Current Affairs 9th Class

Number Systems  

• Rational numbers (Q): The numbers of the form,\[\frac{p}{q}\] where 'p' and 'q' are integers and are called rational     number A numfaer of the form r is a fraction. So all fractions are rational numbers.

            Note: A number of the form \[\frac{\mathbf{a}}{\mathbf{b}}\] is a fraction. So all are rational             numbers. in the fraction, ‘a’     is called the numbers and ‘b’ is called the denominator. e.g.             \[\frac{\mathbf{1}}{\mathbf{2}}\mathbf{-            }\frac{\mathbf{2}}{\mathbf{3}}\mathbf{,}\frac{\mathbf{7}}{\mathbf{6}}\mathbf{,}\frac{\mat            hbf{6}}{\mathbf{11}}\mathbf{,-}\frac{\mathbf{2}}{\mathbf{9}}\mathbf{,}....\]                                     (i) Zero is a rational number.             Note: 0 by 0 is undefined.               (ii) Every integer is a rational number.             (iii) A rational number, may or may not be an integer.             (iv) To write W distinct rational numbers between any two rational numbers 'a' and \['b'\],             we write\[a=\frac{{{P}_{1}}}{q}\] and\[b=\frac{{{P}_{2}}}{q}\] such that \[\left( {{P}_{2}}\text{ }-            \text{ }{{P}_{1}} \right)\]is a positive integer greater than \['n'\],             (Here a < b); \[{{P}_{1}},{{P}_{2}}\]and q are integers\[(q\ne 0)\].             p- +1 p. + 2     Pi + n             A set of n rational numbers can be written as             \[\frac{{{P}_{1}}+1}{q},\frac{{{P}_{2}}+2}{q},......,\frac{{{P}_{1}}+n}{q}\]             i.e.,\[a=\frac{{{p}_{1}}}{q}<\frac{{{p}_{1}}+1}{q}<\frac{{{p}_{1}}+2}{q}.....<\frac{{{p}_{1}}+            n}{q}<\frac{{{p}_{2}}}{q}=b\]             Between two given rational numbers a and b, there are infinitely many rational numbers.  

• Properties of rational number

            (i) If \[\frac{p}{q}\], is a rational number and \['m'\] is a non-zero integer, then \[\frac{p}{q}=\frac{p\times             m}{q\times m}\]                                            

• (ii) If q is a rational number and \['m'\] is a common divisor of p and q, then \[\frac{p}{q}=\frac{p\div m}{q\div m}\]

            (iii) Two rational numbers are equivalent only when the product of the numerator of the first rational             number and the denominator of the second is equal to the product of the denominator of the first and the             numerator of the second.  

• Thus,\[\frac{p}{q}=\frac{r}{s}\operatorname{only}\,if\,p\times s=q\times r\]

            Note \[\frac{\mathbf{-p}}{\mathbf{q}}\mathbf{=}\frac{\mathbf{p}}{\mathbf{-q}}\mathbf{=-            }\frac{\mathbf{p}}{\mathbf{q}}\]  

• Representation of rational numbers on a number line:

            (i) Rational numbers of the form \[\frac{m}{n}\] where m < n are represented on the number line as     shown below.                             (ii) Rational numbers of the form \[\frac{m}{n}\] where m > n are represented on the number line as    shown below.                

• Irrational numbers (Q'): Any number which cannot be expressed in the form of. (Which is neither terminating nor repeating decimal) where p and q are integers and \[q\ne 0\]is said to be an irrational            

            Note:\[\pi \] more...

Polynomials  

• An expression of the form \[p(\operatorname{x})=+{{a}_{n}}{{\operatorname{x}}^{n}}+{{a}_{n-1}}......+{{a}_{2}}{{\operatorname{x}}^{2}}+{{a}_{1}}{{\operatorname{x}}^{2}}+{{a}_{0'}}\,\operatorname{where}{{a}_{0}},{{a}_{1}},a{{ & }_{2}},......,\]are real numbers \['n'\]is a non-negative integer and \[{{a}_{n}}\ne 0\] is called a polynomial of degree.

