PSEB Class 12 Maths cheat sheet

A relation R on A is reflexive if (a,a)∈R for every a∈A, symmetric if (a,b)∈R implies (b,a)∈R, and transitive if (a,b),(b,c)∈R implies (a,c)∈R; an equivalence relation has all three properties.

For finite sets |A|=m and |B|=n: the number of relations from A to B is 2^(mn), the number of functions is n^m, and the number of one-one functions (when m≤n) is nPm = n!/(n−m)!.

A function f:A→B is invertible iff it is one-one and onto; then f⁻¹:B→A, (f⁻¹)⁻¹=f, and (g∘f)⁻¹=f⁻¹∘g⁻¹ when the composite is defined.

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] except π/2, cosec⁻¹:(−∞,−1]∪[1,∞)→[−π/2,π/2] except 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, cosec⁻¹(−x)=−cosec⁻¹x.

Complementary pairs: sin⁻¹x+cos⁻¹x=π/2, tan⁻¹x+cot⁻¹x=π/2, sec⁻¹x+cosec⁻¹x=π/2 for x in the domains where both terms are defined.

tan⁻¹x+tan⁻¹y = tan⁻¹((x+y)/(1−xy)) when xy<1; if x>0,y>0 and xy>1, the sum equals π+tan⁻¹((x+y)/(1−xy)). Also 2tan⁻¹x=tan⁻¹(2x/(1−x²)) for |x|<1.

Matrix multiplication is associative and distributive but not commutative; (AB)ᵀ=BᵀAᵀ and (AB)⁻¹=B⁻¹A⁻¹ whenever the inverses exist.

For an n×n matrix A, det(AB)=det(A)det(B), det(Aᵀ)=det(A), det(kA)=kⁿdet(A), and det(adj A)=(det A)^(n−1).

A square matrix A is invertible iff det A≠0; then A⁻¹=(1/det A)adj A and A·adj A=adj A·A=(det A)I.

For A=[[a,b],[c,d]], det A=ad−bc and A⁻¹=(1/(ad−bc))[[d,−b],[−c,a]] when ad−bc≠0.

A is symmetric if Aᵀ=A and skew-symmetric if Aᵀ=−A; every square matrix decomposes uniquely as (A+Aᵀ)/2 + (A−Aᵀ)/2, a symmetric plus a skew-symmetric matrix.

The system AX=B has a unique solution X=A⁻¹B when det A≠0; if det A=0, it has either no solution or infinitely many solutions.

Area of a triangle with vertices (x₁,y₁), (x₂,y₂), (x₃,y₃) is ½|x₁(y₂−y₃)+x₂(y₃−y₁)+x₃(y₁−y₂)|, the absolute value of a 3×3 determinant.

Cramer's rule: for AX=B with det A≠0, xᵢ=det(Aᵢ)/det(A), where Aᵢ is A with column i replaced by B.

f is continuous at a if lim_{x→a}f(x)=f(a); it is differentiable at a if lim_{h→0}[f(a+h)−f(a)]/h exists.

Differentiability at a point implies continuity there; continuity does not imply differentiability, as f(x)=|x| at x=0 shows.

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

Lagrange's mean value theorem: if f is continuous on [a,b] and differentiable on (a,b), then f′(c)=(f(b)−f(a))/(b−a) for some c∈(a,b).

Standard inverse-trig derivatives: d/dx sin⁻¹x=1/√(1−x²), d/dx cos⁻¹x=−1/√(1−x²), d/dx tan⁻¹x=1/(1+x²), d/dx cot⁻¹x=−1/(1+x²), d/dx sec⁻¹x=1/(|x|√(x²−1)), d/dx cosec⁻¹x=−1/(|x|√(x²−1)).

Exponential and logarithmic derivatives: d/dx(aˣ)=aˣ ln a, d/dx(eˣ)=eˣ, d/dx(log_a x)=1/(x ln a), d/dx(ln|x|)=1/x for x≠0.

For u^v with u>0, logarithmic differentiation gives d/dx(u^v)=u^v(v′ ln u + v u′/u); for f=|x|, f′(x)=x/|x| for x≠0.

