## Monday, October 22, 2007

### STUDY GUIDE TMH JEE MATHEMATICS CH. 1. COMPLEX NUMBERS

JEE SYLLABUS

Algebra of complex numbers, addition, multiplication, conjugation, polar representation, properties of modulus and principal argument, triangle inequality, cube roots of unity, geometric interpretations.
-----------------------

### CH. 2. THEORY OF EQUATIONS

JEE SYLLABUS

Quadratic equations with real coefficients, relations between roots and coefficients, formation of quadratic equations with given roots, symmetric functions of roots.
----------------------------

### STUDY GUIDE CH.3 PROGRESSIONS

JEE SYLLABUS

Arithmetic, geometric and harmonic progressions, arithmetic, geometric and harmonic means, sums of finite arithmetic and geometric progressions, infinite geometric series, sums of squares and cubes of the first n natural numbers.
------------------

I started study of Mathematics(9 Jan 2008). The conceptual portion is very limited in TMH book. But the first problem in the examples itself is complicated. Every problem thereon is a complicated problem.

In Mathematics and Physics, the conceptual portion is going to be limited but the problems are going to be complicated. One has to sit down and do all the problems in examples and exercises to develop the sharp brain that can discover the structure in the problem given in the examination.

### CH. 4. LOGARITHMS

JEE SYLLABUS

Logarithms and their properties.
---------------

### STUDY GUIDE CH. 5 PERMUTATION AND COMBINATION

JEE SYLLABUS

Permutations and combinations,
-----------------
JEE Question 07

The letters of the word COCHIN are permuted and all the permutations are arranged in an alphabetical order as in an English dictionary. The number of words that appear before the word COCHIN is

(A) 360 (B) 192 (C) 96 (D) 48
-------------------------------

JEE question

The number of integral points (integral point means both the coordinates should be integer) exactly in the interior of the triangle with vertices (0, 0), (0, 21) and (21, 0), is

(A) 133 (B) 190
(C) 233 (D) 105

----------------

### CH. 6. BINOMIAL EQUATION

JEE SYLLABUS

Binomial theorem for a positive integral index, properties of binomial coefficients.
----------------

### Study Guide Ch. 7 MATRICES

JEE SYLLABUS

Matrices as a rectangular array of real numbers, equality of matrices, addition, multiplication by a scalar and product of matrices, transpose of a matrix, inverse of a square matrix of order up to three, properties of these matrix operations, diagonal, symmetric and skew-symmetric matrices and their properties, solutions of simultaneous linear equations in two or three variables.
------------------

### CH. 8 DETERMINANTS

JEE SYLLABUS

Determinant of a square matrix of order up to three, inverse of a square matrix of order up to three, properties of these matrix operations, diagonal, symmetric and skew-symmetric matrices and their properties, solutions of simultaneous linear equations in two or three variables.
---------

JEE SYLLABUS

### STUDY GUIDE CH. 10. PROBABILITY

JEE SYLLABUS

Addition and multiplication rules of probability, conditional probability, independence of events, computation of probability of events using permutations and combinations.
------------

### STUDY GUIDE CH.11 ELEMENTARY TRIGONOMETRY

JEE SYLLABUS

Trigonometric functions, their periodicity and graphs, addition and subtraction formulae, formulae involving multiple and sub-multiple angles,
-----------------------

### CH. 12. APPLICATION OF TRIGONOMETRY, TRIANGLES

JEE SYLLABUS

Relations between sides and angles of a triangle, sine rule, cosine rule, half-angle formula and the area of a triangle,
-------------------

### Study Guide Ch. 13. TRIGONOMETRIC EQUATIONS

JEE SYLLABUS

general solution of trigonometric equations.

-------------------------
JEE Question

Let F(x) be an indefinite integral of sin^2 x.

Statement - 1

The function F(x) satisfies F(x +Pi) = F(x) for all real x.

Because

Statement - 2

sin^2 (x+ Pi) = sin^2 x for all real x.

(A) Statement – 1 is True, Statement – 2 is True; Statement – 2 is a correct explanation for statement – 1
(B) Statement – 1 is True, Statement – 2 is True; Statement – 2 is Not a correct explanation for Statement – 1.
(C) Statement – 1 is True, Statement – 2 is False
(D) Statement – 1 is False, Statement – 2 is True

--------------------
JEE 2007 Paper I

The number of solutions of the pair of equations
2sin²θ - cos2θ = 0
2cos²θ - 3sinθ = 0
in the interval [0,2π] is

(A) zero
(B) one
(C) two
(D) four

-----------------------

### CH. 14 INVERSE OF TRIGONOMETRIC FUNCTIONS

JEE SYLLABUS

inverse trigonometric functions (principal value only).
---------------------

JEE 2007 Paper II

Let (x,y) be such that Sin‾¹(ax) + Cos‾¹(y)+Cos‾¹(bxy) = π/2
Match the statements in Column I with statements in Column II and indicate your answer by darkening the appropriate bubbles in the 4 × 4 matrix given in the ORS.

Column I --------------------- Column II
(A) If a = 1 and b = 0,------(p) lies on the circle
then (x, y)----------------- x² + y² = 1
===================================================
(B) If a = 1 and b = 1, ----(q) lies on
then (x, y)----------------- ( x²-1)(y²-1) =0
===================================================
(C) If a = 1 and b = 2,-----(r) lies on y = x
then (x, y)
===================================================

(D) If a = 2 and b = 2,-----(s) lies on
then (x, y)------------------ (4x²-1)(y²-1) = 0 ===================================================
Solution

(A) : (p)
(B) : (q)
(C) : (p)
(D) : (s)
-----------------------------------

### STUDY GUIDE TMH MATHEMATICS CH.15. CARTESIAN SYSTEM

JEE SYLLABUS

Two dimensions: Cartesian coordinates, distance between two points, section formulae, shift of origin.

