Exam: [Course code] [Exam name] with Answer Key

Exam paper template with a separate answer key: candidate instructions, multiple-choice, short-answer and extended problem sections, a point allocation table, worked solutions written in KaTeX math, and marking guidance. The sample questions use introductory calculus and algebra so the format is concrete; replace them with your own. Based on common assessment practice.

1. Instructions to candidates

1. Write your name and student ID on every page.
2. Answer all questions in Sections A, B and C.
3. In Section A, circle one answer per question. No working is required.
4. In Sections B and C, show all working. Correct answers without working may receive limited credit.
5. Give exact answers unless a question asks for a decimal approximation.
6. If you believe a question contains an error, state your assumption and answer the question as you interpret it.
7. Points for each question are shown in brackets.

2. Point allocation

Check: 10 + 12 + 28 = 50.

3. Section A: Multiple choice (10 points)

Q1. [2] What is $\dfrac{d}{dx}\left(x^3\right)$?

(a) $x^2$   (b) $3x^2$   (c) $3x^3$   (d) $\dfrac{x^4}{4}$

Q2. [2] Evaluate $\displaystyle\int_0^3 2x \, dx$.

(a) 3   (b) 6   (c) 9   (d) 18

Q3. [2] What is $\displaystyle\lim_{x \to 0} \frac{\sin x}{x}$?

(a) 0   (b) 1   (c) $\infty$   (d) The limit does not exist

Q4. [2] What is $\dfrac{d}{dx}\left(e^{2x}\right)$?

(a) $e^{2x}$   (b) $2e^{x}$   (c) $2e^{2x}$   (d) $2x\,e^{2x-1}$

Q5. [2] What is the slope of the tangent to $y = x^2$ at $x = 3$?

(a) 3   (b) 6   (c) 9   (d) 2

4. Section B: Short answer (12 points)

Q6. [4] Use the product rule to differentiate $f(x) = x^2 \sin x$.

Q7. [4] Find the critical points of $f(x) = x^3 - 3x$ and classify each as a local maximum or local minimum.

Q8. [4] Solve $2x^2 - 8x + 6 = 0$.

5. Section C: Problems (28 points)

Q9. [14] Let $f(x) = \left(3x^2 + 1\right)^4$.

(a) [6] Find $f'(x)$.

(b) [3] Evaluate $f'(1)$.

(c) [5] Find the equation of the tangent line to $y = f(x)$ at $x = 0$.

Q10. [14]

(a) [8] Find the area of the region enclosed by $y = x^2$ and $y = 2x$.

(b) [6] Solve the system of equations

6. Answer key

6.1 Section A

Worked note for Q2:

6.2 Section B

Q6. Product rule $(uv)' = u'v + uv'$ with $u = x^2$, $v = \sin x$:

Marks: rule stated or applied correctly [2]; each term correct [1 + 1].

Q7. Differentiate and set to zero:

Second derivative $f''(x) = 6x$. At $x = 1$, $f''(1) = 6 > 0$: local minimum, $f(1) = -2$. At $x = -1$, $f''(-1) = -6 < 0$: local maximum, $f(-1) = 2$.

Marks: $f'(x)$ [1]; both critical points [1]; correct classification of each with justification [1 + 1]. Accept a sign chart of $f'$ instead of $f''$.

Q8. Divide by 2 and factor:

Marks: simplification or correct use of the quadratic formula [2]; each root [1 + 1]. Check: $2(1)^2 - 8(1) + 6 = 0$ and $2(9) - 24 + 6 = 0$.

6.3 Section C

Q9 (a). Chain rule with outer function $u^4$ and inner $u = 3x^2 + 1$:

Marks: derivative of outer function [2]; derivative of inner function [2]; correct product and simplification [2].

Q9 (b).

Marks: substitution [1]; arithmetic [2]. Follow-through: award full marks for correct evaluation of an incorrect $f'(x)$ from part (a).

Q9 (c). $f(0) = (0 + 1)^4 = 1$ and $f'(0) = 0$, so the tangent line is

Marks: point $f(0)$ [1]; slope $f'(0)$ [2]; line equation [2].

Q10 (a). Intersections: $x^2 = 2x \implies x(x - 2) = 0 \implies x = 0, 2$. On $[0, 2]$, $2x \ge x^2$, so

Marks: limits of integration [2]; correct integrand with the upper curve first [1]; antiderivative [3]; final value [2]. Accept $\approx 1.33$.

Q10 (b). Adding the equations eliminates $y$:

Marks: valid elimination or substitution [3]; $x$ [1]; $y$ [1]; check in both equations [1].

7. Marking guidance

• Award method marks for correct reasoning even when an arithmetic slip leads to a wrong final answer.
• Apply follow-through: an error in one part is penalised once, and later parts are marked on the student's own earlier value.
• Accept any correct equivalent form (for example $2x\sin x + x^2\cos x$ or $x(2\sin x + x\cos x)$).
• Do not award marks for a correct answer obtained by clearly invalid working.
• Use half-marks only where the scheme shows them.
• Second-mark a sample of [10%] of scripts, including all scripts within [2] points of a grade boundary.

Results summary

Standard and licence note

Based on common assessment practice for written examinations and marking schemes. Original text written for this collection; sample questions and solutions are illustrative and were checked for correctness. Template licence: MIT.