Current Affairs 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 choose any term \[13-5=5\] \[23-18=5\] etc. The constant quantity is said to common difference (c.d.) in A.P.  

• Remember Some Points about A.P.

  (a)        \[{{n}^{th}}\] term whose 1st term and common difference is given in A.P. is written as \[{{t}_{n}}=a+(n-1).d\] where a = 1st term and d = comman difference n = no. of terms in the A.P.   (b)        Sum of the \[{{n}^{th}}\] term in A.P. be   \[Sn=\frac{n}{2}\{2a+(n-1).d\}=\frac{n}{2}\{a+a+(n-1).d\}\]   \[=\frac{n}{2}\{a+{{t}_{n}}\}=\frac{n}{2}\{1st\,\,term+\,last\,\,term\}\]   (c)        Single arithmetic mean between two given quantity is \[A.M=\frac{a+b}{2}\] (d)        For n arithmetic mean between two terms a and b. \[\therefore \] So, the total no. of terms = n+ 2 Last term \[=b=a+(n+2-1).d=a+(n+1).d\] Where d = comman difference in A.P.   (e)        For choosing five term in A.P. We take them as \[a-2b,a-b,a+b\]and \[(a+2b)\] etc.  

• Some very useful results: If the last term, \[{{t}_{n}}\] is in the linear form. i.e. \[{{t}_{n}}=an+b.\] Here, n is considered as variable then the series so formed is said to be in A.P. Similarly if last tern, \[{{t}_{n}}=a{{n}^{2}}+bn+c\] (i.e. in quadratic form), then the series so formed is said to be in A.P.

  If \[{{a}_{1}},\,{{a}_{2}},\,{{a}_{3}},\,{{a}_{4}}....{{a}_{n}}\]is said to be in A.P. Then   (a)        \[{{a}_{1}}\pm \text{k},\,{{a}_{2}}\pm \text{k},....\]will be in A.P.. Where k be any const quantity (b)        Even \[\text{k}.{{a}_{2}},\text{k}.{{a}_{2}},\text{k}.{{a}_{3}}.....\]will be said to be in A.P. (c)        \[\frac{{{a}_{1}}}{\text{k}},\frac{{{a}_{2}}}{\text{k}},\frac{{{a}_{3}}}{\text{k}}...\]is said to be in A.P. (d)        If \[{{a}_{1}},\,{{a}_{2}},\,{{a}_{3}},\,{{a}_{4}},\,{{a}_{5}}....\]be in A.P. and also \[{{b}_{1}},\,{{b}_{2}},\,{{b}_{3}},\,{{b}_{4}},\,{{b}_{5}}....\]be in A.P. then the resulting sequence whose elements are corresponding element addition or multiplication of given sequences is said to be in A.P.             \[\Rightarrow (b-a)=(n+1).d\]             \[d=\frac{b-a}{(n+1)}\]             Hence, 1st A.M. between a and b   \[=a+d\,\,\,\,\,=a+\left( \frac{b-a}{n+1} \right)\] 2nd A.M. between a and \[=a+2d\,\,\,\,\,=a+2.\left( \frac{b-a}{n+1} \right)\] 2nd A.M. between a and b   \[{{n}^{th}}\] A.M. between a and \[b=a+n\frac{(b-a)}{n+1}\]   (e)        Sum of the n-natural number is written as             \[\text{Sn=}\frac{\text{n}(n+1)}{2},\] where \[n\in N\]   (f)         Sum of the square of first n-natural number is written as             \[\text{S}{{\text{n}}^{\text{2}}}\text{=}\frac{\text{n}(n+1)(2n+1)}{6}\]   (g)        Sum of the cube of the 1st n-natural number be             \[\text{S}{{\text{n}}^{3}}\text{=}{{\left( \frac{n(n+1)}{2} \right)}^{2}}\] 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 points \[A({{x}_{1}},{{y}_{1}})\] and \[B({{x}_{2}},y{{  }_{2}})\] internally in the ratio m:n are given by

\[x=\frac{m{{x}_{2}}+n{{x}_{1}}}{m+n},\] \[y=\frac{m{{y}_{2}}+n{{y}_{1}}}{m+n},\]         When P divides AB in the ratio m:n then   \[x=\frac{m{{x}_{2}}-n{{x}_{1}}}{m-n},\] \[y=\frac{m{{y}_{2}}-n{{y}_{1}}}{m-n},\]             When P divides AB in the ratio 1:1 i.e. P is the mid point of AB   \[\therefore \,\,\,\text{P}\equiv \text{(x,y)=}\left( \frac{{{x}_{1}}+{{x}_{2}}}{2},\frac{{{y}_{2}}+{{y}_{1}}}{2} \right)\]  

