Sunday, May 25, 2008
22. Differentiation - Revision points 1
4. Fundamental rules for 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
5. Relation between dy/dx and dx/dy
dy/dx = 1/{dx/dy)
6. 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).
7. 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]
8. Differentiation of parametric form
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)
9. 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)
10. 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
y_{1}, y_{2}
y’, y’’
Dy, D²y
f’(x), f’’(x)
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
5. Relation between dy/dx and dx/dy
dy/dx = 1/{dx/dy)
6. 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).
7. 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]
8. Differentiation of parametric form
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)
9. 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)
10. 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
y_{1}, y_{2}
y’, y’’
Dy, D²y
f’(x), f’’(x)
22. Differentiation of Elementary 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 x^{n} = nx^{n-1}
d/dx of log x = 1/x
d/dx of e^{x} = e^{x}
d/dx of a^{x} = a^{x}log a
d/dx of sin^{-1}x = 1/√(1-x²)
d/dx of tan^{-1} x = 1/(1+x²)
d/dx of cosx = - sin x
d/dx of tanx = sec²x
d/dx of cot x = -cosec²x
d/dx of x^{n} = nx^{n-1}
d/dx of log x = 1/x
d/dx of e^{x} = e^{x}
d/dx of a^{x} = a^{x}log a
d/dx of sin^{-1}x = 1/√(1-x²)
d/dx of tan^{-1} x = 1/(1+x²)
22. Differentiation -2
Leibnitz Theorem and nth derivative
if u(x) and v(x)are functions possessing derivatives up to nth order.
Then
(uv)_{n} = u_{n}(x)v(x)+^{n}C_{1}u_{n-1}(x)v_{1}(x)+...+^{n}C_{k}u_{n-k}(x)v_{k}(x) +...+^{n}C_{n}u(x)v_{n}(x)
where u_{k}(x)= d^{k}(u(x))/dx^{k} (k-th derivative of the function u(x), 1≤k≤n
n-derivatives of some elementary functions
d^{n}(x^{m})/dx^{n} = (m!x^{m-n})/(m-n)! if n≤m
and
d^{n}(x^{m})/dx^{n} = 0 if n>m.
d^{n}(sin x)/dx^{n} = sin (x + nπ/2)
d^{n}(cos x)/dx^{n} = cos (x + nπ/2)
d^{n}(e^{mx})/dx^{n} = m^{n}e^{mx}
if u(x) and v(x)are functions possessing derivatives up to nth order.
Then
(uv)_{n} = u_{n}(x)v(x)+^{n}C_{1}u_{n-1}(x)v_{1}(x)+...+^{n}C_{k}u_{n-k}(x)v_{k}(x) +...+^{n}C_{n}u(x)v_{n}(x)
where u_{k}(x)= d^{k}(u(x))/dx^{k} (k-th derivative of the function u(x), 1≤k≤n
n-derivatives of some elementary functions
d^{n}(x^{m})/dx^{n} = (m!x^{m-n})/(m-n)! if n≤m
and
d^{n}(x^{m})/dx^{n} = 0 if n>m.
d^{n}(sin x)/dx^{n} = sin (x + nπ/2)
d^{n}(cos x)/dx^{n} = cos (x + nπ/2)
d^{n}(e^{mx})/dx^{n} = m^{n}e^{mx}
27 Definite Integrals - Revision points - 1
Properties of definite integrals
1. ∫_{a}^{b} f(x)dx = -∫_{b}^{a} f(x)dx
2. ∫_{a}^{b} f(x)dx = ∫_{a}^{c} f(x)dx + ∫_{c}^{b} f(x)dx
3. ∫_{0}^{a} f(x)dx = ∫_{0}^{a} f(a-x)dx
4. If f(-x) = f(x) (means f is an even function), then
∫_{-a}^{a} f(x)dx = 2∫_{0}^{a} f(x)dx
5. If f(-x) = -f(x) (means f is an odd function), then
∫_{-a}^{a} f(x)dx = 0
6. ∫_{0}^{a}f(x)dx = ∫_{0}^{a}f(a-x)dx and
∫_{a}^{b}f(x)dx = ∫_{a}^{b}f(a+b-x)dx
7. ∫_{0}^{a}f(x)dx = ∫_{0}^{a/2}f(x)dx+∫_{0}^{a/2}f(a-x)dx
Due to the above relation
∫_{0}^{a}f(x)dx = 0 if f(a-x) = -f(x)
∫_{0}^{a}f(x)dx = 2∫_{0}^{a/2}f(x)dx if f(a-x) = f(x)
8. If f is continuous on [a,b], then the integral function defined by g(x) = ∫_{a}^{x}f(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
∫_{a}^{b}f(x)dx = ∫_{a+nT}^{b+nT}f(x)dx, where n is an integer.
In particular
∫_{0}^{nT}f(x)dx = n∫_{0}^{T}f(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)≤∫_{a}^{b}f(x)dx≤M(b-a)
1. ∫_{a}^{b} f(x)dx = -∫_{b}^{a} f(x)dx
2. ∫_{a}^{b} f(x)dx = ∫_{a}^{c} f(x)dx + ∫_{c}^{b} f(x)dx
3. ∫_{0}^{a} f(x)dx = ∫_{0}^{a} f(a-x)dx
4. If f(-x) = f(x) (means f is an even function), then
∫_{-a}^{a} f(x)dx = 2∫_{0}^{a} f(x)dx
5. If f(-x) = -f(x) (means f is an odd function), then
∫_{-a}^{a} f(x)dx = 0
6. ∫_{0}^{a}f(x)dx = ∫_{0}^{a}f(a-x)dx and
∫_{a}^{b}f(x)dx = ∫_{a}^{b}f(a+b-x)dx
7. ∫_{0}^{a}f(x)dx = ∫_{0}^{a/2}f(x)dx+∫_{0}^{a/2}f(a-x)dx
Due to the above relation
∫_{0}^{a}f(x)dx = 0 if f(a-x) = -f(x)
∫_{0}^{a}f(x)dx = 2∫_{0}^{a/2}f(x)dx if f(a-x) = f(x)
8. If f is continuous on [a,b], then the integral function defined by g(x) = ∫_{a}^{x}f(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
∫_{a}^{b}f(x)dx = ∫_{a+nT}^{b+nT}f(x)dx, where n is an integer.
In particular
∫_{0}^{nT}f(x)dx = n∫_{0}^{T}f(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)≤∫_{a}^{b}f(x)dx≤M(b-a)
Differential Equations - Revision Points 1
JEE syllabus
- Formation of ordinary differential equations
- Solution of homogeneous differential equations
- Variables separable method,
- Linear first order differential equations.
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.
Formation of ordinary differential equations
If we are given a relation between variables x and y containing a number of arbitrary constants, we cn form a differntial equation from it by differntiating the given relation enough times so as to eliminate all the arbitrary constants.
- Formation of ordinary differential equations
- Solution of homogeneous differential equations
- Variables separable method,
- Linear first order differential equations.
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.
Formation of ordinary differential equations
If we are given a relation between variables x and y containing a number of arbitrary constants, we cn form a differntial equation from it by differntiating the given relation enough times so as to eliminate all the arbitrary constants.
Differential Equations -Revision points 2
Variables separable method
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
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
Differential Equations - Revision points - 3
Homogeneous Differential Equations
A function f(x,y) is said to be homogeneous of degree n if we can express
f(x,y) = x^{n} .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 = [x^{n} .g1(y/x)]/[x^{n} .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 differential equation is in separable form it can be solved
A function f(x,y) is said to be homogeneous of degree n if we can express
f(x,y) = x^{n} .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 = [x^{n} .g1(y/x)]/[x^{n} .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 differential equation is in separable form it can be solved
Saturday, May 24, 2008
Chapter 5 Complex Numbers
today I studied Chapter Complex Numbers from R D Sharma.
For a first time reading, it seemed to be a hard chapter
1. Introduction
Sqrt(-1) = i
“i” is called imaginary unity
2. Integral powers of IOTA (i)
i³ = i*i² = i*(-1) = -i
To find i^{n} divide n by 4 to get 4m+r where m is the quotient and r is the remainder.
i^{n} will be equal to i^{r}
3. Imaginary quantities
Square root of -3, -5 etc are called imaginary quantities
4. complex numbers
Number of the form a+ib (ex: 4+i3) is called a complex number.
A is called real part (Re(z)) and b is called imaginary part (Im(z)).
