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  • Squares And Square Roots | Study

  • Cubes And Cube Roots | Study

  • Comparing Quantities | Study

  • Algebraic Expressions And Identities | Study

  • Rational Numbers | Study

  • Linear Equations In One Variable | Study

  • CBSE 10 | Science – Study – Free

  • Squares And Square Roots | Study

    Squares And Square Roots | Study

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    Squares and Square Roots | Speed Notes

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    Square: Number obtained when a number is multiplied by itself. It is the number raised to the power 2. 22 = 2 x 2=4(square of 2 is 4).

    If a natural number m can be expressed as n2, where n is also a natural number, then m is a square number. (Scroll down to continue …)

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    All square numbers end with 0, 1, 4, 5, 6 or 9 at unit’s place. Square numbers can only have even number of zeros at the end. Square root is the inverse operation of square.

    There are two integral square roots of a perfect square number.

    Positive square root of a number is denoted by the symbol For example, 32=9 gives

    Perfect Square or Square number: It is the square of some natural number. If m=n2, then m is a perfect square number where m and n are natural numbers. Example: 1=1 x 1=12, 4=2 x 2=22.

    Properties of Square number:

    1. A number ending in 2, 3, 7 or 8 is never a perfect square. Example: 152, 1028, 6593 etc.
    2. A number ending in 0, 1, 4, 5, 6 or 9 may not necessarily be a square number. Example: 20, 31, 24, etc.
    3. Square of even numbers are even. Example: 22 = 4, 42=16 etc.
    4. Square of odd numbers are odd. Example: 52 = 25, 92 = 81, etc.
    5. A number ending in an odd number of zeroes cannot be a perferct square. Example: 10, 1000, 900000, etc.
    6. The difference of squares of two consecutive natural number is equal to their sum. (n + 1)2– n2 = n+1+n. Example: 42 – 32 =4 + 3=7. 122– 112 =12+11 =23, etc.
    7. A triplet (m, n, p) of three natural numbers m, n and p is called Pythagorean

    triplet, if m2 + n2 = p2: 32 + 42 = 25 = 52

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    • Cubes And Cube Roots | Study

      Cubes And Cube Roots | Study

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      Cubes and Cube Root | Speed Notes

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      Cube number: Number obtained when a number is multiplied by itself three times. 23 = 2 x 2 x 2 = 8, 33 = 3 x 3 x 3=27, etc.

      Numbers like 1729, 4104, 13832, are known as Hardy – Ramanujan Numbers. They

      can be expressed as sum of two cubes in two different ways.

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      Numbers obtained when a number is multiplied by itself three times are known as cube numbers. For example 1, 8, 27, … etc.

      If in the prime factorisation of any number each factor appears three times, then the

      number is a perfect cube.

      The symbol

      denotes cube root. For example

      Perfect Cube: A natural number is said to be a perfect cube if it is the cube of some natural number. Example: 8 is perfect cube, because there is a natural number 2 such that 8 = 23, but 18 is not a perfect cube, because there is no natural number whose cube is 18.

      The cube of a negative number is always negative.

      Properties of Cube of Number:

      1. Cubes of even number are even.
      2. Cubes of odd numbers are odd.
      3. The sum of the cubes of first n natural numbers is equal to the square of their sum.
      4. Cubes of the numbers ending with the digits 0, 1, 4, 5, 6 and 9 end with digits 0, 1, 4, 5, 6 and 9 respectively.
      5. Cube of the number ending in 2 ends in 8 and cube of the number ending in 8 ends in 2.
      6. Cube of the number ending in 3 ends in 7 and cube of the number ending in 7

      ends in 3.

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      • Comparing Quantities | Study

        Comparing Quantities | Study

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        Ratio: Comparing by division is called ratio. Quantities written in ratio have the sameunit. Ratio has no unit. (Scroll down to continue …)

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        Equality of two ratios is called proportion. Product of extremes = Product of means
        Percentage: Percentage means for every hundred. The result of any division in
        whichthe divisor is 100 is a percentage. The divisor is denoted by a special
        symbol %, read as percent. Profit and Loss:
        (i) Cost Price (CP): The amount for which an article is bought. (ii) Selling Price (SP): The amount for which an article is sold. Additional expenses made after buying an article are included in the cost price
        and are known as overhead expenses. These may include expenses like amount
        spent onrepairs, labour charges, transportation, etc. Discount is a reduction given on marked price. Discount = Marked Price – Sale
        Price. Discount can be calculated when discount percentage is given. Discount

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        • Algebraic Expressions And Identities | Study

          Algebraic Expressions And Identities | Study

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          Algebraic Expressions and Identities | Speed Notes

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          Expressions are formed from variables and constants.

          Constant: A symbol having a fixed numerical value.

          Example: 2,, 2.1, etc. (Scroll down to continue …)

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          Variable: A symbol which takes various numerical values. Example: x, y, z, etc.

          Algebric Expression: A combination of constants and variables connected by the sign

          +, -, and is called algebraic expression.

          Terms are added to form expressions.

          Terms themselves are formed as product of factors.

