# Current Affairs 11th Class

#### Notes - Mathematics Olympiads - Two Dimensional Geometry

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...

#### Notes - Mathematics Olympiads - Pair Sraight Line

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...

#### Notes - Mathematics Olympiads - Circles

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...

#### Notes - Mathematics Olympiads - Matrices and Determinant

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

• Subtraction of matrix
• Multiplication of matrix
• Inverse of matrixes.
•
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...

#### Notes - Mathematics Olympiads - Three Dimensional Plane

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...

#### Notes - Mathematics Olympiads - Probability

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...

#### Notes - Mathematics Olympiads - Statistics

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...

#### Nucleic Acids

Nucleic acids are the polymers of nucleotide made up of carbon, hydrogen, oxygen, nitrogen and phosphorus and which controls the basic functions of the cell. These were first reported by Friedrich Miescher (1871) from the nucleus of pus cell. Altmann called it first time as nucleic acid. Nuclein was renamed nucleic acid by Altman in (1889). They are found in nucleus. They help in transfer of genetic information. Types of nucleic acids : On the basis of nucleotides i.e., sugars, phosphates and nitrogenous bases, nucleic acids are of two types which are further subdivided. These are DNA (Deoxyribonucleic acid) and RNA (Ribonucleic acid). (1) DNA (Deoxyribonucleic acids) : Term DNA was given by Zacharis.  (i) Types of DNA : It may be linear or circular in eukaryotes and prokaryotes respectively. Palindromic DNA : The DNA helix bears nucleotide in a serial arrangement but opposite in two strands. $-T-T-A-A-C-G-T-T-A-A.......$ $-A-A-T-T-G-C-A-A-T-T......$ Repetitive DNA : This type of arrangement is found near centromere of chromosome and is inert in RNA synthesis. The sequence of nitrogenous bases is repeated several times. Satellite DNA : It may have base pairs up to $160\,\,bp$and are repetitive in nature. Microsatellite has $16\,\,bp$and minisatellite has $1160\,\,bp.$ They are used in DNA matching or finger printing (Jefferey). In eukaryotes, DNA is deutrorotatory and sugars have pyranose configuration. (ii) Chargaff’s rule : Quantitatively the ratio of adenine (A) to thymine (T) and guanine (G) to cytosine (C) is equal. i.e., “Purines are always equal to pyrimidine”. (iii) C value : It is the total amount of DNA in a genome or haploid set of chromosomes. (iv) Sense and Antisense strand : Out of two DNA strand one which carries genetic information in its cistrons is called sense strand while the other strand does not carry genetic information, therefore, doesn’t produce mRNA. The non-functional DNA strand is called antisense strand. (v) Heteroduplex DNA : Hybrid DNA formed as a result of recombination is called heteroduplex DNA. It contains mismatched base pair of heterologous base sequence. X-Ray crystallography study of DNA : It was done by Wilkins. It shows that the two polynucleotide chains of DNA show helical configuration. Single stranded DNA (ssDNA) : It is single helixed circular. And isolated from bacteriophage $\phi \times 174$ by Sinsheimer (1959). It does not follow chargaff’s rule. The replicative form (RF) has plus – minus DNA helix. e.g., parvovirus. Double helical model of DNA: It is also known as Watson and Crick model. (2) RNA or Ribonucleic acid : RNA is second type of nucleic acid which is found in nucleus as well as in cytoplasm i.e., mitochondria, plastids, ribosomes etc. They carry the genetic information in some viruses. They are widely distributed in the cell. Genomic RNA was discovered by Franklin and Conrat (1957).

