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Mathematics | NEB Syllabus and Model Question | Grade 12

Published on: August 12, 2021

Mathematics Syllabus and Model Question | NEB Syllabus Science 12 includes Physics, Chemistry, Biology, Mathematics, Nepali, English, Social Studies

About NEB

National Examinations Board-NEB (formerly Higher Secondary Education Board-HSEB) is the only one education board of Nepal Government.

The Board has a separate seal for its own work implementation purpose.

The examination related tasks of Grade 10 (Secondary Education Examinations (SEE) and class 11 and 12 (School Leaving Certificate Examination (SLCE) have now been affiliated to the jurisdiction of NEB as integrated components.

The examinations of class 10 will be brought into operation in the regional/provincial level.

Compulsory Subjects, Grade 12

Nepali

English

Social Studies and Life Skills Education

Optional Subjects Group 1, Grade 12

Physics

Accounting

Rural Development

Nepalese Legal System

Health and Physical Education

Sports Science

Instructional Pedagogy and Evaluation

Psychology

History

Gender Studies

Hospitality Management

Agronomy

Naturopathy

Human Value Education

Sculpture

Optional Subjects Group 2, Grade 12

Biology

Education and Development

Geography

Procedural Law

Sociology

Ayurveda

Business Studies

Linguistics

Political Science

Philosophy

Population Studies

Horticulture (Fruits, Vegetable, Floriculture, and Mushroom farming)

Food and Nutrition

Dance

Computer Science

Optional Subjects Group 3, Grade 12

Chemistry

Economics

Tourism and Mountaineering Studies

Marketing

Gerontology and Caretaking Education

Yog

Vocal / Instrumental

Sweing and Knitting

Criminal Law and Justice

Culinary Arts

Culture

Fashion Designing

Film and Documentary

Animal Husbandry, Poultry, and Fisheries

Nepali Major

English Major

Maithali

Newari

Hindi

Chinese

German

Japanese

Korean

Urdu

French

Hebrew

Arabic

Sanskrit

Optional Subjects Group 4, Grade 12

Mathematics

Applied Mathematics

Business Mathematics

Human Rights

Library and Information Science

Home Science

Environment Science

General Law

Finance

Co-operative Management

Buddhist Studies

Applied Arts

Singing

Painting

Sericulture and Bee Keeping

Beautician and Hair Dressing

Medicinal Herbals

Plumbing and Wiring

Internal Decoration

Hotel Management

Mass Communication

Mathematics Syllabus, Grade 12

Mathematics

Grades: 12

Subject code: Mat. 402

Credit hours: 5

Working hours: 160

1. Introduction

Mathematics is an indispensable in many fields. It is essential in the field of engineering, medicine, natural sciences, finance and other social sciences. The branch of mathematics concerned with application of mathematical knowledge to other fields and inspires new mathematical discoveries. The new discoveries in mathematics led to the development of entirely new mathematical disciplines. School mathematics is necessary as the backbone for higher study in different disciplines. Mathematics curriculum at secondary level is the extension of mathematics curriculum offered in lower grades (1 to 10).

This course of Mathematics is designed for grade 11 and 12 students as an optional subject as per the curriculum structure prescribed by the National Curriculum Framework, 2075. This course will be delivered using both the conceptual and theoretical inputs through demonstration and presentation, discussion, and group works as well as practical and project works in the real world context. Calculation strategies and problem solving skills will be an integral part of the delivery.

This course includes different contents like; Algebra, Trigonometry, Analytic Geometry, Vectors, Statistics and Probability, Calculus, Computational Methods and Mechanics or Mathematics for Economics and Finance.

Student’s content knowledge in different sectors of mathematics with higher understanding is possible only with appropriate pedagogical skills of their teachers. So, classroom teaching must be based on student-centered approaches like project work, problem solving etc.

2. Level-wise Competencies

On completion of this course, students will have the following competencies:

1. apply numerical methods to solve algebraic equation and calculate definite integrals and use simplex method to solve linear programming problems (LPP).

2. use principles of elementary logic to find the validity of statement.

3. make connections and present the relationships between abstract algebraic structures with familiar number systems such as the integers and real numbers.

4. use basic properties of elementary functions and their inverse including linear, quadratic, reciprocal, polynomial, rational, absolute value, exponential, logarithm, sine, cosine and tangent functions.

5. identify and derive equations or graphs for lines, circles, parabolas, ellipses, and hyperbolas,

6. use relative motion, Newton’s laws of motion in solving related problems.

7. articulate personal values of statistics and probability in everyday life.

8. apply derivatives to determine the nature of the function and determine the maxima and minima of a function and normal increasing and decreasing function into context of daily life.

