Duduli

Laplace transforms help in solving the differential equations with boundary values without finding the general solution and the values of the arbitrary constants.

A square matrix will be called symmetric, if for all values of i & j ,      aij = aji i.e  A' = A Where A' is the transpose of matrix A e.g  

If a matrix has only one row & any number of columns, it is called a row matrix.   e.g     A = [2 7 3 9] Where A is a row matrix 

A square matrix is called a unit matrix if all the diagonal elements are unity & non-diagonal elements are zero. e.g

A matrix, in which the number of rows is equal to the number of columns, is called a square matrix. e.g

The rank of a matrix is said to be r if (a) It has at least one non-zero minor of order r. (b) Every minor of A of order higher than r is zero. Rank = Number of non-zero row in upper triangular matrix.

A matrix ‘A’ wil be called a periodic matrix, if,          A^k+1 = A        Where, K is a positive integer        If K is least positive integer then k is period of A If,     K=1 then A²=A which is idempotent matrix.      

A square matrix, all of whose elements below the leading diagonal are zero, is called an upper triangular matrix. A square matrix, all of whose elements above the leading diagonal are zero, is called a lower triangular matrix. e.g

A matrix, such that A²=A is called idempotent matrix.     e.g            A²=A So given matrix A is an                         idempotent matrix .

A region is said to be connected region if any two points of region D can be connected by a curve which lies completely within the region. It is of two types (i) Simply connected region (ii) multi connected region  (i)Simply connected region A connected region is said to be a Simply connected if all the interior points of a closed curve C drawn in the region D are the points of the region D. e.g (ii) multi connected region A domain or range which is not simply connected is called multi connected region. e.g

A function f(z) is said to be analytic or continuous at a point z0, if 'f' is differentiable not only at z0 but at every point of some neighbourhood of z0.

Let f & g be two permutation then they are called as equal permutation                       iff f(a) = g(a)        ∀ a ∈ S   I.e  image of every elements of S under both 'f' & 'g' are equal.

A group G is said to be a cyclic group if there exists an element a ∈ G such that                        G = [a] I.e   every element of G can be expressed as some integral power of a . → ‘a’ is called the generator of G Example of a cyclic group is G = [ i ]   Where ,  i    = iota

Dark current is a kind of leakage current of a photodiode in the reverse direction when it is exposed to light.

Aldehydes having no alpha hydrogen atom when heated with conc. alkali like NaOH undergo self oxidation and reduction to form sodium d-carboxylic acid & alcohol. It is called cannizziros reaction. It is also called disproportion reaction.

Current transfer ratio β for common emitter operation is defined as the ratio between change in collector current to the change in base current keeping emitter base voltage constant.                           β = Ic/Ib      Where, Ic = collector current                   Ib = base current 

Answer:-

Optical microscopy is a technique used to closely view a sample by an optical microscope through the magnification of the lenses with visible light. An optical microscope is also known as light microscope. It consists of one or a number of lenses to magnify images of small crystalline samples with visible light. The lenses are placed between the sample and the viewer's eye to magnify the image so that the sample can be examined in details.

Resonance energy is the energy that helps in determining the delocalized electrons present in any molecules where a single Lewis structure cannot specifically express the bonding.

Outward osmotic flow of water from a cell containing an aqueous solution through a semipermeable membrane is known as exo osmosis. e.g. an egg (after removing hard shell) placed in conc. NaCl solution is called exo osmosis.

The metal sheet that serves to dissipate the additional heat from the power transistor is known as heat sink.

When the electrical resistance of a substance drops suddenly to zero, when it is cooled below a certain temperature, the phenomenon is known as superconductivity.

Quarks are the basic constituent particles of which elementary particles are composed. Many phenomena in particle physics have been predicted and explained theoretically with the concept of quarks. Of course, their experimental observations remain ambiguous. A quark has no internal structure but it is supposed to be fundamental constituent of matter.