APEX INSTITUTE was conceptualized in May 2008, keeping in view the dreams of young students by the vision & toil of Er. Shahid Iqbal. We had a very humble beginning as an institute for IIT-JEE / Medical, with a vision to provide an ideal launch pad for serious JEE students . We actually started to make a difference in the way students think and approach problems. We started to develop ways to enhance students IQ. We started to leave an indelible mark on the students who have undergone APEX training. That is why APEX INSTITUTE is very well known of its quality of education
Simulation of Magnetically Confined Plasma for Etch Applicationsvvk0
The document describes computational optimization of plasma uniformity in a magnetically enhanced capacitively coupled plasma (CCP) reactor for disk etch applications. Initial simulations using a two-dimensional hybrid plasma equipment model (HPEM) showed non-uniform electron density and radical distributions in a CFP plasma with the magnet placed 125 mm from the substrate. The distance between the magnet and substrate was increased to 113 mm, which improved the uniformity of the electron density, CFx radical densities, and plasma potential above the substrate. Further simulations varying the magnet distance found that plasma density and F radical density decreased with smaller magnet-substrate gaps. The study demonstrates optimization of plasma uniformity through computational modeling of magnetic field and plasma transport parameters.
This document provides an overview of linear impulse and momentum. It defines linear momentum as mass times velocity and describes how the principle of linear impulse and momentum can be derived by integrating Newton's second law over time. Examples are provided to demonstrate how to use this principle to calculate average forces from changes in linear momentum. Impulsive forces that cause large momentum changes over short time intervals are also discussed.
Realizations, Differential Equations, Canonical Quantum Commutators And Infin...vcuesta
1) The document discusses finding different realizations of quantum operators q and p that obey the canonical commutator [q,p]=iħ. It considers cases where p is defined as -iħf(q)∂/∂q and solves for the corresponding q operator.
2) This leads to an infinite number of possible representations, as f(q) can be any function of q. Three specific cases are analyzed.
3) For each case, the Schrodinger equation is derived and solutions are found for a free particle and infinite square well potential. However, some cases cannot yield normalizable wavefunctions or satisfy boundary conditions.
On estimating the integrated co volatility usingkkislas
This document proposes a method to estimate the integrated co-volatility of two asset prices using high-frequency data that contains both microstructure noise and jumps.
It considers two cases - when the jump processes of the two assets are independent, and when they are dependent. For the independent case, it proposes an estimator that is robust to jumps. For the dependent case, it proposes a threshold estimator that combines pre-averaging to remove noise with a threshold method to reduce the effect of jumps. It proves the estimators are consistent and establishes their central limit theorems. Simulation results are also presented to illustrate the performance of the proposed methods.
kinks and cusps in the transition dynamics of a bloch statejiang-min zhang
We discuss the transition dynamics of a Bloch state in a 1D tight binding chain, under two scenarios, namely, weak periodical driving [1] and sudden quench [2]. In the former case, the survival probability of the initial Bloch state shows kinks periodically; it is a piece-wise linear function of time. In the latter, the survival probability (Loschmidt echo in this case) shows cusps periodically; it is a piece-wise quadratic function of time. Kinks in the former case are a perturbative effect, while cusps in the latter are a non-perturbative effect. The kinks and cusps are reminiscent of the so-called dynamical phase transtion termed by Heyl et al. [3].
[1] J. M. Zhang and M. Haque, Nonsmooth and level-resolved dynamics illustrated with the tight binding model, arXiv:1404.4280.
[2] J. M. Zhang and H. T. Yang, Cusps in the quenched dynamics of a bloch state, arXiv:1601.03569.
[3] M. Heyl, A. Polkovnikov, and S. Kehrein, Dynamical quantum phase transitions in the transverse-field Ising model, Phys. Rev. Lett. 110, 135704 (2013).
This document summarizes research on modeling wildfires and turbulent premixed combustion. It discusses how turbulence affects wildfire propagation through turbulent transport of hot air, making the fire front position random. It presents a level set method for modeling deterministic and random fire fronts, accounting for turbulence. It also discusses using a Lagrangian approach to model turbulent premixed combustion, describing how the burned mass fraction evolves due to particle motion, flame front velocity, and curvature.
STUDIES ON INTUTIONISTIC FUZZY INFORMATION MEASURESurender Singh
This document discusses studies on measures of intuitionistic fuzzy information. It begins with introductions and definitions related to fuzzy sets, intuitionistic fuzzy sets, and measures of fuzzy entropy. It then discusses special t-norm operators and proposes a measure of intuitionistic fuzzy entropy based on these t-norms. The measure is defined using a function of the membership, non-membership, and hesitancy degrees of an intuitionistic fuzzy set. Several desirable properties of such a measure are outlined, including sharpness, maximality, resolution, symmetry, and valuation. The document provides mathematical foundations and definitions to propose and analyze a measure of intuitionistic fuzzy entropy.
The document summarizes a study on modeling fatigue damage in solder interconnects using a cohesive zone approach. Key points:
1) A cohesive zone method is used to model fatigue damage accumulation in solder joints, representing the interface between continuum elements.
2) A damage evolution law based on experimental observations is used to model how damage increases with the number of loading cycles and applied load.
3) Results show the predicted damage distribution in a sample solder joint under cyclic loading, as well as the effects of mean strain and load sequencing on damage accumulation.
Simulation of Magnetically Confined Plasma for Etch Applicationsvvk0
The document describes computational optimization of plasma uniformity in a magnetically enhanced capacitively coupled plasma (CCP) reactor for disk etch applications. Initial simulations using a two-dimensional hybrid plasma equipment model (HPEM) showed non-uniform electron density and radical distributions in a CFP plasma with the magnet placed 125 mm from the substrate. The distance between the magnet and substrate was increased to 113 mm, which improved the uniformity of the electron density, CFx radical densities, and plasma potential above the substrate. Further simulations varying the magnet distance found that plasma density and F radical density decreased with smaller magnet-substrate gaps. The study demonstrates optimization of plasma uniformity through computational modeling of magnetic field and plasma transport parameters.
This document provides an overview of linear impulse and momentum. It defines linear momentum as mass times velocity and describes how the principle of linear impulse and momentum can be derived by integrating Newton's second law over time. Examples are provided to demonstrate how to use this principle to calculate average forces from changes in linear momentum. Impulsive forces that cause large momentum changes over short time intervals are also discussed.
Realizations, Differential Equations, Canonical Quantum Commutators And Infin...vcuesta
1) The document discusses finding different realizations of quantum operators q and p that obey the canonical commutator [q,p]=iħ. It considers cases where p is defined as -iħf(q)∂/∂q and solves for the corresponding q operator.
2) This leads to an infinite number of possible representations, as f(q) can be any function of q. Three specific cases are analyzed.
3) For each case, the Schrodinger equation is derived and solutions are found for a free particle and infinite square well potential. However, some cases cannot yield normalizable wavefunctions or satisfy boundary conditions.
On estimating the integrated co volatility usingkkislas
This document proposes a method to estimate the integrated co-volatility of two asset prices using high-frequency data that contains both microstructure noise and jumps.
It considers two cases - when the jump processes of the two assets are independent, and when they are dependent. For the independent case, it proposes an estimator that is robust to jumps. For the dependent case, it proposes a threshold estimator that combines pre-averaging to remove noise with a threshold method to reduce the effect of jumps. It proves the estimators are consistent and establishes their central limit theorems. Simulation results are also presented to illustrate the performance of the proposed methods.
kinks and cusps in the transition dynamics of a bloch statejiang-min zhang
We discuss the transition dynamics of a Bloch state in a 1D tight binding chain, under two scenarios, namely, weak periodical driving [1] and sudden quench [2]. In the former case, the survival probability of the initial Bloch state shows kinks periodically; it is a piece-wise linear function of time. In the latter, the survival probability (Loschmidt echo in this case) shows cusps periodically; it is a piece-wise quadratic function of time. Kinks in the former case are a perturbative effect, while cusps in the latter are a non-perturbative effect. The kinks and cusps are reminiscent of the so-called dynamical phase transtion termed by Heyl et al. [3].
[1] J. M. Zhang and M. Haque, Nonsmooth and level-resolved dynamics illustrated with the tight binding model, arXiv:1404.4280.
[2] J. M. Zhang and H. T. Yang, Cusps in the quenched dynamics of a bloch state, arXiv:1601.03569.
[3] M. Heyl, A. Polkovnikov, and S. Kehrein, Dynamical quantum phase transitions in the transverse-field Ising model, Phys. Rev. Lett. 110, 135704 (2013).
This document summarizes research on modeling wildfires and turbulent premixed combustion. It discusses how turbulence affects wildfire propagation through turbulent transport of hot air, making the fire front position random. It presents a level set method for modeling deterministic and random fire fronts, accounting for turbulence. It also discusses using a Lagrangian approach to model turbulent premixed combustion, describing how the burned mass fraction evolves due to particle motion, flame front velocity, and curvature.
STUDIES ON INTUTIONISTIC FUZZY INFORMATION MEASURESurender Singh
This document discusses studies on measures of intuitionistic fuzzy information. It begins with introductions and definitions related to fuzzy sets, intuitionistic fuzzy sets, and measures of fuzzy entropy. It then discusses special t-norm operators and proposes a measure of intuitionistic fuzzy entropy based on these t-norms. The measure is defined using a function of the membership, non-membership, and hesitancy degrees of an intuitionistic fuzzy set. Several desirable properties of such a measure are outlined, including sharpness, maximality, resolution, symmetry, and valuation. The document provides mathematical foundations and definitions to propose and analyze a measure of intuitionistic fuzzy entropy.
The document summarizes a study on modeling fatigue damage in solder interconnects using a cohesive zone approach. Key points:
1) A cohesive zone method is used to model fatigue damage accumulation in solder joints, representing the interface between continuum elements.
2) A damage evolution law based on experimental observations is used to model how damage increases with the number of loading cycles and applied load.
3) Results show the predicted damage distribution in a sample solder joint under cyclic loading, as well as the effects of mean strain and load sequencing on damage accumulation.
Gauge systems and functions, hermitian operators and clocks as conjugate func...vcuesta
This document summarizes a research article about gauge systems and constraints in physics. It discusses two key problems that can arise: 1) Clocks may not be well-defined over the entire phase space. 2) Quantum operators associated with complete observables may not be self-adjoint. The summary proposes selecting clocks such that their Poisson brackets with constraints are equal to 1. This is shown to solve the two problems for several example systems, including a free particle and a system with two constraints. Clocks and complete observables are constructed for the examples, and it is verified that the operators are self-adjoint.
1) The document provides information on various physics concepts including kinematics equations, forces, energy, and more.
2) It includes 13 key equations related to topics like velocity, acceleration, Newton's laws of motion, Hooke's law, energy, and others.
3) The document also provides details on astrophysics concepts like astronomical units and the Hubble constant, as well as common physics prefixes and units.
This document discusses curvilinear motion and kinematics. It introduces position vectors, path coordinates, velocity vectors, and acceleration vectors for particles moving in three-dimensional space. Key concepts covered include defining the position vector r(t) from a reference point to the particle, the instantaneous velocity vector v as the time derivative of r(t), and the acceleration vector a as the time derivative of v. When working in Cartesian coordinates, the derivatives of vector components are simply the derivatives of the individual x, y, z components.
This chapter discusses refraction of light, including:
- The index of refraction, which is the ratio of the speed of light in a vacuum to its speed in a medium.
- Snell's law, which relates the angles of incidence and refraction based on the indices of refraction of the media.
- Total internal reflection, which occurs when light travels from an optically dense medium to a less dense one at an angle greater than the critical angle.
- Wavelength changes when light moves between media due to the index of refraction.
Several example problems are worked through applying these concepts to compute angles, indices of refraction, wavelengths and speeds of light in various materials.
Nonlinear transport phenomena: models, method of solving and unusual features...SSA KPI
AACIMP 2010 Summer School lecture by Vsevolod Vladimirov. "Applied Mathematics" stream. "Selected Models of Transport Processes. Methods of Solving and Properties of Solutions" course. Part 3.
More info at http://summerschool.ssa.org.ua
This chapter discusses discrete image transforms. It introduces linear transformations and unitary transforms. The discrete Fourier transform (DFT) and discrete cosine transform (DCT) are presented as examples of unitary transforms. The DFT represents an image as a sum of sinusoidal basis images, while the DCT uses cosine basis images. Other transforms discussed include the discrete sine transform (DST), Hartley transform, and Hadamard transform. Orthogonal transforms preserve image properties while changing the representation basis.
