## Sunday, November 23, 2008

### Maximum and Minimum values of a function in its domain

A function f(x) with domain D a sub set of R. Then f(x) is said to attain the maximum value at a point a Є D if f(x) ≤F(a) for all x ЄD.

### Local maxima and local minima

A function f(x) is said to attain a local maximum at x = a if there exists a neighbourhood (a- δ,a+ δ) of a such that
f(x) l.t. f(a) for all x Є (a- δ,a+ δ), x ≠ a

### Higher Order Derivative Test

If f is differentiable function on an interval I and if c is a point in that interval such that

(i) f’(c) = f’’(c) = fn-1 (c) =0
(ii) fn (c) exists and is non zero.

If n is even and ) fn (c) l.t. 0 then x =c is a point of local maximum.
If n is even and ) fn (c) g.t. 0 then x =c is a point of local minimum.

In n is odd then x = c is neither a point of maximum or minimum

### Increasing - Decreasing Functions - Definitions

Strictly increasing functions

A function f(x) is said to be a strictly increasing function on (a,b) if

x1 is less than x2 => f(x1) is l.t.f(x2) for all x1,x2 Є (a,b)

Strictly decreasing functions

A function f(x) is said to be a strictly increasing function on (a,b) if

x1 l.t. x2 => f(x1)g.t.f(x2) for all x1,x2 Є (a,b)

Monotonic function

A function f(x) is said to be monotonic on (a,b) if it is either increasing or decreasing on the interval (a,b)

Increasing or decreasing on [a,b]

A function f(x) is said to be an increasing (decreasing) function on [a,b] if it is increasing (decreasing) on (a,b) and it also increasing (decreasing) at x =a and x =b.

Increasing or decreasing at a point

A function f(x) is said to be a increasing (decreasing) at a point x0 if there is an interval (x0-h, x0+h) containing x0 such that f(x) is increasing (decreasing) on (x0-h, x0+h).

### Necessary and sufficient conditions for Monotonicity of Functions

Necessary condition

f ‘(x) >0 (<0) for x Є (a,b) is the necessary condition for a function f(x) to be increasing (decreasing) on a given interval (a,b)

### Properties of Monotonic Functions

1. If f(x) is strictly increasing function on an interval [a,b], then f ˉ¹ exists and it is also a strictly increasing function.

2. If f(x) is strictly increasing function on an interval [a,b], such that it is continuous, then f ˉ¹ is continuous on [f(a),f(b)].

3. If f(x) is continuous on [a,b] such that f ‘ (c) ≥0 (f ‘ (c)>0) for each c Є(a,b), then f(x) is monotonically increasing function on [a,b].

4. If f(x) is continuous on [a,b] such that f ‘ (c) ≤0 (f ‘ (c)<0) for each c Є(a,b), then f(x) is monotonically decreasing function on [a,b].

5. If f(x) and g(x) are monotonically increasing (or decreasing) functions on [a,b], then

6. If one of the functions is monotonically increasing and the other is monotonically decreasing, then gof is a monotonically decreasing function on [a,b].

## Saturday, November 22, 2008

### Slopes of Tangent and Normal

Slope of tangent at point p = dy/dx at point P

Slope of normal at point p = -[1/(dy/dx)] at point P

### Equations of Tangent and Normal

Equation of tangent at point P(x1,y1).

y-y1 = (dy/dx)(x-x1) {dy/dx is at point P)

Equation of normal at point P(x1,y1).

y-y1 = - [1/(dy/dx)](x-x1) {dy/dx is at point P)

### Angle of Intersection of Two Curves

The angle of intersection of two curves is defined to be the angle between two tangents to the two curves at their point of intersection.

### Lengths of Tangent and Normal

If the tangent to a curve y = f(x) at a point P(x,y) meets x axis at T then PT is called the tangent.

If the normal to a curve y = f(x) at a point P(x,y) meets x axis at N then PN is called the length of the normal.

### Subtangent and Subnormal

If the tangent to a curve y = f(x) at a point P(x,y) meets x axis at T and G is the foot of the ordinate (y-coordinate) at P, then TG is called the cartesian subtangent.

If the normal to a curve y = f(x) at a point P(x,y) meets x axis at N and G is the foot of the ordinate (y-coordinate) at P, then NG is called the cartesian subnormal.

### Rolle's Theorem

When f(x) is a real valued function defined on [a,b] such that

i) it is continuous on [a,b],
ii) it is differentiable on (a,b) and
iii) f(a) = f(b)

there exists a real number c that belongs to (a,b) such that

f'(c) = 0

### Lagrange's Mean value theorem

When f(x) is a function defined on [a,b] such that

i) it is continuous on [a,b] and
ii) it is differentiable on (a,b)

there exists a real number c that belongs to (a,b) such that

f'(c) = (f(b)-f(a))/(b-a)

## Friday, November 21, 2008

### Differentiation - Definition

Let f be a function definedon a neighbourhood of a real number a.
f is said to be differentiable or derivable at a if Lt(x→a) [{f(x)-f(a)}/(x-a)] exists finitely. The limit is called the derivative or differential coefficient of f at a.

It is denoted by f'(a).

### Geometrical meaning of derivative at a point

The derivative of a function at a point x=c is the slope of the tangent to the curve y = f(x) at the point (c,f(c))[x =c and y = f(c)].

### Differentiation of some standard functions

d/dx of sinx = cos x

d/dx of cosx = - sin x

d/dx of tanx = sec²x

d/dx of cot x = -cosec²x

d/dx of xn = nxn-1

d/dx of log x = 1/x

d/dx of ex = ex

d/dx of ax = axlog a

d/dx of sin-1x = 1/√(1-x²)

d/dx of tan-1 x = 1/(1+x²)

### Fundamental Rules of Differentiation

1. d/dx of constant = 0

2. d/dx of (c.f(x)) = c d/dx of f(x)

3. d/dx of (f(x) ±g(x)) = d/dx of f(x) ± d/dx of (g(x))

4. Product rule d/dx of uv = u*dv/dx + v*dv/dx

5. quotient rule d/dx of u/v = [v*du/dx – u*dv/dx]/v²

### Relation between dy/dx and dx/dy

dy/dx = 1/{dx/dy)

### Differentiation of implicit functions

If variables are given as f(x,y) = 0 and if it is not possible to find y as a function of x in the form y = ф(x), then y is said be an implicit function of x.

To find dy/dx in such a case, differentiate both sides of the equation with respect to x, by writing the derivative of g(y) w.r.t. to x as (dg/dy)*(dy/dx).

### Logarithmic differentiation

To find derivatives of the functions of the form [f(x)]g(x)

Procedure is:
Let y = [f(x)]g(x)

Take logarithms on both sides

Log y = g(x)*log [f(x)]

Differentiate w.r.t. x
(1/y)*dy/dx = g(x)*(1/f(x))*d(f(x))/dx + log [f(x)]*d[g(x)]/dx

Therefore dy/dx = (1/y)*[ g(x)*(1/f(x))*d(f(x))/dx + log [f(x)]*d[g(x)]/dx]

### Differentiation of parametric functions

If x = f(t) and y = g(t) are given and we have to find dy/dx

Then, first find dy/dt and dx/dt

dy/dx will be obtained as (dy/dt)/(dx/dt)

### Differentiation of a function with respect to another function

u = f(x) and v = g(x) be two functions. To find the derivative of f(x) with respect of g(x) or du/dv use the formula

du/dv = (du/dx)/(dv/dx)

### Higher order derivatives

Derivative of y w.r.t. x = dy/dx

Derivative of dy/dx w.r.t. x = d²y/dx²

and so on.

The alternative notations of higher order derivatives are

dy/dx, d²y/dx²

y1, y2

y’, y’’

Dy, D²y

f’(x), f’’(x)

## Thursday, November 20, 2008

### Indefinite Integral - Antiderivative - Primitive

a function ф(x) is called a primitive or an antiderivative of a function f(x) if ф'(x) = f(x).

For a function f(x), the collection of all its primitives is called the indefinite integral of f(x0 and is denoted by ∫f(x)dx.

∫f(x)dx = ф(x)+C (where C is a constant)

Here ∫ is the integral sign, f(x) is th integrand, x is the variable of integration and dx is the element of integration or differential of x.

The process of finding an indefinite integral of a given function is called integration of the function.

### Integrals of some standard functions

Basic integrals

Add C (constant) to given ∫f(x)dx

S.no. f(x) ∫f(x)dx

1. 0 ... C (constant)

2. xn (n not equal to -1) ... xn+1/(n+1)

3. 1/x ... ln|x|

4. ex ... ex

5. ax ... ax/ln a

Trigonometric functions

6. sin x ... -cos x

7. cos x ... sin x

8. Cosec²x ...-cot x

9. sec²x ... tan x

### Integration - Some Standard Results

1. ∫kf(x)dx = k∫f(x)dx

2. ∫[f(x)± g(x)]dx = ∫f(x)dx ± ∫g(x)dx

3. d/dx [∫f(x)dx] = f(x)

### Integration by substitution

If ф(x0 si a ocntinuously differntiable function, then to solve

∫f(ф(x))ф'(x)dx; we substitute ф(x) = t and ф'(x)dx will be equal to dt.

Hence the problem is transformed to ∫f(t)dt

## Wednesday, November 19, 2008

### Integrals of the form [f'(x)/f(x)]dx

∫[f'(x)/f(x)]dx = log[f(x)]

Using the formula

∫tan x dx = ∫(sin x/cos x)dx

If f(x) = t, f'(x)dx = dt
cos x = t;
-sin x dx = dt
sin x dx = -dt

∫(sin x/cos x)dx = ∫-dt/t = -log |t|+c = - log|cos x|+C
= log |sec x|+C

### Integrals of the form sin ^m x cos ^n x dx

if m power of sin x is odd, put cosx = t.

If n power of cos x is odd, put sin x = t.

If both m, and n are odd use De'Moivre's theorem.

### Integrals of the form [1/(x²±a²)]dx

∫(1/(x²+a²)dx = (1/a)tanˉ¹(x/a) + C

∫(1/(x²-a²)dx = (1/2a)log|(x-a)/(x+a)|+C

### Integrals of the form [1/(ax²+bx+c)]dx

Make the coefficient of x² as unity. divide by a,

Add and subtract square of half of the coefficient of x to the expression.

### Integrals of the form [1/√(ax²+bx+c)]dx

Make the coefficient of x² as unity. divide by a,

Add and subtract square of half of the coefficient of x to the expression.

### Integrals of the form [(px+q)/(ax²+bx+c)]dx

Express numerator as
(px+q) = λ(derivative of denominator) + µ

### Integrals of the functional form [P(x)/(ax²+bx+c)]dx

P(x) is a polynomial.

Divide P(x) by the denominator to Q(x)+R(x)/(ax²+bx+c)

R(x) will be linear or first degree equation.

