Linear Equation in Three Variables
We can also use the system of linear equation for solving the linear equation in three variables by mere substitution. In this method we find one of the three variables in terms of other two from any one of the equation and substitute it in the second equation. From the second equation we obtained the second variable in terms of the other and substitute it in the third equation and solve it to get the third variable which on re-substituting in the previous steps we get the other variables. It can also be solved by elimination method.
Solve the following system of linear equation: \[3x-y+4z=3,x+2y-3z=0\, and \,6x+5y=-3.\]
(a) \[x=-\frac{39}{10},y=\frac{47}{10},z=\frac{5}{2}\]
(b) \[x=\frac{39}{10},y=\frac{47}{10},z=\frac{5}{2}\]
(c) \[x=\frac{39}{10},y=\frac{7}{10},z=\frac{25}{2}\]
(d) \[x=-\frac{9}{10},y=\frac{17}{10},z=\frac{25}{2}\]
(e) None of more...
Introduction
As we know that the trigonometry is the branch of mathematics which study about the relationship between angles and its sides. All the trigonometrical ratios defined in the special type of a triangle i.e. Right angled triangle. In this chapter we will discuss about these ratios.
Trigonometrical Ratios
In the given right angle triangle ABC, in which right angle at B. Angle C is\[''\theta ''\] (suppose).
Then the trigonometrical ratios are defined as follows:
\[\sin \theta =\frac{\text{Perpendicual}}{\text{Hypotenuse}}=\frac{AB}{AC}\]
\[\cos \theta =\frac{Base}{\text{Hypotenuse}}=\frac{BC}{AC}\]
\[\tan \theta =\frac{\text{Perpendicular}}{Base}=\frac{AB}{BC}\]
\[\cot \theta =\frac{Base}{\text{Perpendicular}}=\frac{BC}{AB}\]
\[\sec \theta =\frac{\text{Hypotenuse}}{Base}=\frac{AC}{BC}\]
\[co\sec \theta =\frac{\text{Hypotenuse}}{\text{Perpendicular}}=\frac{AC}{AB}\] AB
If we represent perpendicular, base and hypotenuse by P, b and h respectively then the ratios can be written as:
Relationship between Ratios
Trigonometrical Identities
In the adjoining figure triangle DEF is a right angled triangle right angle at D. Then the trigonometrical identities are
1. \[{{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1\]
2. \[{{\sec }^{2}}\theta -{{\tan }^{2}}\theta =1\]
3. \[co{{\sec }^{2}}\theta -{{\cot }^{2}}\theta =1\]
We can also derive different relations between identities in different form
(a) \[{{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1\]
or \[{{\sin }^{2}}\theta =1-{{\cos }^{2}}\theta \]
or \[\sin \theta =\pm \sqrt{1-{{\cos }^{2}}\theta }\]
Similarly
\[{{\cos }^{2}}\theta =1-{{\sin }^{2}}\theta \]
or \[\cos \theta =\pm \sqrt{1-{{\sin }^{2}}\theta }\]
(b) \[{{\sec }^{2}}\theta -{{\tan }^{2}}\theta =1\]
or \[{{\sec }^{2}}\theta =1+{{\tan }^{2}}\theta \]
or \[{{\sec }^{2}}\theta =\sqrt{1+{{\tan }^{2}}\theta }\]
Similarly
\[{{\tan }^{2}}\theta ={{\sec }^{2}}\theta -1\]
or \[\tan \theta =\pm \sqrt{{{\sec }^{2}}\theta -1}\]
(c) \[\cos e{{c}^{2}}\theta -{{\cot }^{2}}\theta =1\]
or \[\cos e{{c}^{2}}\theta =1+{{\cot }^{2}}\theta \]
or \[\cos e{{c}^{2}}\theta more...
Values of Ratios
1. \[\text{sin}\theta \]: It always lies between -1 and 1
i.e. \[\text{-1}\le \text{sin}\theta \le 1\]
2. \[\cos \theta \]: It always lies between -1 and 1
i.e. \[\text{-1}\le \cos \theta \le 1\]
3. \[\tan \theta \]: It can take any value in between \[-\infty \]and\[\infty \].
i.e. \[-\infty \le \tan \theta \le \infty \]
4. \[\cot \theta \]: It takes the any value in between \[-\infty \]and\[\infty \].
i.e. \[-\infty \le \cot \theta \le \infty \]
5. \[\sec \theta \]: It can take any value except the values lies between -1 and 1.
i.e. \[\sec \theta \le -1\] and \[\sec \theta \ge 1\]
6. \[co\sec \theta \]: It can take any value except the values lies between -1 and 1.
i.e. \[co\sec \theta \le -1\] and \[co\sec \theta \ge 1\]
Chart for the Sign of Different Trigonometrical
| Quadrant\[\to \] Ratios \[\downarrow \] | I | II | III | IV |
| \[\sin \theta \] | + | + | more...
