Current Affairs 7th Class

  Geometry   In our daily life we observe different geometrical shapes. These geometrical shapes are not only the matter of study of mathematics but are directly related with our daily life. The basic geometrical figures are made up of lines and angles.   Line Segment It is the straight path between two points. In other words we can say that it has two end points and is of finite length.   Ray When a line segment extends infinitely in one direction, it is called a ray. Simply we can say that a ray has one end point and infinite length.   Line When both end of a line segment extended infinitely, it is known as a line. Simply we can say that a line has no end point and infinite length.   Parallel Lines Two lines are said to be parallel if the distance between them always remains same at each and every point. The parallel lines never intersect each other.   In other words we can say that if two lines do not have any common point than they are said to be parallel. In the figure I and m are parallel lines.       Angle If two rays have common end point then the inclination between two rays is called an angle.   Types of Angles The following are different types of angles:

• Acute Angle: The angle whose measure is more than \[{{0}^{o}}\] and less than \[{{90}^{o}}\] is called an acute angle.
• Right Angle: The angle of measure \[{{90}^{o}}\] is called a right angle.
• Obtuse Angle: The angle whose measure is more than \[{{90}^{o}}\] and less than \[{{180}^{o}}\] is called an obtuse angle.

                         

• Straight Angle: The angle whose measure is \[{{180}^{o}}\] is called a straight angle.
• Reflex Angle: The angle whose measure is more than \[{{180}^{o}}\] and less than \[{{360}^{o}}\] is called a reflex angle.
• Complete Angle: The angle whose measure is \[{{360}^{o}}\]is called a complete angle.

                         

• Equal Angles: Two angles are said to be equal if they are of same measure.
• Complementary Angles: If the sum of measures of two angles is \[{{90}^{o}}\] then they are said to be complementary angles.
• Supplementary Angles: If the sum of measures of two angles is \[{{180}^{o}}\] then they are said to be supplementary angles.

  Triangles The word triangle is derived from Greek word, tri means three and hence, it refers to a shape consisting of three internal angles. Obviously the shape consists of three sides. Hence, a triangle can be defined as a more...

  Mensuration   Standard Units of Area The inter relationship among various units of measurement of area are listed below.   \[1\,{{m}^{2}}\]                      =         \[(100\times 100)\,c{{m}^{2}}={{10}^{4}}\,c{{m}^{2}}\] \[1\,{{m}^{2}}\]                      =         \[(10\times 10)\,d{{m}^{2}}=100\,d{{m}^{2}}\] \[1\,d{{m}^{2}}\]        =         \[(10\times 10)\,c{{m}^{2}}=100\,c{{m}^{2}}\] \[1\,da{{m}^{2}}\]       =         \[(10\times 10)\,{{m}^{2}}=100\,{{m}^{2}}\] \[1\,h{{m}^{2}}\]                    =         \[(100\times 100)\,{{m}^{2}}={{10}^{4}}{{m}^{2}}\] \[1\,k{{m}^{2}}\]                     =         \[(1000\times 1000)\,{{m}^{2}}={{10}^{6}}\,{{m}^{2}}\] \[1\,hectare\]      =         \[10000\,{{m}^{2}}\] \[1\,k{{m}^{2}}\]                     =         \[100\,hectare\]   Formula Related to Perimetre and Area

• Area of a triangle \[=\frac{1}{2}\times b\times h\]
• Area of an equilateral triangle \[=\frac{\sqrt{3}}{4}\times {{a}^{2}}\]
• Perimetre of a rectangle \[=2(Length+breadth)\]
• Area of a rectangle \[=Length\times breadth\]
• Diagonal of a rectangle \[=\sqrt{{{(length)}^{2}}+{{(breadth)}^{2}}}\]
• Perimetre of a square \[=4\times side\]
• Area of a square \[=sid{{e}^{2}}=\frac{1}{2}\times {{(diagonal)}^{2}}\]
• Side of a square \[=\sqrt{area}\]
• Diagonal of a square \[=side\times \sqrt{2}\]
• Perimetre of a parallelogram \[=2\times sum\,of\,length\,of\,adjacent\,sides.\]
• Area of a parallelogram \[=base\times corresponding\,height.\]
• Perimetre of a rhombus \[=4\times side\]
• Area of a rhombus \[=base\times vertical\,height.\]
• Area of a rhombus \[=\left( \frac{1}{2} \right)\times product\,of\,diagoanls\]
• Circumference of a circle \[=2\pi r\]
• Area of a circle \[=\pi \,{{r}^{2}}\]
• The volume of a cuboid \[=length\times breadth\times height\]
• The volume of a cube \[={{(length)}^{3}}\]

