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Real numbers Mcqs

 

 

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Sem – 2 Concept – Engineering Mathematics

Sem – 2 Concept – Engineering Mathematics

Sem – 2 Concept

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Engineering Maths  Sem – 2 Concept solutions in PDF Files are designed by the ‘Basics In Maths” Team. These Pdf Files are very useful for students who are prepared for polytechnic examinations.

TS 6th Class Maths Concept


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TS Inter Practice Papers 2021

TS Inter Maths 1A &1B Practice papers (Reduced Syllabus)

TS Inter Maths 1A &1B Practice papers (Reduced Syllabus)

 

TS Inter Maths 1A and 1B Practice papers as per reduced syllabus were designed by the ‘Basics in Maths‘ team.

These Practice papers to do help the intermediate First-year Maths students.

TS Inter Maths 1A and 1B Practice papers as per reduced syllabus are very useful in IPE examinations.

 


 

MATHS 1A PRACTICE PAPER – 1

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TS Inter Maths 1A and 1B Practice papers as per reduced syllabus were designed by the ‘Basics in Maths‘ team.

These Practice papers to do help the intermediate First-year Maths students.

TS Inter Maths 1A and 1B Practice papers as per   syllabus are very useful in IPE examinations.


 

 

MATHS 1B PRACTICE PAPER – 1

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PDF Files

PDF Files || Inter Maths 1A &1B || (New)

Inter Maths 1A &1B|| PDF Files (New)

 

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Inter Mathematics 1A ands 1B Pdf Files|  these   Files were designed by the ‘Basics in Maths‘ team. These notes to do help the intermediate First-year Maths students.

Inter Maths – 1A & 1B   PDF Files are very useful in IPE examinations.

These notes cover all the topics covered in the intermediate First-year Maths syllabus and include plenty of solutions to help you solve all the major types of Math problems asked in the

IPE examinations.  

 


MATHEMATICS 1A

 

Addition Of Vectors SAQ’S

Matrices

 

TS 10th class maths concept (E/M)

 

 

MATHEMATICS 1B 

DC’s and Dr’s

Tangents and Normals

Maxima and Minima

 

 

 

TS 6th Class Maths Concept


 

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Trigonometric Ratios(Qns.& Ans) V.S.A.Q.’S

Trigonometric Ratios(Qns.& Ans) V.S.A.Q.’S designed by the ‘Basics in Maths‘ team.These notes to do help intermediate First-year Maths students.

Inter Maths – 1A two marks questions and solutions are very useful in I.P.E examinations.


Trigonometric Ratios Up to Transformations

 Question 1

Find the value of sin2(π/10) + sin2(4π/10) + sin2(6π/10) + sin2(9π/10)

Sol:

 sin2(π/10) + sin2(4π/10) + sin2(6π/10) + sin2(9π/10)

= sin2(π/10) + sin2(π/2 – π/10) + sin2(π/2+ π/10) + sin2(π – π/10)

= sin2(π/10) + cos2(π/10) + cos2(π/10) + sin2(π/10)

= 1 + 1 = 2

 Question 2

If sin θ = 4/5 and θ not in the first quadrant, find the value of cos θ

Sol:

Given sin θ = 4/5 and θ not in the first quadrant

⇒ θ in the second quadrant

⇒ cos θ < 0

    cos2θ = 1 – sin2 θ

              =1 – (4/5)2

             = 1 – 16/25

∴cos θ   = – 3/5 (∵cos θ < 0)

 Question 3

If 3sin θ + 4 cos θ = 5, then find the value of 4 sin θ – 3cos θ

Sol:

Given, 3sin θ + 4 cos θ = 5

let 4 sin θ – 3cos θ = x

  (3sin θ + 4 cos θ )2 + (4 sin θ – 3cos θ)2 = 52 + x2

 9 sin2 θ + 16 cos2 θ + 12 sin θ cos θ + 16 sin2 θ + 9 cos2 θ – 12sin θ cis θ = 25 + x2

25 sin2 θ + 25 cos2 θ = 25 + x2

25 = 25 + x2

⇒ x2 = 0

 x = 0

∴ 4 sin θ – 3cos θ = 0

 Question 4

If sec θ + tan θ =Trigonometry up to Transformations 1, find the value of sin θ and determine the quadrant in which θ lies

Sol:

Given, sec θ + tan θ =  ———— (1)

 We know that sec2 θ – tan2 θ = 1

⇒ (sec θ + tan θ) (sec θ – tan θ) = 1

 sec θ – tan θ =Trigonometry up to Transformations 2

⇒ sec θ – tan θ = Trigonometry up to Transformations 3———— (2)

 (1) + (2)

⇒ (sec θ + tan θ) + (sec θ – tan θ) = Trigonometry up to Transformations 4

2sec θ =Trigonometry up to Transformations 5

 sec θ =Trigonometry up to Transformations 6

(1) – (2)

⇒ (sec θ + tan θ) – (sec θ – tan θ) = Trigonometry up to Transformations 7

2 tan θ = Trigonometry up to Transformations 8 ⇒ tan θ =Trigonometry up to Transformations 9

Now sin θ = tan θ ÷ sec θ =Trigonometry up to Transformatio

     Sin θ =Trigonometry up to Transformations 11

Since sec θ positive and tan θ is negative θ lies in the 4th quadrant.

 Question 5

Prove that cot (π/16). cot (2π/16). cot (3π/16).… cot (7π/16) = 1

Sol:

cot (π/16). cot (2π/16). cot (3π/16).… cot (7π/16)

= cot (π/16). cot (2π/16). cot (3π/16). cot (4π/16). cot (5π/16) cot (6π/16) cot (7π/16)

= cot (π/16). cot (2π/16). cot (3π/16). cot (π/4). cot (π/2 – 3π/16) cot (π/2 – 2π/16) cot (π/2 – π/16)

= cot (π/16). cot (2π/16). cot (3π/16). cot (π/4). tan (3π/16) tan (2π/16) tan (π/16)

= [cot (π/16). tan (π/16)] [cot (2π/16). tan (2π/16)] [cot (3π/16). tan (3π/16]. cot (π/4)

= 1.1.1.1

 =1

 

 Question 6

If cos θ + sin θ = Trigonometry up to Transformations 12cos θ, then prove that cos θ – sin θ =  sin θ

Sol:

Given, cos θ + sin θ = Trigonometry up to Transformations 12cos θ

Sin θ = Trigonometry up to Transformations 12 cos θ – cos θ

           = ( Trigonometry up to Transformations 12 – 1) cos θ

( Trigonometry up to Transformations 12 + 1) sin θ = ( Trigonometry up to Transformations 12 + 1) ( Trigonometry up to Transformations 12 – 1) cos θ

Trigonometry up to Transformations 12 sin θ + sin θ = cos θ

∴ cos θ – sin θ = Trigonometry up to Transformations 12 sin θ

 Question 7

Find the value of 2(sin6 θ + cos6 θ) – 3 (sin4 θ + cos4 θ)

Sol:

2(sin6 θ + cos6 θ) – 3 (sin4 θ + cos4 θ)

= 2[(sin2 θ)3 + (cos2 θ)3] – 3[(sin2 θ)2 + (cos2)2

= 2[(sin2 θ + cos2 θ)3 – 3 sin2 θ cos2 θ (sin2 θ + cos2 θ)] – 3[(sin2 θ + cos2 θ)2 – 2 sin2 θ cos2 θ]

= 2[1 – 3 sin2 θ cos2 θ] – 3 [1 – 2 sin2 θ cos2 θ]

= 2 – 6 sin2 θ cos2 θ – 3 + 6 sin2 θ cos2 θ

= – 1

 Question 8

If tan 200 = λ, then show thatTrigonometry up to Transformations 13    

Sol:

Given tan 200 = λ

Trigonometry up to Transformations 19 =Trigonometry up to Transformations 14

                               =Trigonometry up to Transformations 15

                              =Trigonometry up to Transformations 16

                              =Trigonometry up to Transformations 18

 Question 9

If sin α + cosec α = 2, find the value of sinn α + cosecn α, n∈ Z

Sol:

Given sin α + cosec α = 2

 ⇒ sin α + 1/ sin α = 2

 ⇒Trigonometry up to Transformations 17  = 2

      sin2 α + 1= 2 sin α

       sin2 α – 2 sin α + 1= 0

  (sin α – 1 )2 = 0

⇒ sin α – 1 = 0

sin α = 1 ⇒ cosec α = 1

 sinn α + cosecn α = (1)n + (1)n =1 + 1 =2

∴ sinn α + cosecn α = 2

 Question 10

Evaluate sin2 Trigonometry up to Transformations 20+ cos2 Trigonometry up to Transformations 21  – tan2   Trigonometry up to Transformations 22

Sol:

 Trigonometry up to Transformations 23

Trigonometry up to Transformations 24

 

 Question 11

Find the value of sin 3300. cos 1200 + cos 2100. Sin 3000

Sol:

 sin 3300. cos 1200 + cos 2100. Sin 3000

=sin (3600 – 300). cos (1800 – 600) + cos (1800 + 300). sin (3600 – 600)

= (– sin 300). (– cos 600) + (– cos300). (– sin600)

= sin 300.  cos 600 + cos300.  Sin600

= sin (600 + 300) = sin 900

=1

 Question 12

Prove that cos4 α + 2 cos2 α Trigonometry up to Transformations 25= (1 – sin4 α)

Sol:

cos4 α + 2 cos2 α Trigonometry up to Transformations 25

= cos4 α + 2 cos2 α (1 – cos2 α)

= (cos2 α)2 + 2 (1 – sin2 α) (sin2 α)

= (1 – sin2 α)2 + 2 sin2 α – 2sin4 α

= 1 + sin4 α – 2 sin2 α + 2 sin2 α – 2sin4 α

= 1 – sin4 α

 Question 13

Eliminate θ from x = a cos3 θ and y = b sin3 θ

Sol:

