Ishu's Smart Notes 📚

• Reflexive: $(a,a) \in R$ for all $a$
• Symmetric: $(a,b) \in R \Rightarrow (b,a) \in R$
• Transitive: $(a,b),(b,c) \in R \Rightarrow (a,c) \in R$
• Equivalence = Reflexive + Symmetric + Transitive
• Empty relation: NOT reflexive. Universal: IS equivalence.

• One-One: $f(a)=f(b) \Rightarrow a=b$
• Onto: Range = Codomain
• Bijective = One-One + Onto (inverse exists!)
• Horizontal line test → One-One
• Onto check: find range and match with codomain

• $(fog)(x) = f(g(x))$ — apply $g$ first!
• $fog \neq gof$ in general
• Inverse function exists only if bijective
• $(f^{-1} \circ f)(x) = x$

• $\sin^{-1}x$: Range $[-\pi/2, \pi/2]$
• $\cos^{-1}x$: Range $[0, \pi]$
• $\tan^{-1}x$: Range $(-\pi/2, \pi/2)$
• Domain of all: $[-1, 1]$ (for sin & cos)

• $\sin^{-1}(-x) = -\sin^{-1}x$
• $\cos^{-1}(-x) = \pi - \cos^{-1}x$

• $\sin(\sin^{-1}x) = x$ always
• $\sin^{-1}(\sin x) = x$ only if $x \in [-\pi/2, \pi/2]$
• For $\tan^{-1}x + \tan^{-1}y$: if $xy > 1$, add $\pi$ to the formula
• $\tan^{-1}1 + \tan^{-1}2 + \tan^{-1}3 = \pi$ — classic question!

• Square matrix: rows = columns
• Diagonal: non-diagonal elements = 0
• Identity (I): diagonal = 1, rest = 0
• Symmetric: $A = A^T$
• Skew-Symmetric: $A = -A^T$ (diagonal = 0)

• $AB \neq BA$ generally
• $(AB)^T = B^T A^T$
• $(A^T)^T = A$
• Any matrix = Symmetric + Skew-Symmetric

• For 2×2: swap $a,d$; negate $b,c$ — divide by $|A|$
• $AA^{-1} = I$
• Inverse exists iff $|A| \neq 0$ (non-singular)

• $|AB| = |A| \cdot |B|$
• $|A^T| = |A|$
• $|kA| = k^n |A|$ for $n \times n$ matrix
• Two identical rows/cols → $|A| = 0$
• Interchange rows/cols → det changes sign

⚠️ If Area = 0 → points are Collinear!

• adj(A) = transpose of cofactor matrix
• $A \cdot \text{adj}(A) = |A| \cdot I$
• System $AX=B$: if $|A|\neq 0$, unique solution
• Consistent system: $|A| = 0$ but $(\text{adj}A)B = 0$
• Inconsistent: $|A| = 0$ and $(\text{adj}A)B \neq 0$

• $f$ is continuous at $c$ if:

• LHL = RHL = $f(c)$ must all be equal
• Differentiable ⟹ Continuous (converse FALSE)
• $|x|$ is continuous but NOT differentiable at $x=0$

• $\frac{d}{dx}(\sin^{-1}x) = \dfrac{1}{\sqrt{1-x^2}}$
• $\frac{d}{dx}(\cos^{-1}x) = \dfrac{-1}{\sqrt{1-x^2}}$
• $\frac{d}{dx}(\tan^{-1}x) = \dfrac{1}{1+x^2}$
• $\frac{d}{dx}(e^x) = e^x$, $\ \frac{d}{dx}(a^x) = a^x \ln a$
• $\frac{d}{dx}(\ln x) = \dfrac{1}{x}$

• Chain rule: $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$
• Implicit: differentiate both sides w.r.t $x$, collect $\frac{dy}{dx}$
• Parametric: $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$
• Log differentiation: use when base and power both are functions of $x$

• Rolle's: $f$ cont on $[a,b]$, diff on $(a,b)$, $f(a)=f(b)$ → $\exists c$ s.t. $f'(c)=0$
• LMVT: $f'(c) = \dfrac{f(b)-f(a)}{b-a}$ for some $c \in (a,b)$

• Put $f'(x) = 0$ → find critical points
• First Derivative Test: sign change of $f'$
• Second Derivative Test:
• $f''(c) < 0$ → Local Maxima
• $f''(c) > 0$ → Local Minima
• Absolute max/min: always check endpoints!

