Tamil Nadu HSE First Year Mathematics cheat sheet

For sets A and B, A∪B={x:x∈A or x∈B}, A∩B={x:x∈A and x∈B}, A−B={x:x∈A and x∉B}; De Morgan: (A∪B)′=A′∩B′, (A∩B)′=A′∪B′.

Inclusion-exclusion: n(A∪B)=n(A)+n(B)−n(A∩B); n(A∪B∪C)=n(A)+n(B)+n(C)−n(A∩B)−n(B∩C)−n(C∩A)+n(A∩B∩C).

If n(A)=m, then A has 2^m subsets, 2^m−1 proper subsets, and |P(A)|=2^m.

Cartesian product: A×B={(a,b):a∈A,b∈B}; n(A×B)=n(A)n(B); A×B=B×A only in special cases.

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}; inverse R⁻¹={(b,a):(a,b)∈R}.

A function f:A→B is one-one if f(a₁)=f(a₂)⇒a₁=a₂; onto if range(f)=B; bijective if both; inverse f⁻¹ exists exactly for bijections.

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

Standard real functions include identity x, constant c, polynomial, rational, modulus |x|, greatest integer [x], signum sgn(x), exponential a^x (a>0,a≠1), and logarithmic log_a x (x>0).

For ax²+bx+c=0, a≠0, Δ=b²−4ac; Δ>0 distinct real roots, Δ=0 equal real roots, Δ<0 complex conjugate roots.

Roots α,β satisfy α+β=−b/a and αβ=c/a; the quadratic with given roots is x²−(α+β)x+αβ=0.

Quadratic formula: x=(−b±√(b²−4ac))/(2a); vertex of y=ax²+bx+c is at x=−b/(2a).

For a>0: |x|<a ⇔ −a<x<a; |x|>a ⇔ x<−a or x>a; |x−h|<a ⇔ h−a<x<h+a.

Laws of logarithms (m,n>0, a>0,a≠1): log_a(mn)=log_a m+log_a n; log_a(m/n)=log_a m−log_a n; log_a m^p=p log_a m; log_b a=log_c a/log_c b.

For rational inequalities, move all terms to one side, factor numerator and denominator, mark zeros and undefined points on a sign chart, and test intervals.

Partial fractions: a proper rational function with distinct linear factors (x−a)(x−b) decomposes as A/(x−a)+B/(x−b); repeated factors require powers up to multiplicity.

Exponent rules: a^m a^n=a^{m+n}; (a^m)^n=a^{mn}; (ab)^n=a^n b^n; a^{−n}=1/a^n; a^0=1 for a≠0.

Degree-radian conversion: π radians=180°; arc length s=rθ and sector area A=½r²θ when θ is in radians.

Pythagorean identities: sin²θ+cos²θ=1; 1+tan²θ=sec²θ; 1+cot²θ=cosec²θ.

Allied angles: sin(π−θ)=sinθ, cos(π−θ)=−cosθ, tan(π−θ)=−tanθ; sin(π/2−θ)=cosθ, cos(π/2−θ)=sinθ.

Compound angles: sin(A±B)=sinAcosB±cosAsinB; cos(A±B)=cosAcosB∓sinAsinB; tan(A±B)=(tanA±tanB)/(1∓tanA tanB).

Double angles: sin2A=2sinAcosA; cos2A=cos²A−sin²A=2cos²A−1=1−2sin²A; tan2A=2tanA/(1−tan²A).

Product-to-sum: 2sinAcosB=sin(A+B)+sin(A−B); 2cosAcosB=cos(A+B)+cos(A−B); 2sinA sinB=cos(A−B)−cos(A+B).

General solutions: sinθ=sinα ⇒ θ=nπ+(−1)^nα; cosθ=cosα ⇒ θ=2nπ±α; tanθ=tanα ⇒ θ=nπ+α, with n∈Z.

Inverse trig principal ranges: sin⁻¹:[−1,1]→[−π/2,π/2]; cos⁻¹:[−1,1]→[0,π]; tan⁻¹:R→(−π/2,π/2).

Fundamental principle: if one operation can be done in m ways and a second in n ways, the two operations in order can be done in mn ways.

Permutation and combination: nPr=n!/(n−r)!; nCr=n!/(r!(n−r)!); nPr=r! nCr; nCr=nC(n−r).

nC0=nCn=1; nC1=n; Pascal identity nCr+nC(r−1)=(n+1)Cr.

Permutations of n objects with repetition types p,q,… total n!/(p!q!…).

Circular permutations: n distinct objects around a circle give (n−1)! arrangements; if clockwise and anticlockwise are identical, divide by 2.

Selecting at least one object from n distinct objects gives 2^n−1 choices; including the empty selection gives 2^n.

Induction: verify P(1) or the first stated integer; assume P(k); prove P(k+1) from P(k).

Induction sum formulas: Σ_{k=1}^n k=n(n+1)/2; Σk²=n(n+1)(2n+1)/6; Σk³=[n(n+1)/2]².

Binomial theorem for positive integer n: (a+b)^n=Σ_{r=0}^n nCr a^{n−r}b^r.