Given x = a cos3 θ and y = b sin3 θ

 cos3 θ = x/a and sin3 θ = y/b

 cos θ = (x/a)1/3 and sin θ = (y/b)1/3

we know that sin2 θ + cos2 θ = 1

 ⇒ [(y/b)1/3]2 + [(x/a)1/3]2 = 1

   (x/a)2/3 + (y/b)2/3 = 1

 Question 14

Find the period of the following functions

Sol:

(i) f(x) = tan 5x

we know that period of tan kx =Trigonometry up to Transformations 26

⇒ period of tan 5x =Trigonometry up to Transformations 27

(ii) f(x) =Trigonometry up to Transformations 28

we know that period of Trigonometry up to Transformations 29= Trigonometry up to Transformations 31

period of Trigonometry up to Transformations 28 = Trigonometry up to Transformations 32

                                      =Trigonometry up to Transformations 33

(iii) f(x) = 2 sinTrigonometry up to Transformations 34 + 2 cosTrigonometry up to Transformations 35

period of sin Trigonometry up to Transformations 34 = Trigonometry up to Transformations 36 = 8

period of cos Trigonometry up to Transformations 35 = Trigonometry up to Transformations 37 = 6

period of given function is = LCM (8, 6) = 24

 

(iv) f(x) = tan (x + 4x + 9x +…. + n2x)

f(x) = tan (x + 4x + 9x +…. + n2x)

       = tan (1 + 4 + 9 + … + n2) x

= tanTrigonometry up to Transformations 38x

we know that period of tan kx =Trigonometry up to Transformations 26

Period of tan  =Trigonometry up to Transformations 39

                        = Trigonometry up to Transformations 40

 Question 15

Prove that sin2(52 ½)0 – sin2 (22 ½)0 =Trigonometry up to Transformations 41

Sol:

We know that sin2 A – sin2B = sin (A +B) sin (A – B)

 ⇒ sin2(52 ½)0 – sin2 (22 ½)0

= sin (52 ½+ 22 ½) sin (52 ½ – 22 ½)

 = sin 750 sin 300

 = Trigonometry up to Transformations 42      

∴ sin2(52 ½)0 – sin2 (22 ½)0 =Trigonometry up to Transformations 41

 

 Question 16

Prove that tan 700 – tan200 = 2 tan 500

Sol:

500 = 700 – 200

Tan 500 = tan (700 – 200)

We know that tan (A –B) =Trigonometry up to Transformations 43

  ⇒ Tan 500 =Trigonometry up to Transformations 44

 ⇒ tan 700 – tan 200 = tan 500 (1 + tan700 tan 200)

     tan 700 – tan 200 = tan 500 [1 + tan700 cot (900 – 200)]

     tan 700 – tan 200 = tan 500 [1 + tan700 cot 700]

     tan 700 – tan 200 = tan 500 [1 + 1]

∴ tan 700 – tan200 = 2 tan 500

 Question 17 

If sin α = Trigonometry up to Transformations 45, sin β =Trigonometry up to Transformations 46  and α, β are acute, show that α + β =Trigonometry up to Transformations 47

Sol:

Given sin α =    Trigonometry up to Transformations 45                                           sin β =Trigonometry up to Transformations 46

Trigonometry up to Transformations 48

 tan α = 1/3                                                          tan β = ½

tan (α + β) =Trigonometry up to Transformations 49

      Trigonometry up to Transformations 50 

  tan (α + β) = 1

∴ α + β =Trigonometry up to Transformations 47

 Question 18

Find tanTrigonometry up to Transformations 51 in terms of tan A

Sol:

 tan Trigonometry up to Transformations 51 =Trigonometry up to Transformations 52

                        =Trigonometry up to Transformations 53

 Question 19

Prove thatTrigonometry up to Transformations 54 = cot 360

Sol:

Trigonometry up to Transformations 54  = Trigonometry up to Transformations 55

(on dividing numerator and denominator by cos 90)

      = Trigonometry up to Transformations 56

   = tan (450 + 90)

    = tan 540

 = tan (900 – 360)

 = cot 360

 ∴Trigonometry up to Transformations 54  = cot 360      

 Question 20

Show that cos 420 + cos 780 + cos 1620 = 0

Sol:

cos 420 + cos 780 + cos 1620

= cos (600 – 180) + cos (600 + 180) + cos (1800 – 180)

=cos 600 cos180 + sin 600 sin 180 + cos 600 cos 180 – sin 600 sin 180 – cos 180

= 2 cos 600 cos 180 – cos 180

= 2 (1/2) cos 180 – cos 180

 = cos 180 – cos 180

= 0

 

 Question 21

Express Trigonometry up to Transformations 57sin θ + cos θ as a single of an angle

Sol:

Trigonometry up to Transformations 57sin θ + cos θ = 2(Trigonometry up to Transformations 58  sin θ + Trigonometry up to Transformations 59cos θ)

                                = 2(cos 300 sin θ + sin 300 cos θ)

                                = 2 sin (θ + 300)

 Question 22

Find the maximum and minimum value of the following functions

(i) 3 sin x –4 cos x

a= 3, b = –4 and c = 0

Trigonometry up to Transformations 60  

                                     = 5

∴ minimum value = –5 and maximum value = 5

(ii) cos (x + ) + 2  sin (x + ) – 3

a= 1, b = 2  and c = – 3

Trigonometry up to Transformations 61

 ∴ minimum value = –6 and maximum value = 0

 Question 23

Find the range of the function f(x) = 7 cos x – 24sin x + 5

Sol:

Given f(x) = 7 cos x – 24sin x + 5

a= 7, b = –24 and c = 5

Trigonometry up to Transformations 62

∴ Range = [–20, 30]    

 Question 24

Prove that sin2α + cos2 (α + β) + 2 sin α sin β cos (α + β) is independent of α

Sol:

sin2α + cos2 (α + β) + 2 sin α sin β cos (α + β)

= sin2α + cos (α + β) [ cos (α + β) +2 sin α sin β]

= sin2α + cos (α + β) [ cos α cos β – sin α sin β +2 sin α sin β]

=sin2α + cos (α + β) [ cos α cos β + sin α sin β]

=sin2α + cos (α + β) cos (α –β)

= sin2 α + cos2 α – sin2 β

=1 – sin2 β

= cos2 β

 Question 25

Simplify Trigonometry up to Transformations 63

Sol:

Trigonometry up to Transformations 63   =Trigonometry up to Transformations 64

                 =Trigonometry up to Transformations 65

                 = tan θ

 

Question 26

For what values of x in the first quadrantTrigonometry up to Transformations 66 is positive?

Sol:

Trigonometry up to Transformations 66 > 0 ⟹ tan 2x > 0

⟹ 0 < 2x < π/2 (∵ x is in first quadrant)

⟹ 0 < x < π/4

Question 27

If cos θ = Trigonometry up to Transformations 67 and π < θ < 3π/2, find the value of tan θ/2.

Sol:

cos θ = Trigonometry up to Transformations 67

π < θ < 3π/2 ⟹ π/2 < θ/2 < 3π/4

tan θ/2 < 0

tan θ/2 =Trigonometry up to Transformations 68

               =– Trigonometry up to Transformations 69 (tan θ/2 < 0)

              =–Trigonometry up to Transformations 70

           = – 2

Question 28

If A is not an integral multiple of π/2, prove that cot A – tan A = 2 cot 2A.

Sol:

cot A – tan A = Trigonometry up to Transformations 71

                         =Trigonometry up to Transformations 72

                          =Trigonometry up to Transformations 73

                          =Trigonometry up to Transformations 74

                           =Trigonometry up to Transformations 75

                           = 2 cot 2A

Question 29

Evaluate 6 sin 200 – 8sin3 200

Sol:

6 sin 200 – 8sin3 200 = 2 (3 sin 200 – 4sin3 200)

                                       = 2 sin 3(200)

                              = 2 sin 600

                              = 2Trigonometry up to Transformations 76 

                              =Trigonometry up to Transformations 77

Question 30

Express cos6 A + sin6 A in terms of sin 2A.

Sol:

cos6 A + sin6 A

= (sin2 A)3 + (cos2 A)3

= (sin2 A + cos2 A)3 – 3 sin2 A cos2 A (sin2 A + cos2 A)

= 1 – 3 sin2 A cos2 A

=1 – ¾ (4 sin2 A cos2 A)

 = 1 – ¾ sin22 A

 

Question 31

If 0 < θ < π/8, show that Trigonometry up to Transformations 79  = 2 cos (θ/2)

Sol:

Trigonometry up to Transformations 79

Trigonometry up to Transformations 80

Trigonometry up to Transformations 81

 =2 cos (θ/2)

Question 32

Find the extreme values of cos 2x + cos2x

Sol:

cos 2x + cos2x = 2cos2 x– 1 + cos2 x

                              =3cos2 x – 1

We know that – 1 ≤ cos x ≤ 1

 ⟹ 0 ≤ cos2 x ≤ 1

      3×0 ≤ 3×cos2 x ≤ 3×1

      0– 1 ≤3 cos2 x – 1≤ 3– 1

   – 1≤3 cos2 x – 1≤ 2

     Minimum value = – 1

     Maximum value = 2

Question 33

Prove that Trigonometry up to Transformations 82 = 4

Sol:

Trigonometry up to Transformations 82

Trigonometry up to Transformations 83

= 4

Question 34

Prove that sin 780 + cos 1320 =Trigonometry up to Transformations 84

Sol:

 sin 780 + cos 1320 = sin 780 + cos (900 + 420)

                                      = sin 780 – sin 420

                                      = 2 cosTrigonometry up to Transformations 85  sinTrigonometry up to Transformations 86

                                     = 2 cos 600 sin 180

                                      = 2Trigonometry up to Transformations 87

                                      =Trigonometry up to Transformations 84

Question 35

Find the value of sin 340 + cos 640 – cos40

Sol:

sin 340 + cos 640 – cos40

= sin 340 –2 sinTrigonometry up to Transformations 88  sinTrigonometry up to Transformations 89

= sin 340 – 2sin 340 sin 300

= sin 340 – 2 sin 340 (1/2)

=sin 340 – sin 340

=0

 

Question 36

Prove that 4(cos 660 + sin 840) =Trigonometry up to Transformations 90

Sol:

4(cos 660 + sin 840)  

=4(cos 660 + sin (900 – 60)  

=4(cos 660 + cos (60)  

= 4[ 2 cosTrigonometry up to Transformations 91  cos Trigonometry up to Transformations 92]

=4[ 2 cosTrigonometry up to Transformations 93  cosTrigonometry up to Transformations 94 ]

=8 cos 360 cos 300

= 8Trigonometry up to Transformations 95

=Trigonometry up to Transformations 90

Question 35

Prove that (tan θ + cot θ)2 = sec2 θ + cosec2 θ = sec2 θ. cosec2 θ

Sol:

tan θ + cot θ =    Trigonometry up to Transformations 96

                         =Trigonometry up to Transformations 97

                         =Trigonometry up to Transformations 98

                         = sec θ. cosec θ

(tan θ + cot θ)2   = sec2 θ. cosec2 θ

sec2 θ + cosec2 θ =Trigonometry up to Transformations 99

                               Trigonometry up to Transformations 100

                              = sec2 θ. cosec2 θ


 

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TS VII CLASS MATHS CONCEPT FEATURE IMAGE

TS 7th Class Maths Concept

7th Class Maths Concept

Studying maths in VII class successfully means that children take responsibility for their learning and learn to apply the concepts to solve problems.