The document provides an overview of key physics equations and concepts for Form 4 students, including equations for relative deviation, prefixes, units for area and volume, equations for average speed, velocity, acceleration, momentum, Newton's laws of motion, and impulse. Key terms are defined for important concepts like displacement, time, mass, force, and velocity. Formulas are presented for calculations involving these fundamental physics quantities and relationships.
Proceedings A Method For Finding Complete Observables In Classical Mechanicsvcuesta
1. The document presents a new method for finding complete observables in classical mechanics, which are gauge invariant quantities.
2. The method starts with partial observables and clocks, which are non-gauge invariant phase space functions. Using constants of motion, the partial observables can be written in terms of the clocks to obtain complete observables.
3. As an example, the method is applied to a particle in a gravitational field, where the Hamiltonian is used as a constant of motion to write the position variable as a function of the momentum and time.
The document provides an overview of key physics equations and concepts related to forces and motion, including equations for relative deviation, prefixes, units of area and volume, average speed, velocity, acceleration, momentum, Newton's laws of motion, and impulse. Key variables and their units are defined for each equation. Examples of displacement-time and velocity-time graphs are also included to illustrate the relationships between displacement, velocity, time, and acceleration.
1. A central force is one that is always directed towards a fixed point. Examples include gravitational force, forces causing uniform circular motion, and simple harmonic motion.
2. To analyze central forces, vectors, differentiation, and vector differentiation must be understood. The differentiation of position, velocity, and acceleration vectors in Cartesian and polar coordinates is examined.
3. For a central force, the radial component of acceleration is related to the magnitude of the force, while the tangential component depends on the angular acceleration and velocity. Examples of central forces producing different types of motion are given.
The document discusses multiple kernel learning approaches for classification. It introduces single kernel learning and describes the challenge of multiple kernel learning when using multiple reproducing kernel Hilbert spaces. The document then summarizes several regularization-based multiple kernel learning approaches, including L1-MKL, L2-MKL, Elasticnet-MKL, and proposes a new Mixed-Norm-Elasticnet-MKL approach.
1. The document outlines key concepts in structural dynamics including idealization of structures as single-degree-of-freedom systems, formulation of the equation of motion, free and forced vibration of undamped and damped systems.
2. Key topics covered include natural frequency determination, Duhamel's integral, damping in structures, and methods for solving dynamic problems.
3. Examples of single-degree-of-freedom systems are presented including lumped mass systems, beams with distributed mass, and determination of effective stiffness.
D. Ishii, K. Ueda, H. Hosobe, A. Goldsztejn: Interval-based Solving of Hybrid...dishii
An approach to reliable modeling, simulation and verification of hybrid systems is interval arithmetic, which guarantees that a set of intervals narrower than specified size encloses the solution. Interval-based computation of hybrid systems is often difficult, especially when the systems are described by nonlinear ordinary differential equations (ODEs) and nonlinear algebraic equations.We formulate the problem of detecting a discrete change in hybrid systems as a hybrid constraint system (HCS), consisting of a flow constraint on trajectories (i.e. continuous functions over time) and a guard constraint on states causing discrete changes. We also propose a technique for solving HCSs by coordinating (i) interval-based solving of nonlinear ODEs, and (ii) a constraint programming technique for reducing interval enclosures of solutions. The proposed technique reliably solves HCSs with nonlinear constraints. Our technique employs the interval Newton method to accelerate the reduction of interval enclosures, while guaranteeing that the enclosure contains a solution.
This document provides a summary of quantum mechanical concepts and solid state physics. It begins with a review of quantum mechanics and the Schrodinger equation. It then discusses the wave nature of electrons and how the Schrodinger equation describes the wavefunction and probability of finding an electron. It also covers energy band diagrams and how the periodic potential in solids leads to the formation of allowed energy bands. It discusses these concepts for isolated atoms, silicon crystals, and the one-dimensional Kronig-Penny model.
Spectroscopic ellipsometry is a technique for investigating the optical properties and electrodynamics of materials. It has several advantages over other optical techniques:
1) It provides an exact numerical inversion with no need for Kramers-Kronig transformations, allowing consistency checks.
2) Measurements are non-invasive and highly reproducible as they do not require reference samples.
3) It is very sensitive to thin film properties due to its ability to measure at oblique angles of incidence.
Ellipsometry has been used to study phenomena like superconductivity in cuprates and pnictides by measuring changes in spectral weight, and collective charge ordering in oxide superlattices.
WAVELET-PACKET-BASED ADAPTIVE ALGORITHM FOR SPARSE IMPULSE RESPONSE IDENTIFI...bermudez_jcm
Presented at IEEE ICASSP-2007:
This paper proposes a wavelet-packet-based (WPB) algorithm for efficient identification of sparse impulse responses with arbitrary frequency spectra. The discrete wavelet packet transform (DWPT) is adaptively tailored to the energy distribution of the unknown system\'s response spectrum. The new algorithm leads to a reduced number of active coefficients and to a reduced computational complexity, when compared to competing wavelet-based algorithms. Simulation results illustrate the applicability of the proposed algorithm.
Saddlepoint approximations, likelihood asymptotics, and approximate condition...jaredtobin
Maximum likelihood methods may be inadequate for parameter estimation in models where many nuisance parameters are present. The modified profile likelihood (MPL) of Barndorff-Nielsen (1983) serves as a highly accurate approximation to the marginal or conditional likelihood, when either exist, and can be viewed as an approximate conditional likelihood when they do not. We examine the modified profile likelihood, its variants, and its connections with Laplace and saddlepoint approximations under both theoretical and pragmatic lenses.
Feedback of zonal flows on Rossby-wave turbulence driven by small scale inst...Colm Connaughton
The document summarizes research on the interaction between large-scale zonal flows and small-scale Rossby wave turbulence. It describes how modulational instability can generate large-scale zonal jets from small-scale Rossby waves through an inverse cascade. The generated jets then provide negative feedback on the small-scale waves by distorting them and inducing spectral diffusion through a nonlocal turbulence theory. Numerical simulations demonstrate this generation of jets and spectral transport between scales.
1) The document provides examples of physics problems involving relativity, the photoelectric effect, and waves and particles. It includes 22 sample problems with calculations related to topics such as relativistic time dilation, relativistic length contraction, photon energy, electron kinetic energy, momentum, and de Broglie wavelength.
2) Problem 38-13 calculates the work function and kinetic energy of photoelectrons emitted from a metal surface illuminated by light of a given wavelength. It finds the work function is 1.04 eV and the kinetic energy of the emitted electrons is 2.07 eV.
3) Problem 38-21 determines the de Broglie wavelength of an electron accelerated through a potential difference. It finds the electron's momentum and calculates
Gauge systems and functions, hermitian operators and clocks as conjugate func...vcuesta
This document summarizes a research article about gauge systems and constraints in physics. It discusses two key problems that can arise: 1) Clocks may not be well-defined over the entire phase space. 2) Quantum operators associated with complete observables may not be self-adjoint. The summary proposes selecting clocks such that their Poisson brackets with constraints are equal to 1. This is shown to solve the two problems for several example systems, including a free particle and a system with two constraints. Clocks and complete observables are constructed for the examples, and it is verified that the operators are self-adjoint.
1) The document provides information on various physics concepts including kinematics equations, forces, energy, and more.
2) It includes 13 key equations related to topics like velocity, acceleration, Newton's laws of motion, Hooke's law, energy, and others.
3) The document also provides details on astrophysics concepts like astronomical units and the Hubble constant, as well as common physics prefixes and units.
This document discusses curvilinear motion and kinematics. It introduces position vectors, path coordinates, velocity vectors, and acceleration vectors for particles moving in three-dimensional space. Key concepts covered include defining the position vector r(t) from a reference point to the particle, the instantaneous velocity vector v as the time derivative of r(t), and the acceleration vector a as the time derivative of v. When working in Cartesian coordinates, the derivatives of vector components are simply the derivatives of the individual x, y, z components.
This chapter discusses refraction of light, including:
- The index of refraction, which is the ratio of the speed of light in a vacuum to its speed in a medium.
- Snell's law, which relates the angles of incidence and refraction based on the indices of refraction of the media.
- Total internal reflection, which occurs when light travels from an optically dense medium to a less dense one at an angle greater than the critical angle.
- Wavelength changes when light moves between media due to the index of refraction.
Several example problems are worked through applying these concepts to compute angles, indices of refraction, wavelengths and speeds of light in various materials.
Nonlinear transport phenomena: models, method of solving and unusual features...SSA KPI
AACIMP 2010 Summer School lecture by Vsevolod Vladimirov. "Applied Mathematics" stream. "Selected Models of Transport Processes. Methods of Solving and Properties of Solutions" course. Part 3.
More info at http://summerschool.ssa.org.ua
This chapter discusses discrete image transforms. It introduces linear transformations and unitary transforms. The discrete Fourier transform (DFT) and discrete cosine transform (DCT) are presented as examples of unitary transforms. The DFT represents an image as a sum of sinusoidal basis images, while the DCT uses cosine basis images. Other transforms discussed include the discrete sine transform (DST), Hartley transform, and Hadamard transform. Orthogonal transforms preserve image properties while changing the representation basis.
The document provides an overview of key physics equations and concepts for Form 4 students, including equations for relative deviation, prefixes, units for area and volume, equations for average speed, velocity, acceleration, momentum, Newton's laws of motion, and impulse. Key terms are defined for important concepts like displacement, time, mass, force, and velocity. Formulas are presented for calculations involving these fundamental physics quantities and relationships.
Proceedings A Method For Finding Complete Observables In Classical Mechanicsvcuesta
1. The document presents a new method for finding complete observables in classical mechanics, which are gauge invariant quantities.
2. The method starts with partial observables and clocks, which are non-gauge invariant phase space functions. Using constants of motion, the partial observables can be written in terms of the clocks to obtain complete observables.
3. As an example, the method is applied to a particle in a gravitational field, where the Hamiltonian is used as a constant of motion to write the position variable as a function of the momentum and time.
The document provides an overview of key physics equations and concepts related to forces and motion, including equations for relative deviation, prefixes, units of area and volume, average speed, velocity, acceleration, momentum, Newton's laws of motion, and impulse. Key variables and their units are defined for each equation. Examples of displacement-time and velocity-time graphs are also included to illustrate the relationships between displacement, velocity, time, and acceleration.
1. A central force is one that is always directed towards a fixed point. Examples include gravitational force, forces causing uniform circular motion, and simple harmonic motion.
2. To analyze central forces, vectors, differentiation, and vector differentiation must be understood. The differentiation of position, velocity, and acceleration vectors in Cartesian and polar coordinates is examined.
3. For a central force, the radial component of acceleration is related to the magnitude of the force, while the tangential component depends on the angular acceleration and velocity. Examples of central forces producing different types of motion are given.
The document discusses multiple kernel learning approaches for classification. It introduces single kernel learning and describes the challenge of multiple kernel learning when using multiple reproducing kernel Hilbert spaces. The document then summarizes several regularization-based multiple kernel learning approaches, including L1-MKL, L2-MKL, Elasticnet-MKL, and proposes a new Mixed-Norm-Elasticnet-MKL approach.
1. The document outlines key concepts in structural dynamics including idealization of structures as single-degree-of-freedom systems, formulation of the equation of motion, free and forced vibration of undamped and damped systems.
2. Key topics covered include natural frequency determination, Duhamel's integral, damping in structures, and methods for solving dynamic problems.
3. Examples of single-degree-of-freedom systems are presented including lumped mass systems, beams with distributed mass, and determination of effective stiffness.
D. Ishii, K. Ueda, H. Hosobe, A. Goldsztejn: Interval-based Solving of Hybrid...dishii
An approach to reliable modeling, simulation and verification of hybrid systems is interval arithmetic, which guarantees that a set of intervals narrower than specified size encloses the solution. Interval-based computation of hybrid systems is often difficult, especially when the systems are described by nonlinear ordinary differential equations (ODEs) and nonlinear algebraic equations.We formulate the problem of detecting a discrete change in hybrid systems as a hybrid constraint system (HCS), consisting of a flow constraint on trajectories (i.e. continuous functions over time) and a guard constraint on states causing discrete changes. We also propose a technique for solving HCSs by coordinating (i) interval-based solving of nonlinear ODEs, and (ii) a constraint programming technique for reducing interval enclosures of solutions. The proposed technique reliably solves HCSs with nonlinear constraints. Our technique employs the interval Newton method to accelerate the reduction of interval enclosures, while guaranteeing that the enclosure contains a solution.
This document provides a summary of quantum mechanical concepts and solid state physics. It begins with a review of quantum mechanics and the Schrodinger equation. It then discusses the wave nature of electrons and how the Schrodinger equation describes the wavefunction and probability of finding an electron. It also covers energy band diagrams and how the periodic potential in solids leads to the formation of allowed energy bands. It discusses these concepts for isolated atoms, silicon crystals, and the one-dimensional Kronig-Penny model.