5. Equality of complex numbers
6. Addition of complex numbers
7. Subtraction of complex numbers
8. Multiplication of complex numbers
(a1+ib1) (a2+ib2) by multiplying and simplifying we get
(a1a2 – b1b2) + i(a1b2+a2b1)
Multiplicative inverse of a+ib = a/(a² + b²) - ib/(a² + b²))
9. Division of complex numbers
z1/z2 = z1* Multiplicative inverse of z2
10. Conjugate of a complex number
conjugate of z (= a+ib) = a-ib (is termed as z bar)
11. Modulus of a complex number
|z| = |a+ib| = SQRT(a² +b²)
12, Reciprocal of a complex number
Multiplicative inverse and reciprocal are same
13. Square root of a complex number
14. Representation of a complex number
Graphical – Argand plane
Trigonometric
Vector
Euler
15. Argument or amplitude of a complex number
16.Eulerian form of a complex number
17. Geometrical representations of fundamental operations
Addition
Subtraction
17a. Modulus and argument of multiplication of two complex numbers
18. Modulus and argument of division of two complex numbers
19. Geometrical representation of conjugate of a complex number
20. Some important results on modulus and argument
For a first time reading, it seemed to be a hard chapter
1. Introduction
Sqrt(-1) = i
“i” is called imaginary unity
2. Integral powers of IOTA (i)
i³ = i*i² = i*(-1) = -i
To find i^{n} divide n by 4 to get 4m+r where m is the quotient and r is the remainder.
i^{n} will be equal to i^{r}
3. Imaginary quantities
Square root of -3, -5 etc are called imaginary quantities
4. complex numbers
Number of the form a+ib (ex: 4+i3) is called a complex number.
A is called real part (Re(z)) and b is called imaginary part (Im(z)).
5. Equality of complex numbers
6. Addition of complex numbers
7. Subtraction of complex numbers
8. Multiplication of complex numbers
(a1+ib1) (a2+ib2) by multiplying and simplifying we get
(a1a2 – b1b2) + i(a1b2+a2b1)
Multiplicative inverse of a+ib = a/(a² + b²) - ib/(a² + b²))
9. Division of complex numbers
z1/z2 = z1* Multiplicative inverse of z2
10. Conjugate of a complex number
conjugate of z (= a+ib) = a-ib (is termed as z bar)
11. Modulus of a complex number
|z| = |a+ib| = SQRT(a² +b²)
12, Reciprocal of a complex number
Multiplicative inverse and reciprocal are same
13. Square root of a complex number
14. Representation of a complex number
Graphical – Argand plane
Trigonometric
Vector
Euler
15. Argument or amplitude of a complex number
16.Eulerian form of a complex number
17. Geometrical representations of fundamental operations
Addition
Subtraction
17a. Modulus and argument of multiplication of two complex numbers
18. Modulus and argument of division of two complex numbers
19. Geometrical representation of conjugate of a complex number
20. Some important results on modulus and argument
Tuesday, May 20, 2008
IIT JEE Revision Material 30. Vectors - 1
1. Introduction
2. Representation of vectors
3. Equality of vectors
4. Types of vectors
5. Parallelogram law of addition of vectors
6. Subtraction of vectors
7. Multiplication of a vector by a scalar
8. Position vector
9. Section formula
10.Linear combination of vectors
2. Representation of vectors
3. Equality of vectors
4. Types of vectors
5. Parallelogram law of addition of vectors
6. Subtraction of vectors
7. Multiplication of a vector by a scalar
8. Position vector
9. Section formula
10.Linear combination of vectors
IIT JEE Revision Material 30. Vectors - 2
17. Angle between two vectors
18. The scalar or dot product
19. Geometrical interpretation of scalar product
20. Properties of scalar product
Property 1 :
The scalar product of two vectors is commutative
a^{v}.b^{v} = b^{v}.a^{v}
Property 2 : Scalar Product of Collinear Vectors :
(i) When the vectors a^{v} and b^{v} are collinear and are in the same direction, then θ = 0
a^{v}.b^{v} = |a^{v}| |b^{v}| = ab
(i) When the vectors a^{v} and b^{v} are collinear and are in the opposite direction, then θ = π
a^{v}.b^{v} = |a^{v}| |b^{v}|(-1) = -ab
Property 3 : Sign of Dot Product
The dot product a^{v}.b^{v} 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.
21. scalar product in terms of components
If a = a1i+a2j+a3k and
b= b1i+b2j+b3k
then a.b = a1b1+a2b2+a3b3
22. 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))
23. Components of a vector b along and 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
24. 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.
This section illustrates the use of scalar product in proving some relations between edges of tetrahedron.
The relation mentioned are:
1. If two pairs of opposite edges of a tetrahedron are perpendicular, then prove that the opposite edges of the third pair are also perpendicular to each other.
2. Show that, 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 penpendicular
25 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.
Example: Forces represented by 6i+2j+3k and 3i-2j+6k respectively act on a particle which gets displaced from the point (2,2,-1) to (4,3,1). Find the work done by the forces.
Resultant force F = (6i+2j+3k) + (3i-2j+6k) = (9i+9k)
Displacement d = (4i+3j+k) – (2i+2j-k) = 2i+j+2k
Total work done = (9i+9k). (2i+j+2k)
= 9*2 +9*2 = 18 +18 = 36 units
18. The scalar or dot product
19. Geometrical interpretation of scalar product
20. Properties of scalar product
Property 1 :
The scalar product of two vectors is commutative
a^{v}.b^{v} = b^{v}.a^{v}
Property 2 : Scalar Product of Collinear Vectors :
(i) When the vectors a^{v} and b^{v} are collinear and are in the same direction, then θ = 0
a^{v}.b^{v} = |a^{v}| |b^{v}| = ab
(i) When the vectors a^{v} and b^{v} are collinear and are in the opposite direction, then θ = π
a^{v}.b^{v} = |a^{v}| |b^{v}|(-1) = -ab
Property 3 : Sign of Dot Product
The dot product a^{v}.b^{v} 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.
21. scalar product in terms of components
If a = a1i+a2j+a3k and
b= b1i+b2j+b3k
then a.b = a1b1+a2b2+a3b3
22. 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))
23. Components of a vector b along and 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
24. 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.
This section illustrates the use of scalar product in proving some relations between edges of tetrahedron.
The relation mentioned are:
1. If two pairs of opposite edges of a tetrahedron are perpendicular, then prove that the opposite edges of the third pair are also perpendicular to each other.
2. Show that, 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 penpendicular
25 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.
Example: Forces represented by 6i+2j+3k and 3i-2j+6k respectively act on a particle which gets displaced from the point (2,2,-1) to (4,3,1). Find the work done by the forces.
Resultant force F = (6i+2j+3k) + (3i-2j+6k) = (9i+9k)
Displacement d = (4i+3j+k) – (2i+2j-k) = 2i+j+2k
Total work done = (9i+9k). (2i+j+2k)
= 9*2 +9*2 = 18 +18 = 36 units
IIT JEE Revision Material 30. Vectors - 3 -Vector Product
Vector product
26. 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.
More about the direction: 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
27. Properties of vector product
a and b are vectors
1. Vector product is not commutative
a×b ≠ b×a
But
a×b = - b×a
2. m is a scalar
m a×b = m(a×b) = a×mb
3. m and n are scalars
m a×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
28. Vector product in terms of components
a = a1i+a2j+a3k
b = b1i+b2j+b3k
a×b =
|i j k|
|a1 a2 a3|
|b1 b2 b3|
26. 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.
More about the direction: 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
27. Properties of vector product
a and b are vectors
1. Vector product is not commutative
a×b ≠ b×a
But
a×b = - b×a
2. m is a scalar
m a×b = m(a×b) = a×mb
3. m and n are scalars
m a×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
28. Vector product in terms of components
a = a1i+a2j+a3k
b = b1i+b2j+b3k
a×b =
|i j k|
|a1 a2 a3|
|b1 b2 b3|
IIT JEE Revision Material 30. Vectors - 4 - Vector Product
Vector product
26. 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.
More about the direction: 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
27. Properties of vector product
a and b are vectors
1. Vector product is not commutative
a×b ≠ b×a
But
a×b = - b×a
2. m is a scalar
m a×b = m(a×b) = a×mb
3. m and n are scalars
m a×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
28. Vector product in terms of components
a = a1i+a2j+a3k
b = b1i+b2j+b3k
a×b =
|i j k |
|a1 a2 a3|
|b1 b2 b3|
26. 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.
More about the direction: 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
27. Properties of vector product
a and b are vectors
1. Vector product is not commutative
a×b ≠ b×a
But
a×b = - b×a
2. m is a scalar
m a×b = m(a×b) = a×mb
3. m and n are scalars
m a×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
28. Vector product in terms of components
a = a1i+a2j+a3k
b = b1i+b2j+b3k
a×b =
|i j k |
|a1 a2 a3|
|b1 b2 b3|
Revision Material 31. Three Dimensional Geometry -1 Direction Cosines
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.
Part 1
Coordinates, Direction cosines and direction ratios
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.
x-coordinate signs and octants
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 formula
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 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
Direction ratios of a line
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.
Part 1
Coordinates, Direction cosines and direction ratios
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.
x-coordinate signs and octants
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 formula
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 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
Direction ratios of a line
Revision Material 31. Three Dimensional Geometry - 3. Plane
Plane
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.
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.
Vector equation of a plane passing through a given point and normal to a given vector
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
In 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.
Equation of a plane in normal 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.
Cartesian form: Im 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
The equation r.n^ = d denotes the distance of the plane from the origin.
Angle between two planes
The angle between two planes is defined as the angle between their normals.