• Area of triangle: A, B & C be the vertices of the triangle ABC such that \[A\equiv ({{x}_{1}},{{y}_{1}}),\] \[B\equiv ({{x}_{2}},{{y}_{2}}),\] & \[C\equiv ({{x}_{3}},{{y}_{3}}),\]

Area of ABC                         \[=\frac{1}{2}\{{{x}_{1}}({{y}_{2}}-{{y}_{3}})+{{x}_{2}}({{y}_{3}}-{{y}_{1}})+{{x}_{3}}({{y}_{1}}-{{y}_{2}})\}\]             [Using determinant form]  

• Area of a quadrilateral: The ara of the quadraleteral, whose vertices are \[A({{x}_{1}},{{y}_{1}}),\] \[B({{x}_{2}},{{y}_{2}}),\] \[C({{x}_{3}},{{y}_{3}}),\]& \[D({{x}_{4}},{{y}_{4}}),\] is

    Note: The rule for writting the area of a quadrilateral is the same as that of a triangle. Similarly, we can find the area of a polygon of n sides with vertices \[{{A}_{1}}({{x}_{1}},{{y}_{1}}),{{A}_{2}}({{x}_{2}},{{y}_{2}})...{{A}_{n}}({{x}_{n}},{{y}_{n}})\]is     If \[{{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}=0\,\,{{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}=0\] and \[{{a}_{3}}x+{{b}_{3}}y+{{c}_{3}}=0\] are the equation of the triangle. Then the area of the triangle be     Where \[{{c}_{1}},\,{{c}_{2}},\,{{c}_{3}}\] be the co-factor of \[{{c}_{1}},\,{{c}_{2}},\,{{c}_{3}}\] in the determinants: \[\therefore \,\,\,{{c}_{1}}={{a}_{2}}{{b}_{3}}-{{a}_{3}}{{b}_{2}},\] \[{{c}_{2}}={{a}_{3}}.{{b}_{1}}-{{a}_{1}}{{b}_{3}}\] & \[{{c}_{3}}={{a}_{1}}.{{b}_{2}}-{{a}_{2}}{{b}_{1}}.\]  

• Condition of collinearity of three points: The three point \[A({{x}_{1}},{{y}_{1}}),\] \[B({{x}_{2}},{{y}_{2}})\] and \[C({{x}_{3}},{{y}_{3}})\] are collinear iff \[\Delta ABC=0\]

  [In determinant form]   Solved Problem  

Prove that the points (a, b + c), (b, c + a) and (c, a + b) are collinear.

Sol.      Let \[A\equiv (a,b+c)\] \[B\equiv (b,c+a)\] \[C\equiv (c,a+b),\] be three point. To show the collinear of the points A, B and C. Area of \[\Delta ABC\] should be zero. Now,         [Two columns are identical]   \[=(a+b+c)\times 0=0\]   Hence 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 angle,   \[\tan \theta =\left| \frac{2\sqrt{{{h}^{2}}}-ab}{a+b} \right|\]   In cosine form   \[\cos \theta =\frac{a+b}{\sqrt{{{(a+b)}^{2}}+4{{h}^{2}}}}\]   Note:    If two lines are coincident             i.e.       \[\theta =0{}^\circ \] or \[180{}^\circ \]      then \[\frac{2\sqrt{{{h}^{2}}-ab}}{a+b}=0\]   Hence \[{{h}^{2}}=ab\Rightarrow {{h}^{2}}=a\]   (b)        If two lines are perpendicular   i.e.        \[\theta =90{}^\circ ,\] i.e. \[\tan 90{}^\circ =\infty \]   then Loosly, it can be written   \[\frac{2\sqrt{{{h}^{2}}-ab}}{a+b}=\infty =\frac{1}{0}\]   \[a+b=0\] i.e. \[\therefore \]Sum of the coefficient of \[{{x}^{2}}\] and \[{{y}^{2}}\] respectively is zero.   (c)        The equation of pair of straight lines passing through the origin and perpendicular to the given equation of pair of straight lines\[/a{{x}^{2}}+2hxy+b{{y}^{2}}=0\] is written as \[b{{x}^{2}}-2hxy+a{{x}^{2}}=0\] The general equation of second degree in x and y be \[a{{x}^{2}}+b{{y}^{2}}+2hxy+2gx+2fy+c=0\] …….(i) represents a pair of straight lines iff \[abc+2fgx-a{{f}^{2}}-b{{g}^{2}}-c{{h}^{2}}=0\]     (a)        If \[\Delta \ne 0\] & \[{{h}^{2}}-ab\ge 0\] then this general equation of second degree in x & y represents the equation of hyperbola. (b)        If \[\Delta \ne 0\] & \[{{h}^{2}}-ab\le 0\] then this pair of lines represent the equation ellipse. (c)        If \[\Delta \ne 0\] & \[{{h}^{2}}-ab=0\]  then this pair of lines represent the equation of parabola (d)        If \[a=b=1\] & \[h=0\] then this represents the equation of circle.  