Spectroscopic ellipsometry is a technique for investigating the optical properties and electrodynamics of materials. It has several advantages over other optical techniques:
1) It provides an exact numerical inversion with no need for Kramers-Kronig transformations, allowing consistency checks.
2) Measurements are non-invasive and highly reproducible as they do not require reference samples.
3) It is very sensitive to thin film properties due to its ability to measure at oblique angles of incidence.
Ellipsometry has been used to study phenomena like superconductivity in cuprates and pnictides by measuring changes in spectral weight, and collective charge ordering in oxide superlattices.
WAVELET-PACKET-BASED ADAPTIVE ALGORITHM FOR SPARSE IMPULSE RESPONSE IDENTIFI...bermudez_jcm
Presented at IEEE ICASSP-2007:
This paper proposes a wavelet-packet-based (WPB) algorithm for efficient identification of sparse impulse responses with arbitrary frequency spectra. The discrete wavelet packet transform (DWPT) is adaptively tailored to the energy distribution of the unknown system\'s response spectrum. The new algorithm leads to a reduced number of active coefficients and to a reduced computational complexity, when compared to competing wavelet-based algorithms. Simulation results illustrate the applicability of the proposed algorithm.
Saddlepoint approximations, likelihood asymptotics, and approximate condition...jaredtobin
Maximum likelihood methods may be inadequate for parameter estimation in models where many nuisance parameters are present. The modified profile likelihood (MPL) of Barndorff-Nielsen (1983) serves as a highly accurate approximation to the marginal or conditional likelihood, when either exist, and can be viewed as an approximate conditional likelihood when they do not. We examine the modified profile likelihood, its variants, and its connections with Laplace and saddlepoint approximations under both theoretical and pragmatic lenses.
Feedback of zonal flows on Rossby-wave turbulence driven by small scale inst...Colm Connaughton
The document summarizes research on the interaction between large-scale zonal flows and small-scale Rossby wave turbulence. It describes how modulational instability can generate large-scale zonal jets from small-scale Rossby waves through an inverse cascade. The generated jets then provide negative feedback on the small-scale waves by distorting them and inducing spectral diffusion through a nonlocal turbulence theory. Numerical simulations demonstrate this generation of jets and spectral transport between scales.
1) The document provides examples of physics problems involving relativity, the photoelectric effect, and waves and particles. It includes 22 sample problems with calculations related to topics such as relativistic time dilation, relativistic length contraction, photon energy, electron kinetic energy, momentum, and de Broglie wavelength.
2) Problem 38-13 calculates the work function and kinetic energy of photoelectrons emitted from a metal surface illuminated by light of a given wavelength. It finds the work function is 1.04 eV and the kinetic energy of the emitted electrons is 2.07 eV.
3) Problem 38-21 determines the de Broglie wavelength of an electron accelerated through a potential difference. It finds the electron's momentum and calculates
IInforme del GPPAN del 1er Periodo del 1er Año de la LXII LegislaturaUNAM
Este documento describe la organización y funcionamiento del Grupo Parlamentario del Partido Acción Nacional en la Cámara de Diputados. Se establecieron una Vicecoordinación y 10 Subcoordinaciones temáticas, así como 10 Consejos Consultivos agrupados por comisiones. El Grupo está representado en la Junta de Coordinación Política y Mesa Directiva, y encabeza 13 comisiones ordinarias y 6 especiales.
O documento discute a importância da educação para o desenvolvimento econômico e social de um país. Ele argumenta que investimentos em educação melhoram a produtividade e capacidade de inovação, levando a maiores ganhos de produto interno bruto a longo prazo. Além disso, uma população mais educada promove sociedades mais igualitárias e democráticas.
This document discusses strategy and strategic planning. It defines strategy as setting choices that position a firm to earn superior long-term returns through uniqueness, value proposition, and sustainable advantage. Common strategic planning involves environmental scans, identifying customer types and needs, developing a value proposition to meet unmet needs, and testing product viability through minimal viable products and metrics like customer acquisition cost. Pivoting based on testing results and repeating the process is emphasized over traditional strategic planning models.
This document contains physics formulae related to mechanics, thermodynamics, electromagnetism, optics, modern physics and more. Some key formulae include:
Density = mass / volume, Force = rate of change of momentum, Kinetic energy = (1/2)mv^2, Ohm's law: V=IR, Index of refraction n=c/v, Half life of radioactive element t1/2=ln(2)/λ, Bohr's model: L=nh/2π.
1. Simple harmonic motion is motion influenced by a restoring force proportional to displacement from equilibrium. For a spring, F=-kx, and for a pendulum, F=mg sinθ.
2. The period T of a spring is the time for one complete oscillation, related to spring constant k and mass m by T=2π√(m/k). Frequency f is the number of oscillations per second, with f=1/T.
3. The displacement y of a spring over time t follows a sinusoidal pattern described by the equation y=A sin(2πft+θ0), where A is the amplitude
Unit 1 Quantum Mechanics_230924_162445.pdfSwapnil947063
1) Quantum mechanics is needed to explain phenomena at the microscopic level that classical physics cannot, such as the stability of atoms and line spectra of hydrogen.
2) According to de Broglie's hypothesis, all matter exhibits wave-particle duality - particles are associated with waves called matter waves. The wavelength of these matter waves is given by de Broglie's equation.
3) In quantum mechanics, the wave function ψ describes the wave properties of a particle. The probability of finding a particle in a region is given by the absolute square of the wave function |ψ|2 in that region.
Classical mechanics failed to explain certain phenomena observed at the microscopic level like black body radiation and the photoelectric effect. This led to the development of quantum mechanics, with key aspects being the wave function Ψ, Schrodinger's time-independent and time-dependent wave equations, and operators like differentiation that act on wave functions to produce other wave functions. The wave function Ψ relates to the probability of finding a particle, with |Ψ|2 representing the probability.
The document provides a list of physics formulas across various topics in mechanics, electricity, thermodynamics, and more. It begins with an introduction on studying physics and understanding concepts through visualization of problems. The bulk of the document then lists key formulas in different areas of physics, providing the formulas and brief explanations. It encourages readers to derive the formulas themselves and find the joy in solving problems independently.
PHYSICS - Chapter 5: Oscillations Exercise SolutionPooja M
1. The document discusses linear simple harmonic motion (S.H.M.) and provides examples and derivations of key equations related to S.H.M. including expressions for velocity, acceleration, and period of oscillation for a simple pendulum and a magnet vibrating in a uniform magnetic field.
2. It is shown that S.H.M. is the projection of uniform circular motion along any diameter of the circle. Graphs of displacement, velocity, and acceleration versus phase angle are provided for a particle performing S.H.M. from the mean and extreme positions.
3. Key conclusions are that the restoring force in S.H.M. is directly proportional to displacement and acts in the
This document provides an overview of natural vibrations with one degree of freedom. It defines key terms like degrees of freedom, simple harmonic motion, angular frequency, and periodic time. Examples of natural vibrations covered include a simple pendulum, a mass on a spring, and beams experiencing transverse vibrations. Methods for analyzing natural vibrations include Rayleigh's energy method and Dunkerley's method for combining frequencies from distributed and point loads. Worked examples are provided to illustrate calculating displacement, velocity, and acceleration of bodies in simple harmonic motion.
This document contains a summary of key physics formulas and concepts for 'O' Level Physics. It includes:
1) Definitions of base SI units for common physical quantities like mass, length, time, current, temperature, and amount of substance.
2) Formulas for measurements, kinematics, dynamics, work, energy, power, gas laws, heat, electromagnetism, optics, and electricity.
3) Descriptions of important physics principles like Newton's laws of motion, conservation of energy, Ohm's law, Kirchhoff's laws, and more.
This document covers concepts in one-dimensional and three-dimensional kinematics, dynamics, work, energy, momentum, rotational motion, and more. Examples are provided to demonstrate how to apply equations for instantaneous and average velocity/acceleration, projectile motion, Newton's laws, work-energy theorem, impulse-momentum, center of mass, moment of inertia, and torque. Problem-solving strategies are outlined for analyzing forces, energy, momentum, and rotational equilibrium.
This document discusses different types of mechanical waves and their properties. It defines mechanical waves as oscillations that transfer energy through a medium. Key points include:
- Mechanical waves can be transverse (perpendicular to direction of travel) or longitudinal (parallel to direction of travel).
- They transport energy through the medium and require a medium, like air or water, to propagate.
- Examples of mechanical waves include water waves, sound waves, and waves on a string or rope.
- Harmonic waves have a sinusoidal shape described by a mathematical function involving amplitude, wavelength, frequency, and phase.
Vibration and WavesT = 2pisqrt(mk)v = lambdafp = A sin(kx.pdfanonakitchen
Vibration and Waves
T = 2pi*sqrt(m/k)
v = lambda*f
p = A sin(kx-wt) w=2*pi*f (pressure in travelling sound wave), k = 2pi/lambda, A = amplitude,
x = displacement.
1. Periodic motion: used to describe a vibration or an oscillation that repeats itself over the same
path.. Displacement: The position of an object. Amplitude: The maximum displacement. Period:
The time in seconds required for one complete cycle. Frequency: The number of cycles, per
second. Simple Harmonic Motion: Sinusoidal motion of a single frequency.
2. Total E = 1/2kA^2. = 1/2kx^2+1/2mv^2. EPE = 1/2kx^2.
3. The equation for the period of a spring is T = 2pi*sqrt(m/k). For example, how long does it
take for a spring with a hanging mass of 5 kg and spring constant 10N/m to exhibit one complete
cycle? T = 2pi*sqrt(5/10). T = 2pi*sqrt(1/2).
4. Since a period is the time it takes for one complete cycle in seconds, frequency is the inverse
of the period. f = 1/T. T = 1/f.
5. For describing the motion generated SHM, you can use x = A sin 2pi*f*t. This is if the motion
starts at equilibrium point. If it starts at the crest/trough of its motion, then use x = A cos 2pi*f*t.
6. For describing the motion generated SHM, you can use x = A sin 2pi*f*t. This is if the motion
starts at equilibrium point. If it starts at the crest/trough of its motion, then use x = A cos 2pi*f*t.
7. The maximum velocity of a particle on SHM can be found by v = 2pi*A*f. Now, for any other
velocity at a point in time, it can be found by v = v0(sqrt(1-x^2/A^2)).
8. The equation for the period of a pendulum is T = 2*pi*sqrt(L/g). L = length of the pendulum.
This only works if its amplitude is small and friction can be ignored.
9. Dampened harmonic motion is the gradual decrease of maximum displacement. This is caused
by the presence of frictional forces. The energy is eventually all transformed to heat.
10. Forced vibration is due to an external source, which applies its own frequency to the spring.
The natural frequency of the spring is actually 1/2pi * sqrt(k/m), which is what it will actually
vibrate at without external force.
11. Amplitude denotes the distance between the equilibrium and the crest or trough of the wave.
Wavelength is the distance from one point of the wave to the corresponding point in the next
cycle. (crest-crest). This can be found be lambda = v/f. Wave velocity is the velocity at which
wave crests move. V = lambda*f. Node is referring to the point of destructive interference.
Antinode is referring to the point of constructive interference.
12. The relationship between velocity, wavelength, and frequency is v = lambda*f, where
lambda is the wavelength and f is the frequency, v is the wave velocity.
13. A transverse wave refers to a wave that causes the particles of the medium it travels in to
move a direction perpendicular to the motion of the wave itself. On the other hand, in a
longitudinal wave, the vibration of the particles of the medium is along the same direction as the
motion of the wa.
Wk 1 p7 wk 3-p8_13.1-13.3 & 14.6_oscillations & ultrasoundchris lembalemba
This document discusses oscillations and simple harmonic motion. It begins by listing learning outcomes related to oscillations, including describing examples of free and forced oscillations. It then provides definitions and equations related to simple harmonic motion, such as the defining equation a=-ω2x. Graphs of displacement, velocity, and acceleration over time for simple harmonic motion are shown. Examples of the simple pendulum and a mass on a spring are provided to illustrate simple harmonic motion. The document also discusses the kinetic and potential energy changes that occur during simple harmonic motion.
PHYSICS (CLASSS XII) - Chapter 5 : OscillationsPooja M
1. Oscillatory motion is a periodic motion that repeats itself after a definite time interval called the period. Linear simple harmonic motion (S.H.M.) is the linear periodic motion of a body where the restoring force is directly proportional to the displacement from the mean position.
