BSEH Class 12 Maths cheat sheet

A relation R from A to B is a subset of A × B; a function is a relation in which every element of the domain has exactly one image.

Types of relations: empty, universal, identity, reflexive (aRa for all a), symmetric (aRb ⇒ bRa), transitive (aRb and bRc ⇒ aRc), equivalence (reflexive + symmetric + transitive).

A function f: A → B is one-one (injective) if f(x₁) = f(x₂) ⇒ x₁ = x₂; it is onto (surjective) if every element of B has a preimage; bijective means both.

A bijection f: A → B has an inverse f⁻¹: B → A with f⁻¹(y) = x ⇔ f(x) = y; the graphs of f and f⁻¹ are mirror images about the line y = x.

Composition: (g∘f)(x) = g(f(x)); composition is associative but generally not commutative, and (g∘f)⁻¹ = f⁻¹∘g⁻¹ when both inverses exist.

Number of relations from a set with m elements to a set with n elements is 2^(mn); number of functions is n^m; number of bijections from an n-set to itself is n!.

Equivalence classes partition the set; the set of all equivalence classes of R on A is a partition of A, and conversely every partition gives an equivalence relation.

An operation ∗ on A is commutative if a∗b = b∗a and associative if (a∗b)∗c = a∗(b∗c); an identity e satisfies a∗e = e∗a = a, and inverses satisfy a∗b = e.

Principal value ranges: sin⁻¹: [−π/2, π/2]; cos⁻¹: [0, π]; tan⁻¹: (−π/2, π/2); cot⁻¹: (0, π); sec⁻¹: [0, π] \ {π/2}; cosec⁻¹: [−π/2, π/2] \ {0}.

Reciprocal identities (only for x in the relevant domain): sin⁻¹(1/x) = cosec⁻¹x, cos⁻¹(1/x) = sec⁻¹x, tan⁻¹(1/x) = cot⁻¹x.

Reflection identities: sin⁻¹(−x) = −sin⁻¹x, tan⁻¹(−x) = −tan⁻¹x, but cos⁻¹(−x) = π − cos⁻¹x and cot⁻¹(−x) = π − cot⁻¹x.

Complementary identities: sin⁻¹x + cos⁻¹x = π/2, tan⁻¹x + cot⁻¹x = π/2, sec⁻¹x + cosec⁻¹x = π/2, valid on the common domain.

Sum formula: tan⁻¹x + tan⁻¹y = tan⁻¹((x+y)/(1−xy)) when xy < 1; if xy > 1 and x, y > 0 the value is π + tan⁻¹((x+y)/(1−xy)).

Difference formula: tan⁻¹x − tan⁻¹y = tan⁻¹((x−y)/(1+xy)) when xy > −1; adjust by π when xy < −1 and x, y > 0.

Double-angle forms: 2tan⁻¹x = tan⁻¹(2x/(1−x²)) for |x| < 1; also 2tan⁻¹x = sin⁻¹(2x/(1+x²)) = cos⁻¹((1−x²)/(1+x²)) for x ≥ 0.

Always simplify the inner expression first, then check whether the resulting argument lies in the principal branch before writing the answer.

Matrix addition requires identical orders and is elementwise; scalar multiplication multiplies every entry by the scalar.

Matrix product AB is defined only when columns of A equal rows of B; the (i, j) entry of AB is the dot product of row i of A with column j of B.

Matrix multiplication is associative and distributive but not commutative: generally AB ≠ BA.

Transpose rules: (Aᵀ)ᵀ = A, (A+B)ᵀ = Aᵀ + Bᵀ, (kA)ᵀ = kAᵀ, (AB)ᵀ = BᵀAᵀ.

Symmetric: Aᵀ = A; skew-symmetric: Aᵀ = −A (diagonal entries of a skew-symmetric matrix are zero).

Every square matrix splits uniquely as A = ½(A + Aᵀ) + ½(A − Aᵀ), a symmetric plus a skew-symmetric matrix.

For square matrices, (A + B)² = A² + AB + BA + B²; the familiar identity (A+B)² = A² + 2AB + B² holds only if AB = BA.

A is invertible only if A is square and |A| ≠ 0; then (AB)⁻¹ = B⁻¹A⁻¹ and (Aᵀ)⁻¹ = (A⁻¹)ᵀ.

Elementary row operations (row swap, row scaling by a nonzero scalar, adding a multiple of one row to another) do not change the solution set of a linear system.

For a 2×2 matrix [[a, b], [c, d]], |A| = ad − bc; for 3×3 expand along any row or column using cofactors, with signs following the + − + pattern.