• Some points to remember

  (a)        Angle between the lines:- If \[\theta \] is the angle between the two lines:             \[a{{x}^{2}}+2hxy+b{{y}^{2}}+2gx+2fy+c=0\]  …… (1)             then \[\tan \theta =\pm \left| \frac{2\sqrt{{{h}^{2}}-ab}}{a+b} \right|\]             \[\Rightarrow \,\,\,\theta ={{\tan }^{-1}}\left| \frac{2\sqrt{{{h}^{2}}-ab}}{a+b} \right|\]   Note:    It is the same form which is obtained by the pair of straight lines passing through the origin.   (b)        Point of intersection of lines: The point of intersection of line (1) is obtained by the partially differentiation of \[\text{f}\equiv a{{x}^{\text{2}}}+b{{y}^{2}}+2hxy+2gx+2\text{f}y+c=0\] w.r.t x and y respective & making             \[\frac{\partial \text{f}}{\partial x}=0\]                            ….. (i)             &  \[\frac{\partial \text{f}}{\partial y}=0\]                        ….. (ii) Then solve these equations and hence, we will obtain the value ofx and y, which are written as             \[(x,y)=\left( \frac{bg-h}{{{h}^{2}}-ab},\frac{a\text{f-gh}}{{{h}^{2}}-ab} \right)\] Here,             \[\frac{\partial \text{f}}{\partial x}=2ax+2hy+2g=0\] & 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, then equation of the circle is written as \[{{x}^{2}}+{{y}^{2}}={{a}^{2}}\] e.g. find the equation of the circle whose centre is (2, 3) and radius is 6 units   Sol.      Let P(x, y) be any point on the circle by distance formula \[{{(x-2)}^{2}}+{{(y-3)}^{2}}={{(6)}^{2}}\] \[{{x}^{2}}+{{y}^{2}}-4x-6y+13=36\] \[{{x}^{2}}+{{y}^{2}}-4x-6y-23=0\] Which is the required equation of the circle.  

• General Equation of the Circle

  Since, the general equation of the second degree be\[a{{x}^{2}}+2hxy+b{{y}^{2}}+2gx+2fy+c=0\]..... (1) Condition for the circle. (i)         a = b should be unity (ii)        product of xy term be zero.   Here equation (1) becomes the general equation of the circle. i.e. The general equation of the circle be written as   \[{{x}^{2}}+{{y}^{2}}+2gx+2fy+c=0\]   Its radius \[=\sqrt{{{g}^{2}}+{{\text{f}}^{2}}-c}\] centre of the circle \[\equiv (-g,-\text{f})\]  

• Nature of the circle

(i)         If \[{{g}^{2}}+{{\text{f}}^{2}}-c>0,\] then equation of the circle represents the real circle with the centre \[(-g,-\text{f}).\] (ii)        If \[{{g}^{2}}+{{\text{f}}^{2}}-c=0\] i.e. radius of the circle is zero. Then the equation of the circle represent point whose co-ordinate be \[(-g,-\text{f}).\] (iii)       If \[{{g}^{2}}+{{\text{f}}^{2}}-c<0\] i.e. radius of the circle is imaginary but its centre, \[(-g,-\text{f}).\]  be real. This type of circle is not possible to draw in the plane.  