2. The differential equation of S.H.M is the second derivative of displacement (x) plus the angular frequency (ω) squared times x equals zero. The expressions for displacement, velocity, and acceleration of a particle in S.H.M involve sinusoidal and cosine functions of angular frequency times time.
3. The amplitude is the maximum displacement, the period is the time for one oscillation, and the frequency is
This document discusses the basic principles of seismic waves. It introduces longitudinal (P) waves and shear (S) waves, and derives the one-dimensional wave equation. It discusses wave phenomena like reflection, transmission, and refraction based on Snell's law at boundaries between layers. It also discusses the different arrivals of direct, reflected, and refracted/head waves that can be measured at the surface for seismic exploration purposes.
This document discusses simple harmonic motion. It begins by introducing simple harmonic motion as the to-and-fro motion of a particle vibrating about a central mean position. It then provides the differential equation that describes simple harmonic motion and discusses key terms like amplitude, period, and frequency. The document also examines simple harmonic motion in pendulums, springs, and other contexts, establishing various governing laws and relationships.
1. Quantum mechanics describes the behavior of matter and light at the atomic scale, which is very different from classical mechanics. Particles have both wave-like and particle-like properties.
2. The de Broglie hypothesis proposed that all particles have an associated wavelength that depends on their momentum. This was confirmed experimentally by observing electron diffraction patterns.
3. Heisenberg's uncertainty principle states that it is impossible to precisely measure both a particle's position and momentum simultaneously. This limits our ability to predict the future behavior of particles.
This document discusses simple harmonic motion (SHM). SHM occurs when an object experiences a restoring force proportional to its displacement from equilibrium. This results in sinusoidal oscillations described by x(t) = Acos(ωt + φ), where A is amplitude, ω is angular frequency, and φ is the phase. SHM includes examples like a mass on a spring and a simple pendulum. The relationships between displacement, velocity, acceleration, period, frequency, and energy in SHM systems are explored.
Oscillations are ubiquitous in nature and occur in many systems when disturbed from equilibrium. The document introduces the simple harmonic oscillator (SHO) model to describe small oscillations near equilibrium. A SHO undergoes sinusoidal oscillations with an angular frequency that depends on the spring constant and mass. Complex numbers provide a useful way to represent the amplitude and phase of oscillations. The SHO model applies to many systems locally, as potentials can often be approximated as quadratic near equilibrium points.
The document summarizes key concepts from elementary quantum physics that will be built upon in the text, including:
1) The time-dependent and time-independent Schrodinger equations, which describe the wave function and energy levels of quantum systems.
2) Observables in quantum physics are represented by operators, and the measurement of an observable leaves the system in an eigenstate of that operator.
3) The Heisenberg uncertainty principle limits the precision with which conjugate variables like position and momentum can be known simultaneously.
4) Angular momentum is quantized and can be decomposed into orbital and spin components, with associated quantum numbers and eigenstates. Operators for total and z-component angular
This document provides a question bank for Physics I with questions related to three topics:
1) Waves and Oscillations: Questions cover types of waves, wave equations, phase and group velocity, standing waves, forced vibrations and resonance.
2) Fields: Questions cover vector and scalar fields, gradient, divergence, curl, Gauss's theorem, Stokes' theorem and their applications.
3) Electromagnetic Theory: Questions cover Gauss's law, electric potential, dielectrics, Ampere's law, Faraday's law, inductance, Maxwell's equations and electromagnetic waves. The document provides 25 questions for the first topic, 9 questions for the second topic and 22 questions for the third topic, for
We at 'Apex Institute' are committed to provide our students best quality education with ethics. Moving in this direction, we have decided that unlike other expensive and 5star facility type institutes who are huge investors and advertisers, we shall not invest huge amount of money in advertisements. It shall rather be invested on the betterment, enhancement of quality and resources at our center.
We are just looking forward to have 'word-of-mouth' publicity instead. Because, there is only a satisfied student and his/her parents can judge an institute's quality and it's faculty members coaching.
Those coaching institutes, who are investing highly on advertisements, are actually, wasting their money on it, in a sense. Rather, the money should be invested on highly experienced faculty members and on teaching gears.
We all at 'Apex' are taking this initiative to improve the quality of education along-with each student's development and growth.
Committed to excellence...
With best wishes.
S . Iqbal
( Motivator & Mentor)
Dear Students/Parents
We at 'Apex Institute' are committed to provide our students best quality education with ethics. Moving in this direction, we have decided that unlike other expensive and 5star facility type institutes who are huge investors and advertisers, we shall not invest huge amount of money in advertisements. It shall rather be invested on the betterment, enhancement of quality and resources at our center.
We are just looking forward to have 'word-of-mouth' publicity instead. Because, there is only a satisfied student and his/her parents can judge an institute's quality and it's faculty members coaching.
Those coaching institutes, who are investing highly on advertisements, are actually, wasting their money on it, in a sense. Rather, the money should be invested on highly experienced faculty members and on teaching gears.
We all at 'Apex' are taking this initiative to improve the quality of education along-with each student's development and growth.
Committed to excellence...
With best wishes.
S . Iqbal
( Motivator & Mentor)
Dear Students/Parents
We at 'Apex Institute' are committed to provide our students best quality education with ethics. Moving in this direction, we have decided that unlike other expensive and 5star facility type institutes who are huge investors and advertisers, we shall not invest huge amount of money in advertisements. It shall rather be invested on the betterment, enhancement of quality and resources at our center.
We are just looking forward to have 'word-of-mouth' publicity instead. Because, there is only a satisfied student and his/her parents can judge an institute's quality and it's faculty members coaching.
Those coaching institutes, who are investing highly on advertisements, are actually, wasting their money on it, in a sense. Rather, the money should be invested on highly experienced faculty members and on teaching gears.
We all at 'Apex' are taking this initiative to improve the quality of education along-with each student's development and growth.
Committed to excellence...
With best wishes.
S . Iqbal
( Motivator & Mentor)
Dear Students/Parents
We at 'Apex Institute' are committed to provide our students best quality education with ethics. Moving in this direction, we have decided that unlike other expensive and 5star facility type institutes who are huge investors and advertisers, we shall not invest huge amount of money in advertisements. It shall rather be invested on the betterment, enhancement of quality and resources at our center.
We are just looking forward to have 'word-of-mouth' publicity instead. Because, there is only a satisfied student and his/her parents can judge an institute's quality and it's faculty members coaching.
Those coaching institutes, who are investing highly on advertisements, are actually, wasting their money on it, in a sense. Rather, the money should be invested on highly experienced faculty members and on teaching gears.
We all at 'Apex' are taking this initiative to improve the quality of education along-with each student's development and growth.
Committed to excellence...
With best wishes.
S . Iqbal
( Motivator & Mentor)
Dear Students/Parents
We at 'Apex Institute' are committed to provide our students best quality education with ethics. Moving in this direction, we have decided that unlike other expensive and 5star facility type institutes who are huge investors and advertisers, we shall not invest huge amount of money in advertisements. It shall rather be invested on the betterment, enhancement of quality and resources at our center.
We are just looking forward to have 'word-of-mouth' publicity instead. Because, there is only a satisfied student and his/her parents can judge an institute's quality and it's faculty members coaching.
Those coaching institutes, who are investing highly on advertisements, are actually, wasting their money on it, in a sense. Rather, the money should be invested on highly experienced faculty members and on teaching gears.
We all at 'Apex' are taking this initiative to improve the quality of education along-with each student's development and growth.
Committed to excellence...
With best wishes.
S . Iqbal
( Motivator & Mentor)
Dear Students/Parents
We at 'Apex Institute' are committed to provide our students best quality education with ethics. Moving in this direction, we have decided that unlike other expensive and 5star facility type institutes who are huge investors and advertisers, we shall not invest huge amount of money in advertisements. It shall rather be invested on the betterment, enhancement of quality and resources at our center.
We are just looking forward to have 'word-of-mouth' publicity instead. Because, there is only a satisfied student and his/her parents can judge an institute's quality and it's faculty members coaching.
Those coaching institutes, who are investing highly on advertisements, are actually, wasting their money on it, in a sense. Rather, the money should be invested on highly experienced faculty members and on teaching gears.
We all at 'Apex' are taking this initiative to improve the quality of education along-with each student's development and growth.
Committed to excellence...
With best wishes.
S . Iqbal
( Motivator & Mentor)
Crash-Course for AIPMT & Other Medical Exams 2016(Essentials heart)APEX INSTITUTE
Dear Students/Parents
We at 'Apex Institute' are committed to provide our students best quality education with ethics. Moving in this direction, we have decided that unlike other expensive and 5star facility type institutes who are huge investors and advertisers, we shall not invest huge amount of money in advertisements. It shall rather be invested on the betterment, enhancement of quality and resources at our center.
We are just looking forward to have 'word-of-mouth' publicity instead. Because, there is only a satisfied student and his/her parents can judge an institute's quality and it's faculty members coaching.
Those coaching institutes, who are investing highly on advertisements, are actually, wasting their money on it, in a sense. Rather, the money should be invested on highly experienced faculty members and on teaching gears.
We all at 'Apex' are taking this initiative to improve the quality of education along-with each student's development and growth.
Committed to excellence...
With best wishes.
S . Iqbal
( Motivator & Mentor)
Crash-Course for AIPMT & Other Medical Exams 2016Target pmt (2)APEX INSTITUTE
Dear Students/Parents
We at 'Apex Institute' are committed to provide our students best quality education with ethics. Moving in this direction, we have decided that unlike other expensive and 5star facility type institutes who are huge investors and advertisers, we shall not invest huge amount of money in advertisements. It shall rather be invested on the betterment, enhancement of quality and resources at our center.
We are just looking forward to have 'word-of-mouth' publicity instead. Because, there is only a satisfied student and his/her parents can judge an institute's quality and it's faculty members coaching.
Those coaching institutes, who are investing highly on advertisements, are actually, wasting their money on it, in a sense. Rather, the money should be invested on highly experienced faculty members and on teaching gears.
We all at 'Apex' are taking this initiative to improve the quality of education along-with each student's development and growth.
Committed to excellence...
With best wishes.
S . Iqbal
( Motivator & Mentor)
Crash-Course for AIPMT & Other Medical Exams 2016 (Essentials cockroach)APEX INSTITUTE
Dear Students/Parents
We at 'Apex Institute' are committed to provide our students best quality education with ethics. Moving in this direction, we have decided that unlike other expensive and 5star facility type institutes who are huge investors and advertisers, we shall not invest huge amount of money in advertisements. It shall rather be invested on the betterment, enhancement of quality and resources at our center.
We are just looking forward to have 'word-of-mouth' publicity instead. Because, there is only a satisfied student and his/her parents can judge an institute's quality and it's faculty members coaching.
Those coaching institutes, who are investing highly on advertisements, are actually, wasting their money on it, in a sense. Rather, the money should be invested on highly experienced faculty members and on teaching gears.
We all at 'Apex' are taking this initiative to improve the quality of education along-with each student's development and growth.
Committed to excellence...
With best wishes.
S . Iqbal
( Motivator & Mentor)
Dear Students/Parents
We at 'Apex Institute' are committed to provide our students best quality education with ethics. Moving in this direction, we have decided that unlike other expensive and 5star facility type institutes who are huge investors and advertisers, we shall not invest huge amount of money in advertisements. It shall rather be invested on the betterment, enhancement of quality and resources at our center.
We are just looking forward to have 'word-of-mouth' publicity instead. Because, there is only a satisfied student and his/her parents can judge an institute's quality and it's faculty members coaching.
Those coaching institutes, who are investing highly on advertisements, are actually, wasting their money on it, in a sense. Rather, the money should be invested on highly experienced faculty members and on teaching gears.
We all at 'Apex' are taking this initiative to improve the quality of education along-with each student's development and growth.
Committed to excellence...
With best wishes.
S . Iqbal
( Motivator & Mentor)
The document provides information about an educational institute called APEX that offers coaching for various competitive exams like IIT-JEE, AIPMT, and NTSE. It highlights some of APEX's strengths such as having experienced and qualified faculty, small student-teacher ratios, regular testing and feedback, and good historical results. It also mentions some questions students should ask before choosing a coaching institute like student-teacher ratios, faculty qualifications, and selection processes.
Dear Students/Parents
We at 'Apex Institute' are committed to provide our students best quality education with ethics. Moving in this direction, we have decided that unlike other expensive and 5star facility type institutes who are huge investors and advertisers, we shall not invest huge amount of money in advertisements. It shall rather be invested on the betterment, enhancement of quality and resources at our center.
We are just looking forward to have 'word-of-mouth' publicity instead. Because, there is only a satisfied student and his/her parents can judge an institute's quality and it's faculty members coaching.