### Integrals of the form [(px+q)/√(ax²+bx+c)]dx

Express numerator as

px + q = λ(derivative of denominator) + µ = λ(2ax+b)+µ

### Integrals of the functional form [1/(a sin²x + b cos²x +c)]dx

Divide numerator and denominator by cos²x

Replace sec²x by (1 + tan² x)

Put tan x = t

dt = sec²xdx

The integral reduces to ∫[1/(At² +Bt +C)]dt

### Integrals of the functional form [1/(a sin x + b cos x +c)]dx

Put sin x = (2 tan x/2)/(1 + tan² (x/2))

cos x = (1 - tan² (x/2))/(1 + tan² (x/2))

Replace (1 + tan² (x/2)) by sec²(x/2)

Put tan (x/2) = t

dt = 1/2 sec²(x/2)dx

The integral reduces to ∫[1/(at² +bt +c)]dt

### Integrals of the functional form [(a sin x + b cos x)/(c sin x + d cos x)]dx

Express numerator as

Numerator = λ(derivative of denominator) + µ(denominator)

## Tuesday, November 18, 2008

### Integrals of [(a sin x+b cos x +c)/(p sin x + q cos x +r)] dx

Express the numerator as

λ(denominator) + µ(Differential of denominator) + υ

The solution will come as λx + µ log|denominator| + υ∫dx/(p sin x + q cos x +r)

### Integration by parts

∫uv dx = u(∫vdx)-∫[du/dx∫vdx]dx

### Integral of e^x [f(x)+f'(x)]dx

∫ex[f(x)+f '(x)]dx = exf(x)+ C

### Integrals of e^ax sinbx dx,e^ax cos bx dx

∫eax sinbx dx = [eax/(a²+b²)[[a sin bx - b cos bx) +C

∫eax cos bx dx = [eax/(a²+b²)[[a cos bx + b sin bx) +C

### Integrals of √(a²±x²) and √(x²-a²)

∫√(a²+x²)dx = (1/2) x√(a²+x²) + (1/2) a²log |x+ √(a²+x²)| +C

∫√(a²-x²) = (1/2) x√(a²-x²) + (1/2) a²sin-1(x/a) + C

∫√(x²-a²) = (1/2) x√(x²-a²)- (1/2) a²log |x+ √(x²-a²)| +C

### Integrals of √(ax²+bx+c)dx

Express (ax²+bx+c) as the sum of difference of two squares and apply appropriate procedure of formula.

### Integrals of the functional form (px+q)[√(ax²+bx+c)]dx

Express px + q = λ.(derivative of ax²+bx+c) + µ

this will give λ = p/2a and µ = q - bp/2a

This transforms the expression into

(p/2a)∫(2ax+b)[√(ax²+bx+c)]dx + [(q-bp)/2a]∫[√(ax²+bx+c)]dx

### Integration of Rational Algebraic Functions by Using Partial Fractions

f(x)/g(x) is a rational algebraic function.

If degree of f(x) is g.t. g(x) convert it into ф(x) +ψ(x)/(g(x)

degree of ψ(x) is l.t. g(x)

Obtian partial fractions from ψ(x)/(g(x)

### Integration of [(x²+1)/(x^4+λx²+1)]dx

Divide the numerator and denominator by x² and put x + 1/x at t or x - 1/x as t as required.

### Integration of Function [G(x)/(P√Q)]dx

∫ [G(x)/(P√Q)]dx

When both P and Q are linear functions of x put Q = t²

For more specific functional form

P√Q = ax+b)√(cx+d)

Hence put cx+d = t²

## Monday, November 17, 2008

### Basic Integration Rules

Basic Integration Rules

Multiplication by a Constant: ∫cf(u) du = c∫f(u) du

Sum Rule: ∫[f(u) + g(u)]du = ∫f(u)du + ∫g(u)du

Difference Rule: ∫[f(u) - g(u)]du = ∫f(u)du - ∫g(u)du

### The definite Integral - Definition

If F(x) is the antiderivative of a function f(x) continuous on (a,b)which means F'(x) = f(x) (a is less than x is less than b), then

∫f(x)dx {from a to b} = F(x) from a to b = F(b) - F(a)

### Evluation of Definite Integrals

To find ∫f(x)dx from a to b

Find indefinite integral of ∫f(x)dx = ф(x)

Evaluate ф(b) and ф(a)

Calculate ф(b) - ф(a)

### Geometric interpretation of definite integral

the definite integral represents the algebraic sum of the areas of the figures bounded by

the graph of the function y = f(x)
the x axis
the straight line x =a and x = b.

if the curve goes above and below the x axis in the interval a to b, the areas of above the x axis enter this sum with a plus sign, while those below the x axis enter it with a minus sign.

### Evaluation of Definite Integrals by Substitution

If a substitution is done in a definite integral, such a substitution needs to be effected at three places.

1. in the integrand

2. in the differential

3. in the limits

### Properties of definite integrals

Properties of definite integrals

1. ∫ab f(x)dx = -∫ba f(x)dx

2. ∫ab f(x)dx = ∫ac f(x)dx + ∫cb f(x)dx

3. ∫0a f(x)dx = ∫0a f(a-x)dx

4. If f(-x) = f(x) (means f is an even function), then
-aa f(x)dx = 2∫0a f(x)dx

5. If f(-x) = -f(x) (means f is an odd function), then
-aa f(x)dx = 0

6. ∫0af(x)dx = ∫0af(a-x)dx and
abf(x)dx = ∫abf(a+b-x)dx

7. ∫0af(x)dx = ∫0a/2f(x)dx+∫0a/2f(a-x)dx

Due to the above relation

0af(x)dx = 0 if f(a-x) = -f(x)
0af(x)dx = 2∫0a/2f(x)dx if f(a-x) = f(x)

8. If f is continuous on [a,b], then the integral function defined by g(x) = ∫axf(t)dt for x Є [a,b]is derivable on [a,b], and g'(x) = f(x) for x Є [a,b].

9. If f(x0 is periodic with period T then

abf(x)dx = ∫a+nTb+nTf(x)dx, where n is an integer.

In particular

0nTf(x)dx = n∫0Tf(x)dx

If m and M are the smallest and greatest values of a function f(x) on an interval [a,b], then m(b-a)≤∫abf(x)dx≤M(b-a)

### Integral Function

If f(x) is a continuous function defined on [a,b]m then a function ф(x) defined by

ф(x) = ∫f(t)dt, (a to x); x belongs to [a,b] is called the integral function of the function f.

### Summation of Series Using Definite Integral as the Limit of a Sum

∫f(x)dx (from a to b) = lim (h→0) h[f(a)+f(a+h)+f(a+2h)+...+f(a+(n-1)h)] where h = (b-a)/n

h→0 implies n→∞

∫f(x)dx (from 0 to 1) = lim (n→∞) (1/n)[Σf(r/n)(from r = to n-1)]

### Gamma Function

If n is a positive number, then the improper integral

0 e-xxn-1dx is defined as a Gamma Function and is denoted by Γn.

### Curve Sketchig - Some Useful Hints

To evaluate the areas of bounded regions, one has to know the rough sketch of the curve.

The principles/ideas that help in visualizing the curve or rough sketching the curve are:

1. Origin and tangents at the origin

Check whether the curve passes through the origin and finding out whether (0,0) satisfies the curve equation.

If the curve passes through (0,0) find the equation of tangent at (0,0)

2. Points of intersection of the curve with the coordinate axes

You can find the point at which the curve crosses the x-axis by putting y = 0 in the curve equation.

Similarly putting x = 0 gives the point at which the curve passes y-axis.

3. Symmetry

If all powers of y in the curve equation are even, then the curve is symmetric about x-axis.

The familiar curve y² = 4ax is symmetric about x-axis.

If all powers x in the equation of curve are even, it is symmetric about y-axis.

If by putting x = -x and y = -y, the equation of the curve remains the same, then it symmetric in opposite directions.

If the equation remains unaltered by interchanging x and y in the equation, then it is symmetric about the line y =x.

4. Regions where the curve does not exist

In the regions where the curve does not exist, x or y values will be imaginary.

In the case of y² = 4ax, when x is negative, y values will be imaginary. Hence curve does not exist on the left side of y-axis.

5. Find points where dy/dx = 0

At these points, the slope of the tangent to the curve is zero and the tangent will be parallel to the x axis.

6. Find maxima and minima points

Find points where dy/dx = 0 and check the sign of d²y/dx² at these points and find out maximum (d²y/dx² is negative) and minimum (d²y/dx² is positive) points of the curve.

Also find the interval in which the dy/dx is g.t. 0. In this interval, the function is monotonically increasing.

In the interval in which dy/dx is l.t. 0, the function is monotonically decreasing.

### Sketching of some common curves

Straight line

General equation = ax+by+c = 0

Circle
General equation = x² + y² + 2gx+2fy+c = 0

Standard parabola

y² = 4ax

General parabola
y = ax²+bx+c

Standard ellipse

x²/a² + y²/b² = 1

### Area of Bounded Regions

if f(x) is a continuous function defined on [a,b]. then, the area bounded by the curve y = f(x), the x-axis and the ordinates x = a and x = b is given by

∫f(x)dx(between limits a to b) or ∫y dx (between limits a to b).

Steps to find the area of bounded regions

1. visualize a rough sketch.

2. Slice the area into horizontal or vertical strips appropriately.

3. Create a formula for area treating the strip as a rectangle.

If the strip is parallel to y-axis, the width of the rectangle will be Δx.

If the strip is parallel to x-axis, the width of the rectangle will be Δy.

4. Find the limits of x (for vertical strips) and limits of y for horizontal strips.

5. Do the integration ∫ydx or ∫xdy
y = f(x) is the given function or x = f(y) is the given function.

## Sunday, November 16, 2008

### Solving differential equation of the type dy/dx = f(x)

dy/dx = f(x)

dy = f(x)dx

Integrating both sides

y = ∫f(x)dx + C

### Solving Differential Equations of the Type dy/dx = f(y)

dy/dx = f(y)

dx/dy = 1/f(y)

dx = [1/f(y)]dy

Integrating both sides

∫dx = ∫[1/f(y)]dy +C

x = ∫[1/f(y)]dy +C

### Solving Differential Equations in Variable Separable Form

General differential equation of first order and first degree is of the form

M + Ndy/dx = 0

Where M and N are any functions of x and y.

If M = f(x) and N = g(y), the equation can be written as

f(x) + g(y)dy/dx = 0

=>
g(y)dy = -f(x)dx

Integrating both sides

∫g(y)dy = ∫-f(x)dx +c

When the M and N can be separated into f(x) and g(y), the equation is called as an equation with separable variable

### Solving Differential Equations Reducible to Variable Separable Form

Differential equations with the form

dy/dx = f(ax+by+c) can be reduced to variable separable form.

