MHT CET Mathematics Syllabus: Check the Latest Syllabus

1. For any two angles A and B, cos (A -B) = cos A cos B + sin A sin B

2. For any two angles A and B, cos (A + B) = cos A cos B − sin A sin B

3. For any two angles A and B, sin (A − B) = sin A cos B − cos A sin B

4. For any two angles A and B, sin (A + B) = sin A cos B + cos A sin B

5. For any two angles A and B, tan (A + B) =` (tan A + tan B)/(1 –tan A tan B)`

6. For any two angles A and B, tan (A -B) = `(tan A -tan B)/(1 + tan A tan B)`

• Problems involving Angle of Elevation
• Problems involving Angle of Depression
• Problems involving Angle of Elevation and Depression

• Slope of a Line Or Gradient of a Line.
• Parallelism of Line
• Perpendicularity of Line in Term of Slope
• Collinearity of Points
• Slope of a line when coordinates of any two points on the line are given
• Conditions for parallelism and perpendicularity of lines in terms of their slopes
• Angle between two lines
• Collinearity of three points

• Introduction of Distance of a Point from a Line
• Distance between two parallel lines

• Locus
• Equation of Locus
• Shift of Origin

• Inclination of a line
• Slope of a line
• Perpendicular Lines
• Angle between intersecting lines
• Different Forms of an equation of a straight line
• General form to other forms

• Point-slope Form
• Slope-Intercept form
• Two-points Form
• Double-Intercept form
• Normal Form

1. The distance of the Origin from a Line
2. The distance of the point (x₁,y₁) from a line
3. The distance between two parallel lines

• Standard form
• Centre-radius form
• Diameter Form

The general equation of a circle is of the form x² + y² + 2gx + 2fy + c = 0, if g² + f² − e > 0.

• The equation of tangent to a standard circle x² + y² = r² at point P(x₁, y₁) on it.

1. Variance and Standard Deviation for raw data
2. Variance and Standard Deviation for ungrouped frequency distribution
3. Variance and Standard Deviation for grouped frequency distribution

• Mean deviation for grouped data
• Mean deviation for ungrouped data

1. Union of two events
2. Exhaustive Events
3. Intersection of two events
4. Mutually Exclusive Events

• Equally likely outcomes
• Probability of an Event
• Elementary Properties of Probability

• Random Experiment
• Outcome
• Sample space
• Favourable Outcome

• Elementary Event
• Certain Event
• Impossible Event
• Algebra of Events

• Union of Two Events
• Exhaustive Events
• Intersection of Two Events
• Mutually Exclusive Events

1. Using the definition of probability
2. Using Venn diagram

• Independent Events

• Partition of a sample space
• Theorem of total probability

• Imaginary number
• Complex Number

• Equality of two Complex Numbers 
• Conjugate of a Complex Number 
• Properties of `barz`
• Addition of complex numbers - Properties of addition, Scalar Multiplication
• Subtraction of complex numbers - Properties of Subtraction
• Multiplication of complex numbers - Properties of Multiplication
• Powers of i in the complex number
• Division of complex number - Properties of Division
• The square roots of a negative real number
• Identities

• Geometrical representation of conjugate of a complex number
• Properties of Complex Conjugates

• Solution of a Quadratic Equation in complex number system

• Properties of Modulus of a complex number
• Square roots of a complex number 

• Modulus of z
• Argument of z
• Argument of z in different quadrants/axes - Properties of modulus of complex numbers, Properties of arguments 
• Polar & Exponential form of C.N.

• Properties of 1, w, w²

• Tree Diagram 
• Addition Principle 
• Multiplication principle

• Properties of the factorial notation:
  For any positive integers m, n.,
  1) n! = n × (n - 1)!
  2) n > 1, n! = n × (n - 1) × (n - 2)!
  3) n > 2, n! = n × (n - 1) × (n - 2) × (n - 3)!
  4) (m + n)! is always divisible by m! as well as by n!
  5) (m × n)! ≠ m! × n!
  6) (m + n)! ≠ m! + n!
  7) m > n, (m - n)! ≠ m! - n! but m! is divisible by n!
  8) (m ÷ n)! ≠ m! ÷ n

• Permutation
• Permutation of repeated things
• Permutations when all the objects are not distinct
• Number of Permutations Under Certain Restricted Conditions
• Circular Permutations

• ⁿC_r , ⁿC_n =1, ⁿC₀ = 1, ⁿC_r = ⁿC_n–r, ⁿC_x = ⁿC_y, then x + y = n or x = y, ⁿ⁺¹C_r = ⁿC_r-1 + ⁿC_r
• When all things are different
• When all things are not different.
• Mixed problems on permutation and combinations.