9. explain anti-derivatives as an inverse process of derivative and use them in various situations.

10. use vectors and mechanics in day to day life.

11. develop proficiency in application of mathematics in economics and finance.

3. Grade-wise Learning Outcomes

On completion of the course, the students will be able to:

1. Algebra

1.1 solve the problems related to permutation and combinations.

1.2 state and prove binomial theorems for positive integral index.

1.3 state binomial theorem for any index (without proof).

1.4 find the general term and binomial coefficient.

1.5 use binomial theorem in application to approximation.

1.6 define Euler’s number.

1.7 Expand ex, ax and log(1+x) using binomial theorem.

1.8 define binary operation and apply binary operation on sets of integers.

1.9 state properties of binary operations.

1.10 define group, finite group, infinite group and abelian group.

1.11 prove the uniqueness of identity, uniqueness of inverse, cancelation law.

1.12 state and prove De Moivre’s theorem.

1.13 find the roots of a complex number by De Moivre’s theorem.

1.14 solve the problems using properties of cube roots of unity.

1.15 apply Euler’s formula.

1.16 define polynomial function and polynomial equation.

1.17 state and apply fundamental theorem of algebra (without proof).

1.18 find roots of a quadratic equation.

1.19 establish the relation between roots and coefficient of quadratic equation.

1.20 form a quadratic equation with given roots.

1.21 sum of finite natural numbers, sum of squares of first n-natural numbers, sum of cubes of first n-natural numbers, intuition and induction, principle of mathematical induction.

1.22 using principle of mathematical induction, find the sum of finite natural numbers, sum of squares of first n-natural numbers, sum of cubes of first n-natural numbers.

1.23 solve system of linear equations by Cramer’s rule and matrix method (rowequivalent and inverse) up to three variables.

2. Trigonometry

2.1 define inverse circular functions. Establish the relations on inverse circular functions.

2.2 find the general solution of trigonometric equations

3. Analytic geometry

3.1 obtain standard equation of ellipse and hyperbola.

3.2 find direction ratios and direction cosines of a line.

3.3 find the general equation of a plane.

3.4 find equation of a plane in intercept and normal form.

3.5 find the equation of plane through three given points.

3.6 find the equation of geometric plane through the intersection of two given planes.

3.7 find angle between two geometric planes.

3.8 write the conditions of parallel and perpendicular planes.

3.9 find the distance of a point from a plane.

4. Vectors

4.1 define vector product of two vectors, interpretation vector product geometrically.

4.2 solve the problems using properties of vector product.

4.3 apply vector product in plane trigonometry and geometry.

5. Statistics and Probability

5.1 calculate correlation coefficient by Karl Pearson’s method.

5.2 calculate rank correlation coefficient by Spearman method.

5.3 interpret correlation coefficient.

5.4 obtain regression line of y on x and x on y.

5.5 solve the simple problems of probability using combinations.

5.6 solve the problems related to conditional probability.

5.7 use binomial distribution and calculate mean and standard deviation of binomial distribution.

6. Calculus

6.1 find the derivatives of inverse trigonometric, exponential and logarithmic functions by definition.

6.2 establish the relationship between continuity and differentiability.

6.3 differentiate the hyperbolic function and inverse hyperbolic function

6.4 evaluate the limits by L’hospital’s rule (for 0/0, ∞/∞).

6.5 find the tangent and normal by using derivatives.

6.6 interpret geometrically and verify Rolle’s theorem and Mean Value theorem.

6.7 find the anti-derivatives of standard integrals, integrals reducible to standard forms and rational function (using partial fractions also).

6.8 solve the differential equation of first order and first degree by separable variables, homogenous, linear and exact differential equation.

7. Computation al methods

7.1 solve algebraic polynomial and transcendental equations by Newton-Raphson methods.

7.2 solve the linear programming problems (LPP) by simplex method of two variables.

7.3 integrate numerically by trapezoidal and Simpson’s rules and estimate the errors.

8. Mechanics

8.1 find the resultant of like and unlike parallel forces/vectors.

8.2 solve the problems related to Newton’s laws of motion and projectile.

Mathematics for Economics and Finance

8.1 use quadratic functions in economics,

8.2 understand input- output analysis and dynamics of market price.

8.3 find difference equations.

8.4 work with Cobweb model and lagged Keynesian macroeconomic model.

8.5 explain mathematically equilibrium and break-even.

8.6 construct mathematical models involving consumer and producer surplus.

8.7 use quadratic functions in economics.

8.8 do input- output analysis.

8.9 analyze dynamics of market.

