Current Affairs 10th Class

Number System   All the numbers we have studied so far are real numbers. The real numbers are divided into two categories which are rational and irrational numbers. All the positive numbers used for counting are called the natural numbers. These start from 1 and end till infinity. The positive numbers which start from zero are called whole numbers. The collections of natural numbers, their negatives along with the number zero are called integers. Rational numbers are the numbers in the form \[\frac{p}{q}\] where \[q\ne 0\] and p, q are integers and irrational q numbers are the numbers which cannot be written in the form p-, where p and q are integers and\[q\ne 0\].   Decimal Expansion of Rational Numbers There are rational numbers which can be expressed as terminating decimals or non- terminating decimals. The non-terminating decimals may be repeating or non- repeating. The rational numbers whose denominators are of the form \[{{2}^{m.}}{{5}^{n}}\] (where m and n are whole numbers) are terminating and rest are non-terminating decimals.   Euclid's Division Lemma For any two positive integers, say x and y (x > y), there exists unique integers say k and r satisfying \[x=ky+r\] where \[0\le r<y\,.\] By using Euclid's division algorithm, we can find the greatest common divisor of two numbers. Note:
  •  A lemma is a proven statement used for proving another statement.
  •  An algorithm is a series of well defined steps which gives a procedure for solving a type of problem.
Example: Find the HCF of 378 and 1260. (a) 252                                                              (b) 126          (c) 378                                                              (d) 63 (e) None of these Ans.     (b) Explanation: By using Euclid's division lemma, we get 1260 = 378 \[\mathbf{\times }\] 3 + 126 now consider the divisor and remainder. So again by applying Euclid's division lemma, we get 378 = 126 \[\mathbf{\times }\] 3 + 0 Now the remainder is zero, so we stop the process. The divisor at this stage is 126. So, HCF of 378 and 1260 is 126.  

Polynomials   Polynomials are algebraic expressions having finite terms. The expression \[{{a}_{n}}{{x}^{n}}+{{a}_{n-1}}{{x}^{n-1}}+{{a}_{n-2}}{{x}^{n-2}}+---+{{a}_{1}}x+{{a}_{0}}\] is called the polynomial of degree n, where \[{{a}_{n}}\ne 0.\] The highest power of the variable in a polynomial is called degree of the polynomial. The polynomials of degree one are called linear polynomial. The polynomials of degree two are called quadratic polynomials. The polynomials of degree three are called cubic polynomials and the polynomials of degree four are called biquadratic polynomials. A real number which satisfies the given polynomial is called zero of the polynomial.   Zeroes of a Polynomial If \[a{{x}^{2}}+\text{ }bx\text{ }+\text{ }c\text{ }=\text{ }0,\text{ }(a\ne o),\]is a quadratic equation whose roots are \[\alpha \] and \[\beta \], then the relation between the roots of the equation and its coefficients is given by: Sum of the roots =\[\alpha +\beta =-\frac{b}{a},\] Product of the roots = \[\alpha \beta =\frac{c}{a}.\]   For a cubic equation \[a{{x}^{3}}+\text{ }b{{x}^{2}}+\text{ }ex\text{ }+\text{ }d\]= 0 (a = 0) roots are a, P and y, the relation between the roots of the equation and its coefficients is given by Sum of roots =\[\alpha +\beta +\gamma =-\frac{b}{a},\] Sum of the product of roots =\[\alpha \beta +\beta \gamma +\gamma \alpha =\frac{c}{a},\] Product of the roots =\[\alpha \beta \gamma =-\frac{d}{a},\]  