Angle between planes in normal form:
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
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]]
Condition of perpendicularity of planes
[a1a2+b1b2+c1c2] = 0
Condition of parallelism
a1/a2 =b1/b2 =c1/c2
Equation of a plane passing through a given point and parallel to two given vectors
Equation for the plane passing though a point having position vector a and parallel to b and c is r = a + λb + μc where λ and μ are scalars. This equation is called a parametric form of the plane as for different points on the plane the values of scalars λ and μ are different.
Nonparametric form: Equation for the plane passing though a point having position vector a and parallel to b and c is
(r-a).(b×c) = 0 or r.(b×c) = a.(b×c)
[rbc] = [abc]
Equation of a plane parallel to a given plane
Vector form: Parallel planes have the common normal. Hence equation of a plane parallel to the plane r .n = d1 is r.n = d2, where d2 is a constant determined by the given conditions.
Cartesian form: Let ax+by+cz+d = 0 be a plane. The direction ratios of its normal are a,b,c. A plane parallel to this plane will also have the same direction ratios. Hence the equation of the plane is ax+by+cz+k = 0 where k is an arbitrary constant and is etermined by the given conditions.
Equation of a plane passing through the intersection of two planes
Vector form: If the given two planes are r .n = d1 and r.n = d2, then the plane passing through the intersection of these two planes is:
(r.n1 – d1) + λ (r.n2 – d2) = 0
Or
r.(n1+ λn2) = d1+ λd2, where λ is an arbitrary constant.
Cartesian form:
Two planes give are:
a1x+b1y+c1z+d1 = 0
a2x+b2y+c2z+d2 = 0
The equation of the plan passing through the intersection of two planes is
(a1x+b1y+c1z+d1)+ λ (a2x+b2y+c2z+d2 = 0)
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.
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.
Vector equation of a plane passing through a given point and normal to a given vector
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
In 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.
Equation of a plane in normal 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.
Cartesian form: Im 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
The equation r.n^ = d denotes the distance of the plane from the origin.
Angle between two planes
The angle between two planes is defined as the angle between their normals.
Angle between planes in normal form:
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
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]]
Condition of perpendicularity of planes
[a1a2+b1b2+c1c2] = 0
Condition of parallelism
a1/a2 =b1/b2 =c1/c2
Equation of a plane passing through a given point and parallel to two given vectors
Equation for the plane passing though a point having position vector a and parallel to b and c is r = a + λb + μc where λ and μ are scalars. This equation is called a parametric form of the plane as for different points on the plane the values of scalars λ and μ are different.
Nonparametric form: Equation for the plane passing though a point having position vector a and parallel to b and c is
(r-a).(b×c) = 0 or r.(b×c) = a.(b×c)
[rbc] = [abc]
Equation of a plane parallel to a given plane
Vector form: Parallel planes have the common normal. Hence equation of a plane parallel to the plane r .n = d1 is r.n = d2, where d2 is a constant determined by the given conditions.
Cartesian form: Let ax+by+cz+d = 0 be a plane. The direction ratios of its normal are a,b,c. A plane parallel to this plane will also have the same direction ratios. Hence the equation of the plane is ax+by+cz+k = 0 where k is an arbitrary constant and is etermined by the given conditions.
Equation of a plane passing through the intersection of two planes
Vector form: If the given two planes are r .n = d1 and r.n = d2, then the plane passing through the intersection of these two planes is:
(r.n1 – d1) + λ (r.n2 – d2) = 0
Or
r.(n1+ λn2) = d1+ λd2, where λ is an arbitrary constant.
Cartesian form:
Two planes give are:
a1x+b1y+c1z+d1 = 0
a2x+b2y+c2z+d2 = 0
The equation of the plan passing through the intersection of two planes is
(a1x+b1y+c1z+d1)+ λ (a2x+b2y+c2z+d2 = 0)
Revision Material Ch 33. Trigonometric Ratios - 1
I.
Function-------------Domain--------------Range
Sin θ------------------R----------------[-1,1]
cos θ------------------R-----------------[-1,1]
tan θ------------R-{(2n+1)π/2, nЄI}------R=(-∞. ∞)
II.Basic relations
sin θ*cosec θ = 1
cos θ*sec θ = 1
tan θ*cot θ = 1
sin² θ + cos² θ = 1
sec²θ - tan²θ =1; 1 + tan² θ = sec² θ
cosec²θ - cot²θ = 1; 1+cot2θ = cosec²θ
III. Allied or Related angles
1. sin (-θ) = -sin θ
2. cos(-θ) = cos θ
IV.Compound angles
1. sin (A+B) = sin A cos B + cosA sin B
2. sin 2A = 2 sin A cos A
3. sin (A-B) = sin A cos B - cos A sin B
4. cos (A+B) = cos A cos B -sin A sin B
5 cos 2A = cos²A - sin²A
6. cos (A-B) = cos A cos B + sin A sin B
Function-------------Domain--------------Range
Sin θ------------------R----------------[-1,1]
cos θ------------------R-----------------[-1,1]
tan θ------------R-{(2n+1)π/2, nЄI}------R=(-∞. ∞)
II.Basic relations
sin θ*cosec θ = 1
cos θ*sec θ = 1
tan θ*cot θ = 1
sin² θ + cos² θ = 1
sec²θ - tan²θ =1; 1 + tan² θ = sec² θ
cosec²θ - cot²θ = 1; 1+cot2θ = cosec²θ
III. Allied or Related angles
1. sin (-θ) = -sin θ
2. cos(-θ) = cos θ
IV.Compound angles
1. sin (A+B) = sin A cos B + cosA sin B
2. sin 2A = 2 sin A cos A
3. sin (A-B) = sin A cos B - cos A sin B
4. cos (A+B) = cos A cos B -sin A sin B
5 cos 2A = cos²A - sin²A
6. cos (A-B) = cos A cos B + sin A sin B
Revision Material Ch 33. Trigonometric Ratios - 2
sin (A+B+C) = sin A cos B cos C + sin B cos A cos C + sin C cos A cos B - sin A sin B sin C
sin 3A = 3 sin A - 4 sin³A
Cos (A+B+C) = cos A cos B cos C - cos A sin B sin C - Cos B sin A sin C - cos C sin A sin B
tan (A+B+C) = [tan A + tan B + tan C - tan A tan B tan C]/[ 1- tan A tan B - tan B tan C - tan C tan A]
Transformation Formulae
sin (A+B) + sin (A-B) = 2 sin A cos B
2 sin A cos B = sin (A+B) + sin (A-B)
sin (A+B) - sin (A-B) = 2cos A sin B
2cos A sin B = sin (A+B) - sin (A-B)
cos (A+B) + cos (A-B) = 2cos A cos B
2cos A cos B = cos (A+B) + cos (A-B)
cos (A+B) - cos (A-B) = 2sin A sin B
2sin A sin B = cos (A+B) - cos (A-B)
Therefore
2 sin A cos B = sin (A+B) + sin (A-B)
2cos A sin B = sin (A+B) - sin (A-B)
2cos A cos B = cos (A+B) + cos (A-B)
2sin A sin B = cos (A+B) - cos (A-B)
The products of two sines or two cosines and one sine and one cosine can be transformed into the sum or differences of two sines or two cosines.
Trigonometric ratios of multiple angles
Trigonometric ratios of angle 2A in terms of an angle A
Sin 2A = 2sin A cos A
Sin 2 A = 2tan A/(1 + tan² A)
Cos 2A = cos² A - sin² A
Cos 2A = 2cos² A – 1
Cos 2A = 1 – 2sin² A
cos 2A = (1- tan² A)/(1+ tan² A)
tan 2A = 2tan A/(1 - tan² A)
Trigonometric ratios of angle 3A in terms of an angle A
Sin 3A = 3sin A - sin³ A
cos 3A = 4cos³ A – 3cos A
tan 3A = (3tan A - tan³ A)/(1-3tan² A)
Trigonometric ratios of sub-multiple angles
Trigonometric ratios of angle A in terms of an angle A/2
sin A = 2sin A/2 cos A/2
sin A = (2tan A/2)/(1 + tan² A/2)
cos A = cos² A/2 - sin² A/2
cos A = 2cos² A/2 – 1
cos A = 1 – 2sin² A/2
cos A = (1- tan² A/2)/(1+ tan² A/2)
tan A = (2tan A/2)/(1 - tan² A/2)
Trigonometric ratios of angle A in terms of an angle A/3
Sin A = 3sin (A/3) - sin³ (A/3)
cos A = 4cos³ (A/3) – 3cos (A/3)
tan A = (3tan (A/3) - tan³ (A/3))/(1-3tan² (A/3))
Trigonometric ratios of angle A/2 in terms of an angle cos A
cos A/2 = ±√[(1+cos A)/2]
sin A/2 = ±√[(1-cos A)/2]
tan A/2 = ±√[(1-cos A)/(1 + cos a)]
sin 3A = 3 sin A - 4 sin³A
Cos (A+B+C) = cos A cos B cos C - cos A sin B sin C - Cos B sin A sin C - cos C sin A sin B
tan (A+B+C) = [tan A + tan B + tan C - tan A tan B tan C]/[ 1- tan A tan B - tan B tan C - tan C tan A]
Transformation Formulae
sin (A+B) + sin (A-B) = 2 sin A cos B
2 sin A cos B = sin (A+B) + sin (A-B)
sin (A+B) - sin (A-B) = 2cos A sin B
2cos A sin B = sin (A+B) - sin (A-B)
cos (A+B) + cos (A-B) = 2cos A cos B
2cos A cos B = cos (A+B) + cos (A-B)
cos (A+B) - cos (A-B) = 2sin A sin B
2sin A sin B = cos (A+B) - cos (A-B)
Therefore
2 sin A cos B = sin (A+B) + sin (A-B)
2cos A sin B = sin (A+B) - sin (A-B)
2cos A cos B = cos (A+B) + cos (A-B)
2sin A sin B = cos (A+B) - cos (A-B)
The products of two sines or two cosines and one sine and one cosine can be transformed into the sum or differences of two sines or two cosines.