These concepts were designed by the ‘Basic in Maths’ team. These notes to do help students fall in love with mathematics and overcome fear.


 1. INTEGERS

 Natural numbers: All the counting numbers starting from 1 are called Natural numbers.

           1, 2, 3… Etc.

 Whole numbers: Whole numbers are the collection of natural numbers including zero.

             0, 1, 2, 3 …

 Integers: integers are the collection of whole numbers and negative numbers.

….,-3, -2, -1, 0, 1, 2, 3,…..

 Integers on a number line:

integers

 Operations on integers:

 addition of integers:

3 + 4 = 7integers addition

-2 + 4 = 2

addion of integers

  Subtraction of integers on a number line:-

6 – 3 = 3subtraction of integers

  Multiplication of integers on a number line:-

2 × 3 ( 2 times of 3) = 6multiplication on number line

           3 × (- 4 ) ( 3 times of -4) = -12

multiplication of itegers

 Multiplication of two negative integers:

  • To multiply two negative integers, first, we multiply them as whole numbers and put plus sign before the result.
  • The multiplication of two negative integers is always negative.

Ex:- -3 × -2 = 6,  -10 × -2 = 20 and so on.

 Multiplication of more than two negative integers:

• If we multiply three negative integers, then the result will be a negative integer.

Ex:- -3 ×   -4 ×   -5 = -60,  -1× -7 × -4 = -28 and so on.

• If we multiply four negative integers, then the result will be a positive integer.

Ex:- -3 ×   -4 ×  -5 × -2  = 120,  -1× -7 × -4  × -2 = 56 and so on.

  Note:-

 1. If the no. of negative integers is even, then the result will be positive.

  2. If the no. of negative integers is odd, then the result will be negative.

multiplication rule

 Division of integers:

  • The division is the inverse of multiplication.
  • When we divide a negative integer by a positive integer or a positive integer by a negative integer, we divide them as whole numbers then put negative signs for the quotient.

Ex:- -3 ÷ 1 = 3, 4 ÷ -2 = -2 and so on.

When we divide a negative integer by a negative integer, we get a positive number as the quotient.

Ex:- -3 ÷ -1 = 3, -4 ÷ -2 = 2 and so on.

       Properties of integers:

      1.Closure property:-

closure

  2.commutative property:-

commutative property

3.associative property:-

associative property

Additive identity:-

1 + 0 = 0 + 1 = 1,   10 + 0 = 0 + 10 = 10

•For any integer ‘a’, a + 0 = 0 + a

•0 is the additive identity.

Additive inverse:-

2 + (-2) = (-2) + 2 = 0,  5 + (-5) = (-5) + 5 = 0

•For any integer ‘a’, a+ (-a) = (-a) + a = 0

•Additive inverse of a = -a and additive inverse of (-a) = a

Multiplicative identity:-

2 × 1 = 1 × 2 = 2,    5 × 1 = 1 × 5 = 5

•For any integer ‘a’, a × 1 = 1 × a = a

•1 is the multiplicative identity.

multiplicative inverse:-

For any integer ‘a’, 1/a × a = a × 1/a = 1

  • multiplicative inverse of a = 1/a
  • Multiplicative inverse of  1/a = a.

distributive property:-

For any three integers a, b and c,    a × (b + c) = (a × b) + (a × c).

3 × (2 + 4) = 18

(3 × 2) + (3 × 4) = 6 + 12 = 18

∴ 3 × (2 + 4) = (3 × 2) + (3 × 4).


2. FRACTIONS, DECIMALS AND RATIONAL NUMBERS

Fraction: A fraction is a number that represents a part of the whole. A group of objects is divided into equal parts, then each part is called a fraction.

 The proper and improper fractions:

In a proper fraction, the numerator is less than the denominator.

Ex: – 1/5, 2/3, and so on.

In an improper fraction, the numerator is greater than the denominator.

Ex: – 5/2,11/5 and so on.

Comparing fractions:

Like fractions: – We have to compare the like fractions with the numerator only because the like fractions have the same denominator. The fraction with the greater numerator is greater and the fraction with the smaller numerator is smaller.

Ex: ,  TS VII maths Fraction, Rational numbers 1  and so on

Unlike fractions: –

With the same numerator: For comparing unlike fractions, we have to compare denominators when the numerator is the same. The fraction with a greater denominator is smaller and the fraction with a smaller denominator is smaller.

Ex: –   TS VII maths Fraction, Rational numbers 2  and so on.

Note: – To find the equivalent fractions of both the fractions with the same denominator, we have to take the LCM of their denominators.

Addition of fractions:

∗ Like Fractions:

vii math addition of fracrtions

∗ Unlike fractions:

vii maths additon of unlike fractions

Subtraction of fractions:

∗ Like fractions:

TS VII maths Fraction, Rational numbers 6

Ex: TS VII maths Fraction, Rational numbers 7

Unlike fractions: – First, we have to find the equivalent fraction of given fractions and then subtract them as like fractions

Ex:  CodeCogsEqn-3

Multiplication of fractions:

Multiplication of fraction by a whole number: –

Multiplication of numbers means adding repeatedly.

Ex: – CodeCogsEqn (2)

multiplying a fraction with whole number

• To multiply a whole number with a proper or improper fraction, we multiply the whole number with the numerator of the fraction, keeping the denominator the same.

2.Multiplication of fraction with a fraction: –

multiplication of two fractions = CodeCogsEqn (3)

Division of fractions:

Ex: – 2 ÷CodeCogsEqn (5) ts vii math division of whole number by a fraction

⇒ 6 one-thirds in two wholes

TS VII maths Fraction, Rational numbers 8

Reciprocal of fraction: reciprocal of a fractionTS VII maths Fraction, Rational numbers 9 is  TS VII maths Fraction, Rational numbers 10 .

Note:

  • dividing by a fraction is equal to multiplying the number by its reciprocal.
  • For dividing a number by mixed fraction, first, convert the mixed fraction into an improper fraction and then solve it.

 Ex: TS VII maths Fraction, Rational numbers 11

1.Division of a whole number by a fraction: –

TS VII maths Fraction, Rational numbers 12

2.Division of a fraction by another fraction: –

TS VII maths Fraction, Rational numbers 13

   Decimal number or fractional decimal:

In a decimal number, a dot(.) or a decimal point separates the whole part of the number from the fractional part.

The part right side of the decimal point is called the decimal part of the number as it represents a part of 1. The part left to the decimal point is called the integral part of the number.

Note: –

  • while adding or subtracting decimal numbers, the digits in the same places must be added or subtracted.
  • While writing the numbers one below the other, the decimal points must become one below the other. Decimal places are made equal by placing zeroes on the right side of the decimal numbers.
Comparison of decimal numbers:

while comparing decimal numbers, first we compare the integral parts. If the integral parts are the same, then compare the decimal part.

Ex: – which is bigger: 13.5 or 14.5

Ans: 14.5

Which is bigger: 13.53 or 13. 25

Ans: 13.53

Multiplication of decimal numbers:

For example, we multiply 0.1 × 0.1For example, we multiply 0.1 × 0.1

TS VII maths Fraction, Rational numbers 14

Multiplication of decimal numbers by 10, 100, and 1000: –

ts vii math multiplication of decimal by 10 100 1000

Here, we notice that the decimal point in the product shifts to the right side by as many zeroes as in 10, 100, and 1000.

Division of decimal number:

Division of decimal number by 10,100 and 1000: –

ts vii division of decimal number by 10 100 1000

Here, we notice that the decimal point in the product shifts to the left side by as many zeroes as in 10, 100, and 1000.

Rational numbers:

 The numbers which are written in the form of p/q, where p, q are integers, and q ≠ 0, are called rational numbers.

Rational numbers are a bigger collection of integers, negative fractional numbers, positive fractional numbers.

Ex: – 1, 2, -1/2, 0 etc.


3. SIMPLE EQUATIONS

Equation: Equation is the condition of a variable. It says that two expressions are equal.

  • An equation has two sides LHS and RHS, on both sides of the equality of sign.
  • One of the expressions of the equation is must have a variable.
  • If we interchange the expressions from LHS to RHS, the equation remains the same

Ex: – x + 2 = 5; 2 = x + 3

Balanced equation:

In an equation, if LHS =RHS, then that equation is balanced.

If the same number is added or subtracted on both sides of the balanced equation, the equation remains will the same.

Ex: 8 + 3 = 11

If add 2 on both sides ⇒ LHS = 8 + 3 + 2 = 13

RHS = 11 + 2 = 13

∴ LHS = RHS

8 +3 = 11 if subtract 2 on both sides

LHS = 8 + 3 – 2 = 9

RHS = 11 – 2 = 9

∴ LHS = RHS

ts vii maths tansposing rules

Using algebraic equations in solving day to day problems:
  1. Read the problem carefully.
  2. Denote the unknown or quantity to be found with some letters such as x, y, z …etc.
  3. Write the problem in the form of an algebraic equation by making a relation among the quantities.
  4. Solve the equation.
  5. Check the solution

4. LINES AND ANGLES

Complimentary angles: When the sum of the angles is 900, the angles are called complementary angles.

