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 these
Answer: (e)
Explanation
We have,
\[3x-y+4z=3-----\left( 1 \right)\]
\[x-2y-3z=-2-----\left( 2 \right)\]
\[6x+5y-5z=-3-----\left( 3 \right)\]
Form equation (1), we have
\[-y=3-3x-4x-----\left( 4 \right)\]
Putting in equation (2) the above value we get,
\[x=\frac{8-13z}{5}-----\left( 5 \right)\]
Putting equation (5) in (4) we get,
\[y=\frac{-9+13z}{5}-----\left( 6 \right)\]
Solve the following system of linear equation: \[5x-7y+z=11,6x-8y-z=15\,and\,3x+2y-6z=7.\]
(a) \[x=2,y=3,z=2\]
(b) \[x=-2,y=5,z=-1\]
(c) \[x=1,y=-1,z=-1\]
(d) \[x=-3,y=1,z=-2\]
(e) None of these
Answer: (c)
Solve the system of the equation: \[6x+y-3z=5,x+3y-2z=5,2x+y+4z=8\]
(a) \[x=1,y=2,z=1\]
(b) \[x=-2,y=5,z=-1\]
(c) \[x=1,y=-1,z=-1\]
(d) \[x=-3,y=1,z=-2\]
(e) None of these
Answer: (a)
Solve the system of equation: \[2y-3z=0,x+3y=-4,3x+4y=3\]
(a) \[x=2,y=3,z=2\]
(b) \[x=5,y=-3,z=-2\]
(c) \[x=4,y=-3,z=-1\]
(d) \[x=-2,y=2,z=-2\]
(e) None of these
Answer: (b)
Find the solution of the system of the equation: \[x+y=8,y+z=10,x+z=12\]
(a) \[x=5,y=3,z=2\]
(b) \[x=5,y=5,z=-1\]
(c) \[x=8,y=-2,z=-1\]
(d) \[x=5,y=3,z=7\]
(e) None of these
Answer: (d)
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
From the above, we conclude that sine of an angle is reciprocal to the cosec of that angle and so - on. 
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 \[\cos e{{c}^{2}}\theta =\pm \sqrt{1+{{\cot }^{2}}\theta }\]
Similarly
\[{{\cot }^{2}}\theta =\cos e{{c}^{2}}\theta -1\]
or \[\cot \theta =\pm \sqrt{\cos e{{c}^{2}}\theta -1}\]
Verification of \[\mathbf{si}{{\mathbf{n}}^{\mathbf{2}}}\mathbf{\theta +co}{{\mathbf{s}}^{\mathbf{2}}}\mathbf{\theta =1}\]
Suppose \[\Delta \text{RST}\] is a right angled triangle in which right angle at R and angle S in formed at\[\theta \].
Here, perpendicular RT represented by "p", base RS represented by "b" and hypotenuse "ST" represented by h.
Then by Pythagoras theorem,
\[{{p}^{2}}+{{b}^{2}}={{h}^{2}}\]
As we know
\[\sin \theta =\frac{p}{h},\] \[\cos \theta =\frac{b}{h}\]
Therefore,
\[{{\sin }^{2}}\theta =\frac{{{p}^{2}}}{{{h}^{2}}}\,\,\text{and}\,\,{{\cos }^{2}}\theta =\frac{{{b}^{2}}}{{{h}^{2}}}\]
\[{{\sin }^{2}}\theta +{{\cos }^{2}}\theta =\frac{{{p}^{2}}}{{{h}^{2}}}+\frac{{{b}^{2}}}{{{h}^{2}}}=\frac{{{p}^{2}}+{{b}^{2}}}{{{h}^{2}}}=\frac{{{h}^{2}}}{{{h}^{2}}}=1\]
Therefore, \[{{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1\]
Similarly we can verify other results.
Values of Different Trigonometrical Ratios in Different Quadrants
Above shows the different quadrants. The following are about the T- Ratios when angles lies in different quadrants
1. When the angle lies in the I quadrant all trigonometrical ratios are positive i.e. The value of \[\text{sin}\theta ,\text{ cos}\theta ,\text{ tan}\theta ,\text{ cot}\theta ,\text{ sec}\theta \]and \[\text{cosec}\theta \] are positive.
2. When the angle\[''\theta ''\] lies in the second quadrant, the value of sine and cosec are positive and other ratios like \[\text{cos}\theta ,\text{ tan}\theta ,\text{ cot}\theta \] and \[\text{sec}\theta \] are negative.
3. When an angle \[''\theta ''\] lies in third quadrant, the value of tan e and cote are positive and other ratios like \[\text{sin}\theta \text{,}\,\text{cos}\theta ,\text{ cosec}\theta \]and \[\sec \theta \] are negative.
4. When an angle \[''\theta ''\] lies in fourth quadrant, the value of \[\text{cos}\theta \text{ sec}\theta \] are positive and other ratios like \[\text{sin}\theta ,\text{ tan}\theta ,\text{ cosec}\theta \] and cote are negative.
From the figure given below you can remember it easily
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 |
| \[\sin \theta \] | + | + | - | - |
| \[\cos \theta \] | + | - | - | + |
| \[\tan \theta \] | + | - | + | - |
| \[\cot \theta \] | + | - | + | - |
| \[\sec \theta \] | + | - | 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.
On the right of O positive integers 1,2,3,4,5 etc. are indicated at the distance of 1 unit, 2 unit, 3 unit ,4 unit ,5 unit etc. respectively.