If f is differentiable and invertible with f′(x)≠0, then (f⁻¹)′(y)=1/f′(x) where y=f(x); the chain rule gives d/dx f(g(x))=f′(g(x))g′(x).

On an interval, f is increasing where f′(x)>0 and decreasing where f′(x)<0; f′(x)≥0 (or ≤0) gives non-decreasing (or non-increasing) behavior.

Critical points occur where f′(x)=0 or f′ is undefined; they are candidates for local extrema but need not be extrema.

First derivative test: at a critical point c, a change from + to − in f′ gives a local maximum, and a change from − to + gives a local minimum.

Second derivative test: if f′(c)=0 and f″(c)>0, then f has a local minimum at c; if f″(c)<0, it has a local maximum; if f″(c)=0, the test is inconclusive.

f is concave up where f″(x)>0 and concave down where f″(x)<0; an inflection point is a point where concavity changes.

On a closed interval [a,b], the absolute maximum and minimum of a continuous f occur at critical points in (a,b) or at the endpoints a and b.

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

If y=f(x), the differential approximation is dy=f′(x)dx and Δy≈f′(x)Δx; for motion with position s(t), velocity is s′(t) and acceleration is s″(t).

Standard integrals: ∫xⁿdx=xⁿ⁺¹/(n+1)+C for n≠−1, ∫(1/x)dx=ln|x|+C, ∫eˣdx=eˣ+C, ∫aˣdx=aˣ/ln a+C.

Trigonometric integrals: ∫sin x dx=−cos x+C, ∫cos x dx=sin x+C, ∫sec²x dx=tan x+C, ∫cosec²x dx=−cot x+C, ∫sec x tan x dx=sec x+C.

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

Integration by parts: ∫u dv=uv−∫v du; the ILATE order (inverse trig, logarithmic, algebraic, trigonometric, exponential) suggests the choice of u.

Fundamental theorem: if F′=f on [a,b], then ∫ₐᵇ f(x)dx=F(b)−F(a); the definite integral is independent of the constant of integration.

Definite integral properties: ∫ₐᵇf=−∫ᵦᵃf, ∫ₐᵇf=∫ₐᶜf+∫ᶜᵇf, and ∫₀ᵃf(x)dx=∫₀ᵃf(a−x)dx.

Symmetry: ∫₋ₐᵃf(x)dx=2∫₀ᵃf(x)dx if f is even, and equals 0 if f is odd.

For a rational integrand, decompose into partial fractions before integrating; for √(a²−x²), √(x²−a²), or √(a²+x²), a trigonometric substitution removes the radical.

The area bounded by y=f(x), the x-axis, and the lines x=a and x=b is ∫ₐᵇ|f(x)|dx; parts below the x-axis are counted positively because of the modulus.

If f(x)≥g(x) on [a,b], the area between the curves y=f(x) and y=g(x) from x=a to x=b is ∫ₐᵇ[f(x)−g(x)]dx.

If the curves cross inside [a,b], split the interval at each intersection and integrate (upper curve − lower curve) on each subinterval.

For a region described with x as a function of y, the area between x=right(y) and x=left(y) for c≤y≤d is ∫ᶜᵈ[right(y)−left(y)]dy.

The area of the circle x²+y²=a² is 4∫₀ᵃ√(a²−x²)dx=πa²; this uses symmetry and the standard integral ∫√(a²−x²)dx.

The area bounded by the parabola y²=4ax and its latus rectum x=a is 8a²/3, found by integrating horizontal strips between the line and the curve.

The area of the ellipse x²/a²+y²/b²=1 is πab, obtained by integrating 2b√(1−x²/a²) from −a to a or by symmetry.

When a region is symmetric about an axis, compute the area of one half and double it; this often simplifies integrals for circles, ellipses, and parabolas.

Order of a differential equation is the highest derivative present; degree is the exponent of that highest derivative after the equation is made polynomial in derivatives.

The general solution of an nth-order differential equation contains n arbitrary constants; a particular solution is obtained by using initial or boundary conditions.

Variable separable: dy/dx=f(x)g(y) gives ∫dy/g(y)=∫f(x)dx, provided g(y)≠0; constant solutions g(y)=0 may be lost and should be checked.

