BSEB Inter Mathematics cheat sheet

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

A function f: A→B assigns each a∈A exactly one b∈B; equal functions need same domain, codomain, and f(x)=g(x) for all x.

Injective (one-one): f(a)=f(b) ⇒ a=b; surjective (onto): range = codomain; bijective: both, hence invertible.

Composition: (g∘f)(x)=g(f(x)), defined when range(f)⊆domain(g); generally g∘f ≠ f∘g.

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

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

Number of one-one functions from an m-set to an n-set is P(n,m)=n!/(n−m)! when n≥m, else 0.

Equivalence relation: reflexive (aRa), symmetric (aRb⇒bRa), transitive (aRb and bRc⇒aRc).

For f(x)=1/x, domain = ℝ{0}; for √x, domain = [0,∞); for log x, domain = (0,∞).

Principal ranges: sin⁻¹: [−π/2, π/2]; cos⁻¹: [0, π]; tan⁻¹: (−π/2, π/2); cot⁻¹: (0, π); sec⁻¹: [0,π]{π/2}; csc⁻¹: [−π/2,π/2]{0}.

sin⁻¹(−x) = −sin⁻¹x; cos⁻¹(−x) = π − cos⁻¹x; tan⁻¹(−x) = −tan⁻¹x.

sin⁻¹x + cos⁻¹x = π/2; tan⁻¹x + cot⁻¹x = π/2; sec⁻¹x + csc⁻¹x = π/2 for admissible x.

tan⁻¹x + tan⁻¹y = tan⁻¹((x+y)/(1−xy)) when xy<1; add π if xy>1 and x,y>0; subtract π if xy>1 and x,y<0.

2tan⁻¹x = tan⁻¹(2x/(1−x²)) for |x|<1; = π + tan⁻¹(2x/(1−x²)) for x>1; = −π + tan⁻¹(2x/(1−x²)) for x<−1.

sin⁻¹(sin x)=x only when x∈[−π/2,π/2]; otherwise reduce x to the principal interval first.

cos⁻¹x = 2 sin⁻¹(√((1−x)/2)) for 0≤x≤1; sin⁻¹x = 2 tan⁻¹(x/(1+√(1−x²))).

Derivatives: d/dx sin⁻¹x = 1/√(1−x²); d/dx cos⁻¹x = −1/√(1−x²); d/dx tan⁻¹x = 1/(1+x²).

Matrix addition requires identical order; multiplication AB requires columns(A)=rows(B); generally AB≠BA.

(AB)ᵀ = BᵀAᵀ; (AB)⁻¹ = B⁻¹A⁻¹; (Aᵀ)⁻¹ = (A⁻¹)ᵀ; (A+B)ᵀ = Aᵀ+Bᵀ.

A is symmetric if Aᵀ=A, skew-symmetric if Aᵀ=−A; every square matrix is the sum of a symmetric and a skew-symmetric matrix.

det(AB)=det(A)det(B); det(kA)=kⁿ det(A) for n×n A; det(Aᵀ)=det(A); det(A⁻¹)=1/det(A).

A is invertible iff det(A)≠0; adj(A) = transpose of cofactor matrix; A·adj(A)=det(A)·I; A⁻¹=adj(A)/det(A).

Cramer/system: AX=B has unique solution X=A⁻¹B when det(A)≠0; no solution or infinite solutions when det(A)=0.

For 3×3 determinant expand along any row/column: |A| = Σ aᵢⱼ Cᵢⱼ with Cᵢⱼ = (−1)^(i+j) Mᵢⱼ.

If two rows/columns are identical or proportional, det=0; interchanging two rows changes sign of det.

Rank via row reduction; consistency: rank(A)=rank([A|B]) for solution to exist.

f is continuous at a if lim(x→a) f(x) = f(a); equivalently LHL = RHL = f(a).

Differentiability at a implies continuity at a; continuity does not imply differentiability (e.g., |x| at 0).

Product rule: (uv)′ = u′v + uv′; quotient rule: (u/v)′ = (u′v − uv′)/v².

Chain rule: d/dx f(g(x)) = f′(g(x))·g′(x).

Derivatives: d/dx eˣ=eˣ; d/dx aˣ=aˣ ln a; d/dx ln x=1/x; d/dx log_a x = 1/(x ln a).

Rolle's theorem: if f continuous on [a,b], differentiable on (a,b), and f(a)=f(b), then ∃c∈(a,b) with f′(c)=0.