Trigonometric ratios of multiple angles
Trigonometric ratios of angle 2A in terms of an angle A
Sin 2A = 2sin A cos A
Sin 2 A = 2tan A/(1 + tan² A)
Cos 2A = cos² A - sin² A
Cos 2A = 2cos² A – 1
Cos 2A = 1 – 2sin² A
cos 2A = (1- tan² A)/(1+ tan² A)
tan 2A = 2tan A/(1 - tan² A)
Trigonometric ratios of angle 3A in terms of an angle A
Sin 3A = 3sin A - sin³ A
cos 3A = 4cos³ A – 3cos A
tan 3A = (3tan A - tan³ A)/(1-3tan² A)
Trigonometric ratios of sub-multiple angles
Trigonometric ratios of angle A in terms of an angle A/2
sin A = 2sin A/2 cos A/2
sin A = (2tan A/2)/(1 + tan² A/2)
cos A = cos² A/2 - sin² A/2
cos A = 2cos² A/2 – 1
cos A = 1 – 2sin² A/2
cos A = (1- tan² A/2)/(1+ tan² A/2)
tan A = (2tan A/2)/(1 - tan² A/2)
Trigonometric ratios of angle A in terms of an angle A/3
Sin A = 3sin (A/3) - sin³ (A/3)
cos A = 4cos³ (A/3) – 3cos (A/3)
tan A = (3tan (A/3) - tan³ (A/3))/(1-3tan² (A/3))
Trigonometric ratios of angle A/2 in terms of an angle cos A
cos A/2 = ±√[(1+cos A)/2]
sin A/2 = ±√[(1-cos A)/2]
tan A/2 = ±√[(1-cos A)/(1 + cos a)]
Revision Ch.34 Properties of Triangles and circles - 1
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)
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)
Revision Ch.34 Properties of Triangles and circles - 2
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.
17. Circum-radius of a cyclic quadrilateral
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)
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.
17. Circum-radius of a cyclic quadrilateral
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)
Revision Ch 36. Inverse Trigonometrical Functions - 1
Definitions
1. y = sin^{-1}x if
(i)-1≤x≤1,
(ii)-π/2≤y≤π/2 and
(iii) x = sin y
2. y = cos^{-1}x if
(i)-1≤x≤1,
(ii) 0≤y≤π and -π/2≤y≤π/2 and
(iii) x = cos y
3. y = tan^{-1}x if
(i)xЄR
(ii) -π/2≤y≤π/2 and
(iii) x = tan y
4. sec^{-1}x = cos^{-1}(1/x)
5. cosec^{-1}x = sin^{-1}(1/x)
1. y = sin^{-1}x if
(i)-1≤x≤1,
(ii)-π/2≤y≤π/2 and
(iii) x = sin y
2. y = cos^{-1}x if
(i)-1≤x≤1,
(ii) 0≤y≤π and -π/2≤y≤π/2 and
(iii) x = cos y
3. y = tan^{-1}x if
(i)xЄR
(ii) -π/2≤y≤π/2 and
(iii) x = tan y
4. sec^{-1}x = cos^{-1}(1/x)
5. cosec^{-1}x = sin^{-1}(1/x)
Sunday, May 18, 2008
Learning status
I read superficially the chapter related to parabola, hyperbola and ellipse.
I also read three dimensional geometry. Made some notes in the blog.
I also read three dimensional geometry. Made some notes in the blog.
Saturday, May 17, 2008
IIT JEE 2010 Mathematics Books
Today I had a look at R.D. Sharma published by Dhanpat Rai. Especially I had a look at indefinite integration and definite integration chapters.
The book has good explanation for various methods in various chapters. Examples are given method wise. This will facilitate understanding the methods.
But many persons write that TMH JEE Mathematics has problems of a more advanced nature and JEE candidates have to answer those questions to develop themselves into capable persons.
I have both the books and of course Dasgupta also to go through.
The book has good explanation for various methods in various chapters. Examples are given method wise. This will facilitate understanding the methods.
But many persons write that TMH JEE Mathematics has problems of a more advanced nature and JEE candidates have to answer those questions to develop themselves into capable persons.
I have both the books and of course Dasgupta also to go through.
Tuesday, May 13, 2008
IIT JEE Mathematics - Useful Book - To Download
Algebra and Trigonometry
http://www.scribd.com/doc/2316477/Beecher-Algebra-and-Trigonometry-3e-HQ
The book has chapters on Circle, Parabola, Hyperbola and Ellipse also.
http://www.scribd.com/doc/2316477/Beecher-Algebra-and-Trigonometry-3e-HQ
The book has chapters on Circle, Parabola, Hyperbola and Ellipse also.
Video Lectures in Mathematics
These are video lectures. But free broadcast is available in USA only.
1. Introduction
An introduction to the series, this program presents several mathematical themes and emphasizes why algebra is important in today’s world.
2. The Language of Algebra
This program provides a survey of basic mathematical terminology. Content includes properties of the real number system and the basic axioms and theorems of algebra. Specific terms covered include algebraic expression, variable, product, sum term, factors, common factors, like terms, simplify, equation, sets of numbers, and axioms. Definitions of these terms lay a foundation for working with the concepts.
3. Exponents and Radicals
This program explains the properties of exponents and radicals: their definitions, their rules, and their applications to positive numbers. The program shows how to use the Rules for Exponents to simplify expressions, demonstrating concepts through a discussion of the O-ring failure of the Challenger Space Shuttle.
4. Factoring Polynomials
This program defines polynomials and describes how the distributive property is used to multiply common monomial factors with the FOIL method. It covers factoring, the difference of two squares, trinomials as products of two binomials, the sum and difference of two cubes, and regrouping of terms.
5. Linear Equations
This is the first program in which equations are solved. It shows how solutions are obtained, what they mean, and how to check them using one unknown. Concepts are worked out in an application problem involving a modern sewage plant near Los Angeles, where a linear equation is set up and solved to determine how long to keep open an inlet pipe.
6. Complex Numbers
To the sets of numbers reviewed in previous lessons, this program adds complex numbers — their definition and their use in basic operations and quadratic equations. Students will learn how to combine like terms, apply the FOIL method, and rationalize the denominator for finding the product or quotient of two complex numbers.
7. Quadratic Equations
This program reviews the quadratic equation and covers standard form, factoring, checking the solution, the Zero Product Property, and the difference of two squares. Environmental and aviation examples provide realistic problems, and the method of Completing the Square is used to solve them.
8. Inequalities
This program teaches students the properties and solution of inequalities, linking positive and negative numbers to the direction of the inequality. The program presents three applications of inequalities: modeling problems of the U.S. Postal Service, finding the cheapest way to travel, and conducting market research in the pizza industry.
9. Absolute Value
In this program, the concept of absolute value is defined, enabling students to use it in equations and inequalities. One application example involves systolic blood pressure, using a formula incorporating absolute value to find a person’s “pressure difference from normal.” The recipe for making fireworks offers another example.
10. Linear Relations
This program looks at the linear relationship between two variables, expressed as a set of ordered pairs. Students are shown the use of linear equations to develop and provide information about two quantities, as well as the applications of these equations to the slope of a line.
11. Circle and Parabola
The circle and parabola are presented as two of the four conic sections explored in this series. The circle, its various measures when graphed on the coordinate plane (distance, radius, etc.), its related equations (e.g., center-radius form), and its relationships with other shapes are covered, as is the parabola with its various measures and characteristics (focus, directrix, vertex, etc.). An earthquake epicenter provides a real-life illustration.
12. Ellipse and Hyperbola
The ellipse and hyperbola, the other two conic sections examined in the series, are introduced. The program defines the two terms, distinguishing between them with different language, equations, and graphic representations. Architecture and surgery provide interesting application examples.
13. Functions
This program defines a function, discusses domain and range, and develops an equation from real situations. The cutting of pizza and encoding of secret messages provide subjects for the demonstration of functions and their usefulness.
14. Composition and Inverse Functions
Graphics are used to introduce composites and inverses of functions as applied to calculation of the Gross National Product. One-to-one functions and the horizontal line test are introduced, and more encoded messages and the hazards of “the bends” in scuba diving provide instructive applications of the functions discussed.
