Karnataka II PUC Mathematics (KSEAB) cheat sheet

A relation R from A to B is a subset of A×B; its domain is {a : (a,b) ∈ R} and range is {b : (a,b) ∈ R}.

R is a function if every element of the domain has exactly one image; equal domains with different codomain elements for one input means not a function.

One-one (injective): f(a)=f(b) ⇒ a=b; many-one otherwise. Onto (surjective): range = codomain. Bijective = both.

Composition: (g∘f)(x)=g(f(x)) exists only when range(f) ⊆ domain(g); in general g∘f ≠ f∘g.

Inverse f⁻¹ exists iff f is bijective; then f⁻¹∘f = identity on domain and f∘f⁻¹ = identity on codomain.

Number of relations from a set of m elements to a set of n elements is 2^(mn); number of functions is n^m; number of bijections (m=n) is n!.

An equivalence relation needs reflexive, symmetric and transitive; a relation that is reflexive, symmetric and transitive partitions the set into disjoint equivalence classes.

For f: R→R, f(x)=x² is neither one-one nor onto; f(x)=x³ is bijective; f(x)=|x| is neither one-one nor onto.

Principal ranges: sin⁻¹:[−1,1]→[−π/2,π/2]; cos⁻¹:[−1,1]→[0,π]; tan⁻¹:R→(−π/2,π/2); cot⁻¹:R→(0,π); sec⁻¹:(−∞,−1]∪[1,∞)→[0,π]∖{π/2}; cosec⁻¹ same domain →[−π/2,π/2]∖{0}.

Reciprocal identities: sin⁻¹(1/x)=cosec⁻¹x for |x|≥1; cos⁻¹(1/x)=sec⁻¹x; tan⁻¹(1/x)=cot⁻¹x for x>0, and cot⁻¹x = tan⁻¹(1/x) − π for x<0.

Reflection identities: sin⁻¹(−x)=−sin⁻¹x; tan⁻¹(−x)=−tan⁻¹x; cos⁻¹(−x)=π−cos⁻¹x; cot⁻¹(−x)=π−cot⁻¹x; sec⁻¹(−x)=π−sec⁻¹x.

Complementary: sin⁻¹x+cos⁻¹x=π/2 for |x|≤1; tan⁻¹x+cot⁻¹x=π/2 for all real x.

Sum formulas valid when xy<1: tan⁻¹x+tan⁻¹y=tan⁻¹((x+y)/(1−xy)); if xy>1 and x,y>0, add π to the result.

tan⁻¹x−tan⁻¹y=tan⁻¹((x−y)/(1+xy)) when xy>−1.

Double-angle: 2tan⁻¹x=tan⁻¹(2x/(1−x²)) for |x|<1; 2sin⁻¹x=sin⁻¹(2x√(1−x²)) only for |x|≤1/√2, else use π−sin⁻¹(...).

Simplification pattern: write sin⁻¹(2x/(1+x²))=2tan⁻¹x for |x|≤1, and cos⁻¹((1−x²)/(1+x²))=2tan⁻¹x for x≥0.

(AB)ᵀ=BᵀAᵀ; (AB)⁻¹=B⁻¹A⁻¹; transpose and inverse reverse the order of multiplication.

A is symmetric if Aᵀ=A, skew-symmetric if Aᵀ=−A; every square matrix is the sum of a symmetric part (A+Aᵀ)/2 and a skew-symmetric part (A−Aᵀ)/2.

For n×n matrices, |kA|=kⁿ|A|; |AB|=|A||B|; |Aᵀ|=|A|; |A⁻¹|=1/|A|; |adj A|=|A|^(n−1).

A is invertible iff |A|≠0; then A⁻¹=(1/|A|)·adj A, and A·adj A=adj A·A=|A|·I.

For a 2×2 matrix [[a,b],[c,d]], adj = [[d,−b],[−c,a]] and A⁻¹ = (1/(ad−bc))·[[d,−b],[−c,a]].

Cramer-style solution of AX=B: x = |A₁|/|A|, y = |A₂|/|A|, z = |A₃|/|A|, where Aᵢ replaces column i by B; unique solution iff |A|≠0.

Area of a triangle with vertices (x₁,y₁),(x₂,y₂),(x₃,y₃) is (1/2)|x₁(y₂−y₃)+x₂(y₃−y₁)+x₃(y₁−y₂)|.

Row/column operations that add a multiple of one row to another leave |A| unchanged; swapping two rows changes the sign; multiplying a row by k multiplies |A| by k.

f is continuous at a iff lim(x→a⁻)f(x)=lim(x→a⁺)f(x)=f(a); all three must exist and agree.

Differentiability implies continuity, but continuity does not imply differentiability (f(x)=|x| at 0).

Standard derivatives: d/dx(xⁿ)=nx^(n−1); d/dx(eˣ)=eˣ; d/dx(aˣ)=aˣ ln a; d/dx(ln x)=1/x; d/dx(sin x)=cos x; d/dx(tan x)=sec²x.

Chain rule: dy/dx = (dy/du)(du/dx); for parametric x=f(t), y=g(t), dy/dx = (dy/dt)/(dx/dt).

Rolle's theorem: f continuous on [a,b], differentiable on (a,b), f(a)=f(b) ⇒ some c∈(a,b) with f′(c)=0.

