Karnataka II PUC Basic Mathematics (KSEAB) flashcards

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

Union cardinality formula

For finite sets A and B, |A union B| equals |A| + |B| minus |A intersection B|; for three sets subtract pairwise intersections and add triple intersection.

Equivalence relation conditions

A relation on a set is an equivalence relation when it is reflexive, symmetric and transitive; it then partitions the set into disjoint equivalence classes.

Bijective function inverse

A function has an inverse exactly when it is both injective and surjective. The inverse reverses each ordered pair, and its graph is the reflection in y = x.

Determinant product rule

For square matrices A and B of the same order, determinant of AB equals determinant of A times determinant of B; determinant of A transpose equals determinant of A.

Inverse by adjugate

For a square matrix A with nonzero determinant, inverse of A equals adjugate of A divided by determinant of A; adjugate is transpose of cofactor matrix.

Cramer unique solution

For AX = B with square coefficient matrix, Cramer's rule gives each variable as determinant with replaced column divided by determinant of A, provided that determinant is nonzero.

Combination formula

nCr equals n factorial divided by r factorial times (n minus r) factorial. It counts r-element subsets chosen from n distinct objects, ignoring order.

Circular permutation count

The number of distinct arrangements of n distinct objects around a circle is (n minus 1) factorial; if clockwise and anticlockwise orders are considered identical, divide by 2.

Binomial general term

In the expansion of (a + b) raised to n, term r plus one equals nCr times a raised to n minus r times b raised to r, for r from zero to n.

AP nth term

For an arithmetic progression with first term a and common difference d, the nth term is a plus (n minus 1)d; the sum of n terms is n/2 times twice a plus (n minus 1)d.

Infinite GP sum

An infinite geometric series with first term a and common ratio r converges only when absolute value of r is less than 1; its sum is a divided by (1 minus r).

AM GM HM inequality

For positive numbers a and b, arithmetic mean is (a+b)/2, geometric mean is square root of ab, harmonic mean is 2ab/(a+b), and AM is at least GM which is at least HM.

Perpendicular line slopes

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

Point-line distance

Distance from point (x1, y1) to line Ax + By + C = 0 is absolute value of Ax1 + By1 + C divided by square root of A squared plus B squared.

Line intercept form

A line with nonzero x-intercept a and nonzero y-intercept b can be written x/a + y/b = 1; this form directly displays where the line crosses each axis.

Circle general equation

Equation x squared plus y squared plus 2gx plus 2fy plus c equals zero represents a circle with center (-g, -f) and radius square root of g squared plus f squared minus c.

Parabola y squared equals 4ax

The parabola y squared = 4ax opens rightward, has vertex at origin, focus (a, 0), directrix x = -a, and latus rectum length 4a.

Ellipse eccentricity

For ellipse x squared/a squared + y squared/b squared = 1 with a greater than b, eccentricity e equals square root of 1 minus b squared/a squared, and foci are at plus or minus ae on x-axis.

Limit sin x over x

As x approaches zero in radians, sine x divided by x approaches 1. This standard limit underlies derivatives of trigonometric functions and many trigonometric limits.

Differentiability implies continuity

If a function is differentiable at a point, then it is continuous there. Continuity does not imply differentiability, as shown by absolute value of x at zero.

Chain rule

For composite function f(g(x)), derivative is f prime evaluated at g(x) multiplied by g prime of x; differentiate the outer function then multiply by inner derivative.

Second derivative test

At a stationary point where first derivative is zero, a positive second derivative gives a local minimum; a negative second derivative gives a local maximum; zero requires another test.

Definite integral evaluation

If F is an antiderivative of continuous f, the definite integral from a to b of f(x) dx equals F(b) minus F(a), by the fundamental theorem of calculus.

Area between curves

For curves y = f(x) above y = g(x) between x = a and x = b, area equals integral from a to b of f(x) minus g(x) dx.

Conditional probability formula

Probability of A given B equals probability of A intersection B divided by probability of B, provided probability of B is greater than zero.

Independent events product rule

Events A and B are independent exactly when probability of A intersection B equals probability of A times probability of B; knowledge of one does not change the other's probability.

Grouped data variance

For grouped data, variance equals sum of f times x squared divided by sum of f, minus the square of the mean; standard deviation is its nonnegative square root.

Corner point theorem

For a linear programming problem with a nonempty bounded feasible region, an optimal value of the objective function occurs at at least one corner point of that region.

Feasible region definition

The feasible region of a linear programming problem is the set of all points satisfying every constraint, including nonnegativity restrictions; it may be bounded, unbounded, or empty.

Compound interest amount

For principal P at annual rate r percent compounded annually for n years, amount A equals P times (1 plus r/100) raised to n; compound interest is A minus P.

Annuity present value

Present value of an ordinary annuity with payment R per period, interest rate i per period, and n periods is R times [1 minus (1+i) raised to -n] divided by i.

Banker's discount

Banker's discount on a bill is face value times rate times time; proceeds equal face value minus banker's discount. True discount uses present value rather than face value.