 

• Each of \[{{a}_{n}}{{\operatorname{x}}^{n}},{{a}_{n-1}},......{{a}_{2}},{{x}^{2}},{{a}_{1}}\operatorname{x}\,and\,{{a}_{n}}\ne 0\]and a with is called a term of the polynomial p(x).

              Note: The power of variable in a polynomial must be a whole number.               

• An expression of the form\[\frac{p\left( \operatorname{x} \right)}{q\left( \operatorname{x} \right)}\] where p(x) and q(x) are polynomials and \[q(\operatorname{x})\ne 0\]is called a rational expression.

              Note: Every polynomial is a rational expression, but every rational expression need not be a polynomial.  

• A polynomial d(x) is called a divisor of a polynomial p(x) if p(x) = d(x).q(x) for some polynomial q(x).

 

• Polynomials of one term, two terms and three terms are called monomial, binomial and trinomial respectively.
• A polynomial of degree one is called a linear polynomial.
• A polynomial of degree two is called a quadratic polynomial.
• A polynomial of degree three is called a cubic polynomial.
• A polynomial of degree four is called a biquadratic polynomial.
• A real number 'a' is a zero of a polynomial p(x) if p (a) = 0. 'a' is also called the root of the equation p(x) = 0.
• Every linear polynomial in one variable has a unique zero.
• A non-zero constant polynomial has no zero.
• Every real number is a zero of the zero polynomial.
• The degree of a non-zero constant polynomial is zero.
• The degree of a zero polynomial is not defined.
• If p(x) and g(x) are two polynomials such that degree of p(x) \[\ge \] degree of g(x) and g(x)\[\ne \]0, then we can find polynomials q(x) and r(x) such that p(x) = g(x) q(x) + r(x).

 

• Factor theorem:
• Let f(x) be a polynomial of degree in > 1 and 'a' be any real number. Then

            (x - a) is a factor of f(x) if (a) = 0.             (a) = 0 if (x - a) is a factor of f(x).             If x - 1 is a factor of a polynomial of degree 'n' then the sum of its coefficients is zero.  

• Remainder theorem:
• If p(x) is any polynomial of degree greater than or equal to 1 and p(x) is divided by the line polynomial x - a (where 'a' is any real number) then the remainder is p (a).
• We can express p(x) as p(x) = (x-a) q(x) +r(x) where q(x) is the quotient and r(x) is U remainder.
• The process of writing an algebraic expression as the product of two or more algebra expressions is called factorization.

 

• Some important identities:

\[{{\left( a+b \right)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}\]

\[{{\left( a\text{ }-\text{ }b \right)}^{2}}=\text{ }{{a}^{2}}-2ab+{{b}^{2}}\])

\[~\left( a+b \right)\text{ }\left( a-b \right)\text{ }=\text{ }{{a}^{2}}-{{b}^{2}}\]

\[{{\left( a\text{ }+\text{ }b\text{ }+\text{ }c more...

Co-ordinate Geometry  

• Co-ordinate Geometry: The branch of mathematics in which geometric problems are solved through algebra by using the coordinate system is known as coordinate geometry.
• In coordinate geometry, every point is represented by an ordered pair, called coordinates of that point.
• A pair of numbers 'a' and V listed in a specific order with 'a' at the first place and 'b' at the second place is called an ordered pair (a, b).

              Note :(i) (a, b) \[\ne \] (b, a)             (ii) If (a, b) = (c, d) then a = a and b=d.            

• The position of a point in a plane is determined with reference to two fixed mutually perpendicular lines called the coordinate axes.

 

• The horizontal line is called X-axis and the vertical line is called Y-axis.

 

• The point of intersection of the coordinate axes is called origin 0(0, 0).

 

• In a point P (a, b), 'a' is called x-coordinate or first coordinate or abscissa and 'b' is called y- coordinate or second coordinate or ordinate.

