Important theorems, proofs, and long-answer patterns for CBSE board exam preparation.
13
Chapters
80
Marks total
11
High-weight chapters
High-Weightage Chapters
π― These chapters carry the most marks β prepare these first:
Ch 5: Continuity and Differentiability β 8 marks
Ch 6: Application of Derivatives β 8 marks
Ch 7: Integrals β 8 marks
Ch 13: Probability β 8 marks
π Ch 3: Matrices
5 marks
Theorem 1
For any square matrix A with real number entries, A + A' is a symmetric matrix and A - A' is a skew symmetric matrix.
Theorem 2
Any square matrix can be expressed as the β of a symmetric and a skew symmetric matrix. Specifically, A = (1/2)(A + A') + (1/2)(A - A').
Theorem 3 (Uniqueness of inverse)
Inverse of a square matrix, if it exists, is unique.
Matrices simplify work compared to straightforward methods
Matrices represent coefficients β systems of linear equations
Matrix notation is used β electronic spreadsheet programs
We follow the notation A = [aα΅’β±Ό]β β β to indicate that A is a matrix of order m x n
π Ch 4: Determinants
5 marks
Theorem 1
If A be any given square matrix of order n, then A(adj A) = (adj A)A = |A|*I, where I is the identity matrix of order n.
Theorem 2
If A and B are nonsingular matrices of the same order, then AB and BA are also nonsingular matrices of the same order.
Theorem 3
The determinant of the β of matrices is equal to β of their respective determinants, that is, |AB| = |A| Β· |B|, where A and B are square matrices of tβ¦
Determinants have wide applications β Engineering, Science, Economics, Social Science, etc.
In this chapter, determinants up to order three with real entries only are studied
Topics covered: properties of determinants, minors, cofactors, applications β finding area of triangle, adjoint and inverse of a square matrix, consistency and inconsistency of systems, solution using inverse of a matrix
For matrix A, |A| is read as determinant of A and not modulus of A
π Ch 5: Continuity and Differentiability
8 marks
Theorem 1
Suppose f and g be two real functions continuous at a real number c. Then: (1) f + g is continuous at x = c, (2) f - g is continuous at x = c, (3) f .β¦
Theorem 2 (Composition)
Suppose f and g are real valued functions such that (f o g) is defined at c. If g is continuous at c and if f is continuous at g(c), then (f o g) is cβ¦
Theorem 3
If a function f is differentiable at a point c, then it is also continuous at that point.
Previously learnt to differentiate polynomial and trigonometric functions
This chapter connects continuity and differentiability
Powerful techniques of differentiation are developed
f is continuous at x = c if: (1) f(c) is defined, (2) lim(x->c) f(x) exists, (3) lim(x->c) f(x) = f(c)
π Ch 6: Application of Derivatives
8 marks
First Derivative Test for Increasing/Decreasing
Let f be continuous on [a, b] and differentiable on (a, b). Then: (a) f is increasing β [a,b] if f'(x) β₯ 0 for each x β (a,b) β (b) f is decreasing β β¦
Necessary condition for local extrema
Let f be a function defined on an open interval I. Suppose c is β I. If f has a local maxima or a local minima at x = c, then either f'(c) = 0 or f isβ¦
First Derivative Test
Let f be a function defined on an open interval I. Let f be continuous at a critical point c β I. Then: (i) If f'(x) changes sign from positive to negβ¦
Chapter 5 covered finding derivatives; this chapter covers applications of those derivatives
Applications include: (i) rate of change, (ii) tangent and normal equations, (iii) turning points for maxima/minima, (iv) increasing/decreasing intervals, (v) approximations
dy/dx is positive if y increases as x increases
dy/dx is negative if y decreases as x increases
π Ch 7: Integrals
8 marks
First Fundamental Theorem of Integral Calculus
Let f be a continuous function on the closed interval [a, b] and let A(x) be the area function. Then A'(x) = f(x), for all x β [a, b].
Second Fundamental Theorem of Integral Calculus
Let f be continuous function defined on the closed interval [a, b] and F be an anti derivative of f. Then β« from a to b of f(x) dx = [F(x)] from a to β¦
Integration is the inverse process of differentiation.
Integral calculus was developed to solve problems of finding functions from derivatives and finding areas under curves.
d/dx[β« f(x) dx] = f(x) and β« f'(x) dx = f(x) + C: differentiation and integration are inverses of each other.
Two indefinite integrals with the same derivative lead to the same family of curves and so they are equivalent.
π Ch 8: Application of Integrals
5 marks
Application of definite integrals to find area under curves is a specific use of integration as the limit of a β.
We also find the area bounded by the above said curves.
If the curve lies below the x-axis, the definite β« gives a negative value; take the absolute value for the area.
If the curve crosses the x-axis, split the β« at the crossing points and take absolute values of the negative parts.
π Ch 9: Differential Equations
5 marks
An equation involving derivative(s) of the dependent variable with respect to independent variable(s) is called a differential equation.
x(dy/dx) + y = 0 is a differential equation because it involves a derivative of y with respect to x.
An ordinary differential equation involves derivatives with respect to only one independent variable.
2(dΒ²y/dxΒ²) + (dy/dx)Β³ = 0 is an ordinary differential equation.
π Ch 10: Vector Algebra
5 marks
Commutative Property of Vector Addition
For any two vectors a and b: a + b = b + a
Associative Property of Vector Addition
For any three vectors a, b, c: (a + b) + c = a + (b + c)
Cauchy-Schwarz Inequality
For any two vectors a and b: |a . b| β€ |a||b|
Scalars are real numbers representing magnitude only
Vectors have both magnitude and direction
This chapter covers basic concepts, operations on vectors, and their algebraic and geometric properties
Since the length is never negative, the notation |a| < 0 has no meaning
π Ch 11: Three Dimensional Geometry
5 marks
In Class XI, Analytical Geometry β two dimensions and introduction to three dimensional geometry used Cartesian methods only
This chapter uses vector algebra for 3D geometry
For any line, if a, b, c are direction ratios, then ka, kb, kc (k β 0) are also direction ratios
Any two sets of direction ratios of a line are proportional
π Ch 12: Linear Programming
5 marks
Theorem 1 (Fundamental Theorem of Linear Programming)
Let R be the feasible region (convex polygon) for a linear programming problem and let Z = ax + by be the objective function. When Z has an optimal vaβ¦
Theorem 2
Let R be the feasible region for a linear programming problem, and let Z = ax + by be the objective function. If R is bounded, then the objective funcβ¦
In earlier classes, systems of linear equations and linear inequalities β two variables were studied
Many applications β mathematics involve systems of inequalities/equations
This chapter applies systems of linear inequalities/equations to solve real life problems
Example: A furniture dealer wanting to maximise profit from buying tables and chairs with investment and storage constraints
If E and F are independent events, then (a) E and F' are independent, (b) E' and F are independent, (c) E' and F' are independent.
Theorem of Total Probability
Let {Eβ, Eβ, ..., Eβ} be a partition of the sample space S, and suppose that each of Eβ, Eβ, ..., Eβ has nonzero probability of occurrence. Let A be aβ¦
In earlier classes, probability was studied as a measure of uncertainty of events β a random experiment
The axiomatic theory and the classical theory of probability are equivalent for equally likely outcomes
Throughout this chapter, experiments have equally likely outcomes unless stated otherwise
When event F is known to have occurred, the sample space reduces from S to F