7th Class

     Simple Interest                   In the case of simple interest, we calculate interest paid by borrower over certain period of time without changing its principle.                         Calculate the simple interest on a sum of $1200 at the rate of 5 % per annum or two years.                 Solution:                 The following steps are to be followed                 Step 1: The interest for first year when                 P =$ 1200, R = 5 % and T = 1 year                 S.I. \[=\frac{P\times R\times T}{100}=\frac{1200\times 5\times 1}{100}=\$60\]                 Therefore, interest for the first year is $ 60.                                 Step 2: Interest for the second year                 P = $ 1200, R = 5 % and T = 1 year                 S.I. \[=\frac{P\times R\times T}{100}=\frac{1200\times 5\times 1}{100}=\$60\]                 Therefore, interest for the second year more...

     Compound Interest                   In this case we observe that the interest paid by borrower is same for every year.                 There are different methods to calculate the interest in case of bank transaction loan etc. In these methods, interest is calculated either quarterly, half yearly or yearly, what so ever may be the case. i.e. The agreement between the lender and borrower on the principle. The amount after that first fixed interval of time will be principle for second interval of time, the amount after second interval of time will be the principle for the third interval and so on. The interest paid by borrower under the above conditions is called compound interest.         Abbreviation used in Interest

•  Principle as P              
•  Time as T
•   Rate as R or r
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     Introduction                   The word triangle is derived from Greek word, triangle means three and hence it to a shape consisting three internal angles. Obviously the shape consists of sides. Hence, a triangle can be defined as a polygon having three sides.  

     Basic Concepts of Triangles                  The general shape of a triangle is shown below:                                 The vertices of a triangle are denoted by the Capital letters of English alphabets.                 In the above figure \[\Delta ABC,\]the sides are AB, BC and CA.                         Altitude                 A Perpendicular drawn from a vertex to the opposite side is called the altitude of the triangle and denoted as h.           Some Basic Facts Related to Triangle

• In any triangle, sum of any two sides is always greater than the 3rd side. i.e \[b+c>a\]or \[a+c>b\]or \[c+b>a.\]
• The sum of all interior angles of a triangle is \[180{}^\circ \]
• The exterior angle at any vertex of the triangle is equal to the more...

     Factorization of the Polynomials   Let us recall that an algebraic expression that is expressed as the product of two or more expressions & each of these is a factor of the given algebraic expression. The process of writing a given algebraic expression as the product of two or more factors is called factorization.   Some of the common methods of factorization of the algebraic expressions are as follows:

•    Factorization by taking out a common factor.
•    Factorization by using identities
•    Factorization by regrouping the terms.  

        Factorization of the Algebraic Expression \[a{{x}^{2}}+bx+c\]   Step 1:   Find the product of constant term and coefficient of \[{{x}^{2}}\] i.e. ac.   Step 2:   Find factors of "ac".   Step 3:   Select the factors of 'ac' in such a way that addition or subtraction of factors must more...

     Algebraic identities   Abatement of equality which holds, for all values of the variable is called algebraic identities. Now we recall some important algebraic identities:

• \[{{(a+b)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}\]
• \[{{(a-b)}^{2}}={{a}^{2}}-2ab+{{b}^{2}}\]
• \[{{a}^{2}}-{{b}^{2}}=(a-b)(a+b)\]
• \[{{(a+b+c)}^{2}}={{a}^{2}}+{{b}^{2}}+{{c}^{2}}+2ab+2bc+2ca\]
• \[{{a}^{3}}+{{b}^{3}}={{(a+b)}^{3}}-3ab(a+b)\]or \[{{(a+b)}^{3}}={{a}^{3}}+{{b}^{3}}+3ab(a+b)\]
• \[{{(a-b)}^{3}}={{a}^{3}}-{{b}^{3}}-3ab(a-b)\]
• \[{{a}^{3}}+{{b}^{3}}={{(a+b)}^{3}}-3ab(a+b)\]or \[(a+b)({{a}^{2}}-ab+{{b}^{2}})\]
• \[{{a}^{3}}-{{b}^{3}}={{(a-b)}^{3}}+3ab(a-b)\]or \[(a-b)({{a}^{2}}+ab+{{b}^{2}})\]
• \[{{a}^{3}}+{{b}^{3}}+{{c}^{3}}-3abc=(a+b+c)\]\[({{a}^{2}}+{{b}^{2}}+{{c}^{2}}-ab-bc-ca)\]

