Tamil Nadu HSE Second Year Business Mathematics and Statistics cheat sheet

A square matrix A is singular when det(A) = 0 and invertible (non-singular) when det(A) ≠ 0; only invertible matrices have an inverse.

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

Cramer's rule for AX = B with det A ≠ 0: x₁ = det A₁/det A, where A₁ is A with column 1 replaced by B; likewise for other variables.

Consistency test: the system AX = B is consistent if rank(A) = rank([A|B]); it has a unique solution when that common rank equals the number of unknowns.

Properties: det(AB) = det A · det B, det(Aᵀ) = det A, det(kA) = kⁿ det A for n×n A, and det(A⁻¹) = 1/det A.

For 3×3 A, det A = a₁₁(a₂₂a₃₃ − a₂₃a₃₂) − a₁₂(a₂₁a₃₃ − a₂₃a₃₁) + a₁₃(a₂₁a₃₂ − a₂₂a₃₁), expanding along the first row.

(AB)ᵀ = BᵀAᵀ and (AB)⁻¹ = B⁻¹A⁻¹; order of reversal matters when both factors are invertible.

A matrix is symmetric if Aᵀ = A and skew-symmetric if Aᵀ = −A; every skew-symmetric matrix of odd order has determinant 0.

If two rows (or columns) of a determinant are identical or proportional, the determinant equals 0; swapping two rows changes only its sign.

Distance between (x₁,y₁) and (x₂,y₂) is √((x₂−x₁)² + (y₂−y₁)²); midpoint is ((x₁+x₂)/2, (y₁+y₂)/2).

Section formula: point dividing P₁P₂ internally in ratio m:n is ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n)); use −n for external division.

Slope of a line through two points is m = (y₂−y₁)/(x₂−x₁); parallel lines have equal slopes and perpendicular lines satisfy m₁m₂ = −1.

Forms of a straight line: slope-intercept y = mx + c, point-slope y − y₁ = m(x − x₁), intercept form x/a + y/b = 1.

Distance from point (x₁,y₁) to line ax + by + c = 0 is |ax₁ + by₁ + c|/√(a² + b²).

Angle θ between lines with slopes m₁, m₂ satisfies tan θ = |(m₁ − m₂)/(1 + m₁m₂)|.

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

A conic's type is fixed by the discriminant B² − 4AC of Ax² + Bxy + Cy² + Dx + Ey + F = 0: negative gives ellipse, zero parabola, positive hyperbola.

Product rule: d(uv)/dx = u′v + uv′; quotient rule: d(u/v)/dx = (u′v − uv′)/v².

Chain rule: if y = f(u) and u = g(x), then dy/dx = f′(u)·g′(x); this underlies all composite-function derivatives.

Standard results: d(eˣ)/dx = eˣ, d(aˣ)/dx = aˣ ln a, d(ln x)/dx = 1/x, d(sin x)/dx = cos x, d(cos x)/dx = −sin x, d(tan x)/dx = sec²x.

A stationary point occurs where dy/dx = 0; the second derivative decides: d²y/dx² < 0 gives a local maximum, > 0 gives a local minimum, = 0 requires further testing.

Marginal cost is MC = d(TC)/dq and marginal revenue is MR = d(TR)/dq; profit is maximised where MR = MC with MC rising.

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

Increasing returns to scale or growth rates: if y = f(t), the proportional rate of change is (1/y)·(dy/dt), and for y = uv this equals the sum of the component rates.

Total cost from marginal cost: TC = ∫MC dq + fixed cost, so differentiation and integration are inverse operations in cost analysis.

Fundamental theorem: ∫ₐᵇ f(x) dx = F(b) − F(a), where F′(x) = f(x); the definite integral is independent of the variable name used.

Standard integrals: ∫xⁿ dx = xⁿ⁺¹/(n+1) + c for n ≠ −1, ∫(1/x) dx = ln|x| + c, ∫eˣ dx = eˣ + c, ∫(1/(a² + x²)) dx = (1/a)tan⁻¹(x/a) + c.

Substitution rule: ∫f(g(x))g′(x) dx = ∫f(u) du with u = g(x); for definite integrals also convert the limits.

Integration by parts: ∫u dv = uv − ∫v du, choosing u by the order inverse, logarithmic, algebraic, trigonometric, exponential.

Area between curve y = f(x), the x-axis, and lines x = a, x = b is ∫ₐᵇ |f(x)| dx; between two curves it is ∫ₐᵇ (upper − lower) dx.

Consumer's surplus at price p₀ is ∫₀^{x₀} D(x) dx − p₀x₀, where x₀ is demand at p₀; producer's surplus is p₀x₀ − ∫₀^{x₀} S(x) dx.

Properties: ∫ₐᵇ f dx = −∫ᵦᵃ f dx, ∫ₐᵃ f dx = 0, and ∫ₐᵇ f dx = ∫ₐᶜ f dx + ∫꜀ᵇ f dx for a < c < b.

For an even function f(−x) = f(x), ∫₋ₐᵃ f dx = 2∫₀ᵃ f dx; for an odd function the integral over symmetric limits is 0.

