11th Class

                                                                                                      Sequence and series    

• Sequence: A sequence is a mapping of or function whose domain is the set of natural number and its range is the set of real number or complex number. But we will study the real number sequence only.

e.g.       (a) \[{{x}_{1}},\,{{x}_{2}},{{x}_{3}}.....{{x}_{n}}.\]   (b)        2, 4, 6, 8, 10..... (c)        1, 4, 7, 10, 13....... etc. These above example is a sequence of real no.   Note:    Sequence is also said to be progression. Generally there are three type of sequence (a)        Arithmatic progression or sequence (A.P.) (b)        Geometric progression (G.P.) (c)        Harmonic prograssion (H.P.)  

• Arithmetic sequence: A sequence (Sn) is said to be an arithmetic sequence if difference between any term and its proceeding term give the constant quantity.

e.g. Sn: 3,8,13,18, 23..... we more...

  Two Dimensional Geometry (Coordinate and Straight Line)   Key Points to Remember  

• Coordinate Geometry: It is the branch of mathematics in which deal with relation between two variable in algebraic form. It is 1st coined by French Mathematician Rene Descarties.

  Let P(x, y) be any point   x\[\to \] abscissa y\[\to \] ordinate  

• Some Basic Formula:
• Distance Formula:

  (a)        The distance between two points \[A({{x}_{1}},\,{{y}_{1}})\] & \[B({{x}_{2}},\,{{y}_{2}})\]   \[Ab=\sqrt{{{({{x}_{2}}-{{x}_{1}})}^{2}}+{{({{y}_{2}}-{{y}_{1}})}^{2}}}\]             (b)        Distance between the origin 0(0, 0) and the point P(x, y) is OP                         \[op=\sqrt{{{x}^{2}}+{{y}^{2}}}\]   e.g.\[A=(5,3)\,\,\,B=(-2,5)\] \[\therefore \,\,\,AB\]             \[=\sqrt{{{(-2-5)}^{2}}{{(5-3)}^{2}}}=\sqrt{49+4}=\sqrt{53}\]  

• Section Formula: The coordinate of the point P(x, y) dividing the line segment joining the two more...

                                                                                     Pair and Straight Line   Key Points to Remember   A hemogenous of equation of second degree of the form \[a{{x}^{2}}+2hxy+b{{y}^{2}}=0\] represents the pair of straight lines which passes through the origin.   (a)        If the lines are real and distinct then \[{{h}^{2}}>ab\]   (b)        If the lines are real and coincidents if \[{{h}^{2}}=ab.\]   (c)        If the lines are imaginary then \[{{h}^{2}}<ab.\] Let \[y={{m}_{1}}x\to (1)\] and \[y={{m}_{2}}x\to (2)\] are two lines which are passing through the origin. Then \[(y-{{m}_{1}}x)(y-{{m}_{2}}x)\equiv a{{x}^{2}}+2hxy+b{{y}^{2}}\]   \[{{y}^{2}}-({{m}_{1}}+{{m}_{2}})xy+{{m}_{1}}{{m}_{2}}{{x}^{2}}={{y}^{2}}+\frac{2h}{b}xy+\frac{a}{b}{{x}^{2}}\]   Equation the coefficient of same variable of the\[\frac{2h}{b}=({{m}_{1}}+{{m}_{2}})\]we have, \[({{m}_{1}}+{{m}_{2}})=\frac{2h}{b}\] & \[{{m}_{1}}.{{m}_{2}}=\frac{a}{b}\]

• Angle between the pair of straight lines

Let q be the angle between the two given pair of straight line \[a{{x}^{2}}+2hxy+b{{y}^{2}}=0,\] which are passing through origin is written as             \[\tan \theta =\pm \frac{2\sqrt{{{h}^{2}}}-ab}{a+b}\] for acute more...

                                                                                       Circles   Key Points to Remember   Circle: A circle is the locus of the points which move in the plane such that the its distance from a fixed point always remain constant, is said to be the circle. The fixed point is said to be the centre of the circle and its distance is said to be the radius of the circle.     Let C (O, r) is a circle with centre 0 & radius r. A be any point it. \[\therefore \]      OA = radius of the circle  

• Standard Equation of the Circle: The standard equation of the circle whose centre be (h, k) and radius, a be \[{{(x-h)}^{2}}+{{(y-k)}^{2}}={{a}^{2}}\]

  When centre be considered as the origin & radius be a, more...

