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
The document describes the midpoint ellipse algorithm. It begins with properties of ellipses and the standard ellipse equation. It then explains the midpoint ellipse algorithm, which samples points along an ellipse using different directions in different regions to approximate the elliptical path. Key steps include initializing decision parameters, testing the parameters to determine the next point, and calculating new parameters at each step. Pixel positions are determined and can be plotted.
This document discusses differential equations. It defines differential equations as equations relating an unknown function and one or more of its derivatives. It describes the order and degree of differential equations. Several examples of first order differential equations are given and solved using techniques like direct integration, variable separation, and solving homogeneous equations. The key steps for solving first order differential equations are outlined.
This document discusses differential equations. It begins by explaining that differential equations are used to model many physical phenomena in areas like economics, engineering, and more. It then provides examples of ordinary and partial differential equations. The rest of the document defines key terms related to differential equations like order, degree, families of curves, and how to derive the differential equation of a family of curves by eliminating parameters. Several examples are provided to illustrate these concepts.
The inverse of a function "undoes" the effect of the function. We look at the implications of that property in the derivative, as well as logarithmic functions, which are inverses of exponential functions.
This document contains tables summarizing formulas for derivatives, trigonometric functions, logarithms. It lists the derivative of common functions like x, x^2, sinx, cosx. It also provides trigonometric formulas for sine, cosine, tangent of sum and difference of angles. Formulas are given for logarithms, including the change of base formula and properties of logarithms.
The document provides an overview of kinks and defects in the sine-Gordon equation. It discusses kink solutions that interpolate between potential vacua and can be boosted. While there are no static multi-soliton solutions, breathers represent a bound kink-antikink state. Adding a defect term breaks translational invariance, preventing analytical solutions. Numerical solutions show a kink scattering off a defect within a certain region. An ansatz is presented for modeling kink scattering.
The inverse of a function "undoes" the effect of the function. We look at the implications of that property in the derivative, as well as logarithmic functions, which are inverses of exponential functions.
The document describes the midpoint ellipse algorithm. It begins with properties of ellipses and the standard ellipse equation. It then explains the midpoint ellipse algorithm, which samples points along an ellipse using different directions in different regions to approximate the elliptical path. Key steps include initializing decision parameters, testing the parameters to determine the next point, and calculating new parameters at each step. Pixel positions are determined and can be plotted.
This document discusses differential equations. It defines differential equations as equations relating an unknown function and one or more of its derivatives. It describes the order and degree of differential equations. Several examples of first order differential equations are given and solved using techniques like direct integration, variable separation, and solving homogeneous equations. The key steps for solving first order differential equations are outlined.
This document discusses differential equations. It begins by explaining that differential equations are used to model many physical phenomena in areas like economics, engineering, and more. It then provides examples of ordinary and partial differential equations. The rest of the document defines key terms related to differential equations like order, degree, families of curves, and how to derive the differential equation of a family of curves by eliminating parameters. Several examples are provided to illustrate these concepts.
The inverse of a function "undoes" the effect of the function. We look at the implications of that property in the derivative, as well as logarithmic functions, which are inverses of exponential functions.
This document contains tables summarizing formulas for derivatives, trigonometric functions, logarithms. It lists the derivative of common functions like x, x^2, sinx, cosx. It also provides trigonometric formulas for sine, cosine, tangent of sum and difference of angles. Formulas are given for logarithms, including the change of base formula and properties of logarithms.
The document provides an overview of kinks and defects in the sine-Gordon equation. It discusses kink solutions that interpolate between potential vacua and can be boosted. While there are no static multi-soliton solutions, breathers represent a bound kink-antikink state. Adding a defect term breaks translational invariance, preventing analytical solutions. Numerical solutions show a kink scattering off a defect within a certain region. An ansatz is presented for modeling kink scattering.
The inverse of a function "undoes" the effect of the function. We look at the implications of that property in the derivative, as well as logarithmic functions, which are inverses of exponential functions.
This document discusses support vector machines (SVMs) and their derivation from an optimization perspective. It begins by formulating maximum margin classification as an optimization problem to minimize a function under constraints. It then provides an overview of convex optimization theory, including concepts like Lagrangian duality. Finally, it applies these concepts to derive the dual formulations of hard and soft-margin SVMs. The key steps are: (1) deriving the Lagrangian for each case, (2) taking derivatives to solve for optimal primal variables, and (3) substituting back to obtain the final dual optimization problem.
This document provides lessons on straight lines and triangles. It defines key concepts such as slope, equations of lines, finding the distance from a point to a line, families of lines, medians, altitudes, angle bisectors, and properties of triangles. Examples are provided for finding equations of lines given points or intercepts, perpendicular lines, and distances between lines and points.
1. The document lists various trigonometric formulae including definitions of radians, trigonometric ratios, domains and ranges, allied angle relations, sum and difference formulae, and solutions to trigonometric equations.
2. Key formulae include the definitions of radians as 180°/π and degrees as π/180 radians, trigonometric ratios in terms of sine and cosine, and multiple angle formulae for sine, cosine, and tangent of doubled angles.
3. Trigonometric functions are also defined over their domains, with ranges between -1 and 1 except for cosecant, secant, and cotangent. Basic trigonometric identities and relations between quadrants are also provided.
This document contains the questions from an Engineering Mathematics examination from December 2012. It covers topics like:
- Using Taylor's series method and Runge-Kutta method to solve initial value problems
- Using Milne's method, Adams-Bashforth method, and Picard's method to solve differential equations
- Properties of analytic functions and bilinear transformations
- Evaluating integrals using Cauchy's integral formula and finding Laurent series
- Expressing polynomials in terms of Legendre polynomials
- Concepts related to probability distributions like binomial, exponential and normal distributions
- Hypothesis testing and confidence intervals
The questions test the students' understanding of numerical methods for solving differential equations, complex analysis topics, orthogonal
This document provides an introduction to inverse problems and their applications. It summarizes integral equations like Volterra and Fredholm equations of the first and second kind. It also describes inverse problems for partial differential equations, including inverse convection-diffusion, Poisson, and Laplace problems. Applications mentioned include medical imaging, non-destructive testing, and geophysics. Bibliographic references are provided.
1. Geodesic sampling and meshing techniques can be used to generate adaptive triangulations and meshes on Riemannian manifolds based on a metric tensor.
2. Anisotropic metrics can be defined to generate meshes adapted to features like edges in images or curvature on surfaces. Triangles will be elongated along strong features to better approximate functions.
3. Farthest point sampling can be used to generate well-spaced point distributions over manifolds according to a metric, which can then be triangulated using geodesic Delaunay refinement.
The document discusses properties and applications of the Z-transform, which is used to analyze linear discrete-time signals. Some key points:
1) The Z-transform plays an important role in analyzing discrete-time signals and is defined as the sum of the signal samples multiplied by a complex variable z raised to the power of the sample's time index.
2) Important properties of the Z-transform include linearity, time-shifting, frequency-shifting, differentiation in the Z-domain, and the convolution theorem.
3) The Z-transform can be used to find the transform of basic sequences like the unit impulse, unit step, exponentials, polynomials, and derivatives of signals.
1. The document provides examples and explanations of concepts in solid geometry including the three dimensional coordinate system, distance formula in three space, and equations for planes, spheres, cylinders, quadric surfaces, and their graphs.
2. Key solid geometry concepts covered include plotting points in three dimensions, finding distances between points and distances from a point to a plane, midpoint formulas, and standard and general equations for planes, spheres, cylinders, ellipsoids, hyperboloids, and paraboloids.
3. Examples are given for graphing equations of a plane, sphere, circular cylinder, parabolic cylinder, and their relation to the standard equations.
This document provides information on differentiation including:
- The definition of the derivative as a limit.
- Rules of differentiation including constant multiples, sums, products, quotients, and chains.
- Derivatives of trigonometric, exponential, and logarithmic functions.
- Examples of calculating derivatives using the definition and rules of differentiation.
This document contains questions from a fourth semester engineering examination on design and analysis of algorithms. It asks students to:
1) Define asymptotic notations and analyze the time complexity of a sample algorithm.
2) Solve recurrence relations for different algorithms.
3) Explain how bubble sort and quicksort work, including tracing quicksort on a sample data set and deriving its worst case complexity.
4) Write the recursive algorithm for merge sort.
The document contains questions assessing students' understanding of algorithm analysis, asymptotic notations, solving recurrence relations, and sorting algorithms like bubble sort, quicksort, and merge sort.
The document discusses digital filter structures. It covers IIR and FIR filter structures. For IIR filters, it describes direct form I and II structures as well as cascade form using biquad sections. Cascade form implements the IIR filter as a product of second-order filter sections in a direct form structure. FIR filters can be implemented using direct form or cascade of direct form filter sections. The choice of structure depends on factors like complexity, memory requirements, and quantization effects.
This document summarizes research investigating the use of local meta-models within the CMA-ES optimization algorithm for large population sizes. It introduces CMA-ES, describes how local meta-models can be used to build surrogate models of the objective function to reduce evaluations, and presents a new variant called nlmm-CMA that uses a more flexible acceptance criterion for the meta-model. Experimental results show nlmm-CMA achieves speedups over lmm-CMA, the prior local meta-model approach for CMA-ES, on benchmark optimization problems.
1) Laplace's equation describes situations where the electric potential (V) or other scalar field satisfies ∇^2V = 0. It can be solved in one, two, or three dimensions using separation of variables.
2) In three dimensions, the general solution is a sum of multipole terms involving associated Legendre polynomials. The leading terms are the monopole and dipole contributions.
3) For a dipole potential, the electric field is proportional to p/r^3 where p is the dipole moment. The field points radially away from a head-to-tail dipole and has no φ dependence.
The document provides information on trigonometric ratios including:
1) It defines the six main trigonometric ratios (sine, cosine, tangent, cotangent, secant, cosecant) using coordinates on a unit circle.
2) It explains the signs of the ratios in the four quadrants of the coordinate plane.
3) It discusses the range of values that the six ratios can take.
4) It provides a table of the exact ratio values for 0, π/6, π/4, π/3, π/2, π, 3π/2, and 2π.
5) It covers fundamental trigonometric
Stuff You Must Know Cold for the AP Calculus BC Exam!A Jorge Garcia
This document provides a summary of key concepts from AP Calculus that students must know, including:
- Differentiation rules like product rule, quotient rule, and chain rule
- Integration techniques like Riemann sums, trapezoidal rule, and Simpson's rule
- Theorems related to derivatives and integrals like the Mean Value Theorem, Fundamental Theorem of Calculus, and Rolle's Theorem
- Common trigonometric derivatives and integrals
- Series approximations like Taylor series and Maclaurin series
- Calculus topics for polar coordinates, parametric equations, and vectors
This calculus cheat sheet provides definitions and formulas for:
1) Integrals including definite integrals, anti-derivatives, and the Fundamental Theorem of Calculus.
2) Common integration techniques like u-substitution and integration by parts.
3) Standard integrals of common functions like polynomials, trigonometric functions, logarithms, and exponentials.
1. This document contains notes and formulas for additional mathematics form 4. It covers topics such as quadratic equations, functions, indices and logarithms, coordinate geometry, and statistics.
2. Quadratic equations are discussed, including finding the roots of a quadratic equation and writing the equation from its roots. Quadratic functions are also covered, specifically the relationship between the sign of b^2 - 4ac and the nature of the roots.
3. Other topics include indices and logarithm laws, coordinate geometry concepts like distance and midpoints, statistics topics such as measures of central tendency (mean, median, mode), and measures of dispersion like standard deviation and interquartile range.
This document provides information on various mathematical topics including:
1. Graphs of polynomial functions in factorized form such as quadratics, cubics, and quartics.
2. Transformations of functions including translations, reflections, dilations, and their effects on graphs.
3. Exponential, logarithmic, and trigonometric functions and their graphs.
4. Relations, functions, and tests to determine if a relation is a function and if a function is one-to-one or many-to-one.
A function is a relation between a set of inputs (domain) and set of outputs (codomain) where each input is mapped to exactly one output. There are different types of functions such as one-to-one, onto, bijective, many-to-one, and inverse functions. Functions can be represented graphically or using function notation such as f(x). Common functions include polynomial, trigonometric, exponential, logarithmic, and composite functions which are the composition of two simpler functions.
This document discusses inverse functions and their derivatives. It defines inverse functions as switching the x- and y-values of a function to "undo" the original function. A function has an inverse only if it passes the horizontal line test. The derivative of an inverse function at a point equals the reciprocal of the derivative of the original function at the corresponding point.
This document discusses different types of relations and functions. It defines equivalence relations, identity relations, empty relations, universal relations, one-to-one functions, onto functions, bijective functions, composition of functions, and invertible functions. It provides examples to illustrate these concepts.
The same function can have different functional expressions in different ranges. Finding maximum/minimum values in these cases becomes very interesting
This document discusses support vector machines (SVMs) and their derivation from an optimization perspective. It begins by formulating maximum margin classification as an optimization problem to minimize a function under constraints. It then provides an overview of convex optimization theory, including concepts like Lagrangian duality. Finally, it applies these concepts to derive the dual formulations of hard and soft-margin SVMs. The key steps are: (1) deriving the Lagrangian for each case, (2) taking derivatives to solve for optimal primal variables, and (3) substituting back to obtain the final dual optimization problem.
This document provides lessons on straight lines and triangles. It defines key concepts such as slope, equations of lines, finding the distance from a point to a line, families of lines, medians, altitudes, angle bisectors, and properties of triangles. Examples are provided for finding equations of lines given points or intercepts, perpendicular lines, and distances between lines and points.
1. The document lists various trigonometric formulae including definitions of radians, trigonometric ratios, domains and ranges, allied angle relations, sum and difference formulae, and solutions to trigonometric equations.
2. Key formulae include the definitions of radians as 180°/π and degrees as π/180 radians, trigonometric ratios in terms of sine and cosine, and multiple angle formulae for sine, cosine, and tangent of doubled angles.
3. Trigonometric functions are also defined over their domains, with ranges between -1 and 1 except for cosecant, secant, and cotangent. Basic trigonometric identities and relations between quadrants are also provided.
