1. Student instructions
- Choose one answer for each question.
- Complete the test independently.
- Work without a calculator for this assessment.
- Use scratch paper for calculations.
- Give exact answers where appropriate.
- If a question takes too long, mark it for review and return to it later.
- Check domain restrictions, signs, sequence indices, and units before submitting.
Suggested timing:
- First attempt: 35 minutes.
- Review unanswered and uncertain questions: 10 minutes.
Start the timer when you begin Question 1.
2. Test questions
Question 1
Find the domain of:
\[ f(x)=\frac{\sqrt{x+2}}{x-1}. \]
A. [−2, ∞)
B. [−2, 1) ∪ (1, ∞)
C. (−2, 1) ∪ (1, ∞)
D. [−2, 1)
Question 2
Find the range of:
\[ f(x)=(x-3)^2-2,\qquad 1\leq x\leq4. \]
A. [−2, 2]
B. [−2, −1]
C. [−1, 2]
D. [1, 4]
Question 3
Which function is even on its domain?
A. \(f(x)=x^3+x\)
B. \(f(x)=x^2+x\)
C. \(f(x)=x^4-3x^2+2\)
D. \(f(x)=x^3+1\)
Question 4
Let:
\[ f(x)=2x-1,\qquad g(x)=x^2. \]
Find:
\[ f(g(-2)). \]
A. −9
B. 25
C. −7
D. 7
Question 5
A function is defined by:
\[ f(x)= \begin{cases} x+2, & x<1,\\ x^2, & x\geq1. \end{cases} \]
Find:
\[ f(0)+f(1). \]
A. 2
B. 3
C. 4
D. 5
Question 6
Which function has all the following features?
- Domain: all real numbers.
- Range: positive real numbers.
- Passes through (0, 1).
- Strictly decreasing.
- Horizontal asymptote: y = 0.
A. \(y=2^x\)
B. \(y=\log_2x\)
C. \(y=\left(\frac12\right)^x\)
D. \(y=-x^2+1\)
Question 7
Solve:
\[ 9^x=27^{x-1}. \]
A. x = 1
B. x = 2
C. x = −3
D. x = 3
Question 8
Solve:
\[ \log_2(x-1)+\log_2(x+1)=3. \]
A. x = 3
B. x = −3
C. x = −3 or x = 3
D. x = 9
Question 9
Solve:
\[ \left(\frac12\right)^x\geq4. \]
A. x ≥ 2
B. x ≤ 2
C. x ≤ −2
D. x ≥ −2
Question 10
Given:
\[ \sin\theta=\frac5{13}, \qquad \frac{\pi}{2}<\theta<\pi, \]
find cos θ.
A. 12/13
B. −12/13
C. −5/13
D. 5/12
Question 11
An arithmetic sequence satisfies:
\[ a_4=11,\qquad a_9=26. \]
Find its general term.
A. aₙ = 3n + 2
B. aₙ = 5n − 9
C. aₙ = 2n + 3
D. aₙ = 3n − 1
Question 12
Find the sum:
\[ 4+7+10+\cdots+31. \]
A. 175
B. 157.5
C. 192.5
D. 350
Question 13
A geometric sequence has:
\[ a_1=5,\qquad r=-2. \]
Find the sixth term.
A. 160
B. −80
C. −160
D. 320
Question 14
Find:
\[ 3+6+12+24+48+96. \]
A. 96
B. 192
C. 381
D. 189
Question 15
A geometric sequence satisfies:
\[ a_2=6,\qquad a_5=162. \]
Find the common ratio.
A. 2
B. 3
C. −3
D. 9
Question 16
A sequence has partial sum:
\[ S_n=n^2+2n, \]
where \(S_n\) is the sum of its first n terms.
Find its nth term.