Substitute ax+by+c = v

### Solving Homogeneous Differential Equations

Homogeneous Differential Equations

A function f(x,y) is said to be homogeneous of degree n if we can express

f(x,y) = xn .g(y/x)

A differential equation of first order and first degree is said to be homogeneous if it is of the form

dy/dx = f1(x,y)/f2(x,y)

where f1 and f2 are homogeneous functions of the same degree n.

Such an equation can be written as

dy/dx = [xn .g1(y/x)]/[xn .g2(y/x)]

=> dy/dx = F(y/x) ....(1)

Put y/x = v
=> y = vx

=> dy/dx = v +x.(dv/dx) .... (2)

From (1) and (2)

F(v) = v +x.(dv/dx)

=> F(v)-v = x.(dv/dx)

=> dx/x = dv/(F(v)-v)

=>dv/(F(v)-v) = dx/x

As this differntial equation is in separable form it can be solved

### Solving Equations Reducible to Homogeneous Forms

If an equation of the form

dy/dx = [a1x+b1y+c1]/[a2x+b2y+c2] is given

Put x = X+h and y = Y+k

where h and k constants.

h and k are to be determined.

Their values are

h/(b1c2-b2c1) = k/(c1a2-c2a1) = 1/(a1b2-a2b1)

This substitution will give the transformed equations

dY/dX = (a1X + b1Y)/(a2X + b2Y)

The above equation is a homogeneous differential equation.

Solve it and substitute X = x-h and Y = y-k

### Solving Linear Differential Equations

In a Linear Differential Equation y and dy/dx appear only in first degree.

The general form is:

dy/dx +Py = Q

P and Q can be functions of x (even constants)

We use integrating factor e∫Pdx.

Multiplying both sides with the integrating factor

e∫Pdx[dy/dx +Py] = Qe∫Pdx

L.H.S. is equal to d/dx of [ye∫Pdx]

d/dx of [ye∫Pdx] = Qe∫Pdx

Integrating both sides w.r.t. x

We get ye∫Pdx = ∫Qe∫Pdx + C

### Solving Linear Differential Equations of the Form dx/dy + Rx = S

dx/dy + Rx = S

R and S are functions of y or constants

Find integration factor e∫Rdy.

Multiply both sides of the given differential equation.

Solution will comes as

xe∫Rdy = ∫Se∫Rdy + C

### Solution of Differential Equations Reducible to Linear Form

When the given diff. equation is of the type

dy/dx + Py = Qyn and P and Q are constants or functions of x alone and value of x is not either zero or one, the equation can be reduced to the linear form.

Procedure:

Divide both sides by yn

y-ndy/dx + Py-n+1 = Q

Substitute y-n+1 = v

we get the transformed equation which is a linear equation

dv/dx + (1-n)Pv = (1-n)Q

### Differential equations - defintions

Differential equation is an equation involving derivatives of a dependent variable with respect to one or more independent variables.

Example: d²y/dx²+ y = x²

Order and degree are two attributes of a differential equation.

The order of a differential equation is the order of the highest differential coefficient involved. If second order derivative is present in the differential equation, the order of the equation is two or it is of second order. The equation give above as an example is a second order equation as d²y/dx² a second order derivative of y is present in the equation.

In the equation, the power to which the higher differential coefficient or derivative is raised is known as the degree of the equation.

Examples (d³y/dx³)4 + (d²y/dx²)² +y² = 0 is a 3rd order and 4th degree equation. 3rd order because d³y/dx³ is present in the equation and it is the highest order derivative in the equation. 4th degree because d³y/dx³ is raised to the fourth power.

### Solution of a differential equation

Solution of a differential equation

It is f(x,y, c1, c2,..cn) which does not involve derivatives and the derivatives of f(x,y,c1,c2...cn) satisify the differential equation.

General solution

It contains as many arbitrary costants as the order of the differential equation.

Particular solution

The arbitrary constants in the general solution are given particular values and the solution so obtained is the particular solution.

### Formation of differential equations

If a polynomial f(x,y,c1, c2...cn) is the solution of a differential equation, where x and y are variables and c1,c2...cn are constants, differentiate the equation n times successively and eliminate the constants.

### Differential equations of first order and first degree

A differential equation that involes x, y and dy/dx is a first order differential equation.

### Geometrical interpretation of the Differential equations of first order and first degree

the equation is f(x,y, dy/dx) = 0

As dy/dx gives the slope of the tangent, this equation gives us a relation between the point and the tangent at that point to the integral curve y = f(x)

### Solution of Differential equations of first order and first degree

First order first degree equations have only x,y and dy/dx terms in them. dy/dx term will be present to the power of one only.

Examples

dy/dx = 1+x+y+xy
y-x(dy/dx) = a(y² +(dy/dx))

It is difficult to solve all equations.

We can solve differential equations when they have certain standard forms.

### Types of vectors

Zero or Null Vector

Unit vector

Like Vector

Unlike vector

Collinear

Parallel vector

Co-initial vector

Co-planar vector

Coterminus vector

Negative of a vector

Reciprocal of a vector

Localized vector

Free vector

## Saturday, November 15, 2008

### Properties of scalar product

Property 1 :
The scalar product of two vectors is commutative
av.bv = bv.av

Property 2 : Scalar Product of Collinear Vectors :
(i) When the vectors av and bv are collinear and are in the same direction, then θ = 0

av.bv = |av| |bv| = ab

(i) When the vectors av and bv are collinear and are in the opposite direction, then θ = π

av.bv = |av| |bv|(-1) = -ab

Property 3 : Sign of Dot Product
The dot product av.bv may be positive or negative or zero.

(i) If the angle between the two vectors is acute (i.e., 0 < θ < 90°) then
cos θ is positive. In this case dot product is positive.
(ii) If the angle between the two vectors is obtuse (i.e., 90 < θ < 180) then
cos θ is negative. In this case dot product is negative.
(iii) If the angle between the two vectors is 90° (i.e., θ = 90°) then
cos θ = cos 90° = 0. In this case dot product is zero.

### Scalar product in terms of components

If a = a1i+a2j+a3k and
b= b1i+b2j+b3k

then a.b = a1b1+a2b2+a3b3

### Angle between two vectors

If θ is the angle between two vectors,
cos θ = a.b/|a||b|
=> θ = cos-1 (a.b/|a||b|)
In component form
If a = a1i+a2j+a3k and
b= b1i+b2j+b3k

θ = cos-1[(a1b1+a2b2+a3b3)/(SQRT(a1²+a2²+a3²)*SQRT(b1²+b2²+b3²))

### Components of a vector b along perpendicular to vector a

Component of vector b along vector a == (a.b/|a|²)aComponent of vector b perpendicular to vector a = b- (a.b/|a|²)a

### Tetrahedron

A tetrahedron is a three dimensional figure formed by four triangles. A tetrahedron in which all edges are equal, is called a regular tetrahedron.

Using scalar product some relations between edges of tetrahedron can be proved.

The relations are:

1. If two pairs of opposite edges of a tetrahedron are perpendicular, then the opposite edges of the third pair are also perpendicular to each other.
2. In a tetrahedron, the sum of the squares of two opposite edges is the same for each pair.
3. Any two opposite edges in a regular tetrahedron are perpendicular.

### Application of scalar product in mechanics to find the work done

Word done by a force is a scalar quantity. This can be calculated by finding the scalar product of force vector and displacement vector.

If the force vector is represented by OA and the displacement vector by OB and if the angle between them is θ work done is

W = OA.OB = |OA||OB|cos θ

|OB|cos θ represents the component of OB in the direction of OA

If a number of forces are acting on a particle, then the sum of works done by the separate forces is equal to the work done by the resultant force.

### Definition of vector product

Definition: a and b are two non-zero non-parallel vectors. Then the vector product a×b is defined as a vector whose magnitude is |a||b| sin θ where θ is the angle between a and b and whose direction is perpendicular to the plane of a and b in such a way that a,b and this direction constitute a right handed system.

The direction of vector product: If η is a unit vector in the direction of a×b, then a,b and η form a system in such a way that , if we rotate vector a into vector b, then η will point in the direction perpendicular to the plane and a and b in which a right handed screw will move if it is turned in the same manner.

Magnitude of a×b = |a||b| sin θ

Geometrical interpretation of vector product

a×b is a vector whose magnitude is equal to the area of the parallelogram having a and b as its adjacent sides.
| a×b| = |a | |b| sin θ |a| is the base and |b| sin θ is the height of the parallelogram

### Properties of vector product

a and b are vectors

1. Vector product is not commutative

a×b ≠ b×a

a×b = - b×a

2. m is a scalar
ma×b = m(a×b) = a×mb

3. m and n are scalars
ma×nb = mn(a×b) = m(a×nb) = n(ma×b)

4. Distributive property over vector addition
a×(b+c) = a×b + a×c (left distributivity)
(b+c) ×a = b×a + c×a (right distributivity)

5. a×(b-c) = a×b - a×c (left distributivity)
(b-c) ×a = b×a - c×a (right distributivity)

6. The vector product of two non-zero vectors is zero is they are parallel or collinear

a = a1i+a2j+a3k
b = b1i+b2j+b3k

a×b =

|i j k|
|a1 a2 a3|
|b1 b2 b3|

## Friday, November 14, 2008

### Coordinates of a point in space

Thee mutually perpendicular lines in space define three mutually perpendicular planes which in turn divide the space into eight parts known as octants and the lines are known as the coordinate axes.

(x,y,z), the coordinates of a point P are the perpendicular distances from P on the three mutually rectangular coordinate planes YOZ, ZOX, and XOY respectively. The coordinates of a point are the distances from the origin of the feet of the perpendiculars from the point on the respective coordinate axes.

Sign convention of three dimensional geometry

All distances measured along OX, OY, and OZ will be positive.
All distances measured along or parallel to OX’, OY’, Oz’ will be negative.

### Signs of Coordinates of a Point in Various Octants

x-coordinate signs and octants

Note the sign '+' or '-'

Octant

OXYZ *** +
OX’YZ *** -
OXY’Z *** +
OX’Y’Z *** -
OXYZ’ *** +
OX’YZ’ *** -
OXY’Z’ *** +
OX’Y’Z’ *** -

y-coordinate signs and octants

Octant

OXYZ *** +
OX’YZ *** +
OXY’Z *** -
OX’Y’Z *** -
OXYZ’ *** +
OX’YZ’ *** +
OXY’Z’ *** -
OX’Y’Z’ *** -

z-coordinate signs and octants

Octant

OXYZ *** +
OX’YZ *** +
OXY’Z *** +
OX’Y’Z *** +
OXYZ’ *** -
OX’YZ’ *** -
OXY’Z’ *** -
OX’Y’Z’ *** -

Equation of xy plane is z = 0.
Equation of yz palne is x = 0.
Equation of xz plane is y = 0.

### Distance formula

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

PQ = √[(x2-x1) ² + (y2-y1) ² + (z2-z1) ²]

### Section Formulas

Internal division: Division of line segment PQ internally.

