Parabola - Definitions - July -Dec Review

Parabola is the locus of a point P which moves in a plane so that its distance from a fixed line of the plane and its distance from a fixed point of the plane, not on the line, are equal.

The fixed point F is called the focus and fixed line is called the directrix of the parabola.

The perpendicular to the directrix from the focus is called the axis of the parabola.

The intersection of the parabola and the axis of the parabola is called vertex.

Vertex is the mid point of axis.

The line joining any two distinct points of the parabola is called a chord.

A chord which passes through the focus is called a focal chord.

The distance between the focus and any point of the focal chord is called focal radius.

The focal chord which is perpendicular to the axis is called the latus rectum

Circle - Definitions- July - Dec Revision

Some definitions

A circle is defined as the locus of a point which moves in a plane such that its distance form a fixed point in that plane is always fixed.

Intercept of the circle on x axis is the length of chord of the circle which is a part of x axis.

Similarly Intercept of the circle on y axis is the length of chord of the circle which is a part of y axis.

Director circle: the locus of the point of intersection of two perpendicular tangents to a given conic is known as its director circle.

Chord of contact: the chord joining the points of contact of the two tangents to a conic drawn from a given point, outside it, is called the chord of contact of tangents.

Pole and Polar:
Polar of a point with respect to a circle: Of through a point P(x1,y1) (inside or outside a circle) there be drawn any straight line to meet the given circle a Q and R, the locus of the point of intersection of the tangents at Q and R is called the polar of point P and P is the called the pole of the polar.

Polar is the locus and pole is a point.

Diameter – definition as a locus: the locus of the middle points of a system of parallel chords of a circle is called a diameter of the circle.

Common chord of two circles: The chord joining the points of intersection of two given circles is called their common chord.

Angle of intersection of two curves: If the two curves C1 and C2 intersect at a point P and PT1 and PT2 be the tangents to the two curves C1 and C2 respectively at P. Then the angle between the tangents at P is called the angle of intersection of the two curves at the point of intersection.

Orthogonal curves: Two curves are said to intersect orthogonally when the two tangents at the common point are at right angles.

Radical axis: the radical axis of two circles is the locus of a point which moves in such a way that the lengths of the tangents drawn from it to the two circles are equal.

Radical centre: The point of concurrence of the radical axes of three circles whose centres are non-collinear, taken in pairs, is called the radical centre of the circles.

Coaxial system of circles: A system of circles, every pair of which has the same radical axis is called a coaxial system of circles.

For formula revision sheet, visit

http://iit-jee-maths.blogspot.com/2008/06/ch-15-circle-concept-review.html

July - December 2008 Revision for IIT JEE 2009

I plan to go through each chapter in Chemistry, Physics and Mathematics in revision mode during July-December 2008 apart from reading or studying chapter which I have not read so far.I am yet to study many chapters in mathematics in details so far.

My thinking is that from 1st January 2009 onwards, the aspirants should focus on memorizing things and a revision during July-Dec at leisurely pace, one chapter per day would help in that. From 1 January 2009, the memorization process should take up three chapters per day.

Ch. 1. Sets - July-Dec Revision

Set is synonymous with the words, ‘collection’, aggregate’, ‘class’, and is comprised of elements.

The words ‘element’, ‘object’, and ‘member’ are synonymous.

Sets designated by specific letters.

N: natural numbers
Z : integers
Z+: positive integers
Q: rational numbers
Q+: positive rational numbers
R: real numbers
R+: positive real numbers
C: complex numbers


Description of a set

Sets can be described by roster method or set-builder method.


Types of sets

Empty set
Singleton set
Finite set
Infinite set
Equivalent set
Equal set
Subset
Universal set
Power set

Theorems on subsets

1. Every set is a subset of itself.
2. The empty set is a subset of every set.
3. The total number of subsets of a finite set containing n elements is 2ⁿ

Laws of algebra of sets


1. Idempotent laws

(i) A U A = A
(ii) A ∩ A = A

2. Identity laws

(i) A U ф = A
(ii) A ∩ U = A

3. commutative law
(i) A U B = B U A
(ii) A ∩ B = B ∩ A

4. Associative laws

(i) (A U B) U C = A U (B U C)
(ii) (A ∩ B) ∩ C = A ∩ (B ∩ C)

5. Distributive laws

(i) A U (B ∩ C) = (A U B) ∩ (A U C)
(ii) A ∩ (B U C) = (A ∩ B) U (A ∩ C)

6. De-morgan’s laws

(i) (A U B)’ = A’ ∩ B’
(ii) (A ∩ B)’ = A’ U B’

Some more theorems

If A and B are two sets

(i) A – B = A ∩ B’
(ii) B – A = B ∩ A’
(iii) A – B = A  A ∩ B = ф
(iv) (A – B) U B = A U B
(v) (A-B) ∩ B = ф
(vi) A is a sub set of B  B’ is a subset of A’
(vii) (A-B) U (B-A) = (A U B) – (A ∩ B)

If A, B and C are three sets, then

(i) A – (B ∩ C) = (A-B) U (A-C)
(ii) A – (B U C) = (A-B) ∩ (A-C)
(iii) A ∩ (B-C) = (A ∩ B) - (A ∩ C)
(iv) A ∩ (B Δ C) = (A∩B) Δ (A∩C)

Some results on Number of elements in sets n(A), n(B), and n(C)

Note union operation and universal set have the same symbol

If A,B and C are finite sets. U is the finite universal set, then

(i) n(A U B) = n(A) +n(B) – n(A∩B)

(ii) n(A U B) = n(A) +n(B)  A, B are disjoint non-void sets.

(iii) n(A-B) = n(A) –n(A∩B)

(iv) n(A ΔB) = Number of elements which belong to exactly one of A or B
= n((A-B) U (B-A))

(v) n(A U B U C) = n(A) + n(B) + n(C) – n(A∩B) – n(B∩C) – n(A∩C)+n(A∩B∩C)

(vi) No. Of elements in exactly two of the sets A,B,C
= n(A∩B) + n(B∩C)+n(C∩A)-3n(A∩B∩C)

(vii) No. Of elements in exactly one of the sets A,B,C
= n(A) +n(B)+n(C)-2n(A∩B)-2n(B∩C)-2n(A∩C)+3n(A∩B∩C)

(viii) n(A’ U B’) = n((A∩B)’) = n(U) – n(A∩B)

(ix) n(A’∩B’) = n((AUB)’) = n(U)-n(A∩B)

Laws of algebra of sets

1. Idempotent laws

A U B = A

A∩A = A

2. Identity laws

3. Commutative laws

4. Associative laws

5. Distributive laws

6. De-Morgan's laws

Theorems - Permutations and Combinations

1. n! = n[9n-1)!]

2. I fone thing can be done in m ways and after it has been done by anyone of these m ways, another thing cna be done in n ways, independent of the ways of dong the first thing, then the total number of ways doing the two things together is mn.

(Chapter : permutations and combinations)

Principle of Mathematical Induction

If a given open sentence involving n is true for n = 1 and its truth for n = kimplies its truth for n = k+1, then it is true for every natural number n

Theorem of Resolved Parts

The sum of the resolved parts of two forces acting through a point along any direction is equal to the resolved part of the resultant of the two forces along that direction.

Explanation

If you find the resolved parts in some direction initially and then find their resultant, it will be equal to the resolved part of the resultant of the two forces in the same direction.

(Topic: Vectors)

Lagrange's Identity in Vector Products

(a × b)² = a²b² - (a.b)

a and b are vectors
a and b are magnitudes of a and b respectively.

Lagrange's identity is a relation between the cross product and the dot product.

(Topic: Vectors)