Saturday, December 09, 2006

Sunday, December 03, 2006

INCOME TAX - TIPS

*CLICK HERE FOR INDEX-TOPIC SEARCH



IMPORTANT TIPS:



  • All governments collect taxes from the citizens of the nation.The amount received in the form of taxes is utilized to create infrastructure, provide better education and maintain public health, etc.


  • Income tax is a type of DIRECT TAX.


  • Income tax is collected from those citizens whose annual income exceeds certain fixed minimum amount.


  • The person who pays tax is called a TAX-PAYER (an ASSESSEE).


  • There are certain rules decided by the government to calculate the amount of tax to be paid by a person based on his annual income.These rules are amended as and when needed (i.e. these rules are not permanent).


  • Income tax is calculated according to the rates prevailing in the year under consideration.


  • Some special SAVINGS made by the person during that year are deducted from his GROSS ANNUAL INCOME before calculating TAXABLE INCOME.


  • The tax amount is calculated on TAXABLE INCOME.


  • Sometimes, the government charges cesses like educational cess and surcharge on the income tax to be paid by the assessee.These are additional liabilities on the tax-payer.


  • For a salaried person, income tax is deducted from his salary every month and deposited to the Income-Tax Department.This is called TAX DEDUCTED AT SOURCE (T.D.S.).


  • Businessman, Industrialists, Professionals and Self-employed persons pay their taxes directly to the I.T.Dept.


  • Each tax payer has to file ANNUAL INCOME TAX RETURN in which he gives statements of his income, expenditure, savings, etc.


  • It is compulsory for the assessee to mention his/her PAN (PERMANENT ACCOUNT NUMBER) in the income tax return filed by him/her.


  • A person whose income is less than the minimum income fixed to calculate tax does not have to pay income tax.


  • A person who has paid more income tax (as T.D.S. or Advanced Tax) than he should actually have paid is refunded the excess amount paid by him.


  • Each ACCOUNTING YEAR starts on 1st April of the year and ends on 31st March of the next year.


  • The next year is called the ASSESSMENT YEAR for the previous accounting year. Thus, for the accounting year 2004 - 2005 (i.e. 1 - 4 - 2004 to 31 - 3 - 2005),the assessment year is 2005 - 2006.


  • Rates of Income Tax for the financial year(accounting year) 2005 - 2006 are:

  • alt=""id="BLOGGER_PHOTO_ID_5005266472666527746" />

      [Here, I.T. = INCOME TAX.The surcharge is calculated on I.T. Thus, if calculated tax is Rs.5000 and surcharge is 10% then it amounts to 10% of 5000 or Rs.500.Similarly, Educational Cess is calculated as 2% of the total of I.T. and surcharge.]

  • For any taxpayer(who is not a woman/senior citizen), the gross income upto Rs.100000/- is free from incone tax.


  • For a woman taxpayer(who is not a senior citizen),the gross income upto Rs.135000/- is free from income tax.


  • For senior citizens (persons above 65 years of age),the gross income upto Rs.185000/- is free from income tax.


  • Savings amount upto Rs.100000/- under section 80C is exempted from tax.Such savings include:LIC(Life Insurance Corporation)Premiums, PPF(Public Provident Fund),GPF(General Provident Fund),PLI(Postal Life Insurance),Certain Government Bonds(RBI Bonds,National Savings Certificate-NSC,Kisan Vikas Patra-KVP,etc).


  • Housing loan interest upto Rs.150000/- is exempted from total income under section 80D.Housing loan principal is exempt under section 80C within the limit of Rs.100000/-.


  • Mediclaim premium upto Rs.10000/- is exempted under section 80D.


  • DO NOT APPLY ANY OTHER INCOME TAX RULE WHICH HAS NOT BEEN MENTIONED IN THE TEXT BOOK.INCOME TAX RULES KEEP ON CHANGING YEAR BY YEAR.IT IS NOT POSSIBLE TO MENTION ALL RULES IN THE BOOK.THE PURPOSE OF THE CHAPTER IS TO GIVE YOU A PRELIMINARY IDEA OF CALCULATING TAX.

Monday, November 27, 2006

INSTALMENTS-TIPS

*CLICK HERE FOR INDEX-TOPIC SEARCH



IMPORTANT TIPS



  • INSTALMENT PURCHASE SCHEME is a contract between the buyer and the seller by which buyer makes part payment of the price of the commodity (called DOWN PAYMENT) at the time of purchase to the seller and agrees to pay remaining amount in fixed instalments and, in return, the seller gives that commodity to the buyer.


  • CASH PRICE : The amount in full (Selling Price) to be paid at the time of purchase is called CASH PRICE of the commodity.


