Assam HS Class 11-12 Business Mathematics & Statistics cheat sheet

Set operations: n(A∪B) = n(A) + n(B) − n(A∩B); for three sets subtract the three pairwise intersections and add back the triple intersection.

De Morgan's laws: (A∪B)′ = A′∩B′ and (A∩B)′ = A′∪B′; useful when counting 'neither' categories in surveys.

Only subsets of A×B are relations from A to B; a relation on A is reflexive if (a,a)∈R for all a, symmetric if (a,b)∈R implies (b,a)∈R, transitive if (a,b),(b,c)∈R implies (a,c)∈R.

An equivalence relation is reflexive, symmetric and transitive; it partitions the set into disjoint equivalence classes.

A function f: A→B assigns exactly one image to each element of A; a relation is a function only if no element of the domain has two images.

Composite function: (f∘g)(x) = f(g(x)), defined only when range of g lies inside domain of f.

Inverse f⁻¹ exists iff f is one-one (injective) and onto (surjective); then f⁻¹(y)=x exactly when f(x)=y.

Business uses: demand and cost as functions of quantity, revenue R(x)=p·x, break-even where R(x)=C(x), and profit P(x)=R(x)−C(x).

Matrix product AB is defined only when columns of A equal rows of B; the result has order (rows of A)×(columns of B), and AB ≠ BA in general.

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

2×2 determinant: |A| = ad − bc for A = [[a,b],[c,d]]; A is singular when |A| = 0 and invertible otherwise.

Inverse of 2×2 matrix: A⁻¹ = (1/|A|)·[[d,−b],[−c,a]], valid only when |A| ≠ 0.

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

Cofactor expansion along any row or column gives the same determinant; sign pattern follows (+ − +) alternating positions.

A system AX = B has a unique solution iff |A| ≠ 0; if |A| = 0 it has no solution or infinitely many.

Input-output (Leontief) model: X = (I − A)⁻¹D, where A is the technology matrix and D final demand; requires |I − A| ≠ 0.

Arithmetic progression: nth term aₙ = a + (n−1)d; sum Sₙ = (n/2)[2a + (n−1)d] = (n/2)(a + l).

Geometric progression: nth term aₙ = arⁿ⁻¹; sum Sₙ = a(rⁿ−1)/(r−1) for r ≠ 1, and S∞ = a/(1−r) only when |r| < 1.

Three terms in AP are best taken as a−d, a, a+d; three terms in GP as a/r, a, ar.

Arithmetic mean of a and b is (a+b)/2; geometric mean is √(ab), defined for positive a, b, and GM ≤ AM with equality when a = b.

A sum of money grows in GP: amounts after successive years at compound interest form a GP with ratio (1+i).

Sinking fund and EMI payment streams are GPs of payments; the accumulated value is a finite GP sum with r = (1+i).

If a, b, c are in GP then b² = ac; if in AP then 2b = a + c - use this to test a given triplet quickly.

Number of terms of a finite AP: n = [(last term − first term)/d] + 1, a frequent step in instalment and depreciation problems.

Fundamental principle: if one task has m ways and another n ways, the two together have m×n ways; add for mutually exclusive choices.

Permutations of n distinct objects taken r at a time: ⁿPᵣ = n!/(n−r)!; with repetition allowed it is nʳ.

Combinations: ⁿCᵣ = n!/[r!(n−r)!]; order is irrelevant, and ⁿCᵣ = ⁿCₙ₋ᵣ.

ⁿPᵣ = r!·ⁿCᵣ; also ⁿCᵣ + ⁿCᵣ₋₁ = ⁿ⁺¹Cᵣ (Pascal-type identity).

Permutations of n objects with p alike of one kind and q of another: n!/(p!q!).

Circular permutations of n distinct objects: (n−1)!; if clockwise and anticlockwise arrangements are indistinguishable, (n−1)!/2.

At least one condition: count = total − none; e.g. selections with at least one woman = all selections − all-men selections.

Number of ways to distribute n distinct things into r distinct boxes with no restriction is rⁿ; selections of any size from n objects total 2ⁿ.

Simple interest: SI = P·i·t; amount A = P(1 + it). Here i is the rate per period and t the number of periods.

Compound interest: A = P(1 + i)ⁿ; compound interest earned = A − P. For m conversions per year use rate i/m over mn periods.

Effective annual rate from nominal rate i compounded m times a year: E = (1 + i/m)ᵐ − 1.

Present value of a single future amount: PV = FV/(1 + i)ⁿ; discounting is the inverse of compounding.

Future value of an ordinary annuity (payments at period end): FV = R·[((1+i)ⁿ − 1)/i], where R is the periodic payment.

Present value of an ordinary annuity: PV = R·[1 − (1+i)⁻ⁿ]/i; for an annuity due multiply by (1+i).

Sinking fund payment: R = S·i/[((1+i)ⁿ − 1)], the payment needed each period to accumulate amount S.

Loan EMI is the annuity-due form of the PV formula: R = P·i(1+i)ⁿ/[(1+i)ⁿ − 1] for loan P over n periods.

Limit rules: limit of a sum, product and quotient equals the corresponding combination of limits, provided the denominator limit is nonzero.

Standard limits: lim(x→0) (sin x)/x = 1, lim(x→0)(eˣ−1)/x = 1, lim(x→0)(1+x)^(1/x) = e, lim(x→∞)(1+1/x)ˣ = e.

Derivative rules: d/dx(xⁿ)=nxⁿ⁻¹, product rule (uv)′=u′v+uv′, quotient rule (u/v)′=(u′v−uv′)/v².

Chain rule: dy/dx = (dy/du)(du/dx); for exponential and log, d/dx(eˣ)=eˣ and d/dx(ln x)=1/x.

