Class 11th PCMB

Class 11 Maths Notes Chapter 14 Mathematical Reasoning

Statements
A statement is a sentence which is either true or false, but not both simultaneously.

Note:
No sentence can be called a statement if

• It is an exclamation.
• It is an order or request.
• It is a question.

Simple Statements
A statement is called simple if it cannot be broken down into two or more statements.

Compound Statements
A compound statement is the one which is made up of two or more simple statement.

Connectives
The words which combine or change simple statements to form new statements or compound statements are called connectives.

Conjunction
If two simple statements p and q are connected by the word ‘and’, then the resulting compound statement “p and q” is called a conjunction of p and q is written in symbolic form as “p ∧ q”.

Note:

• The statement p ∧ q has the truth value T (true) whenever both p and q have the truth value T.
• The statement p ∧ q has the truth value F (false) whenever either p or q or both have the truth value F.

Disjunction
If two simple statements p and q are connected by the word ‘or’, then the resulting compound statement “p or q” is called disjunction of p and q and is written in symbolic form as “p ∨ q”.

Note:

• The statement p ∨ q has the truth value F whenever both p and q have the truth value F.
• The statement p ∨ q has the truth value T whenever either p or q or both have the truth value T.

Negation
An assertion that a statement fails or denial of a statement is called the negation of the statement. The negation of a statement p in symbolic form is written as “~p”.

Note:

• ~p has truth value T whenever p has truth value F.
• ~p has truth value F whenever p has truth value T.

Negation of Conjunction
The negation of a conjunction p ∧ q is the disjunction of the negation of p and the negation of q.
Equivalently we write ~ (p ∧ q) = ~p ∨ ~q.

Negation of Disjunction
The negation of a disjunction p v q is the conjunction of negation of p and the negation of q.
Equivalently, we write ~(p ∨ q) = ~p ∧ ~q.

Negation of Negation
Negation of negation of a statement is the statement itself.
Equivalently, we write ~(~p) = p

The Conditional Statement
If p and q are any two statements, then the compound statement “if p then g” formed by joining p and q by a connective ‘if-then’ is called a conditional statement or an implication and is written in symbolically p → q or p ⇒ q, here p is called hypothesis (or antecedent) and q is called conclusion (or consequent) of the conditional statement (p ⇒ q).

Contrapositive of Conditional Statement
The statement “(~q) → (~p) ” is called the contrapositive of the statement p → q.

Converse of a Conditional Statement
The conditional statement “q → p” is called the converse of the conditional statement “p → q”.

Inverse of Conditional Statement
The Conditional statement “q → p” is called inverse of p → q.

The Biconditional Statement
If two statements p and q are connected by the connective ‘if and only if’, then the resulting compound statement “p if and only if q” is called biconditional of p and q and is written in symbolic form as p ⇔ q.

Quantifier
(i) For all or for every is called universal quantifier.
(ii) There exists is called existential quantifier.

Validity of Statements
A statement is said to valid or invalid according to as it is true or false.
If p and q are two mathematical statements, then the statement
(i) “p and q” is true if both p and q are true.
(ii) “p or g” is true if p is false
⇒ q is true orq is false ⇒ p is true.
(iii) “If p, then q” is true p is true ⇒ q is true
or
q is false
⇒ p is false
or
p is true and q is false less us to a contradiction,
(iv) “p if and only if q” is true, if
(a) p is true ⇒ q is true and
(b) q is true ⇒ p is true.

Class 11 Maths Notes Chapter 13 Limits and Derivatives

Limit
Let y = f(x) be a function of x. If at x = a, f(x) takes indeterminate form, then we consider the values of the function which is very near to a. If these value tend to a definite unique number as x tends to a, then the unique number so obtained is called the limit of f(x) at x = a and we write it as limx→af(x).