  • DOWN PAYMENT : If the buyer does not pay the full price (selling price) of the commodity and prefers to buy it under instalment scheme, then the part payment made by him at the time of purchase is called DOWN PAYMENT.

    For example: Cash price of a TV is Rs. 15000/- and the buyer pays Rs. 5000/- at the time of purchase, then Rs. 5000/- is called DOWN PAYMENT.


  • RATE OF INTEREST : If a commodity is purchased under instalment scheme, the buyer has to pay some extra amount as interest on the unpaid amount which is calculated at some rate called RATE OF INTEREST.


  • INSTALMENT : An amount paid regularly after definite time period (till the total dues,i.e.principal + interest, are paid in full) by the buyer to the seller is called INSTALMENT.


  • For calculating simple interest we use the following formula:

    I = PRN/100

    where I = simple interest, P = principal amount, R = rate of interest( % ) and N = number of years (time period in years).


  • For calculating the rate of interest we use:

    R = (I x 100)/PN.

Sunday, November 26, 2006

ARITHMETIC PROGRESSION-TIPS

*CLICK HERE FOR INDEX-TOPIC SEARCH



IMPORTANT TIPS



  • Observe the following sets of numbers carefully:


    1. 1, 2, 3, 4, 5, ...
    2. 2, 4, 6, 8, 10, ...
    3. 1, 4, 9, 16, 25, ...
    4. 1, 11, 111, 1111, 11111, ...
    5. 1, 1/2, 1/3, 1/4, 1/5, ...



    Here, in each set, we find numbers arranged in some order and there is an obvious (definite) rule by which we can obtain the next number and as many subsequent numbers as we wish to find.

    Such sets are called SEQUENCES ( PROGRESSION ) and each number of the set is called a TERM of the sequence.


  • A sequence refers to an ordered set of numbers in which each number (term) can be obtained by a definite rule.


  • The rule by which a sequence is formed may be written as a formula for nth term of the series but all sequences need not have a formula.


  • The nth term of 1, 2, 3, 4, 5, ... is n.


  • The nth term of 1, 4, 9, 16, 25, ... is n2.


  • The nth term of 3, 4, 7, 12, 19, ... is (n - 1)2 + 3.


  • 2, 3, 5, 7, 11, 13, ... is the progression of prime numbers.


  • The nth term of a progression is denoted by either an or Tn


  • If a progression starts with a definite real number and if any successive term is obtained by adding a constant nonzero real number to the previous term, then the progression is called an ARITHMETIC PROGRESSION (written in short as A.P.)


  • Sequences (a) and (b) in the examples given in the beginning are examples of Arithmetic progression whereas sequences (c), (d) and (e) are progressions but they are not Arithmetic progressions.


  • The difference between any two consecutive terms of an A.P. is a nonzero constant. This difference is called the COMMON DIFFERENCE.


  • The general form of an A.P. whose first term is a (a ∈ R) and the common difference is d (d≠ 0) is:

    a, a + d, a + 2d, a + 3d, ...


  • The nth term (Tn) of an A.P. is

    Tn = a + (n - 1)d


  • The sum of the first n terms of an A.P. is given by

    Sn = (n/2)(a + l)

    where a = first term of the A.P. and l = last term of the A.P.

    Since l = Tn = a + (n - 1)d,

    Sn = n/2 [ 2a + (n - 1)d]


  • Tn - Tn-1 = d


  • Sn - Sn-1 = Tn


  • Tm - Tn = (m - n)d [where m>n,&m,n ∈ N].

Saturday, November 25, 2006

QUADRATIC EQUATION-TIPS

*CLICK HERE FOR INDEX-TOPIC SEARCH




IMPORTANT TIPS



  • ax2 + bx + c is a quadratic polynomial or second degree polynomial where a, b, c are real constants and a ≠ 0.


  • x2 - √ 5 x + 7, 3x2 - 2x + 1, x2 - 3, 4x2, etc. are examples of second degree polynomials.


  • The values of the variable(x) for which the value of the polynomial is zero, are called ZEROS OF A POLYNOMIAL.


  • For a second degree polynomial we can have at the most two values of the variable for which the value of the polynomial is zero, i.e. there are at the most two zeros of a second degree polynomial.


  • For p(x) = x2 - 6x + 8, we have x = 4 and x = 2 for which the value of p(x) is 0. Thus, 4 and 2 are the zeros of p(x).


  • For p(y) = y2 + 4, we cannot find a value of y for which p(y) is 0. Thus, a polynomial may not have zeros.


  • If p(x) is a second degree polynomial, then p(x) = 0 is called a QUADRATIC EQUATION.


