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
Search
Read the Text Version
- 1 - 19
Pages: