# MATHEMATICS (008)

## GRADE XII

## MODEL QUESTION 2078

## SOLUTION

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## MATHEMATICS SYLLABUS

#### 1. Algebra

**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

**2.1 Inverse circular functions.**

**2.2 Trigonometric equations and general values**

#### 3. Analytical Geometry

**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

**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 and Probability

**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

**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**: anti-derivatives, 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

**7.1 Linear programming Problems:**linear programming problems(LPP), solution of LPP by simplex method (two variables)

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

8. Mechanics and Mathematics for Economics and Finance

**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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