  • ax2 + bx + c = 0 is the general form of a quadratic equation, where a, b, c ∈ R and a ≠ 0.


  • The values of x which satisfy the equation ax2 + bx + c = 0 (a ≠ 0) are called the roots of that quadratic equation.


  • Zeros of polynomial ax2 + bx + c are the roots of the quadratic equation ax2 + bx + c = 0.


  • If ax2 + bx + c = (x - α) (x - β) then α and β are the roots of the equation ax2 + bx + c = 0.


  • All polynomials of the form ax2 + bx + c cannot be factorized by normal methods of factorization into factors like (x - α) and (x - β). In such cases, equation ax2 + bx + c = 0 is solved by the METHOD OF PERFECT SQUARE ( also known as the Method of Discriminant or the Method of formula).


  • Method of Perfect Square can be used to solve any quadratic equation which has a real solution, i.e. solution in R.


  • By the method of perfect square, we obtain the roots α and β of the quadratic equation ax2 + bx + c = 0 (a ≠ 0) as

    α = ( - b - √ D)/2a

    β = ( - b + √ D)/2a

    where D = b2 - 4ac and it is called the DISCRIMINANT of the given quadratic equation.


  • If D ≥ 0, the roots of the equation are real.


  • If D > 0, the roots are real and distinct.


  • If D > 0, D is not a perfect square ( for example : D = 22), then the roots are irrational and distinct.


  • If D > 0, D is a perfect square ( for example : D = 25), then the roots are rational and distinct.


  • If D = 0, the roots 0f the equation are equal ( also called 'repeated' or 'identical' roots), i.e. the equation has only one root. The value of this root is ( - b )/2a .


  • If D < 0, ( i.e. D is negative ), then the given equation has no real roots. ( We say real roots do not exist ).

Friday, November 24, 2006

RATIONAL EXPRESSION-TIPS

*CLICK HERE FOR INDEX-TOPIC SEARCH



IMPORTANT TIPS



  • ax + b (a ≠ 0) is a linear polynomial whereas ax2 + bx + c (a ≠ 0)is a polynomial of second degree or a quadratic polynomial.


  • If p(x) and q(x) are polynomials over R and q(x) ≠ 0, then p(x)/q(x) is called a rational expression.


  • If the HCF of p(x) and q(x) is 1, i.e. there is nothing common between p(x) and q(x) except 1 , then p(x)/q(x) is called a rational expression in its REDUCED FORM.


  • If p(x) = h(x) · p'(x) and q(x) = h(x) · q'(x)where h(x) is the HCF of p(x) and q(x), then the rational expression p(x)/q(x) can be reduced to p'(x)/q'(x), i.e. p'(x)/q'(x) is the reduced form of p(x)/q(x).


  • To simplify p(x)/q(x), factorize both p(x) and q(x) into their prime factors; then cancel out like terms in both the numerator and the denominator.


  • x + x = 2x (Some students write x + x = x2. In fact, x · x = x2).


  • 1 + 2 + x = 3 + x (Some students write 1 + 2 + x = 3x which is not true).


  • Two polynomials can be added, subtracted, multiplied, and divided in the same manner as numbers.


  • ADDITION:Two terms of same sign are added and the sign is retained. Thus, (-2) + (-3) = (-5) and (+4) + (+3) = (+7) or 4 + 3 = 7.

    The smaller term is subtracted from the bigger term if two terms have opposite signs and then the sign of the bigger term is retained in the result. Thus, (-7) + (+3) = (-4) and 7 + (-3) = 4.


  • (a - b)2 = (b - a)2 BUT (a - b)3 ≠ (b - a)3. In general, (a - b)n = (b - a)n if n is even and (a - b)n = - (b - a)n if n is odd.


  • x/x = 1. Similarly, (x + 2y)/(x + 2y) = 1. (Remember: x/x ≠ 0).


  • (2x - 4)2 ≠ 2(x - 2)2 BUT (2x - 4)2 = 4(x - 2)2 and (2x - 4)3 = 8(x - 2)3.


  • 16(x + 2)2 ≠ (16x + 32)2 BUT 16(x + 2)2 = (4x + 8)2 and 27(x - 3)3= (3x - 9)3.


  • (2x + 10)/2 ≠ x + 10 (2 cancelled with 2 of 2x).

    (2x + 10)/2 ≠ 2x + 5 (2 cancelled with 2 of 10).

    Actually, (2x + 10)/2 = 2(x + 5)/2 = (x + 5).


  • (4x + 5)/(2x + 5) ≠ 2.[You cannot cancel x + 5].