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

For parametric x=x(t), y=y(t): dy/dx = (dy/dt)/(dx/dt) provided dx/dt≠0.

f is strictly increasing on an interval if f′(x)>0 there; strictly decreasing if f′(x)<0.

Critical points: f′(x)=0 or f′ undefined; evaluate f at critical points and endpoints to find extrema on a closed interval.

First derivative test: sign change + to − gives local maximum; − to + gives local minimum.

Second derivative test: f′(c)=0 and f″(c)<0 ⇒ local maximum; f″(c)>0 ⇒ local minimum; f″(c)=0 inconclusive.

Tangent at (x₀,y₀): y−y₀ = f′(x₀)(x−x₀); normal slope = −1/f′(x₀) when f′(x₀)≠0.

Equation of tangent to y=f(x) at x₀: y − f(x₀) = f′(x₀)(x − x₀).

Rate of change: if y=f(x) and x=x(t), then dy/dt = f′(x)·dx/dt.

Approximation: f(x+Δx) ≈ f(x) + f′(x)Δx for small Δx.

For optimization, express the quantity as a function of one variable using the constraint, then differentiate and test.

∫xⁿ dx = xⁿ⁺¹/(n+1)+C for n≠−1; ∫(1/x)dx = ln|x|+C.

∫eˣ dx=eˣ+C; ∫aˣ dx=aˣ/ln a + C; ∫sin x dx=−cos x+C; ∫cos x dx=sin x+C.

∫sec²x dx=tan x+C; ∫csc²x dx=−cot x+C; ∫sec x tan x dx=sec x+C; ∫csc x cot x dx=−csc x+C.

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

Integration by parts: ∫u dv = uv − ∫v du; choose u by LIATE (Log, Inverse trig, Algebraic, Trig, Exponential).

Substitution: ∫f(g(x))g′(x)dx = ∫f(u)du with u=g(x).

Fundamental Theorem: ∫ₐᵇ f(x)dx = F(b)−F(a) where F′=f; ∫ₐᵇ f(x)dx = −∫ᵦᵃ f(x)dx.

Properties: ∫₀ᵃ f(x)dx = ∫₀ᵃ f(a−x)dx; ∫₋ₐᵃ f(x)dx = 0 if f odd, 2∫₀ᵃ f(x)dx if f even.

Area under y=f(x) from x=a to b (f≥0): A = ∫ₐᵇ f(x)dx.

Area between y=f(x) and y=g(x) from a to b: A = ∫ₐᵇ |f(x)−g(x)|dx.

Area with respect to y: A = ∫_c^d x dy when x is given as a function of y.

Area of region bounded by y=f(x), x-axis, x=a, x=b where f changes sign: split at zeros and add absolute values.

Area of circle x²+y²=r² is πr²; area of ellipse x²/a²+y²/b²=1 is πab.

Area of parabola y²=4ax cut by latus rectum x=a: A = (8a²)/3.

Area bounded by y²=4ax and x²=4ay is (16a²)/3.

Area of triangle with vertices (x₁,y₁),(x₂,y₂),(x₃,y₃) = ½|x₁(y₂−y₃)+x₂(y₃−y₁)+x₃(y₁−y₂)|.

Order = highest derivative present; degree = exponent of highest-order derivative when the equation is polynomial in derivatives.

General solution of an nth-order ODE contains n arbitrary constants; particular solution fixes them via initial conditions.

Variable separable: dy/dx = f(x)g(y) ⇒ ∫dy/g(y) = ∫f(x)dx.

Homogeneous dy/dx = F(y/x): substitute y=vx, so dy/dx = v + x dv/dx, then separate.

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

Exact equation M dx + N dy = 0 requires ∂M/∂y = ∂N/∂x; solution ∫M dx + terms of N free of x = C.

Growth/decay: dy/dt = ky ⇒ y = y₀e^{kt}; Newton cooling: dT/dt = −k(T−T₀).

Family of curves y = mx represents dy/dx = y/x; eliminate parameters to form the ODE.

Dot product: a·b = |a||b|cos θ = a₁b₁+a₂b₂+a₃b₃; a·b=0 ⇒ perpendicular; a·a=|a|².

Cross product: |a×b| = |a||b|sin θ; a×b is perpendicular to both; a×b=0 ⇒ parallel.

Scalar triple product [a b c] = a·(b×c) = determinant of components; volume of parallelepiped = |[a b c]|.

Direction cosines l,m,n satisfy l²+m²+n²=1; direction ratios are proportional to components.

Line through a with direction b: r = a + λb; Cartesian (x−x₁)/l = (y−y₁)/m = (z−z₁)/n.