A. aₙ = n + 2
B. aₙ = 2n − 1
C. aₙ = 2n + 1
D. aₙ = n² + 2
Question 17
Differentiate:
\[ f(x)=2x^3-5x^2+4x-7. \]
A. \(f'(x)=6x^2-10x+4\)
B. \(f'(x)=6x^2-5x+4\)
C. \(f'(x)=2x^2-10x+4\)
D. \(f'(x)=6x^2-10x-7\)
Question 18
For:
\[ f(x)=x^2+\frac3x,\qquad x\neq0, \]
find \(f'(1)\).
A. 5
B. 1
C. −5
D. −1
Question 19
Find the tangent line to:
\[ f(x)=x^2-2x+3 \]
at x = 2.
A. y = 2x + 3
B. y = 2x − 1
C. y = 3x − 3
D. y = 2x − 4
Question 20
A function is differentiable on all real numbers and satisfies:
\[ f'(x)=(x+2)(x-1). \]
On which interval is the function decreasing?
A. (−∞, −2)
B. (1, ∞)
C. (−2, 1)
D. (−∞, 1)
Question 21
Which statement correctly describes:
\[ f(x)=x^3-3x^2? \]
A. It has a local maximum at (0, 0) and a local minimum at (2, −4).
B. It has a local minimum at (0, 0) and a local maximum at (2, −4).
C. It has local maxima at both (0, 0) and (2, −4).
D. It has no stationary points.
Question 22
Find the absolute maximum value of:
\[ f(x)=x^2-4x+1 \]
on the interval [0, 5].
A. −3
B. 1
C. 5
D. 6
Question 23
A rectangle has perimeter 36 meters.
What is its largest possible area?
A. 72 m²
B. 81 m²
C. 36 m²
D. 324 m²
Question 24
A particle’s position is:
\[ s(t)=t^2-6t+5,\qquad 0\leq t\leq5, \]
where position is measured in meters and time in seconds.
Find its total distance traveled over this interval.
A. 5 m
B. 9 m
C. 13 m
D. 4 m
Question 25
A student claims:
Every stationary point is a local maximum or local minimum.
Which function provides a counterexample at x = 0?
A. \(f(x)=x^3\)
B. \(f(x)=x^2\)
C. \(f(x)=-x^2\)
D. \(f(x)=x+1\)
3. Submission check
Before submitting, check that you have:
- Selected exactly one answer for each question.
- Applied all domain restrictions.
- Rejected invalid logarithmic solutions.
- Used the correct sequence indices and term counts.
- Distinguished a function value from a derivative value.
- Checked endpoints where required.
- Distinguished distance from displacement.
Stop the timer and record your completion time.
4. Instructor answer key
| Question | Answer | Question | Answer |
|---|---|---|---|
| 1 | B | 14 | D |
| 2 | A | 15 | B |
| 3 | C | 16 | C |
| 4 | D | 17 | A |
| 5 | B | 18 | D |
| 6 | C | 19 | B |
| 7 | D | 20 | C |
| 8 | A | 21 | A |
| 9 | C | 22 | D |
| 10 | B | 23 | B |
| 11 | D | 24 | C |
| 12 | A | 25 | A |
| 13 | C | — | — |
5. Answers and explanations
Answer 1: B
The square root requires:
\[ x+2\geq0, \]
so x ≥ −2.
The denominator requires x ≠ 1.
Domain: [−2, 1) ∪ (1, ∞).
Answer 2: A
The squared term is smallest at x = 3:
\[ f(3)=-2. \]
Check the endpoints:
\[ f(1)=2,\qquad f(4)=-1. \]
The function is continuous on the interval.
Range: [−2, 2].
Answer 3: C
For the function in option C:
\[ f(-x)=(-x)^4-3(-x)^2+2=x^4-3x^2+2=f(x). \]
Its domain is symmetric about zero, so it is even.
Answer 4: D
Evaluate the inner function first:
\[ g(-2)=(-2)^2=4. \]
Then:
\[ f(4)=2(4)-1=7. \]
Answer: 7.
Answer 5: B
Since 0 < 1, use the first branch:
\[ f(0)=0+2=2. \]
Since 1 belongs to the second branch:
\[ f(1)=1^2=1. \]
Therefore:
\[ f(0)+f(1)=3. \]
Answer 6: C
An exponential function with base between 0 and 1 is strictly decreasing.
