FAD1014: Mathematics II — Proficiency Drill Set
4–5 Day Study Plan
Level: Mid-level proficiency (not mastery)
Topics: Sequences & Series → Finite Summation → Binomial Expansion → Taylor & Maclaurin → Circle & Parabola → Hyperbola
How to Use This Set
This drill set is designed for deep proficiency, not superficial coverage. Each section isolates a topic so you can build solid procedural fluency before tackling the existing mastery set.
Recommended approach:
- Day 1: Sections 1 & 2 (Series fundamentals + summation mechanics)
- Day 2: Section 3 (Binomial expansion — all cases)
- Day 3: Section 4 (Taylor & Maclaurin — derivation and approximation)
- Day 4: Sections 5 & 6 (Circle, Parabola, Hyperbola)
- Day 5: Retry any problems you got wrong without looking at notes
Time estimate: 15–25 minutes per problem. Do not rush.
Formula Reference
Sequences & Summation
| Formula | Expression |
|---|---|
| Sum of first $n$ naturals | $\sum_{r=1}^{n} r = \frac{n(n+1)}{2}$ |
| Sum of squares | $\sum_{r=1}^{n} r^2 = \frac{n(n+1)(2n+1)}{6}$ |
| Sum of cubes | $\sum_{r=1}^{n} r^3 = \frac{n^2(n+1)^2}{4}$ |
| Telescoping | $\sum_{k=1}^{n}[f(k)-f(k-1)] = f(n)-f(0)$ |
Binomial
| Case | Expansion | Validity |
|---|---|---|
| Positive integer $n$ | $(a+b)^n = \sum_{r=0}^{n}\binom{n}{r}a^{n-r}b^r$ | All $x$ |
| General $n$ | $(1+x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \cdots$ | $|x| < 1$ |
| General term | $T_{r+1} = \binom{n}{r}a^{n-r}b^r$ | — |
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 series:
- $e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$
- $\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots$
- $\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots$
- $\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots$ (for $-1 < x \leq 1$)
Geometry
| Shape | Standard Equation | Key Properties |
|---|---|---|
| Circle | $(x-h)^2 + (y-k)^2 = r^2$ | Centre $(h,k)$, radius $r$ |
| Parabola (vertical) | $(x-h)^2 = 4a(y-k)$ | Vertex $(h,k)$, focus $(h,k+a)$, directrix $y = k-a$, LR $= 4a$ |
| Parabola (horizontal) | $(y-k)^2 = 4a(x-h)$ | Vertex $(h,k)$, focus $(h+a,k)$, directrix $x = h-a$, LR $= 4a$ |
| 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 $(h,k\pm c)$, $c^2 = a^2+b^2$, asymptotes $y-k = \pm\frac{a}{b}(x-h)$ |
Section 1: Introduction to Series (L21)
Problem 1.1
Determine whether each sequence converges or diverges. If it converges, state the limit.
(a) $a_k = \frac{3k+2}{k+5}$
(b) $a_k = \frac{(-1)^k}{k^2}$
(c) $a_k = 2^k - 3^k$
Problem 1.2
Express each series using sigma notation.
(a) $5 + 9 + 13 + 17 + 21 + 25$
(b) $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \cdots + \frac{1}{128}$
(c) $1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \frac{1}{11}$
Problem 1.3
A sequence is defined by $a_k = \frac{k^2 + 1}{2k^2 - 3}$.
(a) Write out the first four terms.
(b) Determine whether the sequence converges or diverges as $k \to \infty$.
(c) If the series $S_n = \sum_{k=1}^{n} a_k$ is formed, what can you conclude about the behaviour of $S_n$ as $n \to \infty$? Explain your reasoning.
Problem 1.4
Find the value of:
(a) $\sum_{k=1}^{20} 5$
(b) $\sum_{k=3}^{7} (2k - 1)$
(c) $\sum_{k=1}^{n} (a_{k+1} - a_k)$ where $a_1 = 3$ and $a_{n+1} = 100$. Simplify your answer in terms of $n$.
Section 2: Finite Series and Summation (L22)
Problem 2.1
Evaluate each sum using standard formulas.
(a) $\sum_{r=1}^{12} r$
(b) $\sum_{r=1}^{8} r^2$
(c) $\sum_{r=1}^{5} r^3$
(d) Hence verify that $\sum_{r=1}^{5} r^3 = \left(\sum_{r=1}^{5} r\right)^2$ for this case.
Problem 2.2
Evaluate $\sum_{r=1}^{n} (r+2)(r-1)$. Simplify your answer fully into a single fraction.
