Question

what do you mean by dual nature of matter so that they broke least wavelength is inversely proportional to the velocity of the particle

Answer 1

dual nature. of matter means that matter can exist as particle and also as wave

Answer 2

In case of light some phenomenon like diffraction and interference can be explained on the basis of its wave character. However, the certain other phenomenon such as black body radiation and photoelectric effect can be explained only on the basis of its particle nature. Thus, light is said to have a dual character. Such studies on light were made by Einstein in 1905. Louis de Broglie, in 1924 extended the idea of photons to material particles such as electron and he proposed that matter also has a dual character-as wave and as particle.
 Derivation of de-Broglie EquationThe wavelength of the wave associated with any material particle was calculated by analogy with photon.In case of photon, if it is assumed to have wave character, its energy is given byE = hv   …(i)(According to the Planck’s quantum theory)Where nth frequency of the wave and ‘h’ is is Planck’s constantIf the photon is supposed to have particle character, its energy is given byE = mc2   ….… (ii)(according to Einstein’s equation)where ‘m’ is the mass of photon, ‘c’ is the velocity of light.By equating (i) and (ii) hv = mc2But v = c/λh c/λ   = mc2(or) λ = h /mcThe above equation is applicable to material particle if the mass and velocity of photon is replaced by the mass and velocity of material particle. Thus for any material particle like electron.λ = h/mv   or λ =  where mv = p is the momentum of the particle.
 Derivation of Angular Momentum from de Broglie EquationAccording to Bohr’s model, the electron revolves around the nucleus in circular orbits. According to de Broglie concept, the electron is not only a particle but has a wave character also.If the wave is completely in phase, the circumference of the orbit must be equal to an integral multiple of wave length (λ)Therefore 2πr = nλwhere ‘n’ is an integer and ‘r’ is the radius of the orbitBut λ = h/mv∴ 2πr = nh /mv   or mvr = n h/2π  which is Bohr’s postulate of angular momentum, where ‘n’ is the principal quantum number.“Thus, the number of waves an electron makes in a particular Bohr orbit in one complete revolution is equal to the principal quantum number of the orbit”. Alternatively:  Number of waves ‘n’ = 2πr / λ  = 2πr / h/mv  = 2πmvr / hWhere v and r are the velocity of electron and radius of that particular Bohr orbit in which number of waves are to be calculated, respectively.The electron is revolving around the nucleus in a circular orbit. How many revolutions it can make in one second?Let the velocity of electron be v m/sec. The distance it has to travel for one revolution 2πr, (i.e., the circumference of the circle).Thus, the number of revolutions per second is = v / 2πrCommon unit of energy is electron volt which is amount of energy given when an electron is accelerated by a potential of exactly 1 volt. This energy equals the product of voltage and charge. Since in SI units 
coulombs x volts = joules, 1 eV numerically equals the electronic charge except that joule replaces coulombs.
 Plank’s Quantum Theory When a black body is heated, it emits thermal radiations of different wavelengths or frequency. To explain these radiations, Max Planck put forward a theory known as Planck’s quantum theory. The main points of quantum theory are Substances radiate or absorb energy discontinuously in the form of small packets or bundles of energy. The smallest packet of energy is called quantum. In case of light the quantum is known as photon.  The energy of a quantum is directly proportional to the frequency of the radiation. E µ n (or) E = hn where n is the frequency of radiation and h is Planck’s constant having the value 6.626 X 10–27 erg – sec or 6.626 X 10–34 J–sec. A body can radiate or absorb energy in whole number multiples of a quantum hn, 2hn,3hn………..nhn. Where ‘n' is the positive integer. Neils Bohr used this theory to explain the structure of atom. 
​Photoelectric EffectSir J.J. Thomson, observed that when a light of certain frequency strikes the surface of a metal, electrons are ejected from the metal. This phenomenon is known as photoelectric effect and the ejected electrons are called photoelectrons.A few metals, which are having low ionization energy like Cesium, show this effect under the action of visible light but many more show it under the action of more energetic ultraviolet light.An evacuated tube contains two electrodes connected to a source of variable voltage, with the metal plate whose surface is irradiated as the anode. Some of the photoelectrons that emerge from this surface have enough energy to reach the cathode despite its negative polarity, and they constitute the measured current. The slower photoelectrons are repelled before they get to the cathode. When the voltage is increased to a certain value V0, of the order of several volts, no more photoelectrons arrive, as indicated by the current dropping to zero. This extinction voltage (or also referred as stopping potential) corresponds to the maximum photoelectron kinetic energy i.e., eVo = ½ mv2  I hope this helps!  :)