.
In line 5, the author mentions “the mild Vietnamese
winters” in order to
(A) explain his grandmother’s childhood in Vietnam
(B) recall his past growing up in Vietnam
(C) detail the weather conditions in his home country
(D) describe how much he despises the cold
(E) provide contrast to how cold the author felt in
the new country
P a g e 9 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
2.
Lines 10-13 (“My mother . . . heating supply company”)
suggest that the author’s mother regarded the cold of the
American Midwest as
(A) more drastic than that of Vietnam
(B) an opportunity for economic success
(C) an obstacle to familial happiness
(D) symbolic of other challenges and problems
(E) unimportant to the family’s future
Questions 3-4 refer to the following passage.
Many mammals instinctively raise their fur when they
are cold – a reaction produced by tiny muscles just under
the skin which surround hair follicles. When the muscles
contract the hairs stand up, creating an increased air
(5)
space under the fur. The air space provides more effective
insulation for the mammal’s body, thus allowing it to retain
more heat for longer periods of time. Some animals also
raise their fur when they are challenged by predators or
even other members of their species. The raised fur
(10)
makes the animal appeal slightly bigger, and ideally, more
powerful. Interestingly, though devoid of fur, humans still
retain this instinct. So, the next time a horror movie gives
you “goose bumps,” remember that your skin is following a
deep-seated mammalian impulse now rendered obsolete.
3. The “increased air space under the fur” mentioned in
lines 4-5 serves primarily to
(A) combat cold
(B) intimidate other animals
(C) render goose bumps obsolete
(D) cool overheated predators
(E) make mammals more powerful
4.
Based on the passage, the author would most likely
describe goose bumps on humans as
(A) an unnecessary and unexplained phenomenon
(B) a harmful but necessary measure
(C) an amusing but dangerous feature
(D) a useless but interesting remnant
(E) a powerful but infrequent occurrence
P a g e 10 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
Questions 5-6 refer to the following passage:
Elizabeth Barrett Browning, a feminist writer of the Victorian Era, used her poetry
and prose to take on a wide range of issues facing her society, including “the woman
question.” In her long poem Aurora Leigh, she explores this question as she portrays
both the growth of the artist and the growth of the woman within. Aurora Leigh is
(5)
not a traditional Victorian woman – she is well educated and self-sufficient. In the
poem, Browning argues that the limitation placed on woman in contrast to the freedom
of men enjoy should incite women to rise up and effect a change in their
circumstances. Browning’s writing, including Aurora Leigh, helped to pave the way
for major social change in women’s lives.
5.
It can be inferred from the passage that the author believes the
traditional Victorian woman
(A) wrote poetry
(B) was portrayed accurately in Aurora Leigh
(C) fought for social change
(D) was not well educated
(E) had a public role in society
6.
As used in line 7 “effect” most nearly means
(A) imitate
(B) result
(C) cause
(D) disturb
(E) prevent
Questions 7-8 refer to the following passage.
Each passing evening brings more frustration. Tonight I spent an hour in front of
the typewriter staring at the silent keys, listening to the girl upstairs play the piano
and sing. I’d never noticed her ability before; she is remarkable. It seems everything
she plays is of her own spontaneous creation, an absolute movement of feeling.
(5)
Her music is a painting, the lines so intense and colouring that it manages to exist
above the realm of the material, impressing a desolate image of the pianist upon my
mind. But even the beauty of this pure work of art failed to inspire me and after she
stopped I was again without refuge.
7.
In line 2 the phrase “silent keys” implies that the narrator
(A) can’t play the piano
(B) is suffering from writer’s block
(C) is tone deaf
(D) is searching for clues
(E) is annoyed by the girl upstairs
P a g e 11 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
8.
The narrator uses the description of the girl’s playing in lines 3-4
(“It seems . . . feeling”) mainly to
(A) contrast it with his inability to write
(B) illustrate her talents as a musician
(C) compare it with painting
(D) criticize her lack of skill
(E) indicate his inferiority as an artist
Questions 9-10 refer to the following passage.
Bear Mountain State Park opened in 1916 and rapidly became a popular weekend
destination for many New Yorkers looking for an escape from the city grind. The ensuing unnaturally high volume of visitors to the area caused an upsurge in traffic, and it
was soon apparent that the ferry services used to cross the Hudson were insufficient. In
(5)
1922, the New York State Legislature introduced a bill that authorized a group of private
Investors led by Mary Harriman to build a bridge across the river. The group, known as
the Bear Mountain Hudson Bridge Company, was allotted thirty years to construct and
maintain the bridge, after which the span would be handed over to New York State.
9.
In context, “volume” (line 3) most closely means
(A) loudness
(B) pollution
(C) resentment
(D) capacity
(E) quantity
10.
According to the passage, which is true about the bridge?
I.
It was originally constructed by New York State.
II.
It opened to the public in 1916
III. It was necessitated by inadequate ferry services
(A)
(B)
(C)
(D)
(E)
Statement II
Statement III
Statements I and II
Statements I and III
Statements II and III
Questions 11-12 refer to the following passage.
Like the writers of the Beat Generation almost a half-century before, many of the
original grunge musicians who helped give birth to a movement were horrified at the
final result of their efforts. Grunge music and culture was spawned in Seattle in the
late 1980s as an underground revolt against the shallow values of the time. But with
(5)
Nirvana’s 1991 release of Nevermind and the successive popularity of other Seattle
bands like Pearl Jam and Sound Garden, the once-countercultural grunge movement
skyrocketed into popular consciousness. Many aspects of the culture that were
P a g e 12 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
originally forms of rebellion, such as the hairstyles and fashions worn by grunge
musicians, found their way into the most mainstream of places.
11.
In line 1, the author invokes the Beat Generation in order to
(A) detail an earlier movement in music
(B) introduce the topic by illustrating a similarity
(C) imply the insignificance of grunge
(D) recall an important time in American cultural history
(E) emphasize grunge’s influence upon today’s music
12.
As used in line 2, “movement” most closely means
(A) an organized attempt at change
(B) a specific manner of moving
(C) the changing of location or position
(D) a rhythmic progression or tempo
(E) a mainstream belief
Questions 13-14 refer to the following passage
Bovine spongiform encephalopathy (BSE) is a fatal, transmissible, neurological
disorder found in cattle that slowly attacks a cow’s brain cells, forming what resembles sponge-like holes in the brain. As the disease progresses, the cow begins to
behave abnormally, hence BSE’s more common name, “Mad Cow Disease.”
(5)
(10)
On December 23, 2003, the first case of BSE in the U.S. was detected in a cow from
Washington State. The ensuing national hysteria was largely unfounded; years earlier,
in response to the previous epidemics abroad, the USFDA had implemented preventative measures to contain an outbreak of the disease before it could spread. These
measures were in place for a good reason: there is a causal link between eating
BSE-infected meat and the development of a fatal human brain disorder known as
new variant Creutzfeldt-Jakob Disease (nvCJD).
13.
The first paragraph mostly serves to
(A) explain the origin of the term “Mad Cow Disease”
(B) introduce background information of BSE
(C) warn of BSE’s transmissibility to humans
(D) dissuade people from eating meat
(E) provide in-depth description of BSE’s different stages
P a g e 13 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
14.
Which of the following best describes the “national hysteria” in line 6 of the
passage?
(A) totally justified
(B) necessary for USFDA policy change
(C) rooted in misinformation about BSE’s
(D) completely inexcusable
(E) understandable, but unnecessary
Question 15-16 refer to the following passages.
Ann’s footsteps crunching upon the fallen leaves are amplified by the pre-dusk
serenity that is quickly setting upon the forest. The path before her is quickly dissolving into the growing shadows, yet the fear that would normally be creeping into
her chest is absent. Somewhere secretly inside she finds the prospect of disappearing
(5)
into the woods exhilarating. Liberated, all her daily burdens would go as the daylight
goes, only the empty night, spattered with benevolent stars. But as the trail opens
onto her backyard, Ann is surprised to find herself breaking into a trot, eager to
return to the familiar warmth of her home.
15.
Ann’s absence of fear (lines 3-4) suggests that she
(A) is brave in the face of danger
(B) doesn’t realize she is lost
(C) is an apathetic person
(D) longs for a change in her life
(E) knows exactly where she is
16.
In line 6, “the empty night” is symbolic of
(A) everyday life
(B) a lack of responsibility
(C) being lost
(D) loneliness
(E) death
Question 17-20 refer to the following passages.
Passage 1
Acid rain clouds, formed by the release of gases from burning fossil fuels, join with existing
weather patterns and eventually pour down toxic, highly acidic water droplets that can cause
significant and often irreversible environmental damage. However, nuclear power and
renewable energy technologies – those that take advantage of continuously available
(5)
resources such as the sea, sun and various biofuels – can generate electricity without giving
off the gases that contribute to acid rain, and there are proven ways to effectively sequester
the harmful gases generated by fossil fuel plants. Yet as acid rain continues to seriously
damage countless waterways, forests, crops, and even to erode buildings, senselessly little is
being done to take advantage of these new technologies.
P a g e 14 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
Passage 2
(10)
While the world’s most-developed nations have the luxury of squabbling over the political
and environmental questions raised by those who actually have energy choices, the
developing world usually has only one resource to turn to: coal. One of the cheapest and most
plentiful sources of energy in the world, coal is used to generate nearly 40 percent of the
world’s electricity. But when burned, coal releases large amounts of carbon dioxide - a gas
that, when
(15)
present in excess, can cause a whole host of serious respiratory diseases. So while wealthy
nations can complain about global warming and acid rain, the rest of the world must struggle
to cope with the immediate human damage caused by the only natural resource they can
afford.
17.
In passage 1, the author’s attitude toward the continuing presence of acid rain is best
described as
(A) astonishment that acid rain remains a problem in the developed world
(B) frustration that the use of cleaner technologies is not more widespread
(C) irritation that nothing is being done to curb the creation of acid rain
(D) impatience towards plants that refuse to adopt experimental technologies
(E) scepticism that irreversible damage is really being done to the environment
18.
In passage 2, the author characterizes “the world’s most-developed nations” (line 10)
as which of the following?
(A) Insensitive
(B) Responsible
(C) Privileged
(D) Reckless
(E) Impoverished
19.
How would the author of passage 2 most likely respond to the assertion in passage 1
that “senselessly little” (line 9) is being done to take advantage of new and cleaner
energy-generation technologies?
(A) Wealthier nations have a responsibility to create opportunities for those less
fortunate.
(B) Most countries would adopt these technologies if they were affordable.
(C) The environmental impact of an energy source is just as important as the
cost of energy.
(D) Environmental damage is less significant than damage to humans.
(E) Not all countries can afford these technologies.
P a g e 15 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
20.
The authors of both passages agree that
(A) clean energy technologies are more expensive than conventional methods
(B) acid rain is a problem inevitably created by energy creation
(C) the burning of fossil fuels can release harmful gases
(D) the environmental debate over energy generation is only intensifying
(E) the human and environmental impacts of energy generation are equally
important
Funeral Blues
Wystan Hugh Auden
Stop all the clocks, cut off the telephone,
Prevent the dog from barking with a juicy bone.
Silence the pianos and with muffled drum
Bring out the coffin, let the mourners come.
Let aeroplanes circle moaning overhead
Scribbling on the sky the message: He is Dead,
Put white crépe bows round the necks of the public doves,
Let the traffic policemen wear black cotton gloves.
He was my North, my South, my East and West,
My working week and my Sunday rest,
My noon, my midnight, my talk, my song,
I thought that love would last forever: I was wrong
The stars are not wanted now, put out every one;
Pack up the moon and dismantle the sun;
Pour away the ocean and sweep up the wood.
For nothing now can ever come to any good.
P a g e 16 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
Ontvlugting
Ingrid Jonker
Uit hierdie Valkenburg het ek ontvlug
en dink my nou in Gordonsbaai terug:
Ek speel met paddavissies in ’n stroom
en kerf swastikas aan ’n rooikransboom
Ek is die hond wat op die strande draf
En dom-allenig teen die aandwind blaf
Ek is die seemeeu wat verhongerd daal
En dooie nagte opdis vir ’n maal
Die god wat jou geskep het uit die wind
sodat my smart in jou volmaaktheid vind:
My lyk lê uitgespoel in wier en gras
op al die plekke waar ons eenmaal was.
P a g e 17 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
The More Loving One
Wystan Hugh Auden
Looking up at the stars, I know quite well
That, for all they care, I can go to hell,
But on earth indifference is the least
We have to dread from man or beast.
How should we like it were stars to burn
With a passion for us we could not return?
If equal affection cannot be,
Let the more loving one be me.