 

• The axes divide the plane into four quadrants,

                         (i) \[{{Q}_{1}}\]is the I quadrant? Here both x and y are positive i.e., x > 0 and y > 0. The ordered pair (a, b) belongs to this quadrant.               (ii) \[{{Q}_{2}}\]is the II quadrant. Here x is negative and y is positive i.e., x < 0 and y > 0. The ordered pair (-a, b) belongs to this quadrant.               (iii) \[{{Q}_{3}}\]is the III quadrant. Here both x and y are negative, i.e., x < 0 and y < 0. The ordered pair (-a,- b) belongs to the quadrant,                                                                     (iv) \[{{Q}_{4}}\]is the IV quadrant. Here x is positive and y is negative i.e., x > 0 and y < 0. The ordered pair (a, -b) belongs to this quadrant.                                                 

• The coordinates of any point on X - axis is of the form (a, 0) [y-coordinate zero],

 

•  The coordinates of any point on Y - axis is of the form (0, b) [x-coordinate zero].

   

Linear Equations in Two Variables  

• Equation: A statement of equality of two algebraic expressions involving a variable is called an equation.

 

• Simple linear equation: An equation which contains only one variable of degree 1 is called a simple linear equation.

 

• Solution of an equation: The value of the variable, which when substituted in the given equation, makes the two sides L.H.S (Left Hand Side) and R.H.S (Right Hand Side) of the equation equal is called the solution of that equation.

 

• Transposition: Any term of an equation may be taken to the other side with a change in its sign. This process is called transposition.

 

• Cross multiplication: If \[\frac{\operatorname{ax}+b}{\operatorname{cx}+d}=\frac{p}{q}\]then q (ax + b) = p (cx + d). This process is called cross multiplication.

 

• Linear equation in one variable: An equation of the form ax + b = 0 or ax = c, is a linear equation in one variable x, where \[a\ne 0\]and a, b and c are real numbers.
• Solution of a linear equation in one variable: The value of the variable (x) for which both the sides of the equation become equal is said to be the solution of the equation.

            Note: A solution is also called the ‘root’ of the equation.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      

• Linear equation in two variables:

            (i) An equation of the form ax + by + c = 0, where a, b and c are real numbers, such that 'a' and 'b' are not both zero, is called a linear equation in two variables.             (ii) A linear equation in two variables has infinitely many solutions.             (iii) The graph of every linear equation in two variables is a straight line.             (iv) x = 0 is the equation of Y-axis.             (v) y = 0 is the equation of X-axis.             (vi) The graph of x = a is a straight line parallel to the Y-axis.             (vii) The graph of y = a is a straight line parallel to the X-axis.             (viii) An equation of the type y = mx represents ‘a line passing through the origin.             (ix) Every point on the graph of a linear equation in two variables is a solution of the linear equation. Conversely, every solution of the linear equation is a point on the graph of the linear equation.

Introduction to Euclid's Geometry  

• Axioms: Axioms or postulates are the assumptions which are obvious universal truths and are not to be proved.

 

• Some of the axioms given by Euclid: (i) Things which are equal to the same thing are equal to one another. i.e., if a = c and b = c, then a = b.

              (ii) If equals are added to equals, the wholes are equal. i.e., if a = b and c = d, then a + c = b + d.             Also a = b \[\Rightarrow \]a+c=b+c.             Here, a, b, c and d are same kind of things.               (iii) If equals are subtracted from equals, the remainders are equal.               (iv) The things which coincide with one another are equal to one another.               (v) The whole is greater than the part.             i.e., if a > b, then there exists 'c' such that a = b + c.             Here, 'b' is a part of 'a' and therefore, 'a' is greater than 'b'.             (vi) Things which are double the same things are equal to one another.  

• (vii) Things which are halves of the same things are equal to one another.