On the left of O negative integers -1,-2,-3,-4 etc. are indicated at a distance of 1 unit, 2 unit, 3 unit, 4 unit etc. respectively
Representation of Rational Numbers on a Number Line
Take a line 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. Suppose OP=1 unit.
Let the midpoint of OP is Q, .Therefore, \[OQ=\frac{1}{2}\] unit. Now we mark different points on the number line on right as well as left of O taking OQ =1/2 units.
The above given number line shows different rational numbers with denominator as 2.
Important Points Related to Number Line which Represents Rational Number
(i) A particular point on the number line represents a particular rational number.
(ii) A rational number cannot be represented by two or more than two distinct points on a number line.
(iii) There are infinite points between two distinct points on a line, hence, there are infinite rational numbers between two rational numbers.
Method to find Rational Numbers between Two Rational Numbers
(i) Suppose A and B be two rational number then a rational number which is in between A and B is \[\frac{1}{2}(A+B)\].
(ii) Suppose A and B be two rational number in which A < B, then the n rational numbers between A and B are \[(A+x),(A+2x),\] .............. \[(A+nx)\]. Where \[x=\frac{B-A}{n+1}\]  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. \[\frac{13}{8}=1.625\]
Non-terminating and Repeating Decimals
A decimal in which digit or set of digits repeated in a particular fashion is called repeating decimal.
(1) 2.3333333333333..............
(2) 0.123123123123123....... etc.
In the above given two examples we observe that 3 repeats itself in example 1 and 1, 2, 3 repeat itself in example 2.
The above decimal is also written as \[\text{2}.\overline{\text{3}}\] and \[0.\overline{\text{123}}\]
Method of Conversion of Non-terminating and Repeating Decimal into a Fraction
Step 1: Suppose the given decimal as any variable like \[x,\text{y},.......\] etc.
Step 2: Multiply the given decimal with 10 or power of 10 in such a way that only repeating digits remain on the right of the decimal or all non-repeating terms which are on the right come to left of the decimal.
Step 3: Multiply the decimal obtained in step 2 with 10 or powers of 10 in such a way that repeated digit or a set of digit comes to the left of the decimal. i.e. We multiply by 10 if there is only one digit is repeated, multiply by\[\text{1}{{0}^{\text{2}}}\] or 100 if two digits repeated and so on.
Step 4: Now subtract the decimal obtained in step 2 from the decimal obtained in step 3.
Step 5: Solve the equation whatever get in step 4 and the value of variable in simplified form is the required fraction.
Convert the following into fraction.
(i) \[0.\overline{\text{123}}\]
(ii) \[0.\text{23}\overline{\text{41}}\]
Solution: (i) \[0.\overline{\text{123}}\]
Step 1: Suppose \[x=0.\overline{123}\].
Step 2: Here, there is no non-repeating term on the right of the decimal. Therefore, there is no need to multiply by 10 or powers of 10. \[x=0.\text{123123123123}\] .....(i)
Step 3: Now there are three digits on the right of the decimal which are repeated, that is why\[x\] is multiplied by 103 or 1000. \[x=0.\text{123123123123 }.....)\times \text{1}000\] \[\text{1}000x=\text{123}.\text{123123123123}\] .....(ii)
Step 4: \[\begin{align} & \text{1}000x=\text{123}.\text{123123123123}....... \\ & \underline{-\,\,\,\,\,\,\,\,\,x=~\,\,\,\,\,\,0.\text{123123123123}.......} \\ & \,\,\,999x=123 \\ 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  and  then  which is not irrational.
2. The difference of two rational numbers may or may not be irrational
(i)  then  which is irrational
(ii) Suppose , then  , which is not an irrational number
3. The product of two irrational numbers may or may not be irrational. For example the product of and 5 = 44, which is a rational number.
4. The quotient of two irrational numbers may or may not be irrational. For example,  which is not irrational.
5. The sum of an irrational number and a rational number is irrational.
6. The difference of an irrational number and a rational number is irrational.
7. The product of an irrational number and a rational number may or may not be irrational.
8. The quotient of an irrational number and a rational number is irrational.
Representation of Irrational Numbers on Number Lines
Suppose x' ox be a horizontal line and let 0 be the origin. Take OP as 1 unit and draw so that PQ = 1 unit with centre 0 and OQ as radius draw an arc; meeting at A. Then OA = OQ = unit (by Pythagoras theorem)
Similarly diagrams given below shows 
  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 (a), (-A) is the additive inverse.
Multiplication Properties
(i) The product of two real numbers is always a real number.
(ii) For any two real numbers A and B, AB = BA.
(iii) For any three real numbers A, B and C,  .
(iv) 1 is the multiplicative identity for every real number.
(v) For every non-zero real number A,  is the multiplicative inverse.
(vi) For any three real numbers A, Band C   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 such that \[a=\text{bq}+\text{r},\text{ }0\le \text{r}<\text{b}\]
 Which one of the following statements is true for a rational number?
(a) It is in the form of \[\frac{p}{q}\], Where p and q are integers and\[p\ne 0\]
(b) The decimal representation of a rational number is either terminating or non-terminating and non-repeating decimals
(c) There are five rational number between two given rational numbers
(d) Rational numbers are either terminating or non-terminating and repeating decimals
(e) None of these
Answer: (d)
Match the following:
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