  •               Example:
The zeroes of the polynomials \[\mathbf{6}{{\mathbf{x}}^{\mathbf{2}}}-\text{ }\mathbf{x}\text{ }-\text{ }\mathbf{2}\] are:             (a) \[\frac{1}{3}and\frac{2}{3}\]                                                             (b) \[\frac{-1}{3}and\frac{1}{2}\] (c) \[\frac{2}{3}and\frac{-1}{2}\]                                                            (d) \[\frac{-2}{3}and\frac{1}{2}\] (e) None of these Ans.     (c) Explanation: Let f(x) = \[6{{x}^{2}}-\text{ }x\text{ }-\text{ }2.\] By the method of splitting the middle term, we have \[f\,(x)=\text{ }6{{x}^{2}}-x-2=6{{x}^{2}}-4x+3x-2\] \[=2x\left( 3x-2 \right)+1\left( 3x-2 \right)=\left( 2x+1 \right)\left( 3x-2 \right)\] Zeroes of (x) will be obtained by putting \[2x+\text{ }1=0\]and \[3x-2=0\] \[\Rightarrow x=\frac{-1}{2}\,\,and\,\,x=\frac{2}{3}\]   Division of Polynomials Previously we have studied about the division of the real numbers, in which we obtained quotient and remainder which satisfy the relation: Dividend = Quotient \[\times \] Divisor + Remainder This is also known as Euclid's division lemma. In this section we will discuss about the division of the polynomials which is known as the division algorithm for polynomials. The concept of division of the polynomials can be used for finding the zeroes of the cubic or biquadratic polynomials. For example, when we divide g(x) \[=\text{ }{{x}^{3}}-\text{ }3{{x}^{2}}+\text{ }3x\text{ }-\text{ }5\text{ }by\text{ }h\left( x \right)\text{ }=\text{ }{{x}^{2}}+\text{ }x\text{ }+\text{ }1\]we have,           Here we have quotient \[q\left( x \right)=x-4,\] and remainder \[r\left( x \right)=6x-1\].   Graph of Polynomials In this section we will learn about the construction of linear, quadratic and cubic polynomial graphs. In order to draw a graph of the polynomial f(x), we first find some values of x which satisfy the equation of f(x) and plot these points on a rectangular co-ordinate system and then join these points with free hand curve.   Graph of more...

Co-ordinate Geometry   In this chapter we will discuss about the two as well as three dimensional geometry. We will discuss about the position of the points and locate the point in the plane or on the surface. The three mutually perpendicular lines in the plane are called coordinate axes of the plane. The numbers in a plane which represent the position of a point is called coordinates of the point with reference to the coordinate planes. The eight equal regions into which space is divided by three dimensional axes are called octants.   Distance Formula Let us consider the two points \[A\left( {{x}_{1}},\text{ }{{y}_{1}} \right)\] and B(\[{{x}_{2}},\text{ }{{y}_{2}}\]), in a two dimensional plane, then the distance between the two points is given by \[AB=\sqrt{{{({{x}_{2}}-{{x}_{1}})}^{2}}+{{({{y}_{2}}-{{y}_{1}})}^{2}}.