🔢 FAA MATHEMATICS — The 10-Mark Plan

Section: Mathematics · 10 of 120 marks · 10 syllabus topics · 1 question each Companions: Maths-Analysis-and-Mapping (the evidence) · Maths-Tagged-Answers (all 12 PYQs solved)

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PART 1 — The strategy in one box

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PART 2 — ⭐ HOW DEEP? The depth ladder

This is the single most important table in this plan. For each topic, only one question is coming, so the scope boundary matters more than the content.

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PART 3 — Topic-by-topic: scope, formulas, and time

🟢 EASY TIER — one-step formulas (4.5 hrs → ~3 marks)

(i) Simple / Compound Interest · L2 · 1.5 hrs

Source: any Class 8 arithmetic chapter. Not in NCERT 11–12.

$SI = \frac{P \times R \times T}{100} \qquad A = P\left(1 + \frac{R}{100}\right)^{n} \qquad CI = A - P$

Also know:

• Half-yearly: rate ÷ 2, time × 2 · Quarterly: rate ÷ 4, time × 4
• Difference between CI and SI for 2 years $= P\left(\frac{R}{100}\right)^2$

Depth stop: ✋ no instalments, no compound-interest-with-changing-rates, no depreciation.

(x) Vectors · L2 · 1.5 hrs

Source: NCERT Class 12, Chapter 10.

$|\vec{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2} \qquad \hat{a} = \frac{\vec{a}}{|\vec{a}|} \qquad \text{opposite unit vector} = -\hat{a}$

$\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3 = |\vec a||\vec b|\cos\theta$

Also know: perpendicular ⇒ dot product = 0 · parallel ⇒ cross product = 0 · $\hat{i}, \hat{j}, \hat{k}$ are unit vectors along the axes.

Depth stop: ✋ no triple products, no vector equations of lines/planes, no projection formulae.

(ii) Linear Equations in Two Variables · L3 · 1.5 hrs

Source: NCERT Class 10, Chapter 3.

For $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$:

Depth stop: ✋ no three-variable systems, no graphical solution plotting.

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🟡 MODERATE TIER — a real method (8 hrs → ~4 marks)

(v) Set Theory · L3 · 2 hrs ⭐ asked in BOTH papers

Source: NCERT Class 11, Chapter 1.

$A \cup B \qquad A \cap B \qquad A - B \qquad A' = U - A$ $\textbf{Symmetric difference: } A \triangle B = (A \cup B) - (A \cap B) = (A-B) \cup (B-A)$ $n(A \cup B) = n(A) + n(B) - n(A \cap B)$ $\textbf{De Morgan: } (A \cup B)' = A' \cap B' \qquad (A \cap B)' = A' \cup B'$

⭐ Modulus → interval conversion — this decided 2024·Q85:

• |x| < a \Rightarrow -a < x < a
• |x| > a \Rightarrow x < -a \text{ or } x > a
• $|x - k| \ge a \Rightarrow x \ge k+a \text{ or } x \le k-a$

Also: number of subsets of a set with n elements = $2^n$ · power set.

Depth stop: ✋ no cardinality theory, no infinite sets, no relations-on-sets proofs.

(ix) Coordinate & 3-D Geometry · L2/L3 · 2 hrs ⭐ asked in BOTH papers

Source: NCERT Class 11 Ch 10 (straight lines) + Ch 12 (3-D basics) · Class 12 Ch 11.

$\textbf{Distance: } d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$ $\textbf{Midpoint: } \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right) \qquad \textbf{Section (m:n): } \left(\frac{mx_2+nx_1}{m+n}, \frac{my_2+ny_1}{m+n}\right)$ $\textbf{Area of triangle (2-D): } \tfrac{1}{2}\left|x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)\right|$ $⭐\ \textbf{Area of triangle (3-D): } \tfrac{1}{2}\left|\vec{AB} \times \vec{AC}\right|$ $\textbf{Slope: } m = \frac{y_2-y_1}{x_2-x_1} \qquad \textbf{Line: } y - y_1 = m(x-x_1)$

Collinearity: three points are collinear if the area of the triangle = 0.

