National Benchmark Test (NBT) – South Africa — Questions and Answers

Question 1: The y-intercept of f(x) = 3x² − 5x + 2 is:

−5
0
2 (Correct answer)
3

Correct answer: 2

The y-intercept is found by setting x = 0: f(0) = 0 − 0 + 2 = 2.

Question 2: A ladder leans against a wall making a 90-degree angle between the wall and the ground. The angle between the ladder and the ground is 65 degrees. What is the angle between the ladder and the wall?

65 degrees
15 degrees
35 degrees
25 degrees (Correct answer)

Correct answer: 25 degrees

The three angles of the right triangle sum to 180 degrees: 180 - 90 - 65 = 25 degrees between the ladder and the wall.

Question 3: Make y the subject: 2y + x = 10.

y = (10 - x)/2 (Correct answer)
y = 10 - 2x
y = (x - 10)/2
y = 10 - x

Correct answer: y = (10 - x)/2

Subtract x: 2y = 10 - x. Divide by 2: y = (10 - x)/2.

Question 4: In a right-angled triangle, sin(theta) is defined as:

adjacent / hypotenuse
hypotenuse / opposite
opposite / adjacent
opposite / hypotenuse (Correct answer)

Correct answer: opposite / hypotenuse

By definition, sin(theta) = opposite side / hypotenuse (the 'SOH' in SOH-CAH-TOA).

Question 5: A text notes that two researchers reached 'divergent' conclusions. This means their conclusions were:

Both wrong
Identical
Different from each other (Correct answer)
Never published

Correct answer: Different from each other

'Divergent' means moving apart or differing, so the researchers' conclusions differed from one another.

Question 6: A class has 18 girls and 12 boys. What is the ratio of girls to boys in simplest form?

18:12
6:4
2:3
3:2 (Correct answer)

Correct answer: 3:2

18:12 divides by 6 to give 3:2, the simplest form of the ratio of girls to boys.

Question 7: If f(x) = eˣ, then f′(x) equals:

e
x · eˣ⁻¹
eˣ (Correct answer)
ln(x)

Correct answer: eˣ

The exponential function eˣ is its own derivative: d/dx[eˣ] = eˣ.

Question 8: A R1 200 deposit earns 8% simple interest in one year. How much interest is earned?

R960
R96 (Correct answer)
R80
R120

Correct answer: R96

8% of R1 200 = 0.08 x 1 200 = R96 interest.

Question 9: A table lists test scores: 60, 72, 72, 80, 96. What is the mode of these scores?

96
80
60
72 (Correct answer)

Correct answer: 72

The mode is the most frequent value. 72 appears twice while all others appear once, so the mode is 72.

Question 10: A dataset of five values is 4, 6, 6, 8, 11. What is the mean (average)?

6
8
11
7 (Correct answer)

Correct answer: 7

Sum = 4+6+6+8+11 = 35. Mean = 35 / 5 = 7.

Question 11: When an academic writer uses the phrase 'it is widely assumed that', they are most likely about to:

Introduce a common belief they may then question (Correct answer)
State a proven fact they fully accept
Quote a dictionary definition
End the essay

Correct answer: Introduce a common belief they may then question

Phrases like 'it is widely assumed' often set up a common belief that the writer will examine or challenge, signalling possible disagreement.

Question 12: If 60% of a number is 90, what is the number?

150 (Correct answer)
54
120
180

Correct answer: 150

If 60% = 90, then 1% = 1.5, so 100% = 150. The number is 150.

Question 13: If f′(c) = 0 and f″(c) > 0, the point x = c is:

A local minimum (Correct answer)
A local maximum
An asymptote
A point of inflection

Correct answer: A local minimum

By the second derivative test, f′(c) = 0 with f″(c) > 0 indicates the curve is concave up at c, confirming a local minimum.

Question 14: In the sentence 'The minister's response was evasive,' the word 'evasive' suggests the minister:

Avoided giving a straight answer (Correct answer)
Refused to speak at all
Gave a written report
Answered directly and clearly

Correct answer: Avoided giving a straight answer

'Evasive' describes language that dodges or avoids the question, so the minister avoided a direct answer.