General term: T_{r+1}=nCr a^{n−r}b^r; for a term independent of x, set the exponent of x equal to zero and solve for r.

Middle term: if n is even, one middle term is T_{n/2+1}; if n is odd, two middle terms are T_{(n+1)/2} and T_{(n+3)/2}.

Binomial coefficient sums: C0+C1+…+Cn=2^n; C0−C1+C2−…=0; C0+C2+…=C1+C3+…=2^{n−1}.

(1+x)^n=1+nx+[n(n−1)/2!]x²+… for |x|<1 and rational n, with nCr=n(n−1)…(n−r+1)/r!.

AP: t_n=a+(n−1)d; S_n=n/2[2a+(n−1)d]; a,b,c in AP ⇔ 2b=a+c.

GP: t_n=ar^{n−1}; S_n=a(1−r^n)/(1−r) for r≠1; S_∞=a/(1−r) for |r|<1; a,b,c in GP ⇔ b²=ac.

AM=(a+b)/2, GM=√(ab), HM=2ab/(a+b) for positive a,b; GM²=AM×HM and AM≥GM≥HM.

Special sums: Σn=n(n+1)/2; Σn²=n(n+1)(2n+1)/6; Σn³=[n(n+1)/2]².

Distance between (x₁,y₁) and (x₂,y₂) is √[(x₂−x₁)²+(y₂−y₁)²]; midpoint is ((x₁+x₂)/2,(y₁+y₂)/2).

Section formula for P dividing AB in ratio m:n: ((mx₂+nx₁)/(m+n),(my₂+ny₁)/(m+n)); centroid of triangle is average of vertices.

Slope m=(y₂−y₁)/(x₂−x₁); parallel lines have m₁=m₂; perpendicular lines satisfy m₁m₂=−1.

Line forms: y=mx+c; y−y₁=m(x−x₁); x/a+y/b=1; general ax+by+c=0 has slope −a/b.

Distance from (x₁,y₁) to ax+by+c=0 is |ax₁+by₁+c|/√(a²+b²); angle between lines: tanθ=|(m₁−m₂)/(1+m₁m₂)|.

Circle: (x−h)²+(y−k)²=r²; general x²+y²+2gx+2fy+c=0 has center (−g,−f) and radius √(g²+f²−c).

Line y=mx+c is tangent to x²+y²=r² exactly when c²=r²(1+m²); tangent at (x₁,y₁) is xx₁+yy₁=r².

Parabola y²=4ax: focus (a,0), directrix x=−a, latus rectum 4a; ellipse x²/a²+y²/b²=1 has eccentricity e=√(1−b²/a²) when a>b.

Pair of lines ax²+2hxy+by²=0: angle satisfies tanθ=|2√(h²−ab)/(a+b)|; perpendicular if a+b=0; coincident if h²=ab.

Matrix multiplication AB is defined when columns of A equal rows of B; generally AB≠BA.

Transpose and inverse: (Aᵀ)ᵀ=A; (AB)ᵀ=BᵀAᵀ; (AB)⁻¹=B⁻¹A⁻¹ when inverses exist.

Determinant: |a b; c d|=ad−bc; a 3×3 determinant can be expanded along any row or column using cofactors.

Properties: interchanging two rows changes sign; two identical rows give determinant 0; multiplying one row by k multiplies determinant by k.

A is singular if |A|=0; A⁻¹ exists iff |A|≠0; A⁻¹=(adj A)/|A| and A(adj A)=|A|I.

Adjoint is transpose of cofactor matrix; for 2×2 A=[[a,b],[c,d]], adj A=[[d,−b],[−c,a]].

Symmetry: Aᵀ=A symmetric; Aᵀ=−A skew-symmetric with zero diagonal; every square matrix is symmetric plus skew-symmetric.

Cramer's rule: for AX=B, Δ=|A|; x=Δx/Δ, y=Δy/Δ, z=Δz/Δ when Δ≠0.

Area of triangle with vertices (x₁,y₁),(x₂,y₂),(x₃,y₃) is ½|det[[x₁,y₁,1],[x₂,y₂,1],[x₃,y₃,1]]|.

For vectors a,b: a+b by triangle or parallelogram law; scalar multiplication changes magnitude by |λ| and reverses direction if λ<0.

Position vectors: if A,B have vectors a,b, then AB=b−a; point dividing AB in ratio m:n has vector (mb+na)/(m+n).

Dot product: a·b=|a||b|cosθ; a·b=0 for nonzero perpendicular vectors; |a|²=a·a; projection of a on b is (a·b)/|b|.

Cross product: a×b=|a||b|sinθ n̂; a×b is perpendicular to both a and b; a×b=0 when a and b are parallel; b×a=−a×b.

|a×b| is the area of the parallelogram with sides a,b; area of triangle with sides a,b is ½|a×b|.

Scalar triple product [a b c]=a·(b×c); its absolute value is the volume of the parallelepiped; [a b c]=0 iff a,b,c are coplanar.

Vector triple product: a×(b×c)=b(a·c)−c(a·b); generally a×(b×c)≠(a×b)×c.