}\] If it is a three dimensional plane containing the points \[A\left( {{x}_{1}},\text{ }{{y}_{1}},\text{ }{{z}_{1}} \right)\] and \[B\left( {{x}_{2}},\text{ }{{y}_{2}},\text{ }{{z}_{2}} \right),\]then the distance between the points is given by: \[AB=\sqrt{{{({{x}_{2}}-{{x}_{1}})}^{2}}+{{({{y}_{2}}-{{y}_{1}})}^{2}}+{{({{z}_{2}}-{{z}_{l}})}^{2}}}\]   Section Formula Let us consider the point P(x, y) which divides the line segment joining \[A\left( {{x}_{1}},\text{ }{{y}_{1}} \right)\] and in the ratio k : 1 internally, then the coordinates of the point P(x, y) is given by: \[x=\frac{{{x}_{1}}+k{{x}_{2}}}{k+1}\,\,\,and\,\,y\frac{{{y}_{1}}+k{{y}_{2}}}{k+1}\]           Coordinates of Midpoint The coordinates of the mid-point of a line segment AB with coordinates \[A\left( {{x}_{1}},\text{ }{{y}_{1}} \right)\] and \[B\left( {{x}_{2}},\text{ }{{y}_{2}} \right)\] is given by \[\left( \frac{{{x}_{1}}+{{x}_{2}}}{2},\frac{{{y}_{1}}+{{y}_{2}}}{2} \right)\] Note: (i) If the mid-point of \[\Delta \] ABC are \[P({{x}_{1}},{{y}_{1}}),\]\[Q\,({{x}_{2}},\,\,{{y}_{2}})\] and \[R\,({{x}_{3}},\,\,{{y}_{3}})\] then its vertices will be \[A\,(-{{x}_{1}}+{{x}_{2}}+{{x}_{3}},-{{y}_{1}}+{{y}_{2}}+{{y}_{3}}),\] \[B\,({{x}_{1}}-{{x}_{2}}+{{x}_{3}},\,\,{{y}_{1}}-{{y}_{2}}+{{y}_{3}})\] and \[C\,({{x}_{1}}+{{x}_{2}}-{{x}_{3}},\,\,{{y}_{1}}+{{y}_{2}}-{{y}_{3}})\] (ii) The fourth vertex of a parallegogram whose three vertices in order are \[\left( {{x}_{1}},\text{ }{{y}_{1}} \right),\text{ }\left( {{x}_{2}},\text{ }{{y}_{2}} \right)\text{ }and\text{ }\left( {{x}_{3}},\text{ }{{y}_{3}} \right)\] is \[\left( {{x}_{1}}\text{ }{{x}_{2}}+\text{ }{{x}_{3}},\text{ }{{y}_{1}}-\text{ }{{y}_{2}}+\text{ }{{\text{y}}_{3}} \right)\]   Centroid of a Triangle It is defined as the point of intersection of the medians of the triangle. The coordinates of centroid of a triangle with vertices \[\left( {{x}_{1}},\text{ }{{y}_{1}} \right),\text{ }\left( {{x}_{2}},\text{ }{{y}_{2}} \right)\text{ }and\text{ }\left( {{x}_{3}},\text{ }{{y}_{3}} \right)\] is: \[\left( \frac{{{x}_{1}}+{{x}_{2}}+{{x}_{3}}}{3},\frac{{{y}_{1}}+{{y}_{2}}+{{y}_{3}}}{3} \right)\] Note: in an equilateral triangle orthocentre, centroid, circumcentre, incentre coincide.   Area of a Triangle Let A\[\left( {{x}_{1}},\text{ }{{y}_{1}} \right),\text{ }B\left( {{x}_{2}},\text{ }{{y}_{2}} \right)\text{ }and\text{ }C\left( {{x}_{3}},\text{ }{{y}_{3}} \right)\]be the vertices of a triangle, then the area of the triangle is given by: \[=\frac{1}{2}\left| {{x}_{1}}({{y}_{2}}-{{y}_{3}})+{{x}_{2}}({{y}_{3}}-{{y}_{1}})+{{x}_{3}}({{y}_{1}}-{{y}_{2}}) \right|\]   Conditions for Collinearity  Let the given points be A\[\left( {{x}_{1}},\text{ }{{y}_{1}} \right),\text{ }B\left( {{x}_{2}},\text{ }{{y}_{2}} \right)\text{ }and\text{ }C\left( {{x}_{3}},\text{ }{{y}_{3}} \right)\]. If A, B and C are collinear then. Area of \[\Delta \]ABC = 0. \[\Rightarrow \frac{1}{2}\left| {{x}_{1}}({{y}_{2}}-{{y}_{3}})+{{x}_{2}}({{y}_{3}}-{{y}_{1}})+{{x}_{3}}({{y}_{1}}-{{y}_{2}}) \right|=0.\] Also if ABC are collinear, then slope of AB = slope of BC = slope of CA   Locus The curve described by a point which moves under given condition(s) is called its locus. The equation of the locus of a point is satisfied by the coordinates of every point.   Slope of a Line The trigonometrical tangent of the angle that a line makes with the positive direction of the x-axis in anticlockwise direction is called the slope of the line. more...