Admirer as I think I am
Of stars that do not give a damn,
I cannot, now I see them, say
I missed one terribly all day.
Were all stars to disappear or die,
I should learn to look at an empty sky
And feel its total dark sublime,
Though this might take me a little time.
P a g e 18 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
Uittreksel uit Convivium en die digbundel Mede-wete
Antjie Krog
2.
Maar staafspirale bestaan sê jy
die lieflike vere-rige slanke spiraalarms van ons sterrestelsel bestaan
in ‘n dromende ballet sê jy van waas en wentelbaan
en die stadig-polsende lewensloop van sterre suurstof
en kwasars newelagtige baarmoeders en ontsaglike lig
aan die buitewyke van die heelal sê jy die melkweg neurie
daar’s rustig silwersuiwerende wentelbane die beminning
van mane getye liglyne en ewigdurende ewewig dit alles
bestaan sê jy: is dit nie juis die sterre as ‘n oordaad priemsels op
‘n stil someraand wat ons in wentelbane van uitreiking hou nie?
elke keer as ons in oorgawe na mekaar toe draai
doen ons dit onder ‘n druisende baldakyn van sterre
3.
die univers word anders ontkurk sê jy as bloot deur
oorlewingsgeweld kyk hoe halshemeld vanoggend
die uithaal van lig oor die Mooiberge die liriese teloorgang
die oorskulp-hartbreekliturgie van die herfsmens hurk verwonderd
hierin want welgeluksalig is ons
dat hierdie smal benerige erfsterflike liggaam
soveel oorvloed kan stemvurk
P a g e 19 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
VOCABULARY TEST
1.
Hans found the movie stubs lying on the counter to be ____________ evidence that his friends had gone to
the cinema without him; it was unquestionable that they had seen “Spider-Man”.
(A) immaterial
(B) potential
(C) indisputable
(D) incriminating
(E) nominal
2.
Despite his apparent ____________ lifestyle, the old man was known to drink to excess when visited by
friends.
(A) temperate
(B) laconic
(C) aesthetic
(D) duplicitous
(E) voluble
3.
The waiter performs his job with ____________ and hopes that, if he continues to work ____________ he will
eventually be promoted to maitre’d.
(A) sagacity . . . unscrupulously
(B) leniency . . . decorously
(C) nonchalance . . . tenaciously
(D
acrimony . . .cheerfully
(E) ardor . . . assiduously
4.
Though the film ostensibly deals with the theme of ____________ , the director seems to have
been more interested in its absence – in isolation and the longing for connection
(A) reliance
(B) fraternity
(C) socialism
(D) privation
(E) levity
5.
Everything the candidate said publicly was ____________ ; he manipulated the media in order to present
the image he wanted.
(A) incendiary
(B) calculated
(C) facetious
(D) scrupulous
(E) impromptu
P a g e 20 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
6.
Even though we are supposed to be more evolved than animals, the human tendency toward
____________ and egocentrism, shows that people sometimes can carry a narrow view of the world.
(A) anthropocentrism
(B) humanity
(C) irrationality
(D) temerity
(E) serendipity
7.
Many biologists are critical of the film’s ____________ premise that dinosaurs might one day return.
(A) scientific
(B) tacit
(C) speculative
(D) unwitting
(E) ambiguous
8.
Aristotle espoused a ____________ biological model in which all extant species are unchanging and
eternal and no new species ever came into existence.
(A) paradoxical
(B) morbid
(C) static
(D) holistic
(E) homogeneous
9.
Some critics believe that the ____________ of modern art came with dadaism, while others insist that the
movement was a ____________ .
(A) zenith . . . sham
(B) pinnacle . . . triumph
(C) decline . . . disaster
(D) acceptance . . . success
(E) originality . . . fiasco
10.
The writings of the philosopher Descartes are ____________ ; many readers have difficulty following his
complex, intricately woven arguments.
(A) generic
(B) trenchant
(C) reflective
(D) elongated
(E) abstruse
P a g e 21 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
QUANTITATIVE LITERACY/ MATHS LITERACY
GENERAL ALGEBRA
1.
How many seconds are there in a day?
(A) 5184000
2.
,-.÷0
12(42,,)
6
4.
(D) 𝑏 > 𝑎 > 𝑑 > 𝑐
(E) 𝑎 > 𝑏 > 𝑑 > 𝑐
(E) 14400
8
(D) 70
𝟖
(E) 𝟒𝟓
(C) 𝒃 > 𝑎 > 𝑐 > 𝑑
If 𝑎 = 4√3 , 𝑏 = 5√2 and 𝑐 = 3√5 , which of the following is true?
(B) 𝑎 > 𝑐 > 𝑏
(C) 𝒃 > 𝑎 > 𝑐
(D) 𝑏 > 𝑐 > 𝑎
(E) 𝑐 > 𝑏 > 𝑎
If 3 ≤ 𝑚 ≤ 10 and 12 ≤ 𝑛 ≤ 21 then the difference between the largest and smallest possible
I
values of J is
74
(B) 0
,4
(C) 1M
,4
(D) ,7
,4
(E) 68
In how many ways can 50 be expressed as the sum of two prime numbers?
(A) 1
(B) 2
(C) 3
(D) 𝟒
(E) 5
When the numbers below are arranged in increasing order, which number is in the middle?
(A) 5√6
8.
8
(C) 90
(B) 𝑎 > 𝑐 > 𝑏 > 𝑑
𝟐𝟗
7.
8
(B) 70
(A) 𝑎 > 𝑏 > 𝑐 > 𝑑
(A) 𝟒𝟐
6.
(D) 𝟖𝟔𝟒𝟎𝟎
If 𝑎 = 75, 𝑏 = 36√5 , 𝑐 = 72 and 𝑑 = 32√5 , which of the following is true?
(A) 𝑎 > 𝑏 > 𝑐
5.
(C) 28800
is equal to
(A) 70
3.
(B) 1440
(B) 7√3
(C) 5√5
(D) 3√7
(E) 𝟖√𝟐
If 𝑎 and 𝑏 are composite numbers such that 𝑎 + 𝑏 = 31, what is the biggest possible value of
𝑎𝑏?
(A) 110
(B) 160
(C)180
(D) 𝟐𝟒𝟎
(E) 270
P a g e 22 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
9.
A tap leaks at a rate of one drop per second. A drop contains one-fifth of millilitre of water.
What is the total loss (to the nearest litre) from the dripping tap in one week?
(A) 100
(B) 106
(C) 115
(D) 𝟏𝟐𝟏
(E) 137
DECIMALS AND FRACTIONS
,
10. When 8, is written as a recurring decimal, what is the hundredth digit after the decimal point?
(A) 0
(B) 2
(C) 3
(D) 4
(E) 9
11. If 𝑎 = 0.432432432432 … and 𝑏 = 0.45454545 … are recurring decimals as indicated,
then 𝑎 + 𝑏 is equal to
.04
(A) 8M0
𝟑𝟔𝟏
.9.
(B) 𝟒𝟎𝟕
(C) 8M4
.90
(E) 8,.
(D) 8,,
.91
12. Which of the following numbers is the smallest?
.
(A) 0
13.
T T
2
U V
T T
2
V W
𝟒
(C) .
7
(D) 6
0
(E) 8
𝟒
(C) .
0
(D) 6
,,
(E) 80
1
(E) ,.
(B) 𝟕
−
1
(A) ,7
T T
U V
T T
V W
.
is equal to:
(B) 𝟕
6
14. Which of the following fractions is the biggest?
7
(A) .
𝟑
0
(B) 𝟒
(C) 1
,
4
(D) ,M
,
15. If 𝑎 − 𝑏 = 3 and 𝑎𝑏 = 7 , then Y − Z is equal to:
,
(A) .
𝟑
(B) − 𝟕
1
(C) .
,
(D) − .
,
(E) 1
16. Which of the following CANNOT be a person’s height?
(A)
(B)
(C)
(D)
(2000 × 102. )m
0,0019 km
180 cm
17950 mm
P a g e 23 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
LCD AND FACTORS
17.
,
,
7J-,
,
+ (7J-,)(8J-,) + (8J-,)(9J-,)is equal to:
0
7
(A) (7J-,)(9J-,)
(B) 8J-,
8
𝟑
(C) (7J-,)(8J-,)
(D) 𝟔𝒏-𝟏
.
(E) (7J-,)(8J-,)(9J-,)
18. Factorise (𝑥 7 − 5𝑥)7 − 36
(A) (𝒙 − 𝟑)(𝒙 − 𝟐)(𝒙 − 𝟔)(𝒙 + 𝟏)
(B) (𝑥 + 3)(𝑥 − 2)(𝑥 + 6)(𝑥 + 1)
(C) (𝑥 − 3)(𝑥 + 2)(𝑥 − 6)(𝑥 − 1)
(D) (𝑥 − 3)(𝑥 + 2)(𝑥 + 6)(𝑥 + 1)
(E) (𝑥 + 3)(𝑥 − 2)(𝑥 − 6)(𝑥 − 1)
19.
7
.
d-,
8
− d-7 + d27 is equal to:
𝟑𝒙𝟐 -𝟏𝟓𝒙-𝟔
(A) 𝒙𝟑 -𝒙𝟐 2𝟒𝒙2𝟒
20.
.
.d U 2,0d29
(D) d V-d U28d28
.
(E) d28
7
d-,
22.
.d U2,0d-9
(C) d V -d U -8d28
− d-,is equal to:
d27
𝒙-𝟕
(A) d U 2d27
21.
.d U-,0d29
(B) d V 2d U28d28
ed U2,fed V2,f
(d2,)U
(B) 𝒙𝟐 2𝒙2𝟐
0d2,
(C) d U -d27
d27
(E) d U-d-7
is equal to:
(A) 𝑥 . + 𝑥 7 + 𝑥 + 1
(B) 𝒙𝟑 + 𝟐𝒙𝟐 + 𝟐𝒙 + 𝟏
(D) 𝑥 . − 2𝑥 7 − 2𝑥 + 1
(E) 𝑥 . − 2𝑥 7 + 2𝑥 − 1
d-,
0d2,
(D) d U2d27
(C) 𝑥 . − 𝑥 7 + 𝑥 − 1
d-7
d U -.d-7
+ d U -0d-9 is equal to:
7d-.
(A) 7d U-6d-6
7d-.
(B) d W -,0d U-,7
𝟐𝒙-𝟓
(C) (𝒙-𝟐)(𝒙-𝟑)
.d-7
(D) (d-,)(d-7)(d-.)
7d-.
(E) d V-0d U -4d-9
P a g e 24 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
RATIOS, PERCENTAGES AND AVERAGES
23. Two positive numbers are such that their sum, difference and product are in the ratio 6 ∶ 4 ∶
15. The smaller of the two numbers is:
(A) 15
(B) 9
(C) 4
(D) 𝟑
(C) 60
(D) 61
24. 28% of 75 plus 45% of 80 is equal to:
(A) 55
(B) 𝟓𝟕
(E) 65
25. Thabo buys 𝑝 oranges. He squeezes 𝑞% of them to make fresh orange juice. How many
oranges are left?
kl
(A) ,MM
(B)
𝟏𝟎𝟎𝒑2𝒑𝒒
(C)
𝟏𝟎𝟎
,MMk2l
,MM
k2l
(D) ,MM
(E)
,MMl2kl
,MM
26. There are 200 goats and 50 sheep on a farm and a set amount of food for them. One goat eats
twice as much as one sheep. How many more sheep can he have if he only farms with sheep?
(A) 100
(B) 150
(C) 250
(D) 400
(E) 450
27. There are 5000 trees in a forest. 2000 are pines, the rest are wattles. How many times more
wattles than pines are there?
0
.
(A) .
(B) 0
7
𝟑
(C) .
(D) 𝟐
,
(E) .
J
28. If 75% of n is equal to k % of 25, what is p ?
(A)
(B)
(C)
(D)
8
.
7
.
,
.
7
0
29. If the ratio of red, green and blue balls is 1: g: 5 respectively and there are 150 blue balls out
of a total of 240, what is the value of g?
(A) 2
(B) 3
(C) 4
(D) 5
P a g e 25 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
30. The average mark for a class of boys and girls out of 100 is 55. If the average mark for “b”
boys is 52 and the average mark for “g” girls is 61, then which of the following statements is
true?
