## Tuesday, September 23, 2008

### Properties of Triangles and Circles Associated with them - July Dec Revision

1. Semiperimeter of a triangle is denoted by s.
2. Area of a triangle is denoted by Δ or S.
3. a,b, and c represent sides BC,CA, and AB

4. Sine rule

In any Δ ABC
Sin A/a = Sin B/b = Sin C/c

5. Cosine Formulae

In any Δ ABC

Cos A = [b² + c² -a²]/2bc

Cos B = [c² +a² –b²]/2ac

Cos C = [a² + b² –c²]/2ab

6. Projection formulae

In any Δ ABC

a = b Cos C + C cos B

b= c Cos A + A Cos C

c = a Cos B + b cos A

7. Trigonometrical ratios of half of the angles of a triangle

1. Sin A/2 = √[(s-b)(s-c)/bc]

2. Cos A/2 = √[s(s-a)/bc]

3. tan A/2 = √(s-b)(s-c)/s(s-a)]

8. Area of a triangle

S = ½ ab Sin C = ½ bc sina = ½ ac sin B

9. Napier’s analogy

In any triangle ABC

Tan [(b-c)/2] = [(b-c)cot (A/2)]/(b+c)

10. Circumcircle of a triangle

The circle which passes through the angular points or vertices of a triangle ABC is called its circumcircle.

The centre of this circle can be found by locating the point of intersection of perpendicular bisectors of the sides. It is called circumcentre.

The circumcentre may lie within, outside or upon one of the sides of the triangle.

In a right angled triangle the cicumcentre is vertex where right angle is formed.

The radius of circumcircle is denoted by R.

R = a/(2 Sin A) = b/(2 sin B) = c/(2 sin C)

R = abc/4Δ

11. Inscribed circle or incircle of a triangle

It is the circle touches each of the sides of the triangle.

The centre of the inscribed circle is the point of intersection of bisectors of the angles of the triangle.

The radius of inscribed circle is denoted by r (it is called in-radius) and it is equal to the length of the perpendicular from its centre to any of the sides of the triangle.

Various formulas that give r.

In- radius ( r )= Δ/s

r = (s-a)tan (A/2) = (s-b) tan (B/2) = (s-c) tan (C/2)

r = [a sin B/2 sin C/2/(Cos A/2)

r = 4R sin (A/2) sin (B/2) sin (C/2)

12. Escribed circles of a triangle

The circle which touches the sides BC and two sides AB and AC produced of a triangle ABC is called the escribed circle opposite to the angle A. Its radius is denoted by r1.

Similarly r2 and r3 denote the radii of the escribed circles opposite to the angles B and C respectively.

The centres of the escribed circles are called the ex-centres.

13. Orthocentre and its distances from the angular points of a triangle

In a Δ ABC, the point at which perpendiculars drawn from the three vertices (heights) meet, it called the ortho centre of the ΔABC

14. Regular polygon and Radii of the inscribed and circumscribing circles of a regular polygon
the centre of the polygon will be the in-centre as well as circumcentre of the polygon.

15. Area of a cyclic quadrilateral

a quadrilateral is a cyclic quadrilateral if its vertices lie on a circle.

Area of cyclic quadrilateral = ½ (ab + cd) sin B

16. Ptolemy’s theorem

In a cyclic quadrilateral ABCD, AC.BD = AB.CD + BC.AD

The product of diagonals is equal to the sum of the products of the lengths of opposite sides.

In a cyclic quadrilateral, the circumcircle of the quadrilateral ABCD is also the circumcircle of Δ ABC.

Past IIT questions

1. The perimeter of a Δ ABC is three times the arithmetic mean of the sines of its angle. If the side a is 1, then the angle a is

a. π/6
b. π/3
c. π/2
d. π

(JEE 1992)

2. If the radius of the circumcircle of an isosceles triangle PQR is equal to PQ = PR, the angle P is

a. π/6
b. π/3
c. π/2
d. 2π/3

(JEE 1992)

3. In a Δ ABC, if (cos A)/a = (cos B)/b = (cos C)/c , and the side a =2, the the area of the triangle is

a.1
b. 2
c. (√3)/2
d. √3

(JEE 1993)

4. If in a triangle ABC

2cos A/a = cos B/b + 2 cos C/c = a/bc + b/ca
then the value of the angle A is _____________ degrees

(JEE Screening 1993)

### Free Mathematics Tutorials, Problems and Worksheets

Free mathematics tutorials to help you explore and gain deep understanding of math topics. The site includes several java applets to investigate Graphs of Functions, Equations, and Algebra. Topics explored are: equations of line, ellipse, circle, parabola, hyperbola, polynomials; graphs of quadratic, rational, hyperbolic, exponential and logarithmic functions; one-to-one and inverse functions and inverse trigonometric functions; systems of linear equations; determinants and Cramer's rule; inverse matrix and matrix multiplication; vectors, complex numbers, polar equations; absolute value function; slope of a line; angle in trigonometry, unit circle, solutions to trigonometric equations; graph shifting, stretching, compression and reflection. Applets used as Online Math Calculators and Solvers and Online Geometry Calculators and Solvers are also included.

