Maharashtra HSC Mathematics & Statistics - Commerce (MSBSHSE) cheat sheet

Statement (proposition): a sentence that is either true or false but not both; exclamations, questions and open sentences are not statements.

Truth values are denoted T (1) and F (0). A statement and its negation have opposite truth values; ~p is false exactly when p is true.

Conjunction p ∧ q is true only when both p and q are true. Disjunction p ∨ q is false only when both are false.

Implication p → q is false only in the single case p true, q false; it is true whenever p is false (vacuous truth).

Biconditional p ↔ q is true exactly when p and q have the same truth value; it equals (p → q) ∧ (q → p).

Contrapositive of p → q is ~q → ~p and is logically equivalent to it; the converse q → p and inverse ~p → ~q are not equivalent to it.

De Morgan's laws: ~(p ∧ q) ≡ ~p ∨ ~q and ~(p ∨ q) ≡ ~p ∧ ~q. Also ~(p → q) ≡ p ∧ ~q.

Set operations: A ∪ B = {x : x ∈ A or x ∈ B}; A ∩ B = {x : x ∈ A and x ∈ B}; A − B = {x : x ∈ A and x ∉ B}; A′ = U − A.

n(A ∪ B) = n(A) + n(B) − n(A ∩ B); for three sets subtract the three pairwise intersections and add back n(A ∩ B ∩ C).

A matrix of order m × n has m rows and n columns. Addition is defined only for matrices of the same order; (A + B)ij = aij + bij.

Matrix product AB is defined only when columns of A equal rows of B; if A is m × n and B is n × p, AB is m × p and (AB)ij = Σk aik bkj.

Matrix multiplication is not commutative in general: AB ≠ BA, though both may exist. It is associative and distributive over addition.

Transpose rules: (A′)′ = A, (A + B)′ = A′ + B′, (AB)′ = B′A′.

Determinant of order 2: |a b; c d| = ad − bc. A square matrix is singular when its determinant is 0 and invertible otherwise.

Cramer's rule for ax + by = e, cx + dy = f: x = (ed − bf)/Δ, y = (af − ec)/Δ, where Δ = ad − bc ≠ 0.

Inverse of a nonsingular 2 × 2 matrix: A⁻¹ = (1/|A|) × [d −b; −c a], i.e. adjoint divided by determinant.

Adjoint of a 2 × 2 matrix [a b; c d] is [d −b; −c a]; A · adj A = |A| · I.

Area of a triangle with vertices (x₁,y₁), (x₂,y₂), (x₃,y₃) equals half the absolute value of the determinant with a third column of 1s.

Derivative from first principles: f′(x) = lim(h→0) [f(x + h) − f(x)]/h.

Power rule: d/dx (xⁿ) = n xⁿ⁻¹; d/dx (constant) = 0; d/dx (kx) = k.

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

Chain rule: if y = f(u) and u = g(x), then dy/dx = (dy/du)(du/dx).

Standard derivatives: d/dx(eˣ) = eˣ; d/dx(aˣ) = aˣ log a; d/dx(log x) = 1/x.

Elasticity of demand: η = −(p/x)(dx/dp), usually stated as a positive number; |η| > 1 elastic, |η| < 1 inelastic, |η| = 1 unit elastic.

Marginal function is the derivative of the total function: marginal cost = d(TC)/dq, marginal revenue = d(TR)/dq, marginal utility = dU/dx.

Second derivative d²y/dx² is obtained by differentiating dy/dx again; it measures the rate of change of the slope.

A function is increasing where f′(x) > 0 and decreasing where f′(x) < 0; stationary points satisfy f′(x) = 0.

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

Profit is maximised where marginal revenue equals marginal cost, MR = MC, provided the second-order condition holds.

Cost function C(x) = fixed cost + variable cost; average cost AC = C(x)/x. Minimum AC occurs where AC = MC.

Revenue R = p·x with demand x as a function of price; MR = dR/dx. Total revenue is maximised where MR = 0 and demand is elastic above that output.

Elasticity of demand at price p: η = −(p/x)·(dx/dp). Revenue rises with a price cut when demand is elastic (|η| > 1).