8.10construct difference equations,

8.11understand cobweb model, lagged Keynesian macroeconomics model.

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4. Scope and Sequence of Contents

1. Algebra LH32

1.1 Permutation and combination:

Basic principle of counting, Permutation of (a) set of objects all different (b) set of objects not all different (c) circular arrangement (d) repeated use of the same objects. Combination of things all different, Properties of combination

1.2 Binomial Theorem:

Binomial theorem for a positive integral index, general term. Binomial coefficient, Binomial theorem for any index (without proof), application to approximation. Euler’s number. Expansion of e^x, a^x and log (1+x) (without proof)

1.3 Elementary Group Theory

Binary operation, Binary operation on sets of integers and their properties, Definition of a group, Finite and infinite groups. Uniqueness of identity, Uniqueness of inverse, Cancelation law, Abelian group.

1.4 Complex numbers:

De Moivre’s theorem and its application in finding the roots of a complex number, properties of cube roots of unity. Euler’s formula.

1.5 Quadratic equation:

Nature and roots of a quadratic equation, Relation between roots and coefficient. Formation of a quadratic equation, Symmetric roots, one or both roots common.

1.6 Mathematical induction:

Sum of finite natural numbers, sum of squares of first n-natural numbers, Sum of cubes of first n- natural numbers, Intuition and induction, principle of mathematical induction.

1.7 Matrix based system of linear equation:

Consistency of system of linear equations, Solution of a system of linear equations by Cramer’s rule. Matrix method (row- equivalent and Inverse) up to three variables.

2. Trigonometry LH8

2.1 Inverse circular functions.

2.2 Trigonometric equations and general values

3. Analytic Geometry LH14

3.1 Conic section: Standard equations of Ellipse and hyperbola.

3.2 Coordinates in space: direction cosines and ratios of a line general equation of a plane, equation of a plane in intercept and normal form, plane through 3 given points, plane through the intersection of two given planes, parallel and perpendicular planes, angle between two planes, distance of a point from a plane.

4. Vectors LH8

4.1 Product of Vectors: vector product of two vectors, geometrical interpretation of vector product, properties of vector product, application of vector product in plane trigonometry.

4.2 Scalar triple Product: introduction of scalar triple product

5. Statistics & Probability LH10

5.1 Correlation and Regression: correlation, nature of correlation, correlation coefficient by Karl Pearson’s method, interpretation of correlation coefficient, properties of correlation coefficient (without proof), rank correlation by Spearman, regression equation, regression line of y on x and x on y.

5.2 Probability: Dependent cases, conditional probability (without proof), binomial distribution, mean and standard deviation of binomial distribution (without proof).

6. Calculus LH32

6.1 Derivatives: derivative of inverse trigonometric, exponential and logarithmic function by definition, relationship between continuity and differentiability, rules for differentiating hyperbolic function and inverse hyperbolic function, L’Hospital’s rule (0/0, ∞/∞), differentials, tangent and normal, geometrical interpretation and application of Rolle’s theorem and mean value theorem.

6.2 Anti-derivatives: antiderivatives, standard integrals, integrals reducible to standard forms, integrals of rational function.

6.3 Differential equations: differential equation and its order, degree, differential equations of first order and first degree, differential equations with separable variables, homogenous, linear and exact differential equations.

7. Computational Methods LH10

7.1 Computing Roots: Approximation & error in computation of roots in nonlinear equation, Algebraic and transcendental equations & their solution by bisection and Newton- Raphson Methods

7.2 System of linear equations: Gauss elimination method, Gauss- Seidal method, Ill conditioned systems.

7.3 Numerical integration Trapezoidal and Simpson’s rules, estimation of errors.

8. Mechanics or Mathematics for Economics and Finance LH12

8.1 Statics: Resultant of like and unlike parallel forces.

8.2 Dynamics: Newton’s laws of motion and projectile.

8.3 Mathematics for economics and finance: Consumer and Producer Surplus, Quadratic functions in Economics, Input-Output analysis, Dynamics of market price, Difference equations, The Cobweb model, Lagged Keynesian macroeconomic model.

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5. Practical and Project Activities

The students are required to do different practical activities in different content areas and the teachers should plan in the same way. Total of 34 working hours is allocated for practical and project activities in each of the grades 11 and 12.

The following table shows estimated working hours for practical activities in different content areas of grade 11 and 12

SN	Content area/domain	LH in each of the grades 11 and 12
1	Algebra	10
2	Trigonometry	2
3	Analytic geometry	4
4	Vectors	2
5	Statistics & Probability	2
6	Calculus	10
7	Computational methods	2
8	Mechanics or Mathematics for Economics and Finance	2
	Total	34

Here are some sample (examples) of practical and project activities.