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

Linear first-order equation dy/dx+P(x)y=Q(x): the integrating factor is e^{∫P dx}, and the solution is y·IF=∫Q·IF dx+C.

Exact equation M dx+N dy=0 satisfies ∂M/∂y=∂N/∂x; its solution is ∫M dx + (terms of N not containing x) = C.

For dy/dx=f(ax+by+c), substitute u=ax+by+c to reduce the equation to a separable one in u and x.

Exponential growth/decay: dN/dt=kN gives N=N₀e^{kt}, where N₀ is the initial amount and k>0 models growth, k<0 decay.

Dot product: a·b=a₁b₁+a₂b₂+a₃b₃=|a||b|cosθ; a·b=0 for nonzero vectors iff a⊥b, and a·a=|a|².

Cross product: |a×b|=|a||b|sinθ; its direction is perpendicular to both a and b by the right-hand rule, and a×b=0 iff a and b are parallel.

Area formulas: the parallelogram with adjacent sides a,b has area |a×b|; the triangle with sides a,b has area ½|a×b|.

Scalar triple product [a b c]=a·(b×c) is the signed volume of the parallelepiped with edges a,b,c; [a b c]=0 iff the three vectors are coplanar.

Line in space: vector form r=a+λb; Cartesian form (x−x₁)/l=(y−y₁)/m=(z−z₁)/n, where (l,m,n) are direction ratios.

Plane: vector form r·n=d; Cartesian form ax+by+cz=d, with normal vector n=(a,b,c); the angle between planes uses their normals.

Angle between lines with direction vectors b₁,b₂: cosθ=|b₁·b₂|/(|b₁||b₂|); angle between planes with normals n₁,n₂: cosθ=|n₁·n₂|/(|n₁||n₂|).

Distance from point (x₁,y₁,z₁) to plane ax+by+cz=d is |ax₁+by₁+cz₁−d|/√(a²+b²+c²); shortest distance between skew lines is |(a₂−a₁)·(b₁×b₂)|/|b₁×b₂|.

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

The feasible region is the intersection of all constraint half-planes; it may be bounded, unbounded, or empty (infeasible).

The corner point method evaluates Z at every vertex of the feasible region; for a bounded region, the maximum and minimum occur at vertices.

The fundamental theorem of linear programming: if a linear objective has an optimum on a nonempty bounded feasible region, it occurs at a corner point.

To graph a constraint ax+by≤c, draw the boundary line ax+by=c and shade the side containing a test point that satisfies the inequality.

If the feasible region is unbounded, a maximum or minimum may not exist; compare the corner values with the behavior of Z along the unbounded edges.

A redundant constraint does not change the feasible region; a problem is infeasible if no point satisfies all constraints simultaneously.

If the objective is parallel to a boundary edge of the feasible region, the optimum may occur at every point of that edge, giving multiple optimal solutions.

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

Events A and B are independent iff P(A∩B)=P(A)P(B), equivalently P(A|B)=P(A) when P(B)>0.

Addition rule: P(A∪B)=P(A)+P(B)−P(A∩B); for mutually exclusive events, P(A∩B)=0 and P(A∪B)=P(A)+P(B).

Law of total probability: for a partition E₁,…,Eₙ with P(Eᵢ)>0, P(A)=ΣᵢP(Eᵢ)P(A|Eᵢ).

Bayes' theorem: P(Eᵢ|A)=P(Eᵢ)P(A|Eᵢ)/ΣⱼP(Eⱼ)P(A|Eⱼ), used to reverse conditional probabilities.

For a discrete random variable X with values xᵢ and probabilities pᵢ, mean μ=Σxᵢpᵢ and variance σ²=Σ(xᵢ−μ)²pᵢ=Σxᵢ²pᵢ−μ².

Binomial distribution: X~B(n,p), P(X=r)=C(n,r)p^r(1−p)^{n−r}, mean np, variance np(1−p), standard deviation √(np(1−p)).

Bernoulli trials require a fixed number n of independent trials, two outcomes (success/failure), and constant success probability p; the binomial counts successes.