          Expressions that contain exactly one, two and three terms are called monomials, binomials and trinomials respectively.

          In general, any expression containing one or more terms with non-zero coefficients (and with variables having non- negative exponents) is called a polynomial.

          Like terms are formed from the same variables and the powers of these variables are the same, too.

          Coefficients of like terms need not be the same.

          While adding (or subtracting) polynomials, first look for like terms and add (or subtract) them; then handle the unlike terms.

          There are number of situations in which we need to multiply algebraic expressions: for example, in finding area of a rectangle, the sides of which are given as expressions.

          Monomial: An expression containing only one term. Example: -3, 4x, 3xy, etc.

          Binomial: An expression containing two terms. Example: 2x-3, 4x+3y, xy-4, etc.,

          Polynomial: In general, any expression containing one or more terms with non-zero coefficients (and with variables having non-negative exponents).

          A polynomial may contain any number of terms, one or more than one.

          A monomial multiplied by a monomial always gives a monomial.

          Multiplication of a Polynomial and a monomial:

          While multiplying a polynomial by a monomial, we multiply every term in the polynomial by the mononomial.

          Trinomial: An expression containing three terms.

          Example:

          3x+2y+5z, etc.

          In carrying out the multiplication of a polynomial by a binomial (or trinomial), we multiply term by term, i.e., every term of the polynomial is multiplied by every term in the binomial (or trinomial).

          Note that in such multiplication, we may get terms in the product which are like and have to be combined.

          An identity is an equality, which is true for all values of the variables in the equality.

          On the other hand, an equation is true only for certain values of its variables.

          An equation is not an identity.

          The following are the standard identities:

          (a + b)2 = a2 + 2ab + b2

          (a – b)2 = a2 – 2ab +b2 

          (a + b)(a – b) = a2 –  b2

          (x + a) (x + b) = x2 + (a + b) x + ab

          The above four identities are useful in carrying out squares and products of algebraic expressions.

          They also allow easy alternative methods to calculate products of numbers and so on.

          Coefficients: In the term of an expression any of the factors with the sign of the term is called the coefficient of the product of the other factors.

          Terms: Various parts of an algebraic expression which are separated by + and – signs. Example: The expression 4x + 5 has two terms 4x and 5.

          1. Constant Term: A term of expression having no lateral factor.
          2. Like term: The term having the same literal factors. Example 2xy and -4xy are like terms.

          (iii) Unlike term: The terms having different literal factors.

          Example:

          are unlike terms.

          and 3xy

          Factors: Each term in an algebraic expression is a product of one or more number (s) and/or literals. These number (s) and/or literal (s) are known as the factor of that term. A constant factor is called numerical factor, while a variable factor is known as

          a literal factor. The term 4x is the product of its factors 4 and x.

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          • Rational Numbers | Study

            Rational Numbers | Study

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            Rational numbers are closed under the operations of addition, subtraction and multiplication, But not in division. (Scroll down to continue …)

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            The operations addition and multiplication are
            (i) commutative for rational numbers.

            (ii) associative for rational numbers.

            The rational number 0 is the additive identity for rational numbers.

            The additive inverse of the rational number a/b is -a/b and vice- versa.

            The reciprocal or multiplicative inverse of the rational number
            is if a/b is c/d if (a/b)(c/d) =1

            Distributive property of rational numbers:

            For all rational numbers a, b and c, a(b + c) = ab + ac
            and a(b – c) = ab – ac.

            Rational numbers can be represented on a number line.

            Between any two given rational numbers there are countless rational numbers.

            The idea of mean helps us to find rational numbers between two rational numbers.

            Positive Rationals: Numerator and Denominator both are either positive or negative.

            Example: 2/3, -4/-5

            Positive Rationals: Numerator and Denominator both are either positive or negative.

            Example: -2/3, 4/-5

            .

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            • Linear Equations In One Variable | Study

              Linear Equations In One Variable | Study

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              A statement of equality of two algebraic expressions involving one or more variables. Example: x + 2 = 3

              Linear Equation in One variable: The expressions which form the equation that contain single variable and the highest power of the variable in the equation is one. (Scroll down to continue …)

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              Linear Equations in One Variable

              An algebraic equation is an equality involving variables. It says that the value of the expression on one side of the equality sign is equal to the value of the expression on the other side.

              The equations we study in Classes VI, VII and VIII are linear equations in one variable. In such equations, the expressions which form the equation contain only one variable. Further, the equations are linear, i.e., the highest power of the variable appearing in the equation is 1.

              A linear equation may have for its solution any rational number.

              An equation may have linear expressions on both sides. Equations that we studied in Classes VI and VII had just a number on one side of the equation.

              Just as numbers, variables can, also, be transposed from one side of the equation to the other.

              Occasionally, the expressions forming equations have to be simplified before we can solve them by usual methods. Some equations may not even be linear to begin with, but they can be brought to a linear form by multiplying both sides of the equation by a suitable expression.

              The utility of linear equations is in their diverse applications; different problems on numbers, ages, perimeters, combination of currency notes, and so on can be solved

              using linear equations.

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