Basic Trigonometry || Class 10 || by #cmrohityadav #edata

Applying Pythagoras theorem for the given right-angled triangle, we have:

(Perpendicular)2+(Base)2=(Hypotenuese)2

⇒(P)2+(B)2=(H)2

The Trigonometric properties are given below:

S.no	Property	Mathematical value
1	sin A	Perpendicular/Hypotenuse
2	cos A	Base/Hypotenuse
3	tan A	Perpendicular/Base
4	cot A	Base/Perpendicular
5	cosec A	Hypotenuse/Perpendicular
6	sec A	Hypotenuse/Base

Relation Between Trigonometric Identities:

S.no	Identity	Relation
1	tan A	sin A/cos A
2	cot A	cos A/sin A
3	cosec A	1/sin A
4	sec A	1/cos A

Trigonometric Identities:

1. sin2A + cos2A = 1
2. tan2A + 1 = sec2A
3. cot2A + 1 = cosec2A

NUMBER SYSTEM || BASIC MATHS FOR ENTRANCE EXAM || BY #CMROHITYADAV #eData

Introduction

The collection of numbers is called the Number system. These numbers are of different types such as natural numbers, whole numbers, integers, rational numbers and irrational numbers. Let us see the table below to understand with the examples.

Natural Numbers	N	1, 2, 3, 4, 5, ……
Whole Numbers	W	0,1, 2, 3, 4, 5….
Integers	Z	…., -3, -2, -1, 0, 1, 2, 3, …
Rational Numbers	Q	p/q form, where p and q are integers and q is not zero.
Irrational Numbers		Which can’t be represented as rational numbers

Note: every natural number is an integer and 0 is a whole number which is not a whole number.

Natural Numbers

All the numbers starting from 1 till infinity are natural numbers, such as 1,2,3,4,5,6,7,8,…….infinity. These numbers lie on the right side of the number line and are positive.

Whole Numbers

All the numbers starting from 0 till infinity are whole numbers such as 0,1,2,3,4,5,6,7,8,9,…..infinity. These numbers lie on the right side of the number line from 0 and are positive.

Integers

Integers are the whole numbers which can be positive, negative or zero. They cover rational numbers also but not the irrational numbers.

Example: 2, 33, 0, -67, 9.777, are integers.

Rational Numbers

A number which can be represented in the form of p/q is called a rational number. For example, 1/2, 4/5, 26/8, etc.

Irrational Numbers

A number is called an irrational number if it can’t be represented in a p/q form, where p and q are integers.

Example: √3, √5, √11, etc.

Real Numbers

The collection of all rational and irrational numbers is called real numbers. Real numbers are denoted by R.

Every real number is a unique point on the number line and also every point on the number line represents a unique real number.

Difference between Terminating and Recurring Decimals

Terminating Decimals	Repeating Decimals
If the decimal expression of a/b terminates. I.e comes to an end, then the decimal so obtained is called Terminating decimals.	A decimal in which a digit or a set of digits repeats repeatedly periodically is called a repeating decimal.
Example: ¼ =0.25	Example: ⅔ = 0.666…

Some Special Characteristics of Rational Numbers:

• Every Rational number is expressible either as a terminating decimal or as a repeating decimal.
• Every Terminating decimal is a rational number.
• Every repeating decimal is a rational number.

Irrational Numbers

• The non-terminating, non repeating decimals are irrational numbers.

Example: 0.0100100001001…

• Similarly, if m is a positive number which is not a perfect square, then √m is irrational.

Example: √3

• If m is a positive integer which is not a perfect cube, then 3√m is irrational.

Example: 3√2

Properties of Irrational Numbers

• These satisfy the commutative, associative and distributive laws for addition and multiplication.
• Sum of two irrationals need not be irrational.

Example: (2 + √3) + (4 – √3) = 6

• Difference of two irrationals need not be irrational.

Example: (5 + √2) – (3 + √2) = 2

• Product of two irrationals need not be irrational.

Example: √3 x √3 = 3

• The quotient of two irrationals need not be irrational.

2√3/√3 = 2

• Sum of rational and irrational is irrational.
• The difference of a rational number and an irrational number is irrational.
• Product of rational and irrational is irrational.
• Quotient of rational and irrational is irrational.

Real Numbers

A number whose square is non-negative is called a real number.

• Real numbers follow Closure property, associative law, commutative law, the existence of an additive identity, existence of additive inverse for Addition.
• Real numbers follow Closure property, associative law, commutative law, the existence of a multiplicative identity, existence of multiplicative inverse, Distributive laws of multiplication over Addition for Multiplication.

Rationalisation

If we have an irrational number, then the process of converting the denominator to a rational number by multiplying the numerator and denominator by a suitable number, is called rationalisation.

Example:

3/ √2 = 3/ √2 x √2/ √2 = 3 √2/2

Laws of Radicals:

Let a>0 be a real number, and let p and q be rational numbers, then we have:

i) (ap x aq) = a(p+q)

ii) (ap)q = apq

iii)ap /aq = a(p-q)

iv) ap x bp = (ab)p

Example: Simplify (36)½

Solution: (62)½ = 6(2 x ½) = 61 = 6

Visit for more laws of exponent:
Click here

Laws of Exponents || BASIC MATHS FOR ENTRANCE EXAMS || by #cmrohityadav #edata

Laws of Exponents || BASIC #MATHS FOR #ENTRANCE #EXAMS || by #cmrohityadav #edata

Laws of Exponents

The laws of exponents are demonstrated based on the powers they carry.

• Bases – multiplying the like ones – add the exponents and keep base same. (Multiplication Law)
• Bases – raise it with power to another – multiply the exponents and keep base same.
• Bases – dividing the like ones – ‘Numerator Exponent – Denominator Exponent’ and keep base same. (Division Law)

Let ‘a’ is any number or integer (positive or negative) and ‘m’,  ‘n’ are positive integers, denoting the power to the bases, then;

Multiplication Law

As per the multiplication law of exponents, the product of two exponents with the same base and different powers equals to base raised to sum of the two powers or integers.

           am × an  = am+n

Division Law

When two exponents having same bases and different powers are divided, then it results in base raised to difference of the two powers.

          am ÷ an  = am / an  = am-n

Negative Exponent Law

Any base if has a negative power, then it results in reciprocal but with positive power or integer to the base.

            a-m  = 1/am 

Exponents and Powers Rules

The rules of exponents are followed by the laws. Let us have a look on them with a brief explanation.

Suppose ‘a’ & ‘b’ are the integers and ‘m’ & ‘n’ are the values for powers, then the rules for exponents and powers are given by:

i) a0 = 1

As per this rule, if the power of any integer is zero, then the resulted output will be unity or one.

Example: 50 = 1

ii) (am)n = a(mn)

‘a’ raised to the power ‘m’ raised to the power ‘n’ is equal to ‘a’ raised to the power product of ‘m’ and ‘n’.

Example: (52)3 = 52 x 3

iii) am × bm =(ab)m

The product of ‘a’ raised to the power of ‘m’ and ‘b’ raised to the power ‘m’ is equal to the product of ‘a’ and ‘b’ whole raised to the power ‘m’.

Example: 52 × 62 =(5 x 6)2

iv) am/bm = (a/b)m

The division of ‘a’ raised to the power ‘m’ and ‘b’ raised to the power ‘m’ is equal to the division of ‘a’ by ‘b’ whole raised to the power ‘m’.

Example: 52/62 = (5/6)2

Exponents and Power Solved Questions

Example 1: Write 7 x 7 x 7 x 7 x 7 x 7 x 7 x 7 in exponent form.

Solution:

In this problem 7s are written 8 times, so the problem can be rewritten as an exponent of 8.

7 x 7 x 7 x 7 x 7 x 7 x 7 x 7 = 78.

Example 2: Write below problems like exponents:

1. 3 x 3 x 3 x 3 x 3 x 3
2. 7 x 7 x 7 x 7 x 7
3. 10 x 10 x 10 x 10 x 10 x 10 x 10

Solution: 

1. 3 x 3 x 3 x 3 x 3 x 3 = 36
2. 7 x 7 x 7 x 7 x 7 = 75
3. 10 x 10 x 10 x 10 x 10 x 10 x 10 = 107 

Example 3: Simplify 253/53 

Solution: 

 Using Law: am/bm = (a/b)m

253/53  can be written as (25/5)3  = 53 = 125.