Those coaching institutes, who are investing highly on advertisements, are actually, wasting their money on it, in a sense. Rather, the money should be invested on highly experienced faculty members and on teaching gears.
We all at 'Apex' are taking this initiative to improve the quality of education along-with each student's development and growth.
Committed to excellence...
With best wishes.
S . Iqbal
( Motivator & Mentor)
I.S.C. Class XII MATHEMATICS Sample Papers 2016APEX INSTITUTE
This document provides information about a crash course for IIT-JEE, BITS, UPTU and AIPMT exams, including details about faculty, study material, practice problems, mock tests, and registration. It mentions that the course includes over 240 hours of training by experienced faculty, concise chapter-wise theory, 3000 practice problems, 10 full-length tests, and a 30% discount on registration until March 10th. The batch is scheduled to commence on March 15th, 22nd, and 29th.
The document provides information about a crash course for IIT-JEE, BITS, UPTU and AIPMT exams, including details about the course structure, faculty, study material, practice problems, and test series. It mentions that the course provides over 240 hours of training by experienced faculty, concise chapter-wise theory, 3000 practice problems, expert time management tips, and 10 full-length tests on the IIT-Main exam pattern. It also provides information about course features, registration fees and dates.
Dear Students/Parents
We at 'Apex Institute' are committed to provide our students best quality education with ethics. Moving in this direction, we have decided that unlike other expensive and 5star facility type institutes who are huge investors and advertisers, we shall not invest huge amount of money in advertisements. It shall rather be invested on the betterment, enhancement of quality and resources at our center.
We are just looking forward to have 'word-of-mouth' publicity instead. Because, there is only a satisfied student and his/her parents can judge an institute's quality and it's faculty members coaching.
Those coaching institutes, who are investing highly on advertisements, are actually, wasting their money on it, in a sense. Rather, the money should be invested on highly experienced faculty members and on teaching gears.
We all at 'Apex' are taking this initiative to improve the quality of education along-with each student's development and growth.
Committed to excellence...
With best wishes.
S . Iqbal
( Motivator & Mentor)
Dear Students/Parents
We at 'Apex Institute' are committed to provide our students best quality education with ethics. Moving in this direction, we have decided that unlike other expensive and 5star facility type institutes who are huge investors and advertisers, we shall not invest huge amount of money in advertisements. It shall rather be invested on the betterment, enhancement of quality and resources at our center.
We are just looking forward to have 'word-of-mouth' publicity instead. Because, there is only a satisfied student and his/her parents can judge an institute's quality and it's faculty members coaching.
Those coaching institutes, who are investing highly on advertisements, are actually, wasting their money on it, in a sense. Rather, the money should be invested on highly experienced faculty members and on teaching gears.
We all at 'Apex' are taking this initiative to improve the quality of education along-with each student's development and growth.
Committed to excellence...
With best wishes.
S . Iqbal
( Motivator & Mentor)
Dear Students/Parents
APEX INSTITUTE has been established with sincere and positive resolve to do something rewarding for ENGG. / PRE-MEDICAL aspirants. For this the APEX INSTITUTE has been instituted to provide a relentlessly motivating and competitive atmosphere.
We at 'Apex Institute' are committed to provide our students best quality education with ethics. Moving in this direction, we have decided that unlike other expensive and 5star facility type institutes who are huge investors and advertisers, we shall not invest huge amount of money in advertisements. It shall rather be invested on the betterment, enhancement of quality and resources at our center.
We are just looking forward to have 'word-of-mouth' publicity instead. Because, there is only a satisfied student and his/her parents can judge an institute's quality and it's faculty members coaching.
Those coaching institutes, who are investing highly on advertisements, are actually, wasting their money on it, in a sense. Rather, the money should be invested on highly experienced faculty members and on teaching gears.
We all at 'Apex' are taking this initiative to improve the quality of education along-with each student's development and growth.
Committed to excellence...
With best wishes.
S . Iqbal
( Motivator & Mentor)
Dear Students/Parents
APEX INSTITUTE has been established with sincere and positive resolve to do something rewarding for ENGG. / PRE-MEDICAL aspirants. For this the APEX INSTITUTE has been instituted to provide a relentlessly motivating and competitive atmosphere.
We at 'Apex Institute' are committed to provide our students best quality education with ethics. Moving in this direction, we have decided that unlike other expensive and 5star facility type institutes who are huge investors and advertisers, we shall not invest huge amount of money in advertisements. It shall rather be invested on the betterment, enhancement of quality and resources at our center.
We are just looking forward to have 'word-of-mouth' publicity instead. Because, there is only a satisfied student and his/her parents can judge an institute's quality and it's faculty members coaching.
Those coaching institutes, who are investing highly on advertisements, are actually, wasting their money on it, in a sense. Rather, the money should be invested on highly experienced faculty members and on teaching gears.
We all at 'Apex' are taking this initiative to improve the quality of education along-with each student's development and growth.
Committed to excellence...
With best wishes.
S . Iqbal
( Motivator & Mentor)
Dear Students/Parents
APEX INSTITUTE has been established with sincere and positive resolve to do something rewarding for ENGG. / PRE-MEDICAL aspirants. For this the APEX INSTITUTE has been instituted to provide a relentlessly motivating and competitive atmosphere.
We at 'Apex Institute' are committed to provide our students best quality education with ethics. Moving in this direction, we have decided that unlike other expensive and 5star facility type institutes who are huge investors and advertisers, we shall not invest huge amount of money in advertisements. It shall rather be invested on the betterment, enhancement of quality and resources at our center.
We are just looking forward to have 'word-of-mouth' publicity instead. Because, there is only a satisfied student and his/her parents can judge an institute's quality and it's faculty members coaching.
Those coaching institutes, who are investing highly on advertisements, are actually, wasting their money on it, in a sense. Rather, the money should be invested on highly experienced faculty members and on teaching gears.
We all at 'Apex' are taking this initiative to improve the quality of education along-with each student's development and growth.
Committed to excellence...
With best wishes.
S . Iqbal
( Motivator & Mentor)
Dear Students/Parents
APEX INSTITUTE has been established with sincere and positive resolve to do something rewarding for ENGG. / PRE-MEDICAL aspirants. For this the APEX INSTITUTE has been instituted to provide a relentlessly motivating and competitive atmosphere.
We at 'Apex Institute' are committed to provide our students best quality education with ethics. Moving in this direction, we have decided that unlike other expensive and 5star facility type institutes who are huge investors and advertisers, we shall not invest huge amount of money in advertisements. It shall rather be invested on the betterment, enhancement of quality and resources at our center.
We are just looking forward to have 'word-of-mouth' publicity instead. Because, there is only a satisfied student and his/her parents can judge an institute's quality and it's faculty members coaching.
Those coaching institutes, who are investing highly on advertisements, are actually, wasting their money on it, in a sense. Rather, the money should be invested on highly experienced faculty members and on teaching gears.
We all at 'Apex' are taking this initiative to improve the quality of education along-with each student's development and growth.
Committed to excellence...
With best wishes.
S . Iqbal
( Motivator & Mentor)
Strategies for Effective Upskilling is a presentation by Chinwendu Peace in a Your Skill Boost Masterclass organisation by the Excellence Foundation for South Sudan on 08th and 09th June 2024 from 1 PM to 3 PM on each day.
How to Make a Field Mandatory in Odoo 17Celine George
In Odoo, making a field required can be done through both Python code and XML views. When you set the required attribute to True in Python code, it makes the field required across all views where it's used. Conversely, when you set the required attribute in XML views, it makes the field required only in the context of that particular view.
Main Java[All of the Base Concepts}.docxadhitya5119
This is part 1 of my Java Learning Journey. This Contains Custom methods, classes, constructors, packages, multithreading , try- catch block, finally block and more.
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LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UPRAHUL
This Dissertation explores the particular circumstances of Mirzapur, a region located in the
core of India. Mirzapur, with its varied terrains and abundant biodiversity, offers an optimal
environment for investigating the changes in vegetation cover dynamics. Our study utilizes
advanced technologies such as GIS (Geographic Information Systems) and Remote sensing to
analyze the transformations that have taken place over the course of a decade.
The complex relationship between human activities and the environment has been the focus
of extensive research and worry. As the global community grapples with swift urbanization,
population expansion, and economic progress, the effects on natural ecosystems are becoming
more evident. A crucial element of this impact is the alteration of vegetation cover, which plays a
significant role in maintaining the ecological equilibrium of our planet.Land serves as the foundation for all human activities and provides the necessary materials for
these activities. As the most crucial natural resource, its utilization by humans results in different
'Land uses,' which are determined by both human activities and the physical characteristics of the
land.
The utilization of land is impacted by human needs and environmental factors. In countries
like India, rapid population growth and the emphasis on extensive resource exploitation can lead
to significant land degradation, adversely affecting the region's land cover.
Therefore, human intervention has significantly influenced land use patterns over many
centuries, evolving its structure over time and space. In the present era, these changes have
accelerated due to factors such as agriculture and urbanization. Information regarding land use and
cover is essential for various planning and management tasks related to the Earth's surface,
providing crucial environmental data for scientific, resource management, policy purposes, and
diverse human activities.
Accurate understanding of land use and cover is imperative for the development planning
of any area. Consequently, a wide range of professionals, including earth system scientists, land
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changes, conversion trends, and other related patterns. The spatial dimensions of land use and
cover support policymakers and scientists in making well-informed decisions, as alterations in
these patterns indicate shifts in economic and social conditions. Monitoring such changes with the
help of Advanced technologies like Remote Sensing and Geographic Information Systems is
crucial for coordinated efforts across different administrative levels. Advanced technologies like
Remote Sensing and Geographic Information Systems
9
Changes in vegetation cover refer to variations in the distribution, composition, and overall
structure of plant communities across different temporal and spatial scales. These changes can
occur natural.
Communicating effectively and consistently with students can help them feel at ease during their learning experience and provide the instructor with a communication trail to track the course's progress. This workshop will take you through constructing an engaging course container to facilitate effective communication.
Chapter wise All Notes of First year Basic Civil Engineering.pptxDenish Jangid
Chapter wise All Notes of First year Basic Civil Engineering
Syllabus
Chapter-1
Introduction to objective, scope and outcome the subject
Chapter 2
Introduction: Scope and Specialization of Civil Engineering, Role of civil Engineer in Society, Impact of infrastructural development on economy of country.
Chapter 3
Surveying: Object Principles & Types of Surveying; Site Plans, Plans & Maps; Scales & Unit of different Measurements.
Linear Measurements: Instruments used. Linear Measurement by Tape, Ranging out Survey Lines and overcoming Obstructions; Measurements on sloping ground; Tape corrections, conventional symbols. Angular Measurements: Instruments used; Introduction to Compass Surveying, Bearings and Longitude & Latitude of a Line, Introduction to total station.
Levelling: Instrument used Object of levelling, Methods of levelling in brief, and Contour maps.
Chapter 4
Buildings: Selection of site for Buildings, Layout of Building Plan, Types of buildings, Plinth area, carpet area, floor space index, Introduction to building byelaws, concept of sun light & ventilation. Components of Buildings & their functions, Basic concept of R.C.C., Introduction to types of foundation
Chapter 5
Transportation: Introduction to Transportation Engineering; Traffic and Road Safety: Types and Characteristics of Various Modes of Transportation; Various Road Traffic Signs, Causes of Accidents and Road Safety Measures.
Chapter 6
Environmental Engineering: Environmental Pollution, Environmental Acts and Regulations, Functional Concepts of Ecology, Basics of Species, Biodiversity, Ecosystem, Hydrological Cycle; Chemical Cycles: Carbon, Nitrogen & Phosphorus; Energy Flow in Ecosystems.
Water Pollution: Water Quality standards, Introduction to Treatment & Disposal of Waste Water. Reuse and Saving of Water, Rain Water Harvesting. Solid Waste Management: Classification of Solid Waste, Collection, Transportation and Disposal of Solid. Recycling of Solid Waste: Energy Recovery, Sanitary Landfill, On-Site Sanitation. Air & Noise Pollution: Primary and Secondary air pollutants, Harmful effects of Air Pollution, Control of Air Pollution. . Noise Pollution Harmful Effects of noise pollution, control of noise pollution, Global warming & Climate Change, Ozone depletion, Greenhouse effect
Text Books:
1. Palancharmy, Basic Civil Engineering, McGraw Hill publishers.
2. Satheesh Gopi, Basic Civil Engineering, Pearson Publishers.
3. Ketki Rangwala Dalal, Essentials of Civil Engineering, Charotar Publishing House.
4. BCP, Surveying volume 1
2. PHYSICS
General
1. If x = ambnco, then
2. For vernier calipers, least count = s-v
(s=length of one division on main scale, v=length of one division on vernier scale)
3. Length measured by vernier caliper = reading of main scale + reading of vernier scale ×
least count.