For:
\[ y=\left(\frac12\right)^x, \]
the domain is all real numbers, outputs are positive, the graph passes through (0, 1), and its horizontal asymptote is y = 0.
Answer 7: D
Rewrite using base 3:
\[ 3^{2x}=3^{3x-3}. \]
Therefore:
\[ 2x=3x-3. \]
Answer: x = 3.
Answer 8: A
The original logarithms require x > 1.
Combine:
\[ \log_2\big((x-1)(x+1)\big)=3. \]
Therefore:
\[ x^2-1=8, \]\[ x^2=9. \]
The candidates are −3 and 3, but only 3 satisfies x > 1.
Answer: x = 3.
Answer 9: C
Write 4 using base 1/2:
\[ 4=\left(\frac12\right)^{-2}. \]
The exponential function is decreasing, so:
\[ \left(\frac12\right)^x \geq \left(\frac12\right)^{-2} \]
requires:
\[ x\leq-2. \]
Answer: x ≤ −2.
Answer 10: B
Use:
\[ \sin^2\theta+\cos^2\theta=1. \]
Then:
\[ \cos^2\theta=1-\frac{25}{169} =\frac{144}{169}. \]
Cosine is negative in the second quadrant.
Answer: cos θ = −12/13.
Answer 11: D
There are five steps from the fourth term to the ninth:
\[ 5d=26-11=15. \]
Thus d = 3.
Find the first term:
\[ a_1=11-3(3)=2. \]
Therefore:
\[ a_n=2+3(n-1)=3n-1. \]
General term: aₙ = 3n − 1.
Answer 12: A
Find the number of terms:
\[ 31=4+3(n-1). \]
Thus n = 10.
\[ S_{10}=\frac{10}{2}(4+31)=175. \]
Answer: 175.
Answer 13: C
Use the geometric general term:
\[ a_6=a_1r^5=5(-2)^5=-160. \]
Answer: −160.
Answer 14: D
There are six terms, with first term 3 and ratio 2.
\[ S_6=\frac{3(2^6-1)}{2-1} =3(63)=189. \]
Answer: 189.
Answer 15: B
For a geometric sequence:
\[ a_5=a_2r^3. \]
Therefore:
\[ 162=6r^3, \]\[ r^3=27. \]
Common ratio: r = 3.
Answer 16: C
For n ≥ 2:
\[ a_n=S_n-S_{n-1}. \]
Calculate:
\[ S_{n-1}=(n-1)^2+2(n-1)=n^2-1. \]
Therefore:
\[ a_n=(n^2+2n)-(n^2-1)=2n+1. \]
Check the first term separately:
\[ a_1=S_1=3. \]
The formula also gives 3 when n = 1.
General term: aₙ = 2n + 1.
Answer 17: A
Differentiate each term:
\[ f'(x)=6x^2-10x+4. \]
The derivative of the constant −7 is zero.
Answer 18: D
Differentiate:
\[ f'(x)=2x-\frac3{x^2}. \]
Evaluate:
\[ f'(1)=2-3=-1. \]
Answer: −1.
Answer 19: B
Find the point:
\[ f(2)=4-4+3=3. \]
Find the slope:
\[ f'(x)=2x-2, \]\[ f'(2)=2. \]
The tangent equation is:
\[ y-3=2(x-2). \]
Answer: y = 2x − 1.
Answer 20: C
The derivative is:
\[ f'(x)=(x+2)(x-1). \]
It is negative when the factors have opposite signs:
\[ -2<x<1. \]
Decreasing interval: (−2, 1).
Answer 21: A
Differentiate:
\[ f'(x)=3x^2-6x=3x(x-2). \]
The stationary inputs are 0 and 2.
The derivative signs are:
- Positive for x < 0.
- Negative for 0 < x < 2.
- Positive for x > 2.