Properties: interchanging two rows or columns changes the sign of the determinant; if two rows or columns are identical, the determinant is 0.

Multiplying one row (or column) by k multiplies the determinant by k; hence |kA| = kⁿ|A| for an n×n matrix.

If a row (or column) is a linear combination of the others, the determinant is zero; adding a multiple of one row to another leaves the determinant unchanged.

Product rule: |AB| = |A||B|; also |Aᵀ| = |A| and |A⁻¹| = 1/|A|.

Area of a triangle with vertices (x₁, y₁), (x₂, y₂), (x₃, y₃) is ½|x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|; collinear points give zero.

Cramer/system form: for AX = B with |A| ≠ 0, X = A⁻¹B; the system has a unique solution iff |A| ≠ 0.

Adjoint relation: A(adj A) = (adj A)A = |A|I, and A⁻¹ = (1/|A|) adj A; adj A is the transpose of the cofactor matrix.

Consistency: a square system AX = B has a unique solution if |A| ≠ 0; if |A| = 0 and (adj A)B ≠ 0 there is no solution; if |A| = 0 and (adj A)B = 0 there are infinitely many or no solutions.

f is continuous at a if lim(x→a) f(x) = f(a); equivalently the left limit, right limit and function value all agree.

Differentiability implies continuity, but continuity does not imply differentiability (f(x) = |x| at 0 is the standard counterexample).

Standard derivatives: d/dx(xⁿ) = nxⁿ⁻¹, d/dx(sin x) = cos x, d/dx(cos x) = −sin x, d/dx(tan x) = sec²x, d/dx(sec x) = sec x tan x, d/dx(cot x) = −cosec²x, d/dx(cosec x) = −cosec x cot x.

Exponential and log: d/dx(eˣ) = eˣ, d/dx(aˣ) = aˣ ln a, d/dx(ln x) = 1/x, d/dx(log_a x) = 1/(x ln a).

Inverse trig derivatives: d/dx(sin⁻¹x) = 1/√(1−x²), d/dx(cos⁻¹x) = −1/√(1−x²), d/dx(tan⁻¹x) = 1/(1+x²), d/dx(cot⁻¹x) = −1/(1+x²), d/dx(sec⁻¹x) = 1/(|x|√(x²−1)).

Chain rule: d/dx f(g(x)) = f′(g(x))·g′(x); product rule (uv)′ = u′v + uv′; quotient rule (u/v)′ = (u′v − uv′)/v².

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).

Lagrange's Mean Value Theorem: under continuity on [a, b] and differentiability on (a, b), f′(c) = (f(b) − f(a))/(b − a) for some c in (a, b).

Logarithmic differentiation: take ln of both sides before differentiating products, quotients and variable powers such as xˣ; for parametric x = f(t), y = g(t), dy/dx = (dy/dt)/(dx/dt).

f is strictly increasing on an interval if f′(x) > 0 there and strictly decreasing if f′(x) < 0; f′(x) = 0 at stationary points.

First derivative test: if f′ changes from + to − at c, c is a local maximum; from − to +, a local minimum; no sign change means neither.

Second derivative test: at a stationary point c, f″(c) < 0 gives a local maximum, f″(c) > 0 gives a local minimum, and f″(c) = 0 is inconclusive.

Absolute extrema on [a, b]: evaluate f at all critical points in (a, b) and at the endpoints a and b; the largest and smallest values are the absolute maximum and minimum.

Slope of the tangent to y = f(x) at (x₁, y₁) is f′(x₁); the normal slope is −1/f′(x₁) (normal is vertical when f′(x₁) = 0).

Tangent line at (x₁, y₁): y − y₁ = f′(x₁)(x − x₁); normal line: y − y₁ = −(1/f′(x₁))(x − x₁).

Rate of change: if y = f(x) and x changes at rate dx/dt, then dy/dt = f′(x)·dx/dt; for related quantities, differentiate the relating equation with respect to t.

Approximation: f(x + Δx) ≈ f(x) + f′(x)Δx; the differential dy = f′(x)dx estimates the change in y.

Optimisation procedure: express the quantity as a function of one variable using the constraint, differentiate, solve f′ = 0, then verify with the second derivative or endpoint comparison.

Standard forms: ∫xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1), ∫(1/x) dx = ln|x| + C, ∫eˣ dx = eˣ + C, ∫aˣ dx = aˣ/ln a + C.

Trig integrals: ∫sin x dx = −cos x + C, ∫cos x dx = sin x + C, ∫sec²x dx = tan x + C, ∫cosec²x dx = −cot x + C, ∫tan x dx = ln|sec x| + C.

