FAD1014: Mathematics II — Intensive Single-Session Drill Set

Duration: One sitting (≈ 3–4 hours)
Level: Mid-level proficiency
Objective: Solid fluency across all listed topics in a single intensive pass.

Ground Rules

  • No notes except the formula sheet below.
  • If you get stuck >10 minutes, mark it and move on.
  • Grade strictly after finishing all problems.
  • Target: 9/12 correct for proficiency.

Formula Sheet

Sequences & Summation

$$\sum_{r=1}^{n} r = \frac{n(n+1)}{2}, \quad \sum_{r=1}^{n} r^2 = \frac{n(n+1)(2n+1)}{6}, \quad \sum_{r=1}^{n} r^3 = \frac{n^2(n+1)^2}{4}$$ Telescoping: $\sum_{k=1}^{n}[f(k)-f(k-1)] = f(n)-f(0)$

Binomial

$$(a+b)^n = \sum_{r=0}^{n}\binom{n}{r}a^{n-r}b^r, \qquad (1+x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \cdots ; (|x|<1)$$

Taylor / Maclaurin

$$f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \cdots$$ Standard Maclaurin:

  • $e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$
  • $\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots$
  • $\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots$
  • $\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots$

Geometry

Shape Key Equation / Properties
Circle $(x-h)^2+(y-k)^2=r^2$; tangent $\perp$ radius
Parabola (vertical) $(x-h)^2=4a(y-k)$; focus $(h,k+a)$; directrix $y=k-a$; LR $=4a$
Parabola (horizontal) $(y-k)^2=4a(x-h)$; focus $(h+a,k)$; directrix $x=h-a$
Hyperbola (horizontal) $\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1$; foci $(h\pm c,k)$; $c^2=a^2+b^2$; asymptotes $y-k=\pm\frac{b}{a}(x-h)$
Hyperbola (vertical) $\frac{(y-k)^2}{a^2}-\frac{(x-h)^2}{b^2}=1$; foci $(h,k\pm c)$; asymptotes $y-k=\pm\frac{a}{b}(x-h)$

The Problems

Problem 1 — Series & Limits

(a) Determine whether $a_k = \dfrac{2k^2+3}{k^2-5k+1}$ converges or diverges. If it converges, find the limit.

(b) Express the following in sigma notation: $$\frac{2}{3} + \frac{3}{4} + \frac{4}{5} + \cdots + \frac{11}{12}$$

Problem 2 — Summation

(a) Evaluate $\displaystyle\sum_{r=1}^{20} (3r-1)$.

(b) Using the method of differences, evaluate $\displaystyle\sum_{k=1}^{n} \frac{2}{k(k+2)}$.

(c) Evaluate $\displaystyle\sum_{r=5}^{12} r^2$.

Problem 3 — Binomial Coefficients & Expansion

(a) Evaluate $\dbinom{9}{4}$ and $\dfrac{7!}{4!,3!}$.

(b) Expand $\left(2x - \dfrac{1}{x}\right)^5$ completely.

(c) Find the coefficient of $x^4$ in the expansion of $(x^2 - 3)^6$.

Problem 4 — General Binomial & Approximation

(a) Expand $\sqrt[3]{1+6x}$ in ascending powers of $x$ up to and including the term in $x^3$, stating the range of validity.

(b) By choosing an appropriate value of $x$, use your expansion to approximate $\sqrt[3]{1.06}$ correct to four decimal places.

Problem 5 — Partial Fractions + Series

Express $f(x) = \dfrac{5x+3}{(1+x)(1-2x)}$ in partial fractions. Hence, expand $f(x)$ in ascending powers of $x$ up to the term in $x^2$, stating the range of validity.

Problem 6 — Maclaurin from Definition

Find the first four non-zero terms of the Maclaurin series for $f(x) = e^{x}\cos x$ by computing derivatives at $x=0$.

Problem 7 — Taylor Approximation

Find the Taylor series for $f(x) = \ln x$ about $x = 1$ up to the term in $(x-1)^3$. Use this to approximate $\ln(1.1)$ correct to four decimal places.

Problem 8 — Maclaurin Manipulation

(a) Write down the first four non-zero terms of the Maclaurin series for $\sin(2x^2)$.

(b) Find the first three non-zero terms of the Maclaurin series for $x\ln(1+3x)$.

Problem 9 — Circle Geometry

(a) Find the centre and radius of the circle $x^2 + y^2 + 6x - 8y + 9 = 0$.

(b) Find the equation of the tangent to this circle at the point $(0, 9)$.

(c) Calculate the length of the tangent from the point $(5, 6)$ to this circle.

Problem 10 — Parabola Geometry

(a) Find the vertex, focus, directrix, and length of the latus rectum of the parabola $y^2 + 6y - 8x + 17 = 0$.

(b) A parabola has vertex $(2, -1)$ and directrix $x = -3$. Find its equation in standard form $(y-k)^2 = 4a(x-h)$.

Problem 11 — Hyperbola Geometry

(a) For the hyperbola $\dfrac{(x-1)^2}{16} - \dfrac{(y+2)^2}{9} = 1$, find the centre, vertices, foci, and equations of the asymptotes.

(b) Find the equation of the hyperbola with foci at $(0, \pm 5)$ and vertices at $(0, \pm 3)$.

Problem 12 — Connector: Geometry + Series

A parabolic mirror has cross-section $(x-1)^2 = 8(y-2)$. A light ray parallel to the axis of symmetry strikes the mirror at $x = 5$.

(a) Find the focus of the parabola and the coordinates of the point of reflection.

(b) The arc length element involves $\sqrt{1+\left(\frac{dy}{dx}\right)^2}$. Expand $\sqrt{1+t}$ as a binomial series up to $t^2$, then use this to approximate the arc length from $x=1$ to $x=5$ by setting $t = \left(\frac{dy}{dx}\right)^2$ and integrating term-by-term.

Scoring

Problem Marks Score
1 2 ___
2 3 ___
3 3 ___
4 2 ___
5 2 ___
6 2 ___
7 2 ___
8 2 ___
9 3 ___
10 2 ___
11 2 ___
12 2 ___
Total 27 ___/27

Proficiency: ≥20/27
Strong proficiency: ≥24/27

Post-Session Protocol

  1. Grade immediately. No partial credit — either the answer is correct or it isn't.
  2. For each wrong problem: Write one sentence explaining the exact error (algebra slip? wrong formula? sign error? misidentified focus?).
  3. Re-solve the wrong problems right now, with notes allowed, then close notes and re-solve them again tomorrow.

Related Resources

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