Break-even output is the value of x where total revenue equals total cost, i.e. profit P(x) = R(x) − C(x) = 0.

For a total cost function, the marginal cost at output x approximates the extra cost of producing the next unit.

Integration is the reverse of differentiation: ∫xⁿ dx = xⁿ⁺¹/(n+1) + c for n ≠ −1; ∫(1/x) dx = log|x| + c.

∫eˣ dx = eˣ + c; ∫aˣ dx = aˣ/log a + c; ∫k f(x) dx = k ∫f(x) dx; ∫[f(x) ± g(x)] dx = ∫f(x) dx ± ∫g(x) dx.

Fundamental theorem: ∫ₐᵇ f(x) dx = F(b) − F(a), where F is any antiderivative of f.

Properties: ∫ₐᵇ f dx = −∫ᵦᵃ f dx; ∫ₐᵃ f dx = 0; ∫ₐᵇ f dx = ∫ₐᶜ f dx + ∫꜀ᵇ f dx.

Substitution rule: ∫f(g(x)) g′(x) dx = ∫f(u) du with u = g(x); change limits accordingly in a definite integral.

Integration by parts: ∫u v′ dx = uv − ∫u′ v dx; choose u by the order inverse, logarithmic, algebraic, trigonometric, exponential.

Definite integral of a marginal function over an interval gives the total change: ∫ₐᵇ MC(q) dq = C(b) − C(a), and similarly for MR and marginal propensity.

Consumer's surplus at price p₀ and demand x₀ is ∫₀ˣ⁰ p(x) dx − p₀x₀, the area between the demand curve and the price line.

Area between a curve y = f(x), the x-axis and lines x = a, x = b is ∫ₐᵇ |f(x)| dx; split the interval where f changes sign.

Area bounded by two curves y = f(x) above and y = g(x) below, from x = a to x = b, is ∫ₐᵇ [f(x) − g(x)] dx.

Total cost from marginal cost: C(x) = ∫₀ˣ MC(q) dq + fixed cost, with the constant of integration equal to fixed cost.

Total revenue from marginal revenue: R(x) = ∫₀ˣ MR(q) dq, since R(0) = 0 when no units are sold.

Additional cost of raising output from a to b is ∫ₐᵇ MC(q) dq; this is a definite integral with no constant of integration.

Profit function recovered by integration: P(x) = ∫[MR(q) − MC(q)] dq + c, with c fixed by a known profit at some output.

Consumer's surplus = ∫₀ˣ⁰ (demand price) dx − p₀x₀; producer's surplus = p₀x₀ − ∫₀ˣ⁰ (supply price) dx.

Area under a demand curve between two prices equals the total revenue change implied by moving between those prices, when quantity is expressed as a function of price.

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 containing as many arbitrary constants as the order of the equation is the general solution; a particular solution fixes those constants from given conditions.

Variable separable form: dy/dx = f(x)g(y) is solved by separating, ∫dy/g(y) = ∫f(x) dx, then integrating both sides.

Linear differential equation of first order: dy/dx + Py = Q, with integrating factor e^∫P dx, and solution y·(IF) = ∫Q·(IF) dx + c.

Homogeneous equation dy/dx = f(y/x) is solved by substituting y = vx, so dy/dx = v + x dv/dx, giving a separable equation in v and x.

Exponential growth or decay: dy/dt = ky gives y = y₀e^{kt}; k > 0 growth, k < 0 decay.

Continuous compounding of a principal P at rate r for t years follows dA/dt = rA, giving A = Pe^{rt}.

Verify a proposed solution by substituting it and its derivatives into the equation; the equation must reduce to an identity.

Binomial distribution B(n, p): P(X = r) = ⁿCᵣ pʳ qⁿ⁻ʳ, r = 0, 1, …, n, with q = 1 − p. Conditions: fixed n independent trials, two outcomes, constant p.

Binomial mean = np, variance = npq, standard deviation = √(npq).

Poisson distribution P(X = r) = e^{−m} mʳ / r!, r = 0, 1, 2, …, where m is the mean number of occurrences.

Poisson mean = m and variance = m, so mean equals variance; this distinguishes it from the binomial.