Question 15: Which set of ordered pairs represents a function?

{(1,2),(1,3),(2,4)}
{(1,2),(3,2),(1,5)}
{(1,2),(2,3),(3,4)} (Correct answer)
{(2,1),(2,2),(3,4)}

Correct answer: {(1,2),(2,3),(3,4)}

A function requires each x-value to map to exactly one y-value; only {(1,2),(2,3),(3,4)} satisfies this.

Question 16: Solve for x: x^2 = 49.

x = 7 only
x = 7 or x = -7 (Correct answer)
x = -7 only
x = 24.5

Correct answer: x = 7 or x = -7

Both 7 and -7 squared equal 49, so x = 7 or x = -7.

Question 17: Which of the following equals ∫4x³ dx?

x⁴/4 + C
x⁴ + C (Correct answer)
12x² + C
4x⁴ + C

Correct answer: x⁴ + C

∫4x³ dx = 4 · x⁴/4 + C = x⁴ + C by the power rule for integration.

Question 18: Two friends share R450 in the ratio 5:4. How much does the friend with the larger share receive?

R200
R270
R250 (Correct answer)
R225

Correct answer: R250

Total parts = 5 + 4 = 9. Each part = 450 / 9 = R50. The larger share is 5 x 50 = R250.

Question 19: If an academic passage describes an argument as 'unsubstantiated', the writer is suggesting the argument is:

Too long to read
Translated from another language
Well supported by evidence
Not backed by sufficient evidence (Correct answer)

Correct answer: Not backed by sufficient evidence

'Unsubstantiated' means not supported by evidence or proof, so the writer is questioning the argument's support.

Question 20: Expand: (x + 2)(x + 3).

x^2 + 5x + 5
x^2 + 5x + 6 (Correct answer)
x^2 + 6
x^2 + 6x + 5

Correct answer: x^2 + 5x + 6

FOIL: x^2 + 3x + 2x + 6 = x^2 + 5x + 6.

Question 21: In the AQL Academic Literacy section, a passage states: 'Despite initial scepticism, the municipality eventually endorsed the recycling scheme.' What does this sentence tell you about the municipality's final position?

It proposed an entirely different scheme
It supported the recycling scheme in the end (Correct answer)
It remained undecided about the scheme
It rejected the recycling scheme outright

Correct answer: It supported the recycling scheme in the end

'Eventually endorsed' signals final approval. 'Despite initial scepticism' shows the municipality first had doubts but ultimately supported the scheme, so option B is correct.

Question 22: The circumference of a circle is given by which formula (r = radius)?

pi r^2
r^2 / 2
pi d^2
2 pi r (Correct answer)

Correct answer: 2 pi r

Circumference = 2 pi r (or pi d). The area, by contrast, is pi r^2.

Question 23: A jacket costs R800 and is discounted by 25%. What is the sale price?

R200
R750
R600 (Correct answer)
R775

Correct answer: R600

25% of R800 = R200 discount. Sale price = 800 - 200 = R600.

Question 24: Two data sets have the same mean but Set A's values are tightly clustered while Set B's are widely spread. Which set has the greater range?

Set A
Set B (Correct answer)
They are equal
Range cannot relate to spread

Correct answer: Set B

Range measures spread (highest minus lowest). The more widely spread Set B has the larger range.

Question 25: In academic reading, a 'main idea' of a paragraph is best described as:

The central point the paragraph develops (Correct answer)
An unrelated side comment
The first word of the paragraph
A single supporting example

Correct answer: The central point the paragraph develops

The main idea is the central point a paragraph develops; supporting details and examples explain or back it up.

Question 26: Solve for x: 3x + 7 = 22.

x = 5 (Correct answer)
x = 3
x = 15
x = 7

Correct answer: x = 5

Subtract 7: 3x = 15. Divide by 3: x = 5.

Question 27: A function f(x) is concave down on an interval when:

f(x) < 0
f″(x) > 0
f′(x) < 0
f″(x) < 0 (Correct answer)

Correct answer: f″(x) < 0

A negative second derivative means the rate of change is decreasing, indicating the graph curves downward (concave down).

