Duduli

Three vectors are said to be linearly dependent if their determinant is zero.

Compound interest is the interest  earned on both the initial amount of money and the accumulated  interest from previous periods. It is a concept where the interest you earn or pay grows exponentially over time. This  compounding effect can lead to significant growth in investments or increased costs of borrowing. e.g if you have $100 and it earns 5% interest each year, you will have $105 at the end of the first year. At the end of the second year, you will have $110.25.

Complex number can be written as z=x+iy , where x & y are real numbers, and i=√-1 . This form x+iy , is called the standard form of complex number. When graphing these we can represent them on a coordinate plane called the argand plane or complex plane.

Suppose p(n) is a proposition defined on the positive integer N i.e n€N Step-I (basis of induction) Show p(1) is true.(Generally)  If given  n≥a then show p(a) is true. Step-II (Induction hypothesis) Assume p(K) is true. where k≥a Step-III (Induction step) Show p(k+1) is true using p(k).

The unit step function u(t-a) is defined as follows u(t-a)={0  , when  t<a                             {1   , when t≥a I.e ★Laplace transform of unit step function is e^-as/s ★with the help of unit step functions, we can find the inverse transform of functions.

If the two branches of the curve pass through the double point and the tangent to them are the point is real and coincident, then the double point is called cusp. Shown in the figure

If an algebraic curve passes through the origin, the equation of tangent or tangents at the origin is obtained by equating to zero the lowest degree terms in the equation of the curve.

A function defined by f(x)=1/x is called the reciprocal function or rectangular hyperbola, with coordinate axis as asymptotes. The domain and range of f(x)=1/x is R-{0}. Since, f(x) is odd function, so its graph is symmetrical about opposite quadrants.

A function given by f(x)=x² is called the square function. The domain of square function is R and its range is R^+ U  {0} or [0,♾️). Clearly y=x², is a parabola. Since y=x² is an even function, so its graph is symmetrical about y-axis, shown as:

A function given by f(x)=x³ is called the cube function. The domain and range of cube are both equal to R(i.e. f(x):R→R). Since, y=x³ is an odd function, so its graph is symmetrical about opposite quadrant, i.e., “origin”, shown as:

A relation f:A→B is said to be a function from A to B if (i) dom f = A (ii) No object of A has two or more different images in B.

Let A,B and C be three non-empty set. Let f:A→B and g:B→C be two functions. The composition of of f and g is a function from A to C such that gof(x) = g(f(x)) for every x € A.

A relation R from a set A to a set B (R:A→B) is a subset of A×B . A relation R on a set A is defined to be any subset of A×A.

If S be a finite set consisting of n elements (say) then a composition(usually binary composition) * in S can be described by means of a table consisting of n rows & n columns in which the entry at the intersection of new headed by an element a € S is a*b . such tables are called composition tables. In order to verify group properties of a finite set under some binary operation, conveniently one can do this by the help of composition table.

The power set of a set A is denoted by p(A) and is defined by the set of all subsets of A. e.g if A = {1,2,3}, then   P(A)={∅,{1},{2},{3},{1,2},{2,3},{1,3},{1,2,3}}

A compound statement, which is always true for all possible assignment of truth values to its prime components, is called a Tautology.

A compound statement, which is always false for all possible assignment of truth values to its prime components, is called contradiction.

Let A be a matrix of order m×n .The kernel of a matrix A is the set of all vectors x € Vn such that Ax=O m×1.