TS Inter Maths 1A

ts inter properties of triangles 4 marks important questions 2024

TS Inter Properties of Triangles 4-M Important Questions

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LTS Inter Properties of Triangles 4-M Important Questions  Properties of Triangles Properties of triangles “The four important questions in Properties of Triangles serve as a beacon for students navigating the vast sea of ​​geometric principles. Triangles, being the simplest polygon, contain a wealth of properties that lay the foundation for understanding more complex shapes and spatial relationships.” This curated selection of questions isn’t just about rote memorization; it’s a journey through the intricate tapestry of triangle geometry, where each question unveils a new facet of understanding.   From the classic Pythagorean theorem to the subtle nuances of triangle inequalities, these questions foster a deep appreciation for the elegance and precision of geometric reasoning. By engaging with these questions, students embark on a voyage of discovery, unraveling the mysteries of angle bisectors, medians, altitudes, and more. Moreover, these questions transcend mere academic exercises; they empower students to apply their knowledge to real-world scenarios, from architectural design to navigation. Ultimately, the ‘4 Marks Important Questions in Properties of Triangles’ isn’t just a study aid; it’s a roadmap to mastery, guiding students through the labyrinth of triangle geometry with clarity and confidence. Through diligent practice and thoughtful reflection, students can unlock the secrets of triangles and harness their geometric prowess to solve problems, explore new vistas, and shape the world around them.” Properties of triangles “In the study of geometry, understanding the properties of triangles is essential as they serve as building blocks for more intricate geometric concepts. To master these properties effectively, it’s crucial to focus on certain questions that encapsulate the core principles of triangle geometry. The ‘4 Marks Important Questions in Properties of Triangles’ compilation serves as a strategic guide for students and learners, offering a concise yet comprehensive selection of inquiries that target crucial aspects of triangles. From exploring angle relationships to dissecting the intricacies of triangle congruence and similarity, these questions challenge individuals to apply their knowledge in diverse scenarios. By tackling these questions, students not only solidify their understanding of triangle properties but also hone their problem-solving skills and analytical thinking abilities. Whether preparing for examinations or seeking a deeper grasp of geometry, these questions provide a valuable resource for navigating the intricacies of triangle geometry with confidence and proficiency.”   Properties of triangles “Properties of triangles are fundamental concepts in geometry, forming the basis for understanding more complex geometric relationships. In any examination or study of triangles, certain questions stand out as particularly crucial for understanding these properties deeply. This curated list of ‘4 Marks Important Questions in Properties of Triangles’ highlights key inquiries that not only test comprehension but also encourage critical thinking and application of geometric principles. These questions delve into various aspects of triangles, such as angles, sides, and special properties, ensuring a comprehensive understanding of this foundational geometric shape.” Visit my YouTube channel: Click on the Logo

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ts inter inverse trigonometric functions 4 marks important questions 2024

Ts Inter Inverse Trigonometric Functions 4m Imp Questions

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Ts Inter Inverse Trigonometric Functions 4m Imp Questions Here are some important questions related to inverse trigonometric functions for TS Inter (Telangana State Board of Intermediate Education) exams:     Explain the concept of this topic and discuss its domain and range for each function, such as arcsin(x), arccos(x), and arctan(x). State and Prove the Properties of   Trigonometric Functions. List and prove properties such as the principal value range, periodicity, and relationships between inverse trigonometric functions. Find the Principal Value of an Inverse Trigonometric Expression. Given an expression involving inverse trigonometric functions, determine its principal value within the defined range. Solve equations involving inverse trigonometric functions, ensuring solutions lie within the specified domain and principal value range. Graphical Interpretation of Inverse Trigonometric Functions.   Sketch graphs of inverse trigonometric functions and their principal branches, highlighting key features such as asymptotes, intercepts, and intervals of increase/decrease. Illustrate how inverse trigonometric functions are used in real-world scenarios, such as solving problems related to angles of elevation/depression, trigonometric equations, and geometric constructions. Derivatives and Integrals of Inverse Trigonometric Functions. Discuss techniques for finding derivatives and integrals involving inverse trigonometric functions, emphasising the importance of understanding these functions in calculus. Solving Trigonometric Equations Using this topic. Demonstrate how to solve trigonometric equations by employing inverse trigonometric functions and applying appropriate algebraic techniques. Provide challenging problems that require a deep understanding of inverse trigonometric functions, encouraging students to apply various strategies and techniques to arrive at solutions. These questions cover various aspects of this topic and should prepare students effectively for their TS Inter exams.   Here are a few more specific questions focusing on different aspects of our topic: Given a trigonometric equation involving these topics, find the exact values of the expressions, ensuring solutions lie within the specified range. Verifying Inverse Trigonometric Identities. Provide a set of inverse trigonometric identities and ask students to verify them using algebraic manipulations and properties of trigonometric functions. Applications in Geometry. Present geometric problems that can be solved using inverse trigonometric functions, such as finding the angles or side lengths in triangles or other geometric figures. Solving Equations with Multiple Trigonometric Functions. Construct equations involving multiple trigonometric functions and ask students to solve them using appropriate techniques involving inverse trigonometric functions. Inverse Trigonometric Functions in Calculus. Pose calculus-based questions involving this topic, such as finding limits, derivatives, or integrals that include these functions. Inverse Trigonometric Equations with Constraints. Introduce equations where this topic is subject to certain constraints, such as inequalities or restrictions on the domain, and solve them accordingly. Explore how the graphs of this topic are transformed under operations such as translations, reflections, and dilations, emphasizing changes in amplitude, period, and phase shift. Inverse Trigonometric Equations with Applications. Present real-world problems that can be modelled and solved using this topic, encouraging students to interpret solutions in the context of the problem.   Visit my YouTube channel: Click on the Logo

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ts inter trigonometric equations 4 marks important questions 2024

TS Inter Trigonometric Equations – 4-M Important Questions

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Trigonometric Equations – 4-Marks important Questions Trigonometric Equations “Trigonometric Equations 4-Mark Important Questions” would typically refer to a collection of questions worth four marks each that focus on solving equations involving trigonometric functions. These questions are likely intended for students studying trigonometry at the intermediate level or equivalent. Trigonometric equations involve expressions containing trigonometric functions such as sine, cosine, tangent, cosecant, secant, and cotangent. The goal in solving these equations is typically to find the values of the variable(s) that satisfy the given equation within a specified interval. Trigonometric Equations These types of questions may cover various topics within trigonometric equations, including: Solution of basic trigonometric equations: These equations involve single trigonometric functions and can often be solved using algebraic techniques such as factoring, substitution, or trigonometric identities.  Trigonometric equations involving multiple angles: Equations may involve multiple angles, such as sums, differences, or multiples of trigonometric functions. Students may need to apply trigonometric identities or properties to simplify the equations before solving them. Trigonometric Equations Trigonometric Equations Students may encounter equations where trigonometric identities need to be applied to rewrite the equation in a more simplified form before solving it. Solutions of  Trigonometric Equations with restrictions: Some equations may have restrictions on the domain, such as finding solutions within a specific interval or range of values. Solving Trigonometric Equations involving transformations: Equations may involve transformations of trigonometric functions, such as amplitude changes, phase shifts, or vertical and horizontal translations.   SAQs of Trigonometric Equations serve as a means for students to practice and demonstrate their understanding of trigonometric equations, their ability to apply various problem-solving techniques, and their proficiency in manipulating trigonometric functions to find solutions. Trigonometric Equations Additionally, these questions may also help students prepare for assessments or examinations where solving trigonometric equations is a key component of the curriculum. Visit my YouTube channel: Click on the Logo

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ts inter Trigonometric Ratios Up To Transformations 4 marks important questions

