12th Post Mid Term Math Exam 2022-23

School Name:	Himalaya Public School, Sector 13, Rohini, Delhi 110085 India
Class:	12th Standard (CBSE)
Subject:	Mathematics
Time Duration:	03 Hours
Maximum Marks:	80
Date:	09 / 12 / 2022

General Instructions: 12th Post Mid Term Math Exam

1. This question paper contains five sections A, B, C, D and E. Each section is compulsory. However, there are internal choices in some questions.
2. Section – A has IS MCQ’s and 2 Assertion-Reason questions of 1 mark each.
3. Section – B has 5 very short answer (VSA) type questions of 2 marks each.
4. Section – C has 6 short answer (SA) type questions of 3 Marks each.
5. Section – D has 4 long answer type questions of 5 Marks each.
6. Section – E has 3 case study questions (4 marks each) with sub parts.

Section – A

Multiple Choice Questions

Q.1. A box contains 4 white, 3 black and 5 red balls are drawn one by one with replacement. Then P (4 white balls) is

P.T.O.

12th Class Mathematics Course Structure:

Units	Topics	Marks
I	Relations and Functions	10
II	Algebra	13
III	Calculus	44
IV	Vectors and 3-D Geometry	17
V	Linear Programming	6
VI	Probability	10
Total		100

12th Class Mathematics Course Syllabus:

Unit I: Relations and Functions – 12th Post Mid Term Math Exam 2022-23

Chapter 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
  • Binary operations

Chapter 2: Inverse Trigonometric Functions

• Definition, range, domain, principal value branch
• Graphs of inverse trigonometric functions
• Elementary properties of inverse trigonometric functions

Unit II: Algebra – 12th Post Mid Term Math Exam 2022-23

Chapter 1: Matrices

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

Chapter 2: Determinants

• Determinant of a square matrix (up to 3 × 3 matrices), properties of determinants, minors, co-factors and applications of determinants in finding the area of a triangle
• Ad joint and inverse of a square matrix
• Consistency, inconsistency and number of solutions of system of linear equations by examples, solving system of linear equations in two or three variables (having unique solution) using inverse of a matrix

Unit III: Calculus

Chapter 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

Chapter 2: Applications of Derivatives

• Applications of derivatives: rate of change of bodies, increasing/decreasing functions, tangents and normal, 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)

Chapter 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$\int \frac{dx}{x^2\pm {a^2}’}$, $\int \frac{dx}{\sqrt{x^2\pm {a^2}’}}$, $\int \frac{dx}{\sqrt{a^2-x^2}}$, $\int \frac{dx}{ax^2+bx+c} \int \frac{dx}{\sqrt{ax^2+bx+c}}$

  $\int \frac{px+q}{ax^2+bx+c}dx$, $\int \frac{px+q}{\sqrt{ax^2+bx+c}}dx$, $\int \sqrt{a^2\pm x^2}dx$, $\int \sqrt{x^2-a^2}dx$

  $\int \sqrt{ax^2+bx+c}dx$, $\int \left ( px+q \right )\sqrt{ax^2+bx+c}dx$

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

Chapter 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)

Chapter 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:
  • dy/dx + py = q, where p and q are functions of x or constants
  • dx/dy + px = q, where p and q are functions of y or constants

Unit IV: Vectors and Three-Dimensional Geometry

Chapter 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

Chapter 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:
  • Two lines
  • Two planes
  • A line and a plane
• Distance of a point from a plane

Unit V: Linear Programming

Chapter 1: 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 and unbounded)
  • Feasible and infeasible solutions
  • Optimal feasible solutions (up to three non-trivial constraints)

Unit VI: Probability

Chapter 1: Probability

• Conditional probability
• Multiplication theorem on probability
• Independent events, total probability
• Baye’s theorem
• Random variable and its probability distribution
• Mean and variance of random variable
• Repeated independent (Bernoulli) trials and Binomial distribution