PSEB Class 12 Maths flashcards

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

Equivalence relation

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

Inverse function condition

A function f: A→B has an inverse f⁻¹: B→A exactly when f is both one-one (injective) and onto (surjective).

Principal value of sin⁻¹

The principal value of sin⁻¹x is the value lying in [-π/2, π/2]; for example sin⁻¹(1/2)=π/6 and sin⁻¹(-1/2)=-π/6.

tan⁻¹x + tan⁻¹y formula

Equals tan⁻¹((x+y)/(1-xy)) when xy<1; if x,y>0 and xy>1, add π to keep the angle in (-π/2, π/2).

Inverse of 2×2 matrix

For a 2×2 matrix A = [[a,b],[c,d]] with det A = ad−bc ≠ 0, A⁻¹ = (1/(ad−bc))[[d,−b],[−c,a]].

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.

Adjoint identity

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

Differentiability implies continuity

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

Derivative of tan⁻¹x

d/dx(tan⁻¹x) = 1/(1+x²) for all real x; this follows from the inverse-function derivative formula with tan y = x.

First derivative test

At a critical point c, f′ changes from positive to negative gives a local maximum; negative to positive gives a local minimum.

Concavity from second derivative

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

Absolute extrema on closed interval

The absolute maximum and minimum of a continuous f on [a,b] occur either at critical points inside (a,b) or at the endpoints a and b.

Integration by parts

∫u dv = uv − ∫v du; choose u according to ILATE (inverse trig, logarithmic, algebraic, trigonometric, exponential) to simplify the remaining integral.

Fundamental theorem of calculus

If F is an antiderivative of continuous f on [a,b], then ∫ₐᵇ f(x)dx = F(b) − F(a).

Integral of 1/(x²+a²)

∫ dx/(x²+a²) = (1/a)tan⁻¹(x/a) + C for a≠0; this standard form also gives ∫ dx/(a²+x²) with the same right side.

Area between two curves

If f(x)≥g(x) on [a,b], the area between y=f(x) and y=g(x) is ∫ₐᵇ [f(x)−g(x)] dx; split the interval where they cross.

Area using horizontal strips

When a region is bounded by x=right(y) and x=left(y) for c≤y≤d, its area is ∫ᶜᵈ [right(y)−left(y)] dy.

Area under a curve

The area between y=f(x) and the x-axis from x=a to x=b is ∫ₐᵇ |f(x)| dx; regions below the axis contribute positive area because of the modulus.

Order of a differential equation

The order is the highest derivative that appears in the equation; the degree is the power of that highest derivative after clearing radicals and fractions in derivatives.

Linear differential equation integrating factor

For dy/dx + P(x)y = Q(x), the integrating factor is e^{∫P dx}, and the solution is y·IF = ∫Q·IF dx + C.

Homogeneous differential equation substitution

A first-order equation dy/dx = F(x,y) is homogeneous if F(tx,ty)=F(x,y); substitute y=vx, so dy/dx = v + x dv/dx.

Dot product perpendicular condition

For nonzero vectors a and b, a·b = |a||b|cosθ; hence a·b=0 exactly when a and b are perpendicular (θ=90°).

Cross product area

|a×b| equals the area of the parallelogram with adjacent sides a and b; half of |a×b| is the area of the triangle formed by them.

Scalar triple product

The scalar triple product [a b c] = a·(b×c) gives the signed volume of the parallelepiped; it is zero exactly when a, b, c are coplanar.

Distance from point to plane

For a plane with Cartesian equation ax+by+cz=d, the perpendicular distance from point (x₁,y₁,z₁) is |ax₁+by₁+cz₁−d| / √(a²+b²+c²).

Corner point method

For a linear objective on a bounded feasible region, evaluate Z at every corner (vertex); the largest and smallest values are the maximum and minimum.

Feasible region

The set of all points satisfying every constraint and non-negativity condition of a linear programming problem; its vertices are candidates for the optimum.

Unbounded feasible region

If the feasible region is unbounded, a linear objective may still have an optimum at a vertex; compare the vertex value with the objective's behavior along unbounded edges.

Conditional probability formula

P(A|B) = P(A∩B)/P(B) for P(B)>0; it gives the probability of A when B is known to have occurred.

Bayes' theorem

For a partition Eᵢ, P(Eᵢ|A) = P(Eᵢ)P(A|Eᵢ) / Σⱼ P(Eⱼ)P(A|Eⱼ), updating prior probabilities after observing A.

Binomial distribution mean and variance

If X follows the binomial distribution B(n,p), then P(X=r)=C(n,r)pʳ(1−p)ⁿ⁻ʳ, the mean is np, and the variance is np(1−p).