 

• Euclid's five postulates:

              (i) Postulate 1: A straight line may be drawn from any one point to any other point.             (ii) Postulate 2: A terminated line (i.e., a line segment) can be produced indefinitely on either side to form a line.             (iii) Postulate 3: A circle can be drawn with any centre and any radius.             (vi) Postulate 4: All right angles are equal to one another,                                                                       (v) Postulate 5: If a straight line falling on two straight lines makes the interior angles on the same side of it taken together less than two right angles, then the two straight lines, if produced a indefinitely, meet on that side on which the sum of angles is less than two right angles.  

• Theorems or propositions are the properties which are to be proved, using definitions, axioms/postulates, previously proved statements and deductive reasoning.

 

• Any two distinct straight lines are either intersecting or parallel. If the two lines are intersecting, then they can be oblique to each other or perpendicular to each other.

 

• Intersecting lines: Two distinct straight lines which meet each other at a point are called intersecting straight lines.

 

• Perpendicular lines: Two intersecting lines are perpendicular to each other if one meets the other at a point and the angles made by the first with either side of the second at the point more...

Lines and Angles  

• Angle: An angle is the union of two rays with a common initial point. An angle is denoted by symbol\[\angle \]. It is measured in degrees,

               The angle formed by the two rays \[\overline{AB}\,\,and\,\,\overline{AC}\text{ }is\text{ }\angle BAC\text{ }or\text{ }\angle CAB.~\]called \[\overline{AB}\,\,and\,\,\overline{AC}\]are called the arms and the common initial point ‘A’ is called the vertex of the angle.    

• Bisector of an angle: A line which divides an angle into two equal a parts is alled the bisector of the angle.

                                     e.g., In the adjacent figure, the line OP divides \[\angle \]AOB into two        Equal parts.             \[\angle AOP=\angle POB={{\operatorname{x}}^{o}}\]                                          So, the line OP is ‘called the bisector of \[\angle \]AOB.                                             

• Pairs of angles:

(i) Complementary angles: Two angles are said to be Complementary if the sum of their measures is equal to\[~{{90}^{o}}\]     Here\[\angle x+\angle y={{90}^{o}},\]therefore \[\angle x\,\operatorname{and}\,\angle y\] Complementary angles:   (ii)  Supplementary angles: Two angles are said to be supplementary if the sum of their measures isequal to\[{{180}^{o}}\].                   Here \[\angle x+\angle y={{180}^{o}},\]therefore \[\angle x\,\operatorname{and}\,\angle y\] Supplementary angles.  

• Adjacent angles: Angles having the same vertex and a common arm, and the non-common arms lie on the opposite sides of the common arm are called adjacent angles.

  

• \[\angle AOB\text{ }and\text{ }\angle COB\]with common vertex 0 and common arm OB are adjacent angles,

  Note: \[\angle AOC=\angle AOB+\angle BOC\]

• Linear pair of angles: Two adjacent angles make a linear pair of angles, if the non-common arms of these angles are two opposite rays (with same end point),

                                                              

• In the adjacent figure, \[\angle BAC\text{ }and\text{ }\angle CAD\]form a linear pair of angles because the non -common arms AB and AD of the two angles are two opposite rays.

Moreover,\[\angle \text{ }BAC\text{ }+\text{ }\angle more...

Triangles  

• A triangle is a closed figure bounded by three straight lines. It is denoted by the symbol\[\Delta \].

\[\Delta \]ABC has three sides denoted by AB, BC and CA; three angles denoted by \[\angle ~A,\angle B\text{ }and\text{ }\angle C\,;\]and three vertices denoted by A, B and C.              

• Two geometrical figures are said to be congruent if they have exactly the same shape and size. Congruence is denoted by the symbol =.
• Two triangles are congruent if the sides and angles of one triangle are equal to the corresponding sides and angles of the other triangle.

 

• The congruence of two triangles ABC and PQR under the correspondence \[A\leftrightarrow P,B\leftrightarrow Q\]

and \[C\leftrightarrow R\]is symbolically expressed as \[\Delta ABC=\Delta PQR.\]  

• Two congruent figures are equal in area, but two figures having the same area need not be congruent.

