National Policy on Education (1986) Quiz
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Questions and Answers

What transformation is used to simplify the integral of the form $\frac{(px + q) , dx}{ax^2 + bx + c}$?

  • Transforming it into known standard forms (correct)
  • Using integration by parts
  • Decomposing it into partial fractions
  • Substituting $x = t^2$

In the first example provided, what is the final equivalent expression for $\int (x^2 - 16) , dx$?

  • $\frac{1}{8} \log(x^2 - 16) + C$
  • $\frac{1}{8} \log(x + 4)^2 + C$
  • $\frac{1}{8} \log(x + 4) + C$ (correct)
  • $\frac{1}{8} \log(x - 4) + C$

What substitution is made for the integral of the form $\int (2x - x^2) , dx$?

  • $x = t^2 + 1$
  • $x = t + 1$
  • $x - 1 = t$ (correct)
  • $x + 1 = t$

Why is it necessary to express integrals in terms of known forms?

<p>It makes integrals standard and easier to solve. (C)</p> Signup and view all the answers

What is the primary method used to evaluate the integral $\int (x^2 - 16) , dx$?

<p>Applying logarithmic properties (A)</p> Signup and view all the answers

Who served as the Chief Coordinator for the Textbook Development Committee?

<p>Hukum Singh (A)</p> Signup and view all the answers

Which role did J.V.Narlikar hold in the Textbook Development Committee?

<p>Chairperson, Advisory Group in Science and Mathematics (D)</p> Signup and view all the answers

Which member is associated with the Indian Institute of Science in Bangalore?

<p>C.R.Pradeep (D)</p> Signup and view all the answers

What is the primary focus of the contributions acknowledged in the document?

<p>Academic success of a project (B)</p> Signup and view all the answers

Who is the Chief Advisor mentioned in the committee?

<p>P.K.Jain (C)</p> Signup and view all the answers

Which organization is associated with Hukum Singh?

<p>NCERT, New Delhi (D)</p> Signup and view all the answers

Which member's role is specified as Reader in DESM, NCERT, New Delhi?

<p>R.P.Maurya (A)</p> Signup and view all the answers

Who contributed to the Textbook Review Workshop from the Department of Statistics?

<p>Jagdish Saran (D)</p> Signup and view all the answers

What is one key reason for ignoring other resources and sites of learning?

<p>Strict adherence to the prescribed textbook for examinations (A)</p> Signup and view all the answers

What aspect is highlighted as essential for fostering creativity in children?

<p>Perceiving children as participants in learning (C)</p> Signup and view all the answers

Which of the following is emphasized as necessary besides rigor in the annual calendar?

<p>Flexibility in the daily timetable (B)</p> Signup and view all the answers

How have syllabus designers attempted to address curricular burden?

<p>By restructuring and reorienting knowledge with consideration for child psychology (A)</p> Signup and view all the answers

What approach does the textbook encourage for enhancing the learning experience?

<p>Emphasizing imagination and group discussions (A)</p> Signup and view all the answers

What is recognized as a problem contributing to stress or boredom in schools?

<p>Methods of teaching and evaluation (B)</p> Signup and view all the answers

Who is acknowledged for guiding the development of the textbook?

<p>Professor J.V. Narlikar and Professor P.K. Jain (A)</p> Signup and view all the answers

What is essential for children to generate new knowledge according to the document?

<p>Freedom, space, and time to engage with information (A)</p> Signup and view all the answers

What is an anti-derivative of cos 2x?

<p>sin 2x (B)</p> Signup and view all the answers

What is the anti-derivative of the function 3x^2 + 4x^3?

<p>x^3 + x^4 (A)</p> Signup and view all the answers

If f(x) = log x, what is the derivative f'(x)?

<p>1/x (D)</p> Signup and view all the answers

Which of the following integrals corresponds to ∫ (x^3 - 1) dx?

<p>x^4/4 - x^2/2 + C (D)</p> Signup and view all the answers

What is the result of integrating 2/x with respect to x?

<p>2log(x) + C (B)</p> Signup and view all the answers

What is the integral of x^2 from 0 to 1?

<p>1/3 (D)</p> Signup and view all the answers

Using the property of integrals, what is ∫(x^2 - x) dx?

<p>x^3/3 - x^2/2 + C (B)</p> Signup and view all the answers

What does the derivative of log(-x) yield for x < 0?

<p>-1/x (B)</p> Signup and view all the answers

What is the final result of the integral $\int (2x^2 + 6x + 5) , dx$?