Direction cosines l,m,n of a vector satisfy l²+m²+n²=1; unit vector along a is a/|a|; angle between vectors: cosθ=(a·b)/(|a||b|).

Line: r=a+tb; plane: r·n=d, where n is normal; vector equation of plane through a with normal n is (r−a)·n=0.

Limit exists iff left-hand and right-hand limits both exist and are equal; notation lim_{x→a}f(x)=L.

Algebra of limits: lim(f±g)=lim f±lim g; lim(fg)=lim f lim g; lim(f/g)=lim f/lim g if lim g≠0.

Standard trigonometric limits: lim_{x→0}sinx/x=1; lim_{x→0}tanx/x=1; lim_{x→0}(1−cosx)/x²=1/2.

Standard exponential/log limits: lim_{x→0}(e^x−1)/x=1; lim_{x→0}(a^x−1)/x=log_e a; lim_{x→0}log(1+x)/x=1.

Power limit: lim_{x→a}(x^n−a^n)/(x−a)=n a^{n−1}; also lim_{x→0}(1+x)^{1/x}=e and lim_{n→∞}(1+1/n)^n=e.

Indeterminate forms include 0/0, ∞/∞, 0×∞, ∞−∞, 1^∞, 0^0, ∞^0; factor, rationalize, use standard limits, or rewrite before evaluating.

f is continuous at a iff f(a) is defined, lim_{x→a}f(x) exists, and lim_{x→a}f(x)=f(a).

Polynomials, rational functions where denominator≠0, trigonometric, exponential and logarithmic functions are continuous on their domains; greatest integer function has jumps at integers.

Derivative at a: f′(a)=lim_{h→0}[f(a+h)−f(a)]/h, provided the limit exists; f′(x) is the instantaneous rate of change.

Differentiability at a requires the left derivative and right derivative to exist and be equal; differentiability implies continuity, but continuity does not imply differentiability.

Standard derivatives: d/dx x^n=nx^{n−1}; sinx→cosx; cosx→−sinx; tanx→sec²x; e^x→e^x; logx→1/x.

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

Inverse trig derivatives: (sin⁻¹x)′=1/√(1−x²); (cos⁻¹x)′=−1/√(1−x²); (tan⁻¹x)′=1/(1+x²); (cot⁻¹x)′=−1/(1+x²).

Implicit differentiation: differentiate both sides with respect to x and solve for dy/dx; for y=f(x)^{g(x)}, take logarithms first.

Parametric differentiation: if x=x(t), y=y(t), then dy/dx=(dy/dt)/(dx/dt), provided dx/dt≠0.

Higher derivatives: d²y/dx² is the derivative of dy/dx; notation y′, y″, y‴ or f′(x), f″(x).

Indefinite integrals: ∫x^n dx=x^{n+1}/(n+1)+C for n≠−1; ∫1/x dx=log|x|+C; ∫e^x dx=e^x+C.

Standard integrals: ∫sinx dx=−cosx+C; ∫cosx dx=sinx+C; ∫sec²x dx=tanx+C; ∫1/(1+x²) dx=tan⁻¹x+C; ∫1/√(1−x²) dx=sin⁻¹x+C.

Integration by substitution reverses the chain rule; choose u so that du appears in the integrand, then integrate with respect to u.

Integration by parts: ∫u dv=uv−∫v du; LIATE order for u: logarithmic, inverse trig, algebraic, trigonometric, exponential.

For rational functions, decompose into partial fractions before integrating; terms A/(x−a) integrate to A log|x−a|.

Definite integral: ∫_a^b f(x)dx=F(b)−F(a) for an antiderivative F of continuous f; it represents signed area.

Definite integral properties: ∫_a^a f=0; ∫_a^b f=−∫_b^a f; ∫_a^b f=∫_a^c f+∫_c^b f; ∫_0^a f(x)dx=∫_0^a f(a−x)dx.

Area under y=f(x) from x=a to b is ∫_a^b f(x)dx when f≥0; area between y=f(x) and y=g(x) is ∫_a^b |f−g|dx.

For a finite equally likely sample space S, P(A)=n(A)/n(S); probability satisfies 0≤P(A)≤1, P(S)=1, P(∅)=0.

Addition theorem: P(A∪B)=P(A)+P(B)−P(A∩B); if A and B are mutually exclusive, P(A∪B)=P(A)+P(B).

Complement: P(A′)=1−P(A); P(A∩B′)=P(A)−P(A∩B); P(A′∩B′)=1−P(A∪B).

Conditional probability: P(A|B)=P(A∩B)/P(B) for P(B)>0; multiplication rule P(A∩B)=P(A)P(B|A).

Events A and B are independent iff P(A∩B)=P(A)P(B); then P(A|B)=P(A) and P(B|A)=P(B).

Total probability: if B₁,…,B_n are mutually exclusive and exhaustive, P(A)=ΣP(B_i)P(A|B_i).

Bayes theorem: P(B_i|A)=P(B_i)P(A|B_i)/Σ_j P(B_j)P(A|B_j), for a partition B_i with P(B_i)>0.

Odds in favour of A are a:b, then P(A)=a/(a+b); odds against A are b:a; exhaustive mutually exclusive events have probabilities summing to 1.