15. Variation
In this program, students are given examples of special functions in the form of direct variation and inverse variation, with a discussion of combined variation and the constant of proportionality. These are explored in relation to polynomials and assorted equations, with applications from chemistry, physics, astronomy, and the food industry.
16. Polynomial Functions
This program explains how to identify, graph, and determine all intercepts of a polynomial function. It covers the role of coefficients; real numbers; exponents; and linear, quadratic, and cubic functions. This program touches upon factors, x-intercepts, and zero values. These terms are demonstrated with the baking of pizza.
17. Rational Functions
A rational function is the quotient of two polynomial functions. The properties of these functions are investigated using cases in which each rational function is expressed in its simplified form. The relationship between numerator and denominator is clarified, and sign and other graphs are used to determine intercepts, symmetry, and asymptotes.
18. Exponential Functions
Students are taught the exponential function, as illustrated through formulas. The population of Massachusetts, the “learning curve,” bacterial growth, and radioactive decay demonstrate these functions and the concepts of exponential growth and decay.
19. Logarithmic Functions
This program covers the logarithmic relationship, the use of logarithmic properties, and the handling of a scientific calculator. How radioactive dating and the Richter scale depend on the properties of logarithms is explained. Many rules and tests from previous programs are also incorporated into the lesson.
20. Systems of Equations
The case of two linear equations in two unknowns is considered throughout this program. Elimination and substitution methods are used to find single solutions to systems of linear and nonlinear equations. Consistent, inconsistent, and dependent systems are also explored through examples from ship navigation and garment production.
21. Systems of Linear Inequalities
Elimination and substitution are used again to solve systems of linear inequalities. Linear programming is shown to solve problems in the Berlin airlift, production of butter and ice cream, school redistricting, and other situations while constraints, corner points, objective functions, the region of feasible solutions, and minimum and maximum values are also explored.
22. Arithmetic Sequences and Series
When the growth of a child is regular, it can be described by an arithmetic sequence. This program differentiates between arithmetic and nonarithmetic sequences as it presents the solutions to sequence- and series-related problems. Definitions include sequence, arithmetic sequence, arithmetic series, fixed number, and common difference.
23. Geometric Sequences and Series
This program provides examples of geometric sequences and series (f-stops on a camera and the bouncing of a ball), explaining the meaning of nonzero constant real number and common ratio. Finite and infinite geometric series and the sequence of partial sums are also defined in the discussion.
24. Mathematical Induction
Mathematical proofs applied to hypothetical statements shape this discussion on mathematical induction. This segment exhibits special cases, looks at the development of number patterns, relates the patterns to Pascal’s triangle and factorials, and elaborates the general form of the theorem.
25. Permutations and Combinations
How many variations in a license plate number or poker hand are possible? This program answers the question and shows students how it’s done. Techniques for counting the number of ways in which collections of objects can be arranged, ordered, and combined are demonstrated.
26. Probability
In this final program, students see how the various techniques of algebra that they have learned can be applied to the study of probability. The program shows that games of chance, health statistics, and product safety are areas in which decisions must be made according to our understanding of the odds. It also shows how the subject of probability has evolved to support such fields as genetics, social science, and medicine.
Visit the site. You have to register to view the lectures. Free registration
http://www.learner.org/resources/series66.html?pop=yes&pid=172
1. Introduction
An introduction to the series, this program presents several mathematical themes and emphasizes why algebra is important in today’s world.
2. The Language of Algebra
This program provides a survey of basic mathematical terminology. Content includes properties of the real number system and the basic axioms and theorems of algebra. Specific terms covered include algebraic expression, variable, product, sum term, factors, common factors, like terms, simplify, equation, sets of numbers, and axioms. Definitions of these terms lay a foundation for working with the concepts.
3. Exponents and Radicals
This program explains the properties of exponents and radicals: their definitions, their rules, and their applications to positive numbers. The program shows how to use the Rules for Exponents to simplify expressions, demonstrating concepts through a discussion of the O-ring failure of the Challenger Space Shuttle.
4. Factoring Polynomials
This program defines polynomials and describes how the distributive property is used to multiply common monomial factors with the FOIL method. It covers factoring, the difference of two squares, trinomials as products of two binomials, the sum and difference of two cubes, and regrouping of terms.
5. Linear Equations
This is the first program in which equations are solved. It shows how solutions are obtained, what they mean, and how to check them using one unknown. Concepts are worked out in an application problem involving a modern sewage plant near Los Angeles, where a linear equation is set up and solved to determine how long to keep open an inlet pipe.
6. Complex Numbers
To the sets of numbers reviewed in previous lessons, this program adds complex numbers — their definition and their use in basic operations and quadratic equations. Students will learn how to combine like terms, apply the FOIL method, and rationalize the denominator for finding the product or quotient of two complex numbers.
7. Quadratic Equations
This program reviews the quadratic equation and covers standard form, factoring, checking the solution, the Zero Product Property, and the difference of two squares. Environmental and aviation examples provide realistic problems, and the method of Completing the Square is used to solve them.
8. Inequalities
This program teaches students the properties and solution of inequalities, linking positive and negative numbers to the direction of the inequality. The program presents three applications of inequalities: modeling problems of the U.S. Postal Service, finding the cheapest way to travel, and conducting market research in the pizza industry.
9. Absolute Value
In this program, the concept of absolute value is defined, enabling students to use it in equations and inequalities. One application example involves systolic blood pressure, using a formula incorporating absolute value to find a person’s “pressure difference from normal.” The recipe for making fireworks offers another example.
10. Linear Relations
This program looks at the linear relationship between two variables, expressed as a set of ordered pairs. Students are shown the use of linear equations to develop and provide information about two quantities, as well as the applications of these equations to the slope of a line.
11. Circle and Parabola
The circle and parabola are presented as two of the four conic sections explored in this series. The circle, its various measures when graphed on the coordinate plane (distance, radius, etc.), its related equations (e.g., center-radius form), and its relationships with other shapes are covered, as is the parabola with its various measures and characteristics (focus, directrix, vertex, etc.). An earthquake epicenter provides a real-life illustration.
12. Ellipse and Hyperbola
The ellipse and hyperbola, the other two conic sections examined in the series, are introduced. The program defines the two terms, distinguishing between them with different language, equations, and graphic representations. Architecture and surgery provide interesting application examples.
13. Functions
This program defines a function, discusses domain and range, and develops an equation from real situations. The cutting of pizza and encoding of secret messages provide subjects for the demonstration of functions and their usefulness.
14. Composition and Inverse Functions
Graphics are used to introduce composites and inverses of functions as applied to calculation of the Gross National Product. One-to-one functions and the horizontal line test are introduced, and more encoded messages and the hazards of “the bends” in scuba diving provide instructive applications of the functions discussed.
15. Variation
In this program, students are given examples of special functions in the form of direct variation and inverse variation, with a discussion of combined variation and the constant of proportionality. These are explored in relation to polynomials and assorted equations, with applications from chemistry, physics, astronomy, and the food industry.
16. Polynomial Functions
This program explains how to identify, graph, and determine all intercepts of a polynomial function. It covers the role of coefficients; real numbers; exponents; and linear, quadratic, and cubic functions. This program touches upon factors, x-intercepts, and zero values. These terms are demonstrated with the baking of pizza.
17. Rational Functions
A rational function is the quotient of two polynomial functions. The properties of these functions are investigated using cases in which each rational function is expressed in its simplified form. The relationship between numerator and denominator is clarified, and sign and other graphs are used to determine intercepts, symmetry, and asymptotes.
18. Exponential Functions
Students are taught the exponential function, as illustrated through formulas. The population of Massachusetts, the “learning curve,” bacterial growth, and radioactive decay demonstrate these functions and the concepts of exponential growth and decay.
19. Logarithmic Functions
This program covers the logarithmic relationship, the use of logarithmic properties, and the handling of a scientific calculator. How radioactive dating and the Richter scale depend on the properties of logarithms is explained. Many rules and tests from previous programs are also incorporated into the lesson.
20. Systems of Equations
The case of two linear equations in two unknowns is considered throughout this program. Elimination and substitution methods are used to find single solutions to systems of linear and nonlinear equations. Consistent, inconsistent, and dependent systems are also explored through examples from ship navigation and garment production.
21. Systems of Linear Inequalities
Elimination and substitution are used again to solve systems of linear inequalities. Linear programming is shown to solve problems in the Berlin airlift, production of butter and ice cream, school redistricting, and other situations while constraints, corner points, objective functions, the region of feasible solutions, and minimum and maximum values are also explored.
22. Arithmetic Sequences and Series
When the growth of a child is regular, it can be described by an arithmetic sequence. This program differentiates between arithmetic and nonarithmetic sequences as it presents the solutions to sequence- and series-related problems. Definitions include sequence, arithmetic sequence, arithmetic series, fixed number, and common difference.
23. Geometric Sequences and Series
This program provides examples of geometric sequences and series (f-stops on a camera and the bouncing of a ball), explaining the meaning of nonzero constant real number and common ratio. Finite and infinite geometric series and the sequence of partial sums are also defined in the discussion.