The coordinates of a point R that divides line segment PQ with coordinates P(x1,y1,z1) and Q(x2,y2,z2) in the ratio m1:m2 internally are

[ (m1x2+m2x1)/(m1+m2), (m1y2+m2y1)/(m1+m2), (m1z2+m2z1)/(m1+m2)]

PR:RQ = m1:m2

If the is the mid point that m1:m2 = 1:1 the coordinates are
[(x1+x2)/2, (y1+y2)/2, (z1+z2)/2]

External division: Division of line segment PQ internally.

The coordinates of a point R that divides line segment PQ with coordinates P(x1,y1,z1) and Q(x2,y2,z2) in the ratio m1:m2 externally are

[ (m1x2 -m2x1)/(m1+m2), (m1y2 -m2y1)/(m1+m2), (m1z2 -m2z1)/(m1+m2)]

PR:RQ = m1:m2

### Direction cosines and direction ratios

Direction cosines and direction ratios of a line

The direction cosines of a line: The direction cosines of a line are defined as the direction cosines of any vector whose support is the given line.

If A and B are two points on a given line L, then direction cosines of vectors AB and BA are the direction cosines of line L. If α, β, γ, are the angles which the line L makes with positive directions of x-axis, y-axis, and z-axis respectively, then its direction cosines are either cos α, cos β , cos γ or -cos α, -cos β ,- cos γ.

Therefore if l,m,n are direction cosines of a line, then –l.-m,-n are also its direction cosines.

Also l² +m² + n² = 1

If P and Q are two points with coordinates P(x1,y1,z1) and Q(x2,y2,z2) on line L, it direction cosines (of L or PQ) are

(x2-x1)/PQ, (y2-y1)/PQ, , (z2-z1)/PQ, or (x1-x2)/PQ, (y1-y2)/PQ, , (z1-z2)/PQ

### Plane - Definition and Equation

A plane is a surface such that if any two points are taken on it, the line segment joining them lies completely on the surface. It means every point on the line segment joining any two points on the plane lies on the plane.

Every first degree equation in x,y, and z represents a plane.
ax +by+cz+d = 0 is the general equation of the plane.

We can also write it as

Ax + By+Cz+1 = 0; This is an equation in three unknowns.

### Equation of plane passing through a given point

The general equation of a plane passing through a point (x1,y1, z1) is
a(x – x1)+b(y-y1)+c(z-z1) = 0, where a, b and c are constants.

## Wednesday, November 12, 2008

### Intercept form of plane

The equation of a plane intercepting lengths a, b an c with x-axis, y-axis and z-axis respectively is

x/a +y/b + z/c = 1

This plane intercepts lengths a, b, and c with x, y and z-axis respectively.

Given a plane equation, to find x-intercept, we put y =0 and z = 0. Similarly we do for other intercepts y and z.

Revision topic of
THREE DIMENSIONAL GEOMETRY

### Vector equation of a plane passing through a given point and normal to a given vector - Vector Form

The vector equation of a plane ‘Pi’ passing through a point having position vector a and normal to vector n is (r-a).n = 0 or r.n = a.n.

Where r is the position vector of an arbitrarily chosen point.

It can also be written as r.n = d
Where d = a.n

Revision topic of
THREE DIMENSIONAL GEOMETRY

### Vector equation of a plane passing through a given point and normal to a given vector - Cartesian Form

It is (x-a1)n1 + (y-a2)n2 +(z-a3n3) = 0

Where (a1,a2 and a3) are coordinates of the point a, and n1,n2 and n3 are the direction ratios of vector n normal to the plane.

Therefore, you can remember that coefficients of x,y and z in the Cartesian equation of a plane are the direction ratios of normal to the plane.

Revision topic of
THREE DIMENSIONAL GEOMETRY

### Equation of a plane in normal form - Vector Form

Vector form: the vector equation of a plane normal to unit vector n^ and at a distance d from the origin is r.n^ = d.

Revision topic of
THREE DIMENSIONAL GEOMETRY

### Equation of a plane in normal form - Cartesian Form

If l,m,n are direction cosines of the normal to a given plane which is at a distnance p from the origin, then the equation of the plane is:

lx + my +nz = p

Revision topic of
THREE DIMENSIONAL GEOMETRY

### Angle between two planes

The angle between two planes is defined as the angle between their normals.

Revision topic of
THREE DIMENSIONAL GEOMETRY

### Angle between planes in Vector Form

Given

Plane 1: r.n1 = d1.

Plane 2: r.n2 = d2.

cos θ = (n1.n2)/[ |n1| |n2|]

Condition of perpendicularity of planes

(n1.n2) = 0

Condition of parallelism of planes

n1 = λn2

Revision topic of
THREE DIMENSIONAL GEOMETRY

### Angle between two planes in Cartesian form

The angle θ between the planes a1x +b1y+c1z+d1 = 0, a2x +b2y+c2z+d2 = 0 is given by:

cos θ = [a1a2+b1b2+c1c2]/ [√[a1²+b1²+c1²] √[a2²+b2²+c2²]]

### Systems of Measuring Angles

sexagesimal system or English system

Circular system

French system

First two are commonly used.

In sexagesimal system, a right angle is divided into 90 equal parts called degrees.

In Circular system, the unit of measurement is radian. One radian is the angle made by an arc of length equal to radius of a given circle at its centre.

Topic of
Trigonometric ratios, Identities and Maximum & Minimum Values of Trigonometrical Expressions

### Domain and range of trigonometrical functions

Trigonometric ratios, Identities and Maximum & Minimum Values of Trigonometrical Expressions

### Area of a triangle

Area of a triangle = 1/2 ab sin C = (1/2) bc sin A = (1/2)ac sin B

Area of a triangle = √[s(s-a)(s-b)(s-c)]

Area of a triangle = c² sin A sin B/(2 sin c)

= a² sin B sin C/(2 sin A)

= b² sin C sin A/(2 sin B)

topic of
Properties of Triangles and circles connected with them

## Tuesday, November 11, 2008

### Ptolemy’s theorem

The product of the two diagonals in a cyclic quadrilateral is equal to sum of products of opposite sides.

## Sunday, November 9, 2008

### Periodic Function

A function f(x) is said to be periodic if there exists T>) such that f(x+T)=f(x)for all x in the domain of definition of f(x)

Since

for all n Є Z ( n belongs to Z)

sin (2nπ +x) = sin x
cos (2nπ +x) = cos x
tan (nπ +x) = tan x

sin x, cos x,and tan x are peridic functions.

### Period of a Periodic Function

If T is the smallest positive real number such that f(x+T) = f(s), then it is called the period f f(x)

Since

for all n Є Z ( n belongs to Z)

sin (2nπ +x) = sin x
cos (2nπ +x) = cos x
tan (nπ +x) = tan x

period of sin x = 2π
period of cos x = 2π
period of tan x = 2π

### General Solution of Trigonometrical Equations

sin θ = sin α then θ = nπ + (-1)nα, n Є Z (n belongs to Z)

cos θ = cos α then θ = 2nπ ± α, n Є Z

tan θ = tan α then θ =nπ +α, n Є Z

sin² θ = sin² α then θ =nπ ±α, n Є Z

cos² θ = cos² α then θ =nπ ±α, n Є Z

tan² θ = tan² α then θ =nπ ±α, n Є Z

cos θ = cos α and sin θ = sin α then θ =nπ+α, n Є Z

## Saturday, November 8, 2008

### sin-inv x = cosec-inv(?)

sin-1x = cosec-1(1/x)

cosec-1x = sin-1(1/x)

cos-1x = sec-1(1/x)

sec-1x = cos-1(1/x)

### sin-inv(-x) =

sin-inv(-x) = = sin-inv(x)

cos-inv (-x) = π-cos-inv x

tan-inv (-x) = -tan-inv x

cot-inv (-x) = π - cot-inv x

### 2 sin-inv x =

2 sin-inv x = sin-inv (2x√(1-x²))

2 cons-inv x = cos-inv (2x²-1)

2 tan-inv x = tan-inv(2x/(1-x²)) = sin-inv (2x/(1+x²) = cos-inv [(1-x²)/(1+x²)]

### 3 sin-inv x =

3 sin-inv x = sin-in (3x-4x³)

3 cos-inv x = cons-inv (4x³ - 3x)

3 tan-inv x = tan-inv [(3x-x³)/(1 - 3x²)]

### Tan-inv x + Tan-inv y + Tan-inv z

Tan-inv x + Tan-inv y + Tan-inv z = Tan-inv [(x+y+z-xyz)/(1-xy-yz-zx)]

Tan-inv x1 + Tan-inv x2 + ...+ Tan-inv xn = Tan-inv [(S1-S3+S5...)/(1-S2+S4-S6+...)]
Where Sx denotes the sum of the products of x1,x2,..,xn taken k at a time.

## Friday, November 7, 2008

### Solution of a Triangle - Concept

Triangle has six elements.
Three sides and three angles.

If three of the elements are given, one of them being a side, other three elements can be uniquely determined.

The procedure of finding unknown elements from the known elements of a triangle is termed solving a triangle.

### Useful Results - Solution of Triangles

a. Right angled triangle - orthocentre
In a right angled triangle, orthocentre concides with the vertex containing the right angle.

b. Distance of midpoint of the hypotenuse of a right angled triangle from the vertices of the triangle.
Distance of midpoint of the hypotenuse of a right angled triangle from the vertices of the triangle is equal.

c. Relation between mid-point of the hypotenuse a right angled triangle and its circumcentre.
Mid-point of the hypotenuse a right angled triangle and is the circumcentre of the triangle.

