## Grade 12 Mathematics Syllabus

Complete **Syllabus** of **NEB Class 12 Mathematics** for NEB Exam 2080.

Subject | 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.

**3. Grade-wise Learning Outcomes**

Content Domain/area | Learning Outcomes |
---|---|

1. Algebra | 1.1 Solve the problems related to basic principle of counting. |

1.2 Solve the problems related to permutation and combinations. | |

1.3 State and prove binomial theorems for positive integral index. | |

1.4 State binomial theorem for any integer. | |

1.5 Find the general term and binomial coefficient. | |

1.6 Use binomial theorem in application to approximation. | |

1.7 Define Euler's number. | |

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

1.9 Express complex number in polar form. | |

1.10 State and prove De Moivre's theorem. | |

1.11 Find the roots of a complex number by De Moivre's theorem. | |

1.12 Solve the problems using properties of cube roots of unity. | |

1.13 Apply Euler's formula. | |

1.14 Find the sum of finite natural numbers, sum of squares of first n-natural numbers, sum of cubes of first n-natural numbers. | |

1.15 Find the sum of finite natural numbers, calculate sum of squares of first n-natural numbers, sum of cubes of first n-natural numbers by using mathematical induction. | |

1.16 Solve system of linear equations by Cramer's rule and matrix methods (row-equivalent and inverse) up to three variables. | |

2. Trigonometry | 2.1 Solve the problems using properties of a triangle (sine law, cosine law, tangent law, projection laws, half angle laws) |

2.2 Solve the triangle (simple cases) | |

3. Analytic geometry | 3.1 Solve the problems related to condition of tangency of a line at a point to the circle. |

3.2 Find the equations of tangent and normal to a circle at given point. | |

3.3 Find the standard equation of parabola. | |

3.4 Find the equations of tangent and normal to a parabola at given point. | |

3.5 Obtain standard equation of ellipse and hyperbola. | |

4. Vectors | 4.1 Find scalar product of two vectors, angle between two vectors and interpret scalar product geometrically. |

4.2 Solve the problems using properties of scalar product. | |

4.3 Apply properties of scalar product of vectors in trigonometry and geometry. | |

4.4 Define vector product of two vectors, and interpret vector product geometrically. | |

4.5 Solve the problems using properties of vector product. | |

4.6 Apply vector product in geometry and trigonometry. | |

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. | |

6. Calculus | 6.1 Differentiate the hyperbolic function and inverse hyperbolic function |

6.2 Evaluate the limits by L'hospital's rule (for 0/0, ∞/∞). | |

6.3 Find the tangent and normal by using derivatives. | |

6.4 Find the derivative as rate of measure | |

6.5 Find the anti-derivatives of standard integrals, integrals reducible to standard forms. | |

6.6 Solve the differential equation of first order and first degree by separable variables, homogeneous, linear and exact differential equation. | |

7. Computational methods | 7.1 Solve the system of linear equations by Gauss Elimination method, Gauss Seidel Method (up to 3 variables) |

7.2 Solve the linear programming problems (LPP) by simplex method | |

Or Mechanics | 7.1 Solve the forces/vectors related problems using triangle laws of forces and Lami's theorem. |

7.2 Solve the problems related to Newton's laws of motion and projectile. |

**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- Seidel 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.

**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 | 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.