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 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
Answer (b)
Explanation: By using Euclid's division lemma, we get \[1260=378\times 3+126\]
Now consider the divisor and remainder.
So again by applying Euclid's division lemma, we get \[378=126\times 3+0\]
Now the remainder is zero, so we stop the process.
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 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 \[y={{x}^{3}}-2{{x}^{2}}\] are:
(a) \[\frac{1}{2}\] 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
Answer(c)
Explanation: Let\[f(x)=6{{x}^{2}}-x-2\]
By the method of splitting the middle term, we have =\[f(x)=6{{x}^{2}}-x-2=6{{x}^{2}}-4x+3x-2\]
\[2x(3x-2)+\text{ }1(3x-2)=(2x+1)(3x-2)\]Zeroes of\[f(x)\] will be obtained by putting
\[2x+1=0\,\,and\,\,3x-2=0\] \[\Rightarrow x=\frac{-1}{2}\,\,and\,\,x=\frac{2}{3}\]
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:
\[\mathbf{Dividend}=\mathbf{Quotient\times Divisor+Remainder}\]
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 \[g(x)={{x}^{3}}-3{{x}^{2}}+3x-5\] by\[=\text{ }h(x)={{x}^{2}}+\text{ }x\text{ }+\text{ }1\] we have,
Here we have quotient\[q(x)=x-4\], and remainder \[r(x)=6x-1.\]
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 a Linear Polynomial
We know that \[ax+b\]is a linear polynomial. So the graph of \[y=ax+b\]is as follows:
Example: Draw the graph of the polynomial given by\[\mathbf{f}(\mathbf{x})=\mathbf{3x}-\mathbf{5}\].
Solution: Let\[y=3x-5\], then the following table will represent the values of x and y which satisfy the given equation.
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 \[P(x,\,\,y)\]which divides the line segment joining \[A({{x}_{1}},\,\,{{y}_{1}})\]and \[B({{x}_{2}},\,\,{{y}_{2}})\] 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 \[A({{x}_{1}},{{y}_{1}})\]and\[B({{x}_{2}},{{y}_{2}})\]is given by \[\left( \frac{{{x}_{1}}+{{x}_{2}}}{2},\frac{{{y}_{1}}+{{y}_{2}}}{2} \right)\]
Note:
(i) If the mid-point of a \[\Delta ABC\] 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 whose three vertices in order are\[({{x}_{1}},\,\,{{y}_{1}}),\,\,({{x}_{2}},\,\,{{y}_{2}})\] and \[R({{x}_{3}},\,\,{{y}_{3}})\]is \[({{x}_{1}}-{{x}_{2}}+{{x}_{3}},\,\,{{y}_{1}}-{{y}_{2}}+{{y}_{3}})\]
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\[({{x}_{1}},\,\,{{y}_{1}}),\,\,({{x}_{2}},\,\,{{y}_{2}})\]and \[({{x}_{3}}-{{y}_{3}})\]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\[A({{x}_{1}},\,\,{{y}_{1}})\], \[B({{x}_{2}},\,\,{{y}_{2}})\]and \[C({{x}_{3}},{{y}_{3}})\]be the vertices of a triangle, then the area of the triangle is given by:
\[=\frac{1}{2}\left| {{x}_{1}}({{y}_{2}}-{{y}_{3}})+{{x}_{2}}({{y}_{3}}-{{y}_{1}})+{{x}_{3}}({{y}_{1}}-{{y}_{2}}) \right|=0\]
Conditions for Collinearity
Let the given points be\[A({{x}_{1}},\,\,{{y}_{1}}),B({{x}_{2}},\,\,{{y}_{2}})\]and \[C({{x}_{3}},{{y}_{3}})\]is: If A, B and C are collinear then, Area of \[\Delta ABC=0.\]
\[=\frac{1}{2}\left| {{x}_{1}}({{y}_{2}}-{{y}_{3}})+{{x}_{2}}({{y}_{3}}-{{y}_{1}})+{{x}_{3}}({{y}_{1}}-{{y}_{2}}) \right|=0\]
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. So, slope of a line \[(m)=\tan \theta ,\]where \[\theta \]is the angle made by the line with positive direction of x-axis.
Note: For any two points\[A({{x}_{1}},{{y}_{1}})\]and \[B({{x}_{2}},{{y}_{2}})\], the slope of the line is \[m=\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}\]
Equation of a Line
General equation of a line is ax + by + c = 0, where a, b and c are constants and x and y are variables. The equation of a line with slope m and making an intercept c on y-axis is y =mx + c.
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 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 \[{{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}=0\] and \[{{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}=0\]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:
Find the relation between m and n for which the system of equations
\[4x+6y=7and(m+n)x+(2m-n)y=21\], has unique solution.