TS Inter Trigonometric Ratios up to Transformations – 4-Mark Questions

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TS Inter Trigonometric Ratios up to Transformations – 4-Mark Questions This comprehensive guide is designed to assist students in mastering the fundamental concepts of trigonometric ratios, with a focus on understanding transformations. Tailored to ensure a thorough understanding of the topic, this resource highlights essential questions worth 4 marks each, providing targeted practice for examinations. Through clear explanations and strategic problem-solving techniques, students will gain confidence in manipulating trigonometric functions within various transformations, paving the way for success in both classroom assessments and standardized tests.     “Trigonometric Ratios Up To Transformations: 4 Marks Important Questions” is a meticulously crafted resource aimed at sharpening students’ understanding of trigonometry, specifically focusing on transformations. Within its pages, learners will encounter a curated selection of questions, each worth 4 marks, strategically chosen to reinforce key concepts and test problem-solving skills.   TS  inter Maths 1A Question Papers     Trigonometric ratios upto transformations, maths 1a trigonometric ratios upto transformations important, trigonometric ratios, the product of vectors important questions for ipe, trigonometry 4 marks important questions, trigonometric functions 5marks important questions, most important 4 marks questions,4 marks most important questions in Telugu, most important 4 marks questions 1a and 1b, most important 4 marks questions in Telugu, trigonometric functions important questions 202 Trigonometric Ratios Up To Transformations: 4 Marks Important Questions” is a comprehensive study companion meticulously crafted to aid students in conquering trigonometry. Delving into the pivotal realm of transformations, this resource presents a curated selection of questions, each strategically chosen to reinforce key concepts and hone problem-solving abilities. With clear explanations and step-by-step solutions, learners can navigate through the complexities of trigonometric functions within transformations with confidence. Whether preparing for exams or seeking to deepen comprehension, this guide is an invaluable tool for achieving mastery in trigonometry. These types of questions are likely aimed at testing students’ understanding of trigonometric concepts, their ability to apply these concepts to solve problems, and their proficiency in handling transformations of trigonometric functions.     “TS Inter Trigonometric Ratios up to Transformations – 4-Mark Questions” sounds like a resource or a section within a textbook or study material designed for students studying trigonometry in the Telangana State Intermediate education system. This section likely contains questions that are worth four marks each, focusing on trigonometric ratios and their applications, including transformations. In trigonometry, understanding trigonometric ratios like sine, cosine, and tangent, and how they relate to angles in a right triangle or on the unit circle, is fundamental. Transformations in trigonometry may include changes in amplitude, frequency, phase shift, and vertical or horizontal translations of trigonometric functions.   Visit my Youtube Channel: Click on Below Logo

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how to prepare ts inter maths examination