(A) b = 2g
(B) b = 3g
(C) b = 4g
(D) b = 5g
REMAINDERS AND PROBABILITY
31. A five-digit number 1234𝑋 is dvisible by 9. What is the remainder when it is divided by 7?
(A) 𝟎
(B) 1
(C) 2
(D) 4
(E) 5
(D) 𝟒𝟒𝟒 𝟒𝟒𝟒
(E) 5 555 555
32. Which one of the following numbers is divisible by 7?
(A) 11
(B) 2222
(C) 33 333
SUBJECT OF THE FORMULA, EQUATIONS, SIMULTANEOUS EQUATIONS
,
.
33. If d-. = 4 then d-7 is equal to:
(A) 1
d
(B) -2
(C) 3
(D) -4
(E) 5
.
34. If r = 8 then the incorrect expression in the following is:
d-r
1
r
(A) r = 8
(B) r2d = 4
35. If 1 - 1 = 1 - s
u v
r
then r is equal to:
(A)
( s -1) (v - u) (B) uv(s - 1)
u -v
,
(C)
d-7r
d
,,
= .
𝒙2𝒚
𝟏
(D) 𝒚 = 𝟒
(C) s - 1
u -v
(D) uv ( s - 1)( u - v )
(C) 6
(D) 4
,
36. If 𝑥 + d = 3 , then 𝑥 7 + d U is equal to
(A) 9
(B) 𝟕
67
(E) 3√3
P a g e 26 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
,
,
,
37. If d + r = 7 and 𝑥𝑦 = 6 , what is the value of 𝑥 + 𝑦 ?
(A) 1
(B) 2
(C) 𝟑
(D) 4
(E) 6
38. If 𝑎 + 𝑏 = 22 , 𝑏 + 𝑐 = 25 and 𝑐 + 𝑎 = 23 , then 𝑎𝑏 + 𝑏𝑐 + 𝑐𝑎 is equal to
(A) 387
(B) 𝟒𝟎𝟔
(C) 491
(D) 510
(E) 674
(D) 49
(E) 55
39. If 𝑎 + 𝑏 = 7 and 𝑎𝑏 = 4 , what is the value of 𝑎7 + 𝑏7 ?
(A) 28
(B) 40
(C) 𝟒𝟏
40. If 2𝑎 − 3𝑏 = 4 and 7𝑎 + 2𝑏 = 9, then 𝑎 + 𝑏 is equal to
.
(A) 1
(B) 𝟏
0
(C) − 7
1
(D) 8
(E) −3
(D) 20
(E) 27
41. If 𝑥 + 𝑦 = 12 and 𝑥 . + 𝑦 . = 1188 then the value of 𝑥𝑦 is
(A) 11
(B)𝟏𝟓
(C) 18
42. If 𝑥𝑦 = 7 and 𝑥 + 𝑦 = 3 what is the value of (𝑥 + 1)(𝑦 + 1) ?
(A) 10
(B)𝟏𝟏
(C) 12
(D) 13
(E) 14
43. If 𝑥 = 2 + 3u and 𝑦 = 2 + 32u then 𝑦 expressed as a function of 𝑥 is:
.
.d20
(A) d27
(B) d27
𝟐𝒙2𝟑
7d2,
(C) 𝒙2𝟐
(D) d27
(E) −𝑥
(C) 5
(D) 2√5 − 2
(E) 25 − 2√5
(C) 𝟓
(D) 6
(E) 7
44. If 𝑥 = √5 − 1 then 𝑥 7 − 1 is equal to
(A) √3
,
(B)𝟓 − 𝟐√𝟓
,
,
45. If J + J(J2,) = 8 , then 𝑛 is equal to
(A) 3
(B) 4
46. If 𝑎 − 𝑏 − 𝑐 = 2 , 𝑏 − 𝑐 − 𝑎 = −3 and 𝑐 − 𝑎 − 𝑏 = 5 , then 𝑎 + 𝑏 + 𝑐 is
(A) −1
(B) −2
(C) −3
(D) −𝟒
(E) −5
P a g e 27 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
WORD PROBLEMS
47. A bucket weighs 𝐹 𝑘𝑔 when full of water and 𝐻 𝑘𝑔 when it is half full. What is the weight of
the bucket (in kilograms) when it is empty?
,
(A) 𝐹 − 𝐻
(B) 7 (𝐹 + 𝐻)
,
(E) 𝟐𝑯 − 𝑭
(D) 1
,0
(E) 78
(C) 𝟑𝟎
(D) 9
(E) 12
(C) 2J2, − 1
(D) 27J
(E) 7
(D) 7 (𝐹 − 𝐻)
(C) 𝐹 − 2𝐻
STRANGE SYMBOLS
,
48. If𝑝 ⊛ 𝑞 means k-l then 2⊛ (3 ⊛ 4) is equal to
,
(A) 4
𝟕
(B) 24
(C) 𝟏𝟓
,
49. If 𝑎 ∗ 𝑏 = 𝑎 + 𝑎𝑏 then the value of 3 ∗ (3 ∗ 2) is
(A) 24
(B) 18
MATHEMATICS EXPONENTS
50. 2J − 2J2, is equal to:
(A) 𝟐𝒏2𝟏
(B) 2
,
51. (𝑎𝑏)2, (𝑎2, + 𝑏2, )2, is equal to:
52.
(A) 𝑎 2, + 𝑏 2,
(B) 𝑎𝑏(𝑎2, + 𝑏 2, )
(D) 𝑎𝑏(𝑎 + 𝑏)2,
(E) (𝑎𝑏)2, (𝑎 + 𝑏)
V
(C) (𝒂 + 𝒃)2𝟏
W
√4 × √8 is equal to:
(A) √2
•
(B) 2√12
•
TU
𝟏𝟐
(C) √32
(D) √32
(E) 𝟐 √𝟑𝟐
(A) 𝟑𝒂𝟐 𝒃𝟔 √𝟒𝒂𝟒 + 𝟗𝒃𝟔
(B) √117𝑎,7 𝑏.M
(C) 9𝑎8 𝑏,7 √9𝑎8 + 6𝑏9
(D) €117(𝑎,7 + 𝑏.M )
(E) 9𝑎7 𝑏4 + 6𝑎8 𝑏9
53. √81𝑎8 𝑏,6 + 36𝑎6 𝑏,7 is equal to:
P a g e 28 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
T
54. If [(27J + 1)(27J − 1) + 1]W = 256 then 𝑛 is equal to:
(A) 2
(B) 4
(C) 6
(D) 𝟖
(E) 10
55. (𝑎2, + 𝑏2, )2, is equal to:
,
(A) 𝑎 + 𝑏
,
(B) YU + ZU
Y
Z
(C) Z + Y
𝒂𝒃
(D) 𝒂-𝒃
(E) 𝑎7 + 𝑏7
56. If 𝑚 and 𝑛 are positive integers such that 6I + 2I-J + 3I + 2J = 85, what is
the value of 𝑚7 + 𝑚𝑛 + 𝑛7 ?
(A) 𝟏𝟗
(B) 18
(C) 20
(D) 17
(E) 21
(D) 𝟒𝟎𝟏𝟗
(E) 6029
57. If 3J + 3J + 3J = 97M,M then 𝑛 is equal to:
(A) 2010
(B) 2007
58. One solution to the equation:
(C) 2004
√𝑥 + √𝑥 + √𝑥 + √𝑥 = √𝑥 × √𝑥 × √𝑥 × √𝑥 is:
W
(A) 1
(B) √16
𝟑
(C) √𝟏𝟔
(D) 4
59. Given that 2 x +1 + 2 x = 3 y + 2 - 3 y and that x and y are integers, calculate the value
of x + y:
(A) 6
60. The expression
(B) 5
(D) 3
( x + y ) ( x-1 + y -1 ) is the same as:
-1
(B) 12 + 12
x
y
(A) 1
xy
(2.)(ƒT)
61. The value of (−8)
(A) 24
(C) 4
(C) x 2 + y 2
(D) xy
,
(D) 2
(E) −2
(D) 3
(E) 2
is:
𝟏
(B) − 𝟐
(C) 78
62. If 2I 3J = 64 × 68 , what is the value of 𝑚 − 𝑛?
(A) 𝟔
(B) 5
(C) 4
P a g e 29 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
63.
,
√1-√0
(A)
64.
is equal to:
,
,
√,7
0√.2.√0
√02√.
,
(B) .√7
(C)
(B) −1
(C)√41
√1
,
+
√0
𝟏
√,7
(D) 𝟐 e√𝟕 − √𝟓f
(E)
(D) 49
(E)√𝟏𝟓
√.0
is equal to:
(A) 0
„
65. €𝑎,88J is equal to:
(A) 𝑎,7J
V
(B) 𝑎,7J
W
(C) 𝑎17J
W
(D) 𝒂𝟕𝟐𝒏
𝟔
(E) 𝑎17J
V
66. √64𝑎,9 𝑏,7 + 100𝑎6 𝑏,9 is equal to:
(A) 8𝑎6 𝑏9 + 10𝑎8 𝑏,9
(B) 𝟐𝒂𝟒 𝒃𝟔 √𝟏𝟔𝒂𝟖 + 𝟐𝟓𝒃𝟒
(C) 2𝑎6 𝑏,7 √16𝑎6 + 25𝑏8
(D) 8𝑎8 𝑏. + 10𝑎8 𝑏6
(E) 16𝑎6 𝑏,7 √4𝑎6 + 5𝑏8
67. (𝑎 + 𝑏)2, (𝑎2, + 𝑏 2, ) is equal to:
(A) (𝑎 + 𝑏)7
7
(B) (𝑎 + 𝑏)2,
(C) (𝒂𝒃)2𝟏
(D) 𝑎7 + 𝑏7
(E) 𝑎𝑏
8
68. e√3 − √2f e√3 + √2f is equal to:
(A) 𝟓 + 𝟐√𝟔
(B) 7 − 3√6
(D) 2√3 + 3√2
(E) 1 + 2√2 + 3√3
(C) 4 + 3√6
U
69. €168d is equal to:
(A) 47d
70.
,
Y ƒT -Z ƒT
U
(B) 47d
(C) 167d
𝟐
(D) 𝟏𝟔𝟐𝒙
(E) 168d
is equal to:
(A) 𝑎 + 𝑏
,
(B) Y-Z
Y-Z
(C) YZ
𝒂𝒃
(D) 𝒂-𝒃
(E) −𝑎 − 𝑏
P a g e 30 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
71. √100𝑎8 𝑏,9 + 36𝑎6 𝑏,7 is equal to:
(A) 𝟐𝒂𝟐 𝒃𝟔 √𝟗𝒂𝟒 + 𝟐𝟓𝒃𝟒
(B) 10𝑎7 𝑏6 + 6𝑎8 𝑏9
(C) 2𝑎8 𝑏9 (5𝑎7 + 3𝑏7 )
(D) 10𝑎7 𝑏8 + 6𝑎8 𝑏9
(E) 2𝑎7 𝑏9 (3𝑎7 + 5𝑏7 )
72.
72√.
7√.2.
is equal to:
(A) √3
(B) 2 + √3
𝟏
(C) 3 − √3
(D)
(C) 2€2√6
(D) 3
(E) √3 − 1
√𝟑
73. €5 + 2√6 − €5 − 2√6 is equal to:
(A) 𝟐√𝟐
74.
(B) √10
(E) 2√6
V
€2771d 71
(A) 27.d
…
(B) 27.d
V
(C) 2771d
…
𝟐𝟕
(D) 𝟐𝟕𝟗𝒙
(E) 27.d
U•
75. Which of the following statements is true?
76.
(A) 106 < 5,7 < 278
(B) 𝟐𝟐𝟒 < 𝟏𝟎𝟖 < 𝟓𝟏𝟐
(D) 5,7 < 106 < 278
(E) 278 < 5,7 < 106
If 2.3a = 54 and 5.2b = 80 what is the value of a + b?
(A) 1
77.
(C) 106 < 278 < 5,7
(B) 4
(C) 5
(D) 6
(E) 7
(D) 39
(E) 75
7
What is the value of e√27 − √12f ?
