FAD1014 — Leak-Focused Full Drill
Fully leak-weighted. Maclaurin Series, Ellipse & Differential Equations get ~80% of questions. Everything else gets minimal Part-A-only coverage.
Questions first, then all solutions at the end. Attempt all questions before checking.
Exam structure: Part A = 24 marks (Q1+Q2, answer ALL). Part B = 56 marks (choose 4 of 6).
Leak picks for Part B (in order): ① Maclaurin Series ② Ellipse ③ DEs ④ Integration by Parts or Trig Substitution.
QUESTIONS
SECTION A: MACLAURIN SERIES — LEAKED (Part B)
A1: Standard Expansions — Speed Drill (4 min)
Write first 4 non-zero terms — instant recall, no hesitation:
Q1
$$e^x$$
Q2
$$\sin x$$
Q3
$$\cos x$$
Q4
$$\ln(1+x)$$
Q5
$$\frac{1}{1-x}$$
Q6
$$\sinh x$$
Q7
$$\cosh x$$
A2: Substitution (8 min)
Find the Maclaurin series up to $x^4$:
Q8
$$e^{-3x^2}$$
Q9
$$\sin(2x^2)$$
Q10
$$\cos(x^3)$$
Q11
$$\ln(1 + 5x)$$
Q12
$$x^2 e^{-x}$$
Q13
$$x^3 \cos(x^2)$$
A3: Multiplication (8 min)
Find Maclaurin series up to $x^3$ or $x^4$:
Q14
$$e^x \sin x$$
Q15
$$(\sin x)(\cos x)$$
Q16
$$e^{-x} \ln(1+x)$$
A4: Term-by-Term Integration (8 min)
Q17
Use series to approximate $\displaystyle\int_0^{0.5} e^{-x^2},dx$ correct to 3 d.p.
Q18
Use series to approximate $\displaystyle\int_0^{0.2} \frac{\sin x}{x},dx$ correct to 4 d.p.
Q19
Find series for $\displaystyle\int e^{x^3},dx$ up to $x^7$ term.
A5: Derivation from Definition (10 min)
Q20
Derive Maclaurin series for $\tan x$ up to $x^5$ using direct differentiation (compute $f(0)$ through $f^{(5)}(0)$).
Q21
Derive Maclaurin series for $\sec x$ up to $x^4$.
Q22
Given $f(x) = \ln(1 + \sin x)$, find $f(0), f'(0), f''(0), f'''(0)$ and hence write the series up to $x^3$.
SECTION B: ELLIPSE — LEAKED (Part A & B)
B1: Standard Form — Find Features (6 min)
Q23
For $\dfrac{x^2}{36} + \dfrac{y^2}{16} = 1$, find: centre, vertices, foci, length of major/minor axes. Determine if horizontal or vertical.
Q24
For $\dfrac{(x-2)^2}{9} + \dfrac{(y+1)^2}{25} = 1$, find: centre, vertices, foci.
Q25
For $16x^2 + 9y^2 = 144$, find: centre, vertices, foci, sketch orientation.
B2: Completing the Square (10 min)
Q26
Convert to standard form, find centre and foci: $$x^2 + 9y^2 - 4x + 36y + 4 = 0$$
Q27
Convert to standard form, find centre, vertices, foci: $$4x^2 + y^2 - 8x + 4y - 8 = 0$$
Q28
Convert to standard form: $$9x^2 + 4y^2 - 36x + 8y + 4 = 0$$
B3: Write Equation from Data (6 min)
Q29
Find the equation of the ellipse with foci $(\pm 3, 0)$ and vertices $(\pm 5, 0)$.
Q30
Find the equation of the ellipse with centre $(1,-2)$, focus at $(1,1)$, and vertex at $(1,3)$.
Q31
Find the equation of the ellipse with vertices $(\pm 6, 0)$ and eccentricity $e = \frac{2}{3}$.
Hint: $e = c/a$
B4: Mixed with Leak Pattern (8 min)
Q32
An ellipse has centre at $(2,-1)$, one focus at $(5,-1)$, and passes through $(2, 3)$. Find its equation.
Q33
Sketch the ellipse $\dfrac{(x-1)^2}{16} + \dfrac{(y+2)^2}{9} = 1$. Label centre, vertices, foci.
