CHSE Odisha Business Mathematics & Statistics cheat sheet

For sets A and B: n(A∪B) = n(A) + n(B) − n(A∩B); n(A∪B∪C) = Σn(A) − Σn(A∩B) + n(A∩B∩C).

Power set P(A) has 2^n elements if n(A)=n; number of subsets of size r is C(n,r).

A relation R on A is reflexive if (a,a)∈R for all a∈A, symmetric if (a,b)∈R ⇒ (b,a)∈R, transitive if (a,b),(b,c)∈R ⇒ (a,c)∈R; equivalence if all three hold.

A function f:A→B is injective (one-one) if f(a1)=f(a2) ⇒ a1=a2; surjective (onto) if range = B; bijective if both.

Composition: (g∘f)(x) = g(f(x)); inverse f⁻¹ exists iff f is bijective, and (f⁻¹∘f)(x)=x.

Domain of f(x)=√(g(x)) requires g(x)≥0; domain of f(x)=1/g(x) requires g(x)≠0; log requires argument >0.

Number of relations from a set of m elements to a set of n elements is 2^(mn); number of functions is n^m.

If n(A)=p, n(B)=q, number of one-one functions from A to B is P(q,p) when q≥p, else 0.

De Morgan's laws: (A∪B)′ = A′∩B′ and (A∩B)′ = A′∪B′.

For A=(a_ij){m×n} and B=(b_ij){m×n}, (A+B)_ij = a_ij + b_ij; multiplication AB is defined only if columns of A = rows of B.

Transpose rules: (A′)′ = A, (A+B)′ = A′+B′, (AB)′ = B′A′; inverse (AB)⁻¹ = B⁻¹A⁻¹.

A square matrix A is invertible iff det(A) ≠ 0; then A⁻¹ = adj(A)/det(A).

2×2 determinant: |a b; c d| = ad − bc; 3×3 determinant expands by cofactors along any row or column.

Properties: det(A′)=det(A), det(AB)=det(A)det(B), det(kA)=k^n det(A) for n×n A; det of a matrix with two identical rows is 0.

Cramer's rule for AX=B: x_i = det(A_i)/det(A), where A_i replaces column i by B, provided det(A)≠0.

Consistency: AX=B has unique solution if det(A)≠0; if det(A)=0 and all det(A_i)=0, infinitely many or no solution.

For a symmetric matrix A, A′=A; for skew-symmetric, A′=−A, and diagonal entries of skew-symmetric are 0.

Adjoint property: A·adj(A) = adj(A)·A = det(A)·I.

Number of permutations of n distinct objects taken r at a time: P(n,r) = n!/(n−r)!.

Number of combinations: C(n,r) = n!/[r!(n−r)!]; C(n,r)=C(n,n−r).

Permutations of n objects with repetitions: n!/(p!q!r!…) where p,q,r are frequencies of identical items.

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

Number of ways to select at least one object from n distinct objects is 2^n − 1.

Number of diagonals in a polygon with n sides is C(n,2) − n.

Pascal's identity: C(n,r) + C(n,r−1) = C(n+1,r).

Number of ways to distribute n identical objects into r distinct boxes is C(n+r−1, r−1).

If n is even, C(n,r) is maximum at r=n/2; if n is odd, maximum at r=(n−1)/2 and (n+1)/2.

AP: nth term a_n = a + (n−1)d; sum S_n = n/2[2a + (n−1)d] = n/2(a + l).

Arithmetic mean between a and b is (a+b)/2; inserting m means gives d=(b−a)/(m+1).

GP: nth term a_n = ar^(n−1); sum S_n = a(1−r^n)/(1−r) for r≠1; infinite sum S_∞ = a/(1−r) if |r|<1.

Geometric mean between a and b is √(ab); for positive numbers, AM ≥ GM ≥ HM.

Sum of first n natural numbers: n(n+1)/2; sum of squares: n(n+1)(2n+1)/6; sum of cubes: [n(n+1)/2]^2.

If a,b,c are in AP then 2b=a+c; if in GP then b²=ac; if in HP then 2/b = 1/a + 1/c.

For a GP, the product of terms equidistant from the beginning and end is constant.

Sum of an infinite GP with |r|<1 is a/(1−r); if |r|≥1 the infinite sum does not exist.

Distance between points (x1,y1) and (x2,y2): √[(x2−x1)²+(y2−y1)²]; section formula: ((mx2+nx1)/(m+n), (my2+ny1)/(m+n)).

Slope m = (y2−y1)/(x2−x1); equation through (x1,y1): y−y1 = m(x−x1).

Slope-intercept form y=mx+c; intercept form x/a + y/b = 1; general form Ax+By+C=0 has slope −A/B.

Two lines are parallel if m1=m2; perpendicular if m1m2=−1.

Distance from point (x1,y1) to line Ax+By+C=0 is |Ax1+By1+C|/√(A²+B²).

Angle θ between lines: tanθ = |(m2−m1)/(1+m1m2)|.

Area of triangle with vertices (x1,y1),(x2,y2),(x3,y3) = ½|x1(y2−y3)+x2(y3−y1)+x3(y1−y2)|.

Equation of 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), radius √(g²+f²−c).

Family of lines through intersection of L1=0 and L2=0: L1 + λL2 = 0.

Derivative rules: d/dx(x^n)=nx^(n−1); product: (uv)′=u′v+uv′; quotient: (u/v)′=(u′v−uv′)/v²; chain: dy/dx = dy/du · du/dx.

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

For parametric x=f(t), y=g(t): dy/dx = (dy/dt)/(dx/dt).

