10th Class

 QUADRATIC EQUATION   FUNDAMENTALS

• Quadratic equation: An equation of the form \[a{{x}^{2}}+bx+c=0\] where a, b, and c \[\in \] R and \[a\ne 0\]is called a quadratic equation.

  Note: (i) An equation of degree 2 is called a quadratic equation.    (ii) The quadratic equation is of the form\[a{{x}^{2}}+bx+c=0\].  

• Solution or roots of a quadratic equation: If p(x) = 0 is a quadratic equation, then the zeros of the polynomial p(x) are called the solutions or roots of the quadratic equation P(x) = 0.

  Note: (i) Since the degree of a quadratic equation is 2, if has 2 roots or solutions.    (ii) \[x=a\]is the root of \[p\left( x \right)=0,\]if \[p\left( a \right)=0.\]    (iii) Finding the roots of a quadratic equation is called solving the quadratic equation.  

• Methods of solving more...

QUADRATIC INEQUATION   FUNDAMENTALS Quadratic In equations Consider the quadratic equation\[a{{x}^{2}}+bx+c=0.\,\,\,(a\ne 0)\]where a, b, and c are real numbers. The quadratic in equations related to \[a{{x}^{2}}+bx+c=0\]are \[a{{x}^{2}}+bx+c<0\]and \[a{{x}^{2}}+bx+c>0\]. Assume that a > 0 (This assumption is always valid because if a<0, we can always multiply the in equation by (– 1) to get a > 0.) For example, \[-2{{x}^{2}}+3x+2<0\]can be written as \[-2{{x}^{2}}-3x-2>0\] Note: The change in the sign of the inequality, when it is multiplied by (– 1).   Then following cases arise: Case – 1:    If\[{{b}^{2}}-4ac>0\], then the equation \[a{{x}^{2}}+bx+c=0\]has real and unequal roots. Let \[\alpha \]and \[\beta (\alpha <\beta )\]be the roots. Then,       \[\therefore a{{x}^{2}}+bx+c=a(x-\alpha )(x-\beta )\]

If\[x<\alpha \], then \[\left( x-\alpha \right)<0\] and \[(x-\beta )<0\]

\[\therefore a{{x}^{2}}+bx+c>0\]  

If a\[\alpha <x<\beta \], then \[\left( x-\alpha  \right)>0\] and \[(x-\beta \text{)}<0\]

            \[\therefore more...

 LINEAR EQUATION IN TWO VARIABLES   FUNDAMENTALS While solving the problems, in most cases, first we need to frame an equation. Therefore, we will learn how to frame and solve equations sometimes. Framing an equation is more crucial aspect after which solving the equation may be quite easy.   Algebraic Expression Expression of the form, \[3x,(3x+6),(2x-6y),3{{x}^{2}}+3\sqrt[3]{y},\frac{7{{x}^{6}}}{3}\sqrt{y}\] are algebraic expressions. \[3x\] and 6 are the terms of \[\left( 3x+6 \right)\] and 2x and 6y are the terms of\[2x-6y\]. Algebraic expressions are made of numbers, symbols and the basic arithmetical operations. In the term 3x, 3 is the numerical coefficient of x and x is the variable coefficient of 3.   The following step are involved in solving an equation. Step – 1: Always ensure that the unknown quantities are on the LHS and the known quantities or constants on the more...

STATISTICS   INTRODUCTION Data The word ‘data’ means, information in the form of numerical figures or a set of given facts. For example, the percentage of marks scored by 10 students of a class in a test are: 36, 80, 65, 75, 94, 48, 12, 64, 88 and 98.           Row Data Data obtained from direct observation is called raw data, The marks obtained by 100 students in a. monthly test is an. example of raw data or ungrouped. Intact, little can be inferred from this data. However, arranging the marks in ascending order in the above example is a step towards making raw data more meaningful.   Grouped Data                   To present the data in a more meaningful way, we condense the data into convenient number of classes or groups, generally not exceeding 10 and not less than 5. This helps us in perceiving more...

TRIANGLES   FUNDAMENTALS             Similar figures:

• Figures having the same shape (not necessarily the same size) are called similar figures. Same shapes ensure that the corresponding angles are equal and their corresponding sides are proportional.

  Congruent figures:

• Figures having the same shape and the same size are called congruent figures. Here, apart from angles, corresponding sides are also equal

  Similar Triangles:

• Two triangles are said to be similar, if their corresponding angles are equal and corresponding sides are proportional.

e.g., If in \[\Delta \,ABC\]and \[\Delta \,PQR\] \[\angle A=\angle P,\angle B=\angle Q,\angle C=\angle R\] and       \[\frac{AB}{PQ}=\frac{BC}{QR}=\frac{AC}{PR},\]       then, \[\Delta \text{ }ABC\sim \Delta \,PQR;\] where symbol \[\sim \] is read as ‘is similar to’.   more...

CIRCLE   INTRODUCTION FUNDAMENTALS       

• A circle is the locus of points in a plane which are at a fixed distance from a fixed point.
• The fixed point is called the centre of the circle and the fixed distance is the radius of the circle and is denoted as ‘r’.

• In the figure, OR is a radius of the circle ‘r’.
• PQ is a diameter of the circle. OP and OQ are also the radii of the circle.

• PQ = diameter (d) = 2r                                
• The perimeter of the circle is called the circumference of the circle (C).
• The circumference of the circle is n times the diameter, i.e. \[C=\pi d=2\pi r\]
• Interior and exterior points of a circle.

In the figure more...

AREA OF CIRCLE   FUNDAMENTALS Perimeter and Area of a Circle The distance covered by travelling once around a circle is called its perimeter, and in case of a circle, it is usually called its circumference. The circumference of a circle bears a constant ratio with its diameter. This constant ratio is denoted by the Greek letter \[\pi \] (read as ‘pi’), m other words, \[\frac{circumference}{diameter}=\pi \] Or         \[circumference=\pi \times diameter\] \[=\pi \times 2r\] (where r is the radius of the circle) \[=2\pi r\]   The great Indian mathematician Aryabhatta (A. D. 476 – 550) gave an approximate value of \[\pi \] He stated that \[\pi =\frac{62832}{20000}\], which is nearly equal to 3.1416. It is also interesting to note that using an identity of the great mathematical genius Srinivas Ramanujan (1887 – 1920) of India, mathematicians have been able to calculate the value more...

CO-ORDINATE GEOMETRY   FUNDAMENTALS Co – ordinate geometry is a branch of science which establishes relationship between the position r a point in a plane and pair of algebraic numbers, called its co – ordinates.   Cartesian Co – ordinates Let us draw coordinate axes with ‘O’ as origin. In Cartesian co-ordinates, the position of a point P - determined by knowing its horizontal and vertical distance from origin.   Draw PM and PN perpendiculars on OX and OY respectively. OM is called the x co – ordinate or abscissa of the point P. ON is called the y co – ordinate or the ordinate of the point P. The abscissa and ordinate of a point are known as co – ordinates of the point P. If OM = x, ON = y, then the more...

TRIGONOMETRY   Systems of Measurement of an Angle Circular System In this system, the angle is measured in radians.   Radian: The angle subtended by an arc length APB equal to the radius of a circle at its centre is defined of one radian (see figure). It is written as \[{{1}^{c}}\]. (‘c’ denotes radian) Relation between the Units Look at the circle in the above figure and note that, \[360{}^\circ =2{{\pi }^{c}}\Rightarrow 90{}^\circ =\frac{{{\pi }^{c}}}{2}\] and \[45{}^\circ =\frac{{{\pi }^{c}}}{4}\]; Or, simply, \[90{}^\circ =\frac{\pi }{2},45{}^\circ =\frac{\pi }{4}\] For convenience, the above relation can be written as, \[\frac{D}{90}=\frac{R}{\frac{\pi }{2}}\], where, D denotes degrees, and R radians. Remember

\[1{}^\circ =\frac{\pi }{180}\] radian \[=0.0175\] radians (approximately).

\[1{}^\circ =\frac{180}{\pi }\] degrees \[=57{}^\circ 17'44''\] (approximately).

\[30{}^\circ =\frac{\pi }{6},45{}^\circ =\frac{\pi }{4};60{}^\circ =\frac{\pi }{3};90{}^\circ =\frac{\pi }{2};120{}^\circ =\frac{2\pi }{3},180{}^\circ =\pi \]

  Note: If no more...

PROBABILITY   FUNDAMENTAL Probability - A Theoretical Approach We know, in advance, that the coin can only land in one of two possible ways either head up or tail up (we dismiss the possibility of its ‘landing’ on its edge, which may be possible, for example, if it falls on sand). We can reasonably assume that each outcome, head or tail, is as likely to occur as the other. We refer to this by saying that the outcomes head and tail, are equally likely.   Considering another example of equally likely outcomes, suppose we throw a die once. For us, a die will always mean a fair die. What are the possible outcomes? They are 1, 2, 3, 4, 5, 6. Each number has the same possibility of showing up. So, the equally likely outcomes of throwing a die are 1, 2, 3, 4, more...