• Different form of the circle

  (i)         Circle with centre, (a, b) and which touches the x-axis. Since, the circle touch the x-axis then radius of the circle is equal to the y-ordinate of the centre of the circle. i.e. Radius of the circle = b Hence, equation of the circle is   \[{{({{x}^{2}}-a)}^{2}}+{{(y-b)}^{2}}={{b}^{2}}\] \[{{x}^{2}}+{{y}^{2}}-2ax-2by+{{a}^{2}}+{{b}^{2}}={{b}^{2}}\] \[\Rightarrow {{x}^{2}}+{{y}^{2}}-2ax-2by+{{a}^{2}}=0\]   (ii)        Circle with centre, (a, b) which touches the y-axis. Since, equation of the circle touches the y- axis. i.e. the radius of the circle is equao to the x- ordinate of the centre of the circle. i.e. Radius of the circle is             \[{{({{x}^{2}}+a)}^{2}}+{{(y-b)}^{2}}={{a}^{2}}\]             \[\Rightarrow {{x}^{2}}+{{y}^{2}}-2ax-2by+{{a}^{2}}+{{b}^{2}}={{a}^{2}}\]                \[{{x}_{2}}+{{y}_{2}}-2ax-2by+{{b}_{2}}=0\]   (iii)       Circle with radius a and which touches both the coordinate axis. Since, when centre 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 is represented by [ ] (square bracket) or ( ) etc.   Generally, it is represented as \[A=[{{a}_{ij}}]\]          \[i=1,\,2,\,3,....m\]                                     \[j=1,\,2,\,3,....n\]             Here subscript i denotes no. of row. & subsrcipt j determines no. of column. & \[{{a}_{ij}}\to \] represent the position of element a in the given matrix e.g. \[A=[{{a}_{ij}}]\] & if         \[i\le 3\] and       \[i\le 2\]    

• Order of Matrix: It is the symbol which represent how many rows and columns the matrixs has. In the above example,

Order of matrix \[A=3\times 2\] in which 3 determine number of row & 2 determine no. of column of given matrix.  

• Operation of Matrix

 

Addition of matrics

Subtraction of matrix

Multiplication of matrix

Adjoint of matrix

Inverse of matrixes.

 

• Addition of Matrices

  Let \[A={{[{{a}_{ij}}]}_{m\times n}}\] & \[B={{[{{b}_{ij}}]}_{m\times n}}\] be two matrices, having same order. Then \[A+B\] or \[B+A\] is a matrix whose elements be formed through corresponding addition of elements of two given matrices   \[A+B=B+A\]               Similarly for subtraction operation, we can subtracted two matrices. But \[A-B\ne B-A\] i.e. \[A-B=-(B-A)\]   Note: For addition or substraction operation of two or more than two matrices. They (given matrix) should be the same order.  

• Multiplication operation: Let \[A={{[{{a}_{ij}}]}_{m\times \text{K}}}\] is a matrix of mx k order & \[B={{[{{a}_{ij}}]}_{\text{K}\times \text{P}}}\] is a matrix of \[\text{k}\times p\] order.

For multiplication of two matrices, no. of column of 1st matrix should be equal to no. of row of 2nd matrix. Otherwise multiplication of two matrix does not hold. Then   \[A\times B-[{{c}_{ij}}]\]be a matrix whose order will be\[m\times p\].   e.g.       \[A={{\left[ \xrightarrow[2\,\,\,\,\,\,\,\,\,2\,\,\,\,\,\,\,1]{2\,\,\,\,\,\,\,\,\,1\,\,\,\,\,\,\,\,3} \right]}_{2\times 3}}\]     \[\therefore \,\,\,A\times B\]         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 straight line is represented by two equations where as a plane is represented by single equation in at most three variables.  

• Some basic formula which are used in 3-dimension. The distance between points \[A({{x}_{1}},\,{{y}_{1}},\,{{z}_{1}})\] and \[B({{x}_{1}},\,{{y}_{2}},\,{{z}_{3}})\]be

  \[AB=\sqrt{{{({{x}_{2}}-{{x}_{1}})}^{2}}+{{({{y}_{2}}-{{y}_{1}})}^{2}}+{{({{z}_{2}}-{{z}_{1}})}^{2}}}\]   e.g. Let two points are A (2, 3, 1) & B = (- 5, 2-1)               \[\therefore \,\,\,\,AB=\sqrt{{{(-5-2)}^{2}}+{{(2-3)}^{2}}+{{(-1-1)}^{2}}}\]               \[=\sqrt{49+5}=\sqrt{54}\]  

