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subhasapco123 · 6 months ago

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Understanding Inverse Trigonometric Functions: A Comprehensive Guide

Trigonometry is one of the foundational subjects in mathematics that finds applications in various fields such as physics, engineering, and even computer science. While trigonometric functions like sine, cosine, and tangent describe relationships between the sides and angles of a right triangle, inverse trigonometric functions are equally essential for solving problems that involve angles when the sides of the triangle are known.

Inverse trigonometric functions, as the name suggests, are the reverse of the standard trigonometric functions. This blog will explore the concept of inverse trigonometric functions, their properties, and how they are used in mathematical and real-world applications.

What are Inverse Trigonometric Functions?

The inverse trigonometric functions are the functions that reverse the action of the regular trigonometric functions. In simple terms, while a regular trigonometric function takes an angle and gives a ratio of sides (such as sine giving opposite/hypotenuse), an inverse trigonometric function takes a ratio and gives an angle.

The six trigonometric functions in mathematics are:

Sine (sin)

Cosine (cos)

Tangent (tan)

Cotangent (cot)

Secant (sec)

Cosecant (csc)

Each of these functions has an associated inverse function. For example, the inverse of sine is called arcsine (or sin⁻¹), the inverse of cosine is called arccosine (cos⁻¹), and so on.

Why Do We Need Inverse Trigonometric Functions?

Inverse trigonometric functions are crucial because they allow us to find the angle when we know the value of the trigonometric function. This is particularly useful in fields like navigation, physics, engineering, and computer graphics, where it’s essential to work backward from a ratio of sides to determine the angle.

For instance, if we know the sine of an angle in a right triangle, the inverse sine (sin⁻¹) function can help us determine the measure of the angle. Similarly, inverse functions like arctangent (tan⁻¹) help us find the angle when the ratio of the opposite side to the adjacent side is known.

The Notation of Inverse Trigonometric Functions

The notation for inverse trigonometric functions is a bit different from regular trigonometric functions. Instead of writing "sin(x)" or "cos(x)," the inverse trigonometric functions are denoted with a superscript minus one, such as sin⁻¹(x) or cos⁻¹(x). This notation represents the angle whose sine or cosine is the given value.

Here’s a quick list of the common inverse trigonometric functions:

sin⁻¹(x) or arcsin(x): The inverse of sine, gives the angle whose sine is x.

cos⁻¹(x) or arccos(x): The inverse of cosine, gives the angle whose cosine is x.

tan⁻¹(x) or arctan(x): The inverse of tangent, gives the angle whose tangent is x.

cot⁻¹(x) or arccot(x): The inverse of cotangent, gives the angle whose cotangent is x.

sec⁻¹(x) or arcsec(x): The inverse of secant, gives the angle whose secant is x.

csc⁻¹(x) or arccsc(x): The inverse of cosecant, gives the angle whose cosecant is x.

Domains and Ranges of Inverse Trigonometric Functions

One of the critical aspects of inverse trigonometric functions is that they are restricted to certain domains and ranges to ensure that they are one-to-one functions. A one-to-one function is essential because it ensures that each input corresponds to a unique output.

Arcsin (sin⁻¹):

Domain: -1 ≤ x ≤ 1

Range: -π/2 ≤ y ≤ π/2

The arcsin function gives an angle between -90° and 90°.

Arccos (cos⁻¹):

Domain: -1 ≤ x ≤ 1

Range: 0 ≤ y ≤ π

The arccos function gives an angle between 0° and 180°.

Arctan (tan⁻¹):

Domain: -∞ < x < ∞

Range: -π/2 < y < π/2

The arctan function gives an angle between -90° and 90°.

Arccot (cot⁻¹):

Domain: -∞ < x < ∞

Range: 0 < y < π

The arccot function gives an angle between 0° and 180°.

Arcsec (sec⁻¹):

Domain: |x| ≥ 1

Range: 0 ≤ y ≤ π/2 or π ≤ y ≤ 3π/2

The arcsec function gives an angle between 0° and 90° or between 90° and 180°.