(a) 2m=3n (b) m = 5n
(c)\[2m\ne 3n\] (d) \[m\ne 5n\]
Answer (d)
Explanation:
We have, the system of equations \[4x+6y=7\]and \[(m+n)x+(2m-n)y=21\]
For a unique solution, the required condition is
\[\frac{4}{m+n}\ne \frac{6}{2m-n}\]
\[\Rightarrow 8m-4n\ne 6m+6n\Rightarrow 8m-6m\ne 6n+4n\]\[\Rightarrow 2m\ne 10n\Rightarrow m\ne 5n\]
Quadratic Equation
Quadratic equation is a type of polynomial of degree two. The general form of a quadratic equation is\[a{{x}^{2}}+bx+c=0\], where a, b, c are the constants and\[a\ne 0\]. The quadratic equation which contains both second and first powers of the variable is called a 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 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;
\[\angle OPT=\angle OQT={{90}^{o}},\,\,\angle POT=\angle QOT\]
\[\angle QTO=\angle OTP\] and \[PT=QT\].
Example:
Two tangents PT and QT are drawn to a circle with center 0 from an external point as shown in the following figure, then:
(a)\[\angle QTP=\angle QPO\] (b) \[\angle QTP=2\angle QPO\]
(c)\[\angle QTP=3\angle QPO\] (d) \[\angle QTP=90{}^\circ \]
(e) None of these
Answer (b)
Explanation: In the given figure, we have
TP = TQ [tangents drawn from an external point are equal in length]
\[\Rightarrow \angle TPQ=\angle TQP\]
In\[\Delta TPQ\], 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 degrees.
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
If ‘r’ be the radius and 'h1 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, l 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\[=\frac{4}{3}\pi {{r}^{3}}\]
Hemisphere
If ‘r’ be the radius of a hemisphere, then
Volume of the hemisphere \[=\frac{2}{3}\pi {{r}^{3}}\]
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
Answer (b)
Explanation: Area of the cardboard required
= curved surface area of the toy\[=\pi rl\]
Here, \[l=\sqrt{{{r}^{2}}+{{h}^{2}}}=\sqrt{{{77}^{2}}+{{36}^{2}}}=85\]
\[\therefore \]Curve surface area
\[=\frac{22}{7}\times 77\times 85=22\times 11\times 85=20570c{{m}^{2}}\]
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, 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:
\[\overline{x}=\frac{{{x}_{1}}+{{x}_{2}}+{{x}_{3}}+\,--\,-\,+{{x}_{n}}}{n}\]\[\Rightarrow \overline{x}=\frac{1}{n}\sum\limits_{k=1}^{n}{{{x}_{k}}}\]
Then the deviation of the data from the mean is given by:
\[\left| {{x}_{1}}-\overline{x} \right|,\left| {{x}_{2}}-\overline{x} \right|,\left| {{x}_{3}}-\overline{x} \right|,---,\left| {{x}_{n}}-\overline{x} \right|\]
Now the mean deviation of the data is given by \[\frac{1}{n}\sum\limits_{k=1}^{n}{\left| {{x}_{k}}-\overline{x} \right|}\]
Mean Deviation about Mean of a Grouped Data
Let\[{{x}_{1}},\,\,{{x}_{2}},\,\,{{x}_{3}},---,\,\,{{x}_{n}}\]be the n - observations and \[{{f}_{1}},\,\,{{f}_{2}},\,\,{{f}_{3}},---,\,\,{{f}_{n}}\]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\[\frac{\sum\limits_{k=1}^{n}{{{f}_{k}}\left| {{x}_{k}}-\overline{x} \right|}}{\sum\limits_{k=1}^{n}{{{f}_{k}}}}\]
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 \[{{\left( \frac{n+1}{2} \right)}^{th}}\]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 \[\frac{1}{n}\sum\limits_{k=1}^{n}{\left| {{x}_{k}}-M \right|}\]
Mean Deviation About Median of a Grouped Data
Let be \[{{x}_{1}},\,\,{{x}_{2}},\,\,{{x}_{3}},---,\,\,{{x}_{n}}\]the n-observations and \[{{f}_{1}},\,\,{{f}_{2}},\,\,{{f}_{3}},---,\,\,{{f}_{n}}\] be the corresponding frequencies of the data. Then the mean deviation about the median of the data is given by: \[\frac{\sum\limits_{k=1}^{n}{{{f}_{k}}\left| {{x}_{k}}-M \right|}}{\sum\limits_{k=1}^{n}{{{f}_{k}}}}\]
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\[\alpha \]. The square of standard deviation is called the variance.
Thus for simple distribution, \[\sigma =\sqrt{\frac{\sum\limits_{i=1}^{n}{{{({{x}_{i}}-\overline{x})}^{2}}}}{n}}\]
Note:
(i) The standard deviation of any arithmetic progression is \[\sigma =\left| d \right|\sqrt{\frac{{{n}^{2}}-1}{12}}\]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 as a measure of 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 \[\frac{12}{5}=2.4\] and \[\frac{13}{9}=1.44444\]…, are decimal expansions of rational numbers.
An irrational number is a real number which can not be written as a simple fraction. In other words, a number s is called an irrational number when it can not be written in the form \[\frac{p}{q}\], where p and q are integers and q is not equal to zero.
For example,\[\sqrt{2}\]=1.41421356….,
\[\sqrt{3}=\]1.732058075....,\[\pi =\]3.14159265...... are irrational numbers since they can not be written in the form \[\frac{p}{q}\]
Note:
(i) We use \[\pi =\frac{22}{7}\], which is its approximate value but not accurate.
(ii) The decimal expansion of irrational number is non-terminating non-recurring. For example 1.002000200002 ...... is an irrational number.
For any rational number rand irrational numbers, \[r+s\], \[r-s\]are irrational numbers and for any non zero rational number r and irrational number s, r.s and \[\frac{r}{s}\] are irrational numbers.
When product of two irrational numbers is rational then each one of these factors is called the rationalizing factor of other.
For example,
(i) \[(a+\sqrt{b})\,\,and\,\,(a-\sqrt{b})\]
(ii) \[(a+b\sqrt{m})\,\,and\,\,(a-b\sqrt{m})\]
(iii) \[(\sqrt{m}+\sqrt{n})\,\,and\,\,(\sqrt{m}-\sqrt{n}\]
Are rationalising factors of each other, where a and b are integers and m and n are natural numbers.
For any two rational numbers a and b (when a\[<\]b), \[\frac{a+b}{2}\]is a rational number lying between a and b.
For any two rational numbers a and b (when a \[<\] b), n number of rational numbers
between a and b are,
a + d, a + 2d, a + 3d,…., a + nd, where \[d=\frac{a-b}{n+1}\]
Some Results of Real Number
For all positive real numbers a and b,
(i) \[\sqrt{ab}=\sqrt{a}\times \sqrt{b}\]
(ii)\[\sqrt{\frac{a}{b}}=\sqrt{\frac{a}{b}}\]
(iii) \[(\sqrt{a}+\sqrt{b})\,\,(\sqrt{a}-\sqrt{b})=a-b\]
(iv) \[(a+\sqrt{b)\,\,}(a-\sqrt{b})={{a}^{2}}-b\]
(v) \[{{(\sqrt{a}+\sqrt{b})}^{2}}=a+2\sqrt{ab}+b\]
The Radical Sign and Radicand
A radical expression is an expression of the type \[\sqrt[n]{x}\]. The sign ‘ \[\sqrt[n]{{}}\]’ is called the radical sign the number under this sign ie. ‘x’ is called the radicand and n is called the order of the \[\sqrt{2}\],\[\sqrt{3},\]\[\sqrt{4},\]radical. For example etc. are radicals. Irrational radicals such as etc. \[\sqrt{2}\],\[\sqrt{3},\]\[\sqrt{4},\] are also known as surds.
Laws of Exponents for Real Numbers
If and 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:
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
Trigonometric Identities
The trigonometric ratios of an angle \[\theta \] is said to be a trigonometric identity if it is satisfied for ail values of 9 .In this chapter we will learn about following identities.
Here we have quotient\[q(x)=x-4\], and remainder \[r(x)=6x-1.\]
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 a Linear Polynomial
We know that \[ax+b\]is a linear polynomial. So the graph of \[y=ax+b\]is as follows:
Example: Draw the graph of the polynomial given by\[\mathbf{f}(\mathbf{x})=\mathbf{3x}-\mathbf{5}\].
Solution: Let\[y=3x-5\], then the following table will represent the values of x and y which satisfy the given equation.
\[\angle OPT=\angle OQT={{90}^{o}},\,\,\angle POT=\angle QOT\]
\[\angle QTO=\angle OTP\] and \[PT=QT\].
(a)\[\angle QTP=\angle QPO\] (b) \[\angle QTP=2\angle QPO\]
(c)\[\angle QTP=3\angle QPO\] (d) \[\angle QTP=90{}^\circ \]
(e) None of these
Answer (b)
Explanation: In the given figure, we have
TP = TQ [tangents drawn from an external point are equal in length]
\[\Rightarrow \angle TPQ=\angle TQP\]
In\[\Delta TPQ\], more... 
Then the trigonometrical ratios are defined as follows:
