Board Exam म\" फोड़ना है - A4S Army                 1
Quadratic Equations  Quadratic Polynomials: A polynomial of the form ax2 + bx + c is called a    quadratic polynomial in the variable x. This is a polynomial of the second  degree. In quadratic polynomial ax2 + bx + c, a ≠ 0 is the coefficient of x2, b    is the coefficient of x and c is the constant term (or coefficient of x0).  Quadratic Equation: An equation of the form ax2 + bx + c = 0, a ≠ 0, is  called a quadratic equation in one variable x, where a, b, c are constants.    For example, 4x2 – 3x + 1 = 0 and 3 – x – 7x2 = 0 are quadratic equation in    x.   !      #                                                          #       \"!     \"                                                          \"  But      +     −  5  =  0  is  a  quadratic      equation          in         ,  ������ℎ������������������	������	  ≠  0.    Zeros of a quadratic polynomial or roots of a quadratic  equation    Let ax2 + bx + c be a quadratic polynomial if ������������$ + ������������ + ������ = 0, then ������ is called  a zero of the quadratic polynomial ax2 + bx + c. if ������ is a zero of ax2 + bx + c  i.e., if ������������$ + ������������ + ������ = 0, then we say x = ������ satisfies the equation ax2 + bx + c =  0, and x = ������ is a solution or root of equation ax2 + bx + c = 0.    Methods for solving a quadratic equation             By factorization           By completion of square           By solving quadratic formula    Quadratic Formula  If ax2 + bx + c = 0                                      ������  =  −������  ±  √������������  −  ������������������  		������������	������  =  −������ ± √������                                                    ������������                               ������������    Where ������ = ������������ − ������������������ is known as discriminant. This result is known as quadratic formula or  Sridharacharya formula    Nature of the roots  Case I: When D > 0, i.e., b2 – 4ac > 0. In this case, the roots are real and  distinct.  Case II: When D = 0, i.e., b2 – 4ac = 0. In this case, the roots are real and  equal.  Case III: When D < 0, i.e., b2 – 4ac < 0. In this case, the roots are not real.                                   Board Exam म\" फोड़ना है - A4S Army                                                2
Arithmetic Progressions    Sequence :-Some numbers arranged in definite order, according to a definite rule are said to    form a sequence.    Progression: -Sequences which follow a definite pattern are called progressions.    Arithmetic Progression    An Arithmetic Progression (AP) is a list numbers in which each term is obtained by adding a fixed  number to the preceding term except the first term.  This fixed number is called the common difference (d) of the AP.  Note-It can be positive, negative or zero.    Calculation of ‘d’    an – an-1 = d and so on.    a2 – a1 = d                              a3 – a2 = d  In General Terms in an A.P  a4 – a3 = d    a, a+d, a+2d, a+3d,……. Represent an arithmetic progression.    nth Term of an AP (General Term)    If the first term of an AP is ‘a’ and its common difference is ‘d’ then its nth term is given by the  formula                                                    an = a+(n-1)d    Note-  In an AP, nth term is known as last term of an AP and it is denoted by I, which is given by the formula    Board Exam म\" फोड़ना है - A4S Army                 3
nth Term from the End of an AP    Let ‘a’ be the first term, ‘d’ be the commom difference and ‘l’ be the last term of an AP, then nth  term from the end can be found by the formula    Selection of Terms in an AP    Number  Terms                                         Common difference  of                                                                       ������		  terms          ������ − ������, 	������, 	������ + ������  3    4 ������ − ������������, 	������ − ������, 	������ + ������, 	������ + ������������                  ������������	    5 ������ − ������������, 	������ − ������, 	������, 	������ + ������, 	������ + ������������             ������	    Sum of First n Terms of an AP    If first term of an AP is ‘a’ and its common difference is ‘d’ , then the sum of its first n terms Sn , is  given by the formula                   Sn=  ������       [������������     +  (������     −  ������)������]                      ������                                              Or                                 Sn        =  ������  [������  +  ������]                                            ������                   Board Exam म\" फोड़ना है - A4S Army                                4
Calculation of nth term if sum of n terms is given                            an = Sn- Sn-1    Arithmetic Mean    If a,b and c are in AP, then b is known as arithmetic mean of a and c, i.e. ������ = ������&������������.                       Board Exam म\" फोड़ना है - A4S Army                                     5
Some Applications of Trigonometry    Line of Sight    The line of sight is the line drawn from the eye of an observer to the point where the object is  viewed by the observer.    Horizontal Line    The line which goes parallel from eye to ground, is called horizontal line.    Angle of elevation    The angle of elevation of an object viewed, is the angle formed by the line of  sight with the horizontal when object viewed is above the horizontal level, i.e.  the case when we lower our head to look at the object.    Angle of Depression    The angle of depression of an object viewed, is the angle formed by the line of sight with the  horizontal, when it is below the horizontal level, i.e. the case when we lower our head to look at the  object.    .                       Board Exam म\" फोड़ना है - A4S Army                                     6