Arccsc (csc⁻¹):

Domain: |x| ≥ 1

Range: -π/2 ≤ y ≤ 0 or 0 ≤ y ≤ π/2

The arccsc function gives an angle between -90° and 90°, excluding 0°.

Properties of Inverse Trigonometric Functions

Understanding the properties of inverse trigonometric functions can make working with them much easier. Here are some essential properties:

Inverse of an Inverse: The inverse of an inverse trigonometric function gives the original function back. For example:

sin(sin⁻¹(x)) = x for -1 ≤ x ≤ 1

cos(cos⁻¹(x)) = x for -1 ≤ x ≤ 1

tan(tan⁻¹(x)) = x for all x

Composition of Functions: The inverse and the original trigonometric function can be composed together. For example:

sin⁻¹(sin(x)) = x for -π/2 ≤ x ≤ π/2

cos⁻¹(cos(x)) = x for 0 ≤ x ≤ π

tan⁻¹(tan(x)) = x for -π/2 < x < π/2

Symmetry: Inverse trigonometric functions exhibit symmetry about certain axes. For example, the inverse sine function is symmetric about the y-axis, while the inverse cosine function is symmetric about the line x = 0.

Solving Trigonometric Equations Using Inverse Functions

Inverse trigonometric functions are widely used for solving trigonometric equations. For example, if you are given a problem where you need to find the angle θ, knowing the value of sin(θ) = 0.5, you can use the arcsin function to find the angle:θ=sin⁡−1(0.5)=30∘θ = \sin^{-1}(0.5) = 30^\circθ=sin−1(0.5)=30∘

Similarly, if you are given the tangent value of an angle, you can use the arctan function to find the angle. This process is vital for solving problems in geometry, calculus, and physics.

Real-World Applications of Inverse Trigonometric Functions

Navigation: Inverse trigonometric functions are crucial in navigation and determining bearings. Pilots and sailors use these functions to calculate angles based on given distances and directions.

Physics: In physics, especially in wave motion and optics, inverse trigonometric functions help solve problems involving angles of refraction, angles of incidence, and angular displacement.

Engineering: In electrical engineering and mechanical systems, inverse trigonometric functions are used in control systems, signal processing, and analyzing vibrations.

Computer Graphics: Inverse trigonometric functions are used in computer graphics to rotate and scale objects, especially when working with angles in 3D space.

Conclusion

Inverse trigonometric functions are indispensable tools for solving mathematical and real-world problems involving angles and ratios. From geometry to physics and engineering, they provide a method for determining the angle when the side ratios of a right triangle are known. Understanding the properties and applications of inverse trigonometric functions will undoubtedly help you excel in both theoretical and applied mathematics.

#education #trignometry

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howtofindthedomainofafunctions · 6 years ago

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Mathematics in Plain English - What Is an Inverse Function?

In mathematics, an inverse function essentially undoes what the given function does. If we think in terms of domain and range, then if a given function takes a value x in the domain to a value y in the range, then its inverse function takes that same y value in the range and sends it back to the x value in the domain.

Inverse functions are important in mathematics because in certain problems, we know what the range value is and we need to determine from what domain value this comes. For example, in certain trigonometry problems we know the value that the sine or cosine functions produce and we want to know what angle produces such values. This situation can happen if we want to construct, let us say, a right triangle of given side lengths and we need to know the angle measurement which accommodates those sides.

In order to find the inverse function of a given function, the function at hand mustbe one-to-one. If you have not read my article on this topic, then I repeat the definition of a one-to-one function here: one-to-one functions are such that they pass the horizontal line test; that is, it can never happen that two distinct x values in the domain are sent to the same y value in the range. This condition must be satisfied in order to find an inverse function because if it were not, then there would essentially be two return routes by which to send the y value (we could send the y back to either of the distinct x values from which it came.) This would lead to a poorly defined inverse function.

Once we have a function which is one-to-one, we can find the inverse by switching the x and y values in the equation and solving for y. To see this, let us take a simple example, the linear function y = 3x - 2. All linear functions are one-to-one. Thus we can switch x and y, and solve for y. Doing this we have, x = 3y -2; solving now for y, we have y = (x + 2)/3. That is all there is to it.