How to Prepare well for the TS Inter Maths Examination 1

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How to Prepare well for the TS Inter Maths Examination How to Prepare: Preparing for the TS Inter WellMaths Examination requires a combination of effective study strategies and time management. Here are some tips to help you: Understand the Syllabus: Familiarize yourself with the entire syllabus. Identify the topics that carry more weight and focus on them first. Create a Study Schedule: Plan your study sessions. Allocate dedicated time for each topic, ensuring you cover the entire syllabus before the exam. Practice Regularly: Mathematics requires consistent practice. Solve a variety of problems, including those from previous years’ question papers. This helps you understand the exam pattern and boosts your confidence. Conceptual Clarity: Ensure a strong understanding of the fundamental concepts. If you encounter difficulties, seek help from your teachers, classmates, or online resources. Make Notes: Prepare concise notes for each chapter. These notes can serve as a quick revision tool before the exam. Use Reference Books:   Refer to additional study materials and reference books to gain a deeper insight into complex topics. Choose books that align with your syllabus. Mock Tests: Take mock tests regularly to simulate exam conditions. This helps improve your time management skills and identifies areas that need further attention. Clarify Doubts: Don’t hesitate to clarify doubts with your teachers or classmates. Understanding every concept thoroughly is crucial for success in mathematics. Healthy Lifestyle: Ensure a balance between study and relaxation. Get adequate sleep, maintain a healthy diet, and take short breaks during study sessions to stay focused. Revision: Regularly revise what you’ve studied. Focus on the topics where you feel less confident. Repetition helps reinforce the concepts in your memory. Stay Positive: Maintain a positive mindset. Believe in your abilities and stay confident. Avoid last-minute stress, and trust the efforts you’ve put into your preparation. Remember, consistent and organized preparation is key. Good luck with your TS Inter Maths Examination! Group Study:    How to Prepare Collaborate with classmates for group study sessions. Discussing concepts with others can provide different perspectives and deepen your understanding. Use Technology: Leverage educational apps, online resources, and interactive tutorials to supplement your learning. Many platforms offer practice quizzes and video lessons. Stay Organized: Keep your study materials and notes organized. A well-structured study environment can help you focus better and save time when revisiting topics. Time Management: Practice time management during your study sessions and exams. Allocate specific time limits for each question while solving practice papers to improve efficiency. Focus on Weak Areas: Identify your weaker areas and allocate more time to them. It’s essential to address your weaknesses rather than avoiding them. Stay Updated with Changes: Keep yourself informed about any changes in the exam pattern or syllabus. This ensures that your preparation aligns with the current requirements. Stay Healthy Physically and Mentally: Physical and mental well-being are crucial during exam preparation. Ensure you are getting enough exercise, fresh air, and relaxation to keep stress levels in check. How to Prepare —Teach Others:   Teaching a concept to someone else is a powerful way to reinforce your understanding. It helps solidify your knowledge and identify areas where you may need further clarification. Stay Consistent: Consistency is key in mathematics. Regular, small study sessions are often more effective than cramming. Stick to your schedule and avoid last-minute rushes. Reward Yourself: Celebrate small victories and milestones during your preparation. Rewarding yourself can boost motivation and make the studying process more enjoyable. Previous Years’ Papers: Solve previous years’ question papers to get a sense of the exam pattern and the types of questions asked. This also helps you practice time management. Visual Aids: Use visual aids such as charts, diagrams, and graphs to understand and remember complex concepts. Visual representation can make abstract ideas more tangible. How to Prepare—Stay Positive in Exam Hall:   On the day of the exam, stay calm and positive. Take deep breaths if you feel anxious, and focus on the questions one at a time. Remember, everyone has their unique way of studying, so feel free to adapt these tips to suit your preferences and learning style. Good luck! How to Prepare Visit my YouTube channel: Click on the logo.

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ts inter multiplication of vectors 4 marks important questions

TS inter || Multiplication of Vectors 4m important questions

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TS inter || Multiplication of Vectors 4m important questions Multiplication of Vectors Multiplication of vectors can take different forms depending on the context and the type of multiplication being used. Here are the main types: Scalar Multiplication: In scalar multiplication, a vector is multiplied by a scalar (a single number). Each component of the vector is multiplied by the scalar. For example, if you have a vector v = (x, y, z) and multiply it by a scalar k, you get kv = (kx, ky, kz). Here are some important questions related to the multiplication of Vectors that could be worth 4 marks each. Keep in mind that the specific marking scheme may vary based on the curriculum and exam format.   These questions cover various aspects of the multiplication of vectors, including operations, properties, and applications: Maths IA Two-Mark Questions & Solutions  Maths IB Two Marks Questions & Solutions     Dot Product (Scalar Product): The dot product of two vectors produces a scalar. It is calculated by multiplying the corresponding components of the vectors and summing the results. For two vectors a and b, the dot product is denoted by a · b. The formula is: a · b = a₁b₁ + a₂b₂ + … + aₙbₙ. Geometrically, it represents the projection of one vector onto another. Cross Product (Vector Product): The cross product of two vectors results in another vector that is perpendicular to the plane containing the original vectors. It is denoted by a × b. The formula depends on the dimensionality of the vectors:     For 3-dimensional vectors, the formula is a × b = (a₂b₃ – a₃b₂, a₃b₁ – a₁b₃, a₁b₂ – a₂b₁). The result is a vector perpendicular to both a and b, with a magnitude equal to the area of the parallelogram formed by a and b. These are the fundamental types of vector multiplication used in mathematics and physics. Each type has its properties and applications in various fields. Maths – IA Concept Maths – IB Concept     Maths – IIA Concept Maths – IIB Concept Visit my YouTube channel: Click on the Logo

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ts inter addition of vectors 4 marks important questions 2024

