BSEH Class 12 Maths flashcards

33 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 simultaneously. It partitions the set into disjoint equivalence classes, and every element belongs to exactly one such class.

Bijective function

A function that is both one-one and onto, so each element of the codomain has exactly one preimage. Only bijections possess an inverse function defined on the whole codomain.

Inverse of composition

For bijections f and g whose composition is defined, the inverse of g∘f is f⁻¹∘g⁻¹. The order reverses, mirroring the rule (AB)⁻¹ = B⁻¹A⁻¹ for matrices.

Principal value of sin⁻¹

The value of sin⁻¹x lying in [−π/2, π/2]. For example sin⁻¹(1/2) = π/6, not 5π/6, because the principal branch restricts the output range.

Complementary inverse identity

For x in the common domain, sin⁻¹x + cos⁻¹x = π/2, tan⁻¹x + cot⁻¹x = π/2 and sec⁻¹x + cosec⁻¹x = π/2. These follow from cofunction relationships of the trigonometric ratios.

tan⁻¹ sum formula

tan⁻¹x + tan⁻¹y = tan⁻¹((x+y)/(1−xy)) when xy < 1. If xy > 1 with x, y positive, add π to the right side to stay in the principal range.

Skew-symmetric matrix

A square matrix satisfying Aᵀ = −A. All diagonal entries must be zero, and every square matrix can be written as the sum of a symmetric and a skew-symmetric matrix.

Matrix product non-commutativity

Generally AB ≠ BA even when both products are defined and square. Consequently (A+B)² = A² + AB + BA + B², not A² + 2AB + B², unless A and B commute.

Transpose of a product

For conformable matrices A and B, (AB)ᵀ = BᵀAᵀ. The order of multiplication reverses, exactly as it does for inverses of products.

Determinant under row swap

Interchanging any two rows or two columns of a square matrix multiplies its determinant by −1. A matrix with two identical rows therefore has determinant zero.

Adjoint and inverse

For a square matrix A with |A| ≠ 0, A(adj A) = |A|I, so A⁻¹ = (adj A)/|A|. The adjoint is the transpose of the cofactor matrix.

Triangle area by determinant

The area of a triangle with vertices (x₁,y₁), (x₂,y₂), (x₃,y₃) is half the absolute value of x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂). A zero value means the points are collinear.

Differentiability implies continuity

If f is differentiable at a point then it is continuous there. The converse fails: f(x) = |x| is continuous at 0 but has no derivative at 0.

Rolle's theorem

If f is continuous on [a,b], differentiable on (a,b), and f(a) = f(b), then f′(c) = 0 for some c in (a,b). A horizontal tangent exists somewhere between equal endpoint values.

Derivative of tan⁻¹x

d/dx(tan⁻¹x) = 1/(1+x²) for all real x. The related derivative of cot⁻¹x is −1/(1+x²), differing only in sign because the two functions are complementary.

Second derivative test

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

Absolute extrema on closed interval

To find absolute maximum and minimum of a continuous f on [a,b], evaluate f at all critical points inside (a,b) and at both endpoints, then compare those values.

Normal line slope

At a point where the tangent slope is m = f′(x₁) and m ≠ 0, the normal slope is −1/m. If m = 0 the tangent is horizontal and the normal is vertical.

Integration by parts

∫u v′ dx = uv − ∫u′v dx. Choose u following ILATE order: inverse trigonometric, logarithmic, algebraic, trigonometric, exponential. Useful when the integrand is a product of unlike functions.

Integral of 1/(x²+a²)

∫dx/(x² + a²) = (1/a)tan⁻¹(x/a) + C. The a² in the denominator produces the 1/a factor, obtained by substituting x = a tan θ.

Symmetry property of definite integrals

∫₀ᵃ f(x) dx = ∫₀ᵃ f(a − x) dx. Applying this often turns an awkward integrand into a standard one, especially for trigonometric fractions over [0, π/2].

Area between two curves

If f(x) ≥ g(x) on [a,b], the enclosed area is ∫ₐᵇ [f(x) − g(x)] dx. The limits a and b are usually the x-coordinates of the intersection points.

Area of an ellipse

The region x²/a² + y²/b² ≤ 1 has area πab. For a circle of radius a, where b = a, this reduces to the familiar πa².

Order and degree

Order is the highest derivative appearing in a differential equation. Degree is the exponent of that highest derivative after the equation is cleared of radicals and fractions in derivatives.

Integrating factor for linear ODE

For dy/dx + Py = Q, the integrating factor is e^(∫P dx). Multiplying through makes the left side the derivative of y times the integrating factor.

Homogeneous differential equation substitution

For dy/dx = F(x,y) with F homogeneous of degree zero, substitute y = vx. Then dy/dx = v + x dv/dx, producing a separable equation in v and x.

Cross product magnitude

|a⃗ × b⃗| = |a⃗||b⃗| sin θ, the area of the parallelogram with sides a⃗ and b⃗. Half of this value is the area of the triangle formed by the two vectors.

Scalar triple product

a⃗·(b⃗ × c⃗) gives the signed volume of the parallelepiped with edges a⃗, b⃗, c⃗. It equals zero exactly when the three vectors are coplanar.

Distance from point to plane

For plane ax + by + cz + d = 0 and point (x₁,y₁,z₁), the perpendicular distance is |ax₁ + by₁ + cz₁ + d| / √(a² + b² + c²). The numerator uses the plane equation evaluated at the point.

Shortest distance between skew lines

For lines r⃗ = a⃗₁ + λb⃗₁ and r⃗ = a⃗₂ + μb⃗₂, the distance is |(a⃗₂ − a⃗₁)·(b⃗₁ × b⃗₂)| / |b⃗₁ × b⃗₂|. A zero value means the lines intersect or are parallel.

Corner point theorem

For a linear programming problem with a nonempty bounded feasible region, the optimal value of the objective function occurs at a vertex of the region, or along an entire edge.

Bayes' theorem

P(Bᵢ|A) = P(Bᵢ)P(A|Bᵢ) divided by Σⱼ P(Bⱼ)P(A|Bⱼ), where the Bⱼ partition the sample space. It reverses a conditional probability using prior probabilities.

Binomial mean and variance

For X ~ B(n, p) with q = 1 − p, the mean is np, the variance is npq, and the standard deviation is √(npq). The probabilities are P(X = r) = ⁿCᵣ pʳ qⁿ⁻ʳ.