Tamil Nadu HSE First Year Business Mathematics and Statistics flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Determinant of 2×2 matrix

For matrix [[a,b],[c,d]], the determinant equals ad − bc. A zero determinant means the matrix is singular and has no inverse.

Inverse of a matrix

If A is square and det(A) ≠ 0, then A inverse equals adj(A) divided by det(A). If det(A)=0, the inverse does not exist.

Cramer's rule condition

Cramer's rule solves a square linear system with a unique solution only when the coefficient determinant is nonzero; each variable equals a determinant ratio.

nPr formula

The number of ordered arrangements of r objects chosen from n distinct objects is nPr = n!/(n−r)!, where r is at most n.

nCr formula

The number of unordered selections of r objects from n distinct objects is nCr = n!/[r!(n−r)!], and nCr equals nC(n−r).

General term binomial expansion

In the expansion of (a+b)^n, the term containing b^r is T_(r+1) = nCr a^(n−r) b^r, for r from zero to n.

nth term of an AP

For an arithmetic progression with first term a and common difference d, the nth term is a+(n−1)d; consecutive terms differ by the constant d.

Sum of infinite GP

An infinite geometric series with first term a and common ratio r converges only when |r|<1, and its sum is a/(1−r).

Pythagorean identity

For every angle θ, sin²θ + cos²θ = 1. Dividing by cos²θ gives 1+tan²θ=sec²θ; dividing by sin²θ gives 1+cot²θ=csc²θ.

General solution for sine

If sin θ = sin α, then θ = nπ + (−1)^n α for every integer n. This expression lists all angles having the same sine value.

Cosine rule

In a triangle with sides a, b, c opposite angles A, B, C, a² = b² + c² − 2bc cos A, with analogous forms for other sides.

Distance between two points

The straight-line distance between points (x₁,y₁) and (x₂,y₂) is the square root of (x₂−x₁)²+(y₂−y₁)². This follows from the Pythagorean theorem.

Perpendicular line slopes

Two nonvertical lines are perpendicular exactly when the product of their slopes is −1; a horizontal line is perpendicular to a vertical line.

Circle centre and radius

The circle (x−h)²+(y−k)²=r² has centre (h,k) and radius r. In x²+y²+2gx+2fy+c=0, centre is (−g,−f) and radius squared is g²+f²−c.

Derivative of x^n

For any real exponent n, d/dx(x^n)=n x^(n−1). This power rule also covers constants when n=0, giving derivative zero.

Chain rule

If y=f(u) and u=g(x), then dy/dx = (dy/du)(du/dx). The rule differentiates composite functions by multiplying outer and inner derivatives.

Second derivative test

At a stationary point where f′(x)=0, f″(x)<0 gives a local maximum and f″(x)>0 gives a local minimum; f″(x)=0 is inconclusive.

Power rule for integration

For n ≠ −1, ∫x^n dx = x^(n+1)/(n+1)+C. The separate case ∫x^(−1)dx equals ln|x|+C for nonzero x.

Fundamental theorem of calculus

If F is an antiderivative of continuous f, then the definite integral from a to b of f(x)dx equals F(b)−F(a).

Integration by parts

For differentiable u and v, ∫u dv = uv − ∫v du. Choose u by ILATE: inverse, logarithmic, algebraic, trigonometric, exponential.

Compound interest formula

For principal P, annual rate R percent and n years, amount A=P(1+R/100)^n and compound interest equals A−P when interest is compounded annually.

Effective annual rate

If nominal annual rate r is compounded m times yearly, the effective annual rate is (1+r/m)^m−1, expressing the true yearly growth.

Present value of annuity

The present value of n equal payments A at interest rate i per period, first payment one period away, is A[1−(1+i)^(−n)]/i.

Grouped mean formula

For grouped data with frequencies f and class midpoints x, the mean is Σfx divided by Σf. It weights each midpoint by its class frequency.

Standard deviation

Standard deviation is the positive square root of variance. For ungrouped data it equals √[Σ(x−mean)²/n] or √[Σx²/n−mean²].

Coefficient of variation

Coefficient of variation equals standard deviation divided by mean, often multiplied by 100 percent. It compares relative variability across data sets with different units or means.

Addition rule probability

For events A and B, P(A∪B)=P(A)+P(B)−P(A∩B). For mutually exclusive events, the intersection probability is zero, so probabilities simply add.

Conditional probability

For P(B)>0, P(A|B)=P(A∩B)/P(B). It gives the probability of A after restricting the sample space to event B.

Independent events product rule

Events A and B are independent exactly when P(A∩B)=P(A)P(B), equivalently P(A|B)=P(A) when P(B)>0. Independence means one event's occurrence does not change the other's probability.

Pearson correlation coefficient

Karl Pearson's r equals covariance of x and y divided by the product of their standard deviations; it ranges from −1 to 1 and measures linear association.

Spearman rank correlation

Spearman's rank correlation is r=1−6Σd²/[n(n²−1)], where d is the difference between paired ranks and n is the number of pairs.

Regression line y on x

The regression line of y on x is y−ȳ=byx(x−x̄), where byx=r(σy/σx). It passes through the means and predicts y from x.