Lagrange MVT: under continuity on [a,b] and differentiability on (a,b), some c has f′(c)=(f(b)−f(a))/(b−a).

Derivative of inverse trig: d/dx(sin⁻¹x)=1/√(1−x²); d/dx(tan⁻¹x)=1/(1+x²); d/dx(sec⁻¹x)=1/(|x|√(x²−1)).

Logarithmic differentiation: for y=u(x)^v(x), take ln y = v ln u, then y′ = y(v′ ln u + v u′/u).

f is increasing on an interval if f′(x)>0 there; decreasing if f′(x)<0; solve f′(x)=0 to find critical points for sign analysis.

First-derivative test: sign change from + to − at c means local maximum; − to + means local minimum; no change means neither.

Second-derivative test: at f′(c)=0, f″(c)<0 gives local maximum, f″(c)>0 gives local minimum, f″(c)=0 is inconclusive.

Rate of change: if A(t) is area and r(t) radius, dA/dt = 2πr·dr/dt; related-rate problems differentiate the relation with respect to t.

Slope of tangent at (x₁,y₁) is dy/dx evaluated there; slope of normal = −1/(dy/dx); tangent equation y−y₁=m(x−x₁).

Approximation: f(a+h) ≈ f(a)+h f′(a); differential dy = f′(x) dx.

Absolute extrema on [a,b]: evaluate f at all critical points in (a,b) and at endpoints a, b; largest value is the maximum.

For a rectangle of fixed perimeter, maximum area occurs as a square; for fixed surface area, cube maximises volume.

Standard integrals: ∫xⁿ dx = x^(n+1)/(n+1)+C (n≠−1); ∫1/x dx = ln|x|+C; ∫eˣ dx = eˣ+C; ∫cos x dx = sin x+C; ∫sec²x dx = tan x+C.

∫1/(a²+x²) dx = (1/a)tan⁻¹(x/a)+C; ∫1/√(a²−x²) dx = sin⁻¹(x/a)+C; ∫1/(x²−a²) dx = (1/2a)ln|(x−a)/(x+a)|+C.

Integration by parts: ∫u v dx = u∫v dx − ∫(u′∫v dx)dx; choose u by the ILATE order (inverse trig, log, algebraic, trig, exponential).

∫eˣ[f(x)+f′(x)] dx = eˣf(x)+C; ∫[f(x)]ⁿf′(x) dx = [f(x)]^(n+1)/(n+1)+C for n≠−1.

Definite integral property: ∫ₐᵇf(x)dx = ∫ₐᵇf(a+b−x)dx; use it to handle ∫₀^a f(x)dx when f(a−x) simplifies.

∫₋ₐᵃ f(x)dx = 2∫₀ᵃf(x)dx if f is even, and 0 if f is odd.

Area between y=f(x) and the x-axis from a to b is ∫ₐᵇ|f(x)|dx; area between curves is ∫ₐᵇ(f_upper−f_lower)dx.

Area enclosed by y²=4ax and its latus rectum x=a is (8a²)/3; area of the circle x²+y²=a² is πa².

Order = highest derivative present; degree = power of the highest-order derivative after the equation is made polynomial in derivatives.

Variable separable: write dy/dx = f(x)g(y), then ∫dy/g(y) = ∫f(x)dx.

Homogeneous equation dy/dx = F(x,y) with F homogeneous of degree 0: substitute y=vx, so dy/dx = v+x dv/dx, then separate.

Linear equation dy/dx + Py = Q: integrating factor IF = e^∫P dx; solution y·IF = ∫Q·IF dx + C.

Exact equation M dx + N dy = 0 satisfies ∂M/∂y = ∂N/∂x; solution is ∫M dx (y constant) + ∫(terms of N free of x) dy = C.

Growth/decay: dy/dt = ky gives y = y₀e^(kt); Newton cooling dT/dt = −k(T−T₀) gives T−T₀ = (Tᵢ−T₀)e^(−kt).

Family of curves y = mx + c is represented by y′ = m; eliminating n arbitrary constants needs n differentiations.

Solution of dy/dx = y/x is y = Cx; of dy/dx = −y/x is xy = C, i.e. rectangular hyperbola.

If a = a₁i+a₂j+a₃k and b = b₁i+b₂j+b₃k: a·b = a₁b₁+a₂b₂+a₃b₃; a×b = |i j k; a₁ a₂ a₃; b₁ b₂ b₃|.

|a×b| = |a||b|sin θ gives area of parallelogram; area of triangle = ½|a×b|; a·b = |a||b|cos θ.

a ⊥ b ⇔ a·b = 0; a ∥ b ⇔ a×b = 0 ⇔ a₁/b₁ = a₂/b₂ = a₃/b₃.

Scalar triple product [a b c] = a·(b×c) = volume of parallelepiped; coplanar vectors give [a b c] = 0.

Direction cosines l,m,n of a vector satisfy l²+m²+n²=1; direction ratios are any proportional triple.

Line through point a with direction b: r = a + λb; Cartesian form (x−x₁)/b₁ = (y−y₁)/b₂ = (z−z₁)/b₃.

Angle between lines with directions b₁,b₂: cos θ = |b₁·b₂|/(|b₁||b₂|); distance between skew lines = |(a₂−a₁)·(b₁×b₂)|/|b₁×b₂|.

Plane with normal n through a: (r−a)·n = 0, i.e. lx+my+nz = d; distance of point from plane = |ax₁+by₁+cz₁+d|/√(a²+b²+c²).

