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Test Papers of Class 12 MathematicsIndian Education system primarily consists of two parts one being studying the subjects and the other one is appearing for exams and tests. All these exams are conducted by schools for various classes to gauge how much the students have understood , learned and what is their capability score in any particular subject. But sometimes even after understanding the topics fully there are chances the students don’t do well in exams because of lack of preparation from the examination perspective. The reason being that one needs to prepare for exams in a particular way and for that previous question papers, sample papers, worksheets and unit test papers play a major role.

The important factors in performing well in the exams, apart from, studying day & night, grasping everything and retaining everything, is, to be able to study smart and in a proper disciplined manner, so that all the efforts translate into performance.

When preparing for your exams every expert recommends that students should invest more time and effort into solving question papers and worksheets from previous year

This practice not only will familiarise the students with the format of the question paper, it will also teach them the discipline of answering the entire question paper within the time allotted to you at the examination. The more one solves these, the more confidence will be gained towards achievement of highest score. Often it happens that in the exams the children know the solution of every question but due to lack of time they miss some portions. So it is imperative to practice before hand so that everything is attempted on time in main exams

Unit-I: Relations and Functions

1: Relations and Functions:

Types of relations: reflexive, symmetric, transitive and equivalence relations. One to one and onto functions, composite functions, inverse of a function.

2: Inverse Trigonometric Function

Definition, range, domain, principal value branch, graphs of inverse trigonometric functions. Elementary properties of inverse trigonometric function.

Unit-II: Algebra

1: Matrices

Concept, notation, order equality, types of matrices,zero and identity matrix, transpose of a matrix, symmetric and skew symmetric matrices: Addition and Multiplication and Multiplication with a scalar. Simple properties of addition, multiplication and scalar multiplication. Non-commutability of multiplication of matrices and existence of non-zero matrices whose product in null matrix (restrict to square matrices of order 2). Concept of elementary row and column operations. Invertible matrices and proof of uniqueness of inverse, if it exists; (Here all matrices will have real entries).

2: Determinants

Determinants of square matrix (upto 3 x 3 order), properties of determinants, minors, cofactors and applications of determinants in finding the area of triangle. Adjoint and inverse of a square matrix. Properties of inverse of a matrix. Consistency, inconsistency and number of solutions of system of linear equations in two or three variables (having unique solution ) using inverse of matrix.

Unit-III: Calculus

1: Continuity and Differentiability

Continuity and differentiability, derivative of composite functions, chain rule, derivatives of inverse trigonometric functions.Derivative of implicit functions. Concept of exponential and logarithmic functions. Derivatives of logarithmic and exponential functions. Logarithmic differentiation, derivative of functions expressed in parametric forms. Second order derivatives. Rolle's and Lagrange's Mean Value Theorems (without proof) and their geometric interpretation.

2: Applications of Derivatives

Applications of derivatives: rate of change of bodies, increasing/decreasing functions, tangents and normals, use of derivatives in approximation, maxima and minima (first derivative test motivated geometrically and second derivative test given as a provable tool). Simple problems (that illustrate basic principles and understanding of the subject as well as real-life situations).

3: Integrals

Integration as inverse process of differentiation. Integration of a variety of functions by substitution, by partial fractions and by parts.Evaluation of simple integrals of the following types and problems based on them.

Definite integrals as a limit of a sum, Fundamental Theorem of Calculus (without proof).Basic properties of definite integrals and evaluation of definite integrals.

4: Applications of the Integrals

Applications in finding the area under simple curves, especially lines, circles/parabolas/ellipses (in standard form only), Area between any of the two above said curves (the region should be clearly identifiable).

5: Differential Equations

Definition, order and degree, general and particular solutions of a differential equation. Formation of differential equation whose general solution is given. Solution of differential equations by method of separation of variables solutions of homogeneous differential equations of first order and first degree. Solutions of linear differential equation of the type:

Unit-IV: Vectors and Three-Dimensional Geometry

1: Vectors

Vectors and scalars, magnitude and direction of a vector. Direction cosines and direction ratios of a vector. Types of vectors (equal, unit, zero, parallel and collinear vectors), position vector of a point,negative of a vector, components of a vector, addition of vectors, multiplication of a vector by a scalar, position vector of a point dividing a line segment in a given ratio. Definition, Geometrical Interpretation, properties and application of scalar (dot) product of vectors, vector (cross) product of vectors, scalar triple product of vectors.

2: Three Dimensional Geometry

Direction cosines and direction ratios of a line joining two points. Cartesian equation and vector equation of a line, coplanar and skew lines, shortest distance between two lines. Cartesian and vector equation of a plane. Angle between (i) two lines, (ii) two planes, (iii) a line and a plane. Distance of a point from a plane.

Unit-V: Linear Programming

Linear Programming

Introduction, related terminology such as constraints, objective function, optimization, different types of linear programming (L.P.) problems, mathematical formulation of L.P. problems, graphical method of solution for problems in two variables, feasible and infeasible regions(bounded or unbounded), feasible and infeasible solutions, optimal feasible solutions (up to three non-trivial constraints)

Unit-VI: Probability

Conditional probability, multiplication theorem on probability, independent events, total probability, Bayes' theorem, Random variable and its probability distribution, mean and variance of random variable.