This document contains the questions from an Engineering Mathematics examination from December 2012. It covers topics like:
- Using Taylor's series method and Runge-Kutta method to solve initial value problems
- Using Milne's method, Adams-Bashforth method, and Picard's method to solve differential equations
- Properties of analytic functions and bilinear transformations
- Evaluating integrals using Cauchy's integral formula and finding Laurent series
- Expressing polynomials in terms of Legendre polynomials
- Concepts related to probability distributions like binomial, exponential and normal distributions
- Hypothesis testing and confidence intervals
The questions test the students' understanding of numerical methods for solving differential equations, complex analysis topics, orthogonal
This document provides an introduction to inverse problems and their applications. It summarizes integral equations like Volterra and Fredholm equations of the first and second kind. It also describes inverse problems for partial differential equations, including inverse convection-diffusion, Poisson, and Laplace problems. Applications mentioned include medical imaging, non-destructive testing, and geophysics. Bibliographic references are provided.
1. Geodesic sampling and meshing techniques can be used to generate adaptive triangulations and meshes on Riemannian manifolds based on a metric tensor.
2. Anisotropic metrics can be defined to generate meshes adapted to features like edges in images or curvature on surfaces. Triangles will be elongated along strong features to better approximate functions.
3. Farthest point sampling can be used to generate well-spaced point distributions over manifolds according to a metric, which can then be triangulated using geodesic Delaunay refinement.
The document discusses properties and applications of the Z-transform, which is used to analyze linear discrete-time signals. Some key points:
1) The Z-transform plays an important role in analyzing discrete-time signals and is defined as the sum of the signal samples multiplied by a complex variable z raised to the power of the sample's time index.
2) Important properties of the Z-transform include linearity, time-shifting, frequency-shifting, differentiation in the Z-domain, and the convolution theorem.
3) The Z-transform can be used to find the transform of basic sequences like the unit impulse, unit step, exponentials, polynomials, and derivatives of signals.
1. The document provides examples and explanations of concepts in solid geometry including the three dimensional coordinate system, distance formula in three space, and equations for planes, spheres, cylinders, quadric surfaces, and their graphs.
2. Key solid geometry concepts covered include plotting points in three dimensions, finding distances between points and distances from a point to a plane, midpoint formulas, and standard and general equations for planes, spheres, cylinders, ellipsoids, hyperboloids, and paraboloids.
3. Examples are given for graphing equations of a plane, sphere, circular cylinder, parabolic cylinder, and their relation to the standard equations.
This document provides information on differentiation including:
- The definition of the derivative as a limit.
- Rules of differentiation including constant multiples, sums, products, quotients, and chains.
- Derivatives of trigonometric, exponential, and logarithmic functions.
- Examples of calculating derivatives using the definition and rules of differentiation.
This document contains questions from a fourth semester engineering examination on design and analysis of algorithms. It asks students to:
1) Define asymptotic notations and analyze the time complexity of a sample algorithm.
2) Solve recurrence relations for different algorithms.
3) Explain how bubble sort and quicksort work, including tracing quicksort on a sample data set and deriving its worst case complexity.
4) Write the recursive algorithm for merge sort.
The document contains questions assessing students' understanding of algorithm analysis, asymptotic notations, solving recurrence relations, and sorting algorithms like bubble sort, quicksort, and merge sort.
The document discusses digital filter structures. It covers IIR and FIR filter structures. For IIR filters, it describes direct form I and II structures as well as cascade form using biquad sections. Cascade form implements the IIR filter as a product of second-order filter sections in a direct form structure. FIR filters can be implemented using direct form or cascade of direct form filter sections. The choice of structure depends on factors like complexity, memory requirements, and quantization effects.
This document summarizes research investigating the use of local meta-models within the CMA-ES optimization algorithm for large population sizes. It introduces CMA-ES, describes how local meta-models can be used to build surrogate models of the objective function to reduce evaluations, and presents a new variant called nlmm-CMA that uses a more flexible acceptance criterion for the meta-model. Experimental results show nlmm-CMA achieves speedups over lmm-CMA, the prior local meta-model approach for CMA-ES, on benchmark optimization problems.
1) Laplace's equation describes situations where the electric potential (V) or other scalar field satisfies ∇^2V = 0. It can be solved in one, two, or three dimensions using separation of variables.
2) In three dimensions, the general solution is a sum of multipole terms involving associated Legendre polynomials. The leading terms are the monopole and dipole contributions.
3) For a dipole potential, the electric field is proportional to p/r^3 where p is the dipole moment. The field points radially away from a head-to-tail dipole and has no φ dependence.
The document provides information on trigonometric ratios including:
1) It defines the six main trigonometric ratios (sine, cosine, tangent, cotangent, secant, cosecant) using coordinates on a unit circle.
2) It explains the signs of the ratios in the four quadrants of the coordinate plane.
3) It discusses the range of values that the six ratios can take.
4) It provides a table of the exact ratio values for 0, π/6, π/4, π/3, π/2, π, 3π/2, and 2π.
5) It covers fundamental trigonometric
Stuff You Must Know Cold for the AP Calculus BC Exam!A Jorge Garcia
This document provides a summary of key concepts from AP Calculus that students must know, including:
- Differentiation rules like product rule, quotient rule, and chain rule
- Integration techniques like Riemann sums, trapezoidal rule, and Simpson's rule
- Theorems related to derivatives and integrals like the Mean Value Theorem, Fundamental Theorem of Calculus, and Rolle's Theorem
- Common trigonometric derivatives and integrals
- Series approximations like Taylor series and Maclaurin series
- Calculus topics for polar coordinates, parametric equations, and vectors
This calculus cheat sheet provides definitions and formulas for:
1) Integrals including definite integrals, anti-derivatives, and the Fundamental Theorem of Calculus.
2) Common integration techniques like u-substitution and integration by parts.
3) Standard integrals of common functions like polynomials, trigonometric functions, logarithms, and exponentials.
1. This document contains notes and formulas for additional mathematics form 4. It covers topics such as quadratic equations, functions, indices and logarithms, coordinate geometry, and statistics.
2. Quadratic equations are discussed, including finding the roots of a quadratic equation and writing the equation from its roots. Quadratic functions are also covered, specifically the relationship between the sign of b^2 - 4ac and the nature of the roots.
3. Other topics include indices and logarithm laws, coordinate geometry concepts like distance and midpoints, statistics topics such as measures of central tendency (mean, median, mode), and measures of dispersion like standard deviation and interquartile range.
This document provides information on various mathematical topics including:
1. Graphs of polynomial functions in factorized form such as quadratics, cubics, and quartics.
2. Transformations of functions including translations, reflections, dilations, and their effects on graphs.
3. Exponential, logarithmic, and trigonometric functions and their graphs.
4. Relations, functions, and tests to determine if a relation is a function and if a function is one-to-one or many-to-one.
A function is a relation between a set of inputs (domain) and set of outputs (codomain) where each input is mapped to exactly one output. There are different types of functions such as one-to-one, onto, bijective, many-to-one, and inverse functions. Functions can be represented graphically or using function notation such as f(x). Common functions include polynomial, trigonometric, exponential, logarithmic, and composite functions which are the composition of two simpler functions.
This document discusses inverse functions and their derivatives. It defines inverse functions as switching the x- and y-values of a function to "undo" the original function. A function has an inverse only if it passes the horizontal line test. The derivative of an inverse function at a point equals the reciprocal of the derivative of the original function at the corresponding point.
This document discusses different types of relations and functions. It defines equivalence relations, identity relations, empty relations, universal relations, one-to-one functions, onto functions, bijective functions, composition of functions, and invertible functions. It provides examples to illustrate these concepts.