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1- Interactive Exploring and Tutorials in Precalculus
Java applets are used to explore interactively important topics in precalculus. These tutorials can be used either as complements to topics already studied or to learn a new topic through exploration.

Equations in Two Variables
Slope of a Line
General Equation of a Line: ax + by = c
Slope Intercept Form Of The Equation Of a Line: y = mx + b
Find Equation of a Line - applet
Construct a Parabola
Equation of Parabola
Find Equation of Parabola - applet
Equation of Hyperbola
Equation of a Circle
Find Equation of Circle - applet
Equation of an Ellipse

Functions and Graphs
Graph, Domain and Range of Common Functions
Quadratic Functions (general form)
Quadratic Functions in Standard Form
Horizontal Shift of Graphs
Vertical Shifting or translation of Graphs
Horizontal Stretching and Compression of Graphs
Vertical Stretching and Compression(scaling) of Graphs
Definition of the Absolute Value
Absolute Value Function
Reflection of Graphs In x-axis
Reflection of Graphs In y-axis

Inverse of a Function and One to One Functions
Inverse Function Definition
One-To-One function
Inverse Function

Explore Other Functions
Explore graphs of functions

Exponential and Logarithmic Functions
Exponential Functions
Logarithmic Functions

Hyperbolic Functions
Graphs of Hyperbolic Functions

Trigonometry
Graphs of Basic Trigonometric Functions.
Trigonometric Equations and the Unit Circle.
Angle in Trigonometry
Unit Circle And The Trigonometric Functions sin(x), cos(x) and tan(x)
Inverse Trigonometric Functions
Sine Function
Cosine Function
Tangent Function
Secant Function

Applications Of Trigonometry
Sine Law - Ambiguous Case - applet

Systems Of Equations
Systems of Linear Equations - Graphical Approach

Polar Coordinates And Equations
Polar Coordinates and Equations

Polynomials
Multiplicity of Zeros and Graphs Polynomials
Polynomial Functions

Rational Functions
Rational Functions

Matrix Multiplication
Matrix Multiplication

Geometry
Medians of Triangle - Interactive applet
Perpendicular Bisector- Interactive applet
Central and Inscribed Angles - Interactive applet
Reflection Across a Line
Triangles, Bisectors and Circumcircles - interactive applet

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2- Online Algebra and Trigonometry Tutorials
Tutorials using step by step approach with examples and matched exercises are presented here. Detailed solutions to the examples are also included. Several tutorials have been designed so that they can be used with the applet tutorials in this site.

Graphs of Rational Functions - Tutorials
Identify Graphs of Functions - Tutorials
Domain and Range of Functions
Ellipse Problems
Hyperbola Problems
Solve Linear Inequalities
Solve Polynomial Inequalities
Solve Inequalities With Absolute Value - Tutorials
Solve Rational Inequalities
Solve Equations of the Quadratic Form
Solve Equations with Absolute Value
Slope of a line
Tutorial on Equation of Line
Find inverse function
Tutorial on equation of circle
Tutorial on equation of circle (2)
Solve Systems of Linear Equations
Solve Equations Graphically
Algebra Tutorial
Tutorial on Inequalities
Composition of Functions
Equations with Rational Expressions - Tutorials
Solve Exponential and Logarithmic Equations
Solve Trigonometric Equations

4 - Online Problems and self tests with Answers
Several precalculus problems are included in these pages. The applets are easy to use and answers are provided once the test is finished.

Equations And Inequalities with Absolute Value - Problems
Polynomial and Rational Inequalities - Questions and Answers
Solve Trigonometric Equations
Domain of a Function
Composition of Functions
Solve Exponential and Logarithmic Equations
Graphs of Functions - Questions
Graphs of Polynomial Functions - Questions
Graphs of Rational Functions - Questions
Graphs of Logarithmic and Exponential Functions - Questions
Find Inverse Functions - Questions
Graphs of Trigonometric Functions - Questions
Solve Quadratic Equations With Rational Expressions - Problems
Algebra Problems

http://www.analyzemath.com/

### Mathematics - Practice tests

http://acme-education.learnhub.com/test/take/474-iit-jee-mathematics-test-2

Test 1

http://indiastudycircle.blogspot.com/2008/09/mathematics-entrance-test.html

Test 2

http://indiastudycircle.blogspot.com/2008/09/mathematics-entrance-test-2.html

Test 3

http://indiastudycircle.blogspot.com/2008/09/mathematics-entrance-test-3.html

Test 4
http://indiastudycircle.blogspot.com/2008/09/mathematics-entrance-test-4.html

Test 5
http://indiastudycircle.blogspot.com/2008/09/mathematics-entrance-test-5.html

Test 6
http://indiastudycircle.blogspot.com/2008/09/1-area-enclosed-between-curve-y-loge-x.html

Test 7
http://indiastudycircle.blogspot.com/2008/09/mathematics-entrance-test-7.html

Test 8
http://indiastudycircle.blogspot.com/2008/09/mathematics-entrance-test-8.html

## Saturday, September 6, 2008

### Combinations - July- Dec Revision

Each of the different selections made by taking some or all of a number of objects, irrespective of their arrangement is called a combination.