Pair of Linear Equations in two Variables and Quadratic Equation   Linear Equation in Two Variables A linear equation in two variables is an equation which contains a pair of variables which can be graphically represented in xy-plane by using the coordinate system. For example ax + by = c and dx + e y = f, is a pair of linear equations in two variables' Solutions of the linear equation in two variables are the pair of values of the variables that satisfies the given equation. In other words, we can say that a system of linear equation is nothing but two or more linear equations that are being solved simultaneously. Mostly, the system of equations are used in the business purposes by predicting their future events. They model a real life situation in two system of equations to find the solution and manage their business. We can make an accurate prediction by using system of equations. The solution of the system of equations in two variables is an ordered pair that satisfies each equation.   Graphical Representation of a Pair of Linear Equations in two variables If \[{{a}_{1}}x\text{ }+\text{ }{{b}_{1}}y+\text{ }{{c}_{1}}==\text{ }0\text{ }and\text{ }{{a}_{2}}x+\text{ }{{b}_{2}}y\text{ }+{{c}_{2}}=\text{ }0\]are a pair of linear equations in two variables such that: If \[\frac{{{a}_{1}}}{{{a}_{2}}}\ne \frac{{{b}_{1}}}{{{b}_{2}}}\] then pair of linear equations is consistent with a unique solution. If \[\frac{{{a}_{1}}}{{{a}_{2}}}=\frac{{{b}_{1}}}{{{b}_{2}}}=\frac{{{c}_{1}}}{{{c}_{2}}}\] then the pair of linear equations is consistent and dependent and having infinitely many solutions. If \[\frac{{{a}_{1}}}{{{a}_{2}}}=\frac{{{b}_{1}}}{{{b}_{2}}}\ne \frac{{{c}_{1}}}{{{c}_{2}}},\] then the pair of linear equations is inconsistent and have no solution.   Unique Solution If the lines represented by a pair of linear equations are intersecting each other at one point, then the system is said to have unique solution. The point at which the two lines intersect each other is called the solution of the system of equation.   No Solution If the graph of the system of equation is parallel and does not intersect each other at any point, then it is said to have no solution.   Infinitely Many Solutions If the lines represented by the pair of linear equations in two variables coincides each other, then it is said to have infinitely many solution.   Solving the System of Equations There are different algebraic methods for solving the system of linear equations. The three different methods are:
  •           Elimination Method
  •           Substitution Method
  •           Cross Multiplication Method
 
  •              Example:
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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...

Geometry   In this chapter we will discuss about the similarity of triangles and properties of circles. Two figures having the same shape and not necessarily the same size are called the similar figures. Two polygons of the same number of sides are similar if their corresponding angles are equal and their corresponding sides are in the same ratio. Circle is defined as the locus of a point which is at a constant distance from a fixed point. The fixed point is called the canter of the circle and the fixed distance is called the radius of the circle.   Similar Triangles Two triangles are similar, if their corresponding angles are equal and their corresponding sides are in the same ratio. The ratio of any two corresponding sides in two equiangular triangles is always the same.   Basic Proportionality Theorem It states that if a line is drawn parallel to one side of a triangle to intersect the other two sides in the distinct points, the other two sides are divided in the same ratio. Conversely, if a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side of the triangle.   Area of Similar Triangles It states that the ratio of area of two similar triangles is equal to the square of the ratio of their corresponding sides.   Tangent to a Circle A tangent to a circle is a line which intersects the circle at exactly one point. The point where the tangent intersects the circle is known as the point of contact.   Properties of tangent to a circle Following are some properties of tangent to a circle:
  •          A tangent to a circle is perpendicular to the radius through the point of contact.
  •          A line drawn through the end-point of a radius and perpendicular to it is a tangent to the circle.
  •         The lengths of the two tangents drawn from an external point to a circle are equal.
  •          If two tangents are drawn to a circle from an external point, they subtend equal angles at the centre.
  •          If two tangents are drawn to a circle from an external point, then they are equally inclined to the segment, joining the centre to that point.
