Tamil Nadu HSE Second Year Business Mathematics and Statistics key terms

A square matrix whose determinant equals zero, so it has no multiplicative inverse. In solving linear equations it signals either no solution or infinitely many solutions.

A single number computed from a square matrix that measures whether the matrix is invertible and scales areas or volumes. For a 2×2 matrix it is ad minus bc.

A method of solving a system of linear equations in which each unknown equals a ratio of two determinants: the coefficient determinant in the denominator and a replaced-column determinant in the numerator.

The minor of an element is the determinant left after deleting its row and column; the cofactor is that minor multiplied by (−1) raised to the sum of the element's row and column indices.

The transpose of the cofactor matrix. Multiplying a matrix by its adjoint gives the determinant times the identity matrix, which is the key step in writing the inverse.

For a non-singular square matrix A, the matrix A⁻¹ satisfying A A⁻¹ = I. It equals the adjoint divided by the determinant and is used to solve matrix equations.

The order of the largest non-zero determinant that can be formed from a matrix. It tells how many independent equations a system really contains.

A property of a linear system meaning at least one solution exists. It is tested by comparing the rank of the coefficient matrix with the rank of the augmented matrix.

The tangent of the angle a straight line makes with the positive x-axis, equal to the change in y divided by the change in x between two points on the line.

Any algebraic relation of first degree in x and y whose graph is a straight line, commonly written in slope-intercept, point-slope, two-point or intercept form.

The equation x/a + y/b = 1 of a line, where a is the x-intercept and b is the y-intercept. It is convenient when the two intercepts are known.

Two lines meeting at a right angle. Their slopes multiply to −1, so the slope of a perpendicular is the negative reciprocal of the given slope.

Lines in a plane that never meet. In coordinate geometry they have equal slopes but different intercepts, so their equations differ only in the constant term.

The set of points at a fixed distance from a fixed point. Its standard equation is (x − h)² + (y − k)² = r², with centre (h, k) and radius r.

A curve obtained by cutting a cone with a plane: circle, parabola, ellipse or hyperbola. Each has a standard equation with identifiable centre, focus or directrix.

The ratio of the distance of a point on a conic from the focus to its distance from the directrix. Values below one give an ellipse, one a parabola, above one a hyperbola.

The value a function approaches as its input approaches a stated point, whether or not the function is defined there. It underlies the definition of continuity and the derivative.

A function is continuous at a point when its limit there exists, the function is defined there, and the two are equal. Graphically, the curve has no break at that point.

The instantaneous rate of change of a function with respect to its variable, defined as the limit of the difference quotient. Geometrically it is the slope of the tangent.

The property that a function possesses a derivative at a point. Differentiability at a point implies continuity there, but continuity does not guarantee differentiability.

A straight line touching the curve at a point whose slope equals the derivative there. Its equation uses the point-slope form with that derivative as slope.

The line through a point on a curve perpendicular to the tangent at that point. Its slope is the negative reciprocal of the derivative at the point.

The extra cost of producing one more unit, obtained as the derivative of the total cost function with respect to output. It is used to find the output minimising average cost.

The proportional change in quantity demanded divided by the proportional change in price, written as −(p/x)(dx/dp). Demand is elastic when its magnitude exceeds one.

The family of all antiderivatives of a function, written with an arbitrary constant of integration. Differentiating the result returns the original integrand.

The limit of a sum of areas under a curve between two limits, evaluated by subtracting the antiderivative at the lower limit from its value at the upper limit.

A technique based on the product rule, stated as ∫u dv = uv − ∫v du. It is chosen when the integrand is a product of two unlike functions.

Rewriting a rational expression as a sum of simpler fractions with linear or quadratic denominators, making each piece easy to integrate term by term.

The region bounded by a curve, the x-axis and two ordinates, found by integrating the function between the corresponding limits; below the axis the integral is negative.

The difference between what consumers are willing to pay and what they actually pay, computed as the area between the demand curve and the price line up to the equilibrium quantity.

The order is the highest derivative present; the degree is the power of that highest derivative after the equation is cleared of radicals and fractions in derivatives.

The solution of a differential equation containing as many arbitrary constants as the order of the equation. Particular solutions arise by assigning values to those constants.

A first-order differential equation that can be rearranged so all terms in one variable sit on one side and all terms in the other variable on the other, then integrated directly.

An equation of the form dy/dx + Py = Q with P and Q functions of x only. It is solved using the integrating factor e raised to the integral of P.

A real-valued function assigning a number to each outcome of a random experiment. It may be discrete, taking isolated values, or continuous, taking any value in an interval.

A rule giving the probability that a discrete random variable equals each of its possible values. The probabilities are non-negative and sum to one.

A non-negative function whose integral over an interval gives the probability that a continuous random variable lies in that interval; its total integral over the range is one.

The distribution of the number of successes in n independent trials each with the same success probability p, with mean np and variance npq.

A distribution for counts of rare events in a fixed interval, with a single parameter equal to both its mean and variance. It approximates the binomial when n is large and p small.

A continuous symmetric bell-shaped distribution fixed by its mean and standard deviation. Standardising with z = (x − μ)/σ allows probabilities to be read from tables.

The standard deviation of the sampling distribution of a statistic. For a sample mean it equals the population standard deviation divided by the square root of the sample size.

The probability distribution of a statistic over all possible samples of a given size drawn from a population. Its spread shrinks as the sample size grows.

The result that the sampling distribution of the sample mean tends to normality as sample size increases, whatever the shape of the population, provided the population has finite variance.

A range of values, computed from sample data, that is expected to contain the unknown population parameter with a stated probability such as 95 per cent.

The statement of no effect or no difference that a significance test sets out to examine. It is rejected only when sample evidence is sufficiently unlikely under it.

The claim accepted when the null hypothesis is rejected; it may be one-sided, asserting a direction of difference, or two-sided, asserting only that a difference exists.

The probability, chosen in advance, of rejecting a true null hypothesis, commonly 5 or 1 per cent. It fixes the critical values against which the test statistic is compared.

The boundary value of the test statistic beyond which the null hypothesis is rejected. It is read from the normal or t table using the chosen significance level and degrees of freedom.

A quantity computed from sample data, such as z or t, whose value is compared with a critical value to decide whether to reject the null hypothesis.

The number of independent pieces of information available in a calculation, often sample size minus the number of estimated parameters, used to select the correct statistical table.

A statistical measure expressing the relative change in a variable or group of variables between two periods or places, usually shown as a percentage with a base period equal to 100.

A weighted price index using base-period quantities as weights. It tends to overstate price rises because it ignores consumers substituting away from goods that became dearer.

A weighted price index using current-period quantities as weights. It tends to understate price rises relative to Laspeyres and needs fresh quantity data each period.

The reference period against which other periods are compared in an index number series. It is normally a stable, normal year assigned the value 100.

A sequence of observations of a variable recorded at successive, usually equal, time intervals, analysed as trend, seasonal, cyclical and irregular components.

A series of averages taken over a fixed number of consecutive periods and shifted forward one period at a time, used to smooth fluctuations and reveal the underlying trend.

The long-term general direction of a time series, upward, downward or flat, after short-term and seasonal movements are removed. It is estimated by moving averages or least squares.