SECTION C: DIFFERENTIAL EQUATIONS — LEAKED (Part A & B)
C1: DE Identification — Speed Round (3 min)
State type and method:
Q34
$$\frac{dy}{dx} = \frac{x^2 + 1}{y^2}$$
Q35
$$\frac{dy}{dx} = \frac{x^2 + y^2}{2xy}$$
Q36
$$\frac{dy}{dx} = \frac{x + y + 1}{x + y + 2}$$
Q37
$$\frac{dy}{dx} = \frac{2x + y + 3}{x - y - 1}$$
C2: Separable DE (Part A & B) (10 min)
Q38
Solve: $\displaystyle\frac{dy}{dx} = \frac{x}{y}$, $y(0) = 3$
Q39
Solve: $\displaystyle\frac{dy}{dx} = y^2 e^x$, $y(0) = \frac{1}{2}$
Q40
Solve: $\displaystyle\frac{dy}{dx} = \frac{y\cos x}{1 + y^2}$, $y(0) = 0$
Q41
Solve: $\displaystyle\frac{dy}{dx} = \frac{3x^2}{2y + 1}$, $y(0) = 1$
C3: Homogeneous DE (Part B) (10 min)
Q42
Solve: $\displaystyle\frac{dy}{dx} = \frac{x^2 + y^2}{xy}$
Q43
Solve: $\displaystyle\frac{dy}{dx} = \frac{2xy}{x^2 - y^2}$
Q44
Solve: $\displaystyle\frac{dy}{dx} = \frac{x^2 + 2xy - y^2}{x^2 - 2xy}$, $y(1) = 1$
C4: Non-Homogeneous DE — Leak Pattern (Part A) (8 min)
Q45
Given $(x - 2y + 1),dx + (3x + y - 4),dy = 0$: (a) Test linear dependence/independence using the ratio test. (b) Apply the appropriate method to eliminate constants.
Q46
Given $(2x + y - 1),dx + (4x + 2y + 3),dy = 0$: (a) Test linear dependence/independence. (b) If dependent, find substitution $z = a_2 x + b_2 y$ and write the separable form.
C5: Exponential Growth/Decay — Leak Pattern (6 min)
Q47
A population of bacteria grows at a rate proportional to its size. Initially 1000 bacteria, after 2 hours there are 3000. Find: (a) The population after 6 hours. (b) The time taken for the population to reach 9000.
Q48
A radioactive substance decays at a rate proportional to its mass. Initially 50g, after 10 years 40g remain. Find the half-life.
SECTION D: PART A — Compulsory Foundational Topics
D1: Standard Integration (4 min)
Q49
Evaluate: $\displaystyle\int (4x^3 - 6x + 7),dx$
Q50
Evaluate: $\displaystyle\int \frac{x}{x^2 + 1},dx$
Q51
Evaluate: $\displaystyle\int_0^{\pi/4} \sec^2 x,dx$
Q52
Find area bounded by $y = x^2 + 1$, $x$-axis, $x = 1$, $x = 3$.
D2: Series & Summation (4 min)
Q53
Does $\displaystyle\sum_{n=1}^{\infty} \frac{3n^2 + 2}{5n^2 - 1}$ converge or diverge? Justify.
Q54
Evaluate $\displaystyle\sum_{r=1}^{20} (2r + 3)$.
Q55
Evaluate $\displaystyle\sum_{r=1}^{n} \frac{1}{r(r+1)}$ using method of differences.
D3: Binomial I (3 min)
Q56
Expand $(2x - 1)^5$ completely.
Q57
Find coefficient of $x^4$ in $(x^2 + 3)^6$.
D4: Parabola & Hyperbola — Quick ID (3 min)
Q58
State vertex, focus, directrix of $(x-3)^2 = 12(y+1)$.
Q59
State centre, vertices, asymptotes of $\dfrac{(x-1)^2}{4} - \dfrac{(y+2)^2}{9} = 1$.