Problem 2.3
Find the sum of the series $1 \cdot 3 + 2 \cdot 4 + 3 \cdot 5 + \cdots + n(n+2)$. Express your answer in terms of $n$.
Problem 2.4
Using the method of differences, evaluate:
(a) $\sum_{k=1}^{n} \frac{1}{k(k+1)}$
(b) $\sum_{k=1}^{n} \frac{1}{(2k-1)(2k+1)}$
Problem 2.5
Evaluate $\sum_{r=n}^{2n} (3r - 2)$. Simplify your answer into a quadratic expression in $n$.
Section 3: Binomial Expansion (L23–L24)
Problem 3.1
Evaluate without calculator:
(a) $\dfrac{10!}{7! , 3!}$
(b) $\dbinom{12}{5}$
(c) $\dfrac{4! + 5!}{6!}$
Problem 3.2
Expand completely:
(a) $(2x - 3y)^4$
(b) $\left(x^2 + \dfrac{2}{x}\right)^5$
Problem 3.3
Find the specified term or coefficient.
(a) Find the coefficient of $x^3$ in the expansion of $(1 - 2x)^7$.
(b) Find the middle term in the expansion of $\left(x + \dfrac{1}{x}\right)^8$.
(c) Find the term independent of $x$ in the expansion of $\left(2x^2 - \dfrac{1}{x}\right)^9$.
Problem 3.4
In the expansion of $(1 + ax)^n$, the first three terms are $1 - 12x + 63x^2$. Find the values of $a$ and $n$.
Problem 3.5
Expand each function as a series in ascending powers of $x$ up to and including the term in $x^3$. State the range of validity.
(a) $\sqrt{1 - 4x}$
(b) $\dfrac{1}{(1 + 3x)^2}$
(c) $(1 - 2x)^{\frac{2}{3}}$
Problem 3.6
Expand $(1 + x)^{\frac{1}{2}}$ in ascending powers of $x$ up to the term in $x^3$. By substituting $x = \frac{1}{9}$, find an approximation for $\sqrt{10}$ correct to four decimal places.
Problem 3.7
Express $f(x) = \dfrac{7x - 1}{(1 - x)(1 + 2x)}$ in partial fractions. Hence, expand $f(x)$ in ascending powers of $x$ up to the term in $x^2$, stating the range of values of $x$ for which the expansion is valid.
Section 4: Power Series, Taylor & Maclaurin (L25–L26)
Problem 4.1
Find the Maclaurin series for $f(x) = \ln(1 + 3x)$ up to and including the term in $x^4$.
Problem 4.2
Find the first four non-zero terms of the Maclaurin series for $f(x) = e^{-2x} \sin x$.
Problem 4.3
Find the Taylor series for $f(x) = \sqrt{x}$ about $x = 4$ up to and including the term in $(x-4)^3$. Use this to approximate $\sqrt{4.5}$ correct to three decimal places.
Problem 4.4
Find the first three non-zero terms of the Maclaurin series for:
(a) $f(x) = \cos(2x^2)$
(b) $f(x) = x^2 \ln(1 - x)$
Problem 4.5
The Maclaurin series for a function $f(x)$ begins: $$f(x) = 2 + 5x - 3x^2 + 7x^3 + \cdots$$
(a) Find $f(0)$, $f'(0)$, $f''(0)$, and $f'''(0)$.
(b) Hence find the first three non-zero terms of the Maclaurin series for $f'(x)$.
Problem 4.6
Find the Maclaurin series for $\displaystyle\int_0^x \frac{\sin t}{t} , dt$ up to the term in $x^5$. (Note: define $\frac{\sin t}{t} = 1$ at $t=0$ by continuity.)
Problem 4.7
Find the Taylor series for $f(x) = \sin x$ about $x = \frac{\pi}{6}$ up to the term in $\left(x - \frac{\pi}{6}\right)^3$. Use this to approximate $\sin 35^{\circ}$ correct to four decimal places.
Section 5: Geometry I — Circle & Parabola (L27–L28)
Problem 5.1
For each circle, find the centre and radius.
(a) $x^2 + y^2 - 6x + 4y - 12 = 0$
(b) $3x^2 + 3y^2 + 12x - 9y - 14 = 0$
Problem 5.2
Find the equation of the circle with centre $(-2, 5)$ that passes through the point $(1, 3)$.
Problem 5.3
Show that the line $3x + 4y = 25$ is tangent to the circle $x^2 + y^2 = 25$. Find the point of contact.
Problem 5.4
Find the equation of the tangent and the normal to the circle $x^2 + y^2 - 4x + 6y - 12 = 0$ at the point $(5, 1)$.
