CHSE Odisha Elective Mathematics cheat sheet

A relation R from A to B is a subset of A×B; its domain is the set of first coordinates and its range is the set of second coordinates, while B is the codomain.

A function f: A→B assigns to each x in A exactly one y in B; equal functions must have the same domain, the same codomain, and equal values at every point.

Injective means f(x1)=f(x2) implies x1=x2; surjective means range equals codomain; bijective means both, and then the inverse f⁻¹: B→A exists.

Composition is (g∘f)(x)=g(f(x)), defined only when range(f) is contained in domain(g); composition is generally not commutative.

For a bijection f, (f⁻¹∘f)(x)=x and (f∘f⁻¹)(y)=y; the graph of f⁻¹ is the reflection of y=f(x) across the line y=x.

If A has m elements and B has n elements, 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.

Even function: f(−x)=f(x), graph symmetric about the y-axis; odd function: f(−x)=−f(x), graph symmetric about the origin.

Greatest integer function satisfies [x]≤x<[x]+1; fractional part is {x}=x−[x] with 0≤{x}<1; signum returns −1, 0, or 1 according as x<0, x=0, or x>0.

If f: A→B and g: B→C are bijections, then (g∘f)⁻¹=f⁻¹∘g⁻¹.

sin⁻¹: [−1,1]→[−π/2,π/2], cos⁻¹: [−1,1]→[0,π], tan⁻¹: R→(−π/2,π/2).

cosec⁻¹: |x|≥1→[−π/2,π/2] excluding 0; sec⁻¹: |x|≥1→[0,π] excluding π/2; cot⁻¹: R→(0,π).

sin⁻¹(−x)=−sin⁻¹x, tan⁻¹(−x)=−tan⁻¹x, cosec⁻¹(−x)=−cosec⁻¹x.

cos⁻¹(−x)=π−cos⁻¹x, sec⁻¹(−x)=π−sec⁻¹x, cot⁻¹(−x)=π−cot⁻¹x.

sin⁻¹x+cos⁻¹x=π/2; tan⁻¹x+cot⁻¹x=π/2; cosec⁻¹x+sec⁻¹x=π/2, for x in the relevant domains.

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

2tan⁻¹x=tan⁻¹(2x/(1−x²)) for |x|<1; sin⁻¹x=tan⁻¹(x/√(1−x²)) for |x|<1.

sin(sin⁻¹x)=x for |x|≤1, but sin⁻¹(sin θ)=θ only when θ∈[−π/2,π/2].

Principal value means the value in the stated principal branch; check the branch before simplifying any inverse trigonometric expression.

Matrix addition requires the same order; AB is defined only when columns(A)=rows(B); in general AB≠BA.

Transpose rules: (Aᵀ)ᵀ=A, (AB)ᵀ=BᵀAᵀ, (A+B)ᵀ=Aᵀ+Bᵀ.

Symmetric: Aᵀ=A; skew-symmetric: Aᵀ=−A; every square matrix is the sum of a symmetric part and a skew-symmetric part.

For n×n matrices, |AB|=|A||B|, |Aᵀ|=|A|, |kA|=kⁿ|A|, and |A⁻¹|=1/|A| when A is invertible.

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

AX=B has the unique solution X=A⁻¹B when |A|≠0; when |A|=0 the system has no solution or infinitely many solutions.

For 2×2, |[a b; c d]|=ad−bc and inverse is (1/(ad−bc))[d −b; −c a].

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

Adding a multiple of one row to another leaves a determinant unchanged; swapping two rows changes its sign.

i²=−1; powers of i cycle i, −1, −i, 1 with period 4.

For z=a+bi, conjugate z̄=a−bi and modulus |z|=√(a²+b²); z z̄=|z|².

Division: multiply numerator and denominator by the conjugate of the denominator, then simplify using i²=−1.

Polar form z=r(cos θ+i sin θ), r=|z|, θ=arg z; products multiply moduli and add arguments; quotients divide moduli and subtract arguments.