Ex: 300, 600; 200, 700 and soon.ts vii math complementary angleSupplementary angles: When the sum of the angles is 1800, the angles are called  Supplementary angles.

Ex: 1200, 600; 1100, 700 and soon.

ts vii math supplimentary angles

Adjacent angles: The angle having a common Arm and a common vertex are called Adjacent angles.

TS VII MATH ADJACENT ANGLES

⇒ ∠AOC and ∠BOC adjacent angles.

Vertically opposite angle: If two lines are intersecting at a point, then the angles that are formed opposite to each other at that point are called vertically opposite angles.

TS VII MATH VERTICALLY OPPOSITE ANGLES
Transversal: A line that intersects two or more lines at distinct points is called a transversal.

ts vii math transversal

Angles made by a transversal:

Corresponding angles: 

Two angles that lie on the same side of the transversal and one interior and another one exterior are called corresponding angles.

∠1, ∠5; ∠2, ∠6; ∠3, ∠7 and ∠4, ∠8ts vii math angles made by transversal

Alternate angles: 

Two angles which are the lies opposite side of the transversal and both interior or exterior are called Alternate angles.

∠1, ∠7; ∠2, ∠8 are exterior alternate angles

∠3, ∠5; ∠4, ∠6 are interior alternate angles.

∠3, ∠6; ∠4, ∠5 interior angles same side of the transversal.

Transversal on parallel lines:

If pair of parallel lines are intersected by a transversal then the angles of each pair of corresponding angles are equal

⇒ ∠1, =∠5; ∠2= ∠6; ∠3= ∠7 and ∠4= ∠8ts vii math transversal on parallel lines

•If pair of parallel lines are intersected by a transversal then the angles of each pair of interior alternate angles are equal.

∠3= ∠5; ∠4= ∠6

•If pair of parallel lines are intersected by a transversal then the angles of each pair of exterior alternate angles are equal.

∠1= ∠7; ∠2= ∠8

•If pair of parallel lines are intersected by a transversal then the angles of each pair of interior angles on the same side of the transversal are supplementary.

∠3+∠6= 1800; ∠4+ ∠5 = 1800

Note:

1.If a transversal intersects two lines and the pair of corresponding angles are equal, then the lines are parallel.

2.If a transversal intersects two lines and the pair of alternate angles are equal, then the lines are parallel.

3.If a transversal intersects two lines and the interior angles on the same side of the transversal are supplementary, then the lines are parallel.


5. TRIANGLE AND ITS PROPERTIES


Triangle:
 A closed figure formed by three-line segments is called a triangle.

   In ∆ABC,

  • ts vii math triangleABCThree sides are TS VII maths Triangles 1
  • Three angles are ∠ABC, ∠BCA, ∠ACB
  • Three vertices are A, B, C.

Classification of triangles:

Triangles can be classified according to the properties of their sides and angles.

According to sides:

Based on sides triangles are three types:

  • Scanlan triangle (ii) Isosceles Triangle (iii) equilateral triangle

According to angles:

  • Acute-angled triangle (ii) Right-angled triangle (iii) Obtuse-angled triangle

TS VII MATH CLASSIFICATION OF TRIANGLES

Relationship between the sides of a triangle:

  1. The sum of the lengths of any two sides of a triangle is greater than the third side.

                 ts vii math sum of two sides is grater than the third side

  1. The difference between the lengths of any two sides of a triangle is less than the third side.

            TS vii maths difference of two sides is less than the third side

The altitude of a triangle:

ts vii maths altitude of a triangle
We can draw three altitudes in a triangle.
A perpendicular line drawn from a vertex to its opposite side of a triangle is called the Altitude of the triangle.

Median of a triangle:

ts vii math median of a triangle
In a triangle, a line drawn from the vertex to the mid-point of its opposite side is called the median of the triangle.                                                                                   

Medians of a triangle are concurrent. We can draw three medians in a triangle.ts vii math centroid of triangle

The point of concurrence of medians is called the centroid of the triangle. It is denoted by G

Angle-sum property of a triangle:
Some of the angles in a triangle is 1800

∠A + ∠B + ∠C = 1800

An exterior angle of a triangle:

When one side of the triangle is produced, the angle thus formed is called an exterior triangle.

Exterior angle property:- The exterior angle of a triangle is equal to the sum of two interior opposite angles. ts vii maths exterior angle sum property

x0+ y0 = z0

 


6.RATIO – APPLICATIONS

Ratio: Comparison of two quantities of the same kind is called ‘Ratio.

The ratio is represented by the symbol ‘:’

If the ratio of two quantities ‘a’ and ‘b’ is a : b, then we read this as ‘a is to b’

The quantities ‘a’ and ‘b’ are called terms of the ratio.

Proportion: if two ratios are equal, then they are said to be proportional.
‘a’ is called as first term or antecedent and ‘b’ is called a second term or consequent.

If a: b = c : d, then a, b, c, d are in proportion and TS VII Maths Ratios 1  ⇒ ad = bc.

TS VII Maths Ratios 2

The product of means = the product of extremes

Unitary method:  The method in which we first find the value of one unit and then the value of the required no. of units is known as the unitary method.

Direct proportion: In two quantities, when one quantity increase(decreases) the other quantity also increases(decreases) then two quantities are in direct proportion.

Percentages:

‘per cent’ means for a hundred or per every hundred. The symbol % is used to denote the percentage.

1% means 1 out of 100, 17% means 17 out of 100.

Profit and Loss:  

Selling price = SP; Cost price = CP

If SP > CP, then we get profit

Profit = SP – CP

Profit percentage =

SP = CP + profit

If SP < CP, then we get a loss

Loss = CP – SP

Loss percentage = TS VII Maths Ratios 5

SP = CP – Loss 

Simple interest:

Principle: – The money borrowed or lent out for a certain period is called the Principle.

Interest: – The extra money, for keeping the principle paid by the borrower is called interest.

Amount: – The amount that is paid back is equal to the sum of the borrowed principal and the interest.

Amount = principle + interest

Interest (I) = TS VII Maths Ratios 4  where R is the rate of interest.


7.DATA HANDLING

Data: The information which is in the form of numbers or words and helps in taking decisions or drawing conclusions is called data.

Observations: The numerical entries in the data are called observations.

Arithmetic Mean: The average data is also called an Arithmetic mean.

Arithmetic Mean (A.M) =TS VII Maths Data Handling 1

The arithmetic mean always lies between the highest and lowest observations of the data.

When all the values of the data set are increased or decreased by a certain number, the mean also increases or decreases by the same number.

Mode: The most frequently occurring observation in data is called Mode.

If data has two modes, then it is called bimodal data.

Note: If each observation in a data is repeated an equal no. of times, then the data has no mode.

Median: The middlemost observation in data is called the Median.

Arrange given data in ascending or descending order.

If a data has an odd no. of observations, then the middle observation is the median.

If a data has even no. of observations, then the median is the average of middle observations.

Bar graph:

Bar graphs are made up of uniform width which can be drawn horizontally or vertically with equal spacing between them.

The length of each bar tells us the frequency of the particular item.

Ex:

TS VII Maths Data Handling 2Double bar graph:

It represents two observations side by side.

Ex:

TS VII Maths Data Handling 3

 Pie chart: A circle can be divided into sectors to represent the given data

The angle of each sector =  TS VII Maths Data Handling 4

Ex:

BudgetAmount in rupees
Food1200
Education800
Others2000
Savings5000
Total income9000

S VII Maths Data Handling 5


8.CONGRUENCY OF TRIANGLES

Congruent figures: Two figures are said to be congruent if they have the same shape and size.

TS VII Maths Congruency of Triangles 1

Congruency of line segments:  If two-line segments have the same length, then they are congruent. Conversely, if two-line segments are congruent, then they have the same length.

TS VII Maths Congruency of Triangles 2

 Congruency of Triangles:

Two triangles are said to be congruent if (i) their corresponding angles are equal (ii) their corresponding sides are equal.
Ex: In ∆ ABC, ∆ DEF TS VII Maths Congruency of Triangles 3

∠A ≅ ∠D; ∠B≅ ∠E; ∠C ≅ ∠F

AB ≅ DE; BC≅ EF; AC ≅ DF

∴∆ABC ≅ ∆DEF

The criterion for congruency of Triangles:

1.Side -Side -Side congruency (SSS): –      
If three side of a triangle is equal to the corresponding three sides of another triangle, then the triangles are congruent.

TS VII Maths Congruency of Triangles 4

∴∆ABC ≅ ∆DEF

2.Side -Angle -Side congruency (SAS): –    
If two sides and the angle included between the two sides of a triangle are equal to the corresponding two sides and the included angle of another triangle, then the triangles are congruent.

TS IX Maths Triangles 6

  ∴ ∆ABC ≅ ∆DEF

3.Angle – Side -Angle congruency (ASA): –       
If two angles and included side of a triangle are equal to the corresponding two angles and included side of another triangle, then the triangles are congruent.

TS VII Maths Congruency of Triangles 6

∴ ∆ABC ≅ ∆DEF

4.Right angle – Hypotenuse – Side congruence (RHS): –

 If the hypotenuse and one side of a right-angled triangle are equal to the corresponding hypotenuse and side of the other right-angled triangle, then the triangles are Equal.

TS VII Maths Congruency of Triangles 7

∴∆ABC ≅ ∆DEF


9.CONSTRUCTION OF TRIANGLES

The no. of measurements required to construct a triangle = 3

A triangle can be drawn in any of the situations given below:

  • Three sides of a triangle
  • Two sides and the angle included between them.
  • Two angles and the side included between them.
  • The hypotenuse and one adjacent side of the right-angled triangle.

