Maharashtra HSC Mathematics & Statistics - Arts & Science (MSBSHSE) cheat sheet

Derivative by first principle: f'(x) = lim(h→0) [f(x+h) − f(x)]/h, provided the limit exists and is finite.

Standard derivatives: d/dx(x^n)=nx^(n−1); d/dx(e^x)=e^x; d/dx(a^x)=a^x·log a; d/dx(log x)=1/x; d/dx(sin x)=cos x; d/dx(cos x)=−sin x; d/dx(tan x)=sec²x.

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(sec⁻¹x)=1/(|x|√(x²−1)).

Product rule: (uv)' = u'v + uv'. Quotient rule: (u/v)' = (u'v − uv')/v², valid where v ≠ 0.

Chain rule: dy/dx = dy/du · du/dx; for parametric x=f(t), y=g(t), dy/dx = (dy/dt)/(dx/dt), dx/dt ≠ 0.

Implicit differentiation: differentiate F(x,y)=0 term by term, treating y as a function of x, then solve for dy/dx.

Logarithmic differentiation suits products, quotients and powers: take log of both sides, differentiate, then multiply by the original expression.

Derivative of an inverse function: if y=f(x) has inverse x=g(y), then g'(y)=1/f'(x), provided f'(x) ≠ 0.

Tangent at (x₁,y₁): y − y₁ = f'(x₁)(x − x₁). Normal slope = −1/f'(x₁), valid when f'(x₁) ≠ 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) has f'(c)=0.

Lagrange's mean value theorem: under continuity on [a,b] and differentiability on (a,b), some c in (a,b) satisfies f'(c) = [f(b)−f(a)]/(b−a).

Increasing on an interval where f'(x) > 0; decreasing where f'(x) < 0. Solve f'(x)=0 for critical points and test sign of f' on each subinterval.

First derivative test: f' changes + to − at x=c gives a local maximum; − to + gives a local minimum; no sign change gives neither.

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

Concavity: upward where f''(x) > 0, downward where f''(x) < 0. Inflection points occur where f'' changes sign.

Approximation: f(x+Δx) ≈ f(x) + f'(x)·Δx. Errors: if y=f(x), then dy = f'(x)dx gives the approximate change in y.

Standard integrals: ∫x^n dx = x^(n+1)/(n+1) + c (n ≠ −1); ∫(1/x)dx = log|x| + c; ∫e^x dx = e^x + c; ∫a^x dx = a^x/log a + c.

Trig integrals: ∫sin x dx = −cos x + c; ∫cos x dx = sin x + c; ∫sec²x dx = tan x + c; ∫(1/(1+x²))dx = tan⁻¹x + c; ∫(1/√(1−x²))dx = sin⁻¹x + c.

By parts: ∫u v dx = u∫v dx − ∫(u' ∫v dx)dx; choose u by the ILATE order (inverse trig, log, algebraic, trig, exponential).

Definite integral property: ∫ₐᵇ f(x)dx = −∫ᵦᵃ f(x)dx; ∫ₐᵇ f(x)dx = ∫ₐᶜ f(x)dx + ∫ᶜᵇ f(x)dx; ∫₀ᵃ f(x)dx = ∫₀ᵃ f(a−x)dx.

∫₋ₐᵃ f(x)dx = 2∫₀ᵃ f(x)dx if f is even, and equals 0 if f is odd.

Area under a curve: A = ∫ₐᵇ y dx for y ≥ 0 on [a,b]; where the curve crosses the axis, integrate each piece and add absolute values.

Area between curves: A = ∫ₐᵇ [f(x) − g(x)]dx when f(x) ≥ g(x) on [a,b].

Volume of revolution about the x-axis: V = π∫ₐᵇ y² dx. About the y-axis use V = π∫ x² dy with matching limits.

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

General solution of an nth-order equation contains n arbitrary constants; a particular solution fixes them using given initial conditions.

Variable separable: write as f(y)dy = g(x)dx, integrate both sides, then add the single constant of integration.

Homogeneous equation dy/dx = F(x,y) with F homogeneous of degree 0: substitute y = vx, so dy/dx = v + x(dv/dx), then separate.

Linear equation dy/dx + Py = Q: integrating factor IF = e^(∫P dx), and the solution is y·IF = ∫Q·IF dx + c.

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

Growth/decay model: dy/dt = ky gives y = y₀e^(kt), where k > 0 is growth and k < 0 is decay.

Newton's law of cooling: dT/dt = −k(T − T₀), giving T − T₀ = (T₁ − T₀)e^(−kt).

If a = a₁i + a₂j + a₃k and b = b₁i + b₂j + b₃k, then a·b = a₁b₁ + a₂b₂ + a₃b₃ = |a||b|cos θ.

a × b = |i j k; a₁ a₂ a₃; b₁ b₂ b₃|, and |a × b| = |a||b|sin θ gives the area of the parallelogram; triangle area is half of this.

a·b = 0 means perpendicular vectors; a × b = 0 means parallel vectors. Scalar triple product [a b c] = a·(b × c) gives the parallelepiped volume.