SOLUTIONS
SOLUTIONS A: MACLAURIN SERIES
A1 — Standard Expansions
Q1
$$1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots$$
Q2
$$x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots$$
Q3
$$1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots$$
Q4
$$x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots \quad (-1 < x \leq 1)$$
Q5
$$1 + x + x^2 + x^3 + \cdots \quad (|x| < 1)$$
Q6
$$x + \frac{x^3}{3!} + \frac{x^5}{5!} + \cdots$$
Q7
$$1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \cdots$$
A2 — Substitution
Q8
$$e^{-3x^2} = 1 + (-3x^2) + \dfrac{(-3x^2)^2}{2!} + \cdots = \boxed{1 - 3x^2 + \frac{9x^4}{2} + \cdots}$$
Q9
$\sin(2x^2) = 2x^2 - \dfrac{(2x^2)^3}{3!} + \cdots = \boxed{2x^2}$ (up to $x^4$)
Q10
$\cos(x^3) = 1 - \dfrac{x^6}{2!} + \cdots$ → up to $x^4$: $\boxed{1}$
Q11
$$\ln(1+5x) = 5x - \dfrac{25x^2}{2} + \dfrac{125x^3}{3} - \dfrac{625x^4}{4} + \cdots$$
Q12
$x^2 e^{-x} = x^2\left(1 - x + \dfrac{x^2}{2} - \dfrac{x^3}{6} + \dfrac{x^4}{24}\right) = \boxed{x^2 - x^3 + \frac{x^4}{2} + \cdots}$ (up to $x^4$)
Q13
$x^3 \cos(x^2) = x^3\left(1 - \dfrac{x^4}{2!} + \cdots\right)$ → up to $x^4$: $\boxed{x^3}$
A3 — Multiplication
Q14
$$e^x \sin x = \left(1 + x + \frac{x^2}{2} + \frac{x^3}{6}\right)\left(x - \frac{x^3}{6}\right)$$
$$= x + x^2 + \left(-\frac{1}{6} + \frac{1}{2}\right)x^3 + \cdots = \boxed{x + x^2 + \frac{x^3}{3} + \cdots}$$
Q15
$\sin x \cos x$: by identity $= \frac{1}{2}\sin 2x = \frac{1}{2}\left(2x - \frac{8x^3}{6} + \cdots\right) = \boxed{x - \frac{2x^3}{3} + \cdots}$
Q16
$$e^{-x} \ln(1+x) = \left(1 - x + \frac{x^2}{2} - \frac{x^3}{6}\right)\left(x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4}\right)$$
$= \boxed{x - \frac{3x^2}{2} + \frac{4x^3}{3} + \cdots}$ (up to $x^3$)
A4 — Integration
Q17
$$e^{-x^2} = 1 - x^2 + \frac{x^4}{2!} - \cdots$$
$$\int_0^{0.5} \approx \left[x - \frac{x^3}{3} + \frac{x^5}{10}\right]_0^{0.5} = 0.5 - 0.041667 + 0.003125 = \boxed{0.461}$$
Q18
$$\frac{\sin x}{x} = 1 - \frac{x^2}{6} + \frac{x^4}{120} - \cdots$$
$$\int_0^{0.2} \approx \left[x - \frac{x^3}{18} + \frac{x^5}{600}\right]_0^{0.2} = 0.2 - 0.000444 + 0.0000005 = \boxed{0.1996}$$
Q19
$e^{x^3} = 1 + x^3 + \frac{x^6}{2!} + \cdots$ → $\int e^{x^3},dx = \boxed{x + \frac{x^4}{4} + \frac{x^7}{14} + \cdots + C}$
A5 — Derivation
Q20
$f(0)=0$, $f'(0)=1$, $f''(0)=0$, $f'''(0)=2$, $f^{(4)}(0)=0$, $f^{(5)}(0)=16$
$$\boxed{\tan x = x + \frac{x^3}{3} + \frac{2x^5}{15} + \cdots}$$
Q21
$$\sec x = 1 + \frac{x^2}{2} + \frac{5x^4}{24} + \cdots$$
Q22
$f(0)=0$, $f'(0)=1$, $f''(0)=-1$, $f'''(0)=1$
$$\boxed{\ln(1 + \sin x) = x - \frac{x^2}{2} + \frac{x^3}{6} + \cdots}$$
SOLUTIONS B: ELLIPSE
Q23
Centre $(0,0)$. $a=6$, $b=4$ (horizontal). $c=\sqrt{36-16}=2\sqrt{5}$. Vertices $(\pm6,0)$. Foci $(\pm2\sqrt5,0)$. Major $12$, minor $8$.
Q24
Centre $(2,-1)$. $a=3$, $b=5$ (vertical). $c=\sqrt{25-9}=4$. Vertices $(2,4)$ and $(2,-6)$. Foci $(2,3)$ and $(2,-5)$.