TS Inter || Addition of Vectors 4 Marks Important Questions

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TS Inter || Addition of Vectors 4 Marks Important Questions Addition of Vectors Vector addition is a fundamental operation in mathematics and physics, especially in the study of forces, velocities, and displacements. When you add vectors, you’re essentially combining their magnitudes and directions to find the resultant vector. Here’s how vector addition works:   Magnitude Addition: To add the magnitudes of vectors, simply add their numerical values together. For example, if you have two vectors, A and  B with magnitudes 3 and 4, respectively, their magnitudes add up to 7. Direction Addition: Vectors have both magnitude and direction. To add vectors, you must also consider their directions. You can represent vectors graphically using arrows, with the length of the arrow representing the magnitude and the direction of the arrow representing the direction of the vector.   Here are some important questions related to the addition of vectors that could be worth 4 marks each. Keep in mind that the specific marking scheme may vary based on the curriculum and exam format.   These questions cover various aspects of the addition of vectors, including operations, properties, and applications: Addition of vectors 4 marks important questions         Maths – IA Concept Maths – IB Concept   Resultant Vector: The resultant vector is the sum of the individual vectors. To find the resultant vector, you can use methods like the parallelogram method, triangle method, or component method (using vector components). Parallelogram Method: This method involves constructing a parallelogram using the vectors to be added as adjacent sides. The diagonal drawn from the common point of the vectors represents the resultant vector. Triangle Method: If you have only two vectors, you can use the triangle method. Place the tail of the second vector at the head of the first vector, and draw a vector from the tail of the first vector to the head of the second vector. The resultant vector is the vector from the tail of the first vector to the head of the second vector.     Component Method: You can break down vectors into their horizontal and vertical components. Then add the horizontal components separately and the vertical components separately. Finally, combine the horizontal and vertical components of the resultant vector to get the resultant vector. When adding vectors, it’s essential to maintain the correct signs (positive or negative) and directions. The resultant vector represents the net effect of all the vectors being added together.     Maths – IIA Concept Maths – IIB Concept Visit my YouTube channel: Click on the logo.

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ts inter matrices 4 marks important questions 2024

TS Inter || Matrices 4 Marks Important Questions 2026

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TS Inter || Matrices 4 Marks Important Questions 2026 Matrices   Here are some important questions related to matrices that could be worth 4 marks each. Keep in mind that the specific marking scheme may vary based on the curriculum and exam format. These questions cover various aspects of matrices, including operations, properties, and applications:                     Visit my YouTube channel: Click on the Logo