<p>$4 \log(2x) + 6x + 5 + \tan^{-1}(2x + 3) + C$ (B)</p> Signup and view all the answers

What values are derived for coefficients A and B from the equation $x+3= A(5-4x-x^2) + B$?

<p>$A = -\frac{1}{2}, B = 1$ (D)</p> Signup and view all the answers

What substitution is made in the integral $\int (5-4x-x^2) , dx$ to simplify the expression?

<p>$x + 2 = t$ (C)</p> Signup and view all the answers

What expression is obtained after substituting in the integral $I_1$ where $5 - 4x - x^2 = t$?

<p>$2t + C_1$ (A)</p> Signup and view all the answers

What is the integral representation for $I$ in the equation $I = \int \frac{x+3}{5 - 4x - x^2} , dx$?

<p>$I = -\frac{1}{2} I + I$ (C)</p> Signup and view all the answers

What is the main form of the integral discussed in the content?

<p>$\int \frac{f'(x)}{g(x)} , dx$ (B)</p> Signup and view all the answers

What is the importance of equating the coefficient of x and the constant term in the integral solving process?

<p>To determine the necessary values for coefficients (B)</p> Signup and view all the answers

Which integral evaluates to the expression $2(5 - 4x - x^2) + C_1$?

<p>$I_1$ from $\int (-4 - 2x) , dx$ (C)</p> Signup and view all the answers

What is the result of the integral ∫ x10 + 10x dx?

<p>10x – x10 + C (C)</p> Signup and view all the answers

What does the integral ∫ sin 2x cos2 x equal?

<p>tan x – cot 2x + C (A)</p> Signup and view all the answers

Using the identity cos 2x = 2cos^2 x – 1, how is ∫ cos x dx simplified?

<p>x + sin x + C (C)</p> Signup and view all the answers

What is the outcome of using the identity sin x cos y = 1/2[sin (x + y) + sin (x – y)] in integration?

<p>1/2 * (cos 5x + cos x) + C (D)</p> Signup and view all the answers

What does the identity sin 3x = 3sin x - 4sin^3 x help to find in integration?

<p>An expression for sin 3x in terms of sin x (D)</p> Signup and view all the answers

How is the integral ∫ sin^3 x dx simplified?

<p>By substituting cos x with t (A)</p> Signup and view all the answers

What is the correct integration result for ∫ (1 – cos^2 x) sin x dx?

<p>-cos x + cos^3 x + C (B)</p> Signup and view all the answers

What role do trigonometric identities play in evaluating integrals?

<p>They help rewrite the integrands into simpler forms. (B)</p> Signup and view all the answers

What is the approach used to integrate ∫ sin 2x cos 3x dx?

<p>Applying trigonometric identities (A)</p> Signup and view all the answers

Which of the following represents a correct rearrangement for cos^2 x?

<p>(1 + cos 2x)/2 (A)</p> Signup and view all the answers

Flashcards

Child-Centred Learning

A teaching approach that emphasizes the student's active role in learning, allowing them to explore, question, and discover knowledge independently.

Imaginative Activities

A learning environment that encourages exploration and creativity, allowing students to go beyond prescribed content and explore their own ideas.

Hands-On Experience

The act of integrating learning with real-world applications and experiences.

Inculcating Creativity

The ability to think critically and creatively, generating new insights and solutions.

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Child Psychology

The process of understanding a student's individual learning needs and adapting teaching strategies to cater to them.

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Participants in Learning

The practice of considering a student as an active participant in the learning process, not just a passive recipient of information.

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Flexibility in Timetable

Adjusting the teaching schedule to allow for flexibility in how and when learning occurs.

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Effective Evaluation

The methods used to assess student understanding and progress beyond traditional tests and exams.

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Textbook Development Committee

A group of individuals responsible for the development of a textbook.

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Chief Advisor

The person who provides overall guidance and supervision for the textbook project.

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Chief Coordinator

The person who coordinates the day-to-day activities of the textbook development committee.

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Coordinator

The person who helps coordinate the work of the textbook development committee.

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Emeritus Professor

A person who has a lot of experience and knowledge in a particular subject.

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Professor

A person who teaches at a university.

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Associated

A person who has been involved in a project or task for a long time.

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Teacher

A person who is responsible for teaching in a school.

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Anti-derivative

The process of finding a function whose derivative is a given function.

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Anti-derivative of a function

A function whose derivative is the integrand.

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Indefinite integral

A general form of all anti-derivatives of a function, where 'C' represents an arbitrary constant.

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Integration

Finding the anti-derivative of a function.

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Definite integral

The result of integrating a function from a lower limit to an upper limit.