(A) 3
(B) 13
(C) 15
LOGARITHMS
78. 𝑙𝑜𝑔7 4 × 𝑙𝑜𝑔4 3 is equal to:
(A) 𝑙𝑜𝑔,, 7
(B) 𝒍𝒐𝒈𝟏𝟏 𝟏𝟏
(C) 𝑙𝑜𝑔,, 12
(D) 𝑙𝑜𝑔,, 9
(E) 𝑙𝑜𝑔,, 8
P a g e 31 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
79. (𝑎; 𝑏)is a solution to the simultaneous equations:
𝑙𝑜𝑔7 𝑥 = 𝑙𝑜𝑔8(𝑥 + 12)and 𝑦 − 2𝑥 = 2
The value of 𝑙𝑜𝑔(𝑦) is:
(A) 4
(B) 10
(C) −1
(D) 𝟏
80. The solution of the inequality log 7 𝑥 + log 7 (𝑥 − 3) < 2 is:
(A) −𝟏 < 𝑥 < 4
(B) 3 < 𝑥 < 4
(D) 4 < 𝑥 < 6
(E) 1 < 𝑥 < 5
(C) 0 < 𝑥 < 3
81. If 2Y = 5 and 5Z = 10 then 𝑎𝑏 is equal to:
(A) 𝐥𝐨𝐠 𝟐 𝟏𝟎
(B) 5
(C) 10
(D) log 0 20
(E) 25
82. log 0 500 is equal to:
(A) 𝟑 + 𝟐 𝐥𝐨𝐠 𝟓 𝟐
(B) 5 + log 0 10
(D) log 0 200 + log0 300
(E) 2 log 0 250
(C) 10 + log0 50
83. 𝑙𝑜𝑔. 9 × 𝑙𝑜𝑔8 2is equal to:
(A) log 1 18
(B) log,7 7
(C) log 1 11
(D) log,7 18
(E) 𝐥𝐨𝐠 𝟕 𝟕
84. log 𝑎7 − 𝑙𝑜𝑔𝑏7 is equal to:
(A) log(𝑎7 − 𝑏7 )
(B) 2log (𝑎 − 𝑏)
𝒂
(D) 𝟐 𝐥𝐨𝐠 “𝒃”
(C) (𝑙𝑜𝑔𝑎 − 𝑙𝑜𝑔𝑏)(𝑙𝑜𝑔𝑎 + 𝑙𝑜𝑔𝑏)
(E) 2log (𝑎 + 𝑏)
85. log 7 8 × log 6 64 is equal to:
(A) 2
(B) 4
(C) 5
(D) 𝟔
(E) 8
.
(E) .
86. If log,M 2 = 𝑎 then log,M 500 is equal to:
(A) 𝟑 − 𝒂
(B) 3 + 𝑎
(C) 3𝑎
(D) Y
Y
P a g e 32 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
FACTOR AND REMAINDER THEOREM
87. What is the remainder when 3𝑥 . + 𝑥 7 + 8𝑥 − 7 is divided by 𝑥 7 + 𝑥 + 2?
(A) 𝟒𝒙 − 𝟑
(B) 2𝑥 + 1
(C) 3𝑥 − 2
(D) 𝑥 − 4
(E) 5
88. What is the remainder when 𝑥 8 + 4𝑥 . − 3𝑥 7 − 6 is divided by 𝑥 8 − 1 ?
(A) 4𝑥 . + 12𝑥 7 − 3
(B) 5𝑥 . + 3𝑥 7 + 5
(D) 𝟒𝒙𝟑 − 𝟑𝒙𝟐 − 𝟓
(E) 𝑥 . + 3𝑥 7 − 1
(C) 3𝑥 . + 4𝑥 7 − 1
89. Which of the following is not a factor of 6𝑥 8 − 23𝑥 . + 10𝑥 7 + 29𝑥 − 10 ?
(A) 3𝑥 − 1
(B) 𝑥 − 2
(C) 𝑥 + 1
(D) 𝟐𝒙 − 𝟑
(E) 2𝑥 − 5
90. When 42𝑥 . + 31𝑥 7 − 56𝑥 + 15 is divided by (2𝑥 − 1)(5 + 3𝑥) the quotient is:
(A) 2𝑥 − 3
(B) 3𝑥 − 7
(C) 6𝑥 + 5
(D) 14𝑥 + 1
(E) 𝟕𝒙 − 𝟑
91. If 𝑘𝑥 7 − 𝑘𝑥 − 6 is divisible by both 𝑥 + 1 and 𝑥 + 𝑚, find the value of
𝑘𝑚7 + 𝑘𝑚 + 6
(A) 𝟏𝟐
(B) 15
(C) 18
(D) 21
(E) 20
92. The remainder when 2𝑥 8 + 2𝑥 . − 𝑥 7 − 15𝑥 + 8 is divided by 2𝑥 7 − 4𝑥 + 1 is
(A) 4𝑥 + 5
(B) 3𝑥 − 1
(C) 𝟐𝒙 + 𝟑
(D) −𝑥 + 7
(E) 𝑥 + 4
(D) 𝟕
(E) 1
93. The remainder when 16𝑎7 + 3 is divided by 4𝑎 + 2 is
(A) 0
(B) −7
(C)−1
PATTERNS, SEQUENCES AND SERIES
,
94. The sum of 𝑛 terms of an arithmetic series is 7 𝑛(3𝑛 + 7). The tenth term of
the series is
(A) 32
(B) 47
(C) 29
(D) 38
(E) 51
P a g e 33 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
95. Find the sum of all integers between 100 and 200 which leave a remainder of
3 when divided by 10.
(A) 1684
(B) 1522
(C) 1376
(D) 1480
(E) 1296
96. The sides of a right-angled triangle are in geometric progression and the shortest side has
length 2. What is the length of the hypotenuse?
(A) √11
(B) 𝟏 + √𝟓
(C) 2√3
97. The sum of the first p odd numbers is:
p2 + p
(A) p 3
(B)
(C) p 2
2
(D) √10
(D)
(E) 3√2 − 1
p2 - p
2
,
98. The sum of the first n terms of the sequence is given as: 𝑆J = 𝑛7 (𝑛 + 1)7 . Determine the
8
third term of the sequence.
(A) 36
(B) 𝟐𝟕
(C)18
(D) 9
9
99. What is the sum of the first hundred digits of the decimal expansion of 1 ?
(A) 398
lU
(B) 𝟒𝟓𝟑
lV
(C) 560
(D) 583
(E) 601
lW
100. 𝑞 + 7 + 8 + 6 + ⋯ For which value of 𝑞 does the series converge?
(A) −1 < 𝑞 < 1
(D) −2 ≤ 𝑞 ≤ 2
(B) −1 ≤ 𝑞 ≤ 1
,
,
(E) − 7 < 𝑞 < 7
(C) −𝟐 < 𝑞 < 2
101. Evaluate:
7J
∑,M
J˜,(−1)
(A) 1
(B)−1
(C) 𝟏𝟎
(D) −10
(E) 0
102. The sum of 9 consecutive odd numbers is 189. What is the sum of the smallest value and the
largest value?
(A) 13
(B)16
(C) 29
(D) 𝟒𝟐
(E) 45
P a g e 34 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
103. If 4; 𝑥; 𝑦; is an arithmetic sequence and 𝑥; 𝑦; 18 is a geometric sequence, what is one possible
value of 𝑥?
(A)
(B)
,
8
,
7
(C) 1
(D) 4
FUNCTIONS AND INEQUALITIES
Z
104. The quadratic 𝑥 7 + 𝑎𝑥 + 𝑏 = 0 has equal roots. What is the value of YU ?
(A) 1
𝟏
(B) 2
(C) 𝟒
(D) −4
(E) ±2
105. If 𝑓(𝑥) is a quadratic function that 𝑓 (1) = 2 , 𝑓(2) = 4 and 𝑓(4) = 14,
determine 𝑓(3).
(A) 𝟖
(B) 9
(C) 10
(D) 11
(E) 12
106. The lines 𝑦 = 2𝑥 + 3 and 𝑦 = 𝑘𝑥 − 3 intersect above the line 𝑦 = 5. It follows that:
(A) 𝑘 < −8
(B) −8 < 𝑘 < 0
(D) 2 < 𝑘 < 8
(E) 𝒌 > 8
(C) 0 < 𝑘 < 2
107. If f ( x) = ax + 3 and f ( f (2)) - 3a = 11 , then a possible value of a is:
(A) -2
(B) -1
108.
(C) 3
(D) 4
y
x
A hyperbola with equation y = 6 and a parabola with equation y = x 2 - 7
x
intersect in three points. The x-coordinate of the positive intercept is:
(A)2
(B) 3
(C) 4
(D) 5
P a g e 35 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
109. The line y = a( x + b) + c is reflected in the y-axis. What is the new equation?
(A) y = c - ab - ax
(B) y = ab - c + ax
(C) y = ab + c - ax
(D) y = ax - ab - c
110. Given:
a b
- + c = 0 then x can be expressed as:
x2 x
(A) x =
-b ± b2 + 4ac
2a
(B) x =
(C) x =
b ± b2 - 4ac
2c
(D) x =
2a
-b ± b 2 - 4ac
2c
-b ± b 2 - 4ac
r28
111. What is the range of the function: 7(d-.)U = 1
(A) 𝑦 ≤ 1
(B) 𝑦 ≤ 2
112. Determine 𝑓 2, (𝑥) if 𝑓 (𝑥 ) = T
(C) 𝒚 ≥ 𝟒
(D) 𝑦 ≥ 5
.
-7
•
7d
𝒙
(A) d-.
7d
(B) 𝟑2𝟐𝒙
(C) d2.
(D)
.d-7
d
, d
113. Which graph represents the function 𝑚(𝑥 ) = − “.”
y
y
(A)
x
(B)
y
x
y
(C)
(D)
x
x
P a g e 36 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
d-,
114. If 𝑓(𝑥 ) = d2, and 𝑓 (𝑎) = 5, what is the value of 𝑓(2𝑎)?
(A) 𝟐
(B) 4
(C) 6
(D) 8
(E) 10
115. If 𝑓(𝑥) is a quadratic function x such that 𝑓 (−2) = −6, 𝑓 (0) = −4 and 𝑓 (1) = 0 what is the
value of 𝑓(−1) ?
(A)−5
(B) −4
(C) -𝟔
(D) -3
(E) −2
116. How many real solutions are there of the equation 2d = 𝑥 7 ?
(A) 0
(B) 1
(C) 𝟐
(D) 3
(E) 4
117. A parabolic arch stands on level ground and has a span 𝐴𝐵 of 40 metres. The highest point
(the centre of the arch) is 16 metres above the point 𝑀, the midpoint of 𝐴𝐵. What is the
height (in metres) of the arch above a point on 𝐴𝐵 5 metres from 𝑀?
(A) 14.75
(B) 𝟏𝟓
(C) 15.25
(D) 15.5
(E) 15.75
d2,
118. The solution to the inequality 7d2. < 0 is
𝟑
.
(A) 𝟏 < 𝑥 < 𝟐
(B) 𝑥 < 1 OR 𝑥 > 7
7
(D) 𝑥 < .or𝑥 > 1
7
(C) . < 𝑥 < 1
(E) 𝑥 < 1
,
119. If 𝑓(𝑥 ) = 1 − d and 𝑔(𝑥 ) = 1 − 𝑥 then 𝑓e𝑔(𝑥 )f − 𝑔e𝑓(𝑥 )f is equal to:
(A)
,2d2d U
(B)
d2d U
2,2d2d U
(D) ,2d-d U
7(d2,) U
2𝟏-𝒙2𝒙𝟐
(C) 𝒙(𝟏2𝒙)
d
(E) 0
120. If 𝑓(𝑥) is a quadratic function such that 𝑓(−1) = −3 , 𝑓 (1) = 1 and 𝑓 (2) = 12, find 𝑓(3).
(A) 8
(B) 11
(C) 17
(D) 𝟐𝟗
(E) 35
,
121. The equation 8 (4𝑥 7 − 8𝑥 − 𝑘) = 30 is satisfied when 𝑥 = −5 and when 𝑥 = 7. What is the
value of 2𝑘?