Question 28: An academic text refers to 'the latter' when discussing two methods. If the methods mentioned are 'observation and experimentation', what does 'the latter' refer to?

Experimentation (Correct answer)
Both methods equally
Neither method
Observation

Correct answer: Experimentation

'The latter' always refers to the second of two items mentioned. Here the second item is 'experimentation'.

Question 29: A learner scored 36 out of 45 on a test. What is this as a percentage?

75%
72%
80% (Correct answer)
85%

Correct answer: 80%

36 / 45 = 0.8, which is 80%.

Question 30: A bar graph shows monthly rainfall in Cape Town: Jan 15mm, Feb 17mm, Jun 93mm, Jul 82mm. Which month had the highest rainfall?

January
June (Correct answer)
July
February

Correct answer: June

June recorded 93mm, the largest value shown, so it is the wettest month in the data.

Question 31: A right-angled triangle has legs of 3cm and 4cm. What is the length of the hypotenuse?

7cm
5cm (Correct answer)
12cm
6cm

Correct answer: 5cm

By Pythagoras, hypotenuse = sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5cm.

Question 32: A passage says a policy was 'implemented incrementally'. What can you infer about how the policy was introduced?

Only on paper, never in practice
All at once on a single day
Secretly and without notice
Gradually, in stages over time (Correct answer)

Correct answer: Gradually, in stages over time

'Incrementally' means in small steps or stages, so the policy was rolled out gradually rather than all at once.

Question 33: If f(x) = 2x - 5, what is f(4)?

13
8
3 (Correct answer)
1

Correct answer: 3

Substitute x = 4: f(4) = 2(4) - 5 = 8 - 5 = 3.

Question 34: What is the maximum value of f(x) = −2x² + 8x − 3?

2
5 (Correct answer)
8
−3

Correct answer: 5

The vertex is at x = −b/(2a) = 2; f(2) = −8 + 16 − 3 = 5.

Question 35: If a passage describes a plan as 'feasible', the writer believes the plan is:

Impossible to carry out
Already completed
Capable of being done successfully (Correct answer)
Too expensive to consider

Correct answer: Capable of being done successfully

'Feasible' means practical and capable of being done, so the writer thinks the plan can realistically be carried out.

Question 36: For f(x) = x³ − 3x, the x-values of the turning points are:

x = ±3
x = 0 and x = 3
x = ±1 (Correct answer)
x = ±√3

Correct answer: x = ±1

Setting f′(x) = 3x² − 3 = 0 gives x² = 1, so x = ±1.

Question 37: A passage concludes: 'The evidence, while compelling, is far from conclusive.' What is the author's overall stance toward the evidence?

The evidence is strong but not yet decisive (Correct answer)
The evidence proves the case completely
The evidence is fabricated
The evidence is weak and irrelevant

Correct answer: The evidence is strong but not yet decisive

'Compelling' shows the evidence is strong, but 'far from conclusive' shows it does not fully settle the matter, so it is strong but not decisive.

Question 38: The angles in a triangle are 50 degrees and 60 degrees. What is the third angle?

80 degrees
50 degrees
70 degrees (Correct answer)
110 degrees

Correct answer: 70 degrees

Angles in a triangle add to 180 degrees. 180 - 50 - 60 = 70 degrees.

Question 39: The range of f(x) = −x² + 3 is:

y ≤ 3 (Correct answer)
y ≤ −3
y ≥ 3
All real numbers

Correct answer: y ≤ 3

The parabola opens downward with vertex at (0, 3), so the maximum output is 3 and f(x) ≤ 3 for all x.

Question 40: An author writes that survey results were 'skewed' by a small sample. The word 'skewed' implies the results were:

Perfectly balanced
Doubled in size
Distorted or biased (Correct answer)
Translated incorrectly

Correct answer: Distorted or biased

'Skewed' means distorted away from a true or balanced value, here because of the small sample.

Question 41: The area under f(x) = 2x from x = 0 to x = 3 is:

9 (Correct answer)
18
3
6

Correct answer: 9

∫₀³ 2x dx = [x²]₀³ = 9 − 0 = 9.