### IIT JEE Study Guide 2. Cartesian Product of Sets and Relations

Objective Mathematics by R D Sharma

Contents

2.1 Cartesian product of sets
2.2 Relations
2.3 Types of relations
2.4 Some results on relations
2.5 Composition of relations

Day 1

2.1 Cartesian product of sets

Day 2

2.2 Relations
2.3 Types of relations

Day 3
2.4 Some results on relations
2.5 Composition of relations

Day 4
Objective Exercises 1 to 37

Day 5
Fill in the blanks 1 to 8
True/False 1 to 13

Day 6
Practice exercises 1 to 27

Revision days

Day 7

Day 8

Day 9

Day 10

### JEE Mathamatics Study Guide 4. Binary Operations - Revision Facilitator

4.1 Binary operations
4.2 Types of binary operation
4.3 Identity and inverse elements
4.4 Composition table

Study Plan

Day 1

4.1 Binary operations
4.2 Types of binary operation

Day 2

4.3 Identity and inverse elements
4.4 Composition table

Day 3
Objective type exercises 1 to 12
True/false type exercises 1 to 9

### JEE Study Guide 5. Complex Numbers

Contents of the Chapter

5.1 Introduction
5.2 Integral powers of IOTA(i)
5.3 Imaginary quantities
5.4 Complex numbers
5.5 Equality of complex numbers
5.7 Subtraction of complex numbers
5.8 Multiplication of complex numbers
5.9 Division of complex numbers
5.10 Conjugate of a complex number
5.11 Modulus of a complex number
5.12 reciprocal of a complex number
5.13 Square roots of a complex number
5.14 Representation of a complex number
5.15 Argument or amplitude of a complex number z = x+iy for different signs of x and 5.16 Eulerian form of a complex number
5.17 Geometrical representations of fundamental operations
5.17B Modulus and argument of multiplication of two complex numbers
5.18 Modulus and argument of division of two complex numbers
5.19 Geometrical representation of conjugate of a complex number
5.20 Some important results on modulus and argument
5.21 Geometry of complex numbers
5.22 Affix of a point dividing the line segment joining points having affixes z1 and z2
5.23 Equation of the perpendicular bisector
5.24 Equation of a circle
5.25 Complex number as a rotating arrow in the argand plane
5.25B Some important results
5.26 Some standard loci in the argand plane
5.27 Equation of a straight line
5.28 De-moivere’s theorem
5.29 Roots of a complex number
5.30 Roots of unity
5.31 Cube roots of unity
5.32 Logarithm of a complex number

Study Plan

Day 1

5.1 Introduction
5.2 Integral powers of IOTA(i)
Example 1,2
5.3 Imaginary quantities
Ex 1 to 4
5.4 Complex numbers
Ex 1,2
5.5 Equality of complex numbers
Ex 1,2
Ex 1
5.7 Subtraction of complex numbers
Ex 1

Day 2

5.8 Multiplication of complex numbers
Ex 1, 2
5.9 Division of complex numbers
ex 1
5.10 Conjugate of a complex number
5.11 Modulus of a complex number
ex 1

Day 3
5.12 reciprocal of a complex number

5.13 Square roots of a complex number
ex 1 to 5

Day 4

Exercises

Day 5
5.14 Representation of a complex number

5.15 Argument or amplitude of a complex number z = x+iy for different signs of x and y
ex 1, 1 to 5

Day 6

5.16 Eulerian form of a complex number
ex. 1
5.17 Geometrical representations of fundamental operations
5.17B Modulus and argument of multiplication of two complex numbers
5.18 Modulus and argument of division of two complex numbers

Day 7

5.19 Geometrical representation of conjugate of a complex number
5.20 Some important results on modulus and argument

Day 8

5.21 Geometry of complex numbers
5.22 Affix of a point dividing the line segment joining points having affixes z1 and z2
ex 1 to 5

Day 9

5.23 Equation of the perpendicular bisector
5.24 Equation of a circle
Ex 1 to 10.

Day 10

5.25 Complex number as a rotating arrow in the argand plane
ex 1 to 11
5.25B Some important results
ex 1 to 4

Day 11

5.26 Some standard loci in the argand plane
ex 1,2
5.27 Equation of a straight line
ex 1,1

Day 12

5.28 De-Moivere’s theorem
x 3 to 9

Day 13

5.29 Roots of a complex number
ex 1 to 3
5.30 Roots of unity
5.31 Cube roots of unity
ex 1 to 7

Day 14

5.32 Logarithm of a complex number
ex 1,2

Day 15
Illustrative Obj type Examples 1 to 15

From Day 16 take it as revision period

Day 16
I.O.T.E. 16 to 25

Day 17
I.O.T.E. 26 to 35

Day 18
I.O.T.E. 36 to 45

Day 19
46 to 55

Day 20
Objective Type Exercises 1 to 10

Day 21
O.T.E. 11 to 20

Day 22
O.T.E. 21 to 30

Day 23
O.T.E. 31 to 40

Day 24
O.T.E. 41 to 50

Day 25
O.T.E. 51 to 60

Day 26
O.T.E. 61 to 80 do only odd numbered problems

Day 27
O.T.E. 81 to 100

Day 28
O.T.E. 101 to 120

Day 29
O.T.E. 121 to 140

Day 30
O.T.E. 141 to 160

Still the book has many many probles. You have to do these problems as a special task and also as a revision whenever you find extra time.

### IIT JEE Mathematics Study Plan 6. Sequences and Series

Sections in Chapter

6.1 Sequence
6.2 Arithmetic progression
6.3 General term of an A.P.
Selection of terms in an A.P.
6.5 Sum of n terms of an A.P.
6.6 Properties of arithmetical progressions
6.7 Insertion of arithmetic means

6.8 Geometric Progression
6.9 The nth or general term of a G.P.
6.10 Selection of terms in G.P.
6.11 Sum of n terms of a G.P.
6.12 Sum of an Infinite G.P.
6.13 Properties of geometric progressions
6.14 Insertion of geometric means between two given numbers
6.15 Some important properties of arithmetic and geometric means between two given quantities

6.16 Arithmetico-geometric sequence
6.17 Sum of n terms of an arithmetico-geometric sequence
6.18 Sum to n terms of some special sequences
6.19 Miscellaneous sequences and series

6.20 Harmonic progression
6.21 Properties of arithmetic, geometric, and harmonic means between two given numbers.

Study Plan

Day 1

6.1 Sequence
6.2 Arithmetic progression
Ex. 1 to 7
6.3 General term of an A.P.
Ex. 1 to 7

Day 2

6.4 Selection of terms in an A.P.
6.5 Sum of n terms of an A.P.
Ex. 1 to 12
6.6 Properties of arithmetical progressions
Ex. 1 to 4

Day 3
6.7 Insertion of arithmetic means
Ex. 1 to 5
Revision of A.P.

Day 4

6.8 Geometric Progression
Ex. 1 to 3
6.9 The nth or general term of a G.P.
Ex. 1 to 10
6.10 Selection of terms in G.P.
Ex. 1

Day 5
6.11 Sum of n terms of a G.P.
Ex. 1 to 11
6.12 Sum of an Infinite G.P.
Ex. 1 to 11

Day 6
6.13 Properties of geometric progressions
Ex. 1 to 6
6.14 Insertion of geometric means between two given numbers
Ex. 1,2

Day 7
6.15 Some important properties of arithmetic and geometric means between two given quantities
Ex. 1 to 5

Day 8
6.16 Arithmetico-geometric sequence
Ex. 1 to 3
6.17 Sum of n terms of an arithmetico-geometric sequence
Ex. 1 to 6

Day 9
6.18 Sum to n terms of some special sequences
Ex. 1 to 13
6.19 Miscellaneous sequences and series
Ex. 1

Day 10

6.20 Harmonic progression
Ex. 1 to 3
6.21 Properties of arithmetic, geometric, and harmonic means between two given numbers.

Day 11
Chapter 6
Illustrative Objective Type Questions 1 to 20

Day 12
Chapter 6
I.O.T.P.: 21 to 40

Day 13
Chapter 6
I.O.T.P.: 41 to 48
Objective Type questions 1 to 10

Day 14
Chapter 6
O.T.P. 11 to 30

Day 15
Chapter 6
O.T.P. 31 to 50

Day 15
Chapter 6
O.T.P. 51 to 70

Day 16
Chapter 6 Revision
O.T.P. 71 to 80

Day 17
Chapter 6 Revision
O.T.P. 81 to 90

Day 18
Chapter 6 Revision
O.T.P. 91 to 100

Day 19
Chapter 6 Revision
O.T.P. 101 to 110

Day 20
Chapter 6 Revision
O.T.P. 111 to 120

Day 21
Chapter 6 Revision
O.T.P. 121 to 130

Day 22
Chapter 6 Revision
O.T.P. 131 to 140

Day 23
Chapter 6 Revision
O.T.P. 141 to 150

Day 24
Chapter 6 Revision
O.T.P. 151 to 160

Day 25
Chapter 6 Revision
O.T.P. 161 to 163
Formula revision

Day 26
Chapter 6 Revision
Fill in the blanks 1 to 20

Day 27
Chapter 6 Revision
True or false 1 to 12

Day 28
Chapter 6 Revision
Practice Exercises 1 to 20 even problems

Day 29
Chapter 6 Revision
Practice Exercises 21 to 40 even problems

Day 30
Chapter 6 Revision
Practice Exercises 41 to 67 even problems

### IT JEE Mathematics Study Guide 7. Quadratic Equations and Expressions - Revision Facilitator

R.D. Sharma Objective Mathematics

7.1 Some definitions and results
7.2 Some results on roots of an equation
7.3 Position of roots of a polynomial equation
7.4 Descartes rule of signs
7.5 Relations between roots and coefficients
7.6 Formation of a polynomial equation from given roots.
7.7 Transformation of equations
7.8 Roots of a quadratic equation with real coefficients
7.9 Quadratic expression and its graph
7.10 Sign of a quadratic expression for real values of the variable
7.11 Solution of inequations
7.12 Position of roots of a quadratic equation
7.13 Common roots
7.14 Values of a rational expression P(x)/Q(x) for real values of x, where P(x) and Q(x) are quadratic expressions
7.15 Condition for resolution into linear factors of a quadratic function
7.16 Algebraic interpretation of Rolle’s theorem

Study Plan

Always circle difficult concepts or difficult problems. You need to revise them later on more intensively and make them easy (Any difficult issue becomes easy as you understand the concept and related concepts better.)

Day 1

7.1 Some definitions and results
7.2 Some results on roots of an equation
7.3 Position of roots of a polynomial equation
Ex. 1
7.4 Descartes rule of signs
Ex. 1 to 3

Day 2

7.5 Relations between roots and coefficients
Ex. 1 to 3
7.6 Formation of a polynomial equation from given roots.
7.7 Transformation of equations
Ex. 1 to 4

Day 3

7.8 Roots of a quadratic equation with real coefficients

7.9 Quadratic expression and its graph
7.10 Sign of a quadratic expression for real values of the variable

Day 4

7.11 Solution of inequations
Ex. 1,2 and Ex 1,2

7.12 Position of roots of a quadratic equation
EX. 1 to 4

Day 5

7.13 Common roots
Ex. 1 to 3

Day 6
7.14 Values of a rational expression P(x)/Q(x) for real values of x, where P(x) and Q(x) are quadratic expressions
Ex. 1 to 3

7.15 Condition for resolution into linear factors of a quadratic function
Ex. 1 ro 2

Day 7

7.16 Algebraic interpretation of Rolle’s theorem
Ex. 1,2
Illustrative Objective Type Questions: 1 to 10

Day 8
I.O.T.Q.: 11 to 30

Day 9
I.O.T.Q.: 31 to 50

Day 10.
I.O.T.Q.: 51 to 64

Day 11
Objective Type Questions: 1 to 20

Day 12
O.T.P.: 21 to 60 odd numbered questions
(Persons who can do more problems can do more problems)

Day 13
O.T.P.: 61 to 100 even numbered questions
(Persons who can do more problems can do more problems)

Day 14
O.T.P.: 101 to 140 odd numbered questions
(Persons who can do more problems can do more problems)