The same function can have different functional expressions in different ranges. Finding maximum/minimum values in these cases becomes very interesting
Surjective, Injective, Bijective, etc. If you have heard these terms but do not exactly know what these mean, this is the question for you. If you have not even heard these terms, then start now, hit wikipedia
Math functions, relations, domain & rangeRenee Scott
This document discusses math functions, relations, and their domains and ranges. It provides examples of relations and explains that the domain is the set of first numbers in ordered pairs, while the range is the set of second numbers. A function is defined as a relation where each x-value has only one corresponding y-value. It compares examples of relations that are and are not functions and describes how to evaluate functions by inserting values. Tests for identifying functions like the vertical line test are also outlined.
1) The document discusses inverse trigonometric functions such as sin-1, cos-1, tan-1, cot-1, sec-1, and cosec-1.
2) It explains that the inverse functions are defined by restricting the domains of the original trigonometric functions (sin, cos, etc.) so that they become one-to-one mappings.
3) Graphs of the inverse functions are obtained by reflecting the graphs of the original functions about the line y=x.
This document discusses complex numbers and functions. It introduces complex numbers using Cartesian (x + iy) and polar (r(cosθ + i sinθ)) forms. It describes the Cauchy-Riemann conditions that must be satisfied for a function of a complex variable to be differentiable. A function is analytic if it satisfies the Cauchy-Riemann conditions and its partial derivatives are continuous. Analytic functions have properties like equality of second-order partial derivatives and establishing a relation between the real and imaginary parts.
1) Complex numbers can be represented in Cartesian (x + iy) or polar (r(cosθ + i sinθ)) form, with conversions between the two.
2) The derivative of a complex function f(z) is defined if the Cauchy-Riemann equations are satisfied.
3) A function is analytic if it is differentiable and its partial derivatives are continuous, implying the Cauchy-Riemann equations always hold. Analytic functions have properties like equality of second partial derivatives.
This document discusses inverse trigonometric functions including arcsine, arccosine, and arctangent. It explains that arcsine is the inverse of sine, with domain [-1,1] and range [-π/2, π/2]. Arccosine has domain [-1,1] and range [0,π]. Arctangent has domain (-∞, ∞) and range [-π/2, π/2]. The document also notes that applying the inverse function twice returns the original value, and the outer function's domain takes precedence when functions are composed. It recommends graphing the inverse trig functions to better understand their properties.
1) The document defines and discusses the domains and ranges of inverse trigonometric functions such as sin-1x, cos-1x, and tan-1x.
2) The inverse functions are defined based on reflecting portions of the original trigonometric functions over the line y=x.
3) The domains and ranges of the inverse functions are restricted to ensure each inverse function is a single-valued function.
This document contains the answers to exercises for the third edition of the textbook "Microeconomic Analysis" by Hal R. Varian. The answers are organized by chapter and include solutions to mathematical problems as well as explanations and justifications. Key information provided in the answers includes derivations of production functions, profit functions, cost functions, and factor demand functions for various technologies. Convexity and monotonicity properties of technologies are also analyzed.
Change of variables in double integralsTarun Gehlot
1. The document discusses change of variables for double integrals, introducing the Jacobian determinant which relates the differentials of the original and transformed variables.
2. It provides an example of using a change of variables (u=x-y, v=x+y) to evaluate an integral over a parallelogram region.
3. Polar coordinates are also discussed as a common change of variables technique for double integrals, with an example evaluating an integral over a circular region in polar coordinates.
Change of variables in double integralsTarun Gehlot
1. The document discusses change of variables for double integrals, introducing the Jacobian determinant which relates the differentials of the original and transformed variables.
2. It provides an example of using a change of variables (u=x-y, v=x+y) to evaluate an integral over a parallelogram region.
3. Polar coordinates are also discussed as a common change of variables technique for double integrals, with an example evaluating an integral over a circular region in polar coordinates.
This document provides definitions and notations for 2-D systems and matrices. It defines how continuous and sampled 2-D signals like images are represented. It introduces some common 2-D functions used in signal processing like the Dirac delta, rectangle, and sinc functions. It describes how 2-D linear systems can be represented by matrices and discusses properties of the 2-D Fourier transform including the frequency response and eigenfunctions. It also introduces concepts of Toeplitz and circulant matrices and provides an example of convolving periodic sequences using circulant matrices. Finally, it defines orthogonal and unitary matrices.
The document discusses inverse trigonometric functions. It defines the inverse functions of sine, cosine, tangent, cotangent, secant and cosecant by restricting their domains such that they become one-to-one functions. The principal value branches are defined for each inverse function, with their domains and ranges specified. Graphs are provided to illustrate the inverse relationships between the trigonometric functions and their inverses. Properties like how to obtain the graph of an inverse function from the original are also outlined.
1) For inverse trigonometric functions to exist, the original trigonometric function must be one-to-one. This means that each input has a single, unique output and vice versa.
2) The sine and cosine functions are not one-to-one over their entire domains, so their domains must be restricted for the inverses to be defined. Restricting the domain of sine to [-π/2, π/2] and cosine to [0, π] makes them one-to-one.
3) The inverse functions are denoted as arcsine, arccosine, and arctangent. Their values are restricted to certain quadrants based on the corresponding trigon
This document provides an overview of convex optimization. It begins by explaining that convex optimization can efficiently find global optima for certain functions called convex functions. It then defines convex sets as sets where linear combinations of points in the set are also in the set. Common examples of convex sets include norm balls and positive semidefinite matrices. Convex functions are defined as functions where linear combinations of points on the graph lie below the line connecting those points. Convex functions have properties like their first and second derivatives satisfying certain inequalities, allowing efficient optimization.
This document discusses functions of a complex variable. It introduces complex numbers and their representations. It covers topics like complex differentiation using Cauchy-Riemann equations, analytic functions, Cauchy's integral theorem, and contour integrals. Functions of a complex variable provide tools for physics concepts involving complex quantities like wavefunctions. Cauchy's integral theorem states that the contour integral of an analytic function over a closed path is zero.
An introduction to quantum stochastic calculusSpringer
The document discusses tensor products of Hilbert spaces. It defines positive definite kernels on sets and shows how they can be used to define tensor products. Given Hilbert spaces H1, ..., Hn, it constructs a kernel on the cartesian product of the spaces and shows that its Gelfand pair (H,φ) gives a tensor product of the Hilbert spaces. The map φ from the product space into H is multilinear and H is the completion of the algebraic tensor product of the vector spaces H1, ..., Hn.
1) A radical function is of the form y = f(x) = ax + b, where changing a and b affects the graph.
2) Graphing shows that if a > 0 the graph increases, if a < 0 the graph decreases, larger a makes the graph steeper, and closer to 0 makes the graph flatter.
3) The value of b is the y-intercept, and the domain is all x ≥ 0 while the range is all y above or below b depending on if the graph increases or decreases.
The document provides an overview of functions of a complex variable. Some key points:
1) Functions of a complex variable provide powerful tools in theoretical physics for quantities that are complex variables, evaluating integrals, obtaining asymptotic solutions, and performing integral transforms.
2) The Cauchy-Riemann equations are a necessary condition for a function f(z) = u(x,y) + iv(x,y) to be differentiable at a point. If the equations are satisfied, the function is analytic.
