Tamil Nadu HSE Second Year Business Mathematics and Statistics flashcards

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

Singular matrix

A square matrix whose determinant equals zero. It has no inverse, and any system with it as coefficient matrix is either inconsistent or has infinitely many solutions.

Cramer's rule

A method solving AX = B when det A is non-zero. Each variable equals the determinant with that column replaced by B, divided by det A.

Skew-symmetric matrix

A square matrix satisfying A transpose equals negative A, so all diagonal entries are zero. If its order is odd, its determinant is necessarily zero.

Consistency of linear system

The system AX = B is consistent exactly when the coefficient matrix and augmented matrix have equal rank. Unique solution requires that rank to equal the number of unknowns.

Section formula

Gives coordinates of a point dividing the join of two points in a given ratio. Internal division uses mx₂ plus nx₁ over m plus n; external division replaces n by minus n.

Perpendicular lines condition

Two straight lines with slopes m₁ and m₂ are perpendicular exactly when the product m₁m₂ equals minus one. Parallel lines instead have equal slopes.

Distance from point to line

For line ax + by + c = 0 and point (x₁, y₁), the perpendicular distance is the absolute value of ax₁ + by₁ + c divided by the square root of a² + b².

Circle general equation

The equation x² + y² + 2gx + 2fy + c = 0 represents a circle with centre (−g, −f) and radius equal to the square root of g² + f² − c.

Second derivative test

At a stationary point where the first derivative vanishes, a negative second derivative indicates a local maximum, a positive one a local minimum, and zero is inconclusive.

Marginal cost

The derivative of total cost with respect to output quantity. It approximates the extra cost of producing one more unit and equals average cost plus output times its derivative.

Price elasticity of demand

Defined as minus the ratio of price to quantity multiplied by the derivative of quantity with respect to price. Absolute value above one means elastic demand.

Profit maximisation condition

Profit is maximised where marginal revenue equals marginal cost and marginal cost is rising. Equivalently, the derivative of the profit function changes from positive to negative.

Definite integral evaluation

The definite integral from a to b of f(x) equals F(b) minus F(a), where F is any antiderivative of f. The arbitrary constant cancels in the subtraction.

Integration by parts

The rule integral of u dv equals uv minus integral of v du. Choose u following the order inverse, logarithmic, algebraic, trigonometric, exponential for easiest results.

Consumer's surplus

The excess of what consumers are willing to pay over what they actually pay. It equals the integral of the demand function from zero to equilibrium quantity, minus price times quantity.

Area between two curves

Computed as the definite integral of the upper curve minus the lower curve between the intersection x-limits. Split the interval where the curves cross.

Order and degree

Order is the highest derivative appearing in a differential equation; degree is the power of that derivative once the equation is cleared of radicals and fractions in derivatives.

Integrating factor

For dy/dx + Py = Q, the factor e raised to the integral of P dx. Multiplying through makes the left side the derivative of y times the factor.

Homogeneous differential equation

An equation where dy/dx can be written as a function of y/x alone. The substitution y = vx reduces it to a separable equation in v and x.

Auxiliary equation

For a second-order linear equation with constant coefficients, the quadratic obtained by replacing derivatives with powers of m. Its roots determine the complementary function's form.

Binomial distribution mean

For n independent trials each with success probability p, the mean number of successes is np and the variance is npq, where q equals one minus p.

Poisson distribution formula

Probability of exactly r occurrences equals e to the power minus lambda, times lambda to the power r, divided by r factorial. Mean and variance both equal lambda.

Standard normal variable

Obtained by subtracting the mean from a normal observation and dividing by the standard deviation. Its distribution has mean zero and variance one, allowing table use.

Variance of a random variable

Computed as the expected value of the square minus the square of the expected value. It measures dispersion of the probability distribution about its mean.

Standard error of mean

The standard deviation of the sampling distribution of the sample mean, equal to population standard deviation divided by the square root of sample size.

Stratified sampling

The population is split into internally homogeneous strata, then random samples are drawn from each stratum. Proportional allocation makes each stratum's share match its population share.

Unbiased estimator

An estimator whose expected value equals the parameter being estimated. The sample mean is unbiased for the population mean; sample variance uses divisor n minus one.

Confidence interval

A range of values, built from sample data, that contains the population parameter with a stated probability. A 95% interval for a mean uses x̄ plus or minus 1.96 standard errors.

Type I error

Rejecting a null hypothesis that is actually true. Its probability equals the significance level alpha, commonly set at 0.05 or 0.01 by the investigator.

Chi-square goodness of fit

A test comparing observed frequencies with expected frequencies under a null model. The statistic sums (O − E)²/E over categories, with degrees of freedom categories minus one.

Laspeyres price index

A weighted aggregative index using base-year quantities: sum of current prices times base quantities, divided by sum of base prices times base quantities, multiplied by 100.

Fisher's ideal index

The geometric mean of the Laspeyres and Paasche indices. It satisfies both the time-reversal and factor-reversal tests, which simple weighted indices generally fail.

Moving average

A smoothing device averaging a fixed number of consecutive observations. It removes short-term fluctuations and reveals the underlying secular trend of a time series.