BSEB Inter Mathematics flashcards

34 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 (one-one) if distinct elements of A map to distinct elements of B; equivalently f(a)=f(b) implies a=b.

Surjective function

A function f: A→B is surjective (onto) when every element of the codomain B is the image of at least one element of A, i.e., range(f)=B.

Equivalence relation

A relation on a set that is reflexive, symmetric, and transitive; it partitions the set into disjoint equivalence classes of related elements.

Principal value of sin⁻¹

The unique angle in [−π/2, π/2] whose sine equals the given value; sin⁻¹: [−1,1]→[−π/2,π/2] is the standard principal branch.

Principal value of cos⁻¹

The unique angle in [0, π] whose cosine equals the given value; cos⁻¹: [−1,1]→[0,π] is the standard principal branch.

tan⁻¹x + tan⁻¹y formula

Equals 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.

Skew-symmetric matrix

A square matrix A with Aᵀ = −A; all diagonal entries must be zero. Every square matrix is the sum of a symmetric and a skew-symmetric matrix.

Adjoint and inverse

adj(A) is the transpose of the cofactor matrix; A·adj(A)=det(A)·I, so A⁻¹=adj(A)/det(A) exists exactly when det(A)≠0.

Determinant product rule

For square matrices A and B of the same order, det(AB)=det(A)·det(B); also det(Aᵀ)=det(A) and det(kA)=kⁿdet(A) for n×n A.

Differentiability implies continuity

If f is differentiable at a point, then f is continuous there; the converse fails, as shown by f(x)=|x| at x=0.

Rolle's theorem

If f is continuous on [a,b], differentiable on (a,b), and f(a)=f(b), then some c in (a,b) satisfies f′(c)=0.

Mean value theorem

If f is continuous on [a,b] and differentiable on (a,b), there exists c in (a,b) with f′(c) = (f(b)−f(a))/(b−a).

Second derivative test

At a critical point c with f′(c)=0: if f″(c)<0, c gives a local maximum; if f″(c)>0, a local minimum; if f″(c)=0, the test is inconclusive.

Tangent line equation

The tangent to y=f(x) at x₀ is y − f(x₀) = f′(x₀)(x − x₀); the normal there has slope −1/f′(x₀) when f′(x₀)≠0.

Integration by parts

∫u dv = uv − ∫v du. Choose u by LIATE order: logarithmic, inverse trigonometric, algebraic, trigonometric, exponential functions.

Fundamental theorem of calculus

If F′(x)=f(x), then ∫ₐᵇ f(x)dx = F(b) − F(a); this links definite integrals to antiderivatives.

Even and odd integrals

For f continuous on [−a,a]: ∫₋ₐᵃ f(x)dx = 0 if f is odd, and = 2∫₀ᵃ f(x)dx if f is even.

Area between two curves

For continuous f and g on [a,b], the area between their graphs is ∫ₐᵇ |f(x)−g(x)| dx; split the interval where the curves cross.

Area of an ellipse

The region x²/a² + y²/b² ≤ 1 has area πab; for a circle of radius r (a=b=r) this reduces to πr².

Order and degree of ODE

Order is the highest derivative appearing; degree is the exponent of that highest derivative when the equation is polynomial in all derivatives.

Integrating factor for linear ODE

For dy/dx + Py = Q, the integrating factor is e^∫P dx, and the solution is y·IF = ∫Q·IF dx + C.

Homogeneous differential equation

An equation dy/dx = F(y/x); substitute y=vx so dy/dx = v + x dv/dx, reducing it to a separable equation in v and x.

Scalar triple product

For vectors a, b, c, the scalar triple product a·(b×c) equals the determinant of their components and gives the signed volume of the parallelepiped.

Angle between two planes

For planes with normals n₁ and n₂, cos θ = (n₁·n₂)/(|n₁||n₂|); parallel planes have proportional normals, perpendicular planes have n₁·n₂=0.

Shortest distance between skew lines

For r=a₁+λb₁ and r=a₂+μb₂, the shortest distance is |(a₂−a₁)·(b₁×b₂)| / |b₁×b₂|, provided b₁×b₂≠0.

Corner point theorem

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

Feasible region in LPP

The set of points satisfying all constraints and non-negativity conditions; it is a convex polygon (possibly unbounded) whose vertices are corner points.

Conditional probability

P(A|B) = P(A∩B)/P(B) for P(B)>0; it gives the probability of A given that B has occurred, restricted to the sample space B.

Bayes' theorem

For a partition {Eᵢ}, P(Eᵢ|A) = P(Eᵢ)P(A|Eᵢ) / Σⱼ P(Eⱼ)P(A|Eⱼ), reversing conditional probabilities using prior information.

Binomial distribution mean and variance

For X~B(n,p), P(X=r)=C(n,r)pʳ(1−p)^{n−r}; the mean is np and the variance is np(1−p).

Quadratic roots relations

For ax²+bx+c=0 with roots α,β: α+β=−b/a and αβ=c/a; the discriminant b²−4ac decides whether roots are real, equal, or complex.

Binomial theorem general term

In (a+b)ⁿ, the term with bʳ is T_{r+1}=C(n,r)a^{n−r}bʳ; summing over r=0 to n gives the full expansion.

Standard limit sin x over x

As x→0 (x in radians), sin x/x → 1; this underlies the derivatives of sine and cosine and many trigonometric limits.

L'Hôpital's rule

For a 0/0 or ∞/∞ indeterminate form, lim f(x)/g(x) = lim f′(x)/g′(x) when the latter limit exists and g′≠0 near the point.