--------------------

### STUDY GUIDE CH.16. CIRCLES AND SYSTEMS OF CIRCLES

JEE syllabus

Equation of a circle in various forms, equations of tangent, normal and chord.
Parametric equations of a circle, intersection of a circle with a straight line or a circle, equation of a circle through the points of intersection of two circles and those of a circle and a straight line.

------------
JEE Question

Tangents are drawn from the point (17, 7) to the circle x^2 + y^2 = 169.

Statement - 1
The tangents are mutually perpendicular.

Because

Statement - 2
The locus of the points from which mutually perpendicular tangents can be drawn to the given circle is x^2 + y^2 = 338.

(A) Statement – 1 is True, Statement – 2 is True; Statement – 2 is a correct explanation for statement – 1
(B) Statement – 1 is True, Statement – 2 is True; Statement – 2 is Not a correct explanation for Statement – 1.
(C) Statement – 1 is True, Statement – 2 is False
(D) Statement – 1 is False, Statement – 2 is True

-----------------

### CH. 17. PAIR OF STRAIGHT LINES

JEE SYLLABUS

Equation of a straight line in various forms, angle between two lines, distance of a point from a line. Lines through the point of intersection of two given lines, equation of the bisector of the angle between two lines, concurrency of lines, centroid, orthocentre, incentre and circumcentre of a triangle.
-------------------

### Study Guide TMH Mathematics Ch. 18. CONIC SECTIONS (PARABOLA, ELLIPSE, HYPERBOLA)

JEE SYLLABUS

Equations of a parabola, ellipse and hyperbola in standard form, their foci, directrices and eccentricity, parametric equations, equations of tangent and normal.
Locus Problems.
--------------

### Ch. 19. THREE DIMENSIONAL GEOMETRY

JEE Syllabus

Three dimensions: Direction cosines and direction ratios, equation of a straight line in space, equation of a plane, distance of a point from a plane.

----------------

### Study Guide Ch.20 VECTOR ALGEBRA

JEE Syllabus

Vectors
Addition of vectors, scalar multiplication, scalar products, dot and cross products, scalar triple products and their geometrical interpretations.
------------

JEE Question 2007

If a, b, c be unit vectors such that a + b + c = 0. Which one of the following is correct?
(A) a × b = b × c = c × a = 0
(B) a × b = b × c = c × a is not equal to 0
(C) a × b = b × c = a × c is not equal to 0
(D) a × b, b × c, c × a are mutually perpendicular
------------------------

### Study Guide TMH Mathematics Ch.21. FUNCTIONS

JEE syllabus

Real valued functions of a real variable, into, onto and one-to-one functions, sum, difference, product and quotient of two functions, composite functions, absolute value, polynomial, rational, trigonometric, exponential and logarithmic functions.

------------------

### Ch.22. LIMITS AND CONTINUITY

JEE Syllabus

Limit and continuity of a function, limit and continuity of the sum, difference, product and quotient of two functions, l'Hospital rule of evaluation of limits of functions.

Even and odd functions, inverse of a function, continuity of composite functions, intermediate value property of continuous functions.

----------------

### Study Guide Ch. 23. DIFFERENTIATION

JEE Syllabus

Derivative of a function, derivative of the sum, difference, product and quotient of two functions, chain rule, derivatives of polynomial, rational, trigonometric, inverse trigonometric, exponential and logarithmic functions.

Derivatives of implicit functions, derivatives up to order two.
-----------------------

### Ch. 24 APPLICATIONS OF DERIVATIVES

JEE Syllabus

Geometrical interpretation of the derivative, tangents and normals, increasing and decreasing functions, maximum and minimum values of a function, applications of Rolle's Theorem and Lagrange's Mean Value Theorem.
-----------------

### Study Guide TMH JEE Mathematics Ch. 25 INDEFINITE INTEGRALS

JEE Syllabus

Integration as the inverse process of differentiation, indefinite integrals of standard functions,

### Study Guide Ch.26. DEFINITE INTEGRALS

JEE Syllabus

definite integrals and their properties, application of the Fundamental Theorem of Integral Calculus.
Integration by parts, integration by the methods of substitution and partial fractions, application of definite integrals to the determination of areas involving simple curves.
-------------------
JEE question
If f″(x) = − f(x) and g(x) = f′(x) and F(x) = [f(x/2)]^2 + [g(x/2)]^2 and given that F(5) = 5 then F(10) is equal to

(A) 5
(B) 10
(C) 0
(D) 15

-----------------

### Study Guide TMH JEE Mathematics Ch. 27 DIFFERENTIAL EQUATIONS

JEE syllabus

Formation of ordinary differential equations, solution of homogeneous differential equations, variables separable method, linear first order differential equations.
----------
JEE question

The differential equation dy/dx = [SQRT(1-y^2)]/y
determines a family of circles with

(A) variable radii and a fixed centre at (0, 1)
(B) variable radii and a fixed centre at (0, –1)
(C) fixed radius 1 and variable centres along the x-axis
(D) fixed radius 1 and variable centres along the y-axis