(A) 𝟒𝟎
(B) −40
(C) 20
(D) −20
(E) 0
P a g e 37 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
,
122. If 𝑥 > d 𝑥 ≠ 0, which of the following statements is true?
(A)
(B)
(C)
(D)
–1 < 𝑥 < 1
𝑥 < – 1 or 𝑥 > 1
𝑥 < – 1 or 0 < 𝑥 < 1
𝑥 > 1 or – 1 < 𝑥 < 0
,
,
123. If f(x) = −𝑥 7 + 7 , 𝑥 ≠ 0 then 𝑓(− I) is:
(A)
,
−𝑚
(B) − 1¦𝑚7 − 𝑚
,
,
(C) IU − I
(D) − 1¦𝑚7 − 1¦𝑚
IU
STRAIGHT LINES, CO-ORDINATE GEOMETRY AND CIRCLES
124. Two tangential circles are shown. The smaller circle is also a tangent to the x and
y axes. The larger circle has equation ( x - 23)2 + y 2 = 81.
y
(𝑥 − 23)7 + 𝑦 7 = 81
x
The equation of the smaller circle is:
(A) x 2 - 16 x + y 2 - 16 y + 64 = 0
(B) ( x - 1) + ( y - 1) = 1
(C) x 2 - 10 x + y 2 - 10 y + 25 = 0
(D) ( x - 2) + ( y - 2) = 4
2
2
2
2
125. A regular octagon is shown with OP = 2 units.
y
Q
P
O
R
x
The y-intercept of the line passing through Q and R is:
P a g e 38 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
(B) 4 - 2
(A) 4
(C) 2 + 4 2
(D) 4 + 2 2
126. A circle with equation x 2 + y 2 - 2 x - 4 y = 4 is rotated 900 anticlockwise and then enlarged by
scale factor 2. The equation of the transformed circle is:
(A) ( x - 2) + ( y + 1) = 36
(B) x 2 + y 2 + 8x - 4 y = 16
( x + 2) + ( y -1) = 18
(D) x 2 + y 2 + 4 x - 2 y = 13
2
(C)
2
2
2
127. The smallest possible value of 𝑥 7 + 𝑦 7 − 6𝑦 + 14 is
(A) 1
(B) 2
(C) 3
(D) 4
(E) 𝟓
EUCLIDEAN GEOMETRY
128. In triangle 𝐴𝐵𝐶, ∠𝐴 = 90 and D is the foot of the altitude from 𝐴. If ∠𝐵 = 𝜃 and 𝐵𝐶 = 𝑎 then
𝐴𝐷 is equal to
(A) 𝑎𝑠𝑖𝑛7 𝜃
(B) 𝑎𝑐𝑜𝑠 7 𝜃
(C) 𝑎𝑡𝑎𝑛 𝜃
(D) 𝑎𝑠𝑖𝑛2𝜃
(E) 𝒂𝒔𝒊𝒏𝜽𝒄𝒐𝒔𝜽
129. In triangle 𝐴𝐵𝐶, 𝐴𝐵 = 8, 𝐴𝐶 = 15 and 𝐵𝐶 = 17. What is the radius of the inscribed circle?
(A) 𝟑
(B) 2√2
(C) 1 + √3
(D) √7
(E) √10
130. A circle with centre 𝑂 is inscribed in triangle ABC. If ∠𝐴 = 70° , ∠𝐵 = 80°, determine ∠𝐵𝑂𝐶.
(A) 100°
(B) 110°
(C) 115°
(D) 120°
(E) 𝟏𝟐𝟓°
131. In triangle 𝐴𝐵𝐶 points 𝐷 and 𝐸 lie on 𝐴𝐵 and 𝐴𝐶 respectively, with 𝐷𝐸 parallel to 𝐵𝐶.
If 𝐴𝐷 = 𝑝, 𝐷𝐵 = 𝑞 and 𝐴𝐸 = 𝑟 then 𝐸𝐶 is equal to
(A) 𝑝 + 𝑞 − 𝑟
kl
(B) ¶
(C) 𝑞 + 𝑟 − 𝑝
𝒒𝒓
(D) 𝒑
(E) 𝑞 + 𝑝 − 𝑟
P a g e 39 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
132. In the figure, the circle with centre O passes through A and B, and BD is tangent to the circle. If
∠𝐴𝑂𝐵 = 70°, then ∠𝐷𝐵𝐶 is equal to
D
O
70°
A
(A) 30°
C
B
(B) 𝟑𝟓°
(C) 40°
(D) 45°
(E) 50°
133.
Consider the following diagram:
C
B
30°
A
O
If O is the centre of the circle and AC is a tangent to the circle, what is the value of 𝐴¸?
(A) 15°
(B) 20°
(C) 25°
(D) 𝟑𝟎°
(E) 45°
(D) 40°
(E) 45°
134.
Consider the following diagram:
P
• 160°
C
Q
45°
R
C is the centre of the circle. What is the value of ∠ 𝑄𝑃𝐶?
(A) 25°
(B) 30°
(C) 𝟑𝟓°
P a g e 40 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
GENERAL GEOMETRY AND MENSURATION
135. A circle has a diameter of 4𝑥 + 12. The area of the circle is
(A) 𝟒𝝅𝒙𝟐 + 𝟐𝟒𝝅𝒙 + 𝟑𝟔𝝅
(B) 8𝜋𝑥 7 + 24𝜋𝑥 + 36𝜋
(C) 16𝜋𝑥 7 + 96𝜋𝑥 + 144𝜋
(D) 4𝜋𝑥 7 + 12𝜋𝑥 + 36𝜋
(E) 16𝜋𝑥 7 + 48𝜋𝑥 + 144𝜋
136. In the diagram 𝑃𝑂𝑅 = 120° and ∠𝑄𝑂𝑆 = 145°. What is the size of ∠𝑇𝑂𝑈?
(A) 45°
(B) 60°
(C) 𝟖𝟓°
(D) 90°
(E) 95°
137. The area of an isosceles triangle is 108𝑐𝑚7 and its base is 18𝑐𝑚. What is the perimeter of the
triangle?
(A) 12 𝑐𝑚
(B) 24 𝑐𝑚
(C) 𝟒𝟖 𝒄𝒎
(D) 63 𝑐𝑚
(E) 225 𝑐𝑚
138. A cube has edges of length 6. One of its square faces is 𝐴𝐵𝐶𝐷 and 𝐸 is the centre of the cube.
What us the volume of the pyramid 𝐴𝐵𝐶𝐷𝐸?
(A) 54
(B) 36
(C) 108
(D) 72
(E) 216
139. Circle 𝐴 has radius of 𝜋. Circle 𝐵 has circumference of 8𝜋. Circle 𝐶 has an area of 9𝜋.
List the circles in order, from smallest to largest radius.
(A) 𝐴, 𝐶, 𝐵
(B) 𝐵, 𝐴, 𝐶
(C) 𝐶, 𝐵, 𝐴
(D) 𝐵, 𝐶, 𝐴
(E) 𝑪, 𝑨, 𝑩
140. In triangle 𝐴𝐵𝐶, the bisector of ∠𝐴 meets 𝐵𝐶 in 𝐷. 𝐴𝐵 = 6, 𝐵𝐷 = 3 and 𝐷𝐶 = 4, what is the
length of 𝐶𝐴?
(A) 9
(B) 𝟖
(C) 10
(D) 7
(E) 6
P a g e 41 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
141. Triangle 𝐴𝐵𝐶 is isosceles with 𝐴𝐵 = 𝐵𝐶. Point 𝐷 is the midpoint of both 𝐵𝐶 and
𝐴𝐸, and 𝐶𝐸 = 10. What is the length of 𝐵𝐷?
(A) 8
(B) 𝟓
(C) 10
(D) 6
(E) 7
ˆ = q . 𝐴𝑀 ^ 𝐷𝐶and 𝐶𝑁 ^ 𝐴𝐵.
142. 𝐴𝐵𝐶𝐷 is a rhombus with 𝐴𝐷 = 𝑚 and ADC
A
N
M
C
B
m
D
The area of rectangle 𝐴𝑁𝐶𝑀 is:
(A) m2 sin q .cos q
(C) 12 m 2 (2sin q - sin 2q )
(B) m 2 sin 2q
(D) 12 m2 ( cos 2q .cos q )
143. A right square-based pyramid is shown below.
The base edges are each 10 cm whilst the slant edges are each 13 cm.
13 cm
10 cm
The surface area of the solid is:
(A) 580 cm2
(B) 360 cm2
(C) 620 cm2
(D) 340 cm2
P a g e 42 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
144.
A tunnel has a semi-circular cross-section and a diameter of 10 m. If the roof of the bus just
touches the roof of the tunnel when its wheels are 2 m from one side, the height is the bus is:
(A) 4m
(B)
21 m
(C)
12 m
(D) 2 6 m
0
145. The interior angle of a regular polygon is 1140
.
7
The number of sides of the polygon is:
(A) 21
(B) 19
(C) 17
(D) 13
146. In the diagram AC ^ BD and AD ^ AB . CD = 4 units and BC = 9 units.
A
x
D
4
9
C
B
The value of x is:
(A)
6
(B) 6
(C) 5
(D)
13
147. The area enclosed by the graphs of 𝑦 = 2𝑥 + 4, 𝑥 + 𝑦 = 4 and 𝑦 = 0 is
(A) 24
(B) 𝟏𝟐
(C) 4√10
(D) 2√5
(E) 36
P a g e 43 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
JÅÆ Y¶ÅY
148. The circumference of a circle is doubled. The ratio ǶÈÉÈJYÊ Y¶ÅY is
Ë
(A) 7
7
(B) Ë
(C) 2𝜋
(D) 2
(E) 𝟒
149. A triangle has vertices 𝐴 (3 ; 5), 𝐵 (1 ; 3) and 𝐶 (5 ; 2). What is the area of triangle 𝐴𝐵𝐶?
(A) 4
(B) 𝟓
(C) 6
(D) 7
(E) 8
150. A solid rectangular block of wood measuring 10 𝑐𝑚 by 8 𝑐𝑚 by 7𝑐𝑚 is painted white all over,
then cut into 1 𝑐𝑚 cubes. How many small cubes have paint on none of their faces.
(A) 𝟐𝟒𝟎
(B) 320
(C) 360
(D) 420
(E) 480
151. The circumference of a circle is equal to the perimeter of a square.
What is the ratio Area of square : Area of circle?
𝝅
(A) 𝟒
7
(B) Ë
(C) 4
(D)
Ë
√7
√7
(E) Ë
152. Thirty-six 1 × 1 × 1 cubes are used to make a rectangular prism. How many different
rectangular prisms can be made, using all 36 cubes?
(A) 5
(B) 6
(C) 7
(D) 𝟖
(E) 9
153. In rectangle 𝐴𝐵𝐶𝐷, 𝐴𝐵 = 2 and 𝐴𝐷 = 4. The area of the circle with centre 𝐴 that passes
through 𝐶 is
(A) 25𝜋
(B) 𝟐𝟎𝝅
(C) 16𝜋
(D) 12𝜋
(E) 9𝜋
154. The angles of a triangle are in the ratio 2 ∶ 3 ∶ 5. What is the difference between the largest
and smallest angle?
(A) 9°
(B) 18°
(C) 36°
(D) 45°
(E) 𝟓𝟒°
155. Triangle 𝐴𝐵𝐶 is right angled at 𝐶 and 𝐷 lies on 𝐵𝐶. If 𝐴𝐵 = 14, 𝐴𝐷 = 10 and 𝐶𝐷 = 5, the
perimeter of triangle 𝐴𝐵𝐷 is
(A) 24 + 5√2
(B) 24 + 3√3
(C) 29
(D) 𝟑𝟎
(E) 31
156. A rectangle is made up of three rows of four squares each. The area of each square is 4. What
is the perimeter of the rectangle?
(A) 𝟐𝟖
(B) 30
(C) 36
(D) 40
(E) 48
P a g e 44 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
157. Let 𝑃 be the area of a triangle with sides of length 25, 25and 30. Let 𝑄 be the area of a triangle
with sides of length 25, 25and 40. What is the relationship between 𝑃 and 𝑄?
4
.
(A) 𝑃 = ,9 𝑄
(B) 𝑃 = 8 𝑄
(C) 𝑃 = 𝑄
8
(D) 𝑃 = .
,9
(E) 𝑃 = 4 𝑄
158. In triangle 𝐴𝐵𝐶, ∠𝐴 = 50°. The bisectors of ∠𝐵 and ∠𝐶 meet in 𝑂, inside the triangle. The size
of ∠𝐵𝑂𝐶 is
(A) 100°
(B) 105°
(C) 110°
(D) 𝟏𝟏𝟓°
(E) 120°
159.
120°
The area of the circle is 4𝑥. What is the area of the shaded region?
160.
A)
𝒙(𝟒𝝅2𝟑€𝟑)
B)
d(8Ë2.€.)
C)
d(8Ë2.)
D)
d(8Ë2€.)
E)
None of the above.
𝟑𝝅
.
.
Ë
Consider the following diagram:
6
B
C
P a g e 45 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
ABCD is a rectangle with a perimeter of 30. Find the length of FA.
(A) 3
161.
(B) 𝟑√𝟓
(D) 5√3
(E) 8
The area of a rectangle is 15𝑥 7 + 29𝑥 + 12. The length of one side is 3𝑥 + 4. What is the
value of 𝑥 if the shape needs to be a square?
𝟏
(A) 0
162.
(C) 6
,
(B) 𝟐
(C)1 7
(D)1
(E) −1
Consider the following regular hexagonal prism:
The side lengths of the base are equal to a and the height of the prism is h as shown in the
diagram.
The volume of the prism is 𝑎7 .
h
Determine h.
a
(A)
163.
7
√.
,
(B) .√.