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Mathematical Indunction

Ch. 2 Mathematical Indunction Exerxise Solutions

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Mathematical Indunction (M.I) Exerxise wise Solutions Mathematical Indunction: It is a technique for proving results or establishing statements for natural numbers. This part illustrates the method through a variety of examples. Definition: Mathematical Induction  is a mathematical technique which is used to prove a statement, a formula or a theorem is true for every natural number. The technique involves steps to prove a statement, as stated below − Let P(n) or S(n) be the given statement Step 1: For n =1                  we get LHS = RHS                  then P(n) is true for n= 1 Step 2: Let us assume thet P(n) is true for n =k Step 3: We have to Prove P(n) is true for n= k + 1 Laplace: Laplace was a mathematecian and astronomer whose work was pivotal to the development of mathematical astronomy. His most outstanding work was done in the fields of celestial mechonics, probability, differential equations, and geodesy. His five volume work on celestial mechonics earned him the title of the Newton of France. “Analysis and natural philosophy owe their most important discoveries to this fruitful means, which is called indunction” – Pierr Simon de Laplace Exercise 2(a) Using Mathematical Indunction, Prove each of the following statement for all n ∈ N. 1. 12 + 22 + 32 + …… + n2= Let p(n) be the given statement that  12 + 22 + 32 + …… + n2= For n= 1 LHS = 12 = 1 RHS = =  = 1 LHS = RHS P(n) is true for n = 1 Let us assume that P(n) is true for n = k i.e., 12 + 22 + 32 + …… + k2=  ………… (1) for n = k + 1 add (k +1)2 on both sides of (1) 12 + 22 + 32 + …… + k2 + (k +1)2 =                                                                                                                                 P(n) is true for n = k+ 1 ∴ By the principle of  M.I.  P(n) is true for all n ∈ N ∴ 12 + 22 + 32 + …… + n2= 2.  2.3 + 3.4 + 4.5 + ……… up to n terms = First factors of given series are: 2, 3, 4, 5, …                                   a = 2, d = 1                                  an = a + (n – 1) d                                        = 2 + (n – 1) (1)                                        = 2 + n – 1                                        = n + 1 Second factors of given series are: 3, 4, 5,…                                   a = 3, d = 1                                  an = 3 + (n – 1) d                                        = 3 + (n – 1) (1)                                        = 3 + n – 1                                        = n + 2 nth term of given series is (n + 1) (n + 2) let P(n) be the given statement that 2.3 + 3.4 + 4.5 + ……… + (n + 1) (n + 2) = For n = 1 LHS = 2.3 = 6 RHS =  =  =  = 6 LHS = RHS P(n) is true for n = 1 Let us assume that P(n) is true for n = k i.e., 2.3 + 3.4 + 4.5 + ……… + (k + 1) (k + 2) =  ………… (1) for n = k + 1   add (k + 2) (k + 3) on both sides of (1)  2.3 + 3.4 + 4.5 + ……… + (k + 1) (k + 2) + (k + 2) (k + 3)     =  + (k + 2) (k + 3) P(n) is true for n = k+ 1 ∴ By the principle of  M.I.     P(n) is true for all n ∈ N ∴ 2.3 + 3.4 + 4.5 + ……… up to n terms = 3 .  Sol: let P(n) be the given statement that        For n = 1         LHS =  =         RHS =  =  =        LHS = RHS        P (n) is true for n = 1 Let us assume that P(n) is true for n = k   ………… (1) For n = k + 1 Add on both sides of (1)                                                                                                                                 P (n) is true for n = k + 1 ∴ By the principle of M.I.   P(n) is true for all n ∈ N ∴      4. 43 + 83 + 123 + … up to n terms = 16 n2 (n + 1)2 Sol: let P(n) be the given statement that  4, 8, 12, … are in AP a = 4, d = 4 an = a + (n – 1) d      = 4 + (n – 1)4      = 4 + 4n – 4       = 4n nth term of given series is (4n)3 let P(n) be the given statement that 43 + 83 + 123 + … + (4n)3= 16 n2 (n + 1)2        For n = 1         LHS = 43 = 63         RHS = 16 (1)2 (1 + 1)2 = 16 × 4 = 64        LHS = RHS        P (n) is true for n = 1       Let us assume that P(n) is true for n = k 43 + 83 + 123 + … +