3) Cauchy's integral theorem states that if a function f(z) is analytic in a simply connected region R, the contour integral of f(z) around any closed path in
This document discusses partial derivatives and limits of functions of several variables. It begins by defining functions of two and three variables, and provides examples of finding the domain and range of such functions. It then discusses limits of functions of two variables, noting that the limit can depend on the path taken to approach the point. Examples are provided of calculating limits of functions of two variables along different paths, and showing that the limit does not exist when limits along two paths are not equal.
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)
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.
How to Fix the Import Error in the Odoo 17Celine George
An import error occurs when a program fails to import a module or library, disrupting its execution. In languages like Python, this issue arises when the specified module cannot be found or accessed, hindering the program's functionality. Resolving import errors is crucial for maintaining smooth software operation and uninterrupted development processes.
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.
A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
আমাদের সবার জন্য খুব খুব গুরুত্বপূর্ণ একটি বই ..বিসিএস, ব্যাংক, ইউনিভার্সিটি ভর্তি ও যে কোন প্রতিযোগিতা মূলক পরীক্ষার জন্য এর খুব ইম্পরট্যান্ট একটি বিষয় ...তাছাড়া বাংলাদেশের সাম্প্রতিক যে কোন ডাটা বা তথ্য এই বইতে পাবেন ...
তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
বিসিএস ও ব্যাংক এর লিখিত পরীক্ষা ...+এছাড়া মাধ্যমিক ও উচ্চমাধ্যমিকের স্টুডেন্টদের জন্য অনেক কাজে আসবে ...
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
and water managers, and urban planners, are interested in obtaining data on land use and cover
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.
Executive Directors Chat Leveraging AI for Diversity, Equity, and InclusionTechSoup
Let’s explore the intersection of technology and equity in the final session of our DEI series. Discover how AI tools, like ChatGPT, can be used to support and enhance your nonprofit's DEI initiatives. Participants will gain insights into practical AI applications and get tips for leveraging technology to advance their DEI goals.
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
1. INVERSE CORCULAR FUNCTIONS
1.00 Inverse Function Definition
111
If1 function is one to one and onto from A to B, then function g which associates each element y
a B to
one and only one element x A, such that y = f(x), then g is called the inverse function of f, denoted by x
= g(y). Usually we denote g f –1 {Read as f inverse}
x f –1 ( y).
1.01 Inverse Trigonometric Function
111
We have seen that the trigonometric functions, sin, cos etc. are all periodic and thus, each of them
1
1
achieves the same numerical value at an infinite number of points. Thus, the equation sin x has an
2
infinite number of solutions, viz., x , – etc. If one is to answer the question : “ What is the angle
6 6
1
whose sine is ?”, there is no unique answer. The difficulty arises as the function f : R R defined by
2
f ( x) sin x is not one to one and thus, does not admit of an inverse. To achieve a unique answer to the
aforesaid question we restrict the domain of sin x so that the resulting function is invertible. Thus, the
function g : – , [–1, 1] defined by g ( x) sin x is one to one and onto and admits of an inverse
2 2
(denoted by h sin –1 and read as sin inverse or arc sin) defined as h :[–1, 1] – , where
2 2
h( y) x if y sin x . The function sin –1 is the inverse of the sin function when the sin function is viewed in
a restricted sense.
We similarly define the other inverse trigonometric functions
Important Points
1: sin –1 x is an angle and denotes the smallest numerical angle, whose sine is x.
2: If there are two angles one positive and the other negative having same numerical value. Then we
shall take the positive value.
Progression / APEX INSTITUTE FOR IIT-JEE / AIEEE / PMT, 0120-4901457, +919990495952, +919910817866 Page 1
2. 1.02 Inverse Trigonometric Function
111
Here, sin –1 x,cos ec –1 x, tan –1 x belongs to I and IV quadrant.
1
Here, cos –1 x, sec–1 x, cot –1 x belongs to I and II quadrant.
1. I quadrant is common to all the inverse functions.
2. III quadrant is not used inverse function.
3. IV quadrant is used in the clockwise direction i.e., – / 2 y 0 .
1.03 Domain, Range And Graphs of Inverse Functions
1. If sin y = x, then y sin –1 x under certain condition.
–1 sin y 1; but sin y x
–1 x 1
Again, sin y = –1 y = – /2 and sin y = 1 y = /2
Keeping in mind numerically smallest angles or real numbers.
– /2 y /2
These restrictions on the values of x and y provide us with the domain and range for the function
y sin –1 x .
i.e., Domain : x [–1, 1]
Range : y [– / 2, / 2]
2. Let cos y = x then y cos –1 x under certain condition –1 cos y 1 .
Progression / APEX INSTITUTE FOR IIT-JEE / AIEEE / PMT, 0120-4901457, +919990495952, +919910817866 Page 2
3. –1 x 1
cos y –1 y
cos y 1 y 0
0 y {as cos x is a decreasing function in [0, ]; hence cos cos y cos 0}
These restrictions on the values of x and y provide us the domain and range for the function
y cos –1 x .
i.e., Domain : x [–1, 1]
Range : y [0, ]
3. If tan y = x then y tan –1 x , under certain conditions.
Here, tan y R x R
– tan y – /2 y /2
Thus, domain x R
Range y (– / 2, / 2)
4. If cot y = x, then y cot –1 x (under certain conditions)
cot y R x R;
– cot y 0 y
These conditions on x and y make the function, cot y = x one–one
and onto so that the inverse function exists.
i.e., y cot –1 x is meaningful.
i.e., Domain: x R
Range : y (0, )
5. If sec y = x, then sec–1 x, where x 1 and 0 y ,y /2
Here, Domain : x R – (–1, 1)
Range : y [0, ] – { / 2}
6. If cosec y = x then y cos ec –1 x,
where x 1 and – /2 y / 2, y 0
Here, Domain : R – (–1, 1)
Range : [– / 2, / 2] – {0}
Progression / APEX INSTITUTE FOR IIT-JEE / AIEEE / PMT, 0120-4901457, +919990495952, +919910817866 Page 3
4. 1.04 Principal values & Domains of Inverse Trigonometric / Circular Functions
Function Domain Range
(i ) y sin –1 x where –1 x 1 – y
2 2
–1
(ii ) y cos x where –1 x 1 0 y
(iii ) y tan –1 x where x R – y
2 2
(iv) y cos ec –1 x where x –1 or x 1 – y ,y 0
2 2
(v ) y sec –1 x where x –1 or x 1 0 y ;y
2
(vi ) y cot –1 x where x R 0 y
Note :
(a) 1st quadrant is common to the range of all the inverse functions.
(b) 2nd quadrant is not used in inverse functions.
(c) 4th quadrant is used in the clockwise direction i.e. – y 0.
2
(d) No inverse function is periodic. (See the graphs on page 17)
1 1
Illustration 1 : Find the value of tan cos –1 tan –1 – .
2 3
1 1
Solution : Let y tan cos –1 tan –1 –
2 3
tan – tan
3 6 6
1
y Ans.
3
Illustration 2 : Find the domain of sin –1 (2 x2 –1) .