4. For screw gauge, least count =
where, Pitch =
5. Length measured by screw gauge = Reading of main scale + Reading of circular scale ×
least count.
6. Time period for a simple pendulum =
Where, l is the length of simple pendulum and g is gravitational acceleration.
7. Young’s modulus by Searle’s method, ,
where, L= initial length of the wire, r=radius of the wire and ∆l= change in length.
8. Specific heat of the liquid,
where, m=mass of the solid. m2= mass of the cold liquid
T1=temperature of cold liquid. T2=temperature of hot liquid.
T = final temperature of the system c1= specific heat of the material of
calorimeter and stirrer.
c2= specific heat of material of solid m1 = mass of the calorimeter and stirrer
Mechanics
Vectors
1. = , where is angle between the vectors.
And direction of from ,
2. Two vectors (a1 + a2 + a3 ) and (b1 + b2 + b3 ) are equal if:
a1 = b1 a2= b2 and a3 = b3
3. If angle between two vectors and is ‘ ’ , = ab
x = (ab ) ( is a unit vector perpendicular to both and )
4. Velocity of ‘B’ with respect to ‘A’ , = -
3. Kinematics
t1 = initial time t2 = final time u= initial velocity v = final velocity
v av = average velocity a = acceleration s = net displacement
1.Average speed, Vav = 4. Total distance =
2.Instantaneous speed, 5. a= =v =
V= = 6. When acceleration is constant
v= u +at
3. Displacement =
s= ut + at2 = vt - at2
v2 = u2+2as
Projectile motion
1. Time of flight , t0 =
Range , R=
2. Maximum height H=
3. (x,y)=(ucos )
4. Equation of Projectile ,y =xtan
Forces
1. F - R=ma
2. Frictional force =f, force applied =F
3. =F
4. Circular Motion
1. = = 4. Non- conservative forces : frictional
forces, viscous force..
Α= == Center of Mass, Linear
Momentum, Collision
2. =
1.
AN = 2.
3.
atotal=
4.
3. R at (x,y) =
4. Banking of roads, tan = 5. For perfectly inelastic collision,
6. .If coefficient of restitution is e
(0<e<1),
5. Centripetal force = = mr Velocity of separation = e (Velocity
of approach)
Work and Energy V=
1. Total Energy = Kinetic energy =
potential energy 7. Impulse = =
2. =
3. Conservative forces : spring force,
electrostatic forces…
Rotational Mechanics
η=torque, F=force I=moment of inertia α=angular acceleration L=angular momentum
3.
4. (i) Pure translation
1. Moment of inertia,
(ii) Rotation 0
I= = dm
(iii) Pure rotation 0
(iv) Translation
2. Angular momentum,
(v) Rolling
(vi) Sliding or sliding
L=Iω
5. Gravitation
G = Universal gravitational constant E = Gravitational field
U = Gravitational potential energy F = Gravitational force
V = Gravitational potential
1. F = G (attraction force)
2. Gravitational potential energy, U = -G
3.
4. Gravitational potential, V =
5. Gravitational field, E =
6. Escape velocity, u
Planets and satellites
1. v = T=2 K.E. = , P.E =- => E =-
Simple Harmonic Motion
angular frequency I = moment of inertia T = time period = length of pendulum
1. + =0 4. Physical Pendulum
2.
T= =
3. Angular simple harmonic motion,
T= 2 , =
5. Simple Pendulum,
T= 2 , =
Fluid Mechanics
P= pressure =density V=volume of solid v=volume immersed D= density of solid
d=density of liquid A=cross section area U=upthrust
1) p=
2) Variation of pressure with height, dP = - dh
3) Archimedes Principle mg = v dg (or) VD = vd
4) Equation of continuity, A1v1= A2v2
5) Bernoulli’s equation, P+ p + gh = constant
6. Elasticity
Y= Young’s modulus = stress = strain B=Bulk modulus
F= force A=cross section area =initial length = change in length
1. Y= = 2) B = - 3) Elastic potential energy = × stress × volume
Surface tension
T = surface tension Θ = angle of contact R = radius of the bubble/drop R = radius of the tube
1. Excess pressure inside a drop, ΔP = Excess pressure inside a soap bubble, ΔP =
2. Rise of liquid inside a capillary tube, h =
Viscosity
η=coefficient of viscosity; F=force V=velocity; ζ = density of liquid ; A = Cross
section area
1. F = Stoke’s Law, F = 6πrηv Terminal Velocity, v0 =
Wave Motion
A = Amplitude y = Displacement ΔФ = Phase Difference γ = Frequency λ = wavelength
L = Length of the wire Μ = Mass per unit length ω = Angular frequency Δx = Path difference
1. Equation of a wave, y=
2. Velocity of a wave on a string, V=
3.
4.
Resultant Wave, y = y1 + y2
Constructive interference, Δθ = 2nπ Or Δx = nλ
Destructive interference, Δθ = (2n-1)π Or Δx = (n-1/2)λ
Fundamental Frequency, γ0 =
7. Sound Waves
1. Speed of sound in :
1)Fluids = (b: Bulk Modulus, ρ: Density 2)Solids = (Y: Young’s Modulus, ρ:
Density 3) Gas =
1. Closed organ Pipe, 3. Freq. of beats = |ν1 - ν2|
2. Open organ Pipe, 4. Doppler Effect, ν =
Thermal Physics
P=pressure V=volume n= no. of moles T=temperature R= universal gas constant
co-efficient of linear thermal expansion β=co-efficient of superficial themal expansion
=coefficient of volume thermal expansion
1. Ideal gas equation , PV=nRT . 2.Thermal expansion , α= ;
β= ; =
Kinetic Theory of Gases
1. = Translational kinetic energy , K=
= p; =
2. Vander Waal’s Equation:
( p+ )(v-b)=nRT
Calorimetry
Q=heat taken /supplied ; s=specific heat; m=mass; =change in temperature; L=latent heat of
state change per mass
1. Q=ms 2.Q=mL
Laws of Thermodynamics
W=work done by gas , U=internal energy =initial volume =final pressure =initial pressure
=final pressure
1. dq = dW + dU 2.W = 3. Work done on an ideal gas:
8. 4.Isothermal process , W = nRT ln( ) Isobaric process W = P( )
Isochoric Process, W=0
Adiabatic process, W=
5)Entropy, 6). = 7)
8) = R, = 9) Adiabatic Process ,P =constant
Heat Transfer
e = coefficient of emission =stefan’s constant1 K=coefficient of thermal conductivity
1. =K =-KA 2)Thermal resistance ,R =
2) Heat current , I= ) 4)Series connection ,R=
5) Parallel connection , 6)U=e A (U=energy emitted per second)
2. Newton's law of cooling , f =-K( )( taken in Celsius scale )
Optics :
u=dist. Of the object from the lens/mirror v=dist. Of the image from the lens/mirror
m= magnification i= angle of incidence r=angle of reflection/ refraction
n=refractive index θc=critical angle δ=angle of deviation R= radius of
curvature
P=power
1. Spherical Mirrors, m=-
2. Refraction at plane surfaces n= = real depth / apparent depth
-1
Θc= sin (1/n)
3. Refraction at spherical surface m=
4. Refraction through thin lenses , , m=
5. Prism r + r’ =A , δ = i + i’ – A n=
9. Electricity and Magnetism :
Coloumb’s law: The force between 2 point charges at rest: =
Electric field: =q
Electric potential: ∆V= -E∆r cosѲ V=-
Electrical potential energy: U=
Fields for a particular point: E= . V= .
Gauss law: Net electric flux through any closed surface is equal to the net charge enclosed
by the surface divided by ε0. =
Electric dipole: It is a combination of equal and opposite charges. Dipole moment
,where d is the separation between the 2 point charges.
Electric field due to various charge distribution:
(a) Linear charge distribution: E= , λ is the linear charge density
(b) Plane sheet of charge E=ζ/2ε0 where ζ is the surface charge density
(c) Near a charged conducting surface: E=ζ/ε0
(d) Charged conducting spherical shell:
= , r>R = , r=R
(e) Non conducting charged solid sphere:
= , r>R = , r=R
(f) Facts:
1. In an isolated capacitor, charge does not change.
2. Capacitors in series have equal amt of charges
3. The voltage across 2 capacitors connected in parallel is same.
4. In steady state, no current flows through a capacitor.
5. Sum of currents into a node is zero.
6. Sum of voltages around a closed loop is zero.
7. The temperature coefficient of resistivity is negative for semiconductor.
(g) Electric potential due to various charge distributions
2
Charged ring – V = q/( + x2))
Spherical shell -
(h) Capacitance of a parallel plate capacitor, C = ε0A/d
(i) Ohm’s Law, E = ρJ
10. (j) Force between the plates of a capacitor – Q2/2ε0A
(k) Capacitance of a spherical capacitor, C = ab/(b-a)
(l) Wheatstone Bridge : R1/R2 = R3/R4 , where R1 and R3, R2 and R4 are part of the same
bridges respectively.
Charge on a capacitor in an RC circuit Q(t)= Q0(1-e-t/(RC) ) where Q0 is the charge on the
capacitor at t=0.
Capacitance of a capacitor partially filled with a dielectric of thickness t,
Force between two plates of a capacitor:
Capacitance of a spherical capacitor
Capacitance of a cylindrical capacitor: C=
Grouping of cells:
a) Series combination
If polarity of m cells is reversed,
b) Parallel combination
c) Mixed combination
Current will be maximum when
Heat produced in a resistor=
11. MAGNETISM
Facts:
1.Force on a moving charge in magnetic field is perpendicular to both and
2.Net magnetic force acting on any closed current loop in a uniform magnetic field is zero.
3.Magnetic field of long straight wire circles around wire.
4.Parallel wires carrying current in the same direction attract each other.
Formulae: Force in a charge particle, F=q( ) thus F=qvBsinθ
When a particle enters into a perpendicular magnetic field,it describes a circle.Radius of the
circular path, r= = Time period, T=
Magnetic force on a segment of wire, F=I( )
Force between parallel current carrying wires, =
A current carrying loop behaves as a magnetic dipole of magnetic dipole moment, .
Torque on a current loop: .
Magnetic field on due to a current carrying wire ,
Ampere’s Law : =
Magnetic Field Due to various current disributions :
1)Current in a straight wire :- B=
2)For an infinitely long straight wire :- a=b=π/2. Thus B=
3)On the axis of a circular coil :-
4) At the centre of the circular coil :- B=
5) For a circular arc, B=
6) Along the axis of asolenoid Bc= where n=N/l (No.of turns per unit length )
7) For a very long solenoid Bc=µ0nI 8) At the end of a long solenoid, B= µ0nI/2
12. Electromagnetic – Induction
Induced emf ε=- N where A is the area of the loop . Induced current ,
I=ε/R Induced electric field =- Self –Inducatnce, N =LI ε=-L
Inducatance of a solenoid L=µ0 n2A l where ‘n’ is the number of turns per unit length
Mutual Inducatance, NΦ = MI and ε= - M
Growth of curent in an L-R circuit, I= where = L/R
Decay of current in an L-R circuit I= | Energy stored in a conductor , U = ½ LI2
AC –Circuits
R.M.S current , Irms=I0 /
In RC Circuit Peak current , I0= 0 /Z = 0/
In LCR Circuits –
If 1/ c > ωL, current leads the voltage *If 1/ωc < ωL, current lags behind the voltage.
2
If 1 = ω LC, current is in phase with the voltage.
Power in A.C circuit, P = VrmsIrmscos Ѳ, where co Ѳ is the power factor.
For a purely resistive circuit, Ѳ = 0 * For a purely reactive circuit, Ѳ = 90 or 270. Thus, cosѲ = 0.
Modern Physics :
Problem solving technique ( for nuclear physics)
(a) Balance atomic number and mass number on both the sides.
(b) Calculate the total energy of the reactants and products individually and equate
them.
(c) Finally equate the momenta of reactants and products.
If a particle of mass ‘m’ and charge ‘q’ is accelerated through a potential difference ‘v’,
then wavelength associated with it is given by λ = (h/√(2mq)) x (1/√v))
The de-Broglie wavelength of a gas molecule of mass ‘m’ at temperature ‘T’(in Kelvin) is
given by λ = h/√(3mkt), where k = Blotzmann constant
Mass defect is given by ∆m = [Zmp + (A – Z) mn - mZA] where mp, mn and mZA be the
masses of proton, neutron and nucleus respectively. ‘Z’ is number of protons, (A-Z) is
number of neutrons.