Rational forms: ∫dx/(x² + a²) = (1/a)tan⁻¹(x/a) + C, ∫dx/√(a² − x²) = sin⁻¹(x/a) + C, ∫dx/(x² − a²) = (1/2a)ln|(x−a)/(x+a)| + C.

Integration by substitution: replace the inner function by t so that the integrand becomes f(t)dt; change limits accordingly for definite integrals.

Integration by parts: ∫u v′ dx = uv − ∫u′v dx; choose u by the ILATE order (Inverse trig, Logarithmic, Algebraic, Trigonometric, Exponential).

Partial fractions: decompose proper rational functions into linear and quadratic factors before integrating; reduce improper fractions by division first.

Fundamental Theorem of Calculus: ∫ₐᵇ f(x) dx = F(b) − F(a), where F is any antiderivative of f; the definite integral is independent of the variable name.

Properties: ∫ₐᵇ f = −∫ᵦᵃ f, ∫ₐᵇ f = ∫ₐᶜ f + ∫꜀ᵇ f, and ∫₀ᵃ f(x) dx = ∫₀ᵃ f(a − x) dx, which simplifies many symmetric definite integrals.

Even/odd rule: ∫₋ₐᵃ f(x) dx = 2∫₀ᵃ f(x) dx if f is even, and 0 if f is odd.

Area under y = f(x) from x = a to x = b (with f ≥ 0) is ∫ₐᵇ f(x) dx; areas below the x-axis contribute negative values, so take absolute values or split the interval.

Area between two curves y = f(x) (upper) and y = g(x) (lower) from a to b is ∫ₐᵇ [f(x) − g(x)] dx, where a and b are the intersection x-values.

If the curves cross inside the interval, split at the crossing point and integrate |f − g| over each subinterval.

Area with respect to the y-axis: ∫꜀ᵈ [right curve − left curve] dy, using x expressed as a function of y.

Area of the region bounded by y² = 4ax and its latus rectum x = a is found by integrating 2√(ax) from 0 to a, giving 8a²/3.

Area of the circle x² + y² = a² in the first quadrant is πa²/4; the whole circle area πa² follows by symmetry.

Area of the ellipse x²/a² + y²/b² = 1 is πab, obtained from four times the first-quadrant area.

Area of the triangle bounded by a line and the axes equals ½ × intercept on x-axis × intercept on y-axis, matching the integral of the line.

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

A solution is a function satisfying the equation; the general solution contains as many arbitrary constants as the order, and a particular solution fixes them using initial conditions.

Variable separable: write the equation as f(x)dx = g(y)dy and integrate both sides, adding one constant.

Homogeneous equations of the form dy/dx = F(x, y) where F is homogeneous of degree 0 are solved by substituting y = vx, so dy/dx = v + x(dv/dx).

Linear differential equation dy/dx + Py = Q (P, Q functions of x) has integrating factor IF = e^(∫P dx), and the solution is y·IF = ∫Q·IF dx + C.

For dx/dy + P₁x = Q₁ (treating x as the dependent variable), the integrating factor is e^(∫P₁ dy).

Exact equation M dx + N dy = 0 satisfies ∂M/∂y = ∂N/∂x; its 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.

Growth and decay model: dP/dt = kP gives P = P₀e^(kt); Newton's cooling gives dT/dt = −k(T − T₀).

Position vector of the point dividing AB in ratio m:n internally is (m b⃗ + n a⃗)/(m + n); externally it is (m b⃗ − n a⃗)/(m − n).

Magnitude of a⃗ = a₁î + a₂ĵ + a₃k̂ is |a⃗| = √(a₁² + a₂² + a₃²); the unit vector is a⃗/|a⃗|.

Dot product: a⃗·b⃗ = |a⃗||b⃗|cos θ = a₁b₁ + a₂b₂ + a₃b₃; perpendicular vectors have zero dot product.

Cross product: |a⃗ × b⃗| = |a⃗||b⃗|sin θ, direction given by the right-hand rule; parallel vectors have zero cross product.

In components, a⃗ × b⃗ = (a₂b₃ − a₃b₂)î − (a₁b₃ − a₃b₁)ĵ + (a₁b₂ − a₂b₁)k̂.

Scalar triple product [a⃗ b⃗ c⃗] = a⃗·(b⃗ × c⃗) equals the volume of the parallelepiped with edges a⃗, b⃗, c⃗; coplanar vectors give zero.

Area of a triangle with adjacent sides a⃗ and b⃗ is ½|a⃗ × b⃗|; area of the parallelogram is |a⃗ × b⃗|.