A linear programming problem maximises or minimises a linear objective Z = ax+by subject to linear constraints and non-negativity x,y ≥ 0.

Feasible region = intersection of all constraint half-planes with the first quadrant; it may be bounded or unbounded.

Corner point theorem: if the feasible region is bounded, the optimum of Z occurs at a vertex of the feasible polygon.

Graph each constraint as a line, test a point such as the origin to decide the half-plane, then shade the common region.

Unbounded region: if Z has a maximum at a corner, check whether the open half-plane beyond the boundary keeps increasing Z; if so, no maximum exists.

For constraints like x+2y ≤ 8, 3x+y ≤ 9, x,y ≥ 0, solve pairs of boundary lines to get vertices and evaluate Z at each.

If two corner points give the same optimal Z, every point on the segment joining them is also optimal (multiple optimal solutions).

Slack variables convert ≤ constraints to equations; a binding constraint is one satisfied with equality at the optimum.

P(A∪B) = P(A)+P(B)−P(A∩B); for mutually exclusive events P(A∩B)=0.

Conditional probability P(A|B) = P(A∩B)/P(B), P(B)>0; A and B are independent iff P(A∩B)=P(A)P(B).

Multiplication rule: P(A∩B) = P(A)P(B|A); Bayes' theorem: P(Aᵢ|B) = P(Aᵢ)P(B|Aᵢ)/ΣP(Aⱼ)P(B|Aⱼ).

Total probability: P(B) = Σ P(Aⱼ)P(B|Aⱼ) over a partition {Aⱼ}.

Binomial distribution: P(X=r) = ⁿCᵣ pʳ q^(n−r), q=1−p; mean = np, variance = npq, SD = √(npq).

Bernoulli trials require fixed n, two outcomes per trial, constant p, and independence across trials.

Mean of a discrete random variable: E(X)=Σxᵢpᵢ; variance = Σxᵢ²pᵢ − (E(X))².

P(A′)=1−P(A); P(exactly one of A,B) = P(A)+P(B)−2P(A∩B).

Set identities: A∩(B∪C)=(A∩B)∪(A∩C); (A∪B)′=A′∩B′; |A∪B|=|A|+|B|−|A∩B|.

Number of subsets of an n-element set is 2ⁿ; number of subsets of size r is ⁿCᵣ = n!/(r!(n−r)!); ⁿCᵣ = ⁿC₍ₙ₋ᵣ₎.

ⁿPᵣ = n!/(n−r)!; ⁿCᵣ = ⁿPᵣ/r!; number of ways to arrange n objects with repeats is n!/(p!q!...).

Binomial theorem: (a+b)ⁿ = Σ ⁿCᵣ a^(n−r)bʳ; general term T₍ᵣ₊₁₎ = ⁿCᵣ a^(n−r)bʳ.

Complex numbers: i²=−1; |z₁z₂|=|z₁||z₂|; z·z̄=|z|²; for z=a+bi, modulus √(a²+b²), argument tan⁻¹(b/a) adjusted by quadrant.

Quadratic ax²+bx+c=0 has roots with sum −b/a and product c/a; discriminant D=b²−4ac decides real (D≥0) or complex roots.

Arithmetic progression: aₙ=a+(n−1)d, Sₙ=(n/2)(2a+(n−1)d); geometric: aₙ=ar^(n−1), Sₙ=a(rⁿ−1)/(r−1), S∞=a/(1−r) for |r|<1.

Logarithm rules: log(mn)=log m+log n; log(m/n)=log m−log n; log mᵏ=k log m; log_b a = 1/log_a b.

A relation R from A to B is a subset of A×B; its domain is {a : (a,b) ∈ R} and range is {b : (a,b) ∈ R}.

R is a function if every element of the domain has exactly one image; equal domains with different codomain elements for one input means not a function.

One-one (injective): f(a)=f(b) ⇒ a=b; many-one otherwise. Onto (surjective): range = codomain. Bijective = both.

Composition: (g∘f)(x)=g(f(x)) exists only when range(f) ⊆ domain(g); in general g∘f ≠ f∘g.

Inverse f⁻¹ exists iff f is bijective; then f⁻¹∘f = identity on domain and f∘f⁻¹ = identity on codomain.

Number of relations from a set of m elements to a set of n elements is 2^(mn); number of functions is n^m; number of bijections (m=n) is n!.

An equivalence relation needs reflexive, symmetric and transitive; a relation that is reflexive, symmetric and transitive partitions the set into disjoint equivalence classes.

For f: R→R, f(x)=x² is neither one-one nor onto; f(x)=x³ is bijective; f(x)=|x| is neither one-one nor onto.

Principal ranges: sin⁻¹:[−1,1]→[−π/2,π/2]; cos⁻¹:[−1,1]→[0,π]; tan⁻¹:R→(−π/2,π/2); cot⁻¹:R→(0,π); sec⁻¹:(−∞,−1]∪[1,∞)→[0,π]∖{π/2}; cosec⁻¹ same domain →[−π/2,π/2]∖{0}.

Reciprocal identities: sin⁻¹(1/x)=cosec⁻¹x for |x|≥1; cos⁻¹(1/x)=sec⁻¹x; tan⁻¹(1/x)=cot⁻¹x for x>0, and cot⁻¹x = tan⁻¹(1/x) − π for x<0.