(C)
Consider the following diagram:
a
a
,
𝟐
(D) 𝟑√𝟑
√.
8
(E) .√.
O
45°
•
P
R
4
S
P is the centre of the circle, and RS = 4. <ROS = 45°. What is the circumference of circle P?
(A) 2𝜋
(B) 4√2𝜋
(C) 4𝜋
(D) 8√2𝜋
(E) 8𝜋
P a g e 46 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
164.
What is the area of the shaded part in terms of r?
(A) 16 𝜋 𝑟 7
(B) 12 𝜋 𝑟 7
(C) 11 𝜋 𝑟 7
(D) 7 𝜋 𝑟 7
TRIGONOMETRY
165.
cos7 𝑥 − sin7 𝑥 is equal to
(A) (𝑐𝑜𝑠𝑥 − 𝑠𝑖𝑛𝑥 )7
166.
(C) -1
(D) cos2x
In triangle 𝐴𝐵𝐶, ∠𝐴 = 90 and D is the foot of the altitude from 𝐴. If ∠𝐵 = 𝜃 and
then 𝐴𝐷 is equal to:
(A) 𝑎𝑠𝑖𝑛7 𝜃
167.
(B) 1
(B) 𝑎𝑐𝑜𝑠 7 𝜃
(C) 𝑎𝑡𝑎𝑛 𝜃
𝐵𝐶 = 𝑎
(D) 𝑎𝑠𝑖𝑛2𝜃
(E) 𝒂𝒔𝒊𝒏𝜽𝒄𝒐𝒔𝜽
(D) 𝑐𝑜𝑠55
(E) 𝑐𝑜𝑠65
𝑐𝑜𝑠65. 𝑠𝑖𝑛40 + 𝑐𝑜𝑠25. 𝑐𝑜𝑠40 is equal to:
(A)
𝒄𝒐𝒔𝟏𝟓
(B) 𝑐𝑜𝑠25
(C) 𝑐𝑜𝑠35
P a g e 47 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
168.
,
,
If 𝑡𝑎𝑛𝐴 = 7 and 𝑡𝑎𝑛(𝐴 + 𝐵) = . , then 𝑡𝑎𝑛 (2𝐴 + 𝐵) is equal to:
,
0
(A) 8
169.
(B) 9
,9
(B) ,0
(D) 𝟏𝟔𝟏
(C) 764
𝟐𝟒𝟎
,8.
(E) 41
𝑠𝑖𝑛(90 − 𝑥 ). 𝑐𝑜𝑠(180 − 𝑥 ) − 𝑡𝑎𝑛𝑥. 𝑐𝑜𝑠(−𝑥 ). 𝑠𝑖𝑛 (180 + 𝑥) is equal to:
(A) −1
(B) 1
(C) 𝑐𝑜𝑠2𝑥
(D) 0
(E) −𝒄𝒐𝒔𝟐𝒙
Four points 𝐴, 𝐵, 𝐶 and 𝐷 lie in a plane, with 𝐵 the midpoint of 𝐴𝐶.
If 𝐴𝐵 = 𝐵𝐶 = 𝐵𝐷 = 13 and 𝐶𝐷 = 24, find the length of 𝐴𝐷.
(A) 11
172.
870
1
(E) 9
6
.7,
171.
(D) 𝟏
If 𝑠𝑖𝑛𝛼 = ,1 then 𝑡𝑎𝑛2𝛼 is equal to:
(A) 770
170.
7
(C) 0
(B) 9
(C) 13
(D) 12
(E) 𝟏𝟎
ÐÈJÑ
For 0 < 𝐴 < 90 , ,2ÒÇÐÑ is equal to:
(A) 𝑠𝑖𝑛𝐴 − 𝑡𝑎𝑛𝐴
(B)
,2ÐÈJÑ
ÒÇÐÑ
ÒÇÐÑ
𝟏-𝒄𝒐𝒔𝑨
(C) 𝒔𝒊𝒏𝑨
ÒÇÐÑ
(D) ,2ÐÈJÑ
(E) ,-ÐÈJÑ
173.
The maximum value of 3 − 8𝑠𝑖𝑛𝑥. 𝑐𝑜𝑠𝑥is:
(A) 𝟕
174.
(B) 11
.
(B) 260°
(C) 220°
(D) 340°
7
If 𝑠𝑖𝑛6𝑥 = − 0 , then the value of (𝑠𝑖𝑛3𝑥 − 𝑐𝑜𝑠3𝑥 )7 is:
(A)
176.
(D) −5
One solution to the equation 𝑠𝑖𝑛 “7 𝑥” = 𝑝 is 20°. Which of the following is NOT a solution
to the equation?
(A) -140°
175.
(C) −1
𝟕
𝟓
.
(B) 0
,
(C) − 0
,
(D) 0
.
(E) − 0
The function f ( x) = 2sin 30°.sin x + 2cos150° cos x could be represented by which of the
following graphs?
P a g e 48 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
177.
(A)
(B)
(C)
(D)
If 𝑡𝑎𝑛𝐴 + 𝑠𝑒𝑐𝐴 = 3, then 𝑐𝑜𝑠𝐴 is equal to:
,
𝟑
(A) 9
178.
(B) 𝒕𝒂𝒏𝑨
(E) 6
(C) 𝑐𝑠𝑐𝐴
(D) 𝑠𝑒𝑐𝐴
(E) 𝑐𝑜𝑡𝐴
For all angles 𝐴 and 𝐵, 𝑠𝑖𝑛𝐴𝑐𝑜𝑠𝐵 is equal to:
Ñ
𝟏
(B) 𝟐 (𝐬𝐢𝐧(𝑨 + 𝑩) + 𝐬𝐢𝐧(𝑨 − 𝑩))
,
(C) 7 (cos(𝐴 − 𝐵) + sin(𝐴 + 𝐵))
,
,
(D) 7 sin(𝐴𝐵)
,
(E) 7 sin (𝐴 + 𝐵)
,
If 𝑡𝑎𝑛𝐴 = Y and tan B= Z , then tan (𝐴 + 𝐵) is equal to:
Y-Z
(A) YZ
181.
0
(D) .
8×Y¶ÅYÑÔÕ
(A) tan “Ô”
180.
7
(C) 1
In triangle 𝐴𝐵𝐶, ZU -Ò U 2YU is equal to:
(A) 𝑐𝑜𝑠𝐴
179.
8
(B) 𝟓
7
(B) Y-Z
,
𝒂-𝒃
(C) YZ
(D) 𝒂𝒃2𝟏
YZ
(E) Y-Z2,
If 𝑡𝑎𝑛20 = 𝑡,then 𝑡𝑎𝑛50 is equal to:
𝟏2𝒕𝟐
(A) 𝟐𝒕
(B)
72u U
u
(C)
,-7u U
u
(D)
7-u U
u
(E)
,-u U
u
P a g e 49 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
182.
,
(A) −2
183.
(B) 2
(B) 55𝑐𝑚
𝟏
(D) 48𝑐𝑚
(E) 𝟒𝟓𝒄𝒎
(D) −1
(E) − 7
(D) 2√7
(E) 5√2
√.
(C) 7
In triangle ABC,points 𝑃, 𝑄 and 𝑅 lie on the sides 𝐴𝐵, 𝐵𝐶 and 𝐶𝐴 respectively. If 𝐵𝑃 = 𝐵𝑄,
𝐶𝑄 = 𝐶𝑅 and ∠𝑃𝑄𝑅 = 40° , determine ∠𝑃𝐴𝑅
(B) 70°
(C) 80°
(D) 90°
(E) 𝟏𝟎𝟎°
𝑠𝑖𝑛15°. 𝑐𝑜𝑠15° is equal to:
(B) 𝑡𝑎𝑛30°
√.
(C) 8
(D)
,
√7
𝟏
(E) 𝟒
,
If 𝑡𝑎𝑛𝐴 + ÒÇÐÑ = 2 then 𝑐𝑜𝑠𝐴 is equal to
9
0
(A) ,,
189.
(C) − 7
(B) 𝟔
(A) 𝑡𝑎𝑛15°
188.
(E) √2
In ∆𝐴𝐵𝐶, ∠𝐶 = 2∠𝐴, 𝐴𝐶 = 5, 𝐵𝐶 = 4. Determine 𝐴𝐵
(A) 60°
187.
(C) 54𝑐𝑚
,
(B) 𝟐
(A) 2√10
186.
𝟏
(D) − 𝟐
𝑐𝑜𝑠25°. 𝑐𝑜𝑠35° + 𝑐𝑜𝑠65°. 𝑐𝑜𝑠125° is equal to:
(A) 1
185.
,
(C) 7
A vertical stick of height 84𝑐𝑚 casts a shadow 108𝑐𝑚 long. Next to it is another vertical
stick of
length 35𝑐𝑚. How long is the shadow of the second stick?
(A) 59𝑐𝑚
184.
,
If 𝑡𝑎𝑛𝑥 + ÒÇÐd = 2 then 𝑡𝑎𝑛𝑥 − ÒÇÐd equals
(B) 4
.
(D) 𝟓
8
(D)
(C) 1
𝟒
(E) 6
1
√11
,,
(E) 𝟏𝟐𝟏
1
If sin 2 𝑥 = ,, , then 𝑐𝑜𝑠2𝑥 is equal to:
√1.
(A) ,7,
6
(B) ,,4
(C) ,,
𝟐𝟑
P a g e 50 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
190.
A ladder rest on a wall at an angle of 45° as illustrated in the diagram. If the ladder moves 1
m down the wall, what is the angle that the ladder will now make with the wall?
7
(A) 𝜃 = 𝑐𝑜𝑠 2, “ ”
√7
7√.
(B) 𝜃 = 𝑐𝑜𝑠 2, “ . ”
3m
45°
,
(C) 𝜃 = 𝑐𝑜𝑠 2, “7”
√.
(D) 𝜃 = 𝑐𝑜𝑠 2, “ 7 ”
√𝟐
(E) 𝜽 = 𝒄𝒐𝒔2𝟏 “ 𝟑 ”
191.
If 𝑓(𝑥) = 𝑠𝑖𝑛𝑥 − 1 and 𝑔(𝑥 ) = tan 𝑥, how many times do they intersect, if 𝑥 ∈ [0°; 360°]?
(A) 0
192.
3m
(B) 1
(C) 𝟐
(D) 3
(E) 4
Consider the following diagram:
𝜃
What is the value of 𝜃 in the diagram?
(A) 10°
193.
(B) 30°
(C)45°
(D) 50°
(E) 60°
Consider rhombus ABCD:
A
D
∝
B
m
C
𝐵𝐴¸𝐶 =∝ and BC = m
Find AC.
P a g e 51 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
(A) 2𝑚√1 + 𝑐𝑜𝑠2 ∝
(D) √2𝑚√1 − 𝑐𝑜𝑠2 ∝
194.
(B) √𝟐𝒎√𝟏 + 𝒄𝒐𝒔𝟐 ∝
(E) 1 + 𝑐𝑜𝑠2 ∝
(C) 2𝑚√1 − 𝑐𝑜𝑠2 ∝
d
What is the minimum value of 𝑦 if 𝑦 = 1 + 𝑠𝑖𝑛 7 and 𝑥 ∈ [0°, 360°]?
(A) 0
,
(B)7
,
(C) 𝟏
(D) 1 7
(E) 2
195. What is the value of:
𝑐𝑜𝑠45° + 𝑐𝑜𝑠60°
(A) −1
√0
(B) 7
(C)
𝟏-√𝟐
(D)
𝟐
√7-√.
7
(E)
7√7-,
7
196. Given D ABC and D ABD in the following diagram:
A
15°
4
B
C
30°
D
What is the perimeter of DACD?
(A) 2√6 − 2√2 (B) 4√3
(C) 4 + 2√6 + 2√2 (D) √2 + 4 + √6
(E) 2√6 + 6√2
197. If the graph of cos (𝑥 + 90) is shifted 30° to the left, what is the new graph?
(A) cos (𝑥 + 30)
(B) –sin (𝑥 + 30)
(C) cos (𝑥 - 60)
(D) –sin (60 - 𝑥)
198. The expression sin 𝑥 7 + 𝑥 can also be written as.
(A) sin (2d + 𝑥)
(B) 𝑥 (sin 𝑥 + 1)
(C) (sin 𝑥)7 + 𝑥)
(D) sin (𝑥 7 ) + 𝑥
P a g e 52 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
CALCULUS
199.
,
The maximum value of d U-0d-1 is
.
,
(A) 8
200.