Projection of a⃗ on b⃗ is (a⃗·b⃗)/|b⃗|, and the vector projection is ((a⃗·b⃗)/|b⃗|²)b⃗.

Properties: a⃗ × b⃗ = −(b⃗ × a⃗), a⃗ × a⃗ = 0, and a⃗·(b⃗ × c⃗) = b⃗·(c⃗ × a⃗) = c⃗·(a⃗ × b⃗) under cyclic permutation.

Direction ratios of the line through (x₁, y₁, z₁) and (x₂, y₂, z₂) are x₂ − x₁, y₂ − y₁, z₂ − z₁; direction cosines satisfy l² + m² + n² = 1.

Line through a point with position vector a⃗ and parallel to b⃗: r⃗ = a⃗ + λb⃗; Cartesian form (x − x₁)/l = (y − y₁)/m = (z − z₁)/n.

Angle between two lines with direction vectors b⃗₁ and b⃗₂: cos θ = |b⃗₁·b⃗₂|/(|b⃗₁||b⃗₂|); lines are perpendicular if b⃗₁·b⃗₂ = 0 and parallel if b⃗₁ × b⃗₂ = 0.

Plane with normal n⃗ through point a⃗: (r⃗ − a⃗)·n⃗ = 0, i.e. r⃗·n⃗ = d; Cartesian form ax + by + cz = d with normal (a, b, c).

Distance of point (x₁, y₁, z₁) from plane ax + by + cz + d = 0 is |ax₁ + by₁ + cz₁ + d|/√(a² + b² + c²).

Angle between a line with direction b⃗ and a plane with normal n⃗ satisfies sin θ = |b⃗·n⃗|/(|b⃗||n⃗|).

Shortest distance between skew lines r⃗ = a⃗₁ + λb⃗₁ and r⃗ = a⃗₂ + μb⃗₂ is |(a⃗₂ − a⃗₁)·(b⃗₁ × b⃗₂)|/|b⃗₁ × b⃗₂|.

Two planes are perpendicular if n⃗₁·n⃗₂ = 0 and parallel if n⃗₁ × n⃗₂ = 0; the angle between planes is the angle between their normals.

Distance of a point from a line through a⃗ with direction b⃗ is |(p⃗ − a⃗) × b⃗|/|b⃗|.

A linear programming problem maximises or minimises a linear objective function Z = ax + by subject to linear constraints and non-negativity restrictions x, y ≥ 0.

The feasible region is the intersection of the half-planes defined by the constraints; it is a convex polygon (possibly unbounded).

Corner Point Theorem: if an optimum exists, it occurs at a corner (vertex) of the feasible region; evaluate Z at every vertex and compare.

Bounded feasible region: the maximum and minimum of Z both exist and occur at vertices (possibly along an entire edge if Z is constant there).

Unbounded feasible region: if the objective can be increased without limit, no maximum exists; check whether the open half-plane contains feasible points beyond all bounds.

Graphing method: draw each constraint line, shade the feasible side, identify vertices, then tabulate Z at each vertex.

A constraint of the form x ≥ 0, y ≥ 0 restricts the solution to the first quadrant; equality constraints reduce the feasible region to a line segment.

If two vertices give the same optimal Z, every point on the segment joining them is also optimal.

Conditional probability: P(A|B) = P(A ∩ B)/P(B) for P(B) > 0; rearranged, P(A ∩ B) = P(A|B)P(B).

Independent events: P(A ∩ B) = P(A)P(B); equivalently P(A|B) = P(A).

Multiplication theorem for any two events: P(A ∩ B) = P(A)P(B|A) = P(B)P(A|B).

Total probability: if B₁, …, Bₙ partition the sample space, P(A) = Σ P(Bᵢ)P(A|Bᵢ).

Bayes' theorem: P(Bᵢ|A) = P(Bᵢ)P(A|Bᵢ) / Σⱼ P(Bⱼ)P(A|Bⱼ).

Addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B); for mutually exclusive events the last term is zero.

Random variable X with values xᵢ and probabilities pᵢ: mean E(X) = Σxᵢpᵢ and variance Var(X) = Σxᵢ²pᵢ − (E(X))², with Σpᵢ = 1 and pᵢ ≥ 0.

Binomial distribution B(n, p): P(X = r) = ⁿCᵣ pʳ qⁿ⁻ʳ with q = 1 − p; mean = np, variance = npq, standard deviation = √(npq).

Bernoulli trials require a fixed number n of independent trials, two outcomes (success/failure), and constant success probability p.