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

A first-order first-degree equation of form dy/dx = f(x)g(y) is separable: separate variables and integrate both sides, adding one arbitrary constant.

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

Homogeneous equation dy/dx = f(x,y) with f of degree zero is solved by substituting y = vx, giving dy/dx = v + x(dv/dx).

Exact equation M dx + N dy = 0 satisfies ∂M/∂y = ∂N/∂x; its solution is obtained by integrating M with respect to x and N with respect to y and combining.

Second-order linear equation with constant coefficients: solve the auxiliary equation am² + bm + c = 0; real distinct roots give y = C₁e^{m₁x} + C₂e^{m₂x}.

Repeated root m gives y = (C₁ + C₂x)e^{mx}; complex roots α ± iβ give y = e^{αx}(C₁cos βx + C₂sin βx).

A particular solution of a non-homogeneous equation is added to the complementary function; the number of arbitrary constants in the general solution equals the order.

For a discrete random variable, Σp(x) = 1 and p(x) ≥ 0; mean E(X) = Σx·p(x) and variance = E(X²) − [E(X)]².

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

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

Poisson distribution: P(X = r) = e^{−λ}λʳ/r!, where λ is the mean number of occurrences; mean = variance = λ.

Poisson applies to rare events in a fixed interval of time or space when occurrences are independent and the average rate λ is constant.

Normal distribution with mean μ and standard deviation σ: standardise by z = (x − μ)/σ, then use standard normal tables for areas.

Standard normal curve is symmetric about z = 0, has total area 1, and satisfies P(Z > 0) = 0.5 and P(−z ≤ Z ≤ z) = 2P(0 ≤ Z ≤ z).

Binomial tends to Poisson with λ = np when n is large and p is small; binomial tends to normal when np and nq are both at least 5.

Simple random sampling gives every unit an equal chance of selection; systematic sampling picks every k-th unit after a random start, with k = N/n.

Stratified sampling divides the population into homogeneous strata and samples each; proportional allocation takes nₕ = n·Nₕ/N.

Cluster sampling selects whole groups rather than individuals; it is cheaper but usually less precise than simple random sampling for the same size.

Standard error of the mean is σ/√n when σ is known, or s/√n when estimated from a large sample; it decreases as n increases.

A point estimate is a single value (such as x̄ for μ); an interval estimate gives a range with a stated confidence level.

Large-sample 95% confidence interval for μ is x̄ ± 1.96(σ/√n); the 99% interval uses 1.96 replaced by 2.58.

For a population proportion, the standard error is √(pq/n) and the 95% interval is p̂ ± 1.96√(p̂q̂/n).

An estimator is unbiased if its expected value equals the parameter; sample mean is unbiased for μ, and s² with divisor (n − 1) is unbiased for σ².

Null hypothesis H₀ states no effect or no difference; the alternative H₁ is one-sided or two-sided and determines the critical region.

Type I error rejects a true H₀ (probability α, the significance level); Type II error accepts a false H₀ (probability β), and power is 1 − β.

Large-sample test of mean: z = (x̄ − μ)/(σ/√n), compared with critical values 1.645 (one-tailed 5%), 1.96 (two-tailed 5%), 2.58 (two-tailed 1%).

Test for a proportion: z = (p̂ − p)/√(pq/n), with q = 1 − p taken from H₀.

Difference of two means (large samples): z = (x̄₁ − x̄₂)/√(σ₁²/n₁ + σ₂²/n₂), testing H₀: μ₁ = μ₂.

Small-sample test of a mean uses t = (x̄ − μ)/(s/√n) with n − 1 degrees of freedom; the t distribution is wider-tailed than normal.

Chi-square goodness of fit: χ² = Σ(O − E)²/E with degrees of freedom = number of categories − 1 (minus further estimated parameters).

Decision rule: reject H₀ when the computed statistic falls in the critical region, that is, exceeds the critical value in the direction of H₁.

Simple price index for a commodity is (p₁/p₀) × 100; a price relative uses the same formula for each item before averaging.

Laspeyres index = (Σp₁q₀/Σp₀q₀) × 100 using base-year quantities; Paasche index = (Σp₁q₁/Σp₀q₁) × 100 using current-year quantities.

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

Weighted aggregative index: ΣWR/ΣW, where R = (p₁/p₀) × 100 and W is the chosen weight; the weighted average of relatives uses the same structure.

Cost of living index uses base-year consumption weights: (Σp₁q₀/Σp₀q₀) × 100, with items grouped as food, clothing, fuel, rent and miscellaneous.

Time series 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 of period k smooths a series by averaging k consecutive values; a 4-year moving average for quarterly data needs centring to align with actual periods.

Seasonal index = (seasonal value/average of that year) × 100, or by the ratio-to-moving-average method; indices across seasons should average about 100.

Trend by least squares uses the fitted line y = a + bx with b = Σ(x − x̄)(y − ȳ)/Σ(x − x̄)², coding periods so that Σx = 0 simplifies a = ȳ and b = Σxy/Σx².