Question 42: A recipe uses flour and sugar in the ratio 3:1. If you use 600g of flour, how much sugar do you need?

200g (Correct answer)
300g
100g
600g

Correct answer: 200g

The ratio 3:1 means sugar is one third of the flour. 600 / 3 = 200g of sugar.

Question 43: The transformation that maps f(x) to f(x + 2) − 3 is:

Shift left 2 and down 3 (Correct answer)
Shift right 2 and down 3
Shift right 2 and up 3
Shift left 2 and up 3

Correct answer: Shift left 2 and down 3

Replacing x with x+2 shifts the graph 2 units left; subtracting 3 outside shifts it 3 units down.

Question 44: If f(x) = log₂(x), then f(8) equals:

16
3 (Correct answer)
2
4

Correct answer: 3

log₂(8) = 3 because 2³ = 8.

Question 45: What is the area of a rectangle that is 8m long and 5m wide?

13 m^2
40 m^2 (Correct answer)
45 m^2
26 m^2

Correct answer: 40 m^2

Area of a rectangle = length x width = 8 x 5 = 40 m^2.

Question 46: What is the derivative of f(x) = 3x² − 2x + 5?

6x − 2 (Correct answer)
3x² − 2
3x − 2
6x + 5

Correct answer: 6x − 2

Using the power rule: the derivative of 3x² is 6x, the derivative of −2x is −2, and constants vanish, giving 6x − 2.

Question 47: Petrol increases from R20 to R23 per litre. What is the percentage increase?

13%
23%
3%
15% (Correct answer)

Correct answer: 15%

Increase = R3 on R20. 3 / 20 = 0.15 = 15% increase.

Question 48: Two angles are complementary and one is 35 degrees. What is the other angle?

55 degrees (Correct answer)
145 degrees
325 degrees
45 degrees

Correct answer: 55 degrees

Complementary angles add to 90 degrees. 90 - 35 = 55 degrees.

Question 49: What is lim(x→2) [(x² − 4) / (x − 2)]?

0
Undefined
2
4 (Correct answer)

Correct answer: 4

Factoring gives (x+2)(x−2)/(x−2) = x+2; substituting x=2 yields 4.

Question 50: The average rate of change of f(x) = x² from x = 1 to x = 3 is:

6
8
4 (Correct answer)
2

Correct answer: 4

[f(3) − f(1)] / (3 − 1) = (9 − 1) / 2 = 8/2 = 4.

Question 51: The period of f(x) = sin(2x) is:

π (Correct answer)
4π
2π
π/2

Correct answer: π

The period of sin(bx) is 2π/b; with b = 2 the period is 2π/2 = π.

Question 52: What is the derivative of f(x) = sin(x)?

tan(x)
−sin(x)
−cos(x)
cos(x) (Correct answer)

Correct answer: cos(x)

By the standard trigonometric derivative rule, d/dx[sin(x)] = cos(x).

Question 53: Read this sentence: 'The study's findings were tentative and should not be generalised.' What does 'tentative' most nearly mean here?

Widely accepted
Provisional and uncertain (Correct answer)
Final and certain
Statistically perfect

Correct answer: Provisional and uncertain

'Tentative' means not yet definite or provisional. The clause 'should not be generalised' reinforces that the findings are uncertain.

Question 54: A line graph shows a learner's marks rising from 45% in Term 1 to 63% in Term 4. What is the total percentage-point increase?

63 points
28 points
8 points
18 points (Correct answer)

Correct answer: 18 points

The increase is 63% - 45% = 18 percentage points.

Question 55: A graph's vertical axis is labelled 'Number of learners (thousands)' and a bar reaches 3.5. How many learners does this represent?

3 500 (Correct answer)
350
35
35 000

Correct answer: 3 500

The axis is in thousands, so a value of 3.5 means 3.5 x 1 000 = 3 500 learners.

Question 56: Simplify: 2(x + 4) + 3x.