Depth stop: ✋ no conic sections, no direction cosines/ratios beyond the basics, no plane equations.

(viii) Probability · L3 · 2 hrs ⭐⭐ WORTH TWO MARKS

Source: NCERT Class 11, Chapter 16. You have already written this up — see ST-6-Theory-of-Probability.

$0 \le P(A) \le 1 \qquad P(A) = \frac{\text{favourable}}{\text{total}} \qquad P(A) + P(\bar A) = 1$ $P(A \cup B) = P(A) + P(B) - P(A \cap B) \qquad \text{independent: } P(A \cap B) = P(A)\cdot P(B)$

Standard values worth knowing cold: a card is a King = 4/52 = 1/13 · a diamond = 13/52 = 1/4 · a face card = 12/52 · heads = 1/2 · a six = 1/6.

Depth stop: ✋ Bayes' theorem — read once, don't drill. No random variables, no distributions.

(iii) Permutations, Combinations & Binomial Theorem · L3/L4 · 2 hrs

Source: NCERT Class 11, Ch 7 (P&C) + Ch 8 (Binomial).

$^nP_r = \frac{n!}{(n-r)!} \qquad ^nC_r = \frac{n!}{r!\,(n-r)!} \qquad ^nC_r = {}^nC_{n-r}$ $⭐\ \textbf{Word with repeated letters: } \frac{n!}{p!\,q!\,r!}$ $(a+b)^n = \sum_{k=0}^{n} {}^nC_k\, a^{n-k} b^k \qquad (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$

⭐ The distinction that gets tested: order matters → permutation · order does not matter (a committee, a selection) → combination. n people in a row = $n!$ · n people in a circle = $(n-1)!$

Depth stop: ✋ no general/middle-term formulas, no circular permutations with restrictions, no derangements.

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🔴 HARD TIER — needs precision (7.5 hrs → ~3 marks)

(vii) Matrices & Determinants · L1 · 2 hrs ⭐ highest recall-per-mark in the section

Source: NCERT Class 12, Ch 3 + Ch 4 — properties only.

2×2 determinant: $\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc$

Depth stop: ✋ no adjoint, no inverse computation, no Cramer's rule, no solving equations by matrices. Only properties.

(iv) Limits & Derivatives · L1/L4 · 2.5 hrs

Source: NCERT Class 11, Chapter 13.

$⭐\ \textbf{First principles: } f'(x) = \lim_{h \to 0}\frac{f(x+h) - f(x)}{h}$

Standard limits — memorise these five: $\lim_{x\to a}\frac{x^n - a^n}{x-a} = n a^{n-1} \qquad \lim_{x\to 0}\frac{\sin x}{x} = 1 \qquad ⭐\ \lim_{x\to 0}\frac{e^x - 1}{x} = 1$ $\lim_{x\to 0}\frac{\log(1+x)}{x} = 1 \qquad \lim_{x\to 0}\frac{a^x - 1}{x} = \log_e a$

Derivatives — memorise these: $\frac{d}{dx}(x^n) = nx^{n-1} \qquad ⭐\ \frac{d}{dx}\left(\frac{1}{x}\right) = -\frac{1}{x^2} \qquad \frac{d}{dx}(e^x) = e^x \qquad \frac{d}{dx}(\log x) = \frac{1}{x}$ $\frac{d}{dx}(\sin x) = \cos x \qquad \frac{d}{dx}(\cos x) = -\sin x \qquad \frac{d}{dx}(c) = 0$

Product rule $(uv)' = u'v + uv'$ · Quotient rule $\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$

⭐ The 0/0 trick: if substitution gives 0/0, factorise and cancel. $\frac{x^2-1}{x-1} = x+1 \to 2$ at $x=1$.

Depth stop: ✋ no continuity/differentiability theory, no L'Hôpital, no second derivatives, no applications of derivatives at all.

(vi) Relations and Functions · L4 · 3 hrs — the hardest to judge

Source: NCERT Class 11 Ch 2 + Class 12 Ch 1.