To show that this is actually the function which takes a given x value and sends it back to whence it came, let us verify with a specific example. Let x = 10. In the given function y = 3x - 2,this gives y = 28. The inversefunction should send this value of 28 back to 10. If you plug 28 into the function y = (x + 2)/3, you see that you get 10. Thus the inverse function does what it was designed to do.

For more complicated functions, the inverse can be extremely difficult to find, if not impossible. For these situations, mathematicians rely on more sophisticated tools and methodologies. For the most part though, a given function and its inverse are nothing more than polar opposites: if the function takes a traveler to the North Pole, the inverse takes him back to the South Pole. What a nice guy this inverse function!

To see how his mathematical talent has been used to forge a beautiful collection of love poetry, click below to get the kindle version. You will then see the many connections between mathematics and love.

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frcgame · 6 years ago

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Summary of ON Math Curriculum

Outlined below is a summary of the curriculum of Ontario High School Math that is related to the topic of our game.

Source: http://www.edu.gov.on.ca/eng/curriculum/secondary/math1112currb.pdf

Functions (FRC3U)

Characteristics of Functions

Representing functions:

- Understand the definition of a function. Sample question: determine whether x = y^2 is a function.

- Sketch graphs of y = af(k(x-d))+c by applying one or more transformations to the graphs of f(x) = x, f(x) = x^2, f(x) = \sqrt(x), and f(x) = 1/x; state the domain and range of the transformed functions.

- Determine the roles of the parameters a, k, d, and c in the above equation.

i.e. translations; reflections in the axes, vertical and horizontal stretches and compressions to and from the x and y axes.

- Understand the definition of the inverse of a linear or quadratic function

Problem Solving:

- “determine the number of zeros (i.e., x-intercepts) of a quadratic function, using a variety of strategies (e.g., inspecting graphs; factoring; calculating the discriminant)”

- “solve problems involving the intersection of a linear function and a quadratic function graphically and algebraically (e.g., determine the time when two identical cylindrical water tanks contain equal volumes of water, if one tank is being ﬁlled at a constant rate and the other is being emptied through a hole in the bottom)”

Determining Equivalent Algebraic Expressions

Exponential Functions

- “graph, with and without technology, an exponential relation, given its equation in the form y = a (a > 0, a ≠ 1), deﬁne this relation as the function f(x) = a , and explain why it is a function”

- “simplify algebraic expressions containing integer and rational exponents [e.g., (x^3 ) / (x^(1/2)), (x^6y^3)^(1/3) ]”

- “determine, through investigation, and describe key properties relating to domain and range, intercepts, increasing/decreasing intervals, and asymptotes”

- “determine, through investigation using technology, the roles of the parameters a, k, d, and c in functions of the form y = af(k(x – d))+c, and describe these roles in terms of transformations on the graph of f(x) = a (a > 0, a ≠ 1) (i.e., translations; reﬂections in the axes; vertical and horizontal stretches and compressions to and from the x- and y-axes)”

Trigonometry Functions

- “sketch graphs of y =af(k(x – d)) + c by applying one or more transformations to the graphs of f(x) =sinx and f(x) =cosx, and state the domain and range of the transformed functions”

Logarithmic Functions

- “Determine the graphs of logarithmic functions of the form f(x) = log x, and make connections between the algebraic and graphical representations of these logarithmic function”

#research #week 3 #curriculum

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toppersexam · 5 years ago

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NDA 2020 Age Limit, Syllabus, Online Free Mock Test, MCQ, Books

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rafi1228 · 6 years ago

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jguidein · 7 years ago

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UPSC NDA II Syllabus 2018 Download UPSC NA 1 & NDA NA 2 Syllabus | UPSC NDA IAF Exam Pattern Pdf

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Download UPSC NDA II Syllabus 2018 & Exam Pattern:

Candidates must start Preparing for the Indian Navy, Indian Army, and IAF Exam by downloading the UPSC NDA II Syllabus and Exam Pattern. The willing individual may click the link below to download all State, and Central Govt. Jobs Updates regularly up to date. After successfully applying for UPSC NDA II Recruitment 2018 Notification participants are waiting for UPSC NDA II Syllabus, to know the right topics check the complete web page. UPSC NDA Indian Navy Exam Pattern 2018 UPSC Conducts NDA Exams Which Includes Interviews Covering General Aptitude Team-Building Skills, Psychological Testing, Along With Physical And Social Skills And Medical Tests. It’s Not That Easy At All Subject Code Exam Duration Maximum Marks Mathematics 1 2 ½ Hours 300 General Ability Test 2 2 ½ Hours 600 Total 900 SBS Interview 900 Grand Total 1800 UPSC National Defence Academy II Syllabus 2018 Paper I – Mathematics – 300 Marks Algebra: Concept of a set, operations on sets, Venn diagrams. De Morgan laws, Cartesian product, relation, equivalence relation. Representation of real numbers on a line. Complex numbers—basic properties, modulus, argument, cube roots of unity. Binary system of numbers. Conversion of a number in decimal system to binary system and vice-versa. Arithmetic, Geometric and Harmonic progressions. Quadratic equations with real coefficients. Solution of linear inequations of two variables by graphs. Permutation and Combination. Binomial theorem and its applications. Logarithms and their applications. Matrices and Diterminants Types of matrices, operations on matrices. Determinant of a matrix, basic properties of determinants. Adjoint and inverse of a square matrix, Applications-Solution of a system of linear equations in two or three unknowns by Cramer’s rule and by Matrix Method. Trigonometry Angles and their measures in degrees and radians. Trigonometrical ratios. Trigonometric identities Sum and difference formulae. Multiple and Sub-multiple angles. Inverse trigonometric functions. Applications-Height and distance, properties of triangles. Analyticl Geomentry of To and Three Diamonds Rectangular Cartesian Coordinate system. Distance formula. Equation of a line in various forms. Angle between two lines. Distance of a point from a line. Equation of a circle in standard and in general form. Standard forms of parabola, ellipse and hyperbola. Eccentricity and axis of a conic. Point in a three dimensional space, distance between two points. Direction Cosines and direction ratios. Equation two points. Direction Cosines and direction ratios. Equation of a plane and a line in various forms. Angle between two lines and angle between two planes. Equation of a sphere. Differential Calculas Concept of a real valued function–domain, range and graph of a function. Composite functions, one to one, onto and inverse functions. Notion of limit, Standard limits—examples. Continuity of functions—examples, algebraic operations on continuous functions. Derivative of function at a point, geometrical and physical interpretation of a derivative—applications. Derivatives of sum, product and quotient of functions, derivative of a function with respect to another function, derivative of a composite function. Second order derivatives. Increasing and decreasing functions. Application of derivatives in problems of maxima and minima. Integral Calculas & Differential Equations Integration as inverse of differentiation, integration by substitution and by parts, standard integrals involving algebraic expressions, trigonometric, exponential and hyperbolic functions. Evaluation of definite integrals—determination of areas of plane regions bounded by curves— applications. 18 Definition of order and degree of a differential equation, formation of a differential equation by examples. General and particular solution of a differential equations, solution of first order and first degree differential equations of various types—examples. Application in problems of growth and decay. Vector Algebra Vectors in two and three dimensions, magnitude and direction of a vector. Unit and null vectors, addition of vectors, scalar multiplication of a vector, scalar product or dot product of two vectors. Vector product or cross product of two vectors. Applications—work done by a force and moment of a force and in geometrical problems. Statistics & Probability Statistics: – Classification of data, Frequency distribution, cumulative frequency distribution—examples. Graphical representation—Histogram, Pie Chart, frequency polygon—examples. Measures of Central tendency—Mean, median and mode. Variance and standard deviation—determination and comparison. Correlation and regression. Probability: Random experiment, outcomes and associated sample space, events, mutually exclusive and exhaustive events, impossible and certain events. Union and Intersection of events. Complementary, elementary and composite events. Definition of probability—classical and statistical— examples. Elementary theorems on probability—simple problems. Conditional probability, Bayes’ theorem—simple problems. Random variable as function on a sample space. Binomial distribution, examples of random experiments giving rise to Binominal distribution. UPSC NDA II Syllabus – Paper II(General Ability Test – 600M): Part A: General English The question paper in English will be designed to test the candidate’s understanding of English and workmanlike use of words. Grammar and usage, Vocabulary, Comprehension and cohesion in extended text to test the candidate’s proficiency in English. Part ‘B’—General Knowledge: The question paper on General Knowledge will broadly cover the subjects as follows. Physics, Chemistry, General Science, Social Studies, Geography and Process to Check UPSC NDA II Syllabus 2018: Initially, all candidates need to go on official website of organization which is “www.upsc.gov.in”. Find the appropriate segment or link from home page. Follow the appropriate link for the syllabus. Download the syllabus in the pdf file. Prepare for the exam according to the syllabus. Note: Dear Applicants, if you have any query regarding UPSC NDA II Syllabus 2018 they can comment us through in the comment box. UPSC NDA II Important Links: UPSC NDA II Syllabus & Exam Pattern Official Link Read the full article

