Kerala Plus Two Mathematics (DHSE) 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}; codomain is all of B.

R is a function if every element of the domain has exactly one image; equal domains and equal images at each point means two functions are equal.

One-one (injective): f(x₁)=f(x₂) ⇒ x₁=x₂. Onto (surjective): range = codomain. Bijective = both, hence invertible.

For finite sets, |A|=m, |B|=n: number of relations = 2^(mn); number of functions = n^m; injective functions exist only if m ≤ n.

Composition: (g∘f)(x)=g(f(x)); composition is associative but generally not commutative; (f⁻¹)⁻¹=f and (g∘f)⁻¹=f⁻¹∘g⁻¹.

f is invertible iff it is bijective; then f⁻¹(y)=x ⇔ f(x)=y, and the graph of f⁻¹ is the reflection of the graph of f in y=x.

Equivalence relation needs reflexivity, symmetry and transitivity; a partition of a set corresponds exactly to the equivalence classes of some equivalence relation.

Standard real functions: identity, constant, polynomial, rational, modulus |x|, signum, greatest integer [x], and their domains and ranges.

Binary operation * on A is commutative if ab=ba and associative if (ab)c=a(bc); identity e satisfies ae=ea=a; inverse b satisfies ab=ba=e.

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

Domains: sin⁻¹, cos⁻¹, tan⁻¹, cot⁻¹ defined on ℝ for tan⁻¹/cot⁻¹ and on [−1,1] for sin⁻¹, cos⁻¹; sec⁻¹ and cosec⁻¹ need |x| ≥ 1.

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

Complementary identities: sin⁻¹x+cos⁻¹x=π/2; tan⁻¹x+cot⁻¹x=π/2; sec⁻¹x+cosec⁻¹x=π/2, on respective domains.

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

Addition formula: 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.

Double-angle forms: 2tan⁻¹x=tan⁻¹(2x/(1−x²)) for |x|<1; also 2tan⁻¹x=sin⁻¹(2x/(1+x²))=cos⁻¹((1−x²)/(1+x²)) for x≥0.

Substitution shortcuts: for a>0, sin⁻¹x = tan⁻¹(x/√(a²−x²)) and cos⁻¹x = tan⁻¹(√(a²−x²)/x), valid on the appropriate principal branch.

Inverse trig functions are continuous on their domains; sin⁻¹ and tan⁻¹ are strictly increasing, cos⁻¹ and cot⁻¹ strictly decreasing.

Matrix addition requires equal orders; multiplication AB is defined when columns of A equal rows of B, and then (AB)ᵀ=BᵀAᵀ.

A is symmetric if Aᵀ=A, skew-symmetric if Aᵀ=−A; every square matrix is the sum of a symmetric and a skew-symmetric part: A=½(A+Aᵀ)+½(A−Aᵀ).

Determinant properties: |Aᵀ|=|A|; swapping two rows changes sign; multiplying one row by k multiplies det by k; equal rows give det 0; |AB|=|A||B|.

For order n: |kA|=kⁿ|A|; |adj A|=|A|^(n−1); A(adj A)=(adj A)A=|A|I.

A is invertible iff |A|≠0; then A⁻¹=adj A/|A|. For 2×2, A⁻¹=(1/(ad−bc))[[d,−b],[−c,a]].

Cramer/system form: AX=B has unique solution X=A⁻¹B when |A|≠0; no solution or infinitely many when |A|=0.

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

Singular matrix has |A|=0 and no inverse; non-singular has |A|≠0. Elementary row operations do not change the solution set of AX=B.

Consistency: for AX=B with |A|=0, compare ranks; if rank(A)=rank([A|B]) system is consistent, else inconsistent.

f is continuous at a if lim(x→a)f(x)=f(a); equivalently left limit = right limit = value. Differentiability at a implies continuity, not conversely.

Standard derivatives: (xⁿ)′=nxⁿ⁻¹; (sin x)′=cos x; (cos x)′=−sin x; (tan x)′=sec²x; (eˣ)′=eˣ; (ln x)′=1/x; (aˣ)′=aˣ ln a.