Direction cosines of a vector satisfy l² + m² + n² = 1; direction ratios are any proportional triple.

Line through point a with direction b: r = a + λb. Cartesian form (x−x₁)/l = (y−y₁)/m = (z−z₁)/n.

Plane with normal n through point a: (r − a)·n = 0, i.e. lx + my + nz = d, where d = n·a.

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

Angle between planes equals the angle between their normals: cos θ = |n₁·n₂|/(|n₁||n₂|). Shortest distance between skew lines uses the scalar triple product over |b₁ × b₂|.

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

Graph the constraints as lines, shade the feasible region satisfying all inequalities, and find its corner (vertex) points.

Corner-point method: evaluate Z at every vertex; the optimum of a bounded feasible region occurs at a vertex.

Corner-point evaluation can be replaced by comparing slopes: the optimum vertex is where the objective line's slope first leaves the feasible region.

If the feasible region is unbounded, an optimum found at a vertex is confirmed only if the objective line has no common point with the feasible region beyond that value.

Unbounded region with no finite vertex value means the objective function may have no maximum (or no minimum) on that region.

For two variables the graphical method applies; three or more variables require the simplex method, not the corner-point graph.

A constraint written as x + y ≤ 4 becomes the boundary line x + y = 4; test the origin (0,0) to decide which side to shade.

p → q ≡ ~p ∨ q; its contrapositive ~q → ~p is logically equivalent, but the converse q → p is not.

De Morgan: ~(p ∧ q) ≡ ~p ∨ ~q and ~(p ∨ q) ≡ ~p ∧ ~q. Also p ↔ q ≡ (p → q) ∧ (q → p).

Tautology is true for every assignment of truth values; contradiction is always false; contingency depends on the assignment.

p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r) and p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r) are the distributive laws.

A matrix is symmetric if A' = A and skew-symmetric if A' = −A; every diagonal entry of a skew-symmetric matrix is 0.

(AB)' = B'A'; (A')' = A; (A+B)' = A' + B'. Matrix multiplication is not commutative in general.

A square matrix A is invertible iff |A| ≠ 0, and then A⁻¹ = adj(A)/|A|. For 2×2, adj[[a,b],[c,d]] = [[d,−b],[−c,a]].

Cramer's rule for AX = B: x = D₁/D, y = D₂/D, z = D₃/D, where D = |A| ≠ 0 and Dᵢ replaces column i by B.

Discrete random variable X has probability mass function p(xᵢ) ≥ 0 with Σp(xᵢ) = 1; the cumulative distribution is F(x) = P(X ≤ x).

Expected value E(X) = Σxᵢp(xᵢ); variance Var(X) = E(X²) − [E(X)]²; standard deviation is √Var(X).

For constants a, b: E(aX + b) = aE(X) + b and Var(aX + b) = a²Var(X).

Binomial distribution B(n,p): P(X = r) = C(n,r)p^r q^(n−r), r = 0,1,…,n, with q = 1 − p.

Binomial mean = np, variance = npq, standard deviation = √(npq). Variance is always less than the mean when 0 < p < 1.

Binomial conditions: fixed number n of independent Bernoulli trials, constant success probability p, and only two outcomes per trial.

Mode of B(n,p) is the integer r with (n+1)p − 1 ≤ r ≤ (n+1)p; if (n+1)p is an integer, both that value and one less are modes.

P(X ≥ 1) = 1 − q^n is the standard complement shortcut for at least one success in n binomial trials.

Range = maximum − minimum. Quartile deviation QD = (Q₃ − Q₁)/2, with coefficient of QD = (Q₃ − Q₁)/(Q₃ + Q₁).

Mean deviation about A: MD = (1/n)Σ|xᵢ − A| for raw data; about the median it is minimum. Coefficient = MD/median (or MD/mean).

Variance of raw data: σ² = (1/n)Σxᵢ² − (x̄)²; for grouped data use Σfxᵢ²/Σf − (Σfxᵢ/Σf)². Standard deviation σ = √σ².

Coefficient of variation CV = (σ/x̄)·100%; a larger CV means greater relative variability, and CV is unit-free.

Karl Pearson's coefficient of skewness: Sk = (mean − mode)/σ, or (mean − median)/σ when mode is undefined; sign indicates direction of skew.

Bowley's coefficient of skewness: Sk = (Q₃ + Q₁ − 2Q₂)/(Q₃ − Q₁), which lies between −1 and 1.

For a positively skewed distribution, mean > median > mode; for a negatively skewed distribution, mean < median < mode.

Karl Pearson's correlation coefficient r = Σ(xᵢ−x̄)(yᵢ−ȳ)/√[Σ(xᵢ−x̄)²·Σ(yᵢ−ȳ)²], with −1 ≤ r ≤ 1; r = 0 means no linear correlation.

Regression of y on x: y − ȳ = b_yx(x − x̄), where b_yx = r·(σ_y/σ_x); the two regression lines meet at (x̄, ȳ).