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go-travel-cpap-blog · 8 years ago

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NDA AND NAVAL DEFENCE ACADEMY

1. Algebra

The concept of a set, procedures on sets, Venn layouts. De Morgan laws. Cartesian product, relation, equivalence connection. Read More Representation of real figures on a line. Complicated numbers - basic qualities, modulus, argument, cube origins of unity. Binary program of numbers. Conversion associated with the number in decimal system to binary program and vice-versa. Arithmetic, Geometric and Harmonic progressions. Quadratic equations with real coefficients. A solution of linear inequations of two variables simply by graphs. Permutation and Mixture. Binomial theorem and the application. Logarithms and their own applications. second. Matrices plus Determinants Forms of matrices, procedures on matrices Determinant associated with a matrix, basic qualities of the determinant. Adjoint plus inverse of a sq . matrix, Applications - Answer of a system associated with linear equations in 2 or three unknowns simply by Cramers rule and simply by Matrix Method. 3. Trigonometry Angles and their steps in degrees and within radians. Trigonometrical ratios. Trigonometric identities Sum and distinction formulae. Multiple and Sub-multiple angles. Inverse trigonometric features. Applications - Height plus distance, properties of triangles. 4. Analytical Geometry associated with two and three sizes Rectangular Cartesian Coordinate program. Distance formula. Equation associated with a line in the variety of forms. The position between two lines. The range of a point through a line. Equation associated with the circle in regular and general form. Regular types of the parabola, ellipse plus hyperbola. Eccentricity and axis of the conic. Stage in a 3d area, the distance between two factors. Direction Cosines and path ratios. The equation of the particular plane and a collection in a variety associated with forms. The angle between 2 lines and angle among two planes. Equation associated with the sphere. 5. Gear Calculus The concept of the real-valued function -- domain, range and chart of a function. Amalgamated functions, one to 1, onto and inverse features. Notion of limit, Regular limits - examples. Continuity of functions - good examples, algebraic operations on constant functions. Derivative of the function in a stage, geometrical and physical interpretation of a derivative -- applications. Derivatives of the amount, product and quotient associated with functions, a derivative of the function with respect associated with another function, derivative associated with a composite function. 2nd order derivatives. Increasing plus decreasing functions. The application associated with derivatives in problems associated with maxima and minima. six. Integral Calculus and Gear equations Integration as inverse of differentiation, integration simply by substitution and by components, standard integrals involving algebraic expressions, trigonometric, exponential plus hyperbolic functions. Evaluation associated with definite integrals - dedication of areas of aircraft regions bounded by the figure - applications. Definition associated with order and degree associated with a differential equation, development of a differential formula by examples. General plus the particular solution of the differential equation, solution associated with the first order and 1st-degree differential equations associated with various types - good examples. Application in problems associated with growth and decay. seven. Vector Algebra Vectors within two and three sizes, magnitude and direction associated with a vector. Unit plus null vectors, addition associated with vectors, scalar multiplication associated with the vector, scalar product or even dot product of two vectors. Vector product and mix product of two vectors. Applications-work did with the force and moment associated with the force and within geometrical problems. 8. Data and Possibility Statistics Category of data, Frequency submission, cumulative frequency distribution -- examples Graphical representation -- Histogram, Pie Chart, Rate of recurrence Polygon - examples. Steps of Central tendency -- mean, median and setting. Variance and standard change - determination and assessment. Correlation and regression. Possibility Random experiment, outcomes plus associated sample space, occasions, mutually exclusive and thorough events, impossible and particular events. Union and Intersection of events. Complementary, primary and composite events. Description of probability - traditional and statistical - good examples. Elementary theorems on the possibility - simple problems. Conditional probability, Bayes theorem -- simple problems. Random adjustable as a function on the particular sample space. Binomial submission, examples of random tests giving rise to Binominal distribution. Paper II COMMON ABILITY TEST Part The - ENGLISH