• Section Formula: The coordinate of the point P dividing the line joining \[A({{x}_{1}},\,{{y}_{1}},\,{{z}_{1}})\] & \[({{x}_{2}},\,{{y}_{2}},\,{{z}_{2}})\] in the ratio m:n internally are

  \[P=\left( \frac{m{{x}_{2}}+n{{x}_{1}}}{m+n},\frac{m{{y}_{2}}+n{{y}_{1}}}{m+n},\frac{m{{z}_{2}}+n{{z}_{1}}}{m+n} \right)\]   The co-ordinate of the point P dividing the line joining \[A({{x}_{1}},\,{{y}_{1}},\,{{z}_{1}})\] and \[({{x}_{2}},\,{{y}_{2}},\,{{z}_{2}})\] in the ratio m:n externally are                         \[P=\left( \frac{m{{x}_{2}}-n{{x}_{1}}}{m-n},\frac{m{{y}_{2}}-n{{y}_{1}}}{m-n},\frac{m{{z}_{2}}-n{{z}_{1}}}{m-n} \right)\]   Midpoint of AB be               \[P=\left( \frac{{{x}_{1}}+{{x}_{2}}}{2},\frac{{{y}_{1}}+{{y}_{2}}}{2},\frac{{{z}_{1}}+{{z}_{2}}}{2} \right).\]   e.g.       Find the co-ordinate of the point which divides the line segment joining the point (-2, 3, 5) & (1, - 4, 6) in the ratio (i) 2: 3 internally (ii) 2:3 externally.   Sol.      Here, Let A= (-2, 3, 5) & B= (1, -4, 6) and m:n =2:3 internally Let P divides AB in the ratio m: n internally                                     \[\therefore \,\,\,P=\left( \frac{m{{x}_{2}}+n{{x}_{1}}}{m+n},\frac{m{{y}_{2}}+n{{y}_{1}}}{m+n},\frac{m{{z}_{2}}+m{{z}_{1}}}{m+n} \right)\]               \[=\left( \frac{2.1+3(-2)}{2+3},\frac{2(-4)+3.3}{2+3},\frac{2\times 6-3\times 5}{2+3} \right)\]               \[=\left( \frac{-4}{5},\frac{1}{5},\frac{27}{5} \right)\]   When P divides AB in the ratio m : n externally                                     \[\therefore \,\,\,P=\left( \frac{2.1-3(-2)}{2-3},\frac{2(-4)-3.3}{2-3},\frac{2\times 6-3\times 5}{2-3} \right)\]               \[P=(-8,+17,3).\]  

• Controid of triangle: The co-ordinate of the centroid of the triangle ABC, whose vertices are

  \[A({{x}_{1}},\,{{y}_{1}},\,{{z}_{1}}),\] \[B({{x}_{2}},\,{{y}_{2}},\,{{z}_{2}}),\] & \[C({{x}_{2}},\,{{y}_{2}},\,{{z}_{3}}),\] are   \[\left( \frac{{{x}_{1}}+{{x}_{2}}+{{x}_{3}}}{3},\frac{{{y}_{1}}+{{y}_{2}}+{{y}_{3}}}{3},\frac{{{z}_{1}}+{{z}_{2}}+{{z}_{3}}}{3} \right)\]  

• Centroid of the tetrahedran: If \[({{x}_{1}},\,{{y}_{1}},\,{{z}_{1}}),\] \[({{x}_{2}},\,{{y}_{2}},\,{{z}_{2}}),\] \[({{x}_{3}},\,{{y}_{3}},\,{{z}_{3}})\,a({{x}_{4}},\,{{y}_{4}},\,{{z}_{4}})\] be the vertices of the tetrahedran, then its centroid G is given by

  \[\left( \frac{{{x}_{1}}+{{x}_{2}}+{{x}_{3}}+{{x}_{4}}}{4},\frac{{{y}_{1}}+{{y}_{2}}+{{y}_{3}}+{{y}_{4}}}{4},\frac{z1+z2+z3+z4}{4} \right)\]  

A point R with x-coordinate 4 lies on the segment joining the points \[\mathbf{P(2,-3,4)}\] & \[\mathbf{Q(8,0,1,0)}\mathbf{.}\]

Find the co-ordinate of the point R.   Sol:      Let \[P(2,-3,4)\] and \[Q(8,0,1,0).\] Let R divides PQ in the ratio l: 1 internally   \[\therefore \,\,\,\,R=\left( \frac{8\lambda +2}{\lambda +1},\frac{0.\lambda +(-3)}{\lambda +1},\frac{10\lambda +1\times 4}{\lambda +1} \right)\]…………  (1) Here x co-ordinate =4 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 certain number will be occurred. It cannot be predicted. It is the random experiment and not to consider, throwing dice follows any rule/system.  

• Outcome and Sample Space: A possible result of a random experiment is said to be its outcome/results.

 

• Sample space: The set of all possible outcome of an experiment is called a sample space.

Generally, it is denoted by S. e.g. when a coin is tossed then, whatever, H (Head) or T (Tale) is occurred, i.e. S = {H,T}.  