24. Mathematical Induction
Mathematical proofs applied to hypothetical statements shape this discussion on mathematical induction. This segment exhibits special cases, looks at the development of number patterns, relates the patterns to Pascal’s triangle and factorials, and elaborates the general form of the theorem.
25. Permutations and Combinations
How many variations in a license plate number or poker hand are possible? This program answers the question and shows students how it’s done. Techniques for counting the number of ways in which collections of objects can be arranged, ordered, and combined are demonstrated.
26. Probability
In this final program, students see how the various techniques of algebra that they have learned can be applied to the study of probability. The program shows that games of chance, health statistics, and product safety are areas in which decisions must be made according to our understanding of the odds. It also shows how the subject of probability has evolved to support such fields as genetics, social science, and medicine.
Visit the site. You have to register to view the lectures. Free registration
http://www.learner.org/resources/series66.html?pop=yes&pid=172
Monday, May 12, 2008
Learning status
In the last three days, I went through the chapters on matrices, vector analysis and probability.
Thursday, May 8, 2008
Mathematics IIT JEE 2010 - Study Plan
Strategy for JEE 2010 preparation was outlined in
http://iit-jee-chemistry.blogspot.com/2008/05/jee-2010-aspirants-preparation-strategy.html
Study plan outline for mathematics is given below
The contents Mathematics text for class XI(Maharashtra board syllabus) are given below and based on these contents the chapters to be completed in class XI from JEE syllabus/contents are specified
Text Book:
A New Approach to Mathematics and Statistics Paper I
Chitale et al.
---------
Contents
Angle and its measurement
Trigonometric ratios
Trigonometric ratios of compound angles
Properties of triangles
Inverse circular functions
Locus and its equation
The straight line
Vectors
Linear inequalities
Determinants
Matrices
Measures of dispersion
Bivariate frequency distribution
Text Book - 2
A New Approach to Mathematics and Statistics Paper II
Chitale et al.
------------
Contents
Sets, relations and functions
Logarithms
Complex numbers
Quadratic equations
Sequences and series
Permutations and combinations
Mathematic induction and Binomial theorem
Limits
Derivatives
Integration
Chapters from JEE syllabus to be completed in Class XI (2008-09)
1. Complex numbers
2. Theory of equations
3. Progressions
4. Logarithms
5. Permutations and combinations
6. Binomial theorem
7. Matrices (some portion)
8. Determinants (some portion)
9. Inequalities
10. Elementary trigonometry
11. Solution of triangles and other applications of trigonometry
12. Inverse trigonometric functions
13. Straight line
14. Vectors (some portion)
15. Functions
16. Limits and continuity
17. Differentiation
18. Integration (some portion)
http://iit-jee-chemistry.blogspot.com/2008/05/jee-2010-aspirants-preparation-strategy.html
Study plan outline for mathematics is given below
The contents Mathematics text for class XI(Maharashtra board syllabus) are given below and based on these contents the chapters to be completed in class XI from JEE syllabus/contents are specified
Text Book:
A New Approach to Mathematics and Statistics Paper I
Chitale et al.
---------
Contents
Angle and its measurement
Trigonometric ratios
Trigonometric ratios of compound angles
Properties of triangles
Inverse circular functions
Locus and its equation
The straight line
Vectors
Linear inequalities
Determinants
Matrices
Measures of dispersion
Bivariate frequency distribution
Text Book - 2
A New Approach to Mathematics and Statistics Paper II
Chitale et al.
------------
Contents
Sets, relations and functions
Logarithms
Complex numbers
Quadratic equations
Sequences and series
Permutations and combinations
Mathematic induction and Binomial theorem
Limits
Derivatives
Integration
Chapters from JEE syllabus to be completed in Class XI (2008-09)
1. Complex numbers
2. Theory of equations
3. Progressions
4. Logarithms
5. Permutations and combinations
6. Binomial theorem
7. Matrices (some portion)
8. Determinants (some portion)
9. Inequalities
10. Elementary trigonometry
11. Solution of triangles and other applications of trigonometry
12. Inverse trigonometric functions
13. Straight line
14. Vectors (some portion)
15. Functions
16. Limits and continuity
17. Differentiation
18. Integration (some portion)
Wednesday, May 7, 2008
Mathmatics Book Recommendations
Calculus: I.A Maroon. /Goldstein /Larson
Matrices: Finkbeiner
Statistics: Gupta B.D
Differential Equations: Piaggio H.T.H
Algebra: Hall and Knight / Birkoff & Bartee
N.C.E.R.T Books.
Tata Mc.Graw Hill Problem book
Matrices: Finkbeiner
Statistics: Gupta B.D
Differential Equations: Piaggio H.T.H
Algebra: Hall and Knight / Birkoff & Bartee
N.C.E.R.T Books.
Tata Mc.Graw Hill Problem book
About Preparing for IIT JEE Mathematics
An 2008 article on rediff
http://www.rediff.com//getahead/2008/apr/04jee.htm
Article in March 2008 in Eenadu
http://www.eenadu.net/archives/archive-13-3-2008/pratibhaplus/prati04.pdf
http://www.eenadu.net/archives/archive-13-3-2008/pratibhaplus/prati05.pdf
http://www.rediff.com//getahead/2008/apr/04jee.htm
Article in March 2008 in Eenadu
http://www.eenadu.net/archives/archive-13-3-2008/pratibhaplus/prati04.pdf
http://www.eenadu.net/archives/archive-13-3-2008/pratibhaplus/prati05.pdf
Tuesday, May 6, 2008
Logarithms-1
Given a positive real number y, there is one and only one real number x such that a^{x} = y, we call this number x as logarithm of y to the base a.
Logarithm gives the power to which the base is to raised to get the number.
Logarithms can be defined for any base.Logarithms defined to the base 10 and to the base e are tabulated and used more widely. Logarithms to the base 10 are called common logarithms and logarithms to the base e are called natural logarithms.
Many books follow the convention of writing natural logarithms as ln y and common logarithms as log y.
In mathematics books as logarithms are given for various bases, the base may be specified.
Some properties
1. log_{a}1 = 0, a>0, and a ≠0
2. log_{a}a = 1, a>0, and a ≠0
Good online material
http://tutorial.math.lamar.edu/Classes/Alg/LogFunctions.aspx
Logarithm gives the power to which the base is to raised to get the number.
Logarithms can be defined for any base.Logarithms defined to the base 10 and to the base e are tabulated and used more widely. Logarithms to the base 10 are called common logarithms and logarithms to the base e are called natural logarithms.
Many books follow the convention of writing natural logarithms as ln y and common logarithms as log y.
In mathematics books as logarithms are given for various bases, the base may be specified.
Some properties
1. log_{a}1 = 0, a>0, and a ≠0
2. log_{a}a = 1, a>0, and a ≠0
Good online material
http://tutorial.math.lamar.edu/Classes/Alg/LogFunctions.aspx
Permutations and Combinations-2
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 }C_{r}.
Then P(n,r) = ^{n }C_{r} = n(n-1)(n-2)…(n-(r-1))
Theorem 2
P(n,r) = ^{n }C_{r} = 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-1}C_{r-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-1}C_{r-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-2}C_{r-2}
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 }C_{r}.
Then P(n,r) = ^{n }C_{r} = n(n-1)(n-2)…(n-(r-1))
Theorem 2
P(n,r) = ^{n }C_{r} = 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-1}C_{r-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-1}C_{r-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-2}C_{r-2}
Permutations and Combinations-3 - Revision Points
Revision Points (Text R.D. Sharma)
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 n^{2 }.
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)!..
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 n^{2 }.
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)!..
Revision Points - Permutations and Combinations-4
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 ^{n}C_{r}.
^{n}C_{r} 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
^{n}C_{r} = n!/(n-r)!r!
Results from the theorem
^{n}C_{r} = [n(n-1)(n-2)...(n-r+1)]/(1.2.3...r)
^{n}C_{n} =1
^{n}C_{0} = 1
^{n}C_{r} = ^{n}P_{r}/r!
Properties of ^{n}C_{r} and C(n,r)
1. ^{n}C_{r} = ^{n}C_{n-r}
Note: If x=y = n
^{n}C_{x} = ^{n}C_{y}
2. Let n and r be non-negative integers such that r≤n. Then
^{n}C_{r} + ^{n}C_{r-1} = ^{n+1}C_{r}
3. Let n and r be non-negative integers such that 1≤ r≤n. Then
^{n}C_{r} = (n/r) ^{n-1}C_{r-1}
4. If 1≤ r≤n, then
n.^{n-1}C_{r-1} = (n-r+1)^{n}C_{r-1}
5. ^{n}C_{x} = ^{n}C_{y} implies x+y = n
6. If n is even, then the greatest value of ^{n}C_{r} [0≤ r≤n] is ^{n}C_{n/2}.
7. If n is odd, then the greatest value of ^{n}C_{r} [0≤ r≤n] is ^{n}C_{(n+1)/2} or ^{n}C_{(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
Join Orkut community
Join Orkut community IIT-JEE-Academy for interaction regarding various issues and doubts
http://www.orkut.co.in/Community.aspx?cmm=39291603
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 ^{n}C_{r}.