  Example: Find the area of a right-angled triangle whose sides are 15 cm, 9 cm and 2 cm. (a) \[48\,c{{m}^{2}}\]              (b) \[80\,c{{m}^{2}}\] (c)\[54\,c{{m}^{2}}\]               (d) \[78\,c{{m}^{2}}\] (e) None of these Answer (c)   Explanation: Here, a = 15 cm, b = 9 cm and c = 12 cm Also, \[{{a}^{2}}={{b}^{2}}+{{c}^{2}}\Rightarrow \]The given triangle is a right triangle. \[\therefore \]Area of the right triangle \[=\frac{1}{2}\times 9\times 12=54\,c{{m}^{2}}\] Example: The dimensions of the floor of a room are 15 m and 20 m. How many square tiles each of length 20 cm are required to furnish the floor? (a) 2,400                       (b) 5,200         (c) 7,500                        (d) 5,250 (e) None of these Answer (c)   Explanation: Area of the room \[=15\,m\times 20\,m\] \[=1500\,cm\times 2000\,cm=3\times {{10}^{6}}\,c{{m}^{2}}\] Area of a tile \[=20\,cm\times 20\,cm=400\,c{{m}^{2}}\] Total number of tiles required \[=\frac{3\times {{10}^{6}}}{400}=\frac{30000}{4}=7,500\]    

  Algebraic Expressions   In an algebraic expression constant and variables are linked with arithmetic operations. The value of unknown variable is obtained by simplification of the given expression.   Terms of an algebraic Expression   Variables Alphabetical symbols used in algebraic expressions are called variables a, b, c, d, m, n, x, y, z ........... etc. are some common letters which are used for variables.   Constant Terms The symbol which itself indicate a permanent value is called constant. All numbers are constant. \[6,10,\frac{10}{11},15,-6,\sqrt{3}....\]etc. are constants because, their values are fixed.   Variable Terms A term which contains various numerical values is called variable term. For example. Product of \[\text{X=4 }\!\!\times\!\!\text{ }\,\text{X=4X}\]Product of \[\text{2,X,}{{\text{Y}}^{2\,}}\] and \[\text{Z=}\,\text{2 }\!\!\times\!\!\text{ X }\!\!\times\!\!\text{ }{{\text{Y}}^{\text{2}}}\text{ }\!\!\times\!\!\text{ Z=2X}{{\text{Y}}^{\text{2}}}\text{Z}\] Thus, 4X and \[\text{2X}{{\text{Y}}^{\text{2}}}\text{Z}\] are variable terms   Types of Terms There are two types of terms, like and unlike. Terms are classified by similarity of their variables.   Like and unlike Terms The terms having same variables are called like terms and the terms having different variables are called unlike terms. For example, \[6x,\,x,\,-2x,\,\frac{4}{9}x,\]are like terms and \[6x,\,2{{y}^{2}},\,-9{{x}^{2}}yz,4xy,\]are unlike terms.   Coefficient A number or a symbol multiplied with a variable in an algebraic expression is called its coefficient. In\[-6{{m}^{2}}np\], coefficient of\[n{{m}^{2}}p\] is \[-6\] because \[{{\text{m}}^{\text{2}}}np\]is multiplied with \[-6\] to from \[-6{{m}^{2}}np\]. The variable part of the term is called its variable or literal coefficient. In\[-\frac{5}{4}abc\], variable coefficients are a, b and c. The constant part of the term is called constant coefficient. In term\[-\frac{5}{4}abc\], constant coefficient is\[-\frac{5}{4}\].   Example: Sign of resulting addition of two like terms depends on which one of the following? (a) Sign of biggest term (b) Sign of smallest term (c) Sign of positive term (d) Sign of negative term (e) None of these Answer (a)   Operations on Algebraic Expressions When constant and variables are linked with any of the following fundamental arithmetic operations i.e. addition, subtraction, multiplication and division, then the solution of the expression is obtained by simplification of the expression.   Addition and Subtraction of Terms The addition of two unlike terms is not possible and their addition is obtained in the same form. Addition of \[2x+3x\] is \[5x\] but the addition of \[2x+3y\] is\[2x+3y\]. Subtraction of two like terms is same as the subtraction of whole numbers. For example,\[4x-2x=2x\]   Example: Simplify: \[(2{{x}^{2}}+{{x}^{2}})-(5{{x}^{2}}+11{{x}^{2}})\] (a) \[15{{x}^{3}}\]                   (b) \[15{{x}^{2}}\]      (c) \[-3{{x}^{2}}\]                    (d) \[13{{x}^{2}}\] (e) None of these   Answer (c) Explanation:\[(2{{x}^{2}}+{{x}^{2}})-(5{{x}^{2}}+11{{x}^{2}})\] = \[3{{x}^{2}}-16{{x}^{2}}=-13{{x}^{2}}\]   Example: Evaluate: \[[\{({{x}^{2}}+3{{x}^{2}})-({{x}^{2}}+{{x}^{2}})5\}\div x]\] (a) \[-10x\]                     (b) \[-15x\] (c) \[-6x\]                                   (d) \[10x\] (e) None of these Answer (c) Explanation:\[(4{{x}^{2}}-5\times 2{{x}^{2}})\div x=\]\[\frac{4{{x}^{2}}-10{{x}^{2}}}{x}\] \[=\frac{-6{{x}^{2}}}{x}=-6x\]   Equation An equation is a condition on a variable. For example, the expression \[10x+3=13\] is an equation which describe that the variable is equal to a fixed number. This value itself is called the solution of the equation. Thus,\[10x+3=\]\[13\Rightarrow 10x\]\[=13-3=\]\[10\Rightarrow x=1\] Here the definite value of the variable x in the equation \[10x+3=13\]   Note: (i) An equation has two sides, LHS and RHS, between them more...