Exponents and Powers Applications

Scientific notation uses the power of ten expressed as exponents, so we need a little background before we can jump in. In this concept, we round out your knowledge of exponents, which we studied in previous classes.

The distance between the Sun and the Earth is 149,600,000 kilometers. The mass of the Sun is 1,989,000,000,000,000,000,000,000,000,000 kilograms. The age of the Earth is 4,550,000,000 years. These numbers are way too large or small to memorize in this way.  With the help of exponents and powers these huge numbers can be reduced to a very compact form and can be easily expressed in powers of 10.

Now, coming back to the examples we mentioned above, we can express the distance between the Sun and the Earth with the help of exponents and powers as following:

Distance between the Sun and the Earth 149,600,000 = 1.496× 10 × 10 × 10 × 10 × 10× 10 × 10 = 1.496× 108 kilometers.

Mass of the Sun: 1,989,000,000,000,000,000,000,000,000,000 kilograms = 1.989 × 1030 kilograms.

Age of the Earth:  4,550,000,000 years = 4. 55× 109 years

Some Very Very Standard Algebraic Identities for class 5 to PG ( #class #10 ) by #cmrohityadav #eData

Some Standard Algebraic Identities list are given below:

 1:  (a + b)2 = a2 + 2ab + b2

 2: (a – b)2 = a2 – 2ab + b2

 3: a2 – b2= (a + b)(a – b)

4: (a + b)3 = a3 + b3 + 3ab (a + b)

 5: (a – b)3 = a3 – b3 – 3ab (a – b)

 6: x3 + y3= (x + y) (x2 – xy + y2)

7: x3 – y3 = (x – y) (x2 + xy + y2)

 8: (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca

Similarly

• (x + y – z)2 = x2 + y2 + z2 + 2xy – 2yz – 2xz
• (x – y + z)2 = x2 + y2 + z2 – 2xy – 2yz + 2xz
• (x – y – z)2 = x2 + y2 + z2 – 2xy + 2yz – 2xz

 9: a3 + b3 + c3 – 3abc = (a + b + c)(a2 + b2 + c2 – ab – bc – ca)

 8: (x + a)(x + b) = x2 + (a + b) x + ab

    Similarly

• (x + a)(x – b) = x2 + (a – b)x – ab
• (x – a)(x + b) = x2 + (b – a)x – ab
• (x – a)(x – b) = x2 – (a + b)x + ab

Ad

Revision Notes on Unit and Dimensions by cmrohityadav edata

        

Revision Notes on Unit and Dimensions

In order to make the measurement of a physical quantity we have, first of all, to evolve a standard for that measurement so that different measurements of same physical quantity can be expressed relative to each other. That standard is called a unit of that physical quantity.

System of Units:-


(a) C.G.S (Centimeter-Grand-Second) system.

(b) F.P.S. (Foot-Pound-Second) system.

(c) M.K.S. (Meter-Kilogram--Second) system.

(d) M.K.S.A. (Meter-Kilogram-Second-Ampere) unit.

Dimensional Formula:-


Dimensional formula of a physical quantity is the formula which tells us how and which of the fundamental units have been used for the measurement of that quantity.

How to write dimensions of physical quantities:-


(a) Write the formula for that quantity, with the quantity on L.H.S. of the equation.

(b) Convert all the quantities on R.H.S. into the fundamental quantities mass, length and time.

(c) Substitute M, L and T for mass, length and time respectively.

(d) Collect terms of M,L and T and find their resultant powers (a,b,c) which give the dimensions of the quantity in mass, length and time respectively.

Characteristics of Dimensions:-


(a) Dimensions of a physical quantity are independent of the system of units.

(b) Quantities having similar dimensions can be added to or subtracted from each other.

(c) Dimensions of a physical quantity can be obtained from its units and vice-versa.

(d) Two different physical quantities may have same dimensions.

(e) Multiplication/division of dimensions of two physical quantities (may be same or different) results in production of dimensions of a third quantity.