Solution : Let y sin –1 (2 x 2 – 1)
For y to be defined
– 1 (2 x 2 – 1) 1
0 2x2 2 0 x2 1
x [–1, 1]
Progression / APEX INSTITUTE FOR IIT-JEE / AIEEE / PMT, 0120-4901457, +919990495952, +919910817866 Page 4
5. 1.05 Properties of Inverse Trigonometric Functions
111
Property – 2( A)
1 –1 –1
(i ) sin(sin x) x, –1 x 1 (ii ) cos(cos x) x, –1 x 1
(iii) tan(tan –1 x) x, x R (iv) cot(cot –1 x) x, x R
–1 –1
(v ) sec(sec x) x, x –1, x 1 (vi ) cos ec(cos ec x) x, x –1, x 1
These functions are equal to identity function in their whole domain which may or may not be R.
(See the graphs on page 18)
3
Illustration 3 : Fin the values of cos ec cot cot –1 .
4
3
Solution : Let y cos ec cot cot –1 .........(i )
4
3 3
cot(cot –1 x) x, x R cot cot –1
4 4
from equation (i), we get
3
y cos ec
4
y 2 Ans.
Property – 2( B)
(i ) sin(sin x) x; – x (ii ) cos –1 (cos x) x; 0 x
2 2
(iii ) tan –1 (tan x) x; – x (iv) cot –1 (cot x) x; 0 x
2 2
(v ) sec –1 (sec x) x; 0 x , x (vi) cos ec –1 (cos ecx) x; x 0, – x
2 2 2
These are equal to identity function for a short interval of x only.(See the graphs on page)
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6. 3
Illustration 4 : Find the value of tan –1 tan .
4
3
Solution : Let y tan –1 tan
4
Note tan –1 (tan x) x if x – ,
2 2
3
– ,
4 2 2
3 3 3 3
tan –1 tan ,
4 4 4 2 2
graphs of y = tan –1 (tan x) is as :
3
from the graph we can see that if x ,
2 2
then y tan –1 (tan x) can be written as y x–
3 3
y tan –1 tan – y – 123
4 4 4
Illustration 5 : Find the value of sin –1 (sin 7) .
Solution : Let y sin –1 (sin 7)
Note : sin –1 (sin 7) 7as 7 – ,
2 2
5
2 7
2
graph of y sin –1 (sin x) is as :
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7. 5
From the graph we can see that if 2 x then
2
y sin –1 (sin x) can be written as :
y x–2
–1
sin (sin 7) 7–2
Similarly if we have to find sin –1 (sin(–5)) then
3
–2 –5 –
2
from the graph of sin –1 (sin x), we can say that
sin –1 (sin(–5)) 2 (–5)
2 –5
Property – 2(C )
(i) sin –1 (– x) – sin –1 x; –1 x 1 (ii) tan –1 (– x) – tan –1 x, x R
(iii) cos –1 (– x) – cos –1 x, –1 x 1 (iv) cot –1 (– x) – cot –1 x, x R
The function sin –1 x, tan –1 x and cos ec–1 x are odd functions and rest are neither even nor odd.
Illustration 6 : Find the value of cos –1 sin(–5) .
Solution : Let y cos –1 sin(–5)
cos –1 (– sin 5) cos –1 (– x) – cos –1 x, x 1
– cos –1 cos –5 .........(i)
2
–2 –5 –
2
graph of cos –1 (cos x) is as :
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8. from the graph we can see that if –2 x –
–1
then y=cos (cos x) can be wriiten as y x 2
5 5
from the graph cos –1 cos –5 –5 2 –5
2 2 2
from equation (i), we get
5 3
y – –5 y 5– Ans.
2 2
Property – 2( D)
1 1
(i ) cos ec –1 x sin –1 ; x –1, x 1 (ii) sec –1 x cos –1 ;x –1, x 1
x x
1
tan –1 ; x 0
x
(iii ) cot –1 x
1
tan –1 ; x 0
x
–2
Illustration 7 : Find the value of tan cot –1 .
3
–2
Solution : Let y = tan cot –1 .........(i )
3
cot –1 (– x) – cot –1 x, x R
equation (i) can be written as
–2
y tan – cot –1
3
2 1
y – tan cot –1 cot –1 x tan –1 ifx 0
3 x
3 3
y – tan tan –1 y –
2 2
Property – 2( E )
(i ) sin –1 x cos –1 x , –1 x 1 (ii ) tan –1 x cot –1 x , x R
2 2
(iii ) cos ec –1 x sec –1 , x 1
2
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9. 1
Illustration 8 : Find the value of sin(2cos –1 x sin –1 x) when x .
5
Solution :
Let y sin[2 cos –1 x sin –1 x]
sin –1 x cos –1 x ,x 1
2
y sin 2 cos –1 x – cos –1 x
2
sin cos –1 x
2
1
cos(cos –1 x) x
5
1
y cos cos –1 .........(i )
5
cos(cos –1 x) x if x [–1, 1]
1 1 1
[–1, 1] cos cos –1
5 5 5
1
from equation (i), we get y
Property – 2( F ) 5
1
(i) sin(cos –1 x) cos(sin –1 x) 1 – x 2 , –1 x 1 (ii ) tan(cot –1 x) cot(tan –1 x) , x R, x 0
x
x
(iii ) cos ec(sec –1 x) sec(cos ec –1 x) , x 1
x –12
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10. 3
Illustration 9 : Find the value of sin tan –1 .
4
3
Solutions : Let y sin tan –1 .......(i )
4
Note : To find y we use sin(sin –1 x) x, –1 x 1
For this we convert tan –1 x in sin –1 x
3 3
Let tan –1 tan and 0,
4 4 2
3
sin
5
3
sin –1 (sin ) sin –1 ........(ii )
5
0, sin –1 (sin )
2
equation (ii) can be written as :
3 3 3 3
sin –1 tan –1 tan –1 sin –1
5 4 4 5
3
from equation (i), we get y sin sin –1
5
3
y
5
1 5
Illustration 10 : Find the value of tan cos –1 .
2 3
Solution : 1 5
Let y tan cos –1 ...........(i )
2 3
5 5
Let cos –1 0 0, and cos
3 2 3
equation (i) becomes
y tan .........(ii )
2
5
1–
1 – cos 3 3– 5 (3 – 5) 2
tan 2
2 1 cos 5 3 5 4
1
3
3– 5
tan .........(iii)
2 2
0, 0,
2 2 4
tan 0
2
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11. 3– 5
from equation (iii), we get tan
2 2
3– 5
from equation (ii), we get y
2
1
Illustration 11 : Find the value of cos(2cos–1 x sin –1 x) when x .
5
Solution : Lety cos[2 cos –1 x sin –1 x]
sin –1 x cos –1 x ,x 1
2
y cos 2 cos –1 x – cos –1 x cos cos –1 x
2 2
1
– sin(cos –1 x) x
5
1
y – sin cos –1 ........(i )
5
sin(cos –1 x) 1 – x2 , x 1
1 1 24
sin cos –1 1–
5 25 5
24
from equation (i), we get y –
5
1 1
Aliter : Let cos –1 cos and 0,
5 5 2
24
sin
5
24
sin –1 (sin ) sin –1 .........(ii )
5
0, sin –1 (sin )
2
equation (ii) can be written as
24 1
sin –1 cos –1
5 5
1 24
cos –1 sin –1
5 5
Now equation (i) can be written as
24
y – sin sin –1 ........(iii )
5
24 24 24
[–1, 1] sin sin –1
5 5 5
from equation (iii), we get
24
y –
5
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12. 1.06 Identities of Addition and Subtraction
A.