• Properties of Combinations:
  1. Consider ⁿC_{n - r} = ⁿC_r for 0 ≤ r ≤ n.
  2. ⁿC₀ = `(n!)/(0!(n - 0)!) = (n!)/(n!) = 1, because 0! = 1` as has been stated earlier.
  3. If ⁿC_r = ⁿC_s, then either s = r or s = n - r.
  4. `"" ^nC_r = (""^nP_r)/(r!)`
  5. ⁿC_r + ⁿC_{r - 1} = ^{n + 1}C_r
  6. ⁿC₀ + ⁿC₁ + ......... ⁿC_n = 2ⁿ 
  7. ⁿC₀ + ⁿC₂ + ⁿC₄ + ...... = ⁿC₁ + ⁿC₃ + ⁿC₅ + ....... = 2^{(n - 1)}
  8. ⁿC_r = `"" (n/r) ^(n - 1)C_(r- 1) = (n/r)((n - 1)/(r - 1)) ^(n - 2)C_(r - 2) = ....`
  9. ⁿC_r has maximum value if (a) r = `n/2 "when n is even (b)" r = (n - 1)/2 or (n + 1)/2` when n is odd.

• Function, Domain, Co-domain, Range
• Types of function
  1. One-one or One to one or Injective function
  2. Onto or Surjective function
• Representation of Function
• Graph of a function
• Value of funcation
• Some Basic Functions - Constant Function, Identity function, Power Functions, Polynomial Function, Radical Function, Rational Function, Exponential Function, Logarithmic Function, Trigonometric function

• Composition of Functions
• Inverse functions
• Piecewise Defined Functions
  1) Signum function
  2) Absolute value function (Modulus function)
  3) Greatest Integer Function (Step Function)
  4) Fractional part function

• Definition of Limit
• One-Sided Limit
• Left-hand Limit
• Right-hand Limit
• Existence of a limit of a function at a point x = a
• Algebra of limits:
  Let f(x) and g(x) be two functions such that
  `lim_(x→a) f(x) = l and lim_(x → a) g(x) = m, then`
  1. `lim_(x → a) [f(x) ± g(x)] = lim_(x → a) f(x) ± lim_(x → a) g(x) = l ± m`
  2. `lim_(x → a) [f(x) xx g(x)] = lim_(x→ a) f(x) xx lim_(x→ a) g(x) = l xx m`
  3. `lim_(x → a) [kf(x)] = k xx lim_(x→ a) f(x) = kl, "where" ‘k’ "is a constant"`
  4. `lim_(x → a) f(x)/g(x) = (lim_(x → a) f(x))/(lim_(x → a) g(x)) = l/m "where" m≠ 0`.

1. `lim_(x → 0) ((e^x - 1)/x) = log e = 1`

2. `lim_(x → 0) ((a^x - 1)/x) = log a (a > 0, a ≠ 0)`

3. `lim_(x → 0) [ 1 + x]^(1/x) = e`

4. `lim_(x → 0) (log(1 + x)/x) = 1`

5. `lim_(x → 0) ((e^(px) - 1)/(px)) = 1`, (p constant)

6. `lim_(x → 0) ((a^(px) - 1)/(px)) = log a`, (p constant)

7. `lim_(x → 0) (log(1 + px)/(px)) = 1`, (p constant)

8. `lim_(x → 0) [ 1 + px]^(1/(px)) = e`, (p constant)

• Limit at infinity
• Infinite Limits

• Continuity of a function at a point
• Definition of Continuity
• Continuity from the right and from the left
• Examples of Continuous Functions
• Properties of continuous functions
• Types of Discontinuities
• Jump Discontinuity
• Removable Discontinuity
• Infinite Discontinuity
• Continuity over an interval
• The intermediate value theorem for continuous functions

Discontinuous Function

• Geometric description of conic section
• Degenerate Forms
•  Identifying the conics from the general equation of the conic 

