Tamil Nadu HSE Second Year Mathematics cheat sheet

A homogeneous system AX = O has a non-trivial solution iff det A = 0; if det A ≠ 0 only the trivial solution X = O exists.

Consistency rule: for AX = B, the system is consistent iff rank(A) = rank([A|B]); it has a unique solution when this common rank equals the number of unknowns.

Cramer's rule: x_i = Δ_i / Δ where Δ = det A ≠ 0 and Δ_i is Δ with column i replaced by the constant column B.

Inverse method: when A is non-singular, X = A⁻¹B solves AX = B; for 3×3 matrices use A⁻¹ = adj A / det A.

adj A satisfies A·(adj A) = (adj A)·A = |A|·I; consequently |adj A| = |A|^(n−1) for an n×n matrix.

det(AB) = det A · det B, det(Aᵀ) = det A, and det(kA) = kⁿ det A for an n×n matrix A.

A is singular iff det A = 0; a square matrix is invertible iff its determinant is non-zero.

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

Modulus properties: |z₁z₂| = |z₁||z₂|, |z₁/z₂| = |z₁|/|z₂|, |zⁿ| = |z|ⁿ, and |z|² = z·z̄.

Triangle inequality: |z₁ + z₂| ≤ |z₁| + |z₂|, with equality only when the arguments of z₁ and z₂ are equal.

Polar form z = r(cos θ + i sin θ); Euler form z = re^{iθ}; product multiplies moduli and adds arguments.

De Moivre's theorem: (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ for integral n; use it to expand powers and roots.

nth roots of unity: solutions of zⁿ = 1 are e^{2πik/n}, k = 0, 1, …, n−1; their sum is 0 and their product is (−1)^{n−1}.

Cube roots of unity: 1, ω, ω² with 1 + ω + ω² = 0 and ω³ = 1; useful for simplifying symmetric expressions.

Conjugate rules: (z₁ ± z₂)‾ = z̄₁ ± z̄₂, (z₁z₂)‾ = z̄₁·z̄₂, (z₁/z₂)‾ = z̄₁/z̄₂, and z + z̄ = 2Re(z).

For a quadratic with real coefficients, complex roots occur in conjugate pairs; sum = −b/a and product = c/a still hold.

For ax³ + bx² + cx + d = 0: Σα = −b/a, Σαβ = c/a, αβγ = −d/a (Vieta's formulas).

For a general polynomial, the sum of roots equals −(coefficient of x^{n−1})/(leading coefficient); the product of roots equals (−1)ⁿ·(constant term)/(leading coefficient).

If a polynomial with rational coefficients has a root p + √q (q not a perfect square), then p − √q is also a root.

If a real-coefficient polynomial has root a + ib, then a − ib is also a root; complex roots occur in conjugate pairs.

A cubic with roots α, β, γ has equation x³ − (Σα)x² + (Σαβ)x − αβγ = 0.

Descartes' rule: the number of positive real roots is at most the number of sign changes in f(x), differing by an even number.

To form an equation whose roots are reciprocals of those of f(x) = 0, replace x by 1/x and simplify.

To eliminate the x² term of a cubic, substitute x = y − b/(3a); this converts it to a depressed cubic.

Principal value ranges: sin⁻¹ x ∈ [−π/2, π/2]; cos⁻¹ x ∈ [0, π]; tan⁻¹ x ∈ (−π/2, π/2).

Complementary identity: sin⁻¹ x + cos⁻¹ x = π/2 for x ∈ [−1, 1].

Reciprocal identities hold only for positive arguments: tan⁻¹ x + cot⁻¹ x = π/2 for all real x; sec⁻¹ x + cosec⁻¹ x = π/2 for |x| ≥ 1.

tan⁻¹ x + tan⁻¹ y = tan⁻¹((x + y)/(1 − xy)) when xy < 1; add π when xy > 1 and x, y > 0.

tan⁻¹ x − tan⁻¹ y = tan⁻¹((x − y)/(1 + xy)) when xy > −1.

2 tan⁻¹ x = tan⁻¹(2x/(1 − x²)) for |x| < 1; equals π + tan⁻¹(2x/(1 − x²)) for x > 1.

sin⁻¹(−x) = −sin⁻¹ x, but cos⁻¹(−x) = π − cos⁻¹ x and tan⁻¹(−x) = −tan⁻¹ x.