Day 15
O.T.P.:141 to 184 even numbered questions
(Persons who can do more problems can do more problems)

Revision Period

Day 16
Fill in the blanks exercises: 1 to 20

Day 17
True or false exercises: 1 to 13

Day 18
Practice Exercises: 1 to 10

Day 19
Practice Exercises:11 to 20

Day 20
Practice Exercises: 21 to 30

Day 21
Practice Exercises: 31 to 40

Day 22
Practice Exercises: 41 to 50

Day 23
Practice Exercises: 51 to 60

Day 24
Practice Exercises: 61 to 63

Day 25
Revision of concepts of the chapter

Day 26
O.T.P.: 21 to 60 even numbered questions

Day 27
O.T.P.: 61 to 100 odd numbered questions

Day 28
O.T.P.: 101 to 140 even numbered questions

Day 29
O.T.P.:141 to 184 odd numbered questions

Day 30
Revision of some difficult problems

### IIT JEE Mathematics Study Plan 8. Permutations and Combinations - Revision Facilitator

Chapters in the R.D. Sharma Book

8.1 The factorial
8.2 Exponent of prime p in n!
8.3 Fundamental principles of counting
8.4 Permutations
8.5 Permutations under certain conditions
8.6 Permutations of objects not all distinct
8.7 Permutations when objects can repeat
8.8 Circular permutations
8.9 combinations
8.10 Practical problems on combinations
8.11 Mixed problems on permutations and combinations
8.12 Selection of one or more items
8.13 Division of items into groups
8.14 Division of identical objects into groups
8.15 Some important results

Study plan

Day 1

1. The Factorial
2. Exponent of Prime p in n!
3. Fundamental principles of counting

Day 2

8.4 Permutations
4 theorems and 20 examples

8.5 Permutations under certain conditions
Three theorems and 20 examples

Day 3

8.6 Permutations of Objects not all Distinct
Theorem 1, 11 examples

8.7 Permutations when Objects can Repeat
1 theorem, 6 examples

8.8 Circular Permutations

1 theorem, 10 examples

Day 4

8.9 Combinations
Theorem 1,
Remarks 3
Properties 7
Examples 6

8.10 Practical Problems on Combinations
24 examples

8.11 Mixed Problems on Combinations and Permutations

Examples 6

Total examples for the day 36

Day 5

8.12 Selection of one or more items
Sub topics

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

4 examples

8.13 Division of Items into Groups
Division of items into groups of unequal size
Division of items into groups of equal size

5 examples

8.14 Division of Identical Objects into groups

3 results and 7 examples

8.15 Some important results

4 results

Day 6

Illustrative objective type questions 1 to 10

Attempt objective type exercises 1 to 10

Attempt Practice exercises (given at the end of the chapter)

1 to 4 and 6

Day 7

Illustrative objective type questions 11 to 20

Attempt objective type exercises 11 to 20

Attempt Practice exercises (given at the end of the chapter)
8,23,24,25,28,29

Day 8

Illustrative objective type questions 21 to 40

Attempt objective type exercises 21 to 50

Attempt Practice exercises (given at the end of the chapter)
51 to 55

Day 9

Illustrative objective type questions 41 to 53

Attempt objective type exercises 51 to 80

Attempt Practice exercises (given at the end of the chapter)
56 to 60

Day 10

81-120

Attempt objective type exercises 51 to 80

Attempt Practice exercises (given at the end of the chapter)
31 to 35

Revision period

Day 11

Do 121 – 130 in Objective type exercises.

Day 12

Do 131 – 141 in Objective type exercises.

Day 13

Do the Fill in the blanks type exercise questions 1 to 5

Day 14

Do the Fill in the blanks type exercise questions 6 to 15

Day 15

From Permutations chapter

Do the Fill in the blanks type exercise questions 16 to 20

Day 16

Do the Fill in the blanks type exercise questions 21 to 25

Day 17

Do the Fill in the blanks type exercise questions 26 to 33

Day 18 to 30
Attempt all the practice questions and test paper questions

### IIT JEE Mathematics Study Guide 9. Binomial Theorem

Text Book R D Sharma Objective Mathematics

Day 1

Go through 9.1, 9.2 and 9.3 today

Day 2

Go through
9.3 Multinomial theorem
9.4 Middle terms in a Binomial expansion
9.5 Some important results

Day 3

Go through
9.6 Some problems on applications of Binomial theorem
9.7 Greatest term in the expansion of (x+a)n
9.8 S Properties of the binomial coefficients

Day 4

Go through
9.9 Binomial Theorem for any index
9.10 synopsis

Day 5

Do illustrative objective type examples

1 to 20

Do practice exercises (at the end of the chapter) 1 to 5

Day 6

Do illustrative objective type examples

21 to 40
Do practice exercises (at the end of the chapter) 6 to 10

Day 7

Do illustrative objective type examples

41 to 52

In this, past IIT problems are
45, 52,

Do practice exercises (at the end of the chapter) 11 to 15

Day 8

Do objective type exercises

1 to 20

Do practice exercises (at the end of the chapter) 16 to 20

Q. 19 is a past IIT question.

Day 9

Do objective type exercises

21 to 40

Do practice exercises (at the end of the chapter) 21 to 25

Day 10

Do objective type exercises

41 to 60

Do practice exercises (at the end of the chapter) 26 to 30

Revision Period

Day 11

61-70 in objective exercises

Day 12

71-80 in objective exercises

Day 13

81-90 in objective exercises

day 14

91-100 in objective exercises

Day 15

100-110 in objective exercises

Day 16

111-120 in objective exercises

Day 17

121-130 in objective exercises

Day 18

131-135 in objective exercises

Day 19

Practice exercise problems 1 to 5
Fill in the blank type exercises 1 to 5

Day 20

Practice exercise problems 6 to 15
Do True or False Questions 1 to 7

Day 21

Practice exercise problems 16 to 20 (There are a total of 57 problems in this exercise)

Day 22

Practice exercise problems 21 to 25 (There are a total of 57 problems in this exercise)

Day 23

In the previous chapter Binomial theorem do problems

Practice exercise problems 26 to 30 (There are a total of 57 problems in this exercise)

Day 24

Practice exercise problems 31 to 35

Day 25

Practice exercise problems 36 to 40

Day 26

Concept Revision Facilitator

Try to recollect relevant points on the topic.

If required right click on the topic and click on open in a new window to read the relevant material.

Close the window and come back.

1 Pascal’s Triangle of Binomial coefficients

2 Binomial theorem for positive integral index.

3 Some Deductions from Binomial Theorem

4 Different version of Binomial theorem

5 Multinomial theorem

6 Middle terms in a binomial expansion

7 Results related to Binomial Coefficieints

8 Some problems on applications of binomial theorem

9 Finding greatest term in the expansion of (x+a)^n

10 Properties of the binomial coefficients

11. Binomial theorem for any index

Join IIT JEE Academy Orkut Community for more interaction

http://www.orkut.co.in/Main#Community.aspx?cmm=39291603

### 10. Exponentail and Logarithmic Series - Revision Facilitator

Try to recollect relevant points on the topic.

If required right click on the topic and click on open in a new window to read the relevant material.

Close the window and come back.

1. Number e and Related Ideas

2. Exponential series

3. Exponential theorem

4. Some deductions from exponential series

5. Relations of e (e-1, e-2 etc.)

6. Logarithmic series

### IIT JEE Mathematics Study Guide 12. Determinants

12.1 Definition
12.2 Singular matrix
12.3 Minors and cofactors
12.4 Properties of determinants
12.5 Evaluation of determinants
12.6 Evaluation of determinants by using factor theorem
12.7 Product of determinants
12.8 Differentiation of determinants
12.9 Applications of determinants to coordinate geometry
12.10 Applications of determinants in solving a system of linear equations

Study Plan

Day 1

12.1 Definition
There are three examples in this section
12.2 Singular matrix
12.3 Minors and cofactors

Day 2

12.4 properties of determinants
12.5 Evaluation of Determinants

Day 3

Do all the 19 example problems

Day 4

Study sections

12.6 Evaluation of Determinants using Factor Theorem
2 examples

12.7 Product of determinants
5 examples

Day 5

Do illustrative obj type examples 1 to 20

Day 6

12.8 Differentiation of determinants
12.9 Applications of determinants to coordinate geometry

Day 7
12.10 Applications of determinants in solving a system of linear equations

Day 8
I.O.T.E. 21 to 28
Objective Type Exercises 1 to 10

Day 9
O.T.E. 11 to 30

Day 10
O.T.E. 331 to 50

Day 11
O.T.E. 51 to 72

Day 11
Fill in the blansk 1 to 24

Day 12
True or false 6

Day 13 to 30
Revision Concepts and test paper problems

### IIT JEE Study Guide 13. Cartesian System of Coordinates and Straight Lines - Revision Facilitator

1. Introduction

2. Catesian coordinate system

3. Distance between two points

4. Area of a triangle

5. Section Formulae

6. coordinates of the centroid, in-centre, and ex centre of a triangle

7. Locus and equation to a locus

8. Shifting of origin

9. Rotation of axes

10. Definition of a straight line

11. Slope (Gradient) of a straight line

12. Angle between two straight lines

13. Intercepts of a line on the axes

14. Equations of lines parallel to the coordinate axes

15. Different forms of the equation of a straight line

16. Transformation of general equation in different standard forms

17. point of intersection of two lines

18. Coordination of concurrency of three lines

19. Lines parallel and perpendicular to a given line

20. Angle between two straight lines when their equations are given.

21. Distance of a point from a line

22. Positions of points relative to a line

23. Equations of straight lines passing through a given point and making a given angle with a given line

24. Equations of the bisectors of the angles between two straight lines

25. Some important points of a triangle

26. Family of lines through the intersection of two given lines

Study Plan

Day 1

1. Introduction

2. Catesian coordinate system

3. Distance between two points

Day 2

4. Area of a triangle

5. Section Formulae

6. coordinates of the centroid, in-centre, and ex centre of a triangle

Day 3

7. Locus and equation to a locus

8. Shifting of origin

Day 4

9. Rotation of axes

10. Definition of a straight line

11. Slope (Gradient) of a straight line

12. Angle between two straight lines

Day 5

13. Intercepts of a line on the axes

14. Equations of lines parallel to the coordinate axes

15. Different forms of the equation of a straight line

Day 6

16. Transformation of general equation in different standard forms

17. point of intersection of two lines

18. Coordination of concurrency of three lines

Day 7

19. Lines parallel and perpendicular to a given line

20. Angle between two straight lines when their equations are given.

21. Distance of a point from a line

Day 8

22. Positions of points relative to a line

23. Equations of straight lines passing through a given point and making a given angle with a given line

Day 9

24. Equations of the bisectors of the angles between two straight lines

25. Some important points of a triangle

26. Family of lines through the intersection of two given lines - Up to Example 10

Day 10
26. Family of lines through the intersection of two given lines - Examples 21 to 30

Day 11
Illustrative Objective Type Examples 1 to 19

Day 12
Objective Type Exercise 1 to 20

Day 13
O.T.E. 21 to 40

Day 14
O.T.E. 41 to 60

Day 15
O.T.E. 61 to 80

Revision Period - 30 minutes a day

Day 16
O.T.E. 81 to 90

Day 17
O.T.E. 91 to 100

Day 18
O.T.E. 101 to 110

Day 19
O.T.E. 111 to 120

Day 20
O.T.E. 121 to 130

Day 21
O.T.E. 131 to 142

Day 22
Fill in the blanks type exercise 1 to 10

Day 23
Fill in the blanks type exercise 11 to 20

Day 24
Fill in the blanks type exercise 21 to 30

Day 25
Fill in the blanks type exercise 31 to 40

Day 26
True of False Exercise 1 to 19

Day 27
Practice Exercise 1 to 10

Day 28
Practice Exercise 11 to 20

Day 29
Practice Exercise 21 to 30

Day 30
31 to 41

Should you compulsorily do every problem. Not necessary. If you feel, you know how to do it, you can always skip some problems. But make a note of all difficult problems. You may have to a relook at them sometimes more to remember the complication or complexity in that problem.