De Moivre: (cos θ+i sin θ)ⁿ=cos nθ+i sin nθ; the n roots of zⁿ=w lie equally spaced on a circle.

Triangle and product rules: |z1+z2|≤|z1|+|z2|, |z1z2|=|z1||z2|, |z1/z2|=|z1|/|z2|.

Quadratic ax²+bx+c=0 has roots (−b±√(b²−4ac))/(2a); sum of roots=−b/a; product=c/a.

Discriminant D=b²−4ac: D>0 distinct real roots; D=0 equal real roots; D<0 conjugate complex pair when coefficients are real.

If α and β are roots, the equation is x²−(α+β)x+αβ=0.

nPr=n!/(n−r)!; nCr=n!/(r!(n−r)!); nCr=nC(n−r); nCr+nC(r−1)=n+1Cr; 0!=1.

Arrangements of n objects with p identical of one type and q of another: n!/(p!q!…).

Number of subsets of an n-element set is 2ⁿ; number of nonempty subsets is 2ⁿ−1.

Circular permutations of n distinct objects: (n−1)!; necklace arrangements with clockwise and anticlockwise identical: (n−1)!/2 for n≥3.

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

For even n there is one middle term, T_(n/2+1); for odd n there are two middle terms, T_((n+1)/2) and T_((n+3)/2).

AP: a_n=a+(n−1)d; S_n=n/2[2a+(n−1)d]=n/2(a+l).

GP: a_n=ar^(n−1); S_n=a(1−rⁿ)/(1−r) for r≠1; S∞=a/(1−r) for |r|<1.

If a sequence has a constant difference it is arithmetic; if consecutive terms have a constant ratio it is geometric.

lim_(x→a) f(x)=L exists iff both the left-hand and right-hand limits exist and equal L.

Standard trigonometric limits: lim_(x→0) sin x/x=1, lim_(x→0) tan x/x=1, lim_(x→0) (1−cos x)/x²=1/2.

Standard exponential and logarithmic limits: lim_(x→0)(eˣ−1)/x=1, lim_(x→0) ln(1+x)/x=1, lim_(x→0)(1+x)^(1/x)=e.

Indeterminate forms 0/0, ∞/∞, 0·∞, ∞−∞, 1^∞, 0^0, ∞^0 require factoring, rationalizing, standard limits, or L'Hôpital.

L'Hôpital: for 0/0 or ∞/∞, lim f/g=lim f'/g' provided the latter limit exists.

f is continuous at a iff f(a) is defined, lim_(x→a) f(x) exists, and the limit equals f(a).

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

f'(a)=lim_(h→0)[f(a+h)−f(a)]/h; for the derivative to exist, left and right derivatives must be equal.

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(eˣ)=eˣ, d/dx(ln x)=1/x; also (uv)'=u'v+uv', (u/v)'=(u'v−uv')/v², and dy/dx=dy/du·du/dx.

f is increasing on an interval when f'(x)≥0 there, and strictly increasing when f'(x)>0; decreasing is characterized by f'(x)≤0.

A critical point c satisfies f'(c)=0 or f'(c) is undefined; critical points and endpoints are candidates for extrema.

First derivative test: a sign change from + to − at c gives a local maximum; a sign change from − to + gives a local minimum.

Second derivative test at f'(c)=0: f''(c)<0 gives a local maximum; f''(c)>0 gives a local minimum; f''(c)=0 is inconclusive.

Concave up where f''>0; concave down where f''<0; an inflection point is where concavity changes.

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

If s(t) is position, then velocity v(t)=s'(t) and acceleration a(t)=s''(t).

Absolute extrema on [a,b]: evaluate f at all critical points in (a,b) and at a and b; the largest and smallest values are the absolute maximum and minimum.

Differentials: dy=f'(x)dx, and the approximate change is Δy≈f'(x)Δx.