Reflection identities: sin⁻¹(−x)=−sin⁻¹x; tan⁻¹(−x)=−tan⁻¹x; cos⁻¹(−x)=π−cos⁻¹x; cot⁻¹(−x)=π−cot⁻¹x; sec⁻¹(−x)=π−sec⁻¹x.

Complementary: sin⁻¹x+cos⁻¹x=π/2 for |x|≤1; tan⁻¹x+cot⁻¹x=π/2 for all real x.

Sum formulas valid when xy<1: tan⁻¹x+tan⁻¹y=tan⁻¹((x+y)/(1−xy)); if xy>1 and x,y>0, add π to the result.

tan⁻¹x−tan⁻¹y=tan⁻¹((x−y)/(1+xy)) when xy>−1.

Double-angle: 2tan⁻¹x=tan⁻¹(2x/(1−x²)) for |x|<1; 2sin⁻¹x=sin⁻¹(2x√(1−x²)) only for |x|≤1/√2, else use π−sin⁻¹(...).

Simplification pattern: write sin⁻¹(2x/(1+x²))=2tan⁻¹x for |x|≤1, and cos⁻¹((1−x²)/(1+x²))=2tan⁻¹x for x≥0.

(AB)ᵀ=BᵀAᵀ; (AB)⁻¹=B⁻¹A⁻¹; transpose and inverse reverse the order of multiplication.

A is symmetric if Aᵀ=A, skew-symmetric if Aᵀ=−A; every square matrix is the sum of a symmetric part (A+Aᵀ)/2 and a skew-symmetric part (A−Aᵀ)/2.

For n×n matrices, |kA|=kⁿ|A|; |AB|=|A||B|; |Aᵀ|=|A|; |A⁻¹|=1/|A|; |adj A|=|A|^(n−1).

A is invertible iff |A|≠0; then A⁻¹=(1/|A|)·adj A, and A·adj A=adj A·A=|A|·I.

For a 2×2 matrix [[a,b],[c,d]], adj = [[d,−b],[−c,a]] and A⁻¹ = (1/(ad−bc))·[[d,−b],[−c,a]].

Cramer-style solution of AX=B: x = |A₁|/|A|, y = |A₂|/|A|, z = |A₃|/|A|, where Aᵢ replaces column i by B; unique solution iff |A|≠0.

Area of a triangle with vertices (x₁,y₁),(x₂,y₂),(x₃,y₃) is (1/2)|x₁(y₂−y₃)+x₂(y₃−y₁)+x₃(y₁−y₂)|.

Row/column operations that add a multiple of one row to another leave |A| unchanged; swapping two rows changes the sign; multiplying a row by k multiplies |A| by k.

f is continuous at a iff lim(x→a⁻)f(x)=lim(x→a⁺)f(x)=f(a); all three must exist and agree.

Differentiability implies continuity, but continuity does not imply differentiability (f(x)=|x| at 0).

Standard derivatives: d/dx(xⁿ)=nx^(n−1); d/dx(eˣ)=eˣ; d/dx(aˣ)=aˣ ln a; d/dx(ln x)=1/x; d/dx(sin x)=cos x; d/dx(tan x)=sec²x.

Chain rule: dy/dx = (dy/du)(du/dx); for parametric x=f(t), y=g(t), dy/dx = (dy/dt)/(dx/dt).

Rolle's theorem: f continuous on [a,b], differentiable on (a,b), f(a)=f(b) ⇒ some c∈(a,b) with f′(c)=0.

Lagrange MVT: under continuity on [a,b] and differentiability on (a,b), some c has f′(c)=(f(b)−f(a))/(b−a).

Derivative of inverse trig: d/dx(sin⁻¹x)=1/√(1−x²); d/dx(tan⁻¹x)=1/(1+x²); d/dx(sec⁻¹x)=1/(|x|√(x²−1)).

Logarithmic differentiation: for y=u(x)^v(x), take ln y = v ln u, then y′ = y(v′ ln u + v u′/u).

f is increasing on an interval if f′(x)>0 there; decreasing if f′(x)<0; solve f′(x)=0 to find critical points for sign analysis.

First-derivative test: sign change from + to − at c means local maximum; − to + means local minimum; no change means neither.

Second-derivative test: at f′(c)=0, f″(c)<0 gives local maximum, f″(c)>0 gives local minimum, f″(c)=0 is inconclusive.

Rate of change: if A(t) is area and r(t) radius, dA/dt = 2πr·dr/dt; related-rate problems differentiate the relation with respect to t.

Slope of tangent at (x₁,y₁) is dy/dx evaluated there; slope of normal = −1/(dy/dx); tangent equation y−y₁=m(x−x₁).

Approximation: f(a+h) ≈ f(a)+h f′(a); differential dy = f′(x) dx.

Absolute extrema on [a,b]: evaluate f at all critical points in (a,b) and at endpoints a, b; largest value is the maximum.

For a rectangle of fixed perimeter, maximum area occurs as a square; for fixed surface area, cube maximises volume.

Standard integrals: ∫xⁿ dx = x^(n+1)/(n+1)+C (n≠−1); ∫1/x dx = ln|x|+C; ∫eˣ dx = eˣ+C; ∫cos x dx = sin x+C; ∫sec²x dx = tan x+C.