• Event: An event is the subset of the sample space.
• g. when a dice is thrown the 6 is appeared then i.e. the occurrence of 6 is an event. In other word

  To throw a dice then Sample space S = {1, 2, 3, 4, 5, 6} and an event E = {6}.   Note:    Here, we will study the probability which is based on set theory.  

• Probability of an event: Here, we will define the probability into two ways:

Mathematical (or a priori) definition

Statistical (or empirical) definition.

 

• Mathematical definition of Probability: Probability of an event A, denoted as P(A), is defined as P(A)

  \[=\frac{Number\,\,of\,\,cases\,\,favourable\,\,to\,\,A}{Number\,\,of\,possible\,\,outcome}\]   e.g.       To throw a dice, what is the probability of occurrence of even numbers. Usually, someone can ask this type of question. Then Sample Space, S = {1, 2, 3, 4, 5, 6} E = event of occurrence of even numbers = {2, 4, 6} n(S) = total o. of element/member of sample space = 6 n (E) = no. of element of event = 3 So, P (E) = Probability of occurrence of even no.             \[=\frac{n(E)}{n(S)}=\frac{3}{6}=\frac{1}{2}\]  

A coin is tossed once, what are the all possible outcome? What is the probability of the coin coming of tails?

  Sol.      When, a coin is tossed, as usually, head (H) or Tail (T) can be appeared, i.e. Net consider sample other things that coin will be standard strictly. Only we have to think fruitfully 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 statistics has no roots.   Measure of Central Tendency  

• Central Tendency: The properties of finding and the average value of the data is said to Central Tendency.

  The commanly used measure of central tendency are:   (a) Arithmetic Mean                                (b) Geometric Mean (c) harmonic Mean                                 (d) Median (e) Mode  

• Arithmetic Mean: Mean of unclassified/Raw data/Individual

Let \[{{x}_{1}},\,{{x}_{2}},\,{{x}_{3}}.....{{x}_{n}}\] are n observations. Then their arithmetic mean is written as             \[\overline{x}=\frac{x1+x2+x3+....xn}{n}=\frac{1}{n}.\sum\limits_{i=1}^{n}{xi}\]             \[=\frac{Sum\,\,o\text{f}\,\,\text{observations}}{no.\,\,o\text{f}\,\,\text{observations}}\]  

• Mean of Classified Data: Let \[{{x}_{1}},\,{{x}_{2}},\,{{x}_{3}},\,{{x}_{4}},....{{x}_{n}}\] and let \[{{\text{f}}_{1}},\,{{\text{f}}_{2}},\,{{\text{f}}_{3}},....{{\text{f}}_{n}}\] are their corresponding frequencies. Then

  \[\overline{x}=\frac{\sum{\text{f}\text{.x}}}{\sum{\text{f}}}\]   Weighted Arithmetic Mean: If \[{{w}_{1}},\,{{w}_{2}},\,{{w}_{3}},\,......{{w}_{n}}\] are the weights assigned to the values \[{{x}_{1}},\,{{x}_{2}},\,{{x}_{3}},\,{{x}_{4}}\,......{{x}_{n}}\] respectively. Then the weighted average, or weighted               \[A.M==\frac{{{w}_{1}}{{x}_{1}}+{{w}_{2}}{{x}_{2}}+{{w}_{3}}{{x}_{3}}+......{{w}_{n}}{{x}_{n}}}{{{w}_{1}}+{{w}_{2}}+{{w}_{3}}+.....{{w}_{n}}}\]  

• Combined Mean: If we are given the A.M. of two data sets and their sizes, then the combined

A.M of two data sets can be obtained as.             \[{{\overline{x}}_{12}}=\frac{{{n}_{1}}{{\overline{x}}_{1}}+{{n}_{2}}{{\overline{x}}_{2}}}{{{n}_{1}}+{{n}_{2}}}\]   Where, \[{{\overline{x}}_{12}}=\] combined mean of the two data sets 1 and 2 0 Mean of 1st data \[{{\overline{x}}_{2}}=\]Mean of the 2nd data \[{{n}_{1}}=\]size of the 1st data. \[{{n}_{2}}=\]size of the 2nd data. Some properties about A.M. In statistical data, sum of the deviation of individual values from A.M. is always zero.             i.e.        \[\sum\limits_{i=1}^{n}{\text{f}i}({{x}_{1}}-\overline{x})=0\]   Where \[\text{f}i=\] frequencies of \[xi\,\,\{1\le i\le n\}\] A.M is written s             \[A.M\,\,=\overline{x}=\frac{{{x}_{1}}{{\text{f}}_{\text{1}}}+{{x}_{2}}{{\text{f}}_{2}}+...{{x}_{n}}{{\text{f}}_{n}}}{{{\text{f}}_{1}}+{{\text{f}}_{2}}+{{\text{f}}_{3}}+....{{\text{f}}_{n}}}=\frac{\sum\limits_{i=1}^{n}{{{\text{f}}_{i}}{{x}_{i}}}}{\sum\limits_{i=1}^{n}{\text{f}i}}\]  

• Short-cut Method: For a given data, we suitably choose a term, usually the middle term and call it the assumed mean, to be denoted by A.