                                                                                        Matrices and Determinant   Key Points to Remember  

• Matrices & Determinant

  Let us consider the linear equation   \[{{a}_{1}}x+{{b}_{1}}y={{c}_{1}}\]                                    (i) \[{{a}_{2}}x+{{b}_{2}}y={{c}_{2}}\]                                    (ii)   We have one of the methods to solve these equation by cross multiplication method.   \[\frac{x}{{{b}_{1}}{{c}_{2}}-{{b}_{2}}{{c}_{1}}}=\frac{y}{{{c}_{1}}{{a}_{2}}-{{c}_{2}}{{a}_{1}}}=\frac{1}{{{a}_{1}}{{b}_{2}}-{{a}_{2}}{{b}_{1}}}\]   Now we modify this method & convert this method into standard form (matrix form)            

• Matrix: Actually it is the shorthend of mathematics. It is an operator as addition, multiplication etc. Every matrix has come into existence through the solution of linear equations.

  Given linear equation can be solved by matrix method & it is written as, \[AX=B\]    

• Definition: It is the arrangement of mn things into horizontal (row) vertical (column) wise.

Generally matrix more...

                                                                             Three Dimensional Plane   In three dimensional Geometry, it is not a new geometry though it is the refined or extension form of the two dimension geometry. In 3-dimensional geometry. Three axes i.e. x-axis, y-axis and z-axis are perpendicular to each other is considered. Let \[X'OX',Y'OY\] & \[Z'OZ\] be three mutually perpendicular lines which be intersect at 0. It is called origin.                                   \[X'OX\xrightarrow{{}}x-axis\]   \[Y'OY\xrightarrow{{}}y-axis\]   \[Z'OZ\xrightarrow{{}}z-axis\]   Plane XOY is called xy plane YOZ is called yz plane and ZOX is called zx plane In 3-D, there are 8 quadrents Equation of x-axis be y= 0 & z =0 Equation of y-axis be x = 0 & z = 0 and equation of z-axis be x=0 & y=0   Note: In 3-D, a more...

                                                                                               Probability  

• Probability: Actually, Probability is the mathematical modelling of chances or outcome of the events. In other, it is the branch of mathematics in which we study the occurrence of any element in the numerically form. It always lies between 0 & 1.

i.e.        \[0\le P(E)\le 1\] Where P (E) = Probability of occurrence of the event E.  

• Some basic terms and its concepts

  Random experiment of Trial: An experiment of event or trial of event does not follow any rule of system is said to be random experiments, e.g. throwing a dice in which one of {1, 2, 3, 4, 5, 6} will be occurred. We cannot predict that in it throwing if integer 4 is occurred then the next throwing dice. 3 or 4 or any more...

                                                                                                          Statistics   Statistics history is very old. Early statistics is considered as the imposed form of applied mathematics.  

• Statistics is used as singular and plural: Statistics used as singlular. It is the science in which we collect, analysis, interprete the data.

 

• Statistics used as plural

  (i)         Statistics are aggregate of facts. (ii)        Statistics are affected by a number of factors. (iii)       Statistics are collected in systematic manner. (iv)       Statistics must be reasonable accurate. It is both art and science.  

• Science: Systematised body of knowledge is said to be science.
• Art: Handling of the fact of given information to skill up the knowledge about the matter is said to be art.

  Note: Statistics without science has no fruit and science without more...

                                             SEQUENCE AND SERIES (A.P., G.P. AND H.P.)   INTRODUCTION SEQUENCE                                 A systematic umbers according to a given rule is called a sequence: The sum of terms of a sequence is called a series. The first term of a sequence is denoted as \[{{T}_{1}}\], second term is denoted as \[{{T}_{2}}\], and so on. The nth term, of sequence is denoted by \[{{T}_{n}}\]. It is also referred to as he general term of the sequence.   Finite and Infinite Sequences

A sequence containing finite number of terms is called a finite sequence.

Example: 1, 9, 17, 25, 33, is a finite sequence of 5 terms.

A sequence consisting of infinite numbers of terms is called an infinite sequence.

Example: 3, 6, 9, 12, 15..................... up to infinite number of terms.   If a sequence is given, then we can find its nth term and more...

Sequence and Series   The ordered collection of objects is called sequence. The sequence having specified patterns is called progression. The real sequence is that sequence whose range is a subset of the real numbers. A series is defined as the expression denoting the sum of the terms of the sequence. The sum is obtained after adding the terms of the sequence. If \[{{a}_{1}},\text{ }{{a}_{2}},\text{ }{{a}_{3}},+----,\text{+}{{a}_{n}}\] is a sequence having n terms, then the sum of the series is given by:             \[\sum\limits_{k=1}^{n}{{{a}_{k}}+{{a}_{2}}+{{a}_{3}}+----+{{a}_{n}}}\]   Arithmetic Progression (A.P.) A sequence is said to be in arithmetic progression if the difference between its consecutive terms is a constant. The difference between the consecutive terms of an A.P. is called common difference and nth term of the sequence is called general term. If \[{{a}_{1}},\text{ }{{a}_{2}},\text{ }{{a}_{3}},\text{ }----,\text{ }{{a}_{n}}\]be n terms of the sequence in A.P., then nth term of the more...