Marginal cost is dC/dx, marginal revenue dR/dx, marginal profit dP/dx; marginal revenue equals price plus x times dp/dx under a downward-sloping demand.

Profit is maximised where MR = MC with d²R/dx² < d²C/dx²; minima occur where the second derivative is positive.

Elasticity of demand: η = −(p/x)(dx/dp); demand is elastic if |η| > 1, inelastic if |η| < 1.

Integration basics: ∫xⁿdx = xⁿ⁺¹/(n+1) + C for n ≠ −1; ∫(1/x)dx = ln|x| + C; definite integral gives area under a curve between limits.

Statistics has two senses: plural (numerical data) and singular (methods of collection, presentation, analysis and interpretation).

Data sources: primary data collected first-hand (survey, observation, experiment) and secondary data already published by others.

A population is the whole group under study; a sample is a subset drawn from it, and a census covers every unit.

Classification groups raw data by attributes (qualitative) or variables (discrete/continuous); tabulation arranges it in rows and columns with a clear title and units.

Frequency distribution: exclusive classes (0–10, 10–20) have no gap; inclusive classes (10–19, 20–29) need an adjustment of half the gap to convert.

Class mark = (lower limit + upper limit)/2; class width = upper limit − lower limit; relative frequency = frequency/total.

Graphical presentation: histogram for continuous frequency data, frequency polygon joining class marks, ogive (cumulative curve) for median and quartiles.

Pie chart angle for a component = (component value/total)×360°; good practice requires a source note and consistent scale.

Arithmetic mean: direct method Σx/n; for grouped data x̄ = Σfx/Σf; step-deviation x̄ = A + h(Σfu/Σf).

Median of grouped data: M = L + [(N/2 − cf)/f]·h, with L the lower limit, cf the cumulative frequency before the median class.

Mode of grouped data: Mo = L + [(f₁ − f₀)/(2f₁ − f₀ − f₂)]·h; empirical relation Mode ≈ 3Median − 2Mean.

Quartiles: Q₁ at N/4 and Q₃ at 3N/4 using the same interpolation formula as the median; quartile deviation = (Q₃ − Q₁)/2.

Range = maximum − minimum; mean deviation = Σ|x − A|/n about mean or median; coefficient of mean deviation = MD/mean.

Variance σ² = Σ(x − x̄)²/n (population) or Σ(x − x̄)²/(n−1) (sample); standard deviation is its positive square root.

Coefficient of variation = (σ/x̄)×100; the series with the larger CV is less consistent.

Combined mean of two groups: x̄ = (n₁x̄₁ + n₂x̄₂)/(n₁ + n₂); combined variance also needs the squared deviations of the group means.

Karl Pearson coefficient r = Σ(x−x̄)(y−ȳ)/[√Σ(x−x̄)²·√Σ(y−ȳ)²]; it always lies between −1 and +1.

r = 0 means no linear relation, not necessarily no relation; r = ±1 means perfect linear relation.

Shortcut form: r = [nΣxy − ΣxΣy]/√{[nΣx² − (Σx)²][nΣy² − (Σy)²]}.

Spearman rank correlation: ρ = 1 − 6Σd²/[n(n²−1)], where d is the difference of ranks; use when data are qualitative or ordinal.

Regression of y on x: ŷ = a + bx with b = Σ(x−x̄)(y−ȳ)/Σ(x−x̄)² and a = ȳ − b·x̄.

Regression of x on y has slope 1/b′; the two lines intersect at (x̄, ȳ) and are distinct unless |r| = 1.

r² = b_xy · b_yx, the product of the two regression coefficients; r carries the sign of both slopes.

Estimate from regression: b_xy = r·(σy/σx); the regression line minimises the sum of squared vertical deviations.

Four components: secular trend (T), seasonal variation (S), cyclical variation (C) and irregular variation (I); the multiplicative model is Y = T×S×C×I.

Moving average method: a 4-quarter moving average is centred by taking a 2-period average of successive 4-period averages to align with the middle quarter.

Trend by least squares: fit Y = a + bt with t coded as 0,1,2… or symmetric (−2,−1,0,1,2) so that Σt = 0 and a = ΣY/n, b = ΣtY/Σt².

Seasonal index = (original value/centred moving average)×100; indices average to 100 after adjustment.

Simple aggregative index: P₀₁ = (ΣP₁/ΣP₀)×100; it ignores quantities and is dominated by high-priced items.

Laspeyres index uses base-year quantities: P₀₁ = (ΣP₁q₀/ΣP₀q₀)×100; Paasche uses current-year quantities.

Fisher's ideal index is the geometric mean √(Laspeyres × Paasche) and satisfies the time-reversal and factor-reversal tests.

Real wage or real income = (money wage/consumer price index)×100; a CPI above 100 indicates price rise relative to the base.

Classical probability P(E) = favourable outcomes/total equally likely outcomes; 0 ≤ P(E) ≤ 1 and P(E′) = 1 − P(E).

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

Conditional probability: P(A|B) = P(A∩B)/P(B), P(B) > 0; multiplication rule P(A∩B) = P(A)·P(B|A).

Events are independent when P(A∩B) = P(A)P(B), equivalently P(A|B) = P(A).

Total probability: P(A) = ΣP(Eᵢ)P(A|Eᵢ) over a partition of the sample space.

Bayes' theorem: P(Eᵢ|A) = P(Eᵢ)P(A|Eᵢ)/ΣP(Eⱼ)P(A|Eⱼ); used for revising probabilities after evidence.

Random variable expectation E(X) = Σx·P(x); variance V(X) = E(X²) − [E(X)]².

Binomial distribution: P(X = r) = ⁿCᵣ pʳ qⁿ⁻ʳ with mean np and variance npq; expected monetary value = Σ(outcome × probability).