For implicit F(x,y)=0: dy/dx = −(∂F/∂x)/(∂F/∂y).

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

A function is increasing where f′(x)>0 and decreasing where f′(x)<0.

Equation of tangent at (x1,y1) on y=f(x): y−y1 = f′(x1)(x−x1); normal slope = −1/f′(x1).

Marginal cost = dC/dx; marginal revenue = dR/dx; profit maximum when MR=MC and second derivative condition holds.

Point of inflection occurs where f″(x)=0 and f″ changes sign.

Fundamental theorem: ∫_a^b f(x)dx = F(b)−F(a), where F′(x)=f(x).

Standard integrals: ∫x^n dx = x^(n+1)/(n+1)+C (n≠−1); ∫1/x dx = ln|x|+C; ∫e^x dx = e^x+C; ∫sin x dx = −cos x+C; ∫cos x dx = sin x+C.

Integration by parts: ∫u dv = uv − ∫v du; choose u by ILATE order (Inverse, Logarithmic, Algebraic, Trigonometric, Exponential).

Definite integral properties: ∫_a^b f(x)dx = −∫_b^a f(x)dx; ∫_a^b f(x)dx = ∫_a^c f(x)dx + ∫_c^b f(x)dx.

Area under curve y=f(x) from x=a to x=b: A = ∫_a^b |f(x)| dx.

Area between curves y=f(x) and y=g(x) from a to b: A = ∫_a^b |f(x)−g(x)| dx.

For substitution, ∫f(g(x))g′(x)dx = ∫f(u)du with u=g(x).

Consumer surplus = ∫_0^Q D(q)dq − P·Q; producer surplus = P·Q − ∫_0^Q S(q)dq.

If f is odd, ∫−a^a f(x)dx = 0; if f is even, ∫−a^a f(x)dx = 2∫_0^a f(x)dx.

Mean (ungrouped) = Σx/n; grouped mean = Σfx/Σf where x is class midpoint.

Median: for ungrouped data, middle value after sorting; for grouped, median = L + [(n/2 − cf)/f]·h.

Mode: most frequent value; for grouped, mode = L + [(f1−f0)/(2f1−f0−f2)]·h.

Empirical relation: Mode ≈ 3 Median − 2 Mean.

Range = max − min; quartile deviation = (Q3−Q1)/2.

Mean deviation about a: MD = Σ|x−a|/n; minimum when a is median.

Variance σ² = Σ(x−x̄)²/n; standard deviation σ = √variance; for grouped data use Σfx²/Σf − (Σfx/Σf)².

Coefficient of variation = (σ/mean)×100%; used to compare consistency.

For a frequency distribution, Σf = n and Σf(x−x̄) = 0.

Karl Pearson's correlation coefficient r = [nΣxy − ΣxΣy] / √{[nΣx²−(Σx)²][nΣy²−(Σy)²]}; −1 ≤ r ≤ 1.

Spearman's rank correlation: ρ = 1 − 6Σd²/[n(n²−1)], where d is rank difference.

Regression line of y on x: y − ȳ = b_yx (x − x̄), with b_yx = r σ_y/σ_x = Σ(x−x̄)(y−ȳ)/Σ(x−x̄)².

Regression line of x on y: x − x̄ = b_xy (y − ȳ), with b_xy = r σ_x/σ_y.

Property: r = ±√(b_yx · b_xy); both regression coefficients have the same sign as r.

The two regression lines intersect at (x̄, ȳ).

If r=0, variables are uncorrelated; regression lines are perpendicular.

Coefficient of determination r² gives the proportion of variance in y explained by x.

Correlation does not imply causation; r is independent of change of origin and scale (except sign change for negative scale).

Classical probability P(E) = number of favourable outcomes / total equally likely outcomes; 0 ≤ P(E) ≤ 1.

Addition rule: P(A∪B) = P(A) + P(B) − P(A∩B); for mutually exclusive events, P(A∪B)=P(A)+P(B).

Conditional probability P(A|B) = P(A∩B)/P(B); multiplication rule P(A∩B)=P(A)P(B|A).

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

Bayes' theorem: P(A_i|B) = P(B|A_i)P(A_i) / Σ_j P(B|A_j)P(A_j).

Binomial distribution: P(X=r) = C(n,r) p^r q^(n−r), mean = np, variance = npq.

Poisson distribution: P(X=r) = e^(−λ) λ^r / r!, mean = variance = λ.

Normal distribution: symmetric about mean μ, total area 1; standard normal Z=(X−μ)/σ.

For any distribution, E(aX+b)=aE(X)+b and Var(aX+b)=a²Var(X).

Laspeyres price index: P_L = (Σp1q0 / Σp0q0) × 100; Paasche: P_P = (Σp1q1 / Σp0q1) × 100.

Fisher's ideal index: P_F = √(P_L × P_P); it satisfies both time reversal and factor reversal tests.

Marshall–Edgeworth index uses average quantities: Σp1[(q0+q1)/2] / Σp0[(q0+q1)/2] × 100.

Consumer Price Index (CPI) = (Σp1q0 / Σp0q0) × 100, with base-year quantities as weights.

Time reversal test: P01 × P10 = 1; factor reversal test: P01 × Q01 = Σp1q1/Σp0q0.

Chain index: link relatives multiplied successively; chain index = (current link relative × previous chain index)/100.

Moving average method: for period m, average of m consecutive values; if m is even, centre the averages by two-period moving average.

Least squares trend line: y = a + bx, with b = [nΣxy − ΣxΣy]/[nΣx²−(Σx)²], a = ȳ − b x̄ (x coded as time).

Seasonal index = (seasonal average / overall average) × 100.