Then, we find deviation, \[{{d}_{i}}=({{x}_{i}}-A)\] for each term. Thus \[A.M=\overline{x}=A+\frac{\sum{{{\text{f}}_{i}}{{d}_{i}}}}{{{\text{f}}_{i}}}\]   where A = Assumed Mean, f = frequency  

• Step-Deviation: \[A.M,\,\,\overline{x}=A+\frac{\Sigma {{\text{f}}_{i}}{{d}_{i}}}{N}\times h\]

  Where A = Assumed mean   \[{{d}_{i}}=\frac{{{x}_{i}}-A}{h}=\]deviation of any variate from A   h = width of the class-interval and \[N=\Sigma {{\text{f}}_{i}}\] In a statistical date, the sum of square of deviations of individual values from A.M. is least.   i.e.        \[\sum\limits_{i=1}^{n}{\text{f}i{{(x-\overline{x})}^{2}}=}\] least value             If each of the given observation is doubled then their arithmetic mean is doubled If 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 if the nth term of a sequence is given we can find the terms of the sequence.   Elementary question 1 Find the nth term of the sequence 4, 7, 10, 13, 16,.............. Answer. From observation, \[{{T}_{1}}=3\times 1+1\]                  \[{{T}_{2}}=3\times 2+1\]                    \[\therefore {{T}_{n}}=3n+1\]   SERIES The sum of term of a sequence is called the series of the corresponding sequence. Example: 2 + 4 + 6 +...... + 2n is a finite series of first n even natural number The sum of first n terms of series is denoted by \[{{S}_{n}}\]. Here, \[{{S}_{n}}={{T}_{1}}+{{T}_{2}}+......+{{T}_{n}}\] Here, \[{{S}_{1}}={{T}_{1}}\];\[{{S}_{2}}={{T}_{1}}+{{T}_{2}}\];             \[{{S}_{3}}={{T}_{1}}+{{T}_{2}}+{{T}_{3}}\]; \[{{S}_{4}}={{T}_{1}}+{{T}_{2}}+{{T}_{3}}+{{T}_{4}}\]…………… \[\therefore {{S}_{n}}={{T}_{1}}+{{T}_{2}}+{{T}_{3}}+.....+{{T}_{n}}\] And, we have,    \[{{S}_{2}}-{{S}_{1}}={{T}_{2}}\]                         \[{{S}_{3}}-{{S}_{2}}={{T}_{3}};\]   \[{{S}_{4}}-{{S}_{2}}={{T}_{4}}\]and so on. Similarly,             \[{{S}_{n}}-{{S}_{n-1}}={{T}_{n}}\] Arithmetic mean (A.M.): If a and c are any two terms of an A.P, then the arithmetic mean (A.M.) ‘b’ is given by, \[b=\frac{a+c}{2}\]. When three numbers a, b, c arranged in order of there is creasing values differ by a fixed number ‘d’, then they are said they are said to be in arithmetic progression  (A, P,) and the fixed number ‘d’ is called common difference. \[a+d=b\] \[b+d=c\] Or         \[bacb\]   Or         \[b=\frac{a+c}{2}\] If ‘n’ numbers \[{{x}_{1}},{{x}_{2}}......{{x}_{n}}\] are in A. P. then \[{{x}_{2}}=\text{ }{{\text{x}}_{1}}+d\] \[{{x}_{3}}=\text{ }{{x}_{1}}+2d\] \[{{x}_{n}}=\text{ }{{x}_{1}}(n-1)d\] \[\therefore \]\[{{n}_{th}}\] term of A. P. \['{{X}_{n}}'\] = first term \[+\left( n-1 \right)\times \]common difference Sum of an A. P. up to ‘n’ terms \[{{x}_{1}}={{x}_{1}}\] \[{{x}_{2}}={{x}_{1}}+d\] \[{{x}_{3}}={{x}_{1}}+2d\] \[{{x}_{n}}={{x}_{1}}+(n-1)d\] Adding \[{{S}_{n}}={{x}_{1}}+{{x}_{2}}+.....{{x}_{n}}=n({{x}_{1}})+(0+1+....(n-1)\times d\]                         \[=n{{x}_{1}}+\frac{n(n-1)}{2}d\]                         \[=\frac{n}{2}\left[ 2{{x}_{1}}+(n-1)d \right]\]   GEOMETRIC PROGRESSION (G.P.) A sequence \[{{a}_{1}},{{a}_{2}},{{a}_{3}}.....,{{a}_{n}}\] is said to be in G.P., if the ratio of the consecutive terms is a constant, that is, \[\frac{{{a}_{2}}}{{{a}_{1}}}=\frac{{{a}_{3}}}{{{a}_{2}}}=r\]      \[\Rightarrow {{a}_{2}}={{a}_{1}}.r,\]             \[{{a}_{3}}={{a}_{1}}{{r}^{2}}\] Thus, if ‘r’ is the common ratio, then the nth term of the sequence is given by \[{{a}_{n}}=a{{r}^{n-1}}\]. The of n terms of the G.P. is given by. \[{{S}_{n}}=\frac{a\left( {{r}^{n}}-1 \right)}{r-1};r>1\] and \[{{S}_{n}}=\frac{a\left( 1-{{r}^{n}} \right)}{1-r};r<1\] Infinite G.P.: Sum of infinite G.P. is given by \[{{S}_{\infty }}=\frac{a}{1-r}\] GEOMETRIC MEAN (G.M.) If ‘a’ and ‘b’ are any two terms of G.P., then the geometric mean is given by \[GM=\sqrt{ab}\]   Properties of GP