               If \[(a+b+c)=0\]\[{{a}^{3}}+{{b}^{3}}+{{c}^{3}}=3abc\]       Simplify:  \[{{(2p+3q-+4r)}^{2}}+{{(2p-3q-4r)}^{2}}\] (a) \[2(4{{p}^{2}}+9{{q}^{2}}+16{{r}^{2}}-16rp)\] (b) \[-2(4{{p}^{2}}+9{{q}^{2}}+16{{r}^{2}}-16rp)\] (c) \[2(-4{{p}^{2}}+9{{q}^{2}}-16{{r}^{2}}+16rp)\] (d)\[~2(5{{p}^{2}}+9{{q}^{2}}-16{{r}^{2}}-98rp)\] (e) None of these     Answer: (a) Explanation    Let us first solve, \[{{\left[ 2p+3q+(-4r) \right]}^{2}}={{(2p)}^{2}}+{{(3q)}^{2}}\]\[+{{(-4r)}^{2}}+2(2p)(3q)+2(3q)(-4r)\] \[+2(-4r)(2p)=4{{p}^{2}}+4{{p}^{2}}+9{{q}^{2}}+16{{r}^{2}}\]\[+12pq-24qr-16rp..........(i)\] Now solve, \[~{{(2p-3q-4r)}^{2}}={{\left[ 2p+(-3q)+(-4r) \right]}^{2}}\] \[={{(2p)}^{2}}+{{(-3q)}^{2}}+{{(-4r)}^{2}}+2(2p)(-3q)+2(-3q)\]\[(-4r)+2(-4r)(2p)\] \[=4{{p}^{2}}+9{{q}^{2}}+16r2-12pq+24qr-16rp\text{ }.......\left( ii \right)\] Adding, (i) & (ii) we get, \[{{(2p+3a-4r)}^{2}}+{{(2p-3q-4r)}^{2}}\] \[=4{{p}^{2}}+9{{q}^{2}}+16{{r}^{2}}+12pq-24qr-16rp\]+ \[(4{{p}^{2}}+9{{q}^{2}}+16{{r}^{2}}-12pq+24qr-16rp)\]                 \[=4{{p}^{2}}+9{{q}^{2}}+16{{r}^{2}}+12pq-24qr-16rp+4{{p}^{2}}+9{{q}^{2}}\]    \[+16{{r}^{2}}-12pq+24qr-16rp\]                 \[=8{{p}^{2}}+18{{q}^{2}}+32{{r}^{2}}-32rp\]                 \[=2(4{{p}^{2}}+9{{q}^{2+}}16r2-16rp)\]             The expanded form of \[{{(2x+3y-5z)}^{2}}\] is:                 (a) \[4{{x}^{2}}+9{{y}^{2}}+25{{z}^{2}}+12xy-30yz-20zx\]                 (b) \[5{{x}^{2}}-6{{y}^{3}}+15{{z}^{3}}+12xy-36{{y}^{6}}+21x{{y}^{2}}\]                 (c) \[8{{a}^{2}}+{{a}^{3}}-{{c}^{2}}+7{{x}^{2}}+2{{c}^{2}}+9{{c}^{2}}y\]                 (d) more...

     Operations on Algebraic Expression   Addition and subtraction of an algebraic expressions mean addition and subtraction of like terms.                         Addition of an Algebraic Expression For addition of algebraic expression we may follow any one of the following methods:                                   Row Method In this method, write all expression in a single row then arrange the terms to collect all like terms together and add it.                                   Column Method In this method, arrange each expression in such a way that each like term is placed one below to other in a column.       Add \[3x+2y+3z\]and \[2x-3y+4z\] Solution:                 \[(3x+2y+3z)+(2x-3y+4z)\]          \[=(3x+2x)+(2y-3y)+(3z+4z)=5x-y+7z\]  Column method                 \[3x+2y+3z\]                 \[\frac{+2x-3y+4z}{5x-y+7z}\]                         Subtraction of an Algebraic Expression For subtraction also you may more...