When a radioactive material decays by simultaneous ‘ and ‘ ’ emission, then decay
constant ‘λ’ is given by λ = λ1 + λ2
13. CHEMISTRY
Organic Chemistry
Certain important named reactions
1. Beckmann rearrangement
This reaction results in the formation of an amide(rearrangement product)
2. Diels alder reaction
This reaction involves the addition of 1,4-addition of an alkene to a conjugated
diene to form a ring compound
3. Michael reaction
It’s a base catalysed addition of compounds having active methylene group to
an activated olefinic bond.
Addition reactions
1. Electrophilic addition
14. Markonikov’s rule
Addition to alkynes
2. Nucleophilic addition
3. Bisulphite addition
4. Carbanion addition
Substitution reaction mechanism
1 .Sn1 mechanism
Rate α [R3CX]
First order reaction and rate of hydrolysis of alkyl halides- allyl>benzyl>3⁰>2⁰>1⁰>CH3
2. Sn2 mechanism
Rate α [RX][Nu-] > Rate of hydrolysis-CH3>1⁰>2⁰>3⁰
15. Elimination Reaction mechanisms
1.E1 mechanism(first order)
2. E2 mechanism(second order)
3. E1cB mechanism(elimination, unimolecular)
Comparison between E2 and Sn2 recations
Halogenation
Order of substitution- 3⁰ hydrogen>2⁰ hydrogen>1⁰ hydrogen
RH+X2 -- RX+HX (in presence of UV light or heat)
Reactivity of X2: F2>Cl2>Br2>I2
16. Polymers
Polymer classification
1.based on origin
Natural. Eg-silk, wool, starch etc
Semi-synthetic. Eg-nitrocellulose, cellulose xanthate etc
Synthetic. Eg-teflon, polythene
2. based upon synthesis
Addition polymers. Eg-ethene, polyvinyl chloride
Condensation polymers. Eg- proteins, starch etc
3.based upon molecular forces
Elastomers- they are polymers with very weak intermolecular forces. Eg-vulcanised
rubber
Fibres- used for making long thread like fibres. Eg-nylon-66
Thermoplastics- can be moulded by heating. Eg-polyethylene
Thermosetting polymers- becomes hard on heating. Eg-bakelite
17. Inorganic Chemistry
E. Diagonal Relationship
1. PERIODIC CLASSIFICATION OF It occurs due to similar
ELEMENTS electronegativites and sizes of
participating elements.
A. Atomic radius Disappears after IV group
radius order-van der
waal’s>metallic>covalent 2. TYPES OF COMPOUNDS
in case of isoelectronic species,
when proton number increases, radii A.Fajan’s rules
decreases A compound is more ionic(less
covalent) if it contains larger cation
B. Ionisation potential than anion and has an inert gas
(i)It decreases when configuration.
Atomic size increases. A compound is more covalent if it
Screening effect increases. has small cation and has pseudo
Moving from top to bottom in a inert gas configuration(18 e-
group. configuration).
(ii)It increases when
Nuclear charge increases. B.VSEPR Theory
Element has half filled or fully filled Non-metallic compounds
subshells.
Moving from left to right in a period. B4C3 is hardest artificial substance
(iii)Order of ionization potential is Acidic nature of hydrides of
I1<I2<….<In halogens-HI>HBr>HCl>HF
Acidic nature of oxy-acids of
C. Electron Affinity(E.A) halogens-
2nd E.A is always negative. HClO<HClO2<HClO3<HCIO4
E.A of a neutral atom is equal to the Acidic character decreases with
ionization potential of its anion decrease in electronegativity of
For inert atoms and atoms with fully central halogen atom
filled orbitals,E.A is zero. (CN)2, (SCN)2 etc are called
pseudohalogens
D. Oxidation state Compounds containing C,Cl, Br, F
Oxidation state of s-block elements elements are called halons.
is equal to its group number. Nature of compounds
P-block elements show multivalency.
Common oxidation state of d-block Non-metallic oxides are generally
elements is +2 though they also acidic in nature and metallic oxides
show variable oxidation states. are generally basic in nature.
The common oxidation state of f- Al2O3, SiO2 etc are amphoteric in
block elements is +3. nature and CO,NO etc are neutral.
No element exceeds its group Silicones are polymeric
number in the oxidation state. organosilicon compounds containing
Ru and Os show maximum oxidation Si-O-Si bonds.
state of +8 and F shows only -1
state.
18. 3.EXTRACTIVE METALLURGY
Mineral is naturally occurring 5.CO-ORDINATION CHEMISTRY AND
compound with definite structure and ORGANO-METALLICS
ore is a compound from which an Compounds containing complex
element can be extracted cations are cationic complexes and
economically. anionic complexes.
The worthless impurities sticking to The compounds which dont ionize in
the ore is called gangue and flux is a aqueous solutions are neutral
chemical compound used o remove complexes
non-fusible impurities from the ore. Monodentate ligands that are
Roasting is the process of heating a capable of coordinating with metal
mineral in the presence of air. atom by 2 different sites are called
Calcination is the process of heating ambidentate ligands like nitro etc.
an ore in the absence of air Ligands containing π bonds are
Smelting is the process by which a capable of accepting electron
metal is extracted from its ore in a density from metal atom into empty
fused state antibonding orbital π* of their own
are called π acid ligands.
4. TRANSITION ELEMENTS Coordination number of a metal in
The effective magnetic moment is complex
√n(n+2) B.M(bohr magneton) C.N=1 x no.of monodentate ligands
Color of the compound of transition C.N=2 x no.of bidentate ligands
metals is related to the existence of C.N=3 x no.of tridentate ligands
incomplete d-shell and Charge on complex= O.N of metal
corresponding d-d transition atom+O.N of various ligands
Catalytic behavior is due to variable
oxidation states
They form complexes due to their
small size of ions and high ionic
charges
Physical Chemistry
1. BASIC CONCEPTS Vmp:Vavg:Vrms=1 : 1.128 : 1.224
Van der waal’s equation:
Average atomic mass =
Avogadro’s Law - V
Moles = For 1 molecule, the K.E = 1.5
Atomicity (γ) = = 1.5KT
Cp-Cv = R Critical Pressure :
n = [Molecular Formula Critical Volume : 3b
Weight]/[Empirical formula Weight] Critical Temperature :
Graham’s law of diffusion
2. STATES OF MATTER (effusion) :
Density= Rate of diffusion
Partial pressure= Total
pressure x Mole fraction
19. 3. ATOMIC STRUCTURE 5. CHEMICAL EQUILIBRIUM
Specific charge = e/m = 1.76 x In any system of dynamic
108 c/g equilibrium, free energy change,
Charge on the electron = 1.602 x ∆G = 0
10-19 coulomb The free energy change, ∆G and
Mass of electron = 9.1096 x 10-31 equilibrium constant, K are
kg related as ∆G = -RT lnK.
Radius of nucleus, r = (1.3 x 10- Common Ion Effect : By addition
13
) A1/3 cm, where A = mass of X mole/L of a common ion to a
number of the element weak acid (or weak base),
Energy of electron in the nth orbit becomes equal to Ka/X
= Solubility Product : The Ionic
product (IP) in a saturated
Photoelectric effect : hν = hν0 + solution of the sparingly soluble
0.5mv2 salt = solubility product(SP)
Heisenberg’s Uncertainty IP > SP Precipitation occurs
Principle : ∆x.∆p = h/4π
IP = SP Solution is saturated
Spin Magnetic Moment =
IP < SP Solution is unsaturated
BM ; 1 BM = 9.27 x 10-
24
J/T 6. THERMOCHEMISTRY
The maximum number of
emission lines = Heat of reaction = Heat of
Radioactivity : Decay Constant formation of products – Heat of
formation of reactants = Heat of
combustion of reactants – Heat
of combustion of products =
4. SOLUTIONS
Bond Energy of the reactants –
Bond energy of the products
% by weight of solute = , ∆H = ∆E + ∆n RT
where W = weight of the solution ∆G = ∆H - T∆S (Gibb’s Helmoltz
in g Equation)
Molarity = , where w2 = ∆E = ∆q + w
weight in g of solute whose Order of a reaction = The sum of
molecular weight is M2, V = the indices of the concentration
volume of solution in ml terms in the rate equation. It is
Raoult’s Law (for ideal solutions an experimental value. It can be
of non-volatile solutes): zero, fractional or whole number.
p = p0X1, where Molecularity = the number of
p =Vapour pressure of the molecules involved in the rate
solution, p0 = Vapour pressure of determining step of the reaction.
the solvent and X1 = Mole It is a theoretical value, always a
Fraction of the solvent whole number.
Van’t Hoff factor :
7. ELECTROCHEMISTRY
=
Faraday’s 1st Law – W = Elt,
where W= amount of substance
liberated
20. E = electrochemical equivalent of Triclinic a
the substance, I = current
Hexagonal a=b
strength in amperes and t = time
in seconds. Rhombohedral a=b=c
Faradays 2nd Law – The amount
of substances liberated at the
electrodes are proportional to Superconductivity : A superconductor is
their chemical equivalent, when a material that loses abruptly its
the same quantity of current is
passed through different resistance to the electric current when
electrolytes. 1g eq. wt. of the cooled to a specific characteristic
element will be liberated by temperature. Superconductors are non
passing 96500 coulombs of – stoichiometric compounds consisting
electricity. of rare earthen silicates.
8. NUCLEAR CHEMISTRY
10. SURFACE CHEMISTRY
Isothermal variation of extent of
1 amu = 1.66 x 10-24 g
adsorption with pressure is
1 eV = 1.6 x 10-19 J
1 cal = 4.184 J
Decay Constant λ = Where x is mass of gas adsorbed by
the mass m of adsorbent at pressure P.
K and n are constant for a given pair of
Half life Period t1/2= adsorbant and adsorbate.
Amount N of Substance left after ‘n’ Hardy Schultz Rule
1. The ion having opposite charge to sol
half lives =
particles cause coagulation
2. Coagulating power of an electrolyte
9. SOLID STATE
depends on the valency of the ion, i.e.
CRYSTAL INTERCEP CRYSTAL ANGLES greater the valency more is the
SYSTEM TS coagulating.
Cubic a=b=c Gold Number: The no. of milligrams of
protective colloid required to just
Ortho-rhombic a b
prevent the coagulation of 10ml of red
Tetragonal a=b
gold sol when 1 ml of 10% solution of
Monoclinic a NaCl is added to it.
21. Maths
Complex Numbers
i denotes a quantity such that i2=-1 .
A complex number is represented as a+ib , If a+ib=0 then a=0 and b=0. Also a+ib=c+id
then a=c and b=d.
De Moivre’s Theorem : (cosq +isinq)n=cosnq +sinnq
Euler’s Theorem : e =cosq + isinq And e-iq=cosq-isinq
iq
Therefore cosq=(eiq+e-iq)/2 And sin q = (eiq-eiq)/2
Conjugate complex numbers
and
and
Rotational approach If z1, z2, z3 be vertices of a triangle ABC described in counter-
clockwise sense, then:
or
Properties of Modulus
1) 2) 3) 4)
5) 7)
DeMoivre’s Theorem
If n is a positive or negative integer then (cosA + isinA)n = cos nA + isin nA
Quadratic Equations
A quadratic expression is like ax2+bx+c=0 and its roots are (–b+(b2-4ac)1/2)/2a
and (-b-(b2-4ac)/2a.
If p and q are the two roots of the equation then ax2+bx+c=a(x-p)(x-q)
Also p + q=-b/a. Product of the roots p *q=c/a.
Nature of the roots
D=b2-4ac where D is called discriminant
a) If D>0, roots are real and unequal b) If D=0,roots are real and equal
b) If D<0,then D is imaginary. Therefore the roots are imaginary and unequal.
Conditions for Common Roots
a1x2+b1x+c1=0 a2x2+b2x+c2=0 condition is
a1/a2 = b1/b2= c1/c2
22. Progressions :
Arithmetic Progressions:
1. A general form of AP is a,a+ad,a+2d,.......
2. The nth term of the AP is a+(n-1)d. The sum of n terms are n/2(2a+(n-1)d).
3. Some other sums 1+2+3+4......+n=n(n+1)/2
4. 12+22+32+.....+n2=n(n+1)(2n+1)/6 and 13+23+33.......+n3=n2(n+1)2/4.
Geometric Progressions:
1. A general form of GP is a,ar,ar2,ar3,......
2. The nth term is arn-1 , and the sum of n terms is a(1-rn)/1-r where r>1 And a/(1-r)
where (|r|<1) .
