Last Day Revision | Ishu (My Wife)

~5 Marks

Chapter 1

Relations & Functions

Equivalence = Reflexive + Symmetric + Transitive (ALL 3)
One-One: $f(a)=f(b) \Rightarrow a=b$
Onto: Range = Codomain
Inverse exists ↔ Bijective only
$fog \neq gof$. Apply rightmost function first!

~5 Marks

Chapter 2

Inverse Trigonometry

$\sin^{-1}x$ range: $[-\pi/2,\pi/2]$, $\cos^{-1}x$: $[0,\pi]$
$\sin^{-1}x + \cos^{-1}x = \pi/2$
$\cos^{-1}(-x) = \pi - \cos^{-1}x$ (NOT negative!)
$\tan^{-1}x+\tan^{-1}y$: add $\pi$ if $xy>1,x,y>0$

$2\tan^{-1}x = \sin^{-1}\!\left(\frac{2x}{1+x^2}\right)$

~5 Marks

Chapter 3

Matrices

$(AB)^T = B^TA^T$ — order reverses!
$(AB)^{-1} = B^{-1}A^{-1}$ — order reverses!
$A = \frac{A+A^T}{2} + \frac{A-A^T}{2}$ (Sym + Skew)
Symmetric: $A=A^T$. Skew: $A=-A^T$, diag=0
For 2×2 inverse: swap $a,d$; negate $b,c$; divide by det

~6 Marks

Chapter 4

Determinants

$|kA| = k^n|A|$ for $n\times n$ matrix!
$|AB| = |A||B|$
Cofactor sign: $(-1)^{i+j}$. Checkerboard pattern!
Area of triangle = $\frac{1}{2}|det|$. Always absolute value!
Collinear ↔ det = 0
$A^{-1} = \frac{1}{|A|}\text{adj}A$. AX=B: $X=A^{-1}B$

~8 Marks

Chapter 5

Continuity & Differentiability

Continuous at $c$: LHL = RHL = $f(c)$ (all 3!)
Diff ⟹ Continuous (converse FALSE. $|x|$ example!)
Chain rule: $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$
$x^x$: use log differentiation! Not power rule.
$\frac{d}{dx}(\sin^{-1}x) = \frac{1}{\sqrt{1-x^2}}$
$\frac{d}{dx}(\tan^{-1}x) = \frac{1}{1+x^2}$

~7 Marks

Chapter 6

Application of Derivatives

$f'(x)>0$: increasing. $f'(x)<0$: decreasing.
Maxima: $f''<0$. Minima: $f''>0$.
Absolute max/min: check critical points + ENDPOINTS!
Tangent slope = $f'(x_1)$. Normal slope = $-1/f'(x_1)$
Rate: $\frac{dy}{dt} = \frac{dy}{dx}\cdot\frac{dx}{dt}$

~8 Marks

Chapter 7

Integrals

Always write $+C$ in indefinite integrals! Always!
Change limits when substituting in definite integrals
$\int e^x[f(x)+f'(x)]dx = e^xf(x)+C$
ILATE for by-parts: Inv Trig → Log → Algebra → Trig → Exp
$\int_a^b f = \int_a^b f(a+b-x)dx$ — key property!
Even: $\int_{-a}^a = 2\int_0^a$. Odd: $= 0$

~5 Marks

Chapter 8

Application of Integrals

Area = $\int_a^b|f(x)|dx$. Always positive!
Between curves: find intersection pts first = limits
Circle $x^2+y^2=r^2$: Area $=\pi r^2$
Ellipse: Area $=\pi ab$
Sketch rough graph always — it shows which is upper curve

~6 Marks

Chapter 9

Differential Equations

Order = highest derivative. Degree = power of highest.
General solution MUST have $C$. Particular = find $C$.
Variable separable: $f(x)dx = g(y)dy$, integrate both
Linear: $\frac{dy}{dx}+Py=Q$ → I.F. $= e^{\int Pdx}$

Solution: $y\cdot\text{I.F.} = \int Q\cdot\text{I.F.}\,dx + C$

~6 Marks

Chapter 10

Vector Algebra

Dot: $\vec{a}\cdot\vec{b}=|\vec{a}||\vec{b}|\cos\theta$. ⊥ ↔ dot=0
Cross: $|\vec{a}\times\vec{b}|=|\vec{a}||\vec{b}|\sin\theta$. ∥ ↔ cross=0
$\vec{a}\times\vec{b} = -(\vec{b}\times\vec{a})$ — anti-commutative!
Area of triangle $= \frac{1}{2}|\vec{a}\times\vec{b}|$
Coplanar ↔ scalar triple product $= 0$

~6 Marks

Chapter 11

3D Geometry

DC: normalize DR. $l^2+m^2+n^2=1$ for DCs only!
Angle btw lines: $\cos\theta = |\frac{\vec{b_1}\cdot\vec{b_2}}{|\vec{b_1}||\vec{b_2}|}|$
Pt from plane: $d=\frac{|ax_1+by_1+cz_1+d|}{\sqrt{a^2+b^2+c^2}}$
Skew lines SD: $d=\frac{|(\vec{a_2}-\vec{a_1})\cdot(\vec{b_1}\times\vec{b_2})|}{|\vec{b_1}\times\vec{b_2}|}$

~5 Marks

Chapter 12

Linear Programming

Optimal value is ALWAYS at a corner point!
Make table: list ALL corner points, find $Z$ at each
Shade feasible region: test $(0,0)$ in each constraint
Find vertices by solving pairs of boundary equations

~8 Marks

Chapter 13

Probability

$P(A|B)=\frac{P(A\cap B)}{P(B)}$. Always check $P(B)\neq 0$!
Independent: $P(A\cap B)=P(A)P(B)$
Total Prob.: $P(A)=\sum P(E_i)P(A|E_i)$ — find this FIRST
Bayes': $P(E_i|A)=\frac{P(E_i)P(A|E_i)}{P(A)}$
Binomial: $P(X=r)=\binom{n}{r}p^rq^{n-r}$, Mean$=np$, Var$=npq$
E(X): $\sum x_i P_i$. Var: $E(X^2)-[E(X)]^2$

Last Day Revision ⚡

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