 

• Congruence relation is an equivalence relation:

(i) Congruence relation is reflexive. \[\Delta ABC=\Delta ABC.\] (ii) Congruence relation is symmetric. If \[\Delta ABC\text{ }\cong \text{ }\Delta DEF,\text{ }then\text{ }\Delta DEF\text{ }\cong \text{ }ABC\text{ }.\] (iii)Congruence relation is transitive.  

• If \[\Delta ABC\text{ }=\text{ }\Delta DEF\text{ }and\text{ }\Delta DEF\text{ }=\text{ }\Delta XYZ\text{ }then\text{ }\Delta ABC\text{ }=\text{ }\Delta XYZ.\]
• Criteria for congruence of triangles:

 

• (i) A.S. congruence rule: Two triangles are congruent if two sides and the included angle of

one triangle are equal to the two sides and the included angle of the other triangle.   \[\Delta ABC=\Delta PQR\] \[Since=PQ=7cm,\text{ }\angle C=PR=5cm\,\,and\,\angle A=\angle P=50{}^\circ .\]corresponding sides and angles of the other triangle.   (ii) A.S.A. congruence rule: Two e.g., triangles are congruent if two angles and the included side of one triangle are equal to two   angles and the included side of the other triangle.             \[\Delta ABC\cong \Delta DEF\] \[\angle B=\angle E={{45}^{o}},\angle C=\angle F=30{}^\circ \text{ }and\text{ }BC\text{ }=\text{ }EF\text{ }=\text{ }5cm.\]   (iii) S.S.S. congruence rule: If three sides of one triangle are equal to the three sides of another triangle, then the two triangles are congruent.                     \[\Delta ABC=\Delta XYZ\]   Since AB = XY = 5 cm, BC = YZ = 7 cm and CA = ZX = 6 cm.  

• H.S.congruencerule: fin two right triangles more...

Quadrilaterals  

• A quadrilateral in which the measure of each angle is less than 180° is called a convex quadrilateral.

 

• A quadrilateral in which the measure of at least one of the angles is more than \[{{180}^{o}}\]s known as a concave quadrilateral.

 

• The sum of the angles of a quadrilateral is \[{{360}^{o}}\] (or) 4 right angles.

 

• When the sides of a quadrilateral are produced, the sum of the four exterior angles so formed \[{{360}^{o}}\]

 

• Various types of quadrilaterals:

      (i) Trapezium: (a) A quadrilateral having exactly one pair of parallel sides is called a trapezium. (b) A trapezium is said to be an isosceles trapezium if its non-parallel sides are equal. ABCD is a trapezium in which AB || DC.   This trapezium is said to be an isosceles trapezium if AB || DC and AD = BC.   (ii) Parallelogram: A quadrilateral in which both pairs of opposite sides are parallel is called a parallelogram. ABCD is a parallelogram in which AB || DC and AD II BC.   Properties: (a) In a parallelogram, any two opposite sides are equal. (b) In a parallelogram, any two opposite angles are equal. (c)In a parallelogram, the diagonals bisect each other. (d) In a parallelogram, each diagonal divides it into two congruent triangles. (e) In a parallelogram, any two adjacent angles have their sum equal to \[{{180}^{o}}\] i.e. the adjacent angles are supplementary.   (iii) Rhombus: A quadrilateral having all sides equal is called a rhombus. ABCD is a rhombus in which AB II DC, AD || BC and AB = BC = CD = DA. Properties: (a) The diagonals of a rhombus bisect each other at right angles. (b) Each diagonal of a rhombus divides it into two congruent triangles. (c) Opposite angles of a rhombus are equal and the sum of any two adjacent angles is\[{{180}^{o}}\] (d) The opposite sides of a rhombus are parallel. (e) All the sides of a rhombus are equal.   (iv) Rectangle: A parallelogram whose angles are all right angles is called a rectangle. ABCD is a rectangle in which, AD || BC and AB || CD and \[\angle A=\angle B=\angle C=\angle D={{90}^{o}}\]  

• Properties:

(a) Opposite sides of a rectangle are equal and opposite angles of a rectangle are equal. (b) The diagonals of a rectangle bisect each other. (c) Each diagonal divides the rectangle into two congruent triangles. (d) The diagonals of a rectangle are equal.   (v) Square: A parallelogram having all sides equal and each angle equal to a right angle is called a square. ABCD is a square in which AB || DC, AD more...