Double-angle form: 2 sin⁻¹ x = sin⁻¹(2x√(1 − x²)) for |x| ≤ 1/√2; adjust the range otherwise.

Circle with centre (h, k) and radius r: (x − h)² + (y − k)² = r²; general form x² + y² + 2gx + 2fy + c = 0 with centre (−g, −f) and radius √(g² + f² − c).

Parabola y² = 4ax: vertex at origin, focus (a, 0), directrix x = −a, latus rectum length 4a.

Ellipse x²/a² + y²/b² = 1 (a > b): eccentricity e = √(1 − b²/a²), foci (±ae, 0), latus rectum 2b²/a.

Hyperbola x²/a² − y²/b² = 1: e = √(1 + b²/a²), foci (±ae, 0), asymptotes y = ±(b/a)x.

Tangent to y² = 4ax at (x₁, y₁): yy₁ = 2a(x + x₁); in slope form y = mx + a/m.

Tangent to x²/a² + y²/b² = 1 at (x₁, y₁): xx₁/a² + yy₁/b² = 1; slope form y = mx ± √(a²m² + b²).

Condition for y = mx + c to touch x²/a² + y²/b² = 1: c² = a²m² + b²; for y² = 4ax: c = a/m.

Length of chord of contact and polar relations follow from the T = 0 substitution rule applied to the conic equation.

a·b = |a||b| cos θ; a·b = 0 iff a and b are perpendicular (for non-zero vectors).

a × b = |a||b| sin θ n̂; |a × b| is the area of the parallelogram with sides a and b.

a × b = 0 iff a and b are parallel; a × a = 0 and a × b = −(b × a).

Scalar triple product [a b c] = a·(b × c) equals the volume of the parallelepiped with edges a, b, c.

Coplanarity: a, b, c are coplanar iff [a b c] = 0; four points A, B, C, D are coplanar iff [AB AC AD] = 0.

If i, j, k are unit vectors along axes: i·i = j·j = k·k = 1; i·j = j·k = k·i = 0; i × j = k, j × k = i, k × i = j.

Projection of a on b is (a·b)/|b|; the unit vector along a is a/|a|.

Lagrange's identity: |a × b|² = |a|²|b|² − (a·b)².

Area of a triangle with sides a and b from one vertex: ½|a × b|.

Vector equation of a line through a with direction b: r = a + t b; through two points a and b: r = a + t(b − a).

Vector equation of a plane: r·n = d, where n is normal and d is the perpendicular distance from the origin.

Plane through three non-collinear points a, b, c: r·[(b − a) × (c − a)] = a·[(b − a) × (c − a)].

Shortest distance between skew lines r = a₁ + t b₁ and r = a₂ + s b₂: |(a₂ − a₁)·(b₁ × b₂)| / |b₁ × b₂|.

Distance of point with position vector a from plane r·n = d: |a·n − d| / |n|.

Angle between two planes equals the angle between their normals: cos θ = (n₁·n₂)/(|n₁||n₂|).

Moment of force F about point with position vector r is r × F; its magnitude measures turning effect.

For y = f(x), dy/dx = lim_{h→0} [f(x + h) − f(x)]/h; product rule: (uv)′ = u′v + uv′.

Quotient rule: (u/v)′ = (u′v − uv′)/v²; chain rule: dy/dx = (dy/du)(du/dx).

Standard derivatives: d/dx(sin x) = cos x; d/dx(cos x) = −sin x; d/dx(tan x) = sec² x; d/dx(eˣ) = eˣ; d/dx(ln x) = 1/x.

d/dx(sin⁻¹ x) = 1/√(1 − x²); d/dx(tan⁻¹ x) = 1/(1 + x²); d/dx(sec⁻¹ x) = 1/(|x|√(x² − 1)).

d/dx(xⁿ) = nx^{n−1}; d/dx(aˣ) = aˣ ln a; d/dx(log_a x) = 1/(x ln a).

Implicit differentiation: differentiate both sides with respect to x treating y as a function of x, then solve for dy/dx.

Parametric form: if x = f(t), y = g(t), then dy/dx = (dy/dt)/(dx/dt) provided dx/dt ≠ 0.

Logarithmic differentiation: take ln of both sides when the function is a product, quotient, or power of variable expressions.

A function is increasing where f′(x) > 0 and decreasing where f′(x) < 0 (strict monotonicity on an interval).

Critical points occur where f′(x) = 0 or f′(x) does not exist; these are candidates for local extrema.

First derivative test: sign change from + to − at c gives a local maximum; from − to + gives a local minimum.

Second derivative test: f′(c) = 0 and f″(c) < 0 gives a local maximum; f″(c) > 0 gives a local minimum; f″(c) = 0 is inconclusive.

Concavity: f″(x) > 0 means concave up; f″(x) < 0 means concave down; a change of concavity at a point gives an inflection point.

For a rectangle of fixed perimeter, the square maximises area; for a right circular cylinder of fixed volume, height = diameter minimises surface area.

To optimise a quantity, express it as a function of one variable using the constraint, differentiate, and solve f′(x) = 0.

Use the first derivative to find stationary points, then verify whether they are maxima, minima, or points of inflection.

∫xⁿ dx = x^{n+1}/(n + 1) + C for n ≠ −1; ∫(1/x) dx = ln|x| + C.

∫eˣ dx = eˣ + C; ∫aˣ dx = aˣ/ln a + C; ∫sin x dx = −cos x + C; ∫cos x dx = sin x + C.

∫sec² x dx = tan x + C; ∫cosec² x dx = −cot x + C; ∫(1/(1 + x²)) dx = tan⁻¹ x + C.

Integration by parts: ∫u dv = uv − ∫v du; choose u by the LIATE order (log, inverse, algebraic, trigonometric, exponential).

Substitution rule: ∫f(g(x))g′(x) dx = ∫f(u) du with u = g(x); useful for composite integrands.

Area under y = f(x) between x = a and x = b: ∫ₐᵇ f(x) dx; area between two curves: ∫ₐᵇ [f(x) − g(x)] dx.

Volume of solid of revolution about the x-axis: V = π∫ₐᵇ y² dx; about the y-axis use V = π∫ x² dy.

Length of arc of y = f(x) from a to b: ∫ₐᵇ √(1 + (dy/dx)²) dx.

Definite integral properties: ∫ₐᵇ f(x) dx = −∫_bᵃ f(x) dx; ∫ₐᵇ f(x) dx = ∫ₐᶜ f(x) dx + ∫_cᵇ f(x) dx.

Order of a differential equation is the order of the highest derivative present; degree is the power of the highest derivative after the equation is made polynomial in derivatives.

Variable separable form: dy/dx = f(x)g(y) - separate as dy/g(y) = f(x) dx and integrate both sides.

Linear first-order form: dy/dx + Py = Q; integrating factor IF = e^{∫P dx}; solution is y·IF = ∫Q·IF dx.

Homogeneous equation dy/dx = f(y/x): substitute y = vx so dy/dx = v + x(dv/dx), giving a separable equation in v.

Exact equation M dx + N dy = 0 satisfies ∂M/∂y = ∂N/∂x; solution is ∫M dx + terms of N free of x = C.

Formation of a differential equation: differentiate the given family of curves as many times as the number of arbitrary constants, then eliminate the constants.

A first-order first-degree equation has a general solution containing one arbitrary constant; a particular solution is obtained using an initial condition.

The solution of dy/dx = ky is y = Ce^{kx}; this models exponential growth (k > 0) and decay (k < 0).

Number of binary operations on a set with n elements is n^(n²); number of commutative binary operations is n^(n(n+1)/2).

A group (G, ∗) requires closure, associativity, identity, and inverse for every element; a semigroup needs only closure and associativity.

A group is abelian if a ∗ b = b ∗ a for all a, b; the order of a group is its number of elements.

Cayley's theorem: every finite group of order n is isomorphic to a subgroup of the symmetric group Sₙ.

A ring (R, +, ·) requires (R, +) abelian group, (R, ·) semigroup, and distributivity of · over + on both sides.

In a Boolean algebra: a + a′ = 1, a · a′ = 0, (a′)′ = a, and De Morgan's laws (a + b)′ = a′ · b′ and (a · b)′ = a′ + b′.

A lattice is a partially ordered set in which every pair of elements has a least upper bound and a greatest lower bound.

The number of elements in a finite Boolean algebra is 2ⁿ for some positive integer n; the atoms are the elements covering 0.