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); (sec⁻¹x)′=1/(|x|√(x²−1)).

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

Lagrange MVT: same hypotheses ⇒ ∃c∈(a,b) with f′(c)=(f(b)−f(a))/(b−a).

Logarithmic differentiation: take ln of both sides when the function is a product/quotient of powers or has variable exponent (e.g. xˣ).

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

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

f is strictly increasing on an interval if f′(x)>0 there, strictly decreasing if f′(x)<0; f′(x)=0 at stationary points.

Rate of change: if y=f(x) and x changes at rate dx/dt, then dy/dt=f′(x)·dx/dt (related rates).

Equation of tangent at (x₁,y₁) on y=f(x): y−y₁=f′(x₁)(x−x₁); normal slope = −1/f′(x₁).

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

First derivative test: sign change + to − gives local maximum; − to + gives local minimum; no sign change gives a point of inflection.

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

For a rectangle of fixed perimeter, area is maximum when it is a square; for fixed area, perimeter is minimum for a square.

Optimisation procedure: express the quantity as one-variable function, differentiate, solve f′=0, verify maximum/minimum by second derivative or sign test.

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

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

Integration by parts: ∫u v′dx=uv−∫u′v dx; choose u by the ILATE order (Inverse trig, Log, Algebraic, Trig, Exponential).

Substitution: ∫f(g(x))g′(x)dx=∫f(u)du with u=g(x); for definite integrals also change the limits.

Properties: ∫ₐᵇf=∫ₐᶜf+∫꜀ᵇf; ∫ₐᵇf=∫ₐᵇf(a+b−x)dx; ∫₀ᵃf(x)dx=∫₀ᵃf(a−x)dx; ∫₋ₐᵃf=0 if f is odd, 2∫₀ᵃf if f is even.

Area under y=f(x) from a to b (f≥0) is ∫ₐᵇf(x)dx; between two curves use ∫ₐᵇ|f(x)−g(x)|dx.

Area bounded by x=f(y) with the y-axis between y=c and y=d is ∫꜀ᵈf(y)dy.

Definite integral as limit of sum: ∫ₐᵇf(x)dx=lim(n→∞)h·Σf(a+rh), h=(b−a)/n, r=1 to n.

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

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

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

Homogeneous equation dy/dx=F(y/x): substitute y=vx, giving x dv/dx=F(v)−v, then separate.

Linear equation dy/dx+Py=Q has integrating factor IF=e^(∫P dx) and solution y·IF=∫Q·IF dx.

Exact form M dx+N dy=0 with ∂M/∂y=∂N/∂x has solution ∫M dx+∫(terms of N free of x)dy=C.

Formation: differentiate the family of curves as many times as the number of arbitrary constants and eliminate them.

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

Position vector of point dividing AB in ratio m:n internally is (m b⃗+n a⃗)/(m+n); externally (m b⃗−n a⃗)/(m−n).

Dot product a⃗·b⃗=|a⃗||b⃗|cos θ; perpendicular iff a⃗·b⃗=0; projection of a⃗ on b⃗ is (a⃗·b⃗)/|b⃗|.

Cross product |a⃗×b⃗|=|a⃗||b⃗|sin θ, direction by right-hand rule; parallel iff a⃗×b⃗=0⃗; area of triangle = ½|a⃗×b⃗|.

Scalar triple product [a⃗ b⃗ c⃗]=a⃗·(b⃗×c⃗) gives volume of parallelepiped; coplanar iff it is zero.

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

Line through a⃗ with direction b⃗: r⃗=a⃗+λb⃗; in Cartesian (x−x₁)/l=(y−y₁)/m=(z−z₁)/n.

Plane: r⃗·n̂=d with n̂ unit normal; Cartesian ax+by+cz=d, where (a,b,c) is a normal vector.

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

Angle between line and plane: sin θ=|b⃗·n̂|/(|b⃗|); between two planes use the angle between their normals.

An LPP is: optimise (maximise/minimise) Z=cx+dy subject to linear constraints and x,y≥0 (non-negativity).