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

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

∫sec²x dx=tan x+C; ∫cosec²x dx=−cot x+C; ∫1/(1+x²)dx=tan⁻¹x+C; ∫1/√(1−x²)dx=sin⁻¹x+C.

Fundamental theorem: ∫_a^b f(x)dx=F(b)−F(a), where F is an antiderivative of f.

Properties: ∫_a^b f=−∫_b^a f; ∫_a^a f=0; ∫_a^b f=∫_a^c f+∫_c^b f.

∫_0^a f(x)dx=∫0^a f(a−x)dx; ∫(−a)^a f=0 if f is odd and 2∫_0^a f if f is even.

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

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

Volume of revolution about the x-axis: V=π∫_a^b y²dx; area using the y-axis is ∫_c^d x dy.

Order is the highest derivative present; degree is the exponent of the highest-order derivative after clearing radicals and fractions in derivatives.

The general solution of an order-n differential equation contains n arbitrary constants; initial conditions determine a particular solution.

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

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

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

Exact equation M dx+N dy=0 satisfies ∂M/∂y=∂N/∂x; its solution comes from a potential function whose partial derivatives are M and N.

dy/dx=ky gives y=Ce^(kx), representing exponential growth when k>0 and decay when k<0.

Differentiating a family y=f(x,C) once and eliminating C produces the corresponding first-order differential equation.

A slope field draws at each point (x,y) a short segment whose slope equals dy/dx at that point.

Vector a=a1i+a2j+a3k has magnitude |a|=√(a1²+a2²+a3²); its unit vector is â=a/|a|.

Dot product: a·b=|a||b|cos θ=a1b1+a2b2+a3b3; for nonzero vectors, a·b=0 iff a and b are perpendicular.

Cross product: |a×b|=|a||b|sin θ; a×b is perpendicular to both a and b, and a×b=0 iff the nonzero vectors are parallel.

a×b is given by the determinant with rows i, j, k and the components of a and b; b×a=−a×b.

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

Projection of a on b is (a·b)/|b|; the vector projection is ((a·b)/|b|²)b.

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

Plane: r·n=d or ax+by+cz=d; plane through point a with normal n satisfies (r−a)·n=0.

Angle between vectors: cos θ=(a·b)/(|a||b|); distance from (x1,y1,z1) to ax+by+cz=d is |ax1+by1+cz1−d|/√(a²+b²+c²).

A linear programming problem optimizes a linear objective Z=ax+by subject to linear inequalities and non-negativity restrictions.

The feasible region is the set of points satisfying all constraints; for linear constraints it is convex.

Graphical method: plot each constraint, identify the feasible region, then evaluate the objective at every vertex.

Corner point theorem: for a bounded problem with nonempty feasible region, the optimal objective value occurs at a vertex, possibly along a whole edge.

An unbounded feasible region can still give a finite optimum; check how the objective behaves along unbounded directions.

A slack variable converts a ≤ constraint into an equality; a surplus variable converts a ≥ constraint; artificial variables are used to start the simplex method.

The problem is infeasible when no point satisfies all constraints, and unbounded when the objective can increase or decrease without limit.

If variables must be integers, a vertex optimum may be invalid; examine integer points inside the feasible region.

For equally likely finite outcomes, P(E)=n(E)/n(S); 0≤P(E)≤1 and P(E')=1−P(E).

Addition rule: P(A∪B)=P(A)+P(B)−P(A∩B); mutually exclusive events satisfy P(A∪B)=P(A)+P(B).

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

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

For discrete X, mean μ=ΣxP(x) and variance σ²=Σ(x−μ)²P(x)=Σx²P(x)−μ².

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

Grouped data: mean=Σfixi/Σfi; the median class is where cumulative frequency reaches N/2.

Standard deviation is √variance; adding a constant shifts the mean but not the SD, while multiplying by k multiplies the SD by |k|.

Correlation coefficient r lies in [−1,1]; the regression line of y on x is y−ȳ=r(σy/σx)(x−x̄).