Construction of a triangle when measurements of the three sides are given:

Ex: construct a triangle ABC with sides AB = 4cm, BC = 7cm and AC = 5cm

Step of constructions:

Step -1: Draw a rough sketch of the triangle and label it with the given measurements.

TS VII Maths Construction of Triangles 1

Step -2: Draw a line segment of BC of length 7cm.

TS VII Maths Construction of Triangles 2

Step -3: with centre B, draw an arc of radius 4cm, draw another arc from C with radius 5cm such that it intersects first at A.

Step -4: join A, B and A, C. The required triangle ABC is constructed.

TS VII Maths Construction of Triangles 3

Construction of a triangle when two sides and the included angle given:

EX: construct a triangle ABC with sides AB = 4cm, BC = 6cm and ∠B=600

Step of constructions:

Step -1: Draw a rough sketch of the triangle and label it with the given measurements.

TS VII Maths Construction of Triangles 4

Step -2: Draw a line segment of AB of length 4cm.

TS VII Maths Construction of Triangles 6

Step -3: draw a ray BX making an angle 600 with AB.

TS VII Maths Construction of Triangles 5 new

 

Step -4: draw an arc of radius 5cm from B, which cuts ray BX at C.

TS VII Maths Construction of Triangles 7

Step -5: join C and A, we get the required ∆ABC.

TS VII Maths Construction of Triangles 8

Construction of a triangle when two angles and the side between the angles given:

Ex: construct a triangle PQR with sides QR = 4cm, ∠Q= 1200 and ∠R= 400

Step of constructions:

Step -1: Draw a rough sketch of a triangle and label it with the given measurements.

TS VII Maths Construction of Triangles 23

Step -2: Draw a line segment QR of length 4 cm.

TS VII Maths Construction of Triangles 10

Step -3: Draw a ray RX, making an angle 400 with QR.

TS VII Maths Construction of Triangles 11

Step -4: Draw a ray QY, making an angle 1000 with QR, which intersects ray RX.

Step -5: Mark the intersecting point of the two rays as P. Required triangle PQR is constructed.

TS VII Maths Construction of Triangles 12

Construction of a triangle when two sides and the non-included angles are given:

Ex: construct a triangle MAN with sides MN = 4cm, AM = 3cm and ∠A= 400

Step of constructions:

Step -1: Draw a rough sketch of a triangle and label it with the given measurements.

TS VII Maths Construction of Triangles 13

Step -2: Draw a line segment MA of length 0f 5cm.

TS VII Maths Construction of Triangles 14

Step -3: Draw a ray AX making an angle 400 with the line segment MA.

Step -4: With M as the centre and radius 3 cm draw an arc to cut ray AX. Mark the intersecting point as N.

TS VII Maths Construction of Triangles 16

Step -5: join M, N, then we get the required triangle MAN.

TS VII Maths Construction of Triangles 17

Construction of a right-angled triangle when hypotenuse and sides are given:

Ex: construct a triangle ABC, right angle at B and AB = 4cm, Ac = 5cm  

Step of constructions:
Step -1: Draw a rough sketch of a triangle and label it with the given measurements.

TS VII Maths Construction of Triangles 18

Step -2: Draw a line segment BC of length 0f 4cm.

TS VII Maths Construction of Triangles 19

Step -3: Draw a ray BX perpendicular to BC at B

Step -4: Draw an arc from C with a radius of 5cm to intersect ray BX at A.

TS VII Maths Construction of Triangles 20

Step -5: Join A, C, then we get the required triangle ABC.

TS VII Maths Construction of Triangles 21

 


10.ALGEBRAIC EXPRESSIONS

Variable: It is a dependent term. It takes different value.

Ex: m, x, a, etc.

Constant:  It is an independent term. It has a fixed value.

Ex: 1, 3,TS VII Maths Algebraic Expressions 1  etc.

Like terms and Unlike terms: If the terms contain the same variable with the same exponents, then they are like terms otherwise, unlike terms.

Ex: 3x, –4x, x are like terms

3x, 4y, 4 are unlike terms

Coefficient: Coefficient is a term which the multiple of another term (s)

EX: In 5x. 5 is the coefficient of x and x is the coefficient of 5

5 is a numerical coefficient

x is the literal coefficient

Expression: An expression is a single term or a combination of terms connected by the symbols ‘+’ (plus) or ‘−’ (minus).

Ex: 2x – 3. 3x, 2 +3 – 4 etc.

Numerical Expressions:  If every term of an expression is constant, then the expression is called numerical expression.

Ex: 2 + 3 + 5, 2 – 4 – 7, 1 + 5 – 4 etc.

Algebraic expression:  If an expression at least one algebraic term, then the expression is called an algebraic expression.

Ex: x + y, xy, x – 3, 4x + 2 etc.

Note: Plus (+) and Minus (−) separate the terms

Multiplication (×) and Division (÷) do not separate the terms.

Types of Algebraic expressions:

Monomial: – If an expression has only one term, then it is called a monomial.

Ex: 2x2, 3y, x, y, xyz etc.

Binomial: If an expression has two unlike terms, then it is called binomial.

Ex: 2x+ 3y, x2+ y, x +yz2 etc.

Trinomial: If an expression has three unlike terms, then it is called trinomial.  Ex: 2x+ 3y + 4xy, x2+ y + z, x2 y +yz2 + xy2 etc.

Multinomial: If an expression has more than three unlike terms, then it is called multinomial.

Ex: 2x+ 3y + 4xy +5, x2+ y + z – 4y + 6 ,

x2 y +yz2 + xy2 – 4xy + 8yz etc.

Degree of a monomial: The sum of all exponents of the variables present in a monomial is called the degree of the monomial. 

  Ex: Degree of 5xy3  

An exponent of x is 1 and an exponent of y is 3 

Sum of exponents = 1 + 3 = 4

∴ degree of 5xy3 is 4

Degree of an Algebraic Expression: The highest exponent of all the terms of an expression is called the degree of an Algebraic expression.

      Ex: degree of x2 + 3x + 4x3 is 3

degree of 3xy + 6x2y + 5x2y2 is 4

Addition of like terms:

The sum of two or more like terms is a like term with a numerical coefficient that is equal to the sum of the numerical coefficients of all the like terms in addition.

Ex: 3x + 2x = (3 + 2) x = 5x

4x2y + x2y = (4 + 1) x2y = 5x2y

Subtraction of like terms:  

The difference of two like terms is a like term with a numerical coefficient is equal to the difference between the numerical coefficients of the two like terms.

Ex: 3x − 2x = (3 − 2) x = x

4x2y −2 x2y = (4 −2) x2y = 2x2y

Note: (i) addition and subtraction are not done for unlike terms. (ii) If no terms of an expression are alike then it is said to be in the simplified form.

The standard form of an Expression:

In an expression, if the terms are in such a way that the degree of the terms is in descending order, then the expression is said to be in standard form.

Ex: 5 – 2x2 + 4x +3x3

Standard form is 3x3 – 2x2 + 4x + 5

Finding the value of an expression:

Example: find the value of expression x3 + y + 3, when x = 1 and y = 2

Sol: given expression is x3 + y + 3

Substitute x = 1 and y = 2 in above expression

(1)3 + 2 + 3 = 1 + 2 + 3 = 6

Addition of algebraic expressions:

The addition of expressions can be obtained by adding like terms.

This is in two ways: (i) Column or Vertical method (ii) Row or Horizontal method.

Column or Vertical method:

Step –1: Write the expression in standard form if necessary.

Step –2: write one expression below the other such that the like terms come in the same column.

Step –3: Add the like terms column-wise and write the result just below the concerned column.

Ex: Add x2 + 3x + 5, 3 – 2x + 3x2 and 3x – 2

Sol:         TS VII Maths Algebraic Expressions 2        

Row or Horizontal method.

Step –1: Write the expression in standard form if necessary.

Step –2:  Re-arrange them term by grouping the like terms.

Step –3: Simplify the coefficients.

Step –4: Write the resultant expression in standard form.

   Ex: Add x2 + 3x + 5, 3 – 2x + 3x2 and 3x –2

Sol: (x2 + 3x + 5) + (3 – 2x + 3x2) + (3x –2)

= (x2 + 3x2) + (3x – 2x + 3x) + (5 + 3 – 2)

= (1 + 3) x2 + (3 – 2 + 3) x + 6

=4x2 + 4x + 6

Additive inverse of an expression:

For every algebraic expression there exist another algebraic expression such that their sum is zero. These two expressions are called the additive inverse of each other.

Subtraction of algebraic expressions:

This is in two ways: (i) Column or Vertical method (ii) Row or Horizontal method.

Column or Vertical method:

Step –1: Write the expression in standard form if necessary.

Step –2: write one expression below the other such that the expression to be subtracted comes in the second row and the like terms come one below the other.

Step –3: Change the sign of every term of the expression in the second row to get the additive inverse of the expression.

Step –4: Add the like terms column-wise and write the result just below the concerned column.

Ex: Subtract: x2 + 3x + 5 from 3x2 + 4x – 3

Sol:

 TS VII Maths Algebraic Expressions 3

 

 

Row or Horizontal method:

Step –1: Write the expressions in one row with the expression to be subtracted in a bracket with assigning a negative sign to it.

Step –2:  Add the additive inverse of the second expression to the first expression.

Step –3: Group the like terms and add or subtract.

Step –4: Write the resultant expression in standard form.

 Ex: Subtract: x2 + 3x + 5 from 3x2 + 4x – 3

Sol:   3x2 + 4x – 3 – (x2 + 3x + 5)

= 3x2 + 4x – 3 – x2 – 3x – 5

= (3 – 1) x2 + (4 – 3) x + (– 3 – 5)

= 2x2 + x – 8

 


11.EXPONENTS

We know that,

a × a = a2 (a raised to the power of 2)

a × a × a = a3 (a raised to the power of 3)

a × a × a × a × a × a ×…. m times = am

am is in exponential form

a is called base, m is called exponent or index.

