FAD1014 — 25-Question Minimum Viable Drill

The non-negotiable floor. 25 questions, ~41 minutes. Do these before bed.

Leak focus: Maclaurin Series, Ellipse, Differential Equations.

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Block 1: Maclaurin Standard Forms — MUST BE INSTANT (4 min)

Q1

$$e^x$$

Q2

$$\sin x$$

Q3

$$\cos x$$

Q4

$$\ln(1+x)$$

Q5

$$\frac{1}{1-x}$$

Q6

$$\sinh x$$

Q7

$$\cosh x$$

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Block 2: Maclaurin Substitution — Leak Pattern (6 min)

Q8

Find Maclaurin series up to $x^4$: $$e^{-3x^2}$$

Q9

Find Maclaurin series up to $x^4$: $$\ln(1 + 5x)$$

Q10

Find Maclaurin series up to $x^4$: $$x^2 e^{-x}$$

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Block 3: Ellipse — Completing Square (8 min)

Q11

Convert to standard form, find centre and foci: $$x^2 + 9y^2 - 4x + 36y + 4 = 0$$

Q12

Convert to standard form, find centre, vertices, foci: $$4x^2 + y^2 - 8x + 4y - 8 = 0$$

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Block 4: Ellipse — Write Equation from Data (4 min)

Q13

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

Q14

Find the equation of the ellipse with centre $(1,-2)$, focus at $(1,1)$, and vertex at $(1,3)$.

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Block 5: DE Identification — Decision Triage (3 min)

Q15

$$\frac{dy}{dx} = \frac{x^2 + 1}{y^2}$$

Q16

$$\frac{dy}{dx} = \frac{x^2 + y^2}{2xy}$$

Q17

$$\frac{dy}{dx} = \frac{x + y + 1}{x + y + 2}$$

Q18

$$\frac{dy}{dx} = \frac{2x + y + 3}{x - y - 1}$$

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Block 6: Separable DE (6 min)

Q19

Solve: $\displaystyle\frac{dy}{dx} = \frac{x}{y}$, $y(0) = 3$

Q20

Solve: $\displaystyle\frac{dy}{dx} = y^2 e^x$, $y(0) = \frac12$

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Block 7: Growth/Decay — Leak Pattern (6 min)

Q21

A population grows proportionally 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.

Q22

A radioactive substance decays proportionally to its mass. Initially 50g, after 10 years 40g remain. Find the half-life.

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Block 8: Part A Quickies — Warm-Down (4 min)

Q23

Evaluate: $\displaystyle\int_0^{\pi/4} \sec^2 x,dx$

Q24

Find area bounded by $y = x^2 + 1$, $x$-axis, $x = 1$, $x = 3$.

Q25

Evaluate $\displaystyle\sum_{r=1}^{n} \frac{1}{r(r+1)}$ using method of differences.