Composition: $(f \circ f)(x) = f(f(x))$ — substitute the function into itself.

Ranges worth knowing: $y = x^2 \Rightarrow [0,\infty)$ · $y = x^2 - k \Rightarrow [-k,\infty)$ · $y = k - x^2 \Rightarrow (-\infty, k]$ Monotonicity: $y = x^2$ is increasing on (0, ∞) but not on ℝ.

Depth stop: ✋ no equivalence relations, no binary operations, no inverse functions.

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PART 4 — ⭐ THE COMPLETE FORMULA SHEET (one page)

INTEREST — $SI = \frac{PRT}{100}$ · $A = P(1+\frac{R}{100})^n$ · CI = A − P · months → years!

LINEAR EQ — unique $\frac{a_1}{a_2} \ne \frac{b_1}{b_2}$ · none $\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne\frac{c_1}{c_2}$ · infinite all three equal

P & C — $^nP_r = \frac{n!}{(n-r)!}$ · $^nC_r = \frac{n!}{r!(n-r)!}$ · repeated letters $\frac{n!}{p!q!}$ · row $n!$ · circle $(n-1)!$ $(a+b)^3 = a^3+3a^2b+3ab^2+b^3$

LIMITS — $\frac{x^n-a^n}{x-a}\to na^{n-1}$ · $\frac{\sin x}{x}\to1$ · $\frac{e^x-1}{x}\to1$ · $\frac{\log(1+x)}{x}\to1$ DERIVATIVES — $(x^n)'=nx^{n-1}$ · $(1/x)' = -1/x^2$ · $(e^x)'=e^x$ · $(\log x)'=1/x$ · first principles $\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$

SETS — $A\triangle B=(A\cup B)-(A\cap B)$ · $n(A\cup B)=n(A)+n(B)-n(A\cap B)$ · De Morgan · subsets $=2^n$ $|x|<a \Rightarrow -a<x<a$ · $|x-k|\ge a \Rightarrow x\ge k+a$ or $x\le k-a$

FUNCTIONS — vertical line test = is it a function · horizontal line test = one-one · $f\circ f(x)=f(f(x))$

DETERMINANTS — $\det(I)=1$ · identical rows $=0$ · diagonal $=$ product · $\det(A^T)=\det(A)$ · $\det(AB)=\det A \det B$ · 2×2 $= ad-bc$

PROBABILITY — $0\le P\le1$ · $P(A\cup B)=P(A)+P(B)-P(A\cap B)$ · independent $\Rightarrow P(A)P(B)$ · King 1/13 · diamond 1/4

COORDINATE — $d=\sqrt{\sum(\Delta)^2}$ · midpoint · section formula · 3-D area $=\frac12|\vec{AB}\times\vec{AC}|$ · collinear ⇒ area 0

VECTORS — $|\vec a|=\sqrt{a_1^2+a_2^2+a_3^2}$ · $\hat a = \vec a/|\vec a|$ · opposite $=-\hat a$ · $\perp \Rightarrow$ dot $=0$

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PART 5 — The 8-week schedule

~3 hours a week, alongside your main sections. Maths sits in Week 9 of FAA-MASTER-PLAN but is far better spread thin.

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PART 6 — Exam-day approach

10 questions. Give them ~10 minutes, in your second pass — after GK and theory, before the long Accountancy sums.

Negative marking is −0.25. Break-even is 25%, so attempt anything where you can eliminate even one option. With this section, you should be attempting all 10.

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PART 7 — Traps seen in the two papers

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PART 8 — What to study from

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🎯 IF YOU REMEMBER NOTHING ELSE

This is NCERT 11–12, not aptitude. Do not buy a quant book.

⭐ Probability is two marks — one here, one in Statistics. Do it once, properly.

⭐ Determinant properties are the cheapest mark on the paper — nine rules, pure recall, two hours, and 2024 tested five of them in a single question.

⭐ Set Theory and Coordinate Geometry appeared in BOTH papers. If you are short on time, these two go first.

The difficulty is the format, not the maths. Six of ten questions make you judge four separate claims. Practise reading statements, not just solving sums.

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