Mathematics Question Paper Design - Class XII

Typology of Questions

Remembering: Exhibit memory of previously learned material by recalling facts, terms, basic concepts and answers.

Number of Questions Asked that are Very Short Answer Type Question = 4

Number of Questions Asked that are Short Answer Type Questions = 1

Number of Questions Asked that are Long Answer Type Questions Asked = 2

Total Marks = 16

Total Weightage = 20%

Typology of Questions

Understanding: Demonstrate understanding of facts and ideas by organizing, comparing, translating, interpreting, giving descriptions and stating main ideas

Number of Questions Asked that are Very Short Answer Type Question = 6

Number of Questions Asked that are Short Answer Type Questions = 2

Number of Questions Asked that are Long Answer Type Questions Asked = 4

Total Marks = 28

Total Weightage = 35%

Typology of Questions

Applying: Solve problems to new situations by applying acquired knowledge, facts, techniques and rules in a different way

Number of Questions Asked that are Very Short Answer Type Question = 6

Number of Questions Asked that are Short Answer Type Questions = 2

Number of Questions Asked that are Long Answer Type Questions Asked = 2

Total Marks = 20

Total Weightage = 25%

Typology of Questions

Analysing : Examine and break information into parts by identifying motives or causes. Make inferences and find evidence to support generalizations Evaluating: Present and defend opinions by making judgments about information, validity of ideas, or quality of work based on a set of criteria. Creating: Compile information together in a different way by combining elements in a new pattern or proposing alternative solutions

Number of Questions Asked that are Very Short Answer Type Question = 4

Number of Questions Asked that are Short Answer Type Questions = 1

Number of Questions Asked that are Long Answer Type Questions Asked = 2

Total Marks = 16

Total Weightage = 20%

Class 12 Maths Marks Distribution

Unit I : Relations and Functions

Marks : 8

Unit II : Algebra

Marks : 10

Unit II : Calculus

Marks : 35

Unit III : Vectors and Three - Dimensional Geometry

Marks : 14

Unit IV : Linear Programming

Marks : 5

Unit V : Probability

Marks : 8

Total : 80

Internal Assessment : 20

Topic : Relations & Functions

Equivalence relation : No. of maximum marks allotted on the board exam = 4

Binary operations : No. of maximum marks allotted on the board exam = 1

Topic : Inverse Trigonometric Functions

Principle value : No. of maximum marks allotted on the board exam = 1

Some simple problems based on

tan-1x + tan-1y = tan-1 𝑥+𝑦 1−𝑥.𝑦

: No. of maximum marks allotted on the board exam = 4Topic : Matrices

Problems based on elementary transformation for finding inverse of a square matrix. : No. of maximum marks allotted on the board exam = 6

Types and Algebra of Matrices : No. of maximum marks allotted on the board exam = 2(1 Mark each ) or 1 (4Marks)

Topic : Determinants

Evaluation of determinants : No. of maximum marks allotted on the board exam = 1

Solution of Linear Equations by Matrix Inverse Method : No. of maximum marks allotted on the board exam = 6

Topic : Continuity and Differentiability

Continuity at a point Logarithmic and Parametric diff. : No. of maximum marks allotted on the board exam = 1

Few Questions based on second order derivatives & Mean value theorem : No. of maximum marks allotted on the board exam = 4

Topic : Applications of differentiation

Increasing / Decreasing based on polynomials : No. of maximum marks allotted on the board exam = 4

Error & Approximation : No. of maximum marks allotted on the board exam = 4

Selected Problems based on Maxima & Minima : No. of maximum marks allotted on the board exam = 6

Topic : Integrals

Integration as limit of a sum : No. of maximum marks allotted on the board exam = 4

Question based on properties like

∫𝑎𝑓(𝑥)dx = ∫𝑎𝑓(𝑎−𝑥)dx 00

: No. of maximum marks allotted on the board exam = 6/4Topic : Application of integral

Simple questions for finding area under a curve : No. of maximum marks allotted on the board exam = 6

Topic : Differential Equations

Solution of Homogeneous differential equations : No. of maximum marks allotted on the board exam = 4

Solution of Linear differential equations : No. of maximum marks allotted on the board exam = 4

Topic : Vectors

Angle between two vectors/projection formula. : No. of maximum marks allotted on the board exam = 1

Unit vector condition for perpendicular & parallel vectors and some selected questions : No. of maximum marks allotted on the board exam = 4

Topic : Three Dimensional Geometry

Direction Cosines/ Direction Ratios : No. of maximum marks allotted on the board exam = 1

Shortest Distance between two lines, Condition for perpendicularity of two lines : No. of maximum marks allotted on the board exam = 4/6

Topic : Linear Programming Problems

All types of problems related to L.P.P. : No. of maximum marks allotted on the board exam = 6

Topic : Probability

Baye’s Theorem : No. of maximum marks allotted on the board exam = 6

Probability Distribution & Binomial Distribution : No. of maximum marks allotted on the board exam = 4