Left Hand and Right-Hand Limits
If values of the function at the point which are very near to a on the left tends to a definite unique number as x tends to a, then the unique number so obtained is called the left-hand limit of f(x) at x = a, we write it as

Existence of Limit

Some Properties of Limits
Let f and g be two functions such that both limx→af(x) and lim limx→ag(x) exists, then

Some Standard Limits

Derivatives
Suppose f is a real-valued function, then

Fundamental Derivative Rules of Function
Let f and g be two functions such that their derivatives are defined in a common domain, then

Some Standard Derivatives

Class 11 Maths Notes Chapter 12 Introduction to Three Dimensional Geometry

Coordinate Axes
In three dimensions, the coordinate axes of a rectangular cartesian coordinate system are three mutually perpendicular lines. These axes are called the X, Y and Z axes.

Coordinate Planes
The three planes determined by the pair of axes are the coordinate planes. These planes are called XY, YZ and ZX plane and they divide the space into eight regions known as octants.

Coordinates of a Point in Space
The coordinates of a point in the space are the perpendicular distances from P on three mutually perpendicular coordinate planes YZ, ZX, and XY respectively. The coordinates of a point P are written in the form of triplet like (x, y, z).
The coordinates of any point on

• X-axis is of the form (x, 0,0)
• Y-axis is of the form (0, y, 0)
• Z-axis is of the form (0, 0, z)
• XY-plane are of the form (x, y, 0)
• YZ-plane is of the form (0, y, z)
• ZX-plane are of the form (x, 0, z)

Distance Formula
The distance between two points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) is given by

The distance of a point P(x, y, z) from the origin O(0, 0, 0) is given by
OP = x2+y2+z2−−−−−−−−−−√

Section Formula
The coordinates of the point R which divides the line segment joining two points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) internally or externally in the ratio m : n are given by

The coordinates of the mid-point of the line segment joining two points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) are

The coordinates of the centroid of the triangle, whose vertices are (x₁, y₁, z₁), (x₂, y₂, z₂) and (x₃, y₃, z₃) are

Class 11 Maths Notes Chapter 11 Conic Sections

Circle
A circle is the set of all points in a plane, which are at a fixed distance from a fixed point in the plane. The fixed point is called the centre of the circle and the distance from centre to any point on the circle is called the radius of the circle.
The equation of a circle with radius r having centre (h, k) is given by (x – h)² + (y – k)² = r².

The general equation of the circle is given by x² + y² + 2gx + 2fy + c = 0 , where, g, f and c are constants.

• The centre of the circle is (-g, -f).
• The radius of the circle is r = g2+f2−c−−−−−−−−−√

The general equation of the circle passing through origin is x² + y² + 2gx + 2fy = 0.

The parametric equation of the circle x² + y² = r² are given by x = r cos θ, y = r sin θ, where θ is the parametre and the parametric equation of the circle (x – h)² + (y – k)² = r² are given by x = h + r cos θ, y = k + r sin θ.

Note: The general equation of the circle involves three constants which implies that at least three conditions are required to determine a circle uniquely.

Parabola
A parabola is the set of points P whose distances from a fixed point F in the plane are equal to their distance from a fixed line l in the plane. The fixed point F is called focus and the fixed line l is the directrix of the parabola.

Main Facts About the Parabola

Ellipse
An ellipse is the set of all points in a plane such that the sum of whose distances from two fixed points is constant.
or
An ellipse is the set of all points in the plane whose distances from a fixed point in the plane bears a constant ratio, less than to their distance from a fixed point in the plane. The fixed point is called focus, the fixed line a directrix and the constant ratio(e) the eccentricity of the ellipse. We have two standard forms of ellipse i.e.

Main Facts about the Ellipse

Hyperbola
A hyperbola is the locus of a point in a plane which moves in such a way that the ratio of its distance from a fixed point in the same plane to its distance from a fixed line is always constant which is always greater than unity. The fixed point is called the focus, the fixed line is called the directrix and the constant ratio, generally denoted bye, is known as the eccentricity of the hyperbola.
We have two standard forms of hyperbola i.e.

Main Facts About Hyperbola