Tamil Nadu HSE First Year Business Mathematics and Statistics cheat sheet

A matrix of order m×n has m rows and n columns; addition is defined only for equal orders, while AB is defined when columns of A equal rows of B.

For A=[[a,b],[c,d]], det(A)=ad−bc; a square matrix is singular when its determinant is zero and invertible when det(A)≠0.

For an n×n matrix A, det(AB)=det(A)det(B), det(Aᵀ)=det(A), and det(kA)=kⁿ det(A).

A⁻¹=adj(A)/det(A) when det(A)≠0; adj(A) is the transpose of the cofactor matrix, and A·adj(A)=det(A)I.

Cramer's rule applies to n equations in n unknowns with coefficient determinant D≠0; each variable is Dᵢ/D, where Dᵢ replaces one column by constants.

Matrix multiplication is not generally commutative; (AB)ᵀ=BᵀAᵀ and (AB)⁻¹=B⁻¹A⁻¹ when both inverses exist.

A symmetric matrix satisfies Aᵀ=A; a skew-symmetric matrix satisfies Aᵀ=−A, so its diagonal entries are zero.

For a 3×3 skew-symmetric matrix, det(A)=0; for any square matrix, det(A)=0 means rows or columns are linearly dependent.

A rational fraction is proper when numerator degree is less than denominator degree; only proper fractions are decomposed directly into partial fractions.

For distinct linear factors, use A/(x−a)+B/(x−b); for a repeated factor (x−a)², include A/(x−a)+B/(x−a)².

For an irreducible quadratic factor ax²+bx+c, use (Ax+B)/(ax²+bx+c) in the partial-fraction decomposition.

nPr=n!/(n−r)! counts ordered arrangements; nCr=n!/[r!(n−r)!] counts unordered selections; nPr=r!·nCr.

nCr=nC(n−r), nC0=nCn=1, and Pascal's identity is nCr+nC(r−1)=(n+1)Cr.

Binomial theorem: (a+b)ⁿ=Σ_{r=0}^{n} nCr a^(n−r)b^r; the general term is T_{r+1}=nCr a^(n−r)b^r.

If n is even, (a+b)ⁿ has one middle term T_{n/2+1}; if n is odd, it has two middle terms T_{(n+1)/2} and T_{(n+3)/2}.

The sum of all binomial coefficients is 2ⁿ; the alternating sum Σ(−1)^r nCr is zero for n>0.

An arithmetic progression has nth term a+(n−1)d and sum Sₙ=n/2[2a+(n−1)d]=n/2(first+last).

A geometric progression has nth term ar^(n−1); its finite sum is a(1−rⁿ)/(1−r) for r≠1.

An infinite geometric series converges exactly when |r|<1, with sum a/(1−r); it diverges when |r|≥1.

For positive a and b, AM=(a+b)/2, GM=√(ab), HM=2ab/(a+b), and GM²=AM×HM.

Σk=n(n+1)/2, Σk²=n(n+1)(2n+1)/6, and Σk³=[n(n+1)/2]².

Three numbers are in GP when the square of the middle term equals the product of the other two.

In an AP with an odd number of terms, the middle term equals the arithmetic mean of all terms.

A sequence is an ordered list indexed by natural numbers; its series is the sum of its terms.

Fundamental identities: sin²θ+cos²θ=1, 1+tan²θ=sec²θ, and 1+cot²θ=csc²θ.

Addition formulas: sin(A±B)=sinAcosB±cosAsinB; cos(A±B)=cosAcosB∓sinAsinB.

tan(A±B)=(tanA±tanB)/(1∓tanA tanB), provided the denominator is nonzero.

Double-angle formulas: sin2A=2sinAcosA; cos2A=cos²A−sin²A=1−2sin²A=2cos²A−1; tan2A=2tanA/(1−tan²A).

General solutions: sinθ=sinα gives θ=nπ+(−1)ⁿα; cosθ=cosα gives θ=2nπ±α; tanθ=tanα gives θ=nπ+α.

Sine rule: a/sinA=b/sinB=c/sinC=2R; cosine rule: a²=b²+c²−2bc cosA.

Arc length s=rθ and sector area A=½r²θ when θ is measured in radians; π radians equals 180°.

ASTC signs: all positive in quadrant I, sine in II, tangent in III, cosine in IV.

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

Internal section point dividing AB in ratio m:n is ((mx₂+nx₁)/(m+n),(my₂+ny₁)/(m+n)).

Slope m=(y₂−y₁)/(x₂−x₁); parallel lines have equal slopes, and perpendicular nonvertical lines satisfy m₁m₂=−1.

Line forms: y=mx+c, y−y₁=m(x−x₁), and ax+by+c=0 has slope −a/b.

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

Angle θ between lines satisfies tanθ=|(m₁−m₂)/(1+m₁m₂)|, with the acute angle taken when needed.

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

Parabola y²=4ax has focus (a,0) and directrix x=−a; ellipse x²/a²+y²/b²=1 has major axis 2a when a>b.

Derivative at x is f′(x)=lim_{h→0}[f(x+h)−f(x)]/h when the limit exists.

Power rule: d/dx(xⁿ)=nx^(n−1); derivative of a constant is zero, and differentiation is linear.

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

Trig derivatives: (sinx)′=cosx, (cosx)′=−sinx, (tanx)′=sec²x, (cotx)′=−csc²x, (secx)′=secx tanx, (cscx)′=−cscx cotx.