Harmonic Progressions:
1. A general forms of a HP is 1/a, 1/(a+d),1/(a+2d),......
Means : Let Arithmetic mean be A, Geometric mean be G and Harmonic Mean be H,
between two positive numbers a and b, then
A=a+b/2 , G=(ab)2, H=2ab/(a+b).
A,G,H are in GP i.e. G2=HA Also A>=G>=H.
Series of Natural Numbers
Logarithms
(i) loga(mn) = logam + logan (ii) loga(m/n) = logam - logan
p
(iii) loga(m ) = plogam
(vi) logba logcb = logca
(vii) logba = 1/logab (a≠1.b≠1,a>0,b>0) (viii) logba logcb logac = 1
Permutations :
1. Multiplication Principle : There are m ways doing one work and n ways doing another
work then ways of doing both work together = m.n
2. Addition Principle: There are m ways doing one work and n ways doing another work
then ways doing either m ways or n ways = m+n.
23. n
3. pr=n(n-1)(n-2)(n-3).......(n-r+1) =
4. The number of ways of arranging n distinct objects along a circle is (n-1)!
5. The number of permutations of n things taken all at a time,p are alike of one kind,q
are alike of another kind and r are alike of a third kind and the rest n-(p+q+r) are all
different is
6. The number of ways of arranging n distinct objects taking r of them at a time where
any object may be repeated any number of times is n-r
7. The coefficient of xr in the expansion of (1-x)-n = n+r-1Cr
8. The number of ways of selecting at least one object out of ‘n’ distinct objects = 2n-1
9. npr = n-1pr + n-1pr-1
10. The number of permutations of n different objects taken r at a time is nr.
11. n unlike bjects can be arranged in a circle in n-1pr .
Combinations :
1. A selection of r objects out n different objects without reference to the order of placing
is given by nCr.
* nCr = npr/r! * nCr= nCn-r . * nCr= n-1Cr-1 + n-1Cr
Some important results
1)nC0= nCn = 1. nC1=n 2) nCr + nCr-1 = n+1Cr
3)2n+1C0+ 2n+1C1+.....+ 2n+1Cn = 22n 4)nCr = n/r . n-1Cr-1
5) nCr/ nCr-1 = n-r+1/r 6)nCn + n+1Cn+ n+2Cn+......+ 2n-1Cn= 2nCn+1
Probability :
Let A and B be any two events. Then A or B happening is said to be A union B(A+B) and
A and B happening at the same time is said to be A intersection B(AB).
1.P(A)=P(AB) + P(AB’) 2. P(B)=P(AB)+P(A’B)
3. P(A+B)=P(AB)+P(AB’)+P(A’B) 4. P(A+B)=P(A)+P(B)-P(AB)
5.P(AB)=1-P(A’+B’) 6.P(A+B)=1-P(A’B’)
24. Trigonometry :
sin(A+B)=sinAcosB + cosAsinB sin(A-B)=sinAcosB-cosAsinB
cos(A+B)=cosAcosB – sinAsinB cos(A-B)=cosAcosB – sinAsinB
tan(A+B)=(tanA + tanB)/(1-tanAtanB) tan(A-B)=(tanA-tanB)/(1+tanAtanB)
Transformations:
sinA +sinB = 2 sin(A+B/2)cos(A-B/2) sinA – sinB=2 cos(A+B/2)sin(A-B/2)
cosA + cosB=2cos(A+B/2)cos(A-B/2) cosA – cosB=2sin(A+B/2)sin(A-B/2)
Relations between the sides and angles of a triangle
(Here a,b,c are three sides of a triangle , A,B,C are the angles , R is the circumradius and
s is semi-perimeter of a triangle)
a/sinA =b/sinB=c/sinC=2R(Sine Formula) cosC=(a2+b2- c2)/2ab(Cosine Formula)
a=c cosB + b cosC(Projection Formula)
Half Angle Formulas
sinA/2={(s-b)*(s-c)/bc}(1/2) cosB/2={s(s-b)/ca}1/2
tanA/2={(s-b)(s-c)/s(s-a)}1/2
Inverse Functions :
sin-1x+cos-1x=π/2 tan-1x+cot-1x=π/2
tan-1x + tan-1y= tan-1(x+y/1-xy) if xy<1 tan-1x + tan-1y= π- tan-1(x+y/1-xy) if xy>1
sin-1x+sin-1y=sin-1[x (1-y2)1/2 + y(1-x2)1/2] sin-1x-sin-1y=sin-1[x (1-y2)1/2 - y(1-x2)1/2]
cos-1x+cos-1y=cos-1[xy- (1-x2)1/2(1-y2)1/2] cos-1x-cos-1y=cos-1[xy+(1-x2)1/2(1-y2)1/2]
Analytical Geometry :
Points : let A,B,C be respectively the points (x1,y1),(x2,y2) and (x3,y3).
The centroid of the triangle is [x1+x2+x3/3 , y1+y2+y3/3]
In centre of triangle [ax1+bx2+cx3/a+b+c , ay1+by2+cy3/a+b+c ]
The Area is given by ½[y1(x2-x3)+y2(x3-x1)+y3(x1-x2)]
Locus: When a point moves in accordance with a geometric law, its path is called a locus.
Line: Standard form: ax+by+c=0 Slope form :y=(tanq)x+c where q is the angle the line
makes with the x-axis and ‘c’ is the intercept on y-axis.
Intercept form : x/a + y/b =1 where a and b are intercepts on x and y axes.
Normal form :xcosq +ysinq=p
Line passing through the points (x1,y1) and (x2,y2) is (x-x1)/(x1-x2)= (y-y1)/(y1-y2)
Length of a perpendicular from a point (x1,y1) to the line ax+by+c=0 is |(ax1+by1+c)/(a2+b2)(1/2) |
25. Circle :
1. Equation of a circle: A circle having centre(h,k) and radius r (x-h)2+(y-k)2=r2.
2. General equation of a circle : x2+y2+2gx + 2fy+c=0 here the centre is (-g,-f) and radius is
(g2+f2-c)
3. Equation of circle described by joining the points (x1,y1) and (x2,y2) as diameter is:
(x-x1)(x-x2)+(y-y1)(y-y3)
4.
Length of tangent : From (x1,y1) to the circle x2+y2+2gx + 2fy+c=0 is
[x12+y12+2gx1 + 2fy1+c]1/2
5. Equation of tangent: the equation of a tangent to x2+y2+2gx + 2fy+c=0 at(x1,y1) is
xx1+yy1+g(x+x1)+f(y+y1)+c=0.
6. Condition for the line y=mx + c to touch a circle is x2+y2=a2 is c2=a2(1+m2).
7. Condition for orthogonal intersection of two circles S= x2+y2+2gx + 2fy+c=0 and
S1=x12+y12+2gx1 + 2fy1+c = 0 is given by 2gg1+2ff1=c+c1
Parabola:
1. The standard equation is y2=4ax where x-axis is axis of parabola and y-axsi is tangent at
the vertex. Vertex is A(0,0) and Focus is S(a,0) and Directrix is x+a=0
2. Parametric Form of a point on y2=4ax is P(at2,2at). At P the slope of tangent is 1/t.
3. Equation of tangent is x-yt+at2=0. Equation of normal is y+tx-2at-at3=0.
2 2
4. If P(at1 ,2at1) and Q(at2 ,2at2) then the slope of chord PQ is 2x-y(t1+t2)+2at1t2=0
Ellipse:
1. Standard equation is x2/a2+ y2/b2=1 ; x-axis is major axis length 2a y-axis is minor axis
length 2b And b2= a2 (1-e2) [ e is eccentricity and e < 1]
2. There are two foci S(ae,0) and S’(-ae,0). And the two directrices are x=a/e and x=-a/e.
3. If P is any point on ellipse then i) SP + S’P=2a ii)SP. SP’=CD2 where CD is semi-diameter
parallel to the tangent at P.
4. Parametric Form of a point P on x2/a2+ y2/b2=1 is P(acosq, bsinq) . The equation of the
tangent is x/a cosq + y/b sinq -1=0 . Equation of normal is ax/cosq - by/sinq=a2-b2
5. The locus of points of intersection of perpendicular tangents of the ellipse x2/a2+ y2/b2-1=0 is
called the director circle and is given by x2+y2=a2+b2 .
Hyperbola:
1. Standard equation of Hyperbola is x2/a2- y2/b2=1 . x-axis- transverse axis length -2a ,
y-axis conjugate axis, length 2b where e =1+ b2/a2.
2
2. Parametric equation of a point on x2/a2- y2/b2=1 are x=asecq and y=b tanq where q is the
parameter.
3. Auxiliary circle : The circle described on the transverse axis of the hyperbola as diameter is
called auxiliary circle and is given by x2+y2=a2.
4. Condition for tangency : A line y=mx+c is a tangent to x2/a2- y2/b2=1 iff c2=a2m2-b2 and the
equation is xx1/a2- yy1/b2 =1
5. Asymptotes of a Hyperbola: Asymptotes of hyperbola x2/a2- y2/b2=1 are given by x2/a2 –
y2/b2 = 0.
26. 6. Conjugate hyperbola : x2/a2- y2/b2=1 is the conjugate hyperbola of y2/b2-x2/a2 = 1. If e1 and e2
are their eccentricities then 1/e12 + 1/e22 =1.
7.Rectangular Hyperbola: It is denoted by xy=c2.A point on xy=c2 is represented in the
parametric form by( ct, c/t ). At P(ct, c/t) , the slope of the tangent is -1/t2 . Equation of tangent is
x + yt2-2ct =0. Slope of normal is xt3-yt+c-ct4=0
Coordinate Geometry :
Let α, β and γ be the angles made by the plane with the X, Y and z axes respectively. Then
cosα , cosβ and cosγ and are denoted by l, m and n respectively and
are called direction cosines of the plane or line.
If P(x, y, z) is the point and if Op=r, then x/r = cosα, y/r= cosβ and γ
z/r= cosγ . Also cos2α + cos2β + cos2γ =1 β
Standard Form of the equation of a plane:
α
1) If p is the length of the normal from the origin on the plane then the
equation of the plane is lx+my+nz=Φ .
2) The equation of the plane parallel to ax+by+cy+d=0 and passing through (x1, y1, z1 ) is given
by a(x-x1) + b(y-y1) + c(z-z1) +d =0
3) The equation of a plane parallel to the z-axis is ax + by + d= 0 etc.
4) a,b,c are direction ratios of the normal to plane ax+by+cz+d=0
5) The perpendicular distance between point P(x1,y1,z1) on the plane ax+by+cz+d =0 is given
by (ax1+by1+cz1+d)/ .
Differential Calculus:
A polynomial of x is given as a0xn+a1xn-1 +......... +an-1x + an . Here a0,a 1,a2...... are constants .
Laws of limits :
1) = 2) =
3) = 4) =
5) = nan-1 6) = 7)
Differentiation:
1) f’(x) =
27. Some derivatives of common functions :
Function Derivative Function Derivative
C 0 Cu C* du/dx
u+v du/dx + dv/dx uv u dv/dx + vdu/dx
u/v ( v du/dx - udv/dx) /v2 xn nxn-1
ex ex log x 1/x
sin x cos x cos x - sin x
tan x sec2 x cosec x -cosec x cot x
sec x sec x tan x cot x - cosec2 x
sin-1 x 1/ cos-1 x - 1/
-1 2
tan x 1 / 1+ x
Geometric Meaning of Derivative : If the tangent at x=a to the curve y=f(x) makes an angle of
θ of with the x-axis then tan θ = the value of dy/dx at at x=a i.e. f’(a).
Maxima and Minima : f(x) attains a maximum at x=a if f’(a) = 0 and f’’(a) is negative . Also f(x)
attains a minimum at x=a if f’(a) = 0 and f’’(a) is positive .
Rolle’s Theorem : If a function f(x) is differentiable in the interval (a,b) then there exists at least
one value of x1 of x in the interval (a,b) such that f’(x1) = 0.
Le’ Hospitals Rule : 0/0 form:
= and so on
Integral Calculus :
If
= F(b) – F(a) this is known as Definite integral .
Methods of Intigeration :
a) By Substitution
28. Some Standard Substitution
Integral Substitutions
(i) t=ax+b
(ii) t= xn
(iii) t=f(x)
(iv) f(x)
(v)
(b) Integration By Parts :
(c) Integration of
(i) If n be odd and m be even put t=cos x
(ii)If n be even and m be odd, put t=sinx
(iii) If m and n are both odd then put t=cos x or sin x
(d) Properties of Definite Integrals :
(a) =-
(c) function
and =0 if
(2a-x)
ALL THE BEST