Poisson approximation to binomial is used when n is large and p is small, with m = np (typically n ≥ 50, p ≤ 0.05).

For either distribution, P(X ≥ r) = 1 − P(X ≤ r − 1) and probabilities over all r sum to 1.

For a Poisson variable, P(X = 0) = e^{−m}; this is the probability of no occurrence in the interval.

Binomial is used for counts of successes in a fixed number of trials; Poisson for counts of rare events per unit time, length or area.

A discrete random variable takes values with probabilities p(x) satisfying p(x) ≥ 0 and Σ p(x) = 1.

Expected value (mean): E(X) = Σ x·p(x). For a function, E[g(X)] = Σ g(x)·p(x).

Variance: Var(X) = E(X²) − [E(X)]², where E(X²) = Σ x²·p(x). Standard deviation is the positive square root of variance.

Shortcut rules: E(aX + b) = aE(X) + b and Var(aX + b) = a²Var(X); adding a constant does not change variance.

Cumulative distribution F(x) = P(X ≤ x) is a non-decreasing step function with F(x) = 0 below the smallest value and 1 at or above the largest.

If X and Y are independent, Var(X ± Y) = Var(X) + Var(Y) and E(XY) = E(X)E(Y).

Coefficient of variation = (standard deviation / mean) × 100%; it compares variability of distributions with different means.

Price relative = (p₁/p₀) × 100; quantity relative = (q₁/q₀) × 100, where subscript 0 is base year and 1 is current year.

Simple aggregative price index: P₀₁ = (Σp₁/Σp₀) × 100. It ignores quantity weights and is sensitive to units.

Laspeyres price index uses base-year quantities: P₀₁ = (Σp₁q₀/Σp₀q₀) × 100.

Paasche price index uses current-year quantities: P₀₁ = (Σp₁q₁/Σp₀q₁) × 100.

Fisher's ideal index is the geometric mean of Laspeyres and Paasche: P₀₁ = √(L × P), and it satisfies both time-reversal and factor-reversal tests.

Weighted aggregative index with explicit weights w: P₀₁ = (Σpw/Σw) × 100; weighted average of price relatives uses w = p₀q₀.

Cost of living index numbers use a fixed consumption basket; dearness allowance adjustments are based on such index numbers.

Chain index numbers link year-to-year relatives; the product of link relatives divided by 100 repeatedly gives the index relative to the original base.

Components of a time series: secular trend (T), seasonal variation (S), cyclical variation (C) and irregular or random variation (I).

Multiplicative model: Y = T × S × C × I. Additive model: Y = T + S + C + I.

Moving average method smooths a series; a 4-quarterly moving average is centred by taking a 2-period moving average of the 4-period averages.

Method of least squares fits a straight-line trend y = a + bx with b = Σ(x − x̄)(y − ȳ)/Σ(x − x̄)² and a = ȳ − b x̄; coding x about the middle makes Σx = 0 and simplifies to b = Σxy/Σx².

Seasonal index for a season = (average of that season's values / overall average) × 100 in the simple ratio-to-moving-average approach.

The sum of seasonal indices should be 400 for four quarters or 1200 for twelve months; adjust proportionally if it is not.

Trend values from a fitted line are used for short-term forecasting by substituting the required period number into y = a + bx.

In a multiplicative model, deseasonalised value = observation / seasonal index; this isolates trend and cycle.

Real (deflated) wage or income = (money income / price index) × 100; this converts nominal values to base-year purchasing power.

Purchasing power of money = 100 / price index (as a multiple); a rise in the index lowers purchasing power.

If a price index rises from 100 to 125, a money income must rise by 25% to keep real income unchanged.

Deflation of a value series by a price index uses V_real = (V_nominal / index) × 100 for each year, with the same base.

A time series of sales deflated by a price index gives volume of sales in constant prices, useful for comparing real growth.

Index numbers are not additive: an index for a group is not the simple sum of subgroup indices but a weighted aggregate.

Forecast from a trend line: substitute the coded time value into y = a + bx, then adjust by the seasonal index if quarterly data are modelled multiplicatively.