Value of trigonometric ratios of standard angles              ������������ ������������������ ������������������ ������������������ ������������������                     1   1 √3 1  sin ������    0    cos ������    1         2 √2   2                 0                     √3 1    1    tan ������    0        2 √2 2                     1 1 √3 Not                                         defined  cot ������    Not      √3  1   10                     √3            defined                             √3  sec ������    1        2 √2 2 Not                       √3 defined  cosec ������  Not      2       2                 1                         √2            defined                             √3              Board Exam म\" फोड़ना है - A4S Army                           7
Chapter–10 Circles    Circles-A circle is a collection of all those points in a plane which are at a constant    distance (radius) from a fixed point of that plane. Constant distance is length of radius and  fixed point is centre.    Note- Two or more circles having the same centre are called concentric circles.  Secant: A line which intersects circle in two distinct points is called a secant of the circle.  Tangent: The tangent to a circle is a line that meets the circle at exactly one point.    Length of the Tangent        • The length of the segment of the tangent from the external point and the point of           contact with the circle is called the length of the tangent from the external point to the           circle.    In the above figure, AB is called the length of tangent.                       Board Exam म\" फोड़ना है - A4S Army                                     8
Number of Tangent from a Point on a Circle    (i) There are exactly two tangents to a circle through a point lying outside the circle.  circle ,i.e. PT1 and PT2  (ii) There is one and only one tangent to a circle passing through a point lying on the circle.  (iii) There is no tangent to a circle passing through a point lying inside the circle.                       Board Exam म\" फोड़ना है - A4S Army                                     9
Some Important Terms Related to Chapter 12 Area Related to Circles  Chord :-A line segment joining any two points on the circumference of the circle is called a chord    of the circle. If this chord passes through the centre, then the chord (or diameter) is the longest  chord of the circle    Semi – Circle    A diameter of a circle divides it into two equals parts or in two equal arcs. Each of these two arcs is  called a semi-circles    Circumference    The length of the complete circle is called the circumference of the circle.    Arc (Minor and Major)    A continuous piece of a circle is called an arc. In adjoining figure, P and Q are two points on a circle  which divide it into two parts, called the arcs. The larger part is called the major arc QRP and the  smaller part is called the minor arc PMQ    Sector    The region between an arc and the two radii, joining the ends of the arc to the centre, is called a  sector.    The sector formed by minor arc, is called minor sector and  The sector formed by major arc, is called major sector                       Board Exam म\" फोड़ना है - A4S Army                                    10
Segment    The region between a chord and either of its arc is called a segment of the circular region or simply a  segment of the circle. The segment formed by minor arc along with chord, is called minor segment  and the segment formed by major arc, is called the major segment.    Important Results based on Class 9th    (i) The perpendicular drawn from the centre of a circle to a chord bisects it and vice-versa.  (ii) Equal chords of a circle are equidistant from the centre  (iii) The angle subtended by an arc (or corresponding chord ) at the centre of the circle is twice the  angle subtended by the same arc at any point on the remaining part of the circle    (iv) Equals chords of a circle subtend equal angles at the centre.  (v) The angle in a semi-circle is a right angle.  (vi) Angles in the same segment of a circle are equal.  (vii) The sum of any pair of opposite angle of a cyclic quadrilateral is 180o.  (viii) If two circles intersect at two points, then the line through the centres in the perpendicular  bisector of the common chord.    Theorem Related to Tangent of circle (Class 10th Theorem on Latest Syllabus)    Theorem 1 The Tangent at any point of a circle is perpendicular to the radius through the point of  contact.    Here, O is centre of circle and AB is tangent of circle at P and it is point of contact and OP is radius.                                                        ∴ ������������ ⊥ ������������.                       Board Exam म\" फोड़ना है - A4S Army                                    11
Theorem 2 The lengths of two tangents drawn from an external point to a circle are equal.  Here,P is exterior point and PA and PB are tangents PA=PB    Important Results    (i) If two circles touch internally or externally, then point of contact lies on the straight line through  the two centres .    (ii) The opposite sides of a quadrilateral circumscribing acircle subtend supplementary angles at the  centre of the circle.                       Board Exam म\" फोड़ना है - A4S Army                                    12
Surface Areas and Volumes    Cuboid:  TSA(Total Surface Area) = 2(lb + bh + hl)  Lateral Surface Area (LSA) = 2h (l + b)    Volume = lbh.  Diagonal of cuboid = √������$ + ������$ + ℎ$  Cube: LSA = 4a2             TSA = 6a2             Volume = a3,           Diagonal of cube = √3������                       Board Exam म\" फोड़ना है - A4S Army                                    13
