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 numbers are the numbers which cannot be written in the form\[\frac{p}{q}\], 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 more...

  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\[a{{x}^{2}}+bx+c=0\], \[(a\ne 0)\]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\[a{{x}^{3}}+b{{x}^{2}}+cx+d=0(a\ne 0)\]roots are \[\alpha ,\,\,\beta \]and\[\gamma \], the relation 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\[A({{x}_{1}},\,\,{{y}_{1}})\]and\[B({{x}_{2}},\,\,{{y}_{2}})\] 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\[A({{x}_{1}},{{y}_{1}},{{z}_{1}})\]and\[B({{x}_{2}},{{y}_{2}},{{z}_{2}})\] then the distance between the points is given by: \[AB=\sqrt{{{({{x}_{2}}-{{x}_{1}})}^{2}}+{{({{y}_{2}}-{{y}_{1}})}^{2}}+{{({{z}_{2}}-{{z}_{1}})}^{2}}}\]   Section Formula Let us consider the point 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+ ey=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 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 centre 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 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 radian\[={{\left( \frac{180}{\pi } \right)}^{o}}\]and \[{{1}^{o}}=\left( \frac{\pi }{180} \right)\] 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\[-\text{ }\theta ,\,\,90{}^\circ \pm \theta ,\,\,180{}^\circ \pm \theta \], etc are angles allied to the angles \[\theta \]where \[\theta \]is measured in 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 "h" 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 more...

  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, give 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 \[{{x}_{1}},\,\,{{x}_{2}},\,\,{{x}_{3}},---,\,\,{{x}_{n}}\]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 more...

  Real Numbers   In this chapter we will learn about real numbers. A real number can be any positive or negative numbers. All the rational and irrational numbers are real numbers. In other words we can say that real numbers are the set of rational and irrational numbers.   Important Points Related to Real Numbers

• A rational number is a real number which can be written as a simple fraction (i.e. in a ratio of two integers). In other words, a number r is called a rational number when it can be written in the form \[\frac{p}{q}\] where p and q are integers and q is not equal to zero. For example \[\frac{3}{5}\], 0, 3, \[\frac{1}{100}\] are rational numbers.

 

• The decimal expansion of a rational number is either terminating or non-terminating recurring.

                      For example more...

  Introduction to Trigonometry   As we know, the trigonometry is the branch of Mathematics in which we study about the relationship between angles and its sides. In this chapter, we will discuss about trigonometric ratios which are defined in a right-angled triangle.   Trigonometrical Ratios In the given triangle ABC, \[\angle B=90{}^\circ \]and let angle C is\[\theta \].     Then the trigonometrical ratios are defined as follows:

• \[\sin \theta =\frac{Perpendicular}{Hypotenuse}=\frac{AB}{AC}\]
• \[\cos \theta =\frac{Base}{Hypotenuse}=\frac{BC}{AC}\]
• \[\tan \theta =\frac{Perpendicular}{Base}=\frac{AB}{BC}\]
• \[\cot \theta =\frac{Base}{Perpendicular}=\frac{BC}{AB}\]
• \[\sec \theta =\frac{Hypotenuse}{Base}=\frac{AC}{BC}\]
• \[cosec\theta =\frac{Hypotenuse}{\operatorname{Perpendicular}}=\frac{AC}{AB}\]

  Relationship between T-ratios \[\sin \theta =\frac{1}{\operatorname{cosec}\theta },\,\,\cos \theta =\frac{1}{\sec \theta }\,\,and\,\,\cot \theta =\frac{1}{\tan \theta }\]   From the above, we conclude that sine of an angle is reciprocal to the cosec of that angle and so on.   Tringonometric Ratios of Complementary Angles

• \[\sin more...