If each term of GP is multiplied or divided 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 sequence is given by \[{{a}_{n}}\]= a + (n - 1)d, where 'a' is the first term of the sequence, 'd' is the common difference and 'n' is the number of terms in the sequence. For example 10th term of the sequence 3, 5, 7, 9, --- is given by:             \[{{a}_{10}}=a+9d\]    \[\Rightarrow \]   \[{{a}_{10}}=3+9\times 2=21\]   Sum of n terms of the A.P. If \[{{a}_{1}},\text{ }{{a}_{2}},\text{ }{{a}_{3}}--,\text{ }{{a}_{n}}\]be n terms of the sequence in A.P., then the sum of n-terms of the sequence is given by \[{{S}_{n}}=\frac{n}{2}[2a+(n-1)\,d]\] For example the sum of first 10 terms of the sequence 3, 5, 7, 9,......... is given by: \[{{S}_{10}}=\frac{10}{2}[2\times 3+9\times 2d]\Rightarrow {{S}_{10}}=120\] If \[{{S}_{n}}\] is the sum of the first n terms of an AP, then its \[{{n}^{th}}\]term is given by \[{{a}_{n}}\]= \[{{S}_{n}}-\text{ }{{S}_{n}},\]   Geometric Progression (G.P.) A sequence is said to be in G.P., if the ratio between its consecutive terms is constant. The sequence\[{{a}_{1}},\text{ }{{a}_{2}},\text{ }{{a}_{3}}\],.... an is said to be in G.P. If the ratio of its consecutive terms is a constant, the constant term is called common ratio of the G.P. and is denoted by r. For example any sequence of the form 2, 4, 8, 16,... is a G.P. Here the common ratio of any two consecutive terms is 2. If 'r' is the common ratio, then the nth term of the sequence is given by \[{{a}_{n}}a{{r}^{n-1}}\] The sum of n terms of a G.P. is given by             \[{{S}_{n}}=\frac{a({{r}^{n}}-1)}{r-1},if\] \[r>1\,\,and\,\,{{S}_{n}}=\frac{a(1-{{r}^{n}})}{1-r}\,\,if\,\,r<1\] Sum to infinity of a G.P. is given by \[{{S}_{\infty }}=\frac{a}{1-r}\]   Harmonic Progression (H.P.) A sequence is said to be in H.P. If the reciprocal of its consecutive terms are in A.P. It has got wide application in the field of geometry and theory of sound. These progressions are generally solved by inverting the terms and using the property of arithmetic progression. Three numbers a, b, c are said to be in H.P. if \[\frac{1}{a},\frac{1}{b}\] are \[\frac{1}{c}\] in A.P.   Some Useful Results             (i) Sum of first n natural numbers i.e. 1 + 2 + 3 + ...... n =\[\frac{(n+1)n}{2}\] (ii) Sum of the more...