From the above points we conclude that in the following figure; more...

Trigonometry and Its Application   The word trigonometry is a Greek word consists of two parts 'trigonon' and 'metron' which means measurements of the sides and angles of a triangle. This was basically developed to find the solutions of the problems related to the triangles in the geometry. Initially we use to measure angles in terms of degree, but now we will use another unit of measurement of angles called radians. The relation between the radian and degree measure is given by: 1 radians \[={{\left( \frac{180}{\pi } \right)}^{o}}\]and 1o=\[{{\left( \frac{\pi }{180} \right)}^{o}}\]radians or \[{{\pi }^{c}}\]=\[{{180}^{o}}\]   Trigonametric ratios of allied angles Two angles are called allied angles when their sum or difference is either zero or a multiple of\[90{}^\circ \]. The angles\[-\,\theta ,\text{ }90{}^\circ \pm \text{ }\theta ,\]\[180{}^\circ +\text{ }\theta ,\] etc. are angles allied to the angles \[\theta \] where \[\theta \] is measured in degrees.
\[\theta \] \[\sin \theta \] \[\cos \theta \] \[\tan \theta \] \[\text{cosec}\,\theta \] \[\sec \theta \] \[\cot \theta \]
\[-\,\theta \] \[-\,\sin \theta \] \[\cos \theta \] \[-\,\tan \theta \] \[-\,\text{cosec}\,\theta \] \[\sec \theta \] \[-\cot \theta \]
\[90{}^\circ -\,\theta \] more...
Mensuration   We are familiar with some of the basic solids like cuboid/ cone, cylinder and sphere. In this chapter we will discuss about how to find the surface area and volume of these figures. In our daily life, we come across number of solids made up of combinations of two or more of the basic solids.   Surface Area of Solids We may get the solids which may be combinations of cylinder and cone or cylinder and hemisphere or cone and hemisphere and so on. In such cases we find the surface area of each part separately and add them to get the surface area of entire solid.   Cylinder If 'r' is the radius and 'h1 is the height of a cylinder, then Curved surface area of the cylinder = \[2\pi rh\] Total surface area of the cylinder = \[2\pi r(r+h)\]   Cone If 'r' be the radius and 'h' be the height of a cone, then Curved surface area of the cone = \[\pi rl\] Total surface area of the cone = \[\pi r(r+l)\]   Where, I is the slant height of the cone and is given by \[l=\sqrt{{{r}^{2}}+{{h}^{2}}}\]   Sphere If 'r' be the radius of a sphere, then Surface area of the sphere \[4\pi {{r}^{2}}\]   Hemisphere If 'r' be the radius of a hemisphere, then Curved surface area of the hemisphere = \[2\pi {{r}^{2}}\] Total Surface area of the hemisphere =\[3\pi {{r}^{2}}\]                                        Volume of Solids The volume of the combined figures is obtained by finding the volume of each part separately and then adding them together.   Cylinder If 'r' be the radius and 'h' be the height of a cylinder, then Volume of the cylinder = \[\pi {{r}^{2}}h\]   Cone If 'r' be the radius and 'h' be the height of a cone, then Volume of the cone =\[\frac{1}{3}\pi {{r}^{2}}h\]   Sphere If 'r' be the radius of a sphere, then Volume of a the sphere = \[\frac{4}{3}\pi {{r}^{2}}h\]   Hemisphere If 'r' be the radius of a hemisphere, then Volume of the hemisphere = \[\frac{2}{3}\pi {{r}^{2}}h\]  
  •       Example:
A toy is in the form of a cone of radius 77 cm and height 36 cm. Find the area of the cardboard required to make the toy. (a) 18720\[c{{m}^{2}}\]                                               (b) 20570\[c{{m}^{2}}\]    (c) 21426\[c{{m}^{2}}\]                                               (d) 22480\[c{{m}^{2}}\] (e) None of these Ans.     (b) Explanation: Area of the cardboard required = curved surface area of the toy = \[\pi rl\] Here, \[I{{=}^{+}}\sqrt{{{r}^{2}}+{{h}^{2}}}=\sqrt{{{77}^{2}}+{{36}^{2}}}=85\] \[\therefore ~Curve\text{ }surface\text{ }area\text{ }=\text{ }\frac{22}{7}\text{ }\times \text{ }77\text{ }\times \text{ }85\text{ }=\text{ }22\text{ }\times \text{ }11\text{ }\times \text{ }85\text{ }=\text{ }20570\text{ }c{{m}^{2}}\]    

Statistics and Probability   Statistics Statistics is the branch of Mathematics which deals with the collection and interpretation of data. The data may be represented in different graphical forms such as bar graphs, histogram, ogive curve, and pie chart. This representation of data reveals certain salient features of the data. These values of the data are called measure of central tendency. The various measures of central tendencies are mean, median and mode. A measure of central tendency gives us the rough idea of where data points are centered. But in order to make more accurate interpretation of central values of the data, we should also have an idea of how the data are scattered around the measure of central tendency.   Mean Deviation about Mean of an Ungrouped Data Let \[{{x}_{1}},\text{ }{{x}_{2}},\text{ }{{x}_{3}},\text{ }---,\text{ }{{x}_{n}}\]be the n observations, then the mean of the data is given by: \[\overline{x}=\frac{{{x}_{1}}+{{x}_{2}}+{{x}_{3}}+---+{{x}_{n}}}{n}\]\[\Rightarrow \overline{x}=\frac{1}{n}\sum\limits_{k\,=\,1}^{n}{{{X}_{k}}}\] Then the deviation of the data from the mean is given by: \[|{{x}_{1}}-\overline{x}|,|{{x}_{2}}-\overline{x}|,|{{x}_{3}}-\overline{x}|,---|{{x}_{n}}-\overline{x}|\] Now the mean deviation of the data is given by \[\frac{1}{n}\sum\limits_{k=1}^{n}{|{{X}_{k}}-\overline{x}|}\]   Mean Deviation about Mean of a Grouped Data Let \[{{x}_{1}},\text{ }{{x}_{2}},\text{ }{{x}_{3}},\text{ }---,\text{ }{{x}_{n}}\]be the n - observations and \[{{f}_{1}},\text{ }{{f}_{2}},\text{ }{{f}_{3}},---,\text{ }{{f}_{n}}\]be the corresponding frequencies of the data. Then the mean of the data is given by: \[\overline{x}=\frac{{{x}_{1}}{{f}_{1}}+{{x}_{2}}{{f}_{2}}+---+{{x}_{n}}{{f}_{n}}}{{{f}_{1}}+{{f}_{2}}+---+{{f}_{n}}}\] or, \[\overline{x}=\frac{\sum\limits_{k\,=\,1}^{n}{{{X}_{k}}}{{f}_{k}}}{\sum\limits_{k\,=\,1}^{n}{{{f}_{k}}}}\] Then the mean deviation about mean is given by \[\frac{\sum\limits_{k\,=\,1}^{n}{{{f}_{k}}|{{x}_{k}}-\overline{x}|}}{\sum\limits_{k\,=\,1}^{n}{{{f}_{k}}}}\]   Mean Deviation About Median of an Ungrouped Data The median of an ungrouped data is obtained by arranging the data in the ascending order. If the data contains odd number of terms, then the median is \[{{\left( \frac{n+1}{2} \right)}^{th}}\]I term of the data and if the data contains even number of terms, then the median is the average of \[{{\left( \frac{n}{2} \right)}^{th}}\]and\[{{\left( \frac{n}{2}+1 \right)}^{th}}\]terms i.e., \[\frac{{{\left( \frac{n}{2} \right)}^{th}}term+{{\left( \frac{n}{2}+1 \right)}^{th}}term}{2}\].   If M is the median of the data, then mean deviation about M is given by \[\frac{1}{n}\sum\limits_{k\,=\,1}^{n}{|{{x}_{k}}-M}|\] Mean Deviation About Median of a Grouped Data Let \[{{x}_{1}},\text{ }{{x}_{2}},\text{ }{{x}_{3}},\text{ }---,\text{ }{{x}_{n}}\] be the n -observations and \[{{f}_{1}},\text{ }{{f}_{2}},\text{ }{{f}_{3}},---,\text{ }{{f}_{n}}\] be the corresponding frequencies of the data. Then the mean deviation about the median of the data is given by: \[\frac{\sum\limits_{k\,=\,1}^{n}{{{f}_{k}}|{{x}_{k}}-M|}}{\sum\limits_{k\,=\,1}^{n}{{{f}_{k}}}}\] For the grouped data the median can