Difference between Combinations and Permutations

In a combination, the ordering of the selected objects is immaterial whereas in a permutation, the ordering is essential. For example AB and BA are same as combinations, but different as permutations.

Associate the word selection for combinations and arrangement for permuations.

Notation

The number of all combinations of n objects, taken r at a time is denoted by C(n,r) or nCr.

nCr is defined when n and r are non-negative numbers.

Theorem

The number of all combinations of n distinct objects, taken r at a time is given by

nCr = n!/(n-r)!r!

Results from the theorem

nCr = [n(n-1)(n-2)...(n-r+1)]/(1.2.3...r)

nCn =1

nC0 = 1

nCr = nPr/r!

Properties of nCr and C(n,r)

1. nCr = nCn-r

Note: If x=y = n

nCx = nCy

2. Let n and r be non-negative integers such that r≤n. Then

nCr + nCr-1 = n+1Cr

3. Let n and r be non-negative integers such that 1≤ r≤n. Then
nCr = (n/r) n-1Cr-1

4. If 1≤ r≤n, then

n.n-1Cr-1 = (n-r+1)nCr-1

5. nCx = nCy implies x+y = n

6. If n is even, then the greatest value of nCr [0≤ r≤n] is nCn/2.

7. If n is odd, then the greatest value of nCr [0≤ r≤n] is nC(n+1)/2 or nC(n-1)/2.

Selection of one or more items

Selection from different items

The number of ways of selecting one or more items from a group of n distinct items is 2ⁿ - 1.

Selection from identical items

1. The number of ways of selecting r items out of n identical items is 1.
2. The total number of ways of selecting zero or more i.e. at least one item from a group of n identical items is (n+1).
3. The total number of selections of some or all out of p+q+r items where p are alike of one kind, q are alike of second kind, and rest are alike of third kind is {(p+1)(q+1)(r+1)}-1.

Selection of items from a group containing both identical and different items

1. the total number of ways of selecting one or more items from p identical items of one kind; q identical items of second kind, r identical items of third kind and n different items is

[(p+1)(q+1)(r+1) 2ⁿ]-1

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## Wednesday, September 3, 2008

### Permutations - July Dec Revision

8.6 Permutations of Objects not all Distinct

Theorems and Formulas

Theorem
The number of mutually distinguishable permutations of n things, taken all at a time, of which p are alike of one kind, q alike of second such that p+q = n, is

n!/p!q!

Formulas based on the above theorem

1. The number of mutually distinguishable permutations of n things, taken all at a time, of which p1 are alike of one kind, p2 alike of second,…, pr alike of of rth kind such that p1+p2+…pr = n, is

n!/p1!p2!…pr!

2. The number of mutually distinguishable permutations of n tings, of which p are alike of one kind, q alike of second and remaining all are distinct is
n!/p!q!

3. suppose there are r things to be arranged, allowing repetitions. Let further p1,p2,…,pr be the integers such that the first object occurs exactly p1 times, the second occurs exactly p2 times, etc. Then the total number of permutations of these r objects to the above condition is

(p1+p2+…+pr)!/p1!p2!…pr!

8.7 Permutations when Objects can Repeat
Theorem
The number of permutations of n different things, taken r at a time, when each may be repeated any number of times in each arrangement is n2 .

8.8 Circular Permutations
If we arrange objects along a closed curve for example a circle, the permutations are known as circular permutations. In a circular permutation, we have to consider one object as fixed and the remaining are arranged as in case of linear arrangement.

Linear arrangement is arrangement in a row.

Theorem
The number of circular permutations of n distinct objects is (n-1)!.

Anti-clock wise and clockwise order of arrangements are considered as distinct permutations in the above theorem.

If the anticlockwise and clockwise order is not distinct as in case of a garland which can be turned over easily, the number of distinct permutations will be ½ (n-1)!..

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## Monday, September 1, 2008

### Permutations - July-Dec Revision

Each of the arrangement which can be made by taking some or all of a number of things is called a permutation.

Theorem 1

Let r and n be positive integers such that 1≤r≤n. then the number of all permutations of n distinct things taken r at a time is given by

n(n-1)(n-2)…(n-(r-1))

Notation: Let r and n be positive integers such that 1≤r≤n. then the number of all permutations of n distinct things taken r at a time is denoted by the symbol P(n,r) or n Cr.

Then P(n,r) = n Cr = n(n-1)(n-2)…(n-(r-1))

Theorem 2

P(n,r) = n Cr = n!/(n-r)!

Theorem 3

The number of all permutations of n distinct things taken all at a time is n!.

Theorem 4

0! = 1

8.5 Permutations under certain conditions
Three theorems

Theorem 1
The number of all permutations of n different objects taken r at a time, when a particular object is to be always included in each arrangement is r.n-1Cr-1

Theorem 2

The number of all permutations of n different objects taken r at a time, when a particular object is never taken in each arrangement is, n-1Cr-1

Theorem 3

The number of all permutations of n different objects taken r at a time, when two specified objects always occur together is 2!(r-1) n-2Cr-2