• $f'(x) > 0$ on $(a,b)$ → Strictly Increasing
• $f'(x) < 0$ on $(a,b)$ → Strictly Decreasing
• Find intervals by solving $f'(x) = 0$ then checking signs

• Slope of tangent at $(x_1,y_1)$: $m = f'(x_1)$
• Slope of normal: $-\frac{1}{m}$
• Tangent: $y - y_1 = m(x - x_1)$
• If $f'=0$ → tangent is horizontal
• Rate of change: $\frac{dy}{dt} = \frac{dy}{dx} \cdot \frac{dx}{dt}$

• $\int x^n dx = \dfrac{x^{n+1}}{n+1} + C$ (n ≠ -1)
• $\int e^x dx = e^x + C$
• $\int \frac{1}{x} dx = \ln|x| + C$
• $\int \sin x\, dx = -\cos x + C$
• $\int \cos x\, dx = \sin x + C$
• $\int \sec^2 x\, dx = \tan x + C$

• $\int_a^b f(x)dx = \int_a^b f(a+b-x)dx$
• $\int_0^a f(x)dx = \int_0^a f(a-x)dx$
• $\int_{-a}^{a} f(x)dx = 2\int_0^a f(x)dx$ if even
• $\int_{-a}^{a} f(x)dx = 0$ if odd function

ILATE Rule — choose $u$ as: Inverse Trigo → Log → Algebraic → Trig → Exponential

• Area between curves: $\int_a^b [f(x) - g(x)]\, dx$ where $f \geq g$
• Always take absolute value if area is asked

• Circle $x^2+y^2=a^2$: Area = $\pi a^2$
• Area of half circle via integration: $\int_0^a \sqrt{a^2-x^2}\, dx = \dfrac{\pi a^2}{4}$
• Parabola $y^2=4ax$: Use symmetry, integrate wrt $y$
• Ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$: Area $= \pi ab$

• Order: highest derivative present
• Degree: power of highest order derivative (after clearing radicals)
• General solution: has arbitrary constants
• Particular solution: constants fixed using initial conditions

• Form: $\frac{dy}{dx} = f(x) \cdot g(y)$
• Separate: $\frac{dy}{g(y)} = f(x)\, dx$
• Integrate both sides separately

• Form: $\frac{dy}{dx} + Py = Q$ (P, Q in terms of x)

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

• Component form: $a_1b_1 + a_2b_2 + a_3b_3$
• Perpendicular ↔ $\vec{a} \cdot \vec{b} = 0$
• Projection of $\vec{a}$ on $\vec{b}$: $\dfrac{\vec{a}\cdot\vec{b}}{|\vec{b}|}$

• Result is a vector ⊥ to both $\vec{a}$ and $\vec{b}$
• Parallel ↔ $\vec{a} \times \vec{b} = \vec{0}$
• Area of parallelogram = $|\vec{a} \times \vec{b}|$
• Area of triangle = $\frac{1}{2}|\vec{a} \times \vec{b}|$

• $[\vec{a}\ \vec{b}\ \vec{c}] = \vec{a} \cdot (\vec{b} \times \vec{c})$
• Volume of parallelepiped = $|[\vec{a}\ \vec{b}\ \vec{c}]|$
• Coplanar ↔ $[\vec{a}\ \vec{b}\ \vec{c}] = 0$

• Vector form: $\vec{r} = \vec{a} + \lambda\vec{b}$
• Cartesian: $\dfrac{x-x_1}{l} = \dfrac{y-y_1}{m} = \dfrac{z-z_1}{n}$
• Direction cosines: $l^2+m^2+n^2=1$
• Angle between lines: $\cos\theta = |\dfrac{\vec{b_1}\cdot\vec{b_2}}{|\vec{b_1}||\vec{b_2}|}|$

• Equation: $ax+by+cz+d=0$
• Normal vector: $(a, b, c)$

• Angle between planes: $\cos\theta = \dfrac{|\vec{n_1}\cdot\vec{n_2}|}{|\vec{n_1}||\vec{n_2}|}$

• Skew lines: non-parallel, non-intersecting
• Shortest distance between skew lines:

If $\vec{b_1}\times\vec{b_2} = \vec{0}$, lines are parallel.

• Objective Function: $Z = ax + by$ to maximize/minimize
• Constraints: inequalities defining the region
• Feasible Region: set of all feasible solutions (convex polygon)
• Corner Point Theorem: optimal value is always at a vertex!

• 1. Write objective function $Z = ax + by$
• 2. Convert inequalities to equations, plot lines
• 3. Find feasible region (shade it)
• 4. Find all corner points (vertices)
• 5. Evaluate $Z$ at each corner → pick max/min

• $P(A \cap B) = P(A|B) \cdot P(B)$
• $P(A \cap B) = P(B|A) \cdot P(A)$
• Independent: $P(A \cap B) = P(A) \cdot P(B)$

Always verify: $\sum P(E_i) = 1$

• $n$ = no. of trials, $p$ = success prob, $q = 1-p$

• Mean $= np$, Variance $= npq$
• Binomial if: fixed trials, 2 outcomes, independent

• Probability distribution table: $\sum P_i = 1$
• Mean (E(X)): $\sum x_i P_i$
• Variance: $E(X^2) - [E(X)]^2$
• $E(X^2) = \sum x_i^2 P_i$