PHYSICAL QUANTITY

SYMBOL

DIMENSION

MEASUREMENT UNIT

UNIT


Length

s

L

Meter

m


Mass

M 

M

Kilogram

Kg


Time

t

T

Second

Sec


Electric charge

q

Q

Coulomb

C


luminous intensity

I

C

Candela

Cd


Temperature

T

K

Kelvin

oK


Angle



q

none

Radian

None


Mechanical Physical Quantities (derived)




PHYSICAL QUANTITY

SYMBOL

DIMENSION

MEASUREMENT  UNIT

UNIT













Area

A

L2

square meter

m2


Volume



V

L3

cubic meter 

m3


velocity



v

L/T

meter per second

m/sec


angular velocity



w

T-1

radians per second

1/sec


acceleration



a

LT-2

meter per square second

m/sec2


angular acceleration

a

T-2

radians per square

second 

1/sec2


Force



F

MLT-2

 Newton

Kg m/sec2


Energy



E

ML2T-2

 Joule

Kg m2/sec2


Work



W

       

ML2T-2

Joule

Kg m2/sec2


Heat



Q

     

ML2T-2

Joule

Kg m2/sec2


Torque



t

ML2T-2

Newton meter

Kg m2/sec2


Power



P

ML2T-3

watt  or  joule/sec

Kg m2/sec3


Density



D or ρ

ML-3

kilogram per

cubic meter

Kg/m3


pressure



P

    ML-1T-2

Newton per square meter

Kg m-1/sec2


impulse



J

MLT-1

Newton second

Kg m/sec


Inertia



I

ML2

Kilogram square meter



Kg m2 




luminous 

flux

f



C

lumen (4Pi candle for point source)

cd sr 


illumination



E

CL-2

lumen per

square meter

cd sr/m2


entropy



S

       ML2T-2K-1

joule per degree

Kg m2/sec2K


Volume

rate of flow

Q

L3T-1

cubic meter

per second

m3/sec


kinematic

viscosity

n

L2T-1

square meter

m2/sec


per second




dynamic

viscosity

m



      ML-1T-1



Newton second

per square meter

Kg/m sec        




specific

weight

g

  ML-2T-2

Newton

per cubic meter

Kg m-2/sec2


Electrical Physical Quantities (derived)


Electric

current

I

QT-1

Ampere

C/sec


emf, voltage,

potential

E

ML2T-2Q-1

Volt

Kg m2/sec2C


resistance or 

impedance 

R



ML2T-1Q-2



ohm



Kgm2 /secC2




Electric 

conductivity

s

M-2L-2TQ2

mho 



secC2/Kg m3 




capacitance



C

M-1L-2T2 Q2

Farad

sec2C2/Kgm2


inductance



L

ML2Q-2

Henry

Kg m2 /C2


Current density



J



QT-1L-2



ampere per

square meter

C/sec m2




Charge density

r

QL-3

coulomb per cubic meter

C/m3


magnetic flux,

Magnetic induction

B

MT-1Q-1

weber per

square meter

Kg/sec C


magnetic

intensity

H

QL-1T-1

ampere per meter

C/m sec


magnetic vector

potential

A

MLT-1Q-1

weber/meter

Kg m/sec C


Electric

field intensity

E

MLT-2Q-1

volt/meter or

newton/coulomb

Kg m/sec2 C


Electric displacement

D

QL-2

coulomb per square meter

C/m2


permeability



m

MLQ-2

henry per meter

Kg m/C2


permittivity,

e

T2Q2M-1L-3

farad per meter

sec2C2/Kgm3




dielectric constant

                     

K

    M0L0T0

 None

 None


frequency



f or n

T-1

Hertz

sec-1


angular frequency

W

T-1

radians per second         

sec-1




Wave length



l

L

Meters

M







Principle of homogeneity:-



It states that “ the dimensional formulae of every term on the two sides of a correct relation must be same.”


Types of error:-



(a) Constant errors:- An error is said to be constant error if it affects, every time, a measurement in a similar manner.

(b) Systematic errors:- Errors which come into existence by virtue of a definite rule, are called systematic errors.

(c) Random error or accidental error:- Error which takes place in a random manner and cannot be associated with a systematic cause are called random or accidental errors.

(d) Absolute error:-    

Relative Error:-


 

Percentage Error:-