(i ) sin –1 x sin –1 y sin –1 x 1 – y 2 y 1 – x2 , x 0, y 0 & ( x2 y2 ) 1
– sin –1 x 1 – y 2 y 1 – x2 , x 0, y 0 & x2 y2 1
Note that : x 2 y2 1 0 sin –1 x sin –1 y
2
x2 y2 1 sin –1 x sin –1 y
2
(ii ) cos –1 x cos –1 y cos –1 xy – 1 – x 2 1 – y 2 , x 0, y 0
x y
(iii ) tan –1 x tan –1 y tan –1 ,x 0, y 0 & xy 1
1 – xy
x y
tan –1 ,x 0, y 0 & xy 1
1 – xy
,x 0, y 0 & xy 1
2
Note that : xy 1 0 tan –1 x tan –1 y ; xy 1 tan –1 x tan –1 y
2 2
B.
(i ) sin –1 x – sin –1 y sin –1 x 1 – y 2 – y 1 – x 2 , x 0, y 0
(ii ) cos –1 x – cos –1 y cos –1 xy – 1 – x 2 1 – y 2 , x 0, y 0, x y
x– y
(iii ) tan –1 x – tan –1 y tan –1 ,x 0, y 0
1 xy
Note : For x 0 and y 0 those identities can be used with the help of preperties 2(C )
i.e. change x and y to – x and – y which are positive.
3 15 84
Illustration 12 : Show that sin –1 sin –1 – sin –1 .
5 17 85
2 2
3 15 3 15 8226
Solution : 0, 0 and 1
5 17 5 17 7225
3 15 3 225 15 9
sin –1 sin –1 – sin –1 1– 1–
5 17 5 289 17 25
3 8 15 4
– sin –1 . .
5 17 17 5
84
– sin –1
85
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13. 12 4 63
Illustration 13 : Evaluate cos –1 sin –1 – tan –1 .
13 5 16
12 4 63
Solution : Let z cos –1 sin –1 – tan –1
13 5 16
4 4
sin –1 – cos –1
5 2 5
12 4 63
z cos –1 – cos –1 – tan –1
13 2 5 16
4 12 63
z – cos –1 – cos –1 – tan –1 .......(i )
2 5 13 16
4 12 4 12
– 0, 0 and
5 13 5 13
4 12 4 12 16 144 63
cos –1 – cos –1 cos –1 1– 1– cos –1
5 13 5 13 25 169 65
equation (i) can be written as
63 63
z – cos –1 – tan –1
2 65 16
63 63
z sin –1 – tan –1 .........(ii)
65 16
63 63
sin –1 tan –1
65 16
from equation (ii), we get
63 63
z tan –1 – tan –1 z 0 Ans.
65 65
5
Illustration 14 : Evaluate tan –1 9 tan –1 .
4
5 5
Solution : 9 0, 0 and 9 1
4 4
5
9
–15 –1 –1 4
tan 9 tan tan
4 5
1 – 9.
4
tan –1 (–1) –
4
5 3
tan –1 9 tan –1 .
4 4
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14. C.
1
2sin –1 x if x
2
1
(i ) sin –1 2 x 1 – x 2 – 2sin –1 x if x
2
1
–( 2sin –1 x) if x –
2
2cos –1 x if 0 x 1
(ii) cos –1 (2x 2 –1) =
2 – 2cos –1 x if –1 x 0
2 tan –1 x if x 1
2x
(iii) tan –1 = 2 tan –1 x if x 1
1 – x2
–( 2 tan –1 x) if x –1
1 – x2 2 tan –1 x if x 0
(iv) cos –1 `
1 x2 –2 tan –1 x if x 0
Illustration 15 : Define y cos –1 (4 x3 – 3x) in terms of cos –1 x and also draw its graph.
Solution : Let y cos –1 (4 x 3 – 3 x)
Note Domain : [–1,1] and range : [0, ]
Let cos –1 x [0, ] and x cos
y cos –1 (4cos 3 – 3cos )
y cos –1 (cos 3 ) .........(i)
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15. [0, ] 3 [0,3 ]
to define y cos –1 (cos 3 ), we consider the graph of cos –1 (cos x) in the interval [0,3 ].
Now from the above graphs we can see that
(i ) if 0 3 cos –1 (cos 3 ) 3
from equation (i), we get
y 3 if 3
y 3 if 0
3
1
y 3cos –1 x if x 1
2
(ii ) if 3 2 cos –1 (cos 3 ) 2 –3
from equation (i), we get
y 2 –3 if 3 2
2
y 2 –3 if
3 3
1 1
y 2 – 3cos –1 x if – x
2 2
(iii ) 2 3 3 cos –1 (cos 3 ) –2 3
from equation (i), we get
y –2 3 if 2 3 3
2
y –2 3 if
3
1
y –2 3cos –1 x if –1 x –
2
from (i), (ii) & (iii), we get
1
3cos –1 x ; x 1
2
1 1
y cos –1 (4 x 3 – 3 x) 2 – 3cos –1 x ; – x
2 2
1
–2 3cos –1 x ; –1 x –
2
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16. Graph :
For y cos –1 (4 x3 – 3 x)
domain :[–1,1]
range :[0, ]
1
(i ) if x 1, y 3cos –1 x
2
dy –3
–3(1 – x 2 ) –1/ 2 ..........(i )
dx 1 – x2
dy 1
0 if x ,1
dx 2
1
decreasing if x ,1
2
again if we differentiate equation (i) w.r.t. ' x ', we get
d2y 3x
–
dx 2 (1 – x 2 )3/ 2
d2y 1 1
0 if x ,1 concavity downwards if x ,1
dx 2 2 2
1 1
(ii ) if – x , y 2 – 3cos –1 x.
2 2
dy 3 dy 1 1
0 if x – ,
dx 1– x 2 dx 2 2
1 1 d2y 3x
increasing if x – , and
2 2 dx 2 (1 – x 2 )3/ 2
1 d2y
(a) if x – , 0 then 0
2 dx 2
1
concavity downwards if x – ,0
2
1 d2y
(b) if x 0, then 0
2 dx 2
1
concavity downwards if x 0,
2
1 dy d2y
(iii ) Similarly if –1 x – then 0 and 0.
2 dx dx 2
the graph of y cos –1 (4 x 3 – 3 x) is as
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17. D.
x y z – xyz
If tan –1 x tan –1 y tan –1 z tan –1 if , x 0, y 0, z 0 & ( xy yz zx) 1
1 – xy – yz – zx
NOTE :
(i ) If tan –1 x tan –1 y tan –1 z then x y z xyz
(ii) If tan –1 x tan –1 y tan –1 z then xy yz zx 1
2
(iii) If tan –11 tan –1 2 tan –1 3
1 1
(iv) tan –1 1 tan –1 tan –1
2 3 2
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18. Inverse Trigonometric Functions
Some Useful Graphs
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19. Progression / APEX INSTITUTE FOR IIT-JEE / AIEEE / PMT, 0120-4901457, +919990495952, +919910817866 Page 19
20. Progression / APEX INSTITUTE FOR IIT-JEE / AIEEE / PMT, 0120-4901457, +919990495952, +919910817866 Page 20
21. Progression / APEX INSTITUTE FOR IIT-JEE / AIEEE / PMT, 0120-4901457, +919990495952, +919910817866 Page 21
22. Progression / APEX INSTITUTE FOR IIT-JEE / AIEEE / PMT, 0120-4901457, +919990495952, +919910817866 Page 22
23. 1.07 General Definitions
111
1 1. sin –1 x, cos –1 x, tan –1 x etc. denote angles or real numbers whose sine is x , whose cosine is x and
whose tangent is x, provided that the answers given are numerically smallest available. These are
also written as arc sin x, arc cos x etc.