∫1/(a²+x²) dx = (1/a)tan⁻¹(x/a)+C; ∫1/√(a²−x²) dx = sin⁻¹(x/a)+C; ∫1/(x²−a²) dx = (1/2a)ln|(x−a)/(x+a)|+C.

Integration by parts: ∫u v dx = u∫v dx − ∫(u′∫v dx)dx; choose u by the ILATE order (inverse trig, log, algebraic, trig, exponential).

∫eˣ[f(x)+f′(x)] dx = eˣf(x)+C; ∫[f(x)]ⁿf′(x) dx = [f(x)]^(n+1)/(n+1)+C for n≠−1.

Definite integral property: ∫ₐᵇf(x)dx = ∫ₐᵇf(a+b−x)dx; use it to handle ∫₀^a f(x)dx when f(a−x) simplifies.

∫₋ₐᵃ f(x)dx = 2∫₀ᵃf(x)dx if f is even, and 0 if f is odd.

Area between y=f(x) and the x-axis from a to b is ∫ₐᵇ|f(x)|dx; area between curves is ∫ₐᵇ(f_upper−f_lower)dx.

Area enclosed by y²=4ax and its latus rectum x=a is (8a²)/3; area of the circle x²+y²=a² is πa².

Order = highest derivative present; degree = power of the highest-order derivative after the equation is made polynomial in derivatives.

Variable separable: write dy/dx = f(x)g(y), then ∫dy/g(y) = ∫f(x)dx.

Homogeneous equation dy/dx = F(x,y) with F homogeneous of degree 0: substitute y=vx, so dy/dx = v+x dv/dx, then separate.

Linear equation dy/dx + Py = Q: integrating factor IF = e^∫P dx; solution y·IF = ∫Q·IF dx + C.

Exact equation M dx + N dy = 0 satisfies ∂M/∂y = ∂N/∂x; solution is ∫M dx (y constant) + ∫(terms of N free of x) dy = C.

Growth/decay: dy/dt = ky gives y = y₀e^(kt); Newton cooling dT/dt = −k(T−T₀) gives T−T₀ = (Tᵢ−T₀)e^(−kt).

Family of curves y = mx + c is represented by y′ = m; eliminating n arbitrary constants needs n differentiations.

Solution of dy/dx = y/x is y = Cx; of dy/dx = −y/x is xy = C, i.e. rectangular hyperbola.

If a = a₁i+a₂j+a₃k and b = b₁i+b₂j+b₃k: a·b = a₁b₁+a₂b₂+a₃b₃; a×b = |i j k; a₁ a₂ a₃; b₁ b₂ b₃|.

|a×b| = |a||b|sin θ gives area of parallelogram; area of triangle = ½|a×b|; a·b = |a||b|cos θ.

a ⊥ b ⇔ a·b = 0; a ∥ b ⇔ a×b = 0 ⇔ a₁/b₁ = a₂/b₂ = a₃/b₃.

Scalar triple product [a b c] = a·(b×c) = volume of parallelepiped; coplanar vectors give [a b c] = 0.

Direction cosines l,m,n of a vector satisfy l²+m²+n²=1; direction ratios are any proportional triple.

Line through point a with direction b: r = a + λb; Cartesian form (x−x₁)/b₁ = (y−y₁)/b₂ = (z−z₁)/b₃.

Angle between lines with directions b₁,b₂: cos θ = |b₁·b₂|/(|b₁||b₂|); distance between skew lines = |(a₂−a₁)·(b₁×b₂)|/|b₁×b₂|.

Plane with normal n through a: (r−a)·n = 0, i.e. lx+my+nz = d; distance of point from plane = |ax₁+by₁+cz₁+d|/√(a²+b²+c²).

A linear programming problem maximises or minimises a linear objective Z = ax+by subject to linear constraints and non-negativity x,y ≥ 0.

Feasible region = intersection of all constraint half-planes with the first quadrant; it may be bounded or unbounded.

Corner point theorem: if the feasible region is bounded, the optimum of Z occurs at a vertex of the feasible polygon.

Graph each constraint as a line, test a point such as the origin to decide the half-plane, then shade the common region.

Unbounded region: if Z has a maximum at a corner, check whether the open half-plane beyond the boundary keeps increasing Z; if so, no maximum exists.

For constraints like x+2y ≤ 8, 3x+y ≤ 9, x,y ≥ 0, solve pairs of boundary lines to get vertices and evaluate Z at each.

If two corner points give the same optimal Z, every point on the segment joining them is also optimal (multiple optimal solutions).

Slack variables convert ≤ constraints to equations; a binding constraint is one satisfied with equality at the optimum.

P(A∪B) = P(A)+P(B)−P(A∩B); for mutually exclusive events P(A∩B)=0.

Conditional probability P(A|B) = P(A∩B)/P(B), P(B)>0; A and B are independent iff P(A∩B)=P(A)P(B).

Multiplication rule: P(A∩B) = P(A)P(B|A); Bayes' theorem: P(Aᵢ|B) = P(Aᵢ)P(B|Aᵢ)/ΣP(Aⱼ)P(B|Aⱼ).

Total probability: P(B) = Σ P(Aⱼ)P(B|Aⱼ) over a partition {Aⱼ}.

Binomial distribution: P(X=r) = ⁿCᵣ pʳ q^(n−r), q=1−p; mean = np, variance = npq, SD = √(npq).

Bernoulli trials require fixed n, two outcomes per trial, constant p, and independence across trials.