The query paper in English will certainly be designed to check the candidates understanding associated with English and workman such as utilisation of words. The syllabus covers various aspects such as Grammar and usage, language, comprehension and cohesion within extended text to check the candidate's proficiency within English.

Part B -- GENERAL UNDERSTANDING

The query paper on General Understanding will broadly cover the particular subjects Physics, Chemistry, Common Science, Social Studies, Location and Current Events. The particular syllabus given below will be designed to indicate the particular scope of these topics included in this papers. The topics mentioned are certainly not to be regarded because exhaustive and questions on topics of similar character not specifically mentioned in the syllabus can furthermore be asked. Candidates solutions are likely to show their own knowledge and intelligent knowing of the subject. Area A (Physics) Physical Qualities and States of Issue, Mass, Weight, Volume, Denseness and Specific Gravity, Theory of Archimedes, Pressure Measure. The motion of objects, Speed and Acceleration, Newton's Laws and regulations of Motion, Force plus Momentum, Parallelogram of Causes, Stability and Equilibrium associated with bodies, Gravitation, elementary suggestions of work, Power plus Energy. Effects of Warmth, Measurement of temperature plus heat, change of Condition and Latent Heat, Settings of transference of Warmth. Sound waves and their own properties, Simple musical devices. Rectilinear propagation of Gentle, Reflection and refraction. Circular mirrors and Lenses. Human being Eye. Natural and Synthetic Magnets, Properties of the Magnet, Earth like a Magnets. Static and Current Electrical power, conductors and nonconductors, Ohms Law, Simple Electrical Circuits, Heating, Lighting and Magnet effects of Current, Dimension of Electrical Power, Main and Secondary Cells, Make use of X-Rays.

#NDA #Vector Algebra

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pathfinderdefenceacademyposts · 5 years ago

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Best NDA Coaching Classes in Lucknow | Best NDA Coaching Center in Lucknow

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ELIGIBILITY CRITERIA

Nationality: Apart from Indian origin, candidates from other countries can also appear for NDA 2019 Candidates must either be.

India citizen.

Citizen of Bhutan.

Citizen of Nepal.

Tibetan refugee who came over to India before January 1, 1962, with the intention of permanently settling in India.

Indian origin person migrated from Pakistan, Burma, Sri Lanka and East African Countries of Kenya, Uganda, the United Republic of Tanzania, Zambia, Malawi, Zaire, and Ethiopia or Vietnam with the intention of permanently settling in India.

AGE LIMITS

Nationality: Apart from Indian origin, candidates from other countries can also appear for NDA. Candidates must either be.

Minimum Age: 15-1/2 years. (For Form Filling)

Maximum Age: 18-1/2 years. (For Form Filling)

The date of birth will be calculated as it is entered in the Matriculation or Secondary School Leaving Certificate or in a certificate recognized by an Indian University as equivalent to Matriculation.

Marital Sex: Candidates must be unmarried.

Gender: Only male candidates are eligible to apply for NDA.

EDUCATIONAL QUALIFICATION

Army Wing of Pathfinder Defence Academy: Candidates applying for the Indian Army must have passed class 12/HSC in 10+2 pattern of School Education or equivalent examination conducted by a State Education Board or a University.