Problem 5.5
Find the length of the tangent from the point $(6, -2)$ to the circle $x^2 + y^2 + 2x - 8y + 8 = 0$.
Problem 5.6
For each parabola, find the vertex, focus, directrix, and length of the latus rectum. Sketch the curve.
(a) $x^2 = 12y$
(b) $y^2 = -8x$
(c) $(x - 2)^2 = -16(y + 1)$
Problem 5.7
Find the equation of the parabola with vertex at $(3, -1)$ and focus at $(3, 2)$.
Problem 5.8
Find the equation of the parabola with focus at $(-1, 4)$ and directrix $y = -2$.
Problem 5.9
A parabola has its axis parallel to the $y$-axis, passes through the points $(1, 0)$, $(2, 1)$, and $(3, 4)$. Find its equation in the form $y = ax^2 + bx + c$.
Problem 5.10
The circle $x^2 + y^2 = 25$ and the parabola $y^2 = 12x$ intersect at two points.
(a) Find the coordinates of the intersection points.
(b) Find the equation of the line joining these two points.
(c) Show that this line is parallel to the directrix of the parabola.
Section 6: Hyperbola (L31–L32)
Problem 6.1
For each hyperbola, find the centre, vertices, foci, and equations of the asymptotes. Sketch the curve.
(a) $\dfrac{x^2}{9} - \dfrac{y^2}{16} = 1$
(b) $\dfrac{(y-1)^2}{4} - \dfrac{(x+2)^2}{9} = 1$
Problem 6.2
Find the equation of the hyperbola with vertices at $(\pm 4, 0)$ and foci at $(\pm 5, 0)$.
Problem 6.3
Find the equation of the hyperbola with foci at $(2, -1)$ and $(2, 7)$, and transverse axis of length $6$.
Problem 6.4
A hyperbola has asymptotes $y = 2x + 1$ and $y = -2x + 5$, and passes through the point $(2, 5)$. Find its equation in standard form.
Problem 6.5
For the hyperbola $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$, prove that the difference of the distances from any point $P(x,y)$ on the hyperbola to the two foci is equal to $2a$.
Cross-Topic Connectors (Optional — for deeper proficiency)
Connector C.1 [Series + Binomial]
Using the binomial expansion of $(1+x)^{\frac{1}{2}}$, show that: $$\sum_{r=0}^{n} \binom{1/2}{r} x^r$$ provides a convergent approximation for $\sqrt{1+x}$ when $|x| < 1$. By integrating the series for $(1+x)^{-1/2}$ term-by-term from $0$ to $x$, deduce the first four terms of the series for $\sin^{-1} x$ (up to $x^5$).
Connector C.2 [Parabola + Series]
A parabolic arch has equation $y = 8 - \frac{x^2}{2}$ (in metres). Expand $\sqrt{1 + \left(\frac{dy}{dx}\right)^2}$ as a binomial series up to the term in $x^2$. Use this to set up an approximation for the arc length from $x = 0$ to $x = 2$.
Connector C.3 [Hyperbola + Binomial]
For the hyperbola $\frac{x^2}{4} - \frac{y^2}{9} = 1$, express $y$ as a function of $x$ for the upper branch. Expand $y$ as a binomial series in $\frac{1}{x}$ up to the term in $\frac{1}{x^3}$, valid for $|x| > 2$. What does this tell you about the behaviour of the hyperbola as $x \to \infty$?
Progress Tracker
| Section | Problems | Completed | Score |
|---|---|---|---|
| 1. Introduction to Series | 1.1 – 1.4 | ☐ | ___/4 |
| 2. Finite Series & Summation | 2.1 – 2.5 | ☐ | ___/5 |
| 3. Binomial Expansion | 3.1 – 3.7 | ☐ | ___/7 |
| 4. Taylor & Maclaurin | 4.1 – 4.7 | ☐ | ___/7 |
| 5. Circle & Parabola | 5.1 – 5.10 | ☐ | ___/10 |
| 6. Hyperbola | 6.1 – 6.5 | ☐ | ___/5 |
| Total | 38 problems | ___/38 |
Proficiency target: 28/38 (≈75%) on first attempt, 34/38 (≈90%) on review.
Related Resources
- FAD1014 L21 — Introduction to Series
- FAD1014 L22 — Finite Series and Summation
- FAD1014 L23-L24 — Binomial Expansion
- FAD1014 L25-L26 — Power Series, Taylor & Maclaurin
- FAD1014 L27-L28 — Geometry I (Circle & Parabola)
- FAD1014 L31-L32 — Hyperbola
- FAD1014 Mastery Set — Interleaved Mathematics II — attempt after scoring ≥32/38 here
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