Mean of a discrete random variable: E(X)=Σxᵢpᵢ; variance = Σxᵢ²pᵢ − (E(X))².

P(A′)=1−P(A); P(exactly one of A,B) = P(A)+P(B)−2P(A∩B).

Set identities: A∩(B∪C)=(A∩B)∪(A∩C); (A∪B)′=A′∩B′; |A∪B|=|A|+|B|−|A∩B|.

Number of subsets of an n-element set is 2ⁿ; number of subsets of size r is ⁿCᵣ = n!/(r!(n−r)!); ⁿCᵣ = ⁿC₍ₙ₋ᵣ₎.

ⁿPᵣ = n!/(n−r)!; ⁿCᵣ = ⁿPᵣ/r!; number of ways to arrange n objects with repeats is n!/(p!q!...).

Binomial theorem: (a+b)ⁿ = Σ ⁿCᵣ a^(n−r)bʳ; general term T₍ᵣ₊₁₎ = ⁿCᵣ a^(n−r)bʳ.

Complex numbers: i²=−1; |z₁z₂|=|z₁||z₂|; z·z̄=|z|²; for z=a+bi, modulus √(a²+b²), argument tan⁻¹(b/a) adjusted by quadrant.

Quadratic ax²+bx+c=0 has roots with sum −b/a and product c/a; discriminant D=b²−4ac decides real (D≥0) or complex roots.

Arithmetic progression: aₙ=a+(n−1)d, Sₙ=(n/2)(2a+(n−1)d); geometric: aₙ=ar^(n−1), Sₙ=a(rⁿ−1)/(r−1), S∞=a/(1−r) for |r|<1.

Logarithm rules: log(mn)=log m+log n; log(m/n)=log m−log n; log mᵏ=k log m; log_b a = 1/log_a b.

A relation R from A to B is a subset of A×B; its domain is {a : (a,b) ∈ R} and range is {b : (a,b) ∈ R}.

R is a function if every element of the domain has exactly one image; equal domains with different codomain elements for one input means not a function.

One-one (injective): f(a)=f(b) ⇒ a=b; many-one otherwise. Onto (surjective): range = codomain. Bijective = both.

Composition: (g∘f)(x)=g(f(x)) exists only when range(f) ⊆ domain(g); in general g∘f ≠ f∘g.

Inverse f⁻¹ exists iff f is bijective; then f⁻¹∘f = identity on domain and f∘f⁻¹ = identity on codomain.

Number of relations from a set of m elements to a set of n elements is 2^(mn); number of functions is n^m; number of bijections (m=n) is n!.

An equivalence relation needs reflexive, symmetric and transitive; a relation that is reflexive, symmetric and transitive partitions the set into disjoint equivalence classes.

For f: R→R, f(x)=x² is neither one-one nor onto; f(x)=x³ is bijective; f(x)=|x| is neither one-one nor onto.

Principal ranges: sin⁻¹:[−1,1]→[−π/2,π/2]; cos⁻¹:[−1,1]→[0,π]; tan⁻¹:R→(−π/2,π/2); cot⁻¹:R→(0,π); sec⁻¹:(−∞,−1]∪[1,∞)→[0,π]∖{π/2}; cosec⁻¹ same domain →[−π/2,π/2]∖{0}.

Reciprocal identities: sin⁻¹(1/x)=cosec⁻¹x for |x|≥1; cos⁻¹(1/x)=sec⁻¹x; tan⁻¹(1/x)=cot⁻¹x for x>0, and cot⁻¹x = tan⁻¹(1/x) − π for x<0.

Reflection identities: sin⁻¹(−x)=−sin⁻¹x; tan⁻¹(−x)=−tan⁻¹x; cos⁻¹(−x)=π−cos⁻¹x; cot⁻¹(−x)=π−cot⁻¹x; sec⁻¹(−x)=π−sec⁻¹x.

Complementary: sin⁻¹x+cos⁻¹x=π/2 for |x|≤1; tan⁻¹x+cot⁻¹x=π/2 for all real x.

Sum formulas valid when xy<1: tan⁻¹x+tan⁻¹y=tan⁻¹((x+y)/(1−xy)); if xy>1 and x,y>0, add π to the result.

tan⁻¹x−tan⁻¹y=tan⁻¹((x−y)/(1+xy)) when xy>−1.

Double-angle: 2tan⁻¹x=tan⁻¹(2x/(1−x²)) for |x|<1; 2sin⁻¹x=sin⁻¹(2x√(1−x²)) only for |x|≤1/√2, else use π−sin⁻¹(...).

Simplification pattern: write sin⁻¹(2x/(1+x²))=2tan⁻¹x for |x|≤1, and cos⁻¹((1−x²)/(1+x²))=2tan⁻¹x for x≥0.

(AB)ᵀ=BᵀAᵀ; (AB)⁻¹=B⁻¹A⁻¹; transpose and inverse reverse the order of multiplication.

A is symmetric if Aᵀ=A, skew-symmetric if Aᵀ=−A; every square matrix is the sum of a symmetric part (A+Aᵀ)/2 and a skew-symmetric part (A−Aᵀ)/2.

For n×n matrices, |kA|=kⁿ|A|; |AB|=|A||B|; |Aᵀ|=|A|; |A⁻¹|=1/|A|; |adj A|=|A|^(n−1).