Air Force, Navy and Naval Academy of PathFinder Defence Academy: Candidates applying for Air Force, Navy and Naval Academy must have passed class 12/HSC in 10+2 pattern of School Education with Physics and Mathematics conducted by a State Education Board or a University.

Physical standard required for NDA: Candidates appearing in NDA must be physically and mentally fit according to the prescribed physical standards. A candidate recommended by the Services Selection Board (SSB) will undergo a medical examination by a Board of Service Medical Officers. Only those candidates will be declared qualified NDA and admitted to the academy who is declared fit by the medical board. The candidate must be in good physical and mental health and free from any disease/disability which is likely to interfere with the efficient performance of military duties. The minimum acceptable height is 157 cm for Army, Navy and Naval Academy while 162.5 cm for Air Force. For Gurkhas and individuals belonging to hills of North-Eastern, the minimum acceptable heights will be 5 cm less. For more detail required physical standards candidates are advised to read the NDA notification.

SYLLABUS

Paper-I Mathematics (Maximum Marks – 300) :

Algebra: Concept of set, operations on sets, Venn diagrams. De Morgan laws. Cartesian product, relation, equivalence relation. Representation of real numbers on a line. Complex numbers – basic properties, modulus, argument, and cube roots of unity. Binary system of numbers. Conversion of a number in decimal system to binary system and vice-versa. Arithmetic, Geometric and Harmonic progressions. Quadratic equations with real coefficients. The solution of linear inequations of two variables by graphs. Permutation and Combination. Binomial theorem and its application. Logarithms and their applications.

Matrices and Determinants: Types of matrices, operations on matrices Determinant of a matrix, basic properties of determinant. Adjoint and inverse of a square matrix, Applications – Solution of a system of linear equations in two or three unknowns by Cramer’s rule and by Matrix Method.

Trigonometry: Angles and their measures in degrees and in radians. Trigonometrically ratios. Trigonometric identities Sum and difference formulae. Multiple and Sub-multiple angles. Inverse trigonometric functions. Applications – Height and distance, properties of triangles.

Analytical Geometry of two and three dimensions: Rectangular Cartesian Coordinate system. Distance formula. Equation of a line in various forms. The angle between the two lines. The distance of a point from a line. Equation of a circle in standard and in general form. Standard forms of parabola, ellipse, and hyperbola. Eccentricity and axis of a conic. The point in a three-dimensional space, distance between two points. Direction Cosines and direction ratios. Equation of a plane and a line in various forms. The angle between the two lines and the angle between the two planes. Equation of a sphere.

Differential Calculus: Concept of a real-valued function – domain, range, and graph of a function. Composite functions, one to one, onto and inverse functions. Notion of limit, Standard limits – examples. Continuity of functions – examples, algebraic operations on continuous functions. Derivative of a function at a point, geometrical and physical interpretation of a derivative – applications. Derivatives of sum, product, and quotient of functions, a derivative of a function with respect of another function, derivative of a composite function. Second-order derivatives. Increasing and decreasing functions. Application of derivatives in problems of maxima and minima.

Integral Calculus and Differential equations: Integration as inverse of differentiation, integration by substitution and by parts, standard integrals involving algebraic expressions, trigonometric, exponential and hyperbolic functions. Evaluation of definite integrals – determination of areas of plane regions bounded by curves – applications. Definition of order and degree of a differential equation, formation of a differential equation by examples. General and particular solution of a differential equation, solution of the first order and first-degree differential equations of various types – examples. Application in problems of growth and decay.

Vector Algebra: Vectors in two and three dimensions, magnitude and direction of a vector. Unit and null vectors, the addition of vectors, scalar multiplication of vector, scalar product or dot product of two vectors. Vector product and cross product of two vectors. Applications-work did by a force and moment of a force, and in geometrical problems.

Statistics and Probability:

Statistics: Classification of data, Frequency distribution, cumulative frequency distribution – examples Graphical representation – Histogram, Pie Chart, Frequency Polygon – examples. Measures of Central tendency – mean, median and mode. Variance and standard deviation – determination and comparison. Correlation and regression.