^{n}C_{r} 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
^{n}C_{r} = n!/(n-r)!r!
Results from the theorem
^{n}C_{r} = [n(n-1)(n-2)...(n-r+1)]/(1.2.3...r)
^{n}C_{n} =1
^{n}C_{0} = 1
^{n}C_{r} = ^{n}P_{r}/r!
Properties of ^{n}C_{r} and C(n,r)
1. ^{n}C_{r} = ^{n}C_{n-r}
Note: If x=y = n
^{n}C_{x} = ^{n}C_{y}
2. Let n and r be non-negative integers such that r≤n. Then
^{n}C_{r} + ^{n}C_{r-1} = ^{n+1}C_{r}
3. Let n and r be non-negative integers such that 1≤ r≤n. Then
^{n}C_{r} = (n/r) ^{n-1}C_{r-1}
4. If 1≤ r≤n, then
n.^{n-1}C_{r-1} = (n-r+1)^{n}C_{r-1}
5. ^{n}C_{x} = ^{n}C_{y} implies x+y = n
6. If n is even, then the greatest value of ^{n}C_{r} [0≤ r≤n] is ^{n}C_{n/2}.
7. If n is odd, then the greatest value of ^{n}C_{r} [0≤ r≤n] is ^{n}C_{(n+1)/2} or ^{n}C_{(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
Join Orkut community
Join Orkut community IIT-JEE-Academy for interaction regarding various issues and doubts
http://www.orkut.co.in/Community.aspx?cmm=39291603
Matrices - 1
Matrices will be an easy topic, but speed is required in doing various arithmetic calculations and some calculations to solve equations to get the answers.
For example a problem is given in 2006 in comprehension where matrix A is given.
From AX1 = B1 you are asked to find X1, from AX2 = B2, and Ax3 = B3 you have to find X2 and X3. You are asked to make matrix [X1 X2 X3], then find its determinant, and then its inverse and then [R1][X][C1]
All questions are straight forward but computation is involed and you have to do fast.
Concepts
1. A matrix is a rectangular array of numbers [a_{0}]
2. A matrix with m rows and n columns is called an m×n matrix and the size or dimension of this matrix is said to be m×n.
3. Tow matrices are said to be equal provided they are of the same dimension and corresponding elements of the two matrices are equal.
4. A matrix is termed as square matrix if m = n or its size is m×m.
5. A matrix is termed as row matrix if m = 1
6. A matrix is termed as column matrix if n = 1
7. A matrix is termed as null or zero matrix if a_{0}] = 0 for all i and j.
8. A matrix is termed as diagonal matrix if a_{0}] = 0 for all ij where i is not equal to j.
9. A matrix is termed as scalar matrix if a_{0}] = 0 for all ij where i is not equal to j and a_{0}] = constant (k) for all i and j.
10. A matrix is termed as identity matrix or unit matrix if a_{0}] = 0 for all ij where i is not equal to j and a_{0}] = 1 for all i and j.
For example a problem is given in 2006 in comprehension where matrix A is given.
From AX1 = B1 you are asked to find X1, from AX2 = B2, and Ax3 = B3 you have to find X2 and X3. You are asked to make matrix [X1 X2 X3], then find its determinant, and then its inverse and then [R1][X][C1]
All questions are straight forward but computation is involed and you have to do fast.
Concepts
1. A matrix is a rectangular array of numbers [a_{0}]
2. A matrix with m rows and n columns is called an m×n matrix and the size or dimension of this matrix is said to be m×n.
3. Tow matrices are said to be equal provided they are of the same dimension and corresponding elements of the two matrices are equal.
4. A matrix is termed as square matrix if m = n or its size is m×m.
5. A matrix is termed as row matrix if m = 1
6. A matrix is termed as column matrix if n = 1
7. A matrix is termed as null or zero matrix if a_{0}] = 0 for all i and j.
8. A matrix is termed as diagonal matrix if a_{0}] = 0 for all ij where i is not equal to j.
9. A matrix is termed as scalar matrix if a_{0}] = 0 for all ij where i is not equal to j and a_{0}] = constant (k) for all i and j.
10. A matrix is termed as identity matrix or unit matrix if a_{0}] = 0 for all ij where i is not equal to j and a_{0}] = 1 for all i and j.
Differentiation - Online Material
Very good online material
http://tutorial.math.lamar.edu/Classes/CalcI/DerivativeIntro.aspx
List of the topics covered in this chapter.
The Definition of the Derivative In this section we will be looking at the definition of the derivative.
Interpretation of the Derivative Here we will take a quick look at some interpretations of the derivative.
Differentiation Formulas Here we will start introducing some of the differentiation formulas used in a calculus course.
Product and Quotient Rule In this section we will took at differentiating products and quotients of functions.
Derivatives of Trig Functions We’ll give the derivatives of the trig functions in this section.
Derivatives of Exponential and Logarithm Functions In this section we will get the derivatives of the exponential and logarithm functions.
Derivatives of Inverse Trig Functions Here we will look at the derivatives of inverse trig functions.
Derivatives of Hyperbolic Functions Here we will look at the derivatives of hyperbolic functions.
Chain Rule The Chain Rule is one of the more important differentiation rules and will allow us to differentiate a wider variety of functions. In this section we will take a look at it.
Implicit Differentiation In this section we will be looking at implicit differentiation. Without this we won’t be able to work some of the applications of derivatives.
Related Rates In this section we will look at the lone application to derivatives in this chapter. This topic is here rather than the next chapter because it will help to cement in our minds one of the more important concepts about derivatives and because it requires implicit differentiation.
Higher Order Derivatives Here we will introduce the idea of higher order derivatives.
Logarithmic Differentiation The topic of logarithmic differentiation is not always presented in a standard calculus course. It is presented here for those how are interested in seeing how it is done and the types of functions on which it can be used.
http://tutorial.math.lamar.edu/Classes/CalcI/DerivativeIntro.aspx
List of the topics covered in this chapter.
The Definition of the Derivative In this section we will be looking at the definition of the derivative.
Interpretation of the Derivative Here we will take a quick look at some interpretations of the derivative.
Differentiation Formulas Here we will start introducing some of the differentiation formulas used in a calculus course.
Product and Quotient Rule In this section we will took at differentiating products and quotients of functions.
Derivatives of Trig Functions We’ll give the derivatives of the trig functions in this section.
Derivatives of Exponential and Logarithm Functions In this section we will get the derivatives of the exponential and logarithm functions.
Derivatives of Inverse Trig Functions Here we will look at the derivatives of inverse trig functions.
Derivatives of Hyperbolic Functions Here we will look at the derivatives of hyperbolic functions.
Chain Rule The Chain Rule is one of the more important differentiation rules and will allow us to differentiate a wider variety of functions. In this section we will take a look at it.
Implicit Differentiation In this section we will be looking at implicit differentiation. Without this we won’t be able to work some of the applications of derivatives.
Related Rates In this section we will look at the lone application to derivatives in this chapter. This topic is here rather than the next chapter because it will help to cement in our minds one of the more important concepts about derivatives and because it requires implicit differentiation.
Higher Order Derivatives Here we will introduce the idea of higher order derivatives.
Logarithmic Differentiation The topic of logarithmic differentiation is not always presented in a standard calculus course. It is presented here for those how are interested in seeing how it is done and the types of functions on which it can be used.
Probability - 1
Random experiment
Any action whihc gives one or more results is called a random experiment or trial.Each result of the experiment ic called an outcome of the experiment.
Sample space: The set of all possible outcomes of an experiment is called the sample space of that experiment and is denoted by S.
When the number of possible outcomes are limited or finite it is called finite sample space.
n(s) represents the number of possible outcomes of S.
Example: If we toss a coin once, Head (H) or Tail (T) are possible.
So sample space S is {H,T}
n(S) = 2.
Unbiased experiment
In an unbiased experiment each of the outcomes in the sample space are equally likely to occur.
Event
An event is a subset of a sample space. We can define probabilities for specified events.
An event will be a null set, when the outcomes specified for the event to happen are not in the sample space.
Definition of probability
If S is the finite sample space of an experiment and every outcome of S is equally likely and if E is an event (i.e. E is contained in S), the the probability that E takes place is defined as
P(E) = n(E)/n(S)
Problems on probability
Drawing cards from a pack of cards.
Whenever a number of cards are drawn from a pack of cards, it is understood that, whenever a card is drawn from the pack, it is not replaced in the pack unless otherwise stated in the problem.
1. Two cards are drawn from a well shuffled pack of 52 cards. Find the probability that both are Hearts.
From the pack two cards can drawn in ^{52}C_{2} ways.
Therefore n(S) = ^{52}C_{2} = 52*51/2
The event two hearts can occur in ^{13}C_{2} ways as 13 Heart cards are there in the pack. If we identify the event with symbol 2H.
n(2H) = ^{13}C_{2} = 13*12/2
So P(2H) = n(2H)/n(S) = (52*51)/(13*12) [denominators cancel each other]
= 1/17
The probability of drawing two hearts is 1/17.