 Introduction
Previously we have studied about various types of numbers like natural numbers, whole numbers, integers, fractions and its decimal representation, rational numbers along with its different operations and properties. In this chapter, we will study about a new number system known as REAL NUMBER which includes rational and irrational numbers.
 Representation of Numbers on Number Lines
It is a way to represent numbers on a line with the help of diagram.
Representation of Integers on Number Line
Take a line AB extended infinitely in both direction. Take a point O on it and represent it as zero (0). Mark points on line at equal distances on both sides of O. Equal distances are taken as per our convenience and take it as a unit.
 Decimal Representation of Numbers
There are three different types of decimal representation of numbers.
i. Terminating
ii. Non terminating and Repeating
iii. Non terminating and Non-Repeating
 Terminating Decimal
If the decimal representation of \[\frac{a}{b}\] comes to an end then it is called terminating decimal.
If the prime factor of denominator having 2, 5 or 2 and 5 only then decimal representation of \[\frac{a}{b}\] (which is in the lowest form) must be terminating.
As for example\[2\frac{4}{5}\] is a terminating decimal i.e. \[2\frac{4}{5}=\frac{14}{5}=2.8\]
Check whether \[\frac{39}{24}\] is terminating or non-terminating
Solution:
To convert \[\frac{39}{24}\] into the lowest form, we get \[\frac{39}{24}=\frac{13}{8}\], here denominator of \[\frac{13}{8}\] is 8 whose prime factor is \[~\text{2}\times \text{2}\times \text{2}\] which contains only 2 as a factor.
Therefore, it is a terminating decimal. more...
 Irrational Number
The decimal representation of an irrational number is non-terminating and non-repeating. In other words we can say that non-terminating and non-repeating decimals are called irrational numbers.
(i) 10.0202002000200002................
(ii) The square of any positive integer which is not a perfect square is irrational  are irrational number
(iii)  is an irrational number
 Properties of Irrational Numbers
1. The sum of two irrational numbers may or may not be irrational.
(i) Suppose  ,  then  , which is irrational
(ii) Suppose two irrational numbers  more...
 Real Number
Real numbers are the collection of rational and irrational numbers or In other words we can say that a number whose square is always non-negative, is know as real numbers.
Note:
(i) There are infinite real numbers between any two distinct real numbers.
(ii) 
 Properties of Real Numbers
In this section we will study about addition and multiplication properties of real number.
Addition Properties
(i) The sum of two real numbers is always real.
(ii) For any two real numbers A + B = B + A.
(iii) For any three real numbers A, B and C (A + B) + C = A + (B + C).
(iv) O is the additive identity of real number.
(v) For a real number more...
 Rationalization
The process of making denominator of a irrational number to a rational by multiplying with a suitable number is called rationalization. This process is adopted when the denominator of a given number is irrational. The number by which we multiply the denominator or convert it into rational is called rationalizing factor.
Rationalize the denominator of \[\frac{6}{\sqrt{7}+\sqrt{2}}\].
Solution:
We have:
\[\frac{6}{\sqrt{7}+\sqrt{2}}=\frac{6\times (\sqrt{7}-\sqrt{2})}{(\sqrt{7}+\sqrt{2})(\sqrt{7}-\sqrt{2})}=\frac{6\times (\sqrt{7}-\sqrt{2})}{7-2}\]
Here, \[(\sqrt{7}-\sqrt{2})\] is rationalizing factor.
Therefore, \[\frac{6}{\sqrt{7}+\sqrt{2}}=\frac{6}{5}(\sqrt{7}-\sqrt{2})\]
 Laws of Radicals Let
\[x>0\] be any real number if a and b rational number then
(i) \[({{x}^{a}}\times {{x}^{b}})={{x}^{a+b}}\]
(ii) \[{{({{x}^{a}})}^{b}}={{x}^{ab}}\]
(iii) \[\frac{{{x}^{a}}}{{{x}^{b}}}={{x}^{a-b}}\]
(iv) \[{{x}^{a}}\times {{y}^{a}}={{(xy)}^{a}}\]
Euclid's Division Lemma
Let a and b be any two positive integer. Then, there exist unique integers q and r more...
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