Try to recollect relevant points on the topic.

If required right click on the topic and click on open in a new window to read the relevant material.

Close the window and come back.

Euclidean Geometry and Analytic Geometry - Difference

Distance between two points

Area of a triangle

Section Formulae

coordinates of the centres related to triangle

Definition of a straight line

Slope (Gradient) of a straight line

Angle between two straight lines

Intercepts of a line on the axes

Joint equation of a pair of straight lines

### IIT JEE Mathematics Study Plan 14. The Family of Lines - - Revision Facilitator

14.1 Introduction
14.2 Joint equation of a pair of straight lines
14.3 Pair of straight lines through the origin
14.4 Angle between pair of lines given by ax² +2hxh+by² = 0
14.5 Bisectors of the angle between the lines given by a homogeneous equation
14.6 General equation of a second degree
14.7 Equations of the bisectors of the angles between the lines represented by the equation ax² +2hxh+by²+2gx+2fy+c = 0
14.8 Lines joining the origin to the points of intersection of a line and curve

Study Plan

10 Days for both main study and revision

Day 1

14.1 Introduction
14.2 Joint equation of a pair of straight lines
14.3 Pair of straight lines through the origin

Day 2

14.4 Angle between pair of lines given by ax² +2hxh+by² = 0
14.5 Bisectors of the angle between the lines given by a homogeneous equation
14.6 General equation of a second degree

Day 3

14.7 Equations of the bisectors of the angles between the lines represented by the equation ax² +2hxh+by²+2gx+2fy+c = 0
14.8 Lines joining the origin to the points of intersection of a line and curve

Day 4
Illustrative Objective Type Examples 1 to 7
Objective Type Exercise 1 to 10

Day 5
O.T.E. 11 ro 32

day 6
Fill in the blanks type Exercise 1 to 5
practce exercise 1 to 10

Day 7
Practice exercise 11 to 28

Day 8 to 10 Revision
Concepts, Formulas, and test paper problems

### IIT JEE Mathematics Study Plan 15. The Circle

Sections in the chapter

1. Definition
2. Standard equation of a circle
3. Some particular cases of the central form of the equation of a circle
4. General equation of a circle
5. Equation of a circle when the co-ordinates of end points of a diameter are given
6. Intercepts on the axis
7. Position of a point with respect to a circle
8. Equation of a circle in parametric form
9. Intersection of a straight line and a circle
10. The length of the intercept cut off from a line by a circle
11. Tangent to a circle at a given point.
12. Normal to a circle at a given point
13. Length of the tangent from a point to a circle
14. Pair of tangents drawn from a point to a given circle
15. Combined equation of pair of tangents
16. Director circle and its equation
17. Chord of contact of tangents
18. Pole and polar
19. Equation of the chord bisected at a given point
20. Diameter of a circle
21. Common tangents to two circles
22. Common chord of two circles
23. Angle of intersection of two curves and the condition of orthogonality of two circles.
25. Equation of a circle through the intersection of a circle and line
26. Circle through the intersection of two circles
27. Coaxial system of circles

Study Plan

Sections in the chapter

Day 1

1. Definition
2. Standard equation of a circle
3. Some particular cases of the central form of the equation of a circle
4. General equation of a circle

Day 2

5. Equation of a circle when the co-ordinates of end points of a diameter are given
6. Intercepts on the axis
7. Position of a point with respect to a circle
8. Equation of a circle in parametric form

Day 3

9. Intersection of a straight line and a circle
10. The length of the intercept cut off from a line by a circle
Illustrative Objective Type Examples: 1 to 8

Day 4

11. Tangent to a circle at a given point.
12. Normal to a circle at a given point
13. Length of the tangent from a point to a circle

day 5

14. Pair of tangents drawn from a point to a given circle
15. Combined equation of pair of tangents
16. Director circle and its equation
17. Chord of contact of tangents

Day 6

18. Pole and polar
19. Equation of the chord bisected at a given point
20. Diameter of a circle
21. Common tangents to two circles

Day 7
Concept revision
I.O.T.E. 7 to 25 (If some problems cannot be understood leave them for later days)

Day 8

22. Common chord of two circles
23. Angle of intersection of two curves and the condition of orthogonality of two circles.

Day 9
25. Equation of a circle through the intersection of a circle and line
26. Circle through the intersection of two circles
27. Coaxial system of circles

Day 10
I.O.T.E. 26 to 53

Revision Period

Day 11

Objective Type Exercises 1 to 10

Day 12
Objective Type Exercises 11 to 20

Day 13
Objective Type Exercises 21 to 30

Day 14
Objective Type Exercises 31 to 40

Day 15
Objective Type Exercises 41 to 50

Day 16
Objective Type Exercises 51 to 60

Day 17
Objective Type Exercises 61 to 70

Day 18
]Objective Type Exercises 71 to 80

Day 19
Objective Type Exercises 81 to 90

Day 20
Objective Type Exercises 91 to 100

### IIT JEE Mathematics Study Plan 16. Parabola - Revision Facilitator

Ch. 16. Parabola-RDS-
Sections

1. Conic sections: Definition

2. The parabola

3. Equation of parabola in its standard form

4. Some other standard forms of parabola

5. Position of a point with respect to a parabola

6. Equation of a parabola in parametric form

7. Equation of the chord joining any two points on the parabola

8. Intersection of a straight line and a parabola

9. Equation of tangent in different forms

10. Equation of normal in different forms

11 Number of normals drawn from a point to a parabola

12. Some results in conormal points

13 Number of tangents drawn from a point to a parabola

13a. Equation of the pair of tangents from a point to a parabola

14. Equation of the chord of contacts of tangents to a parabola

15. Equation of the chord bisected at a given point

16. Equation of diameter of a parabola

17. Length of tangent, subtangent, normal and subnormal

18. Pole and Polar

19. Some important results at a glance

Study Plan

1. Conic sections: Definition

2. The parabola

3. Equation of parabola in its standard form

4. Some other standard forms of parabola

5. Position of a point with respect to a parabola

6. Equation of a parabola in parametric form

7. Equation of the chord joining any two points on the parabola

8. Intersection of a straight line and a parabola

9. Equation of tangent in different forms

10. Equation of normal in different forms

11 Number of normals drawn from a point to a parabola

12. Some results in conormal points

13 Number of tangents drawn from a point to a parabola

13a. Equation of the pair of tangents from a point to a parabola

14. Equation of the chord of contacts of tangents to a parabola

15. Equation of the chord bisected at a given point

16. Equation of diameter of a parabola

17. Length of tangent, subtangent, normal and subnormal

18. Pole and Polar

19. Some important results at a glance

### IIT JEE Mathematics Study Plan 17. Ellipse

Ch. 17. Ellipse Parabola-RDS-

Sections

1. Introduction

2. Equation of ellipse in its standard form

3. Second focus and second directrix of the ellipse

4. Vertices, major and minor axes, foci, directrices and centre of the ellipse

5. Ordinate, double ordinate and latus rectum of the ellipse

6. Focal distances of a point on the ellipse

7. Equation of ellipse in other forms

8. Position of a point with respect to an ellipse

9. Parametric equations and parametric coordinates

10. Equation of the chord joining any two points on an ellipse

11. Condition of a line to be a tangent to an ellipse

12. Equation of tangent in terms of its slope

13. Equation of tangent at a point

14. Number of tangents drawn from a point to an ellipse

15. Equation of normal in different forms

16. Number of normals

17. Properties of eccentric angles of the conormal points

18. Equation of the pair of tangents from a point to an ellipse

19. Equation of the chord of contacts of tangents

19a. Equation of the chord bisected at a given point

20. Equation of diameter of an ellipse

21. Some properties of ellipse

Study Plan

Day 1

1. Introduction

2. Equation of ellipse in its standard form

3.Second focus and second directrix of the ellipse

4. Vertices, major and minor axes, foci, directrices and centre of the ellipse

5. Ordinate, double ordinate and latus rectum of the ellipse

Day 2

6. focal distances of a point on the ellipse

7. equation of ellipse in other forms

Day 3

8. Position of a point with respect to an ellipse

9. Parametric equations and parametric coordinates

10. Equation of the chord joining any two points on an ellipse

11. Condition of a line to be a tangent to an ellipse

Day 4

12. Equation of tangent in terms of its slope

13. Equation of tangent at a point
Ex. 1 to 11

Day 5

14 Number of tangents drawn from a point to an ellipse

15. Equation of normal in different forms

16 Number of normals

Day 6

17. Properties of eccentric angles of the conormal points

18. Equation of the pair of tangents from a point to an ellipse

19. Equation of the chord of contacts of tangents

19a. Equation of the chord bisected at a given point

Day 7

20. Equation of diameter of an ellipse

21. Some properties of ellipse

Day 8
Illustrated Objective Type Examples 1 to 18

Day 9
Objective Type Exercises 1 to 20

Day 10
O.T.E. 21 to 40

Revision

Day 11
O.T.E. 41 to 50

Day 12
O.T.E. 51 to 60

Day 13
O.T.E. 61 to 70

Day 14
O.T.E. 71 to 73

Day 15
Concept revision

Day 16
Formula revision

Day 17 to 20
Test paper problems

### IIT JEE Mathematics Study Plan 18. Hyperbola

Sections in the R D Sharma Chapter

1. Hyperbola - Definitions

2. Equation of hyperbola in its standard form

3. Second focus and second directrix of the hyperbola

4. Vertices, major and minor axes, foci, directrices and centre of the hyperbola

5. Eccentricity

6. Length of latus rectum

7. Focal distances of a point

8. Conjugate hyperbola

9 .Parametric equations and parametric coordinates

10. Equation of the chord joining any two points on a hyperbola

11. Intersection of a line and a hyperbola

12. Condition of a line to be a tangent to a hyperbola

13. Equation of tangent in different forms

14. Number of tangents drawn from a point to a hyperbola

15. Equation of the pair of tangents from a point to a hyperbola

16. Equation of the chord of contacts of tangents

17. Equation of normal in different forms

18 Number of normals

19. Equation of the chord of a hyperbola bisected at a given point

20 Asymptotes of a hyperbola

21. Rectangular hyperbola

Study Plan

Day 1

1. Hyperbola - Definitions

2. Equation of hyperbola in its standard form

3. Second focus and second directrix of the hyperbola

4. Vertices, major and minor axes, foci, directrices and centre of the hyperbola

5. Eccentricity

day 2

6. Length of latus rectum

7. Focal distances of a point

8. Conjugate hyperbola

Day 3

9. Parametric equations and parametric coordinates

10. Equation of the chord joining any two points on a hyperbola

11. Intersection of a line and a hyperbola.

Revision of sections covered so far.