Plane: r·n = d; Cartesian ax+by+cz=d with normal (a,b,c).

Angle between planes = angle between normals: cos θ = (n₁·n₂)/(|n₁||n₂|).

Distance from point (x₁,y₁,z₁) to plane ax+by+cz+d=0 is |ax₁+by₁+cz₁+d|/√(a²+b²+c²).

Shortest distance between skew lines r=a₁+λb₁ and r=a₂+μb₂ is |(a₂−a₁)·(b₁×b₂)|/|b₁×b₂|.

LPP: optimize (maximize/minimize) a linear objective Z = ax+by subject to linear constraints and non-negativity.

Feasible region is the intersection of half-planes defined by constraints; it is convex.

Corner Point Theorem: if an optimum exists, it occurs at a vertex of the feasible region.

Graphical method: plot constraints, shade feasible region, evaluate Z at each corner, pick max or min.

Bounded feasible region ⇒ objective has both max and min on vertices; unbounded may have no finite optimum.

If two corner points give the same optimal Z, every point on the segment between them is also optimal.

Non-negativity x≥0, y≥0 confines the region to the first quadrant.

For constraints ax+by≤c, shade the side containing the origin when c≥0; test (0,0).

P(E) = favourable outcomes / total equally likely outcomes; 0≤P(E)≤1; P(E′)=1−P(E).

Addition: P(A∪B)=P(A)+P(B)−P(A∩B); mutually exclusive ⇒ P(A∩B)=0.

Conditional probability: P(A|B)=P(A∩B)/P(B), P(B)>0.

Multiplication: P(A∩B)=P(A)P(B|A); independent events satisfy P(A∩B)=P(A)P(B).

Total probability: P(A)=ΣP(Eᵢ)P(A|Eᵢ) for a partition {Eᵢ}.

Bayes' theorem: P(Eᵢ|A)=P(Eᵢ)P(A|Eᵢ)/ΣP(Eⱼ)P(A|Eⱼ).

Binomial distribution: P(X=r)=C(n,r)pʳ(1−p)^{n−r}; mean=np; variance=np(1−p).

Random variable mean μ=Σxᵢpᵢ; variance = Σxᵢ²pᵢ − μ².

Quadratic ax²+bx+c=0: roots α,β satisfy α+β=−b/a, αβ=c/a; discriminant D=b²−4ac determines nature of roots.

Complex numbers: i²=−1; modulus |a+bi|=√(a²+b²); conjugate a−bi; (a+bi)(a−bi)=a²+b².

De Moivre: (cos θ+i sin θ)ⁿ = cos nθ + i sin nθ.

AP: aₙ=a+(n−1)d; Sₙ=n/2[2a+(n−1)d]. GP: aₙ=ar^{n−1}; Sₙ=a(rⁿ−1)/(r−1) for r≠1; S∞=a/(1−r) for |r|<1.

Binomial theorem: (a+b)ⁿ=Σ C(n,r)a^{n−r}bʳ; general term T_{r+1}=C(n,r)a^{n−r}bʳ.

Permutations P(n,r)=n!/(n−r)!; combinations C(n,r)=n!/[r!(n−r)!]; C(n,r)=C(n,n−r).

nPr = r!·nCr; number of arrangements of n objects with repetitions = n!/(p!q!…).

Sum of first n natural numbers = n(n+1)/2; sum of squares = n(n+1)(2n+1)/6; sum of cubes = [n(n+1)/2]².

Limit exists iff LHL=RHL; standard limits: lim(x→0) sin x/x = 1; lim(x→0)(1−cos x)/x² = 1/2; lim(x→0)(eˣ−1)/x = 1; lim(x→0) ln(1+x)/x = 1.

L'Hôpital's rule applies to 0/0 or ∞/∞ forms: lim f/g = lim f′/g′ when the latter exists.

Standard derivatives: d/dx xⁿ=nx^{n−1}; d/dx sin x=cos x; d/dx cos x=−sin x; d/dx tan x=sec²x; d/dx cot x=−csc²x.

d/dx sec x = sec x tan x; d/dx csc x = −csc x cot x.

Indeterminate forms: 0/0, ∞/∞, 0·∞, ∞−∞, 1^∞, 0⁰, ∞⁰; rewrite to apply L'Hôpital or standard limits.

For 1^∞ form: lim f^g = e^{lim g(f−1)} when f→1, g→∞.

Continuity of composite: if f continuous at a and g continuous at f(a), then g∘f is continuous at a.

Sandwich theorem: if g(x)≤f(x)≤h(x) and lim g=lim h=L, then lim f=L.