#### Cell Division / Cell Reproduction / Cell Cycle

It is the process by which a mature cell divides and forms two nearly equal daughter cells which resemble the parental cell in a number of characters. In unicellular organisms, cell division is the means of reproduction by which the mother cell produces two or more new cells. In multicellular organism also, new individual develop from a single cell. Cell division is central to life of all cell and is essential for the perpetuation of the species. Discovery : Prevost and Dumans (1824) first to study cell division during the cleavage of zygote of frog. Nagelli (1846) first to propose that new cells are formed by the division of pre-existing cells. Rudolf Virchow (1859) proposed “omnis cellula e cellula” and “cell lineage theory”. A cell divides when it has grown to a certain maximum size which disturb the karyoplasmic index (KI)/Nucleoplasmic ratio (NP)/Kernplasm connection. Cell cycle : Howard and Pelc (1953) first time described it. The sequence of events which occur during cell growth and cell division are collectively called cell cycle. Cell cycle completes in two steps: (1) Interphase,                  (2) M-phase/Dividing phase (1) Interphase : It is the period between the end of one cell division to the beginning of next cell division. It is also called resting phase or not dividing phase. But, it is actually highly metabolic active phase, in which cell prepares itself for next cell division. In case of human beings it will take approx 25 hours. Interphase is completed in to three successive stages. G1 phase/Post mitotic/Pre-DNA synthetic phase/gap Ist : In which following events take place. (i) Intensive cellular synthesis. (ii) Synthesis of rRNA, mRNA ribosomes and proteins. (iii) Metabolic rate is high. (iv) Cells become differentiated. (v) Synthesis of enzymes and ATP storage. (vi) Cell size increases. (vii) Decision for a division in a cell occurs. (viii) Substances of G stimulates the onset of next S – phase. (ix) Synthesis of NHC protein, carbohydrates, proteins, lipids. (x) Longest and most variable phase. (xi) Synthesis of enzyme, amino acids, nucleotides etc. but there is no change in DNA amount. S-phase/Synthetic phase (i) DNA replicates and its amount becomes double $(2C-4C).$ (ii) Synthesis of histone proteins and NHC (non-histone chromosomal proteins). (iii) Euchromatin replicates earlier than heterochromatin. (iv) Each chromosome has 2 chromatids. G2-phase/Pre mitotic/Post synthetic phase/gap-IInd (i) Mitotic spindle protein (tubulin) synthesis begins. (ii) Chromosome condensation factor appears. (iii) Synthesis of 3 types of RNA, NHC proteins, and ATP mole. (iv) Duplication of mitochondria, plastids and other cellular macromolecular complements. (v) Damaged DNA repair occur. (2) M-phase/Dividing phase/Mitotic phase : It is divided in to two phases, karyokinesis and cytokinesis. Duration of cell cycle : Time period for ${{G}_{1}},S,{{G}_{2}}$ and M-phase is species specific under specific environmental conditions. e.g., 20 minutes for bacterial cell, 8-10 hours for intestinal epithelial cell, and onion root tip cells may take 20 hours. ${{\mathbf{G}}_{\mathbf{0}}}\mathbf{-}$phase (Lajtha, 1963) : The cells, which are not to divide further, do more...

#### Metabolically Inactive Cell Inclusions

Within the cytoplasm of a cell there occur many different kinds of non-living structures which are called inclusions or ergastic / Deutoplasmic substances. (1) Vacuoles : The vacuole in plants was discovered by Spallanzani. It is a non-living reservoir, bounded by a differentially or selectively permeable membrane, the tonoplast. The vacuole is filled with cell sap or tonoplasm. They contain water, minerals and anthocyanin pigments. Some protozoans have contractile vacuoles which enlarge by accumulation of fluid or collapse by expelling them from the cell. The vacuoles may be sap vacuoles, contractile vacuoles or gas vacuoles (pseudo vacuoles). Function of vacuoles : Vacuole maintains osmotic relation of cell which is helpful in absorption of water. Turgidity and flaccid stages of a cell are due to the concentrations of sap in the vacuole. (2) Reserve food material The reserve food material may be classified as follows : (i) Carbohydrates : Non-nitrogenous, soluble or non- soluble important reserve food material. Starch cellulose and glycogen are all insoluble. (a) Starch : Found in plants in the form of minute solid grains. Starch grains are of two types : Assimilation starch : It is formed as a result of photosynthesis of chloroplasts. Reserve starch : Thick layers are deposited around an organic centre called hilum. (b) Glycogen : Glycogen or animal starch occurs only in colourless plants like fungi. (c) Inulin : It is a complex type of polysaccharide, soluble and found dissolved in cell sap of roots of Dahlia, Jaruslem, Artichoke, Dandelion and members of compositae. (d) Sugars : A number of sugars are found in solution of cell sap. These include glucose, fructose, sucrose, etc. (e) Cellulose : Chemical formula is ${{({{C}_{6}}{{H}_{10}}{{O}_{5}})}_{n}}.$The cell wall is made up of cellulose. It is insoluble in water. (ii) Fats and Oils : These are important reserve food material. These are always decomposed into glycerol and fatty acids by enzymatic action. Fat is usually abundant in cotyledons than in the endosperm. e.g., flax seed produce linseed oil, castor produce castor oil, cotton seeds produce cottonseed oil, etc. (iii) Proteins and Amides (Aleurone grains) : Storage organ usually contain protein in the form of crystalline bodies known as crystalloids (potato). Proteins may be in the form of aleurone grains as in pea, maize, castor, wheat, etc. (3) Excretory Products : The organic waste products of plants are by-product of metabolism. They are classified as : (i) Resins : They are believed to be aromatic compounds consisting of carbon, hydrogen and oxygen and are acidic in nature. Sometimes they are found in combination with gums and are called gum resin. e.g., Asafoetida (heeng). (ii) Tannins : They are complex nitrogenous compounds of acid nature having an astringent taste. Presence of tannin in plants makes its wood hard durable and germ proof. (iii) Alkaloids : These are organic, basic, nitrogenous substance. They occur in combination with organic acids and most of them are poisonous. From plants, cocaine, hyoscine, morphine, nicotine, quinine, atropine, strychnine and daturine etc. are extracted. (iv) Glucosides more...

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