Laws of exponents:

  • am × an = am + n
  • TS VII Maths Algebraic Expressions 1
  • (am)n = amn
  • am = an ⇒ m = n
  • (ab)m = am.an
  • a0 = 1
  • TS VII Maths Algebraic Expressions 2
  • TS VII Maths Algebraic Expressions 3

Standard form:  A number that is expressed as the product of the largest integer exponent of 10 and a decimal number between 1 and 10 is said to be in standard form.

Ex: 1324 in standard form is 1.324 × 103.

 


12.QUADRILATERALS

Quadrilateral: A Quadrilateral is a closed figure with four sides, four angles and four vertices.

In Quadrilateral ABCD

TS VII Maths Quadrilaterals1

  • AB, BC, CD, and AD are sides.
  • A, B, C and D are the vertices.
  • ∠ABC, ∠BCD, ∠CDA and ∠DAC are the angles.

Diagonal of a Quadrilateral:

The line segment joining the opposite vertices of a quadrilateral are called the diagonals of the Quadrilateral. In the above figure AC, BD is the diagonals.

Adjacent sides of a Quadrilateral:

The two sides of a Quadrilateral that have a common vertex are called the adjacent sides of the Quadrilateral. From the above figure, AB, BC; BC, CD; CD, DA and DA, AB are the adjacent sides.

Adjacent angles of a Quadrilateral:

The two angles of a Quadrilateral that have a common side are called the adjacent angles of the Quadrilateral. From the above figure, ∠A, ∠B; ∠B, ∠C; ∠C, ∠D and ∠D, ∠A is the adjacent angles.

Opposite sides of a Quadrilateral:

The two sides of a quadrilateral, which do not have a common vertex are called opposite sides of a quadrilateral. From the above figure, AB, CD; BC, DA are the opposite sides.

Opposite angles of a Quadrilateral:

The two angles of a quadrilateral, which do not have a common side are called opposite angles of a quadrilateral. From the above figure, ∠A, ∠C; ∠B, ∠D are the opposite angles.

Interior and exterior of a Quadrilateral:

TS VII Maths Quadrilaterals2

In a Quadrilateral ABCD, S, N are interior points, M, P are exterior points and A, B, C, D and Q are lies on the Quadrilateral.

Convex Quadrilateral:

A Quadrilateral is said to be a convex Quadrilateral if all line segments joining points in the interior of the Quadrilateral also lie in the interior of the Quadrilateral.

TS VII Maths Quadrilaterals3

Concave Quadrilateral:

A Quadrilateral is said to be a concave Quadrilateral if all line segments joining points in the interior of the Quadrilateral not lie in the interior of the Quadrilateral.

TS VII Maths Quadrilaterals4

Angle sum property of a quadrilateral:

The Sum of the angle in a Quadrilateral is 3600

TS VII Maths Quadrilaterals 5

In a Quadrilateral ABCD, ∠A + ∠B + ∠C + ∠D = 3600

Types of Quadrilaterals:

1.Trapezium:

In a Quadrilateral, one pair of opposite sides are parallel then it is Trapezium.

TS VII Maths Quadrilaterals6

In a Trapezium ABCD, AB∥ DC; AC, BD are diagonals.

2.Kite:

In a Quadrilateral two distinct consecutive pairs of sides are equal in length then it is called a Kite.

TS VII Maths Quadrilaterals 7

In a Kite ABCD, AB = BC; AD = DC AC, BD are diagonals.

3.Parallelogram:

In a Quadrilateral, two pairs of opposite sides are parallel then it is Parallelogram.

TS VII Maths Quadrilaterals 8

In a Parallelogram ABCD, AB∥ DC, AD∥ BC; AD, BD are diagonals.

Properties of parallelogram: –

  • The opposite sides of a parallelogram are equal in length.
  • The opposite angles are equal in measure.
  • The sum of the adjacent angles is 1800
  • Diagonals are bisected to each other and not equal in length.

4.Rhombus:

In a parallelogram in which two adjacent sides are equal, then it is a Rhombus.

TS VII Maths Quadrilaterals 9

In a Parallelogram ABCD, AB∥ DC, AD∥ BC; AD, BD are diagonals.

Properties of Rhombus: –

  • All sides of a Rhombus are equal in length.
  • The opposite angles are equal in measure.
  • The sum of the adjacent angles is 1800
  • Diagonals are bisected to each other perpendicularly and not equal in length.

5.Rectangle:

In a parallelogram all angles are equal, then it is a Rectangle.

TS VII Maths Quadrilaterals 10

Properties of Rectangle: –

  • The opposite sides are equal in length.
  • Each angle is 900.
  • The sum of the adjacent angles is 1800
  • Diagonals are bisected to each other and not equal in length.
  • Each diagonal divides the rectangle into two congruent triangles.

6.Square:

In a rectangle adjacent sides are equal, then it is a Square.

TS VII Maths Quadrilaterals 11

Properties of Square: –

  • All sides of a square are equal in length.
  • Each angle is 900.
  • The sum of the adjacent angles is 1800
  • Diagonals are bisected to each other and equal in length.
  • Each diagonal divides the square into two congruent triangles.

 


13.AREA AND PERIMETER

Area of a parallelogram:

TS VII Maths Area and Perimeter 1
Area of parallelogram (A) = b × h square units.

The area of the parallelogram is equal to the product of its base (b) and the height(h)

Area of a Triangle:

TS VII Maths Area and Perimeter 2
Area of triangle = ½ b × h square units.

The area of the triangle is equal to half the product of its base (b) and height (h).

In a Right-angled triangle, two of its sides can be the height.
Area of a Rhombus:

TS VII Maths Area and Perimeter 3

The area of the Rhombus is equal to half the product of its diagonals

Area of rhombus = ½ d1 × d2 square units.

Circumference of the circle:

TS VII Maths Area and Perimeter 4

Circumference of circle = 2πr = πd

Area of the rectangular path:

TS VII Maths Area and Perimeter 5

Area of Rectangular path = area of the outer rectangle – are of the inner rectangle

 


14. UNDERSTANDING 2D AND 3D SHAPES

Net: Net is a short of skeleton-outline in 2d, which when folded the result in 3d shape.

Nets of 3D shapes:

1.Cube:

TS VII maths Understanding 2D and 3D Shapes1               TS VII maths Understanding 2D and 3D Shapes 2

2.Cylinder:

TS VII maths Understanding 2D and 3D Shapes 3                TS VII maths Understanding 2D and 3D Shapes 4

 3.Pyramid:

TS VII maths Understanding 2D and 3D Shapes5    TS VII maths Understanding 2D and 3D Shapes6

Oblique Sketches:

Oblique sketches are drawn on a grid paper to visualise 3D shapes.

Ex: Draw an oblique sketch of a 3×3×3 cube

TS VII maths Understanding 2D and 3D Shapes1

Step-1: Draw the front face

TS VII maths Understanding 2D and 3D Shapes 7

Step-2: Draw the opposite face, which is the same as the front face. The sketch is somewhat offset from Step-1

TS VII maths Understanding 2D and 3D Shapes 8

Step-3: Join the corresponding corners.

TS VII maths Understanding 2D and 3D Shapes 9

Step-4: Redraw using dotted lines for hidden edges.

TS VII maths Understanding 2D and 3D Shapes 10

Isometric Sketches:

Isometric sketches are drawn on a dot isometric paper to visualise 3D shapes.

Ex: Draw an oblique sketch of a 2×3×4 cuboid

TS VII maths Understanding 2D and 3D Shapes 14

Step-1: Draw a rectangle to show the front face.

TS VII maths Understanding 2D and 3D Shapes 11

Step-2:  Draw four parallel line segments of length 3cm.

TS VII maths Understanding 2D and 3D Shapes 12

 Step-3: Connect the corresponding corners with appropriate line segments

TS VII maths Understanding 2D and 3D Shapes 13

 Step-4: This is an isometric sketch of a cuboid

TS VII maths Understanding 2D and 3D Shapes 15

 

 


15.SYMMETRY

Line of symmetry: The line which divides a figure into two identical parts is called the line of symmetry or axis of symmetry.

TS VII maths Symmetry 1

An object can have one or more than one lines of symmetry.

Regular polygon:

If a polygon has equal sides and equal angles, then the polygon is called a Regular polygon.

Lines of symmetry for Regular polygons:

Regular polygonNo. of sidesNo. of axes of symmetry
Triangle33
Square44
Pentagon55
Polygonnn

 Rotational symmetry: If we rotate a figure, about a fixed point by a certain angle and the figure looks the same as before, then the figure has rotational symmetry.

The angle of rotational symmetry:  The minimum angle of rotation of a figure to get the same figure as the original is called the angle of rotational symmetry or angle of rotation.

The angle of rotation of the equilateral triangle is 1200

The angle of rotation of a square is 900

Order of rotational symmetry:

The no. of times a figure, rotated through its angle of rotational symmetry before it comes to the original position is called the order of rotational symmetry.

The order of rotational symmetry for an equilateral triangle is 3.

The order of rotational symmetry for a square is 4.

Note: All figures have rotational symmetry of order 1, as can be rotated completely through 3600 to come back to its original position.

An object has rotational symmetry, only when the order of symmetry is more than 1.

• Some shapes have a line of symmetry and some have rotational symmetry and some have both.

Square, Equilateral triangle and Circle have both line and rotational symmetry.

  

 


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TS VI CL;ASS MATHS CONCEPT FEATURE IMAGE

TS 6 th class Maths Concept || Basics In Maths


6th maths notes|| TS 6 th class Maths Concept

Studying maths in the 6th  class successfully meaning that children take responsibility for their own learning and learn to apply the concepts to solve problems.

This note is designed by the ‘Basics in Maths’ team. These notes to do help students fall in love with mathematics and overcome fear. 


1. KNOWING OUR NUMBERS

•  Number: A number is a mathematical object used to count and measure.1,2,3…….etc.

Comparing numbers:

• We can compare the numbers by counting the digits in the numbers.

• Now Compare   5432 and 4678…

5432 is greater as the digits at the ten thousand place in 5432 is greater than that in  4678.

Order of numbers:

• Ascending Order: –

arrange the numbers from smallest to the greatest; this order is called Ascending order.