be obtained by \[l+\left( \frac{\frac{N}{2}-C}{f} \right)\times h\] Where,  I = lower limit of the median class N = sum of all frequencies c = cumulative frequency of preceding median class h = class width f = frequency of the median class   Standard Deviation and Variance Standard deviation is the square root of the arithmetic mean of the squares of deviations of the terms from their arithmetic mean and it is denoted by o. The square of standard deviation is called the variance. Thus for simple distribution, \[\sigma =\sqrt{\frac{\sum\limits_{i\,=\,1}^{n}{{{({{x}_{1}}-\overline{x})}^{2}}}}{n}}\]   Note: (i) The standard deviation of any arithmetic progression is \[\sigma =\,\,|d|\sqrt{\frac{{{n}^{2}}-1}{12}}\]where d = common difference and n = number of terms of the A.P. (ii) Coefficient of variation (C.V.) =\[\frac{\sigma }{x}\]\[\times \]100   Probability We have studied about the probability more...

Verbal and Non-verbal Reasoning   In this chapter, we will solve problems related to reasoning and aptitudes as we know that Reasoning and logic skills are an integral part of Mathematics.   Find the Missing Number In such types of problems, we have to choose a missing number (or character) in the figure out of the given options.  
  •                   Example:
Find the missing number in the following figure.             (a) 1728                                                (b) 1331 (c) 729                                                              (d) 512 (e) None of these Ans.     (b) Explanation: Here the pattern is ________ \[{{(18+10+8)}^{\frac{3}{2}}}=216,\] \[{{(15+12+22)}^{\frac{3}{2}}}=343,\] \[{{(57+43+21)}^{\frac{3}{2}}}=1331\]   Direction Sense Problems In such types of problems, we draw a diagram by using the given information. The following diagram shows all the directions in a proper manner.  
  •             Example:
If P is to the north of Q and R is to the west of Q, then in which direction is P with respect to R? (a) North-east                                                     (b) South-west    (c) North                                                               (d) South (e) None of these Ans.     (a)        Explanation: Clearly from the diagram it is clear that P is in north-east direction with respect to R.                          Blood-Relations Problems In such types of problems, we should first be clear on the relations given in the following table: Mother's father \[\to \] Maternal grand father Mother's mother \[\to \] Maternal grand mother Husband of aunt \[\to \]Uncle Wife of maternal uncle \[\to \] Maternal aunt Mother's brother \[\to \] Maternal uncle Mother's sister \[\to \] Maternal aunt Children of maternal uncle/aunt \[\to \]Cousins Father's father \[\to \]Paternal grand father Father's mother \[\to \]Paternal grand mother Father's brother \[\to \] Uncle Father's sister \[\to \] Aunt Children of uncle \[\to \]Cousins  
  •             Example:
Pointing at a photo, A man said, "His father is the only son of my mother". The man has a relation from the person in the photo is of _____ (a) Uncle                                                           (b) Grandfather (c) Father                                                           (d) Cousin (e) None of these Ans.     (c) Explanation:             Clearly the photo belongs to man's son/so the man is the father of the person shown in the photo.   Figure Based Problems In such types of problems, a series of figures is given which proceeds with a certain rule or pattern.  
  •              Example:
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