Exponential and logarithmic derivatives: (eˣ)′=eˣ, (aˣ)′=aˣ ln a, (lnx)′=1/x, and (log_a x)′=1/(x ln a).

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

Stationary points satisfy f′(x)=0; f″(x)<0 gives a local maximum, f″(x)>0 gives a local minimum, and f″(x)=0 is inconclusive.

Tangent at (x₁,y₁) is y−y₁=f′(x₁)(x−x₁); normal slope there is −1/f′(x₁) when f′(x₁)≠0.

Power rule: ∫xⁿdx=x^(n+1)/(n+1)+C for n≠−1; ∫x⁻¹dx=ln|x|+C.

Standard integrals: ∫eˣdx=eˣ+C, ∫aˣdx=aˣ/ln a+C, ∫sinx dx=−cosx+C, ∫cosx dx=sinx+C, ∫sec²x dx=tanx+C.

Definite integral: if F′=f, then ∫_a^b f(x)dx=F(b)−F(a); the constant C cancels.

Definite integral properties: ∫_a^b=−∫_b^a, ∫_a^a=0, and ∫_a^b=∫_a^c+∫_c^b when f is integrable on the intervals.

Integration by parts: ∫u dv=uv−∫v du; ILATE orders u as inverse, logarithmic, algebraic, trigonometric, exponential.

Substitution: ∫f(g(x))g′(x)dx=∫f(u)du when u=g(x), then substitute back to x.

Area under y=f(x) from a to b above the x-axis is ∫_a^b f(x)dx; area between curves is ∫(upper−lower)dx.

Standard inverse-trig integrals: ∫dx/(x²+a²)=(1/a)tan⁻¹(x/a)+C and ∫dx/√(a²−x²)=sin⁻¹(x/a)+C.

Simple interest I=PNR/100 for principal P, annual rate R percent and time N years; amount is P+I.

Compound amount A=P(1+R/100)ⁿ for annual compounding; compound interest is A−P.

For compounding m times yearly, A=P(1+R/(100m))^(mn); half-yearly uses m=2 and quarterly uses m=4.

Effective annual rate for nominal rate r compounded m times is (1+r/m)^m−1.

Present value of a single amount A due after n periods at rate i per period is A/(1+i)ⁿ.

Present value of an ordinary annuity of n payments A at rate i per period is A[1−(1+i)^(−n)]/i.

Future value of an ordinary annuity is A[(1+i)ⁿ−1]/i; each payment earns interest until the final period.

Depreciation at fixed annual rate r gives book value P(1−r/100)ⁿ after n years.

Ungrouped mean is Σx/n; grouped mean is Σfx/Σf, using class midpoints x and frequencies f.

Median of ordered data is the middle value for odd n and the average of the two middle values for even n.

Grouped median is L+[(N/2−cf)/f]h, where L is lower boundary of the median class, cf cumulative frequency before it, f its frequency, h class width.

Mode is the most frequent value; grouped mode is L+[(f₁−f₀)/(2f₁−f₀−f₂)]h.

Range=max−min; coefficient of range=(max−min)/(max+min).

Mean deviation about mean is Σ|x−x̄|/n for ungrouped data; coefficient is mean deviation divided by the mean.

Variance=Σ(x−x̄)²/n=Σx²/n−x̄²; standard deviation is its positive square root.

Coefficient of variation=(standard deviation/mean)×100%; it compares relative dispersion.

For moderately skewed data, mode≈3 median−2 mean; quartile deviation=(Q₃−Q₁)/2.

For equally likely outcomes, P(A)=number favourable to A/total outcomes; 0≤P(A)≤1, P(S)=1, P(∅)=0.

Complement rule: P(A′)=1−P(A); odds in favour a:b give probability a/(a+b).

Addition rule: P(A∪B)=P(A)+P(B)−P(A∩B); mutually exclusive events have P(A∩B)=0.

Conditional probability: P(A|B)=P(A∩B)/P(B) for P(B)>0.

Multiplication rule: P(A∩B)=P(A)P(B|A); independent events satisfy P(A∩B)=P(A)P(B).

Total probability: for a partition B₁,...,Bₙ, P(A)=ΣP(Bᵢ)P(A|Bᵢ).

Bayes theorem: P(Bᵢ|A)=P(Bᵢ)P(A|Bᵢ)/ΣP(Bⱼ)P(A|Bⱼ).

Counting uses nPr for ordered selections and nCr for unordered selections; probability can be computed by favourable and total counts.

Karl Pearson correlation r=cov(x,y)/(σxσy); with deviations, r=Σxy/(nσxσy), and −1≤r≤1.

A positive r indicates positive linear association; negative r indicates inverse association; r=0 indicates no linear association.

Spearman rank correlation: r=1−6Σd²/[n(n²−1)], where d is the difference between paired ranks; average ranks handle ties.

Regression line y on x: y−ȳ=byx(x−x̄), where byx=r σy/σx=Σxy/Σx² using deviations.

Regression line x on y: x−x̄=bxy(y−ȳ), where bxy=r σx/σy=Σxy/Σy².

The two regression lines intersect at (x̄,ȳ); byx and bxy have the same sign.

r²=byx·bxy; the coefficient of determination r² gives the proportion of variation in one variable explained by the other.

Correlation measures association, not causation; a zero correlation does not rule out a nonlinear relationship.