5x + 8 (Correct answer)
5x + 4
2x + 7
6x + 8

Correct answer: 5x + 8

Expand: 2x + 8 + 3x. Combine like terms: 5x + 8.

Question 57: A passage describes a finding as 'counterintuitive'. This suggests the finding:

Was expected by everyone
Goes against what most people would expect (Correct answer)
Matches what most people expect
Is impossible to measure

Correct answer: Goes against what most people would expect

'Counterintuitive' means contrary to intuition or common expectation, so the finding surprises most people.

Question 58: The asymptotes of f(x) = 2/(x − 3) + 1 are:

x = 3 and y = −1
x = −3 and y = 1
x = 1 and y = 3
x = 3 and y = 1 (Correct answer)

Correct answer: x = 3 and y = 1

The vertical asymptote occurs where the denominator is zero (x = 3) and the horizontal asymptote is the constant being added (y = 1).

Question 59: Factorise fully: x^2 - 9.

(x + 9)(x - 1)
x(x - 9)
(x + 3)(x - 3) (Correct answer)
(x - 3)(x - 3)

Correct answer: (x + 3)(x - 3)

x^2 - 9 is a difference of two squares: x^2 - 3^2 = (x + 3)(x - 3).

Question 60: A survey of 200 students shows 30% prefer online classes. How many students prefer online classes?

30
200
70
60 (Correct answer)

Correct answer: 60

30% of 200 = 0.30 x 200 = 60 students.

Question 61: What is the domain of f(x) = √(4 − x)?

All real numbers
x ≥ 4
x < 4
x ≤ 4 (Correct answer)

Correct answer: x ≤ 4

For the square root to be defined, the expression inside must be non-negative: 4 − x ≥ 0 gives x ≤ 4.

Question 62: The gradient of the tangent to f(x) = x³ at x = 2 is:

12 (Correct answer)
6
3
8

Correct answer: 12

f′(x) = 3x², so f′(2) = 3(4) = 12.

Question 63: A passage states: 'Attendance dwindled over the term.' From 'dwindled', you can infer attendance:

Was never recorded
Stayed exactly the same
Grew steadily
Decreased gradually (Correct answer)

Correct answer: Decreased gradually

'Dwindled' means became gradually smaller, so attendance decreased over time.

Question 64: What is the value of cos(60 degrees)?

0
1
0.87
0.5 (Correct answer)

Correct answer: 0.5

cos(60 degrees) = 0.5, a standard value worth memorising for the NBT Maths paper.

Question 65: A pie chart shows a household budget: Rent 40%, Food 25%, Transport 20%, Other 15%. If the total monthly income is R12 000, how much is spent on rent?

R2 400
R3 000
R4 800 (Correct answer)
R6 000

Correct answer: R4 800

Rent is 40% of R12 000. 0.40 x 12 000 = R4 800.

Question 66: Solve the inequality: x - 6 < 2.

x < 8 (Correct answer)
x > 8
x > -4
x < 4

Correct answer: x < 8

Add 6 to both sides: x < 8.

Question 67: A reading passage uses the word 'consequently' to begin a sentence. What relationship does this signpost word indicate?

Cause and effect (Correct answer)
Comparison
Contrast
Sequence in time only

Correct answer: Cause and effect

'Consequently' introduces a result or effect that follows from a previously stated cause, making it a cause-and-effect connector.

Question 68: If f(x) = 2x + 1 and g(x) = x², what is f(g(3))?

37
7
13
19 (Correct answer)

Correct answer: 19

g(3) = 9, then f(9) = 2(9) + 1 = 19.

Question 69: A text states: 'Whereas urban schools reported improvements, rural schools saw little change.' The word 'whereas' signals:

A cause of the change
A chronological order
A contrast between the two groups (Correct answer)
Agreement between the two groups

Correct answer: A contrast between the two groups

'Whereas' is a contrast connector, here highlighting the difference between urban and rural school outcomes.

National Benchmark Test (NBT) – South Africa

The NBT assesses the academic readiness of South African school leavers for higher education across Academic Literacy and Mathematics, helping universities determine admission and support needs.

135

Questions

180 min

Time Limit

70%

To Pass

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