Any action whihc gives one or more results is called a random experiment or trial.Each result of the experiment ic called an outcome of the experiment.
Sample space: The set of all possible outcomes of an experiment is called the sample space of that experiment and is denoted by S.
When the number of possible outcomes are limited or finite it is called finite sample space.
n(s) represents the number of possible outcomes of S.
Example: If we toss a coin once, Head (H) or Tail (T) are possible.
So sample space S is {H,T}
n(S) = 2.
Unbiased experiment
In an unbiased experiment each of the outcomes in the sample space are equally likely to occur.
Event
An event is a subset of a sample space. We can define probabilities for specified events.
An event will be a null set, when the outcomes specified for the event to happen are not in the sample space.
Definition of probability
If S is the finite sample space of an experiment and every outcome of S is equally likely and if E is an event (i.e. E is contained in S), the the probability that E takes place is defined as
P(E) = n(E)/n(S)
Problems on probability
Drawing cards from a pack of cards.
Whenever a number of cards are drawn from a pack of cards, it is understood that, whenever a card is drawn from the pack, it is not replaced in the pack unless otherwise stated in the problem.
1. Two cards are drawn from a well shuffled pack of 52 cards. Find the probability that both are Hearts.
From the pack two cards can drawn in ^{52}C_{2} ways.
Therefore n(S) = ^{52}C_{2} = 52*51/2
The event two hearts can occur in ^{13}C_{2} ways as 13 Heart cards are there in the pack. If we identify the event with symbol 2H.
n(2H) = ^{13}C_{2} = 13*12/2
So P(2H) = n(2H)/n(S) = (52*51)/(13*12) [denominators cancel each other]
= 1/17
The probability of drawing two hearts is 1/17.
Probability - 2
Probability of combination of events
Event definitions
Union of events: If A and B are two events of the sample space S then A U B or A+B is a union of events and is the event that either A or B or both take place.
Intersection of events: If A and B are two events of the sample space S, the A ∩ B or AB is an intersection of events is the event that both A and B take place.
Mutually Exclusive events: Two events A and B of the sample space S are said to be mutually exclusive events if they cannot occur simulataneously. If A occurs B does not occur or if B occurs A does not occur.
It means A ∩ B , the event that both will occur is a null set.
Exhausive events: If two events A and B of a sample space are said to be exhausive events, if A U B contains all the points of the sample space.
Theorem regarding combination of events
If A and B are two events of sample space S, then
P(A U B) = P(A) + P(B) - P(A ∩ B)
Event definitions
Union of events: If A and B are two events of the sample space S then A U B or A+B is a union of events and is the event that either A or B or both take place.
Intersection of events: If A and B are two events of the sample space S, the A ∩ B or AB is an intersection of events is the event that both A and B take place.
Mutually Exclusive events: Two events A and B of the sample space S are said to be mutually exclusive events if they cannot occur simulataneously. If A occurs B does not occur or if B occurs A does not occur.
It means A ∩ B , the event that both will occur is a null set.
Exhausive events: If two events A and B of a sample space are said to be exhausive events, if A U B contains all the points of the sample space.
Theorem regarding combination of events
If A and B are two events of sample space S, then
P(A U B) = P(A) + P(B) - P(A ∩ B)
Trigonometric Equations - 1
Trigonometric equations are of the form sin θ = 1/2.
The general solution to the equation is θ = nπ + (-1)ⁿπ/6
General solutions list
1. If sin θ = sin α
the θ = nπ + (-1)ⁿα (n ∈ I)
2. If Cos θ = cos α
then θ = 2nπ ± α (n ∈ I)
3. If tan θ = tan α, then θ = nπ + α (n ∈ I)
4. If sin θ = sin α
and Cos θ = cos α
then θ = 2nπ + α (n ∈ I)
The general solution of the problem is to be given as an answer to a trigonmetric equations problem, unless the solution required is specified over a specific interval or range.
Very good online lessons
http://tutorial.math.lamar.edu/Classes/CalcI/TrigEquations.aspx
http://tutorial.math.lamar.edu/Classes/CalcI/TrigEquations_CalcI.aspx
http://tutorial.math.lamar.edu/Classes/CalcI/TrigEquations_CalcII.aspx
The general solution to the equation is θ = nπ + (-1)ⁿπ/6
General solutions list
1. If sin θ = sin α
the θ = nπ + (-1)ⁿα (n ∈ I)
2. If Cos θ = cos α
then θ = 2nπ ± α (n ∈ I)
3. If tan θ = tan α, then θ = nπ + α (n ∈ I)
4. If sin θ = sin α
and Cos θ = cos α
then θ = 2nπ + α (n ∈ I)
The general solution of the problem is to be given as an answer to a trigonmetric equations problem, unless the solution required is specified over a specific interval or range.
Very good online lessons
http://tutorial.math.lamar.edu/Classes/CalcI/TrigEquations.aspx
http://tutorial.math.lamar.edu/Classes/CalcI/TrigEquations_CalcI.aspx
http://tutorial.math.lamar.edu/Classes/CalcI/TrigEquations_CalcII.aspx
Mathematics online material
Saturday, May 3, 2008
Trigonometry - Inverse Functions - 1
Sin(sin‾¹x) = x
sin‾¹(sinθ) = θ
Similar relations hold for cos, tan, cosec, sec and cot.
sin‾¹x = cosec‾¹(1/x)
cos‾¹x = sec‾¹(1/x)
tan‾¹x = cot‾¹(1/x)
Relation between f(-x) and f(x)
sin‾¹(-x) = -sin‾¹x
cosec‾¹(x) = -cosec‾¹x
cos‾¹(x) = π - cos‾¹x
secπ -(-x) = π - sec‾¹x
tan‾¹(x) = -tan‾¹x
cot‾¹(-x) = π-cot‾¹x
sin‾¹x + cos‾¹x = π/2
tan‾¹x + cot‾¹x = π/2
sec‾¹x + cosec‾¹x = π/2
sin‾¹x + sin‾¹y = sin‾¹[x*SQRT(1-y²) + y*SQRT(1-x²)]
sin‾¹x-sin‾¹y = sin‾¹[x*SQRT(1-y²) - y*SQRT(1-x²)]
cons‾¹x +‾cos‾¹y = cos‾¹[xy - SQRT(1-x²) SQRT(1-y²)]
cos‾¹x-cos‾¹y = cos‾¹[xy + SQRT(1-x²) SQRT(1-y²)]
tan‾¹x + tan‾¹y =tan‾¹[(x+y)/(1-xy)] if xy<1
tan‾¹x -tan‾¹y = tan‾¹[(x-y)/(1+xy)] if xy>-1
Multiples of inverse functions
2sin‾¹x
2cos‾¹x
2tan‾¹x
3sin‾¹x
3cos‾¹x
3tan‾¹x
tan‾¹ [(1+x)/(1-x)] = π/4+ tan‾¹x
tan‾¹ [(1-x)/(1+x)] = π/4 - tan‾¹x
sin‾¹(sinθ) = θ
Similar relations hold for cos, tan, cosec, sec and cot.
sin‾¹x = cosec‾¹(1/x)
cos‾¹x = sec‾¹(1/x)
tan‾¹x = cot‾¹(1/x)
Relation between f(-x) and f(x)
sin‾¹(-x) = -sin‾¹x
cosec‾¹(x) = -cosec‾¹x
cos‾¹(x) = π - cos‾¹x
secπ -(-x) = π - sec‾¹x
tan‾¹(x) = -tan‾¹x
cot‾¹(-x) = π-cot‾¹x
sin‾¹x + cos‾¹x = π/2
tan‾¹x + cot‾¹x = π/2
sec‾¹x + cosec‾¹x = π/2
sin‾¹x + sin‾¹y = sin‾¹[x*SQRT(1-y²) + y*SQRT(1-x²)]
sin‾¹x-sin‾¹y = sin‾¹[x*SQRT(1-y²) - y*SQRT(1-x²)]
cons‾¹x +‾cos‾¹y = cos‾¹[xy - SQRT(1-x²) SQRT(1-y²)]
cos‾¹x-cos‾¹y = cos‾¹[xy + SQRT(1-x²) SQRT(1-y²)]
tan‾¹x + tan‾¹y =tan‾¹[(x+y)/(1-xy)] if xy<1
tan‾¹x -tan‾¹y = tan‾¹[(x-y)/(1+xy)] if xy>-1
Multiples of inverse functions
2sin‾¹x
2cos‾¹x
2tan‾¹x
3sin‾¹x
3cos‾¹x
3tan‾¹x
tan‾¹ [(1+x)/(1-x)] = π/4+ tan‾¹x
tan‾¹ [(1-x)/(1+x)] = π/4 - tan‾¹x
Learning status
Today I read all the chapters related to the calculus portion.
Regarding indefinite integrals, I prepared a written notes for various methods of evaluating integrals to write examples for each method after some time.
In case of definite integrals and differential equations, I posted some material in the blog.
Regarding indefinite integrals, I prepared a written notes for various methods of evaluating integrals to write examples for each method after some time.
In case of definite integrals and differential equations, I posted some material in the blog.
Subscribe to:
Posts (Atom)