Therefore, x = 0 gives a local maximum and x = 2 gives a local minimum.
\[ f(0)=0,\qquad f(2)=8-12=-4. \]
Local maximum point: (0, 0).
Local minimum point: (2, −4).
Answer 22: D
Differentiate:
\[ f'(x)=2x-4. \]
The interior stationary input is x = 2.
Compare its value with both endpoint values:
\[ f(0)=1, \]\[ f(2)=-3, \]\[ f(5)=6. \]
Absolute maximum value: 6, attained at x = 5.
Answer 23: B
Let the sides be x and y:
\[ 2x+2y=36. \]
Therefore:
\[ y=18-x. \]
Area:
\[ A(x)=x(18-x)=18x-x^2, \qquad 0<x<18. \]
Differentiate:
\[ A'(x)=18-2x. \]
The stationary input is x = 9. The derivative changes from positive to negative there.
Then y = 9.
Maximum area: 81 m².
Answer 24: C
Find the velocity:
\[ v(t)=s'(t)=2t-6. \]
It changes from negative to positive at t = 3, so the particle reverses direction there.
Calculate:
\[ s(0)=5, \]\[ s(3)=9-18+5=-4, \]\[ s(5)=25-30+5=0. \]
Total distance:
\[ |-4-5|+|0-(-4)|=9+4=13. \]
Total distance: 13 m.
The displacement is −5 m.
Answer 25: A
For:
\[ f(x)=x^3, \]
we have:
\[ f'(x)=3x^2. \]
At x = 0, the derivative is zero, so the point is stationary.
However, the function increases through the origin. There is no local maximum or minimum.
Option D is not a counterexample because its derivative is 1, so the point is not stationary.
6. Topic diagnosis
Record your score in each area.
| Assessment area | Questions | Available marks | Revision focus |
|---|---|---|---|
| Function concepts and properties | 1–5 | 5 | Domain, range, parity, composition, and piecewise functions. |
| Elementary functions | 6–10 | 5 | Exponential graphs, equations, inequalities, logarithms, and trigonometry. |
| Sequences and summation | 11–16 | 6 | Common differences, ratios, general terms, finite sums, and partial sums. |
| Derivatives and applications | 17–25 | 9 | Differentiation, tangents, monotonicity, extrema, optimization, and motion. |
A total score can hide a weak topic. Check the topic scores as well as the overall result.
7. Interpreting your result
| Score | Interpretation | Next action |
|---|---|---|
| 22–25 marks: 88–100% | Strong performance on this test. | Correct mistakes and check topic scores before proceeding. |
| 20–21 marks: 80–84% | Meets the overall course target. | Complete targeted revision and check topic scores before proceeding. |
| 15–19 marks: 60–76% | Understanding is developing, with gaps requiring revision. | Review weak topics and complete a fresh assessment. |
| 0–14 marks: 0–56% | Foundational revision is required. | Revisit the relevant lessons and worked examples before retesting. |
Recommended progression criteria
Before proceeding to the next module, aim to satisfy all of the following:
- Overall score: At least 20 out of 25.
- Function concepts: At least 3 out of 5.
- Elementary functions: At least 3 out of 5.
- Sequences and summation: At least 4 out of 6.
- Derivatives and applications: At least 6 out of 9.
- Corrections: Explain and independently solve every missed question.
Correct guesses should also be reviewed. Repeating the same test after reading its solutions checks correction, but a fresh set of questions gives better evidence of independent mastery.
8. Post-test correction task
For each incorrect or guessed answer, write:
- Question number.
- My original method.
- Why my answer was wrong or uncertain.
- The correct method and solution.
- A check that would prevent the same mistake.
Classify the error as one of the following:
- Concept misunderstanding.
- Incorrect method selection.
- Algebra or arithmetic error.
- Missed domain restriction.
- Incorrect sign.
- Sequence indexing or term-count error.
- Misreading the requested quantity.
- Time-management problem.
Module completion task: Finish the corrections, revise weak areas, and meet the progression criteria before starting the next module.