Feasible region is the intersection of the half-planes defined by the constraints; it may be bounded or unbounded.

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

Graphical method: draw each constraint as a line, shade the feasible side, find vertices, evaluate Z at every vertex.

For an unbounded feasible region, if Z has a maximum it still occurs at a vertex; check whether the region is actually unbounded in the improving direction.

Bounded feasible region ⇒ both maximum and minimum exist; open/unbounded may give no finite optimum.

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

Constraints are of form ax+by ≤ c or ≥ c; the test point (0,0) quickly identifies which side to shade when c≠0.

P(A)=n(A)/n(S) for equally likely outcomes; 0≤P(A)≤1; P(A′)=1−P(A).

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

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

Independent events: P(A∩B)=P(A)P(B), equivalently P(A|B)=P(A).

Bayes' theorem: P(Eᵢ|A)=P(Eᵢ)P(A|Eᵢ)/ΣP(Eⱼ)P(A|Eⱼ), where Eᵢ partition the sample space.

Total probability: P(A)=ΣP(Eⱼ)P(A|Eⱼ) over a partition {Eⱼ}.

Binomial distribution B(n,p): P(X=r)=ⁿCᵣ pʳ(1−p)ⁿ⁻ʳ; mean=np, variance=np(1−p).

Random variable X has mean μ=Σxᵢpᵢ and variance σ²=Σxᵢ²pᵢ−μ²; Σpᵢ=1.

Statement is a sentence that is either true or false but not both; questions, orders and exclamations are not statements.

Negation of 'p and q' is 'not p or not q'; negation of 'p or q' is 'not p and not q' (De Morgan).

Implication p⇒q is false only when p is true and q is false; its contrapositive ~q⇒~p is equivalent to it.

p⇒q is equivalent to ~p∨q; converse is q⇒p; inverse is ~p⇒~q; only contrapositive matches the original.

'For all x, P(x)' negates to 'there exists x such that not P(x)'; 'there exists' negates to 'for all ... not'.

Mean of ungrouped data x̄=Σxᵢ/n; for grouped data x̄=Σfᵢxᵢ/Σfᵢ with class marks as xᵢ.

Variance σ²=Σ(xᵢ−x̄)²/n=Σxᵢ²/n−x̄²; standard deviation σ=√variance; adding a constant to all observations leaves variance unchanged.

Coefficient of variation CV=(σ/x̄)×100, used to compare variability of two series; larger CV means more variable.

Set laws: A∪(B∩C)=(A∪B)∩(A∪C); (A∪B)′=A′∩B′; n(A∪B)=n(A)+n(B)−n(A∩B).

Power set P(A) has 2ⁿ elements if n(A)=n; number of subsets of size r is ⁿCᵣ.

Permutations: ⁿPᵣ=n!/(n−r)!; arrangements of n objects with repetitions n₁,n₂,… give n!/(n₁!n₂!…).

Combinations: ⁿCᵣ=n!/(r!(n−r)!); ⁿCᵣ=ⁿCₙ₋ᵣ; ⁿCᵣ+ⁿCᵣ₋₁=ⁿ⁺¹Cᵣ.

Circular arrangements of n distinct objects: (n−1)!; necklace (clockwise=anticlockwise) gives (n−1)!/2.

Binomial theorem: (a+b)ⁿ=ΣⁿCᵣ aⁿ⁻ʳbʳ; general term Tᵣ₊₁=ⁿCᵣ aⁿ⁻ʳbʳ.

Middle term: if n even, single middle term is Tₙ/₂₊₁; if n odd, two middle terms T₍ₙ₊₁₎/₂ and T₍ₙ₊₃₎/₂.

AP: aₙ=a+(n−1)d, Sₙ=n/2[2a+(n−1)d]; GP: aₙ=arⁿ⁻¹, Sₙ=a(rⁿ−1)/(r−1), S∞=a/(1−r) for |r|<1.

Arithmetic mean of a and b is (a+b)/2; geometric mean is √(ab); AM≥GM for positive a,b.