• Creating a Set
• Creating Set using List or Tuple
• Set Operations
• Programs using Sets

• Representation of a set

1. Roster Method
2. Set-Builder Method
3. Venn Diagram

• Number of elements of a set
• Types of Sets

1. Empty Set
2. Singleton set
3. Finite set
4. Infinite set
5. Subset
6. Superset
7. Proper Subset
8. Power Set
9. Equal sets
10. Equivalent sets
11. Universal set

• Operations on sets

1. Complement of a set
2. Union of Sets
3. Intersection of sets
   - De Morgan's Laws
4. Difference of Sets

•  Intervals

1. Open Interval
2. Closed Interval
3. Semi-closed Interval
4. Semi-open Interval

• Definition of Relation
• Domain
• Co-domain and Range of a Relation

• Empty Relation
• Universal Relation
• Trivial Relations
• Identity relation
• Symmetric relation
• Transitive relation
• Equivalence Relation
• Antisymmetric relation
• Inverse relation
• One-One Relation (Injective)
• Many-one relation
• Into relation
• Onto relation (Surjective)

• Finite sequence 
• Infinite sequence
• Progression

• General form of Geometric Progression
• General term of Geometric Progression
• The General term or the n^th term of a G.P.
• Sum of the first n terms of a G.P. (S_n)

• Expressing recurring decimals as rational numbers

• Arithmetic mean (A. M.)
• Geometric mean (G. M.) 
• Harmonic mean (H. M.)

• n^th term of A.G.P.
• Sum of n terms of A.G.P.
• Properties of Summation

• Concept of Statements

1. Conjunction (∧)
2. Disjunction (∨)
3. Conditional statement (Implication) (→)
4. Biconditional (Double implication) (↔) or (⇔)
5. Negation (∼)

• Universal quantifier (∀)
• Existential quantifier (∃)

• Negation of conjunction
• Negation of disjunction
• Negation of implication
• Negation of biconditional

• Idempotent law
• Associative law
• Commutative law
• Distributive law
• Identity law
• Complement law
• Involution law
• DeMorgan’s laws

• Two switches in series
• Two switches in parallel

• Interchange of any two rows or any two columns
• Multiplication of the elements of any row or column by a non-zero scalar
• Adding the scalar multiples of all the elements of any row (column) to corresponding elements of any other row (column)

•  Inverse of a nonsingular matrix by elementary transformation
•  Inverse of a square matrix by adjoint method

• Method of Inversion
• Method of Reduction

• Trigonometric equation
• Solution of Trigonometric equation
• Principal Solutions
• The General Solution

• Polar co-ordinates
• Relation between the polar co-ordinates and the Cartesian co-ordinates
• Solving a Triangle
• The Sine rule
• The Projection rule
• Applications of the Sine rule, the Cosine rule and the Projection rule

• Introduction of Inverse Trigonometric Functions

• Degree of a term
• Homogeneous Equation

• The necessary conditions for a general second degree equation
  ax² + 2hxy + by² + 2gx + 2fy + c = 0

1. abc + 2fgh - af² - bg² - ch² = 0
2. h² - ab ≥ 0

• Equation of a line through a given point and parallel to a given vector `vec b`
• Equation of a line passing through two given points

• Magnitude of a Vector

• Zero Vector
• Unit Vector
• Co-initial and Co-terminus Vectors
• Equal Vectors
• Negative of a Vector
• Collinear Vectors
• Free Vectors
• Localised Vectors

• Addition of Two Vectors
  - Parallelogram Law
  - Triangle Law of addition of two vectors
• Subtraction of two vectors
• Scalar multiplication of a vector

• Co-ordinates of a point in space
• Co-ordinates of points on co-ordinate axes
• Co-ordinates of points on co-ordinate planes
• Distance of P(x, y, z) from co-ordinate planes
• Distance of any point from origin
• Distance between any two points in space
• Distance of a point P(x, y, z) from coordinate axes

• Vector addition using components
• Components of a vector in two dimensions space
• Components of a vector in three-dimensional space

• Section formula for internal division
• Midpoint formula
• Section formula for external division

• Finding angle between two vectors
• Projections
• Direction Angles and Direction Cosine
• Direction ratios
• Relation between direction ratios and direction cosines

• Angle between two vectors
• Geometrical meaning of vector product

• Equation of a line passing through a given point and parallel to given vector
• Equation of a line passing through given two points