     The value of an Algebraic Expression     Step 1:   If possible simplify the given algebraic expression.   Step 2:   Replace variable with given numerical value.   Step 3:   Simplify it.     Find the value of \[\frac{{{x}^{3}}+{{y}^{3}}+{{z}^{3}}-3xyz}{{{x}^{2}}+{{y}^{2}}+{{z}^{2}}-xy-yz-zx},\] lf \[x=1,y=2\] and \[z=-1.\] (a) 24                                                     (b) 14 (c) 7                                                       (d) 2 (e) None of these     Answer: (d) Explanation \[\frac{{{x}^{3}}+{{y}^{3}}+{{z}^{3}}-3xyz}{{{x}^{2}}+{{y}^{2}}+{{z}^{2}}-xy-yz-zx}=\frac{(x+y+z+)({{x}^{2}}+{{y}^{2}}+{{z}^{2}}-xy-yz-zx)}{({{x}^{2}}+{{y}^{2}}+{{z}^{2}}-xy-yz-zx)}\] \[=(x+y+z).\] Now putting the values of \[x,\text{ }y\]and \[z\] we get = 2             Find the value of \[4\text{ }xy\text{(}x-y\text{)}-6{{x}^{2}}\text{(}y-{{y}^{2}}\text{)}-3{{y}^{2}}(2{{x}^{2}}-x)+2xy(x-y)\] for \[x=5\]and \[y=13.\] (a) - 195                               (b) 2535 (c) - 2535                                             (d) 215 (e) None of these     Answer: (c)   Explanation \[4xy(x-y)-6{{x}^{2}}(y-{{y}^{2}})-3{{y}^{2}}(2{{x}^{2}}-x)+2xy(x-y)\]\[4{{x}^{2}}y-4x{{y}^{2}}-6{{x}^{2}}y+6{{x}^{2}}{{y}^{2}}-6{{y}^{2}}{{x}^{2}}+3x{{y}^{2}}+2{{x}^{2}}y-2x{{y}^{2}}\] After simplification, we get \[-\text{ }3x{{y}^{2}}=-3\times 5\times more...

     Introduction   We have discussed about the addition, subtraction, multiplication and division of the arithmetic expression into previous chapter. In this chapter, we will discuss about the operation on algebraic expression.          Algebraic Expression   It is the combination of constants and variables along with the fundamental operations \[(+,\,-,\,\,\times ,\,\,\div )\] Terms: It is the part of an algebraic expression which is separated by the sign of addition and subtraction. \[5{{x}^{4}}{{y}^{2}},35{{x}^{4}}{{y}^{2}}-13{{x}^{2}}y,6xy,-3\] is an algebraic expression having \[8{{x}^{3}}{{y}^{2}},-4{{x}^{2}}y,6xy,-3\] as its term.                         Like and Unlike Terms The terms having similar variable(s) are called like terms otherwise it is unlike. In an algebraic expression \[5{{x}^{4}}{{y}^{2}},-13{{x}^{2}}y+6xy-3-35{{x}^{4}}{{y}^{2}};5{{x}^{4}}{{y}^{2}},35{{x}^{4}}{{y}^{2}}\]are like terms and are unlike terms.           Types of Algebraic Expression   Monomials An algebraic expression which contains one term is more...

     Negative Rational Number as Exponent   Let us consider 'a' be a positive rational number and \[x\left( i.e{{.}^{\frac{-m}{n}}} \right)\] be a negative rational exponent then it is defined as \[{{a}^{x}}\left( i.e.{{a}^{\frac{-m}{n}}} \right).\]       Find the value of \[{{4}^{\frac{-3}{2}}}\] and \[{{\left( 512 \right)}^{\frac{-2}{9}}}\] Solution: \[{{4}^{\frac{-3}{2}}}=\frac{1}{^{{{4}^{\frac{3}{2}}}}}=\frac{1}{^{\left( {{4}^{3}} \right)\frac{1}{2}}}=\frac{1}{{{\left( 64 \right)}^{\frac{1}{2}}}}=\frac{1}{8}\]and \[\frac{1}{{{\left( 512 \right)}^{\frac{-2}{9}}}}=\frac{1}{{{\left( {{512}^{2}} \right)}^{\frac{-2}{9}}}}\]\[=\frac{2}{\left( {{\left\{ {{\left( {{2}^{9}} \right)}^{2}} \right\}}^{\frac{1}{9}}} \right)}=\frac{1}{^{_{2}\cancel{9}\times 2\times \frac{1}{\cancel{9}}}}=\frac{1}{4}\]             Evaluate : \[{{\left( \frac{4}{9} \right)}^{\frac{3}{2}}}\times {{\left( \frac{4}{9} \right)}^{\frac{1}{2}}}\] (a) \[{{\left( \frac{4}{9} \right)}^{2}}\]                                    (b) \[{{\left( \frac{81}{16} \right)}^{4}}\] (c) \[{{\left( \frac{4}{9} \right)}^{4}}\]                                     (d) \[\frac{4}{27}\]                 (e) None of these     Answer: (a) Solution: \[{{\left( \frac{4}{9} \right)}^{\frac{3}{2}}}\times {{\left( \frac{4}{9} \right)}^{\frac{1}{2}}}={{\left( \frac{4}{9} \right)}^{\frac{3}{2}+\frac{1}{2}}}={{\left( \frac{4}{9} \right)}^{\frac{3+1}{2}}}={{\left( \frac{4}{9} \right)}^{\frac{\cancel{4}}{\cancel{2}}}}={{\left( \frac{4}{9} \right)}^{2}}\]       more...