 Ex:- 23, 44, 65, 79, 100

• Descending Order: –

arrange the numbers from greatest to the smallest, this order is called Ascending order.

 Ex:- 100,79, 65, 33, 23

Formations of numbers

• Form the largest and smallest possible numbers using the digits 3, 2, 4, 1 without repetition

• Largest number formed by arranging the given digits in descending order _ 4321. 

• Smallest number formed by arranging the given digits in ascending order _ 1234.

• Greatest two-digit number is 99.

• Greatest three-digit number is 999.

• Greatest four-digit number is 9999.

 Place value

• Place value is the positional notation, which defines the position of a digit.

  Ex:- 3458     

8 is one place, 5 is tens place, 4 is hundreds place and 3 is thousands place.

Expanded form

• It refers to expand the numbers to see the value of each digit.

Ex :- 3458 = 3000 + 400 + 50 + 8

                    = 3×1000 + 4×100 + 5×10 + 8×1

• Note:-

        1 hundred = 10 tens

       1 thousand = 10 hundreds

      1 lakh = 100 thousands = 1000 hundreds   

Reading and Writing the numbers

Place value table for Indian system :

place value table 1

Example: Represents the number in 6,35,21,892 in place value table

placevalue table 2

Place value table for International system :

place value table 3 Ex:- represents the number in 635,218,924 in place value table

place value table 4

Use of commas:

• Indian system of numeration:- in the Indian system of numeration we use ones, tens, hundreds, thousands, lakhs and crores. The first comma comes after three digits from the right, the second comma comes two digits latter and the third comma comes after another two digits.E

Ex:-  “three crores thirty-five lakh seventeen thousand four hundred thirty” can be written as.3,35,17,430

• International system of numeration:- in the International system of numeration we use ones, tens, hundreds, thousands, millions and billions. 

Ex:- “ six hundred thirty-five million two hundred eighteen thousand nine hundred twenty-four” can be written as 635,218,924.

       Note:-10 millimetres = 1centimeter

                     100 centimetres = 1 meter

                    1000 meters = 1 kilometer

                   1000 milligrams = 1 gram

                    1000 grams = 1 kilo gram


2. WHOLE NUMBERS

Natural numbers: All the counting numbers starting from 1 are called Natural numbers.

                   1, 2, 3… Etc.

 Successor and Predecessor: If we add 1 to any natural number, we get the next number, which is called the Successor. If we subtract 1 from any natural number, we get the previous number, which is called Predecessor.

   Ex: – successor of 23 is 24 and predecessor of 32 is 31.

Note:- There is no predecessor of 1 in natural numbers.

Whole numbers: Whole numbers are the collection of natural numbers.

     0, 1, 2, 3 …

Representation of whole number on the number line:

• Draw a line mark a point on it.

number line for whole numbers

• Label it as ‘0’

• Mark as many points at equal distance to the right of 0.

• Label the points as 1, 2, 3, 4, … respectively.

• The distance between any two consecutive points is the unit distance.

  Addition on the number line:
  •  The distance between 2 and 4 is 2 units, like as the distance between 2 and 6 is 4 units
  • The number on the write is always greater than the number on the left
  • The number on the left of any number is always smaller than that number

        Addition of the whole number can represent on the number line

        Ex:-  3 + 2 = 5

       Start from three, we add 3 to 2. We make two jumps to the right of the number line as shown above. We reach at 5.

 Subtraction on the number line:

      Subtraction of the whole number can be represented on the number line


        Ex :-5 – 3 = 2

     Start from 5, we subtract 3 from 5. We make three jumps to the left of the number line shown as above. We reach at 2.

Multiplication on the number line:

NUMBER LINEFor multiplying 2 and 3, start from 0, make 2 jumps using 3 units at a time to the right, as you reach to 6. Thus, 2 × 3 =6.

Properties of whole numbers

Closer property: Two whole numbers are said to be closed if their operation (+, -, ×,÷) is always closed.

Addition:-Whole numbers are closed under addition.

Ex: 3, 2 are whole numbers ⟹ 3 + 2 = 5 ( 5 is whole number)

Subtraction:- Whole numbers are not closed under subtraction as their difference not always a whole number.

Ex:- 2 – 3 = −1 ( −1 is not a whole number)

Multiplication:- Whole numbers are closed under multiplication.

Ex:- 3 × 2 = 6, 6 is a whole number.

Division:- Whole numbers are not closed under division, as their division is not always a whole number.

Ex:-  3 ÷ 2 is not a whole number.

Commutative property: Two whole numbers are said to be commutative if the result is the same when we change their position.

Addition:-Whole numbers are commutative under addition.

Ex: 3, 2 are whole numbers ⟹ 3 + 2 = 5 and 2 + 3 = 5 ( 3 + 2 = 2 + 3).

Subtraction:- Whole numbers are not commutative under subtraction.

Ex:- 2 – 3 = −1 and 3 – 2 = 1( 2 −3 ≠ 3 – 2 ).

Multiplication:- Whole numbers are commutative under multiplication.

Ex:- 3 × 2 = 6 and 2 ×3 = 6 (3 × 2 = 2 ×3)

Division:- Whole numbers are not commutative under division.

Ex:-  3 ÷ 2  ≠ 2 ÷ 3.

Associative property: For any three whole numbers a, b and c if (a ⨀ b)⨀ c = a ⨀ (b ⨀ c), then whole numbers are associative under operation ⨀. [ ⨀ = +, –, × and ÷ ].

Addition:-Whole numbers are associative under addition.

Ex: ( 3 + 2) + 5 = 10 and  3 + (2 + 5) = 10 ⟹ ( 3 + 2) + 5 =   3 + (2 + 5)

Subtraction:- Whole numbers are not associative under subtraction.

Ex:- : ( 3 − 2) − 5 = −4  and  3 − (2 − 5) = 6 ⟹ ( 3 + 2) + 5 ≠3 + (2 + 5)

Multiplication:- Whole numbers are associative under multiplication.

Ex:- (3 × 2) ×5 = 30 and 3 ×(2 × 5) = 30  ⟹  (3 × 2) ×5 =  3 ×(2 × 5)

Division:- Whole numbers are not associative under division.

Ex:-  ( 3 ÷ 2) ÷ 5 ≠3 ÷ (2 ÷ 5).

Distributive property:

For any three whole numbers a, b and c, a×(b + c) = (a × b) +( a × c).

Note :Division by zero is not defining.

Identity under addition and multiplication:

        2 +0 = 2, 5 + 0 = 5 and so on.

      Thus, 0 is the additive identity.

      2 ×1 = 2, 4 × 1 = 4 and so on.

      Thus, 1 is a multiplicative identity.

Patterns:

  • Every number can be arranged as a line. The number 2 is shown as       patter2

         The number 3 as shown aspattern e

 

  • Some numbers can be shown as rectangles. 8 can be shown as

pattern 8

 

  • Some numbers can be arranged as squares. 9 can be shown as

pattern9

  • Some numbers can be shown as triangles.

       3 can be shown as          triangle number 3               6 can be shown as

triangle form 6

 


3. PLAYING WITH NUMBERS

Divisibility Rule:

The process of checking whether a number is divisible by a given number or not without actual division is called divisibility rule for that number.

Divisibility by 2:- a number is divisible by 2 if its once place is either 0, 2, 4, 6 or 8.

Ex:- 26 is divisible by 2. 35 not divisible by 2.

Divisibility by 3:- if the sum of the digits of a number is divisible by 3, then that number is divisible by 3.

Ex:- 231 → 2 + 3 +1 =6, 6 is divisible by 3

        ∴ 231 is divisible by 3

        436 → 4 + 3 + 6 = 13, 13 is not divisible by 3

∴ 436 is not divisible by 3.

Divisibility by 4:- if the last two digits of a number is divisible by 4, then that number is divisible by 4.

Ex:- 436, 36 is divisible by 4            ∴ 436 is divisible by 4

        623, 23 is not divisible by 4      ∴ 623 is not divisible by 4.

Divisibility by 5:- a number is divisible by 5, if its once place is either 0 or 5.

Ex:- 20, 25 are divisible by 5. 22, 46 are not divisible by 5.

Divisibility by 6:- a number is divisible by 6, if it is divisible by both 3 and 2.

Ex:-  242 is divisible by both 2 and 3     ∴ 242 is divisible by 6

          232 is divisible by 3 but not 2        ∴ 232 is not divisible by 6

Divisibility by 8:- if the last three digits of a number is divisible by 8, then that number is divisible by 8.

Ex:- 4232, last three digits 232 are divisible by 8

                     ∴ 4232 is divisible by 8.

Divisibility by 9:- if the sum of the digits of a number is divisible by 9, then that number is divisible by 9.

Ex:-  459, 4 + 5 + 9 = 18 → 18 is divisible by 9       ∴ 459 is divisible by 9

          532, 5 + 3 + 2 = 10 → 10 is not divisible by 9       ∴ 532 is not divisible by 9.

Divisibility by 10:- a number is divisible by 10 if its once place is 0.

Ex:- 20 is divisible by 10. 22, 45 are not divisible by 10.

Divisibility by 11:- A number is divisible by 11 if the difference between the sum of the digits at odd places and the sum of the digits at even places is either 0 or 11.

Ex:- 6545

     Sum of the digits at odd places = 5 + 5 = 10

    Sum of the digits at even places = 4 + 6 = 10

    Now difference is 10 – 10 = 0

     ∴ 6545 is divisible by 11.

Factors: a number which divides the other number exactly is called a factor of that number.

6 = 1×6

   = 2×3      ⟹  factors of 6 are: 1, 2, 3 and 6

Note- 1)1 is a factor of every number.

             2) Every number is a factor of itself.

             3) Every factor is less than are equal to the given number.

             4) Factors of a given number are countable.

Prime numbers: The numbers, which have only two factors 1, and itself are called prime numbers.

2, 3, 5, 7, …. Are prime numbers

Composite numbers: The number, which has more than two factors are called composite numbers.

4, 6,8,9….. are composite numbers.

  • Note: – 1) 1 is neither prime nor composite

             2) 2 is the smallest prime number

             3) 4 is the smallest composite number.