Probability: Random experiment, outcomes, and associated sample space, events, mutually exclusive and exhaustive events, impossible and certain events. Union and Intersection of events. Complementary, elementary and composite events. Definition of probability – classical and statistical – examples. Elementary theorems on probability – simple problems. Conditional probability, Bayes’ theorem – simple problems. Random variable as function on a sample space. Binomial distribution, examples of random experiments giving rise to Binominal distribution.

PAPER-II

General Ability Test (Maximum Marks-600)

PART – A

ENGLISH (Maximum Marks 200).

The question paper in English will be designed to test the candidate’s understanding of English and workmanlike use of words. The syllabus covers various aspects like Grammar and usage, vocabulary, comprehension, and cohesion in extended text to test the candidate’s proficiency in English.

PART – B

SYLLABUS OF PHYSICS

Physical properties and states of matter. Mass Weight, Volume, Density, and Specific Gravity, Principle of Archimedes, Pressure Barometer.

The motion of objects: Velocity and acceleration. Newton’s Laws of motion. Force and Momentum. Parallelogram of Forces. Stability and equilibrium of bodies. Gravitation, elementary ideas of work, Power and Energy.

Effects of heat: Measurement of temperature and heat. Change of state and latent heat. Modes of transference of heat. Sound Waves and their properties. Simple musical instruments. Rectilinear propagation of light. Reflection and Refraction. Spherical mirrors and lenses. Human eye.

Natural and artificial magnets: properties of a magnet. Earth as a magnet.

Static and current electricity: Conductors and non-conductors. Ohm’s law. Simple electrical circuits. Heating, lighting and magnetic effects of current. Measurement of electrical power. Primary and Secondary Cells. Use of X-rays.

General principles in the working of the following: Simple pendulum, Simple Pulleys, Siphon, Levers, Balloon. Pumps, Hydrometer, Pressure Cooker, Thermos Flask, Gramophone, Telegraphs. Telephone, Periscope, Telescope, Microscope, Mariner's Compass, Lightning Conductors and Safety Flues.

Syllabus of General Science :

Basis of Life – Cells, Protoplasms and Tissues, Elementary knowledge of human body and its important organs, Food – Source of Energy for man, Constituents of food, Balanced Diet, Achievements of Eminent Scientists, Difference between the living and non-living, Growth and Reproduction in Plants and Animals, Common Epidemics, their causes and prevention, The Solar System – Meteors and Comets, Eclipse.

History: Freedom Movement in India, elementary knowledge of five-year plans of India, Bhutan, Sarvodaya, National Integration and Welfare State, Basic Teachings of Mahatma Gandhi, A broad survey of Indian History, with emphasis on Culture and Civilisation, Elementary study of Indian Constitution and Administration, Panchayati Raj, Co-operatives and Community Development, Forces shaping the modern world; Renaissance, Exploration, and Discovery; War of American Independence. French Revolution, Industrial Revolution, and the Russian Revolution. Impact of Science and Technology on Society. Concept of one World, United Nations, Panchsheel, Democracy. Socialism and Communism. Role of India in the present world.

Geography: Origin of Earth, Rocks, and their classification; Weathering – Mechanical and Chemical, Earthquakes and volcanoes, Atmosphere and its composition; Temperature and Atmospheric Pressure, Planetary Winds, cyclones, and Anti-cyclones; Humidity; Condensation and Precipitation; Types of Climate. Major Natural regions of the World, Important Sea ports and main sea, land, and air routes of India. Main items of Imports and Exports of India, The Earth, its shape and size. Lattitudes and Longitudes, Concept of time, International Date Line, Movements of Earth and their effects, Ocean Currents and Tides, Regional Geography of India – Climate, Natural vegetation. Mineral and Power resources; location and distribution of agricultural and industrial activities.

Current Events: Current important world events, Knowledge of Important events that have happened in India in recent years, prominent personalities – both Indian and International including those connected with cultural activities and sports.

#nda #nda coaching #coaching classes

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