Grade 12 Mathematics

Learning Outcomes

Permutation and combination
1. Solve the problems related to basic principle of counting.
2. Solve the problems related to permutation and combinations.

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Binomial Theorem:
3. State and prove binomial theorems for positive integral index.
4. State binomial theorem for any integer.
5. Find the general term and binomial coefficient.
6. Use binomial theorem in application to approximation.
7. Define Euler's number.
8. Expand \(e^x, a^x, \log(1+x) \) using binomial theorem.

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Complex numbers
9. Express complex number in polar form.
10. State and prove De Moivre's theorem.
11. Find the roots of a complex number by De Moivre's theorem.
12. Solve the problems using properties of cube roots of unity.
13. Apply Euler's formula.

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Sequence and series:
14. Find the sum of finite natural numbers, sum of squares of first n-natural numbers, sum of cubes of first n-natural numbers.
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.

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

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Trigonometry
17. Solve the problems using properties of a triangle (sine law, cosine law, tangent law, projection laws, half angle laws)
18. Solve the triangle (simple cases)

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Analytic geometry
19. Solve the problems related to condition of tangency of a line at a point to the circle.
20. Find the equations of tangent and normal to a circle at given point.
21. Find the standard equation of parabola
22. Find the equations of tangent and normal to a parabola at given point
23. Obtain standard equation of ellipse and hyperbola.

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Vectors
24. Find scalar product of two vectors, angle between two vectors and interpret scalar product geometrically.
25. Solve the problems using properties of scalar product
26. Apply properties of scalar product of vectors in trigonometry and geometry.
27. Define vector product of two vectors, and interpret vector product geometrically.
28. Solve the problems using roperties of vector product.
29. Apply vector product in geometry and trigonometry

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Statistics and Probability
30. Calculate correlation coefficient by Karl Pearson's method.
31. Calculate rank correlation coefficient by Spearman method.
32. Interpret correlation coefficient.
33. Obtain regression line of y on x and x on y
34. Solve the simple problems of probability using combinations
35. Solve the problems related to conditional probability.

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Calculus
36. Differentiate the hyperbolic function and inverse hyperbolic function
37. Evaluate the limits by L'hospital's rule (for \( \frac{0}{0}, \frac{\infty}{\infty}\))
38. Find the tangent and normal by using derivatives.
39. Find the derivative as rate of measure
40. Find the anti-derivatives of standard integrals, integrals reducible to standard forms .
41. Solve the differential equation of first order and first degree by separable variables, homogenous, linear and exact differential equation

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Computational methods
42. Solve the system of linear equations by Gauss Elimination method, Gauss Seidel Method (up to 3 variables)
43. Solve the linear programming problems (LPP) by simplex method

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Mechanics
44. Solve the forces/vectors related problems using triangle laws of forces and Lami’s theorem
45. Solve the problems related to Newton's laws of motion and projectile.

Scope and Sequence of Contents

1: Algebra (44)

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
2. Binomial Theorem:Binomial theorem for a positive integer, general term, Binomial coefficient, Binomial theorem for any index (without proof), application to approximation, Euler's number, Expansion of \( e^x,a^x \log(1+x)\) (without proof)
3. Complex numbers: Polar form of 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.
4. Sequence and series:Sum of finite natural numbers, sum of squares of first n-natural numbers, Sum of cubes of first n-natural numbers, principle of mathematical induction and use it to find sum.
5. Matrix based system of linear equation: Solution of a system of linear equations by Cramer's rule and matrix method (row-equivalent and inverse) up to three variables

2: Trigonometry (12)

1. Properties of a triangle (Sine law, Cosine law, tangent law, Projection laws, Half angle laws)
2. Solution of triangle (simple cases)

3: Analytic Geometry (20)

1. Conic section: Condition of tangency of a line at a point to the circle, Tangent and normal to a circle, Standard equation of parabola, equations of tangent and normal to a parabola at a given point, Standard equations of Ellipse and hyperbola.

4: Vectors (12)

1. Product of Vectors: Scalar product of two vectors, angle between two vectors, geometric interpretation of scalar product, properties of scalar product, application of scalar product in geometry and trigonometry, vector product of two vectors, geometrical interpretation of vector product, properties of vector product, application of vector product in geometry and trigonometry.

5:Statistics and Probability (12)

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 (only elementary concept), regression equation, regression line of y on x and x on y
2. Probability: Dependent cases, conditional probability (without proof).

7:Computational Methods (12) or Mechanics

1. Numerical computation: Roots of algebraic and transcendental equation (bisection and Newton-Raphson method)
2. Numerical integration:Trapezoidal rule and Simpson's rule

7:Mechanics(12) or Computational Methods

1. Statics: Forces and resultant forces, parallelogram law of forces, composition and resolution of forces, Resultant of coplanar forces acting on a point.
2. Dynamics: Motion of particle in a straight line, Motion with uniform acceleration, motion under the gravity, motion down a smooth inclined plane.

Sample project works/practical works 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. Verify the sine law by taking particular triangle in four quadrants.
3. Verifications of a) Cosine law b) Projection law
4. Construction of ellipse by using a piece of pencil, rope and nails
5. Prepare a concrete material to show parabola by using thread and nail in wooden panel.
6. Construct an ellipse using a rectangle.
7. Express the area of triangle and parallelogram in terms of vector.
8. 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.
9. 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.
10. Find how many peoples will be there after 5 years in your districts by using the concept of differentiation.
11. Verify that the integration is the reverse process of differentiation with examples and curves.
12. Identify different applications of Newton's law of motion and related cases in our daily life.
13. Investigate a daily life problem on projectile motion. Solve that problem and present in the classroom.
14. Write any one real life problem related to linear programming problem and solve that problem by using simplex method.

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 evaluation

Internal assessment includes classroom participation, terminal examinations, and project work/practical work (computer works and lab work) and presentation. The scores of evaluation will be used for providing feedback and to improve their learning. Individual and group works are assigned as projects.

The basis of internal assessment is as follows:

Classroom participation	Marks from terminal examinations	project work/practical work	Total
3	6	16	25

(i) Classroom participation

The mark for classroom participation is 3 which is given on the basis of attendance and participation of students in activities in each grade.

(ii) Marks from trimester examinations

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

(iii) Project work/practical work

Each Student should do at least one project work/practical work from each of seven content areas and also be required to give a 15 minutes presentation for each project work and practical work in classroom. These seven project works/practical works will be documented in a file and will be submitted at the time of practical evaluation. Out of seven projects/practical works from each area any one project work/practical work should be presented at the time of practical evaluation by student.

b. Final/External Examination

Final/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.