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Property V of Integrals

A property of integrals allowing us to split a function into separate terms, and integrate each term individually.

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Integration by substitution

The ability to integrate functions by rewriting them in different forms.

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Constant of Integration

A method used to evaluate definite integrals by substituting 'C' with constants of integration.

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Integration of (px + q) / (ax² + bx + c)

A method of integrating a function containing a linear expression in the numerator and a quadratic expression in the denominator.

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Integration by Standard Forms

A general technique used to simplify integrals by transforming them into standard forms.

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Evaluating Integrals

The process of finding the integral of a function, often involving techniques like substitution, integration by parts, and partial fractions.

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Transforming Integrals into Standard Forms

A specific strategy used to evaluate integrals of the type (px + q) / (ax² + bx + c) by transforming the integrand into standard integral forms.

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Known Standard Forms

Integrals that can be readily calculated without using complex integration techniques.

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Integral of cos^2(x)

The integral of cos^2(x) can be found using the identity: cos(2x) = 2cos^2(x) - 1. Rearranging this identity, we get cos^2(x) = (1 + cos(2x))/2. We can then integrate this expression.

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Integral of sin(2x)cos(3x)

The integral of sin(2x)cos(3x) can be found using the product-to-sum identity: sin(x)cos(y) = (1/2)[sin(x + y) + sin(x - y)]

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Integral of sin^3(x)

The integral of sin^3(x) can be found using the identity: sin(3x) = 3sin(x) - 4sin^3(x). Rearranging this identity, we get sin^3(x) = (3sin(x) - sin(3x))/4

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Integrating sin^3(x) using substitution

Substituting u = cos(x) and du = -sin(x)dx, we can express the integral of sin^3(x) in terms of u. Then, we can integrate the resulting polynomial in u and substitute back u = cos(x) to get the final result.

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Integrating trigonometric functions using identities and substitution

The integral of a trigonometric function involving a product of sine and cosine can be evaluated using a combination of trigonometric identities, substitution, and integration by parts.

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Integration by parts

Integration by parts is a technique for integrating the product of two functions using the formula: ∫u dv = uv - ∫v du.

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Trigonometric identities

Trigonometric identities are equations that are true for all values of the variables involved. They can be used to simplify trigonometric expressions and functions.

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Substitution

Substitution is a technique for evaluating integrals by replacing a variable with a new expression. This can simplify the integral and make it easier to evaluate.

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Study Notes

National Policy on Education (1986) and Child-Centred Education

  • A child-centered education system is envisioned, building on the National Policy on Education (1986).
  • Success depends on educators encouraging reflection on learning, promoting creative activities, and fostering imaginative questions.
  • Children generate new knowledge through interaction with information from adults, given space, time, and freedom.
  • Reliance on textbooks alone discourages use of diverse learning resources.

Textbook Usage and Learning

  • Treating children as participants, not passive receivers of knowledge, is crucial for fostering creativity and initiative.
  • School routines and functioning must adapt significantly.
  • Timetable flexibility and rigorous annual calendars are vital for dedicated teaching time.
  • Teaching and evaluation methods directly affect the effectiveness of the textbook in creating a positive learning experience for students, avoiding stress or boredom.
  • Syllabus designers are restructuring and reorienting knowledge through the lens of child psychology and available teaching time.

Textbook Development and Acknowledgements

  • The textbook prioritizes opportunities for reflection, discussion, and hands-on activities.
  • The National Council of Educational Research and Training (NCERT) acknowledges the development committee's hard work.
  • Specific thanks to advisory group chair Professor J.V. Narlikar and Chief Advisor Professor P.K. Jain.
  • Gratitude for contributions from various teachers, principals, institutions, and organizations.

Textbook Committee Members (Partial List)

  • J.V. Narlikar, Emeritus Professor, Inter-University Centre for Astronomy and Astrophysics (IUCAA)
  • P.K. Jain, Professor, Department of Mathematics, University of Delhi
  • Hukum Singh, Professor and Head, DESM, NCERT
  • Other names and affiliations of committee members listed.

Additional Information (Integration concepts)

  • Examples and explanations of integration techniques involving trigonometric functions, algebraic equations and other math concepts are presented. Various examples and solved problems illustrate how to evaluate integrals are presented, including details on substitution, and applying trigonometric identities to solving integrals.

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Description

Test your knowledge on the National Policy on Education (1986) and its emphasis on child-centered education. This quiz explores concepts such as creative learning, teacher roles, and the importance of diverse resources in education. Assess your understanding of how these elements contribute to a positive educational experience.

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