• Introduction of Distance of a Point from a Line
• Distance between two parallel lines

• Distance between skew lines
• Distance between parallel lines

• Passing through a point and perpendicular to a vector
• Passing through a point and parallel to two vectors
• Passing through three non-collinear points
• In normal form
• Passing through the intersection of two planes

• Angle between two planes
• Angle between a line and a plane

• Convex Sets
• Graphical representation of linear inequations in two variables
• Graphical solution of linear inequation

• Meaning of Linear Programming Problem
• Mathematical formulation of a linear programming problem
• Familiarize with terms related to Linear Programming Problem

• Graphical method of solution for problems in two variables
• Feasible and infeasible regions and bounded regions
• Feasible and infeasible solutions
• Optimum feasible solution

• Rule of Differentiation
• Introduction

• First and Second Derivative test
• Determine critical points of the function
• Find the point(s) of local maxima and local minima and corresponding local maximum and local minimum values
• Find the absolute maximum and absolute minimum value of a function

• ∫ tan x dx = log | sec x |  + C
• ∫ cot x dx = log | sin x | + C
• ∫ sec x dx = log | sec x + tan x | + C
• ∫ cosec x dx = log | cosec x – cot x | + C

• `int(u.v) dx = u intv dx - int((du)/(dx)).(intvdx) dx`
• Integral of the type ∫ e^x [ f(x) + f'(x)] dx = e^xf(x) + C
• Integrals of some more types

1. `I = int sqrt(x^2 - a^2) dx = x/2 sqrt(x^2 - a^2) - a^2/2 log | x + sqrt(x^2 - a^2 )| + C`
2. `I = int sqrt(x^2 - a^2) dx = x/2 sqrt(x^2 - a^2) - a^2/2 log | x + sqrt(x^2 - a^2 )| + C`
3. `I = int sqrt(x^2 - a^2) dx = x/2 sqrt(x^2 - a^2) - a^2/2 log | x + sqrt(x^2 - a^2 )| + C`

1. `int1/(x^2 + a^2) dx = 1/a tan^-1 (x/a) + c`
2. `int1/(x^2 - a^2) dx = 1/(2a) log ((x - a)/(x + a)) + c`
3. `int1/(a^2 - x^2) dx = 1/(2a) log ((a + x)/(a - x)) + c`
4. `int1/sqrt(a^2 - x^2) dx = sin^-1 (x/a) + c`
5. `int1/sqrt(x^2 - a^2) dx = log ( x + sqrt(x^2 - a^2))+ c`
6. `int1/sqrt(x^2 + a^2) dx = log ( x + sqrt(x^2 + a^2))+ c`
7. `int1/(xsqrt(x^2 - a^2)) dx = 1/a sec^-1(x/a) + c`

If ∫ f(x) dx = g(x) + c, then

`int_a^b f(x) dx = [g(x) + c]_a^b = g(b) - g(a)`.

• `int(u.v) dx = u intv dx - int((du)/(dx)).(intvdx) dx`

1. `int_a^a f(x) dx = 0`
2. `int_a^b f(x) dx = - int_b^a f(x) dx`
3. `int_a^b f(x) dx = int_a^b f(t) dt`
4. `int_a^b f(x) dx = int_a^c f(x) dx + int_c^b f(x) dx`
   where a < c < b, i.e., c ∈ [a, b]
5. `int_a^b f(x) dx = int_a^b f(a + b - x) dx`
6. `int_0^a f(x) dx = int_0^a f(a - x) dx`
7. `int_0^(2a) f(x) dx = int_0^a f(x) dx + int_0^a f(2a - x) dx`
8. `int_(-a)^a f(x) dx = 2. int_0^a f(x) dx`, if f(x) even function
   = 0,  if f(x) is odd function

• Simple curves: lines, parabolas, polynomial functions

• Formation of Differential equations from Physical Situations
• Formation of Differential Equations from Geometrical Problems

• Population Growth and Growth of bacteria
• Ratio active Decay
• Newton's Law of Cooling
• Surface Area

• Discrete random variable
• Continuous random variable
• Probability Mass Function
• Cumulative Distribution Function or Distribution Function
• Cumulative Distribution Function from Probability Mass function
• Probability Mass Function from Cumulative Distribution Function