Right circular cylinder    CSA(Curve Surface Area) = 2������������ℎ  TSA = 2������������ℎ + 2������������$ = 2������������(������ + ℎ)  Volume = ������������$ℎ    Hollow cylinder    Thickness of cylinder = R – r  External CSA = 2������������������    Internal CSA = 2������������ℎ  TSA = External curved area + internal curved area + area of two ends             = 2������������ℎ + 2������������ℎ + 2������(������$ − ������$)           = 2������(������ℎ + ������ℎ + 	 ������$ − ������$)  Volume of material = ������������$ℎ − ������������$ℎ = ������(������$ − ������$)ℎ                       Board Exam म\" फोड़ना है - A4S Army                                    14
Cone    CSA = ������������������ = 	������������	√������$ + ℎ$  TSA = ������������������ + 	������������$ = ������������(������ + ������)	             #  Volume  =  !  ������������$ℎ    Slant height = ������ = 	 <(������$ + ℎ$)                                            Sphere    CSA = ������������������������  TSA = ������������������������    Volume  =  )  ������������!,					             !                               Board Exam म\" फोड़ना है - A4S Army                                           15
Spherical Shell  Thickness = R – r    Volume  =  )  ������	(������!  −   ������!).             !    Hemisphere  CSA = 2������������$  TSA = 3������������$	             $  Volume  =  !  ������������!,					    Hemispherical Shell    External CSA = 2������������$  Internal CSA = 2������������$  TSA = 2������������$ + 2������������$ + ������(������$ − ������$) = ������(3������$ + ������$)                                $  Volume  of  material       =  !  ������(������!  −  ������!)                                     Board Exam म\" फोड़ना है - A4S Army                                                 16
Statistics                 Statistics measures of central tendency    Mean – the arithmetic mean (or, simply mean) is the sum of the values of all  the observations divided by the total number of observations.                         Mean of ungrouped data    The mean of n numbers x1, x2, x3, … xn denoted by ������ (read as X bar) is    defined as:             ������#  +  ������$  +  ������!  +	  …  + 	 ������*     ∑ ������                                          ������                       ������                   ������  =                                       =    Where Σ is a Greek alphabet called sigma. Thus, ∑ ������ means sum of all x.                         Mean of grouped data  Direct method: if the variates observations x1, x2, x3, … xn have frequencies    f1, f2, f3, …. Fn respectively, then the mean is given by:  Mean 	                       ������#������# + ������$������$ + 	 … + 	 ������*������*           ∑ ������+������+               ������  =       ������# + ������$ + 	 … + 	 ������*             =   ∑ ������+    	    Short cut method: in some problems, where the number of variates is large  or the values of xi or fi are larger, then the calculations become tedious. To  overcome this difficulty, we use short cut or deviation method.    Assumed mean method  Find the class mark or mid-value of each class, as:    Xi = class mark = G,-./0	,+2+3&$455/0	,+2+3H  In this method, an approximate mean, called assumed mean or provisional  mean is taken. This assumed mean is taken preferably near the middle, say  A and the deviation di = xi – A for each variate xi.  The mean is given by the formula:                            Mean         ������  =   ������  +  ∑ 7\"8\"                                                       ∑ 7\"                 Board Exam म\" फोड़ना है - A4S Army                             17
Mode – The mode of a distribution is the value of observation with highest  frequency. In a continuous frequency distribution with equal class interval,    mode is obtained by locating a class with the maximum frequency.                                                        ������# − ������9                        ������������������������           =  ������  +  2������# − ������9 −  ������$  ×  ℎ    Where,    l = lower limit of the modal class    f1 = frequency of the modal class  f0 = frequency of the class preceding the modal class    f2 = frequency of the class succeeding the modal class.    h = size of the modal class.    Median    The median gives the value of the middle – most observation in the data.  Median of ungrouped data    For finding median of ungrouped data, we first arrange the data in    ascending order.    • If n is odd, median = G*&$#H ������ℎ observation    •  If   n  is  even,  median  =  #  O*$  ������ℎ  +  G*$  +  1H  ������ℎ	Q  observation.                                   $    •                                  Median of grouped data    To find median class, we locate the class whose cumulative frequency is  greater than (nearest to) /2 median of a grouped or continuous frequency  distribution                                             Median= l + R#!:7;7S × ℎ    Where, l = lower limit of the median class  n = number of observations  f = frequency of the median class  h = size of the median class (assuming class size to be equal)  cf = cumulative frequency of the class preceding the median class       Empirical relationship between the three measure of central tendency.                               3 median = mode + 2 mean                          Board Exam म\" फोड़ना है - A4S Army                                      18
Board Exam म\" फोड़ना है - A4S Army                19
                                
                                
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