Tamil Nadu HSE Second Year Mathematics flashcards

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

Cramer's rule

Solution of AX = B with det A ≠ 0 given by x_i = Δ_i/Δ, where Δ is det A and Δ_i replaces column i with B.

Homogeneous system condition

AX = O has a non-trivial solution iff det A = 0. If det A ≠ 0, only the trivial solution X = O exists.

Adjoint determinant identity

For an n×n matrix A, A·(adj A) = |A|·I, so |adj A| = |A|^(n−1). Useful for computing determinants of adjoints quickly.

De Moivre's theorem

(cos θ + i sin θ)ⁿ = cos nθ + i sin nθ for any integer n. Used to compute powers and roots of complex numbers in polar form.

Cube roots of unity

The solutions of z³ = 1 are 1, ω, ω², where ω = e^{2πi/3}. They satisfy 1 + ω + ω² = 0 and ω³ = 1.

Triangle inequality for complex numbers

For any complex numbers z₁ and z₂, |z₁ + z₂| ≤ |z₁| + |z₂|. Equality holds only when the arguments of z₁ and z₂ are equal.

Vieta's formulas for cubic

For ax³ + bx² + cx + d = 0 with roots α, β, γ: Σα = −b/a, Σαβ = c/a, αβγ = −d/a.

Conjugate root pair theorem

If a polynomial with real coefficients has a + ib as a root, then a − ib is also a root. Similarly, p + √q and p − √q occur together for rational coefficients.

Descartes' rule of signs

The number of positive real roots of f(x) is at most the number of sign changes in f(x), differing by an even integer. Apply to f(−x) for negative roots.

Complementary inverse trig identity

sin⁻¹ x + cos⁻¹ x = π/2 for x in [−1, 1]; tan⁻¹ x + cot⁻¹ x = π/2 for all real x; sec⁻¹ x + cosec⁻¹ x = π/2 for |x| ≥ 1.

Principal value of sin⁻¹

sin⁻¹ x is defined for x in [−1, 1] and returns a value in [−π/2, π/2]. Its graph is the reflection of y = sin x restricted to that interval.

Circle general form

x² + y² + 2gx + 2fy + c = 0 represents a circle with centre (−g, −f) and radius √(g² + f² − c), provided g² + f² − c > 0.

Ellipse eccentricity

For x²/a² + y²/b² = 1 with a > b, eccentricity e = √(1 − b²/a²). Foci are at (±ae, 0) and the latus rectum has length 2b²/a.

Hyperbola asymptotes

For x²/a² − y²/b² = 1, the asymptotes are y = ±(b/a)x. The conjugate hyperbola shares the same asymptotes with equation y²/b² − x²/a² = 1.

Tangency condition for ellipse

The line y = mx + c touches x²/a² + y²/b² = 1 iff c² = a²m² + b². For the parabola y² = 4ax, tangency requires c = a/m.

Scalar triple product

For vectors a, b, c, the scalar triple product [a b c] = a·(b × c) gives the signed volume of the parallelepiped with edges a, b, c.

Coplanarity condition

Three vectors a, b, c are coplanar iff their scalar triple product is zero. Four points A, B, C, D are coplanar iff [AB AC AD] = 0.

Cross product magnitude

|a × b| = |a||b| sin θ, where θ is the angle between a and b. This equals the area of the parallelogram with sides a and b.

Shortest distance between skew lines

For lines r = a₁ + t b₁ and r = a₂ + s b₂, the shortest distance is |(a₂ − a₁)·(b₁ × b₂)| / |b₁ × b₂|.

Distance from point to plane

The perpendicular distance from a point with position vector a to the plane r·n = d is |a·n − d| / |n|.

Vector equation of a plane

A plane with unit normal n̂ at perpendicular distance d from the origin has equation r·n̂ = d. In Cartesian form this becomes lx + my + nz = d.

Chain rule

If y = f(u) and u = g(x), then dy/dx = (dy/du)·(du/dx). This rule extends to any number of nested functions.

Derivative of tan⁻¹ x

d/dx(tan⁻¹ x) = 1/(1 + x²). Similarly, d/dx(sin⁻¹ x) = 1/√(1 − x²) for |x| < 1.

Logarithmic differentiation

For y = u(x)^{v(x)} or products and quotients of variable expressions, take ln of both sides, differentiate implicitly, then multiply by y to obtain dy/dx.

Second derivative test

If f′(c) = 0 and f″(c) < 0, then f has a local maximum at c. If f″(c) > 0, it has a local minimum. If f″(c) = 0, the test is inconclusive.

Point of inflection

A point where the concavity of f changes sign. At such a point f″(x) = 0 or fails to exist, and f″ changes sign across it.

Integration by parts

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

Volume of solid of revolution

Revolving y = f(x) about the x-axis from a to b gives volume V = π∫ₐᵇ y² dx. The cross-sections are circles of radius y.

Area between two curves

If f(x) ≥ g(x) on [a, b], the area enclosed is ∫ₐᵇ [f(x) − g(x)] dx. Determine intersection points to find limits of integration.

Integrating factor for linear ODE

For dy/dx + Py = Q, the integrating factor is e^{∫P dx}. Multiplying through makes the left side d/dx(y·IF), so y·IF = ∫Q·IF dx.

Order and degree of ODE

Order is the order of the highest derivative in the equation. Degree is the power of that highest derivative after clearing radicals and fractions in derivatives.

Number of binary operations

On a set with n elements, there are n^(n²) binary operations in total and n^(n(n+1)/2) commutative binary operations.