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Functions Exercise 1c

Functions Exercise 1c Solutions ||TS|| Basics In MAths

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Functions Exercise 1c Solutions  Functions Exercise 1c The famous mathematician ” Lejeune Dirichlet”  defined a function. Function: A variable is a symbol which represents any one of a set of numbers, if two variables x and y so related that whenever a value is assigned to x there is autometically assigned by some rule or correspondence a value to y, then we say y is a function of x. Chapter 1 Functions Exercise 1c Solutions for inter first year  students, prepared by Mathematics expert of www.basicsinmaths.com Functions Exercise 1c   I. 1.Find the domains of the following real valued functions   (i) f (x) = given function is f (x) =  f (x) is defined when (x2 – 1) (x + 3) ≠ 0   ⟹ (x2 – 1) ≠ 0 or (x + 3) ≠ 0   ⟹ (x + 1) (x – 1) ≠ 0 or (x + 3) ≠ 0   ⟹ x ≠ 1, x ≠ – 1 or x ≠ – 3 ∴ Domain of f(x) is R – {– 1, – 3, 1} (ii) f (x) =  Given function is f (x) =  f (x) is defined when (x – 1) (x – 2) (x – 3) ≠ 0       ⟹ x ≠ 1, x ≠ 2 or x ≠ 3 ∴ Domain of f(x) is R – {1, 2, 3} (iii) f (x) = Given function is f (x) = f (x) is defined when 2 – x > 0 and 2 – x ≠ 1                           ⟹ 2 > x  and  2 – 1 ≠ x                         ⟹ 2 > x  and  x ≠ 1                         ∴ Domain of f(x) is (– ∞, 2) – {1} (iv) f (x) = Given function is f (x) = f (x) is defined when x ∈ R       ∴ Domain of f(x) is R Functions Exercise 1c (v) f (x) = Given function is f (x) = f (x) is defined when 4x – x2 ≥ 0       ⟹ x (4 – x) ≥ 0 ⟹ x (x – 4) ≤ 0 ⟹ (x – 0) (x – 4) ≤ 0 ⟹ x ∈ [0, 4] ∴ Domain of f(x) is [0, 4] (vi)  f (x) =  Given function is f (x) =  f (x) is defined when 1 – x2 > 0     ⟹ x2 – 1 < 0 ⟹ (x – 1) (x + 1) < 0 ⟹ x ∈ (– 1, 1) ∴ Domain of f(x) is (– 1, 1) (vii) f (x) =  Given function is f (x) =   f (x) is defined when x + 1≠ 0      ⟹ x ≠ – 1     ∴ Domain of f(x) is R – {– 1} (viii) f(x) =   Given function is f (x) =   f (x) is defined when x2 – 25 ≥ 0      ⟹ (x – 5) (x + 5) ≥ 0     ⟹ x ∈ (–∞, –5] ∪ [5, ∞)     ⟹ x ∈ R – (– 5, 5)  ∴ Domain of f(x) is R – (– 5, 5) Functions Exercise 1c (ix) f(x) = Given function is f (x) = f (x) is defined when x – [x] ≥ 0      ⟹ x ≥ [x]  ⟹ x ∈ R    ∴ Domain of f(x) is R   (x) f(x) = Given function is f (x) = f (x) is defined when [x] – x ≥ 0      ⟹ [x] ≥ x  ⟹ x ∈ Z    ∴ Domain of f(x) is Z Functions Exercise 1c 2. find the ranges of the following real valued functions   (i) f(x) = Given function is f (x) = Let y =   ⟹  |4 – x2| = ey ∵ ey > 0 ∀ y ∈ R ∴ Range of f(x) is R (ii)  f(x) = Given function is f (x) = f (x) is defined when [x] – x ≥ 0      ⟹ [x] ≥ x  ⟹ x ∈ Z    Domain of f(x) is Z Range of f = {0} (iii)  f(x) = Given function is f (x) = f (x) is defined when x ∈ R                 Domain of f(x) is R            For x ∈ R   [x] is an integer            Since sin nπ = 0, ∀ n ∈ z             ⟹ sin π[x] = 0           ∴ Range of f = {0} (iv)  f (x) =            Given function is f (x) =           f (x) is defined when x – 2 ≠ 0           ⟹ x ≠ 2          Domain of f(x) is R – {2}          Let y =             =              = x + 2        If x = 2 ⟹ y = 2 + 2 = 4         ∴ Range of f(x) is R – {4} Functions Exercise 1c (v)  f (x) = let y =       y2 = 9 + x2           x2 = y2 – 9        x =      it is defined when y2 – 9 ≥ 0       ⟹ (y – 3) (y + 3) ≥ 0     y ∈ (– ∞, – 3] ∪ [3, ∞) but y = ≥ 0 ∴ Range of f(x) is [3, ∞) 3. If f and g are real valued functions f(x) = 2x – 1 ang g (x) = x2 then        find Sol:     Given f and g are real valued functions f(x) = 2x – 1 ang g (x) = x2 (i) (3f – 2g) (x) = 3 f(x) – 2g (x)                          = 3 (2x – 1) – 2(x2)                          = 6x – 3 – 2×2                           = – 2×2 + 6x – 3                    ∴ (3f – 2g) (x) =– 2×2 + 6x – 3 (ii) (fg) (x) = f (x) g (x)          = (2x – 1) (x2)                 = 2×3 + x2     ∴ (fg) (x) = 2×3 + x2   (iii)     4. If f = {(1, 2), (2, – 3) (3, – 1)} then find (i) 2f (ii) (fog)

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