(B) 1
,
(D) 7
,
(E) 8
For which positive value of 𝑘 is the line 𝑦 = 𝑘𝑥 − 9 tangent to the parabola 𝑦 = 𝑥 7 ?
(A) 2
201.
𝟒
(C) 𝟑
(B)3
(C) 𝟔
(D) 8
(E) 9
d-,
If 𝑥 is a real number, the maximum value of d U -. is equal to
,
,
(A) .
(B) 9
,
(C) − 9
𝟏
,
(D) 𝟐
(E) − 7
PROBABILITY
202.
Ann and Bob each throw coins. What is the probability that Ann throws more heads than
Bob?
,
𝟏
(A) 7
(B) 𝟒
,
(C) 6
0
(D) ,9
.
(E) 6
TRICK QUESTIONS AND WORD PROBLEMS
203.
A girl cycles to school at a constant speed of x km/h and returns at a constant speed of y
km/h. An expression for her average speed is:
(A)
204.
x+ y
2
(B) x - y
2
(C)
2xy
x+ y
(D)
2( x + y)
xy
I am in a queue with 𝑛 fewer people behind me than in front of me. There are three times
as many people in the queue as there are behind me. How many people are behind me (in
terms
of 𝑛)?
(A) 𝑛
(B) 2𝑛
(C) 𝒏 + 𝟏
(D) 2𝑛 – 1
P a g e 53 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
205.
pI
A man drives at 60 à for 50 𝑘𝑚, and then drives at 40 𝑘𝑚/ℎ for another 50 𝑘𝑚. What is
his average speed, in 𝑘𝑚/ℎ, for the whole journey?
(A) 46
206.
(E) 54
(C) 7.75
(B) 𝟑
(D) 8
,
T
lã
(C) 4
(E) 8.25
,M
= 1 then 𝑟 is equal to:
(D) 5
(E) 6
(B) 𝟏𝟖𝟎𝟗𝟔
(C) 18097
(D) 18098
(E) 18099
If 5k = 7 , 7l = 9 , 9¶ = 11 and 11Ð = 25 , what is the value of 𝑝𝑞𝑟𝑠?
(A) 1
210.
(D) 52
If the number 107M,, − 2011 is written as an integer, the sum of its digits is
(A) 18095
209.
(B) 𝟕. 𝟓
If 𝑝, 𝑞 and 𝑟 are positive integers such that 𝑝 +
(A) 2
208.
(C) 50
Jacob ran for 10 𝑘𝑚 at 6 𝑘𝑚/ℎ and then ran back along the same route at 10 𝑘𝑚/ℎ. What
was his average speed, in 𝑘𝑚/ℎ, for the whole route?
(A) 7
207.
(B) 𝟒𝟖
(B) 𝟐
(C) 3
(D) 4
(E) 5
A boy runs from his home to school at 9 𝑘𝑚/ℎ then walks back home at 5 𝑘𝑚/ℎ. What was
his average speed, to one decimal place, for the trip to school and back?
(A) 6.3 𝑘𝑚/ℎ
(B) 𝟔. 𝟒 𝒌𝒎/𝒉
(C) 6.6 𝑘𝑚/ℎ
(D) 6.9 𝑘𝑚/ℎ
(E) 7.1 𝑘𝑚/ℎ
P a g e 54 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
NBT COURSE
QUANTITATIVE LITERACY
PRACTICE TEST 1
20 MINUTES
SIMULTANEOUS EQUATIONS
SYMBOLS
1.
4.
If 𝑥 + 𝑦 = 8 and 𝑦 − 𝑥 = −2,
then 𝑦 =
(A)
(B)
(C)
(D)
(E)
(A)
(B)
(C)
(D)
(E)
-2
3
5
-8
10
B
If 4𝑎 + 3𝑏 = 19 and 𝑎 + 2𝑏 = 6,
then 𝑎 + 𝑏 =
(A)
(B)
(C)
(D)
(E)
-8
-2
2
20
40
SPECIAL TRIANGLES
SIMULTANEOUS EQUATIONS
2.
If 𝑟 ∗ 𝑠 = 𝑟(𝑟 − 𝑠)for all integers 𝑟 𝑎𝑛𝑑 𝑠,
then 4 ∗ (3 ∗ 5) equals
4
5
6
7
8
60°
150°
A
C
Note: Figure not
drawn to scale.
SYMBOLS
3.
5.
,
If 𝑥 ≠ 0 let ¬𝑥 be defined by ¬𝑥 = 𝑥 − d
,
What is the value of ¬2 − ¬ 7 ?
(A)
(B)
(C)
(D)
(E)
0
3/2
3
6
9/2
In triangle 𝐴𝐵𝐶 above, if 𝐴𝐵 = 4,
then 𝐴𝐶 =
(A)
(B)
(C)
(D)
(E)
6
7
8
9
10
P a g e 55 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
SPECIAL TRIANGLES
RATES
B
9.
5
A
6
(A)
(B)
(C)
(D)
(E)
C
Note: Figure not drawn to scale.
6.
If the perimeter of triangle 𝐴𝐵𝐶 above is
16, what is its area?
(A)
(B)
(C)
(D)
(E)
8
9
10
12
15
10.
In a group of 24 people who are either
homeowners or renters, the ratio of
homeowners to renters is 5:3. How many
homeowners are in the group?
(A)
(B)
(C)
(D)
(E)
8
9
12
14
15
RATIOS
8.
Magazine A has a total of 28 pages, 16 of
which are advertisements and 12 of which
are articles.
Magazine B has a total of 35 pages, all of
them either advertisements or articles. If
the ratio of the number of pages of
advertisements to the number of pages of
articles is the same for both magazines,
then Magazine B has how many more
pages of advertisements than Magazine A?
(A)
(B)
(C)
(D)
(E)
à
æ
ℎ𝑑
æ
ℎ+7
ℎ−𝑑
æ
à
RATES
RATIOS
7.
If David paints at the rate of ℎ houses per
day, how many houses does he paint in 𝑑
days, in terms of ℎ and 𝑑?
Bill has to type a paper that is 𝑝 pages
long, with each page containing 𝑤 words.
If Bills types an average of 𝑥 words per
minute, how many hours will it take him
to finish the paper?
(A)
(B)
(C)
(D)
(E)
60𝑤𝑝𝑥
Æd
9Mk
9MÆk
d
Ækd
9M
Æk
9Md
REMAINDERS
11.
When 𝑧 is divided by 8, the remainder is 5.
What is the remainder when 4𝑧 is divided
by 8?
(A)
(B)
(C)
(D)
(E)
1
3
4
5
7
2
3
4
5
6
P a g e 56 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
REMAINDERS
PERCENTAGES
12.
15.
When 𝑛 is divided by 12, the remainder
is 0. What is the remainder when 2𝑛 is
divided by 6?
(A)
(B)
(C)
(D)
(E)
0
1
2
3
4
(A)
(B)
(C)
(D)
(E)
AVERAGES
13.
The average (arithmetic mean) of six
numbers is 16. If five of the numbers are
15, 37, 16, 9 and 23, what is the sixth
number?
(A)
(B)
(C)
(D)
(E)
Eighty-five percent of the members of a
student organization are registered to
attend a certain field trip. If 16 of the
members who registered were unable to
attend, resulting in only 65 percent of the
members making the trip, how many
members are in the organization?
64
68
72
80
96
PERCENTAGES
16.
-20
-4
0
6
16
If a sweater sells for R48 after a 25 percent
markdown, what was its original price?
(A)
(B)
(C)
(D)
(E)
R56
R60
R64
R68
R72
AVERAGES
14.
The average (arithmetic mean) of five
numbers is 8. If the average of two of these
numbers is -6, what is the sum of the other
three numbers?
(A)
(B)
(C)
(D)
(E)
28
34
46
52
60
MULTIPLE AND STRANGE FIGURES
17.
P
Q
S
R
In the figure above, square 𝑃𝑄𝑅𝑆 is
inscribed in a circle. If the area of
square 𝑃𝑄𝑅𝑆 is 4, what is the radius of the
circle?
(A)
(B)
(C)
(D)
(E)
1
√2
2
2√2
4√2
P a g e 57 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
MULTIPLE AND STRANGE FIGURES
COMBINATIONS
20.
Note: Figure not drawn to scale.
18.
In the figure above, the quarter circle with
center 𝐷 has a radius of 4 and rectangle
𝐴𝐵𝐶𝐷 has a perimeter of 20. What is the
perimeter of the shaded region?
(A)
(B)
(C)
(D)
(E)
Three people stop for lunch at a hotdog
stand. If each person orders one item and
there are three items to choose from, how
many different combinations of food could
be purchased? (Assume that order doesn’t
matter; e.g. a hotdog and two sodas are
considered the same as two sodas and a
hotdog.)
(A)
(B)
(C)
(D)
(E)
6
9
10
18
27
20 − 8𝜋
10 + 2𝜋
12 + 2𝜋
12 + 4𝜋
4 + 8𝜋
COMBINATIONS
19.
Five people attend a meeting. If each
person shakes hands once with every
other person at the meeting, what is the
total number of handshakes that take
place?
(A)
(B)
(C)
(D)
(E)
10
15
25
120
3125
P a g e 58 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
NBT COURSE
MATHEMATICS INTRODUCTION
PRACTICE TEST 2
20 MINUTES
ABSOLUTE VALUE
RATIONAL EQUATIONS AND
INEQUALITIES
1.
d V 28d
If d V 20d U -9d = 0 , and 𝑥 ≠ 0, 2, or 3,
what is the value of 𝑥?
(A)
-3
(B)
-2
(C)
0
(D)
1
(E)
4
RADICAL EQUATIONS
2.
If €𝑥 + 2𝑦 − 2 = 15, what is the value of
𝑦 in terms of 𝑥?
(A)
(B)
(C)
(D)
(E)
Which of the following equations best
represents the graph above?
(A)
(B)
(C)
(D)
(E)
7642d
7
289 − 𝑥
,12d
7
17 − 𝑥
289
𝑦 = |𝑥|
𝑦 = |𝑥| − 1
𝑦 = |𝑥 − 1|
𝑦 = |𝑥 − 1| − 1
𝑦 = |𝑥 − 2|
FUNCTION NOTATION
MANIPULATION WITH INTEGER AND
RATIONAL EXPONENTS
3.
4.
𝟏
V
What is the value of 4𝟐 + 4U ?
(A)
4
(B)
8
(C)
10
(D)
16
(E)
64
5.
(d U24)
If 𝑓(𝑥 ) = (d-.) , What is the value of
𝑓(−4)?
(A)
(B)
(C)
(D)
(E)
-7
,
−8
0
,
8
7
P a g e 59 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
CONCEPTS OF DOMAIN AND RANGE
6.
If 𝑔(𝑥 ) = 2 − √𝑥 − 7, and 𝑔(𝑥) is a real
number, which of the following cannot
be the value of 𝑥?
(A)
4
(B)
7
(C)
11
(D)
102
(E)
496
LINEAR FUNCTONS –
EQUATIONS AND GRAPHS
FUNCTIONS AS MODELS
8.
The graph shows the function 𝑔(𝑥 ).
What is the value of 𝑔(0)?
(A)
(B)
(C)
(D)
(E)
7.
The graph represents the annual tuition
for college 𝑌 from 2000 -2003. Based on
the graph, what was most likely the
tuition for college for 𝑌 in 1999?
(A)
(B)
(C)
(D)
(E)
−1
,
−7
0
1
3
QUADRATIC FUNCTIONS –
EQUATIONS AND GRAPHS
$6,000
$9,000
$15,000
$18,000
$21,000
9.
Which of the following equations best
describes the curve above?
(A)
(B)
(C)
(D)
(E)
𝑦 = 𝑥7 + 4
𝑦 = 𝑥7 − 1
𝑦 = −𝑥 7 + 4
𝑦 = −𝑥 7 + 1
𝑦 = −𝑥 7 − 1
P a g e 60 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
QUALITATIVE BEHAVIOR OF
GRAPHS AND FUNCTIONS
PROPERTIES OF TANGENT LINES
12.
10.
The figure above shows the graph of
𝑔(𝑥). What is the largest value of 𝑔(𝑥)
shown in this figure?
(A)
(B)
(C)
(D)
(E)
−2
2
4
6
6.5
ëëëë is 8 units long
In the figure above, 𝐵𝐷
and tangent to the circle at point C. ëëëë
𝐴𝐶 is
a diameter of the circle. If the
circumference of the circle is 6𝜋 ,what is
the area of ∆𝐴𝐵𝐷?
(A)
(B)
(C)
(D)
(E)
9
12
24
9𝜋
10𝜋
ANALYTICAL GEOMETRY
SEQUENCES INVOLVING
EXPONENTIAL GROWTH
11.