A is invertible iff |A|≠0; then A⁻¹=(1/|A|)·adj A, and A·adj A=adj A·A=|A|·I.

For a 2×2 matrix [[a,b],[c,d]], adj = [[d,−b],[−c,a]] and A⁻¹ = (1/(ad−bc))·[[d,−b],[−c,a]].

Cramer-style solution of AX=B: x = |A₁|/|A|, y = |A₂|/|A|, z = |A₃|/|A|, where Aᵢ replaces column i by B; unique solution iff |A|≠0.

Area of a triangle with vertices (x₁,y₁),(x₂,y₂),(x₃,y₃) is (1/2)|x₁(y₂−y₃)+x₂(y₃−y₁)+x₃(y₁−y₂)|.

Row/column operations that add a multiple of one row to another leave |A| unchanged; swapping two rows changes the sign; multiplying a row by k multiplies |A| by k.

f is continuous at a iff lim(x→a⁻)f(x)=lim(x→a⁺)f(x)=f(a); all three must exist and agree.

Differentiability implies continuity, but continuity does not imply differentiability (f(x)=|x| at 0).

Standard derivatives: d/dx(xⁿ)=nx^(n−1); d/dx(eˣ)=eˣ; d/dx(aˣ)=aˣ ln a; d/dx(ln x)=1/x; d/dx(sin x)=cos x; d/dx(tan x)=sec²x.

Chain rule: dy/dx = (dy/du)(du/dx); for parametric x=f(t), y=g(t), dy/dx = (dy/dt)/(dx/dt).

Rolle's theorem: f continuous on [a,b], differentiable on (a,b), f(a)=f(b) ⇒ some c∈(a,b) with f′(c)=0.

Lagrange MVT: under continuity on [a,b] and differentiability on (a,b), some c has f′(c)=(f(b)−f(a))/(b−a).

Derivative of inverse trig: d/dx(sin⁻¹x)=1/√(1−x²); d/dx(tan⁻¹x)=1/(1+x²); d/dx(sec⁻¹x)=1/(|x|√(x²−1)).

Logarithmic differentiation: for y=u(x)^v(x), take ln y = v ln u, then y′ = y(v′ ln u + v u′/u).

f is increasing on an interval if f′(x)>0 there; decreasing if f′(x)<0; solve f′(x)=0 to find critical points for sign analysis.

First-derivative test: sign change from + to − at c means local maximum; − to + means local minimum; no change means neither.

Second-derivative test: at f′(c)=0, f″(c)<0 gives local maximum, f″(c)>0 gives local minimum, f″(c)=0 is inconclusive.

Rate of change: if A(t) is area and r(t) radius, dA/dt = 2πr·dr/dt; related-rate problems differentiate the relation with respect to t.

Slope of tangent at (x₁,y₁) is dy/dx evaluated there; slope of normal = −1/(dy/dx); tangent equation y−y₁=m(x−x₁).

Approximation: f(a+h) ≈ f(a)+h f′(a); differential dy = f′(x) dx.

Absolute extrema on [a,b]: evaluate f at all critical points in (a,b) and at endpoints a, b; largest value is the maximum.

For a rectangle of fixed perimeter, maximum area occurs as a square; for fixed surface area, cube maximises volume.

Standard integrals: ∫xⁿ dx = x^(n+1)/(n+1)+C (n≠−1); ∫1/x dx = ln|x|+C; ∫eˣ dx = eˣ+C; ∫cos x dx = sin x+C; ∫sec²x dx = tan x+C.

∫1/(a²+x²) dx = (1/a)tan⁻¹(x/a)+C; ∫1/√(a²−x²) dx = sin⁻¹(x/a)+C; ∫1/(x²−a²) dx = (1/2a)ln|(x−a)/(x+a)|+C.

Integration by parts: ∫u v dx = u∫v dx − ∫(u′∫v dx)dx; choose u by the ILATE order (inverse trig, log, algebraic, trig, exponential).

∫eˣ[f(x)+f′(x)] dx = eˣf(x)+C; ∫[f(x)]ⁿf′(x) dx = [f(x)]^(n+1)/(n+1)+C for n≠−1.

Definite integral property: ∫ₐᵇf(x)dx = ∫ₐᵇf(a+b−x)dx; use it to handle ∫₀^a f(x)dx when f(a−x) simplifies.

∫₋ₐᵃ f(x)dx = 2∫₀ᵃf(x)dx if f is even, and 0 if f is odd.

Area between y=f(x) and the x-axis from a to b is ∫ₐᵇ|f(x)|dx; area between curves is ∫ₐᵇ(f_upper−f_lower)dx.

Area enclosed by y²=4ax and its latus rectum x=a is (8a²)/3; area of the circle x²+y²=a² is πa².

Order = highest derivative present; degree = power of the highest-order derivative after the equation is made polynomial in derivatives.

Variable separable: write dy/dx = f(x)g(y), then ∫dy/g(y) = ∫f(x)dx.

Homogeneous equation dy/dx = F(x,y) with F homogeneous of degree 0: substitute y=vx, so dy/dx = v+x dv/dx, then separate.

Linear equation dy/dx + Py = Q: integrating factor IF = e^∫P dx; solution y·IF = ∫Q·IF dx + C.

Exact equation M dx + N dy = 0 satisfies ∂M/∂y = ∂N/∂x; solution is ∫M dx (y constant) + ∫(terms of N free of x) dy = C.