Sample project works/mathematical activities for grade 12

1. Represent the binomial theorem of power 1, 2, and 3 separately by using concrete materials and generalize it with n dimension relating with Pascal’s triangle.

2. Take four sets R, Q, Z, N and the binary operations +, ‒, ×. Test which binary operation forms group or not with R, Q, Z, N.

3. Prepare a model to explore the principal value of the function sin–1x using a unit circle and present in the classroom.

4. Draw the graph of sin‒1x, using the graph of sin x and demonstrate the concept of mirror reflection (about the line y = x).

5. Fix a point on the middle of the ceiling of your classroom. Find the distance between that point and four corners of the floor. 6. Construct an ellipse using a rectangle.

7. Express the area of triangle and parallelogram in terms of vector.

8. Verify geometrically that: ?⃗ × (?⃗ + ?) = ?⃗ × ?⃗ + ?⃗ × ?⃗ ?

9. Collect the grades obtained by 10 students of grade 11 in their final examination of English and Mathematics. Find the correlation coefficient between the grades of two subjects and analyze the result.

10. Find two regression equations by taking two set of data from your textbook. Find the point where the two regression equations intersect. Analyze the result and prepare a report.

11. Find, how many peoples will be there after 5 years in your districts by using the concept of differentiation.

12. Verify that the integration is the reverse process of differentiation with examples and curves.

13. Correlate the trapezoidal rule and Simpson rule of numerical integration with suitable example.

14. Identify different applications of Newton’s law of motion and related cases in our daily life.

15. Construct and present Cobweb model and lagged Keynesian macroeconomic model.

6. Learning Facilitation Method and Process

Teacher has to emphasis on the active learning process and on the creative solution of the exercise included in the textbook rather than teacher centered method while teaching mathematics. Students need to be encouraged to use the skills and knowledge related to maths in their house, neighborhood, school and daily activities. Teacher has to analyze and diagnose the weakness of the students and create appropriate learning environment to solve mathematical problems in the process of teaching learning. The emphasis should be given to use diverse methods and techniques for learning facilitation. However, the focus should be given to those method and techniques that promote students’ active participation in the learning process. The following are some of the teaching methods that can be used to develop mathematical competencies of the students:

· Inductive and deductive method

· Problem solving method

· Case study

· Project work method

· Question answer and discussion method

· Discovery method/ use of ICT

· Co-operative learning

7. Student Assessment

Evaluation is an integral part of learning process. Both formative and summative evaluation system will be used to evaluate the learning of the students. Students should be evaluated to assess the learning achievements of the students. There are two basic purposes of evaluating students in Mathematics: first, to provide regular feedback to the students and bringing improvement in student learning-the formative purpose; and second, to identify student’s learning levels for decision making.

a. Internal Examination/Assessment

i. Project Work:

Each Student should do one project work from each of eight content areas and has to give a 15 minute presentation for each project work in classroom. These seven project works will be documented in a file and will be submitted at the time of external examination. Out of eight projects, any one should be presented at the time of external examination by each student.

ii. Mathematical activity:

Mathematical activities mean various activities in which students willingly and purposefully work on Mathematics. Mathematical activities can include various activities like (i) Hands-on activities (ii) Experimental activities (iii) physical activities. Each student should do one activity from each of eight content area (altogether seven activities). These activities will be documented in a file and will be submitted at the time of external examination. Out of eight activities, any one should be presented at the time of external examination by each student.

iii. Demonstration of competency in classroom activity:

During teaching learning process in classroom, students demonstrate 10 competencies through activities. The evaluation of students’ performance should be recorded by subject teacher on the following basis.

· Through mathematical activities and presentation of project works.

· Identifying basic and fundamental knowledge and skills.

· Fostering students’ ability to think and express with good perspectives and logically on matters of everyday life.

· Finding pleasure in mathematical activities and appreciate the value of mathematical approaches.

· Fostering and attitude to willingly make use of mathematics in their lives as well as in their learning.

iv. Marks from trimester examinations:

Marks from each trimester examination will be converted into full marks 3 and calculated total marks of two trimester in each grade.

The weightage for internal assessment are as follows:

Classroom participation	Project work /Mathematical activity (at least 10 work/activities from the above mentioned project work/mathematical activities should be evaluated)	Demonstration of competency in classroom activity	Marks from terminal exams	Total
3	10	6	6	25

b. External Examination/Evaluation

External evaluation of the students will be based on the written examination at the end of each grade. It carries 75 percent of the total weightage. The types and number questions will be as per the test specification chart developed by the Curriculum Development Centre.