13.
If point 𝑅 is (2,4) and point 𝑆 is (7,7),
ëëëë?
what is the length of 𝑅𝑆
(A)
(B)
(C)
(D)
(E)
2
√7
√34
9
√202
A scientist is running an experiment
with two species of bacteria that grow
exponentially . If species A doubles in
population every two days, species B
doubles in population every five days,
and each species began the experiment
with a population of 50 bacteria, what
will the difference be between the
populations of the two species after ten
days?
(A)
(B)
(C)
(D)
(E)
200
800
1,200
1,400
1,500
P a g e 61 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
TRANSFORMATIONS AND THEIR EFFECT ON GRAPHS OF FUNCTIONS
14.
The figure above shows the graph of the function ℎ(𝑥). Which of the following figures shows
the graph of the function ℎ(𝑥 + 1)?
(A)
(C)
(E)
(B)
(D)
P a g e 62 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
17.
SETS
15.
If set 𝐴 = {2,3,5,7,10} and set
𝐵 = {3,4,5,6,7}, how many elements are
in the intersection of the two sets?
(A)
(B)
(C)
(D)
(E)
2
3
5
7
10
DIRECT AND INVERSE VARIATION
16.
The rate at which a certain balloon
travels is inversely proportional to the
amount of weight attached to it. If the
balloon travels 10 inches per second
when there is a 2-gram weight attached
to it, approximately how much weight
must be attached to the balloon for it to
travel 18 inches per second?
(A)
(B)
(C)
(D)
(E)
,
(A)
𝑦 = 7𝑥
(B)
(C)
(D)
(E)
𝑦 = 7𝑥 +4
𝑦=𝑥
𝑦 = 2𝑥 − 4
𝑦 = 2𝑥
,
GEOMETRIC NOTATION FOR
LENGTH, SEGMENTS, LINES, RAYS,
AND CONGRUENCE
3
18.
0.4 grams
1.0 grams
1.1 grams
3.6 grams
10.0 grams
DATA INTERPRETATION, SCATTER
PLOTS, AND MATRICES
Which of the following equations best
fits these points?
In the figure above, ∆𝐴𝐵𝐶 ≅ ∆𝐸𝐹𝐷.
What is the area of ∆𝐴𝐵𝐶?
(A)
(B)
(C)
(D)
(E)
6
7.5
6√2
6√3
12
TRIGONOMETRY
19.
In the figure above, ∆𝐴𝐵𝐶 ≅ ∆𝐸𝐹𝐷.
What is the value of 𝑥?
(A)
(B)
(C)
(D)
(E)
3
4
5
6
7
P a g e 63 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
20.
The figure above shows a square
inscribed in a square inscribed in
another square. What is the probability
that a point selected at random from the
interior of the largest figure will fall
within the shaded region?
(A)
,
(B)
(C)
(D)
(E)
0
,
8
,
.
8
4
,
7
P a g e 64 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
NBT COURSE
MATHEMATICS ADVANCED
PRACTICE TEST 3
15 MINUTES
3.
1.
2.
(A)
(B)
(C)
(D)
(E)
In the figure above, what is the maximum
number of nonoverlapping regions into
which the shaded area can be divided
using exactly two straight lines?
(A)
(B)
(C)
(D)
(E)
3
4
5
6
7
A certain school event was open only to
juniors and seniors. Half the number of
juniors who had planned to attend
actually attended. Double the number of
seniors who had planned to attend
actually attended. If the ratio of the
number of juniors who had planned to
attend to the number of seniors who had
planned to attend was 4 to 5, then juniors
were what fraction of attendees?
(A)
(B)
(C)
(D)
(E)
,
9
,
0
8
,4
8
,0
It cannot be determined from the
information given.
If 𝑝 − 𝑞 = 4 and 𝑟 is the number of
integers less than 𝑝 and greater than 𝑞,
then which of the following could be
true?
I.
𝑟=3
II.
𝑟=4
III.
𝑟=5
4.
Volumes 12 through 30 of a certain
encyclopedia are located on the bottom
shelf of a book case. If the volumes of the
encyclopedia are numbered
consecutively, how many volumes of the
encyclopedia are on the bottom shelf?
(A)
(B)
(C)
(D)
(E)
5.
I only
II only
III only
I and II
I, II and III
17
18
19
29
30
A reservoir is at full capacity at the
beginning of summer. By the first day of
fall, the level in the reservoir is 30
percent below full capacity. Then during
the fall, a period of heavy rains increases
the level by 30 percent. After the rains,
the reservoir is what percent of its full
capacity?
(A)
(B)
(C)
(D)
(E)
60%
85%
91%
95%
100%
P a g e 65 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
6.
Two classes, one with 50 students and
the other with 30, take the same exam.
The combined average of both classes is
84.5. If the larger class averages 80, what
is the average of the smaller class?
(A)
(B)
(C)
(D)
(E)
7.
8.
7:3
6:5
13:8
21:11
35:18
11.
,
𝑥 is 33 . % less than 𝑦 and 33 . % greater
than 𝑧. What percentage of 𝑧 is 𝑦?
(A)
(B)
(C)
(D)
(E)
,
33 . %
40%
50%
7
66 . %
75%
4
5
8
9
10
A shop assistant increases the price of a
sweater by 20%, increases it by another
25%, and then finally decreases it by
50%. If the sweater originally cost R100,
what is the final price of the sweater?
(A)
(B)
(C)
(D)
(E)
13.
5:3
4:3
10:9
16:15
12:11
How many multiples of 5 are there from
-20 to 20, inclusive?
(A)
(B)
(C)
(D)
(E)
12.
4
5
6
7
8
,
The farmers market has oranges, apples
and bananas . If the ratio of bananas to
oranges is 5:4, and the ration of bananas
to apples is 4:3, what is the ratio of
oranges to apples?
(A)
(B)
(C)
(D)
(E)
Michael mows lawns for his summer job.
He only averages three lawns a week for
the first eight weeks, but then averages
six lawns a week for the last four weeks.
What is the average number of lawns he
mowed per week during the entire
summer?
(A)
(B)
(C)
(D)
(E)
9.
87.2
89.0
92.0
93.3
94.5
In a pet shop, the ratio of puppies to
kittens is 7:6, and the ratio of kittens to
guinea pigs is 5:3. What is the ratio of
puppies to guinea pigs?
(A)
(B)
(C)
(D)
(E)
10.
R60
R75
R80
R95
R105
What is the greatest value of 𝑥 so that
3𝑥 + 7 < 21?
(A)
(B)
(C)
(D)
(E)
3
4
5
6
7
P a g e 66 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
14.
The ratio of boys to girls in the school
play is 2:5. Three new boys join the
group, for a total of nine boys. What is
the new ratio of boys to girls?
(A)
(B)
(C)
(D)
(E)
15.
2:3
3:4
3:5
4:5
5:5
Jim averaged 30 kilometres per hour for
the first four hours of his trip, then
averaged 42 kilometres per hour for the
next two hours. What was Jim’s average
speed for the entire trip?
(A)
(B)
(C)
(D)
(E)
32 km.h-1
34 km.h-1
36 km.h-1
38 km.h-1
None of the above
P a g e 67 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
NBT COURSE
QUANTITATIVE LITERACY PRACTICE TEST 4
15 MINUTES
INTERPRETATION OF VISUAL INFORMATION
Study the following information given in the table and graph on the nutritional facts of various
types of alcoholic beverages:
CALORIES
(cal)
CARBS
(g)
PROTEIN
(g)
ALCOHOL
CONTENT
BY VOLUME
(ABV) %
Glass of white wine
120
0
0
12
Glass of
red wine
102
0
0
14
Light beer
102
5
0
4
Lager beer
128
10
1,4
7
Wheat beer
165
16
2
7
Stout beer
170
6
0
7
Cider
210
30
0
12
Cosmopolitan
100
24
0,5
20
Mojito
242
40
0
20
Martini
70
17
0
20
Margarita
174
19
0
18
Long Island Ice Tea
454
62
0
30
Gin and Tonic
70
18
0
10
Whiskey and Soda
100
18
0,2
16
Brandy and coke
250
32
0
16
ALCOHOL
VITAMINS
COMMONLY
DEPLETED
A, B1, B2, B6,
C, D, E, K
A, B1, B2, B6,
C, D, E, K
A,
B2, B6,
C, D, E, K
A, B1, B2, B6,
C, D, E, K
A, B1, B2, B6,
C, D, E, K
A, B1, B2, B6,
C, D, E, K
A, B1, B2, B6,
D, E, K
A, B1, B2, B6,
D, E, K
A, B1, B2, B6,
D, E, K
A, B1, B2, B6,
C, D, E, K
A, , B2, B6,
D, E, K
A, B1, B2, B6,
D, E, K
A,
B6,
C,
E, K
A, B1, B2, B6,
C, D, E, K
A, B1, B2, B6,
C, D, E, K
P a g e 68 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
P a g e 69 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
1.
What is the ratio to the beverage that
contains the most calories to the
beverage that contains the least
calories?
A)
B)
C)
D)
E)
2.
6.
7.
A, B1, B2, B6, C, D, E, K
A, B6, C, D, E, K
A, B2, B6, C, D, E, K
A, B1, B6, C, D, E, K
A, B6, E, K
Which type of vitamin is depleted the
least by ALL the alcoholic beverages?
A)
B)
C)
D)
E)
Vitamin B1
Vitamin C
Vitamin B2
Vitamin B6
Vitamin K
20%
27%
73%
80%
100%
According to what criterion are the
following drinks arranged?
Red Wine, Cider, Brandy and Coke,
Mojito, Long Island Ice Tea
A)
B)
C)
D)
E)
8.
Margarita
Brandy and Coke
Light Beer
Long Island Ice Tea
Cosmopolitan
What percentage of alcoholic beverages
do NOT contain proteins?
A)
B)
C)
D)
E)
Which type of vitamins will be depleted
regardless of the drink that is
consumed?
A)
B)
C)
D)
E)
4.
White wine
Light beer
Cider
Mojito
Long Island Ice Tea
Which type of alcoholic beverage will
most likely cause intoxication?
A)
B)
C)
D)
E)
Which of the following beverages has
the biggest calories : alcohol by content
volume ratio?
A)
B)
C)
D)
E)
3.
227:35
35:227
25:7
7:25
121:60
5.
Alcohol by content volume
Proteins
Carbohydrates
Calories
No fixed arrangement
What is the average calories contained
in the beverages?
A)
B)
C)
D)
E)
164
201
214
223
246
P a g e 70 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
9.
What is the median value of the calories
contained in all the beverages?
A)
B)
C)
D)
E)
10.
11.
120
128
146
165
170
How many Mojitos must you drink in
order to consume 1 kg of
carbohydrates?
A)
B)
C)
D)
E)
B)
C)
D)
E)
If 100 g of sugar contains 387 calories,
which of the following beverages
contain 500 g of sugar?
A)
B)
C)
D)
E)
13.
2
6
10
16
25
In South Africa the legal limit is a breath
alcohol content of 0,24 g of 1000 ml.
This means that brandy and whiskey
drinkers can have one tot of 25 ml of
alcohol every hour, and approximately
one glass of red wine per hour.
In order to stay within the legal limit,
how much of the following may be
consumed per hour?
A)
12.
According to the graph, what percentage
of beverages contain more than 150
calories?
A)
B)
C)
D)
E)
14.
7,7 brandy and coke beverages
10,4 whiskey and sodas
10 Cosmopolitans
5 Mojitos
10 glasses of white wine
50%
47%
45%
40%
37%
What fraction of beverages contain less
than 100 calories and less than 10 g of
carbohydrates?
A)
B)
C)
D)
E)
None of the beverages
2/15
3/15
4/15
5/15
Two Long Island Ice Teas per
hour
A half a Long Island Ice Tea per
hour
Two glasses of white wine per
hour
One Cosmopolitan per hour
Two ciders per hour
P a g e 71 | 72
NATIONAL BENCHMARK TESTS
PREPARATION COURSE
15.
Which of the following combinations list
an increase in the proteins contained in
the beverages?
A)
B)
C)
D)
E)
Glass of white wine, Glass of red
wine, Light Beer, Wheat Beer,
Lager Beer
Glass of white wine, Glass of red
wine, Light Beer, Lager Beer,
Wheat Beer
Glass of white wine, Glass of red
wine, Whiskey and Soda,
Cosmopolitan, Wheat Beer
Glass of white wine, Whiskey and
Soda, Cosmopolitan, Lager Beer,
Wheat Beer
None of the above
P a g e 72 | 72