  • While simplifying rational expression, strictly follow the rule of the order of operations: BODMAS → Brackets of Division, Multiplication, Addition and Subtraction.

Thursday, November 23, 2006

HCF AND LCM OF POLYNOMIALS-TIPS

*CLICK HERE FOR INDEX-TOPIC SEARCH



IMPORTANT TIPS




  • Make sure that you know how to find HCF(Highest Common Factor) and LCM(Lowest Common Multiple) of given set of numbers.


  • p(x)=anxn + an-1xn-1 + an-2xn-2 + ... + a1x+a0 is called the general form of a polynomial in x, where ai∈ R(i=0,1,2,3,...) and an≠ 0.


  • n is called the degree of polynomial.


  • p(x) is the symbol for a polynomial over x and q(y) is the symbol for a polynomial over y.Similarly, we use r(x), m(x), n(y), p(t), s(t), etc. to denote polynomials symbolically.


  • If p(x)= 0, then it is called the ZERO POLYNOMIAL.


  • If p(x)= k, (k∈ R, a constant), then it is called a CONSTANT POLYNOMIAL.


  • If n = 1, the polynomial is called a LINEAR POLYNOMIAL. Thus, p(x)= 3x + 2 is a linear polynomial.


  • If n = 2, the polynomial is called a SECOND-DEGREE POLYNOMIAL or a QUADRATIC POLYNOMIAL. Thus, p(x)=2x2 - 3x - 3 is a second-degree polynomial.


  • The highest exponent(index) of the variable denotes the degree of a polynomial.Thus, m(y)= 6y5 - 4y3 + x2 - 3x + 1 is a polynomial of degree 5.


  • The standard method of writing a polynomial is to write it either in the ascending order or the descending order of the exponent of its variable.Thus, p(x)= x4 - 3x3 + 2x2 + 5x + 6 (descending order) or p(x)= 6 + 5x + 2x2 - 3x3 + x4 (ascending order).


  • In the general form of a polynomial, ai is the coefficient of xi (i=1,2,3,...). Thus, for i = 0, we have a0x0 = a0 ( because x0 = 1). ∴ a0 is called the CONSTANT TERM.


  • If a polynomial p(x) is the product of polynomials g(x) and h(x), then g(x) and h(x) are aclled the factors of p(x).


  • HCF (GCF) OF POLYNOMIALS:If h(x) is a common factor of the given set of polynomials and every common factor of the given polynomials is a factor of h(x), then h(x) is said to be the Highest(Greatest) Common Factor of the given set of polynomials.


  • LCM OF POLYNOMIALS:If m(x) is a common multiple of the given set of polynomials and every common multiple of given polynomials is also a multiple of m(x), then m(x) is said to be the Least Common Multiple of the given set of polynomials.


  • If h(x) and m(x) are the HCF and LCM, respectively, of two polynomials p(x) and q(x), then

    p(x) · q(x) = ± h(x) · m(x)


  • ALERT:

    p(x) · q(x) · r(x) ≠ ± h(x) · m(x)



    The relation holds only for two polynomials.

Monday, October 30, 2006

LINEAR EQUATIONS OF TWO VARIABLES-IMPORTANT TIPS

*CLICK HERE FOR INDEX-TOPIC SEARCH



IMPORTANT TIPS:




  • ax+b = 0 (a,b are real numbers and a is nonzero) is a linear equation of one variable. Thus, 2x+3 = 0 and 0.4x-5 = 0 are linear equations of one variable.

  • ax+by+c = 0(a,b,c are real numbers and a and b are not equal to zero simultaneously)is a linear equation of two variables. Thus, 3x-2y+4 = 0 and(2x/3)+(4y/5)-6 = 0 are linear equations of two variables.

  • y = (-a/b)x-(c/b) or y = (-ax-c)/b is called y-form of the equation ax+by+c = 0

  • If we have only one equation, ax+by+c = 0, then it has got infinite solutions because for each value of x we get some value of y which satisfies the equation.

  • If we have a pair of equations a1x+b1y+c1 = 0 and a2x+b2y+c2 = 0,then there are three possibilities:
    1. The equations have a particular(unique) solution where unique values of x and y satisfy both equations. This happens when a1b2-a2b1≠ 0

    2. The equations do not have a solution, i.e. there is no set of values of both x and y which can satisfy both equations. This happens when a1b2-a2b1 = 0 and either b1c2-b2c1≠ 0 or c1a2-c2a1≠ 0

    3. The pair of equations has infinite solutions.This happens when a1b2-a2b1 = b1c2-b2c1 = c1a2-c2a1 = 0


  • METHOD OF CROSS MULTIPLICATION:

    Let the given pair of equations be:

    a1x+b1y+c1 = 0
    a2x+b2y+c2 = 0

    (a1, b1 and a2, b2 are not zero simultaneously)

    To understand the procedure READ THE DETAILS IN THE FOLLOWING IMAGE and
    To read the details clearly ENLARGE IT by opening it in another window.