Q25
$\frac{x^2}{9} + \frac{y^2}{16} = 1$. Centre $(0,0)$, vertical. $a=3$, $b=4$, $c=\sqrt7$. Vertices $(0,\pm4)$. Foci $(0,\pm\sqrt7)$.
Q26
$(x-2)^2 + 9(y+2)^2 = 36$ → $\frac{(x-2)^2}{36} + \frac{(y+2)^2}{4} = 1$. Centre $(2,-2)$, $a=6$, $b=2$, $c=\sqrt{36-4}=4\sqrt2$. Foci $(2\pm4\sqrt2,-2)$.
Q27
$4(x-1)^2 + (y+2)^2 = 16$ → $\frac{(x-1)^2}{4} + \frac{(y+2)^2}{16} = 1$. Centre $(1,-2)$, $a=2$, $b=4$ (vertical), $c=2\sqrt3$. Vertices $(1,2),(1,-6)$. Foci $(1,-2\pm2\sqrt3)$.
Q28
$9(x-2)^2 + 4(y+1)^2 = 36$ → $\frac{(x-2)^2}{4} + \frac{(y+1)^2}{9} = 1$. Centre $(2,-1)$.
Q29
$c=3$, $a=5$, $b^2=25-9=16$ → $\boxed{\frac{x^2}{25} + \frac{y^2}{16} = 1}$
Q30
Centre $(1,-2)$, vertical. $c=3$, $b=5$, $a^2=25-9=16$ → $\boxed{\frac{(x-1)^2}{16} + \frac{(y+2)^2}{25} = 1}$
Q31
$a=6$, $e=\frac23 \to c=4$, $b^2=36-16=20$ → $\boxed{\frac{x^2}{36} + \frac{y^2}{20} = 1}$
Q32
Centre $(2,-1)$, $c=3$ (horizontal), $b=4$ from point $(2,3)$. $a^2=9+16=25$ → $\boxed{\frac{(x-2)^2}{25} + \frac{(y+1)^2}{16} = 1}$
Q33
Centre $(1,-2)$, $a=4$ (horizontal), $b=3$, $c=\sqrt7$. Vertices $(-3,-2)$ and $(5,-2)$. Foci $(1\pm\sqrt7,-2)$.
SOLUTIONS C: DIFFERENTIAL EQUATIONS
C1 — ID
| Q | Type | Method |
|---|---|---|
| 34 | Separable | $y^2,dy = (x^2+1),dx$ |
| 35 | Homogeneous | $y=vx$, all degree 2 |
| 36 | Non-Homogeneous (Dependent) | $\frac11=\frac11$, $z=x+y$ |
| 37 | Non-Homogeneous (Independent) | $\frac21 \neq \frac{1}{-1}$, solve for $h,k$ |
C2 — Separable
Q38
$y,dy=x,dx$ → $y^2=x^2+C$. $y(0)=3$ → $C=9$. $\boxed{y^2=x^2+9}$
Q39
$\frac{dy}{y^2}=e^x,dx$ → $-y^{-1}=e^x+C$. $y(0)=\frac12$ → $-2=1+C$ → $C=-3$. $\boxed{y=\frac{1}{3-e^x}}$
Q40
$\frac{1+y^2}{y},dy=\cos x,dx$ → $\ln|y|+\frac{y^2}{2}=\sin x+C$. $y(0)=0$ → $y=0$ is a trivial solution (satisfies both sides). $\boxed{y=0}$
Q41
$(2y+1),dy=3x^2,dx$ → $y^2+y=x^3+C$. $y(0)=1$ → $C=2$. $\boxed{y^2+y=x^3+2}$
C3 — Homogeneous
Q42
$y=vx$: $v+xv'=\frac{1+v^2}{v}$ → $xv'=\frac{1}{v}$ → $v,dv=\frac{dx}{x}$ → $\frac{v^2}{2}=\ln|x|+C$
$$\boxed{y^2=2x^2(\ln|x|+C)}$$
Q43
$y=vx$: $v+xv'=\frac{2v}{1-v^2}$ → $xv'=\frac{2v}{1-v^2}-v=\frac{v+v^3}{1-v^2}$
Separate: $\frac{1-v^2}{v(1+v^2)},dv=\frac{dx}{x}$ → integrate → back-substitute $v=y/x$.