Growth/decay: dy/dt = ky gives y = y₀e^(kt); Newton cooling dT/dt = −k(T−T₀) gives T−T₀ = (Tᵢ−T₀)e^(−kt).

Family of curves y = mx + c is represented by y′ = m; eliminating n arbitrary constants needs n differentiations.

Solution of dy/dx = y/x is y = Cx; of dy/dx = −y/x is xy = C, i.e. rectangular hyperbola.

If a = a₁i+a₂j+a₃k and b = b₁i+b₂j+b₃k: a·b = a₁b₁+a₂b₂+a₃b₃; a×b = |i j k; a₁ a₂ a₃; b₁ b₂ b₃|.

|a×b| = |a||b|sin θ gives area of parallelogram; area of triangle = ½|a×b|; a·b = |a||b|cos θ.

a ⊥ b ⇔ a·b = 0; a ∥ b ⇔ a×b = 0 ⇔ a₁/b₁ = a₂/b₂ = a₃/b₃.

Scalar triple product [a b c] = a·(b×c) = volume of parallelepiped; coplanar vectors give [a b c] = 0.

Direction cosines l,m,n of a vector satisfy l²+m²+n²=1; direction ratios are any proportional triple.

Line through point a with direction b: r = a + λb; Cartesian form (x−x₁)/b₁ = (y−y₁)/b₂ = (z−z₁)/b₃.

Angle between lines with directions b₁,b₂: cos θ = |b₁·b₂|/(|b₁||b₂|); distance between skew lines = |(a₂−a₁)·(b₁×b₂)|/|b₁×b₂|.

Plane with normal n through a: (r−a)·n = 0, i.e. lx+my+nz = d; distance of point from plane = |ax₁+by₁+cz₁+d|/√(a²+b²+c²).

A linear programming problem maximises or minimises a linear objective Z = ax+by subject to linear constraints and non-negativity x,y ≥ 0.

Feasible region = intersection of all constraint half-planes with the first quadrant; it may be bounded or unbounded.

Corner point theorem: if the feasible region is bounded, the optimum of Z occurs at a vertex of the feasible polygon.

Graph each constraint as a line, test a point such as the origin to decide the half-plane, then shade the common region.

Unbounded region: if Z has a maximum at a corner, check whether the open half-plane beyond the boundary keeps increasing Z; if so, no maximum exists.

For constraints like x+2y ≤ 8, 3x+y ≤ 9, x,y ≥ 0, solve pairs of boundary lines to get vertices and evaluate Z at each.

If two corner points give the same optimal Z, every point on the segment joining them is also optimal (multiple optimal solutions).

Slack variables convert ≤ constraints to equations; a binding constraint is one satisfied with equality at the optimum.

P(A∪B) = P(A)+P(B)−P(A∩B); for mutually exclusive events P(A∩B)=0.

Conditional probability P(A|B) = P(A∩B)/P(B), P(B)>0; A and B are independent iff P(A∩B)=P(A)P(B).

Multiplication rule: P(A∩B) = P(A)P(B|A); Bayes' theorem: P(Aᵢ|B) = P(Aᵢ)P(B|Aᵢ)/ΣP(Aⱼ)P(B|Aⱼ).

Total probability: P(B) = Σ P(Aⱼ)P(B|Aⱼ) over a partition {Aⱼ}.

Binomial distribution: P(X=r) = ⁿCᵣ pʳ q^(n−r), q=1−p; mean = np, variance = npq, SD = √(npq).

Bernoulli trials require fixed n, two outcomes per trial, constant p, and independence across trials.

Mean of a discrete random variable: E(X)=Σxᵢpᵢ; variance = Σxᵢ²pᵢ − (E(X))².

P(A′)=1−P(A); P(exactly one of A,B) = P(A)+P(B)−2P(A∩B).

Set identities: A∩(B∪C)=(A∩B)∪(A∩C); (A∪B)′=A′∩B′; |A∪B|=|A|+|B|−|A∩B|.

Number of subsets of an n-element set is 2ⁿ; number of subsets of size r is ⁿCᵣ = n!/(r!(n−r)!); ⁿCᵣ = ⁿC₍ₙ₋ᵣ₎.

ⁿPᵣ = n!/(n−r)!; ⁿCᵣ = ⁿPᵣ/r!; number of ways to arrange n objects with repeats is n!/(p!q!...).

Binomial theorem: (a+b)ⁿ = Σ ⁿCᵣ a^(n−r)bʳ; general term T₍ᵣ₊₁₎ = ⁿCᵣ a^(n−r)bʳ.

Complex numbers: i²=−1; |z₁z₂|=|z₁||z₂|; z·z̄=|z|²; for z=a+bi, modulus √(a²+b²), argument tan⁻¹(b/a) adjusted by quadrant.

Quadratic ax²+bx+c=0 has roots with sum −b/a and product c/a; discriminant D=b²−4ac decides real (D≥0) or complex roots.

Arithmetic progression: aₙ=a+(n−1)d, Sₙ=(n/2)(2a+(n−1)d); geometric: aₙ=ar^(n−1), Sₙ=a(rⁿ−1)/(r−1), S∞=a/(1−r) for |r|<1.

Logarithm rules: log(mn)=log m+log n; log(m/n)=log m−log n; log mᵏ=k log m; log_b a = 1/log_a b.