Day 4

12. Condition of a line to be a tangent to a hyperbola

13. Equation of tangent in different forms

14. Number of tangents drawn from a point to a hyperbola

15. Equation of the pair of tangents from a point to a hyperbola

16. Equation of the chord of contacts of tangents

Day 5

17. Equation of normal in different forms

18 Number of normals

19. Equation of the chord of a hyperbola bisected at a given point

20 Asymptotes of a hyperbola

Day 6

21. Rectangular hyperbola

Illustrative Objective Type Examples 1 to 5

Day 7
I.O.T.P. 6 to 17

Day 8
Objective Type Exercise: 1 to 20

Day 9
o.T.E. 21 to 44

Day 10
Concept Review
Formula Review

Revision Period

Day 11 to 20
Revision of concepts and test paper problem solving

Revision Facilitator

1. Hyperbola - Definitions

2. Equation of hyperbola in its standard form

3. Second focus and second directrix of the hyperbola

4. Vertices, major and minor axes, foci, directrices and centre of the hyperbola

5. Eccentricity

6. Length of latus rectum

7. Focal distances of a point

8. Conjugate hyperbola

9 .Parametric equations and parametric coordinates

10. Equation of the chord joining any two points on a hyperbola

11. Intersection of a line and a hyperbola

12. Condition of a line to be a tangent to a hyperbola

13. Equation of tangent in different forms

14. Number of tangents drawn from a point to a hyperbola

15. Equation of the pair of tangents from a point to a hyperbola

16. Equation of the chord of contacts of tangents

17. Equation of normal in different forms

18 Number of normals

19. Equation of the chord of a hyperbola bisected at a given point

20 Asymptotes of a hyperbola

21. Rectangular hyperbola

### IIT JEE Mathematics Study Plan 19. Real functions & Revision Facilitator

Sections in the Chapter

1.Introduction

2 Description of real functions

3 Intervals (Closed and open)

4 Domains and ranges of real functions

5 Real functions - Examples

6 Operations on real functions

7 Even and odd functions

8 Extension of a function

9 Periodic function

Study Plan

Day 1

1.Introduction

2 Description of real functions
Ex 1 to 9

Day 2

3 Intervals (Closed and open)

4 Domains and ranges of real functions

5 Real functions - Examples

Day 3

6 Operations on real functions

7 Even and odd functions

Day 4

8 Extension of a function

9 Periodic function

Day 5
Objective Type Exercises: 1 to 20

Day 6
O.T.E. 21 to 40

Day 7
O.T.E. 41 to 60

Day 8
O.T.E. 61 to 80

Day 9
O.T.E. 81 to 100

Day 10
O.T.E. 101 to 116

Day 11
Fill in the blanks type exercise 1 to 11

Day 12 concept Revision

Day 13 to 20
Revision and test paper problem solving.

1 Description of real functions

2 Intervals (Closed and open)

3 Domains and ranges of real functions

4 Real functions - Examples

5 Operations on real functions

6 Even and odd functions

7 Extension of a function

8 Periodic function

### IIT JEE Mathematics Study Plan - 20. Limits - Revision Facilitator

Sections in the Chapter

20.1 Informal approach to limit
20.2 Formal approach to limit
20.3 Evaluation of left hand and right hand limits
20.4 Difference between the value of a function at a point and the limit at a point
20.5 The algebra of limits
20.6 Evaluation of limits

Study Plan

Day 1

20.1 Informal approach to limit
20.2 Formal approach to limit
20.3 Evaluation of left hand and right hand limits

Day 2
20.4 Difference between the value of a function at a point and the limit at a point
20.5 The algebra of limits
20.6 Evaluation of limits - Upto Evaluation of Limits using standard results

Day 3
20.6 Evaluation of limits

Day 4
Objective Type Exercises: 1 to 20

Day 5
O.T.E.: 21 to 40

Day 6
O.T.E.: 41 to 60

Day 7
O.T.E.: 61 to 80

Day 8
O.T.E.: 81 to 100

Day 9
O.T.E.: 101 to 120

Day 9
O.T.E.: 121 to 133

Day 10
Fill in the blanks type exercise 1 to 16

Revision Period
Day 11
True or Fales Questions 1 to 15

Day 12
Practice Exercises : 1 to 10

Day 13
Practice Exercises : 11 to 20

Day 14
Practice Exercises : 21 to 30

### IIT JEE Mathematics Study Plan 21. Continuity and Differentiability - Revision Facilitator

Sections in the Chapter

21.1 Introduction
21.2 Continuity at a point
21.3 Continuity functions in an interval
21.4 Continuous functions
21.5 Cauchy’s definition of continuity
21.6 Heine’s definition of continuity
21.7 Discontinuous functions
21.8 Properties of continuous functions
21.9 Differentiability at a point
21.10 Relation between continuity and differentiability
21.11 Differentiability in a set
21.12 Some results on differentiability

Study Plan

Day 1

21.1 Introduction

21.2 Continuity at a point

21.3 Continuity functions in an interval

21.4 Continuous functions

21.5 Cauchy’s definition of continuity

Day 2

21.6 Heine’s definition of continuity
21.7 Discontinuous functions
21.8 Properties of continuous functions

Day 3
21.9 Differentiability at a point
21.10 Relation between continuity and differentiability
21.11 Differentiability in a set
21.12 Some results on differentiability

Day 4
Objective Type Exercises: 1 to 20

Day 5
O.T.E.: 21 to 40

Day 6
O.T.E.: 41 to 60

Day 7
O.T.E.: 61 to 80

Day 8
O.T.E.: 81 to 100

Day 9
O.T.E.: 101 to 127

Day 10
Fill in the blank Exercises 1 to 21

Day 11
True/False Type Exercises 1 to 11

Day 12
Practice Exercises 1 to 7

Day 13
Practice Exercises 8 to 14

21.1 Introduction

21.2 Continuity at a point

21.3 Continuity functions in an interval

21.4 Continuous functions

21.5 Cauchy’s definition of continuity
21.6 Heine’s definition of continuity
21.7 Discontinuous functions
21.8 Properties of continuous functions
21.9 Differentiability at a point
21.10 Relation between continuity and differentiability
21.11 Differentiability in a set
21.12 Some results on differentiability

### IIT JEE Mathematics Study Plan 22. Differentiation and Revision Facilitator

Sections in the Chapter

22.1 Differentiation

22.2 Geometrical meaning of derivative at a point

22.3 Differentiation of some standard functions

22.4 Fundamental Rules of Differentiation

22.5 Relation between dy/dx and dx/dy

22.6 Differentiation of implicit functions

22.7 Logarithmic differentiation

22.8 Differentiation of parametric functions

22.9 Differentiation of a function with respect to another function

22.10 Higher order derivatives

Study Plan

Day 1

22.1 Differentiation

22.2 Geometrical meaning of derivative at a point

22.3 Differentiation of some standard functions

22.4 Fundamental Rules of Differentiation

22.5 Relation between dy/dx and dx/dy

22.6 Differentiation of implicit functions

Day 2
Ilustrative Objective Type Examples 1 to 10 (Ignore if they are on topics not covered so far)
Objective Type Exercises 1 to 10 (Ignore if they are on topics not covered so far)

Day 3

22.7 Logarithmic differentiation

22.8 Differentiation of parametric functions

22.9 Differentiation of a function with respect to another function

22.10 Higher order derivatives

Day 4
I.O.T.E.: 11 to 21
O.T.E.: 11 to 20

Day 5
O.T.E.: 21 to 40

Day 6
O.T.E.: 41 to 56

Day 7
Fill in the blanks type exercise 1 to 11
True/False type exercise 1 to 4

Day 8
Practice Exercises 1 to 20

Day 9
Practice Exercise 21 to 26
Concept Revision

Day 10
Formula Revision

Revision Period

Day 11 to 20
Test Paper problem solving

Revision Facilitator

Try to recollect relevant points on the topic.

If required right click on the topic and click on open in a new window to read the relevant material.

Close the window and come back for the next topic.

22.1 Differentiation - definition

22.2 Geometrical meaning of derivative at a point

22.3 Differentiation of some standard functions

22.4 Fundamental Rules of Differentiation

22.5 Relation between dy/dx and dx/dy

22.6 Differentiation of implicit functions

22.7 Logarithmic differentiation

22.8 Differentiation of parametric functions

22.9 Differentiation of a function with respect to another function

22.10 Higher order derivatives

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### IIT JEE Mathematics Study Plan 23. Tangents, Normals and other Applications of Derivatives - Revision Facilitator

Sections in the Chapter

1 Slopes of the tangent and the norm

2 Equations of tangent and normal

3 Angle of intersection of two curves

4 Lengths of tangent, normal

5. Subtangent, and subnormal

6. Rolle’s theorem

7. Lagrange’s mean value theorem

Study Plan

Day 1

1 Slopes of the tangent and the norm

2 Equations of tangent and normal

3 Angle of intersection of two curves

Day 2

4 Lengths of tangent, normal, Subtangent, and subnormal

5. Rolle’s theorem

6. Lagrange’s mean value theorem

Day 3

Illustrative Objective Type Examples 1 to 13
Objective type exercise 1 to 10

Day 4

O.T.E.: 11 to 30

Day 4

O.T.E.: 31 to 50

Day 5

O.T.E.:51 t 61
Bulletin board: 1 to 8

Day 6

Practice Exercise 1 to 20

Day 7

Practice Exercise 21 to 40

Revision Facilitator

Try to recollect relevant points on the topic.

If required right click on the topic and click on open in a new window to read the relevant material.

Close the window and come back.

1 Slopes of the tangent and the norm

2 Equations of tangent and normal

3 Angle of intersection of two curves

4 Lengths of tangent, normal

5.Subtangent, and subnormal

6.Rolle’s theorem

7. Lagrange’s mean value theorem

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