    What we have done while going from step (A) to step (B) is :

    Write the product of b1 andc2 (downward arrow) minus the product of b2 and c1 (upward arrow) below x. Follow the same procedure for y and 1.

    So, we get (B). NEXT ?

    x = b1c2 - b2c1
          a1b2 - a2b1

    and

    y = c1a2 - c2a1
          a1b2 - a2b1



    Thus, we get the values of x and y which satisfy the given pair of equations.

  • PRACTICAL PROBLEMS:

    In solving practical problems, one must not forget that it is the lack of proper understanding of the language that makes the solution difficult.So, while solving practical problems with the help of linear equations :
    1. Read the statement carefully.
    2. Analyse it thoroughly.
    3. Form correct equations.


    This requires great patience and exhaustive practice.More you practise, more you understand the method.



DESSERT

2 is the only prime which is even!

Friday, October 27, 2006

INTRODUCTORY NOTE

*CLICK HERE FOR INDEX-TOPIC SEARCH



It is a general impression among highschool students and their parents that Mathematics is a very tough subject to digest.In fact, about 40% of a student's study time is utilized for mathematics only.It is obvious and essential because mathematics is not a novel or a collection of short stories which you finish in one go. It is a subject which asks for visualization, computing skills,power of analysis and persistence in effort.

MATHEMATICS differs from other subjects in the sense that you cannot afford to forget basic concepts and methods you have studied previously.You can and sometimes you do forget the story of a king and his adventures which you were tought in Class VI or Class VII. You cannot afford to forget Methods of Factorization or Mathematical Identities when you enter Class X.The story of Class VII has no connection to the one you are going to study in Class X, but the topics of mathematics do have basic connections as you move forward. This aspect makes Mathematics an entirely different subject.

REMEMBER:


MATHEMATICS, AS FAR AS I HAVE EXPERIENCED AND ENJOYED, IS NOT A TOUGH SUBJECT AT ALL

WHAT YOU MUST REMEMBER



The following is a list of some important mathematical rules, formulas, methods, etc. ,which you have already studied uptil now and which you cannot afford to forget.Most of the students ignore this very important fact and it is too late when they realise what they missed.You must remember:

  • Tables of 1-20(at least)

  • Squares of 1-25(at least)

  • Cubes of 1-12

  • Decimal Calculations

  • Percentage Calculations

  • Calculation of HCF/GCF and LCM

  • Conversion of Units(e.g. 1 m= 100 cm)

  • All important Mathematical Identities and their Proper Use.

  • All methods of factorization

  • Rules of Indices

  • Rules and Methods connected to Ratio and Proportion

  • All Geometrical Definitions and Concepts(point,ray,line.angle,plane,etc.)

  • Types of Triangles and Quadrilaterals

  • Properties of Triangles and Quadrilaterals

  • Formulas to calculate Perimeter, Area, Volume,etc.

  • Preliminary Trigonometry

  • Preliminary Set Theory and Set Operations.


Once you have mastered these all, your studies will become much easier and more interesting.

A PUZZLE FOR FUN



Harsh went to a departmental store to purchase something.He spent half of the total amount he had when he entered the store.When he came out he had just as many paise as he had rupees while entering the store and half as many rupees as he had paise while entering the store.What amount actually Harsh carried when he entered the store?

WELCOME TO MY MATHS BLOG

Welcome my dear students,
Your MANOJSIR is now available on net to solve your problems of mathematics.I will be posting topics of mathematics of Class 10 of Gujarat Secondary Education Board(GSEB).I hope that this will be of help to every student of Class 10 studying in any EDUCATION BOARD OF INDIA.

This is my humble effort to serve my students.


INDEX-TOPIC SEARCH
*INTRODUCTORY NOTE

*LINEAR EQUATIONS OF TWO VARIABLES-TIPS

*HCF AND LCM OF POLYNOMIALS-TIPS

*RATIONAL EXPRESSION

*QUADRATIC EQUATION-TIPS

*ARITHMETIC PROGRESSION-TIPS

*INSTALMENTS-TIPS

*INCOME TAX-TIPS

*MEAN-TIPS

*SIMILAR TRIANGLES-TIPS

*CONDITIONS FOR SIMILARITY-TIPS

*SIMILARITY AND PYTHAGORAS' THEOREM-TIPS