Q44
$y=vx$: $v+xv'=\frac{1+2v-v^2}{1-2v}$ → $xv'=\frac{1+v+v^2}{1-2v}$
$\frac{1-2v}{1+v+v^2},dv=\frac{dx}{x}$. Integrate: $\frac{4}{\sqrt3}\tan^{-1}\frac{2v+1}{\sqrt3}-\ln|v^2+v+1|=\ln|x|+C$. Back-substitute $v=y/x$.
C4 — Non-Homogeneous
Q45
(a) $\frac{1}{3} \neq \frac{-2}{1}$ → Independent
(b) Solve $x-2y+1=0$ and $3x+y-4=0$ → $x=1$, $y=1$. Substitute $X=x-1$, $Y=y-1$ → homogeneous.
Q46
(a) $\frac{2}{4}=\frac{1}{2}$ → Dependent
(b) $z=2x+y$, $\frac{dz}{dx}=2-\frac{z-1}{2z+3}=\frac{3z+7}{2z+3}$ → $\boxed{\frac{2z+3}{3z+7},dz=dx}$
C5 — Applications
Q47
$P(t)=1000e^{kt}$, $P(2)=3000$ → $e^{2k}=3$ → $k=\frac{\ln3}{2}$
(a) $P(6)=1000e^{6k}=1000(3)^3=\boxed{27,000}$
(b) $1000e^{kt}=9000$ → $e^{kt}=9$ → $t=\frac{\ln9}{k}=\frac{2\ln3}{\ln3/2}=\boxed{4\text{ hrs}}$
Q48
$m(t)=50e^{-kt}$, $m(10)=40$ → $e^{-10k}=0.8$ → $k=-\frac{\ln0.8}{10}\approx0.02231$
Half-life: $t_{1/2}=\frac{\ln2}{k}=\frac{10\ln2}{-\ln0.8}\approx\boxed{31.1\text{ years}}$
SOLUTIONS D: PART A COMPULSORY
D1 — Integration
Q49
$$\boxed{x^4-3x^2+7x+C}$$
Q50
$$\boxed{\frac12\ln(x^2+1)+C}$$
Q51
$$\left[\tan x\right]_0^{\pi/4}=\boxed{1}$$
Q52
$$\int_1^3(x^2+1),dx=\left[\frac{x^3}{3}+x\right]_1^3=(9+3)-(\frac13+1)=\boxed{\frac{32}{3}}$$
D2 — Series
Q53
$\lim_{n\to\infty}\frac{3n^2+2}{5n^2-1}=\frac35\neq0$ → Diverges (nth term test)
Q54
$$2\sum_{1}^{20}r+3\cdot20=2\cdot\frac{20\cdot21}{2}+60=420+60=\boxed{480}$$
Q55
$\frac{1}{r(r+1)}=\frac1r-\frac1{r+1}$ → $1-\frac1{n+1}=\boxed{\frac{n}{n+1}}$
D3 — Binomial
Q56
$$32x^5-80x^4+80x^3-40x^2+10x-1$$
Q57
$$\binom{6}{2}(x^2)^2(3)^4=15\cdot81=\boxed{1215}$$
D4 — Parabola & Hyperbola
Q58
Vertex $(3,-1)$. $4a=12\to a=3$. Opens up. Focus $(3,2)$. Directrix $y=-4$.
Q59
Centre $(1,-2)$, horizontal. $a=2$, $b=3$, $c=\sqrt{13}$. Vertices $(3,-2),(-1,-2)$. Asymptotes $y+2=\pm\frac32(x-1)$.
APPENDIX: NOT TESTED (Skip)
- Integration of powers of trig functions
- Integration by partial fractions
- Area between curves / Volume of revolution
- Linear DE (integrating factor)
- Bernoulli DE
- Binomial II (non-positive $n$)
- Taylor series (about $a \neq 0$)
- Circle geometry
PART B STRATEGY
Choose 4 of 6. Recommended order:
- Maclaurin Series (leaked) — substitution, multiplication, integration
- Ellipse (leaked) — completing square is mechanical
- Differential Equations (leaked) — identify type first
- Integration by Parts or Trig Sub — whichever you drilled more
Time: 20 min per Part B. If stuck >5 min, switch.