If there are two angles one positive & the other negative having same numerical value, then
positive angle should be taken.
EXERCISE-3
Part : (A)
1. If cos –1 cos –1 cos –1 v 3 then v v is equal to .
(a) –3 (b) 0 (c) 3 (d) –1
2. Range of f ( x) sin –1 x tan –1 x sec –1 x is.
3 3 3
(a) , (b) , (c) , (d) none of these
4 4 4 4 4 4
3
3. The solution of the equation sin –1 tan – sin –1 – 0 is.
4 x 6
(a) x = 2 (b) x = –4 (c) x = 4 (d) none of these
4. The value of sin –1[cos{cos –1 (cos x) sin –1 (sin x)}], where x , is
2
(a) (b) (c) – (d) –
2 4 4 2
5. The set of values of k for which x 2 – kx + sin –1 (sin 4) > 0 for all real x is
(a) {0} (b) (2, 2) (c) R (d) none of these
6. sin –1 (cos(sin –1 x)) cos –1 (sin(cos –1 x)) is equal to
3
(a) 0 (b) 4 (c) (d)
2 4
1 2 x2 x
7. cos –1 x 1– x 2 . 1– cos – – cos –1 x holds for
2 4 2
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24. (a) x 1 (b) x R
(c) 0 x 1 (d) –1 x 0
8. tan –1 a tan –1 b, where a 0, b 0, ab 1, is equal to
a b a b
(a) tan –1 (b) tan –1 –
1 – ab 1 – ab
a b a b
(c) tan –1 (d) – tan –1
1 – ab 1 – ab
–1
9. The set of values of „x‟ for which the formula 2sin x sin –1 2 x 1– x2 is true, is.
(a) (–1, 0) (b) [0, 1]
3 3 1 1
(c) – , (d) – ,
2 2 2 2
2
10. The set of values of „a‟ for which x ax sin –1 ( x 2 – 4 x 5) cos –1 ( x 2 – 4 x 5) 0 has at
least one solution is
– , – 2 2 , – , – 2 2 ,
(a) (b)
(c) R (d) none of these
–1 3
11. All possible values of p and q for which cos p cos –1 1– p cos –1 1– q holds, is
4
1 1 1
(a) p 1, q (b) q 1, p (c) 0 p 1, q (d) none of these
2 2 2
[cot –1 x] [cos –1 x] 0
12. If , where [.] denotes the greatest integer function, then complete set of
values of „x‟ is
(a) cos1, 1 (b) (cot 1, cos 1) (c) cot1, 1 (d) none of these
–1 2 –1
13. The complete solution set of the inequality [cot x] – 6[cot x] 9 0 , where [.] denotes
greatest integer function, is
(a) – , cot 3 (b) [cot 3, cot 2] (c) cot 3, (d) none of
these
1 1
14. tan cos –1 x tan – cos –1 x , x 0 is equal to
4 2 4 2
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25. 2 x
(a) x (b) 2x (c) x (d) 2
1 –1 3sin 2
15. If sin , then tan is equal to .
2 5 4 cos 2 4
(a) 1/3 (b) 3 (c) 1 (d) –1
u
16. If u cot –1 tan – tan –1 tan , then tan – is equal to .
4 2
(a) tan (b) cot (c) tan (d) cot
–1 1 – sin x 1 sin x
17. The value of cot 1 – sin x – 1 sin x
,
2
x , is :
x x x x
– 2 –
(a) 2 (b) 2 2 (c) 2 (d) 2
–1 –1 –1
18. The number of solutions of the equation, sin x cos (1– x) sin (– x), is/are .
(a) 0 (b) 1 (c) 2 (d) more than 2
–1 1 1 2
19. The number of solutions of the equation tan 2 x 1 tan –1 tan –1 is .
4x 1 x2
(a) 0 (b) 1 (c) 2 (d) 3
1 1 1 1
If tan –1 tan –1 tan –1 ........ tan –1 tan –1 , then is equal to.
20. 1 2 1 2.3 1 3.4 1 n(n 1)
n n n 1 1
(a) n 2 (b) n 1 (c) n (d) n
n
21. If cot –1 ,n N , then the maximum value of 'n' is :
6
(a) 1 (b) 5 (c) 9 (d) none of these
( C )9
–1
22. The number of real solutions of (x, y) where, (C )3 sin x, y cos (cos x), – 2
y x 2 , is :
(a) 2 (b) 1 (c) 3 (d) 4
–1 1 1
23. The value of cos 2 cos 8 is equal to
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26. (a) 3/4 (b) –3/4 (c) 1/16 (d) ¼
Part : (B)
1 1
24. , and are three angles given by 2 tan –1 ( 2 –1), 3sin –1 sin –1 – and
2 2
1
cos –1 . Then
3
(a) (b) (c) (d)
cos–1 x tan –1 x then
25.
5 –1 5 1
x2 x2
(a) 2 (b) 2
5 –1 5 –1
sin(cos –1 x) tan(cos –1 x)
(c) 2 (d) 2
2x tan(2 tan –1 a) 2 tan(tan –1 a tan –1 a3 )
26. For the equation , which of the following is invalid?
a2 x 2a x a2 2ax 1 0 a 0 a –1, 1
(a) (b) (c) (d)
4n
The sum of tan –1 is equal to
27. n 1 n4 – 2n2 2
tan –1 2 tan –1 3 4tan –1 1 /2 sec –1 – 2
(a) (b) (c) (d)
tan(cos –1 (4 / 5)) is a/b then.
28. If the numerical value of
(a) a + b = 23 (b) a – b = 11 (c) 3b = a + 1 (d) 2a = 3b
x 2 – x – 2 0,
29. If satisfies the in equation then a value exists for
sin –1 cos–1 sec–1 cosec–1
(a) (b) (c) (d)
x 1
If f ( x) cos –1 x cos –1 3 – 3x2 then:
30. 2 2
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27. 2 2 2
f f 2cos –1 –
(a) 3 3 (b) 3 3 3
1 1 1
f f 2cos –1 – m
(c) 3 3 (d) 3 3 3
1.01 SINE RULE
111
In1a triangle ABC, the sides are proporticnal to the sines of the angles opposite to them i.e.
a b c
sin A sin B sin C
Illustration 1 : In any ABC, prove that
EXERCISE-3
Q1: In a ABC, prove that a cot A b cot B c cot C 2( R r ) .
s s s r
Q2: In a ABC, prove that 4 –1 –1 –1 .
a b c R
Q3: If , , are the distances of the vertices of a triangle from the corresponding points of contact with
y
the incircle, then prove that r2 .
y
Q4: In a ABC prove that, r1r2 r2 r3 r3 r1 s2 .
Q5: In a ABC prove that rr1 rr2 rr3 ab bc ca – s 2 .
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