CHSE Odisha Elective Mathematics flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Injective function

A function f: A→B is injective when f(x1)=f(x2) implies x1=x2, so distinct elements of the domain always have distinct images.

Bijective function

A function is bijective when it is both injective and surjective; then every element of the codomain has exactly one preimage, so an inverse function exists.

Composition of functions

(g∘f)(x)=g(f(x)); the composite is defined only when the range of f lies inside the domain of g, and composition is not generally commutative.

Principal value of sin⁻¹

The principal value of sin⁻¹ x lies in [−π/2, π/2]; it is the unique angle in that interval whose sine equals x, for −1≤x≤1.

Domain of sec⁻¹

sec⁻¹ x is defined for |x|≥1, and its principal range is [0, π] excluding π/2, because secant is undefined where cosine is zero.

Identity sin⁻¹x + cos⁻¹x

For every x in [−1,1], sin⁻¹x + cos⁻¹x = π/2; this follows because the principal angles are complementary within their respective ranges.

Determinant of product

For square matrices A and B of the same order, |AB|=|A||B|; the determinant of a product equals the product of the determinants.

Adjugate matrix

The adjugate adj(A) is the transpose of the cofactor matrix; it satisfies A(adj A)=(adj A)A=|A|I, and A⁻¹=adj(A)/|A| when |A|≠0.

Skew-symmetric matrix

A square matrix A is skew-symmetric when Aᵀ=−A; all diagonal entries are then zero, and every square matrix splits into symmetric plus skew-symmetric parts.

Modulus of complex number

For z=a+bi, the modulus is |z|=√(a²+b²); it is the distance from the origin to the point (a,b) and satisfies z z̄=|z|².

Argument of complex number

The argument θ of z=a+bi is the angle its vector makes with the positive real axis; tan θ=b/a, with quadrant chosen from signs of a and b.

Quadratic discriminant rule

For ax²+bx+c=0 with real coefficients and D=b²−4ac: D>0 gives distinct real roots, D=0 equal real roots, and D<0 a conjugate complex pair.

nCr recurrence

nCr + nC(r−1) = (n+1)Cr; this Pascal-type identity underlies binomial coefficients and lets combinations be built row by row.

General term of binomial

In (a+b)^n, the general term is T_(r+1)=nCr a^(n−r) b^r; setting r=0,1,…,n lists every term in the expansion.

Sum of infinite GP

For a geometric progression with first term a and ratio r, if |r|<1 then S∞=a/(1−r); if |r|≥1 the infinite sum does not converge.

Continuity at a point

f is continuous at a when f(a) is defined, lim_(x→a) f(x) exists, and the limit equals f(a); all three conditions must hold.

L'Hôpital's rule

For a 0/0 or ∞/∞ indeterminate form, lim f/g = lim f'/g' provided the latter limit exists; verify the form before differentiating.

Differentiability implies continuity

If f is differentiable at a, then f is continuous at a; the converse fails, as f(x)=|x| is continuous but not differentiable at 0.

Critical point

A point c in the domain of f where f'(c)=0 or f'(c) is undefined; critical points are candidates for local maxima, minima, or neither.

Second derivative test

At a stationary point c with f'(c)=0: f''(c)<0 gives a local maximum, f''(c)>0 gives a local minimum, and f''(c)=0 is inconclusive.

Fundamental theorem of calculus

If F is an antiderivative of continuous f on [a,b], then ∫_a^b f(x)dx=F(b)−F(a); it links definite integrals with antiderivatives.

Integration by parts

∫u dv = uv − ∫v du; choose u using the ILATE order-inverse trigonometric, logarithmic, algebraic, trigonometric, exponential-to simplify the new integral.

Integrating factor

For dy/dx + P(x)y = Q(x), the integrating factor is IF=e^(∫P dx); multiplying the equation by IF makes the left side the derivative of y·IF.

Order of differential equation

The order is the highest derivative appearing in the differential equation; the general solution of an order-n equation contains n arbitrary constants.

Dot product

a·b=|a||b|cos θ=a1b1+a2b2+a3b3; it is a scalar, and for nonzero vectors a·b=0 exactly when a and b are perpendicular.

Cross product

a×b is a vector perpendicular to both a and b with magnitude |a||b|sin θ; it is zero exactly when the nonzero vectors are parallel.

Scalar triple product

[a b c]=a·(b×c); its absolute value equals the volume of the parallelepiped with edges a,b,c, and it is zero when the vectors are coplanar.

Feasible region

The set of all points satisfying every constraint and non-negativity restriction in a linear programming problem; for linear constraints it is a convex polygon or polyhedron.

Corner point theorem

For a bounded linear programming problem with a nonempty feasible region, the optimal value of the objective occurs at a vertex of the feasible region.

Conditional probability

P(A|B)=P(A∩B)/P(B), defined when P(B)>0; it gives the probability of A after restricting the sample space to outcomes in B.

Binomial distribution mean

If X~B(n,p), then E(X)=np and Var(X)=np(1−p); for r=0,1,…,n the probability mass function is P(X=r)=nCr p^r(1−p)^(n−r).

Variance of discrete variable

For a discrete random variable X, Var(X)=Σ(x−μ)²P(x)=Σx²P(x)−μ², where μ=E(X); variance is nonnegative and its square root is standard deviation.