Areas of Parallelograms and Triangles  

• A polygonal region is the union of a polygon and it’s interior. For e.g., the union of a triangle and its interior is called the triangular region.

 

• Every polygonal region has area. Area of a figure is a number associated with the part of the plane enclosed by that figure.

 

• Two congruent figures have equal areas but figures with equal areas need not be congruent.

 

• If ABCD is a rectangle with AB = \[l\] m and BC = b m, then the area of the rectangular region ABCD is\[lb\text{ }sq.\text{ }m\text{ }or\text{ l}b\text{ }{{m}^{2}}\].

 

• If A and B are two regions having at the most a line segment common between them two, then the area of their combined region S is equal to the sum of their areas taken separately. i.e, ar (S) = ar (A) +ar (B).

 

• The area of a parallelogram is the product of any of its sides and the corresponding altitude.

 

• Two figures are said to be on the same base and between the same parallels, if they have a common base (side) and the vertices, (or the vertex) opposite to the common base of each figure lie on the same line parallel to the base.

 

• A diagonal of a parallelogram divides it into two triangles of equal areas.

 

• Parallelograms on the same (or equal) base and between the same parallel lines are equal in area.

 

• Parallelograms on the same base (or equal bases) and having equal areas lie between the same parallels.

 

• Parallelograms on equal bases and between the same parallel lines are equal in area.

   

• Area of a triangle is half the product of any of its sides and the corresponding altitude.

 

• Triangles on the same base and between the same parallel lines are equal in area.
• Two triangles having the same base (or equal bases) and equal areas lie between the same parallel lines.
• If a triangle and a parallelogram are on the same base and between the same parallel lines, then the area of the triangle is equal to half that of the parallelogram.

 

• The area of a trapezium is half the product of its height and the sum of the parallel sides.

 

• Triangles with equal areas and having one side of one triangle, equal to one side of the other, have their corresponding altitudes equal.

 

• A rectangle and a parallelogram on the same base and between the same parallels have more...

Circles  

• A circle is a closed figure in a plane formed by

            The collection of all the points in the plane which centre are at a constant distance from a fixed point in the plane. The fixed point is called the centre of length of radlus the circle and the constant distance is called the radius of the circle.                                                          

• The plane region inside the circle is called tne interior of the circle,

 

• If a circle is drawn in the plane X (infinite dimensions), then the part of the plane region outside the circular region is called the exterior of the circle.                                                    

                                                          

• The circumference of a circle is the length of the complete circular curve constituting the circle, given by circumference, \[C=\text{2}\pi r\]where r is the radius of the circle.                                            

 

• Any two points A and B of a circle, divide the circle into two parts. The smaller part is called a minor arc of the circle denoted by \[\overset\frown{AB}\](read as     arc AB).The larger part is called a major arc denoted by q\[\overset\frown{APB}\]or BA (read as arc\[\overset\frown{BA}\]). 

                                                 

• A line segment joining two points on the circumference of the circle, is called a chord of the circle. In the figure, AB is a chord.

               

• A chord passing through the centre O of the circle is called a diameter of the circle.

  A diameter of a circle is the longest chord of the circle and its length is twice that of the radius of the circle. Diameter, d = 2r where r is the radius of the circle.                                                                                                     

• A diameter of a circle divides it into two equal arcs. Each of these two arcs is called a semicircle. In the figure AB is a semicircle.

     

• A chord more...