Co – prime number: The number which has no common factor except 1 is called co-prime number.

Ex:- (2, 3), (4,5) ……

Twin – primes: If the difference of two prime numbers is 2, then those numbers are called twin prime numbers.

Ex:- (2,3), (3,5), (17,19)…..

Factorization: When a number is expressed as the product of its factors, we say that the number has been factorized. The process of finding the factors is called Factorisation.

Ex:-  factors of 24 are: 1, 2, 3, 4, 6, 8, 12 and 24

           24 = 1 × 24 = 2 × 12 = 3 × 8 = 4 × 6

Prime factorisation: The process of finding the prime factors is called prime factorisation.

Ex:- 24 = 2 × 12

                   2 × 3 × 4

                   2 × 3 × 2 × 2

∴ Prime factorisation of 24 is 2 × 2 × 2 × 3.

Methods of prime factorization: 

Division method:- Prime factorization of 12 using the division method,      

fallow the procedure.                                                                                                 

                                                                                                                                                  DIVISION METHOD

Start dividing by the least prime factor. Continue division till the resulting number to be divided is 1.

The prime factorization of 12 is 2 × 2 × 3.

Factor tree method:- To find the prime factorization of 24, using the factor tree method we proceed as follows:

  • Express 24 as a product of two numbers.
  • FACTOR TREE METHODFactorise 4 and 6 further, since they are composite numbers.               
  • Continue till all factors are prime numbers.                                                                                                                            
  •  The prime factorization of 24 is 2 × 2 × 2 × 3.

Common factors: Common factors are those numbers, which are factors of all the given numbers.

Ex:- 12, 9

          Factors of 12 are:  1, 2, 3, 4, 6 and 12

           Factors of 9 are:  1, 3 and 9

∴ Common factors of 12, 9 are 1,3

Highest Common Factor (H.C.F):- The highest common factor of two or more numbers is the highest of their common factors. It is also called ad Greatest Common Divisor(G.C.D).

Ex:- H.C.F of 12, 9

         Factors of 12 = 1, 2, 3,4, 6, 12

         Factors of 9 = 1, 3, 9

Common factors of 12, 9 = 1,3

Highest common factor is 3

∴ H.C.F of 12, 9 is 3

Method of finding HCF:
Prime factorization method:

The HCF of 9 , 12 can be found by the prime factorization method as follows.

9 = 3  × 3

     12 =3 ×  2× 2                                                                                                                              PRIME FACTORISATION METHOD

The common factor of 12, 9 is 3

 ∴ H.C.F of 12, 9 is 3

Continue division method:

Euclid invented this method. Divide the larger number by smaller and then divide the previous divisor by the remainder until the remainder zero. The last divisor is the HCF of given numbers.

HCF OF DIVISION METHOD

Common multiple multiples of 3 are 3, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39,42…

Multiples of  4 are 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52….

Common multiples of 3 and 4 are  12, 24, 36….

Least common multiple (LCM):- The least common multiple of two or more given numbers is the lowest of their common multiple.

Ex:- LCM of 3 and 4

  Multiples of 3 =    3, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39,42…

 Multiples of 4 =   4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52….

Common multiples of 3 and 4 =  12, 24, 36….

∴ LCM of 3, 4 is 12.

Methods of finding LCM:

1.    Prime factorization method:-  the LCM of  6, 15 by using prime factorization method is as follows:

i)  Express each number as the product of prime factors

Prime factors of 6 =   2 × 3

Prime factors of 15 = 5× 3

ii)  Take the common factors both: 3

iii)  Take the extra factors of both 6 and 15 i.e., 2 and 5

iv)  The product of all common factors of two numbers and extra common factors of both finds LCM.

           ∴ LCM of 6 and 15 = (3) × 2 × 5 = 30.

2.    Division method:- To find LCM of 6 and 15:

i.    Arrange the given numbers in a row.

ii.  Then divide the least prime number, which divides at least two of the given numbers, and carry forward the numbers, which are not divisible by that number if any.

iii.Repeat the process till no numbers have a common factor other than 1.

iv. LCM is the product of the divisors and the remaining numbers.

Ex:-  

LCM1

                            ∴ LCM of 6, 15 = 3 × 2 × 5 = 30.

 Note:-  the product of LCM and HCF of given numbers = the product of given numbers.

            Ex:- 6 × 15 = 30 × 3 = 90.


4.BASIC GEOMETRICAL IDEAS

The term ‘geometry’ is derived from the Greek word ‘geometron’.

  Geo means Earth and metron means measurement.

Point: Point is a location or position on the surface of the plane. It is denoted by capital letters of the English alphabet. 

ts vi math poin

Line: It is made up of infinitely many points with infinity length.

ts vi math line

It is denoted by TS vi maths line AB

Ray: Ray is a part of a line. It begins at a point and goes on endlessly n a specific direction.

ts vi math ray

It is denoted by TS vi maths ray AB                                                
Line segment: It is a part of the line with the finite length.   ts vi math linesegment

It is denoted by TS vi maths line segment AB

Intersecting lines: If two lines are meeting at the same point, then those lines are called intersecting lines. That pint is called the point of intersection.ts vi math intersecting lines

Parallel lines: The lines, which are never meet at any point, are called parallel lines. ts vi math paralel lines

Curve: Anything, which is not straight, is called Curve.

Simple curve: – A curve that does not cross itself.ts vi math simple curve

 Open curve: – A curve in which its endpoints do not meet.ts vi math open curve

 Closed curve: – A curve that has no endpoint is called a closed curve.ts vi math closed curve

∗ A closed curve has three parts

The Interior of the curve: – It refers to the inside area of the curve. (B)ts vi math parts of closed curve

The exterior of the curve: – It refers to the outside area of the curve. (A)

On the curve: – It refers to the inside area of the curve. (C)

Polygon: – polygon is a simple closed figure made by line segments.

ts vi class polygon

Angle: the figure formed by two rays having a common end is called an angle.ts vi math angle

Here two rays OA, OB are arms of the angle

O is the Vertex. It is denoted by ∠AOB or ∠ BOA.

Triangle: A simple closed figure formed by the three line segments is a triangle. The line segments are called sides of the triangle.

ts vi math triangle

  • AB, BC and AC are sides of a triangle.
  • A,  Band C are vertices of a triangle.
  • ∠ABC, ∠BAC and ∠ACB are angles of the triangle.
  • This triangle is denoted by ∆ABC.

Quadrilateral: A simple closed figure formed by the four-line segments is a Quadrilateral.

  • AB, BC, CD and DA are the sides of the quadrilateral.ts vi math quadrilateral
  • A, B, C and D are the vertices of the quadrilateral.
  • ∠A, ∠B, ∠C and ∠D are the angles of quadrilateral.
  • AB, DC and BC, AD are opposite sides of the quadrilateral.
  • AB, BC; AD, DC; DC, BC and AD, AB are adjacent sides( the sides which have common vertex are called adjacent sides)
  • A, C and B, D are opposite vertices and also opposite angles.
  • AC and BD diagonals of a rectangle (A line segment joining opposite vertices is called diagonal).

Circle: The set of points that are at a constant distance from a fixed point is called a circle. The fixed point is called the centre of the circle and the constant distance is called the radius of the circle.ts vi math circle

  • O is the center of the circle.
  • OA, OB, and OC radii of the circle
  • AB is the diameter of the circle.
  • PQ is a chord.

Circumference of the circle: – the length of the boundary of the circle is called the circumference of the circle.

Arc: – The part of the circumference is called Arc. From the above fig.  is an arc.APisarc.s vi math sector and segment

Sector: – Region enclosed by an arc and two radii is called a sector.

Segment: – The region enclosed by arc and chord is called a segment of the circle.

 


5. MEASURES OF LINES AND ANGLES

Measure of line segment:

  • A line segment is a part of the line with two endpoints.
  • This makes it possible to measure a line segment.
  • This measure of each line segment is its ‘length’.
  • We use length to compare line segments.
  • We can compare the length of two line segments by: (i) simple observation (ii) tracing on a paper and (iii) using instruments.

Simple observation: – We can tell which line segment is greater than other just by observing the two-line

segments but it is not sure.vi math compare line segments

Here we can clearly say that CD > AB but sometimes it is difficult to tell which one is greater.

Tracing on a paper: – In this method we have to trace one line on paper then put the traced line segment on the other line to check which one is greater.

But this is a difficult method because every time to measure the different size of line segments we have to make a separate line segment.

Comparing by instruments: – To compare any two-line segments accurately, we use ruler(scale) and divider.

∗ We can use a ruler to measure the length of a line segment.vi math comparing by istruments

Put the zero mark at point A and then move toward l to measure the length of the line segment, but it may have some errors based on the thickness of the ruler.

∗ This could be made accurate by using a Dividervi math comparing by istruments-1

  • Put the one end of the divider on point A and open it to put another end on point B.
  • Now pick up the divider without disturbing the opening and place it on the ruler so that one end lies on “0”.
  • Read the marking on the other end and we can compare the two line.

Measure of an angle: Angle is formed two rays or two-line segments.

  • We can understand the concept of right and straight angles by directions.vi math measure of angle
  • There are four directions-North, South, East and West.
  • When we move from North to East then it forms an angle of 90°, which is called Right Angle.

 

  • When we move from North to South then it forms an angle of 180°, which is called Straight Angle.
  • When we move four right angles in the same direction then we reach to the same position again i.e. if we make a clockwise turn from North to reach to North again then it forms an angle of 360°, which is called a Complete Angle. This is called one revolution.

∗ In a clock, there are two hands i.e. minute hand and hour hand, which moves clockwise in every minute. When the clock hand moves from one position to another then turns through an angle.

  • When a hand starts from 12 and reaches to 12 again then it is said to be completed avi math measure of angle - clock
  • s were the ray moves in the opposite direction of the hands of a clock are called anti – clockwise angles. These are denoted by positive measure.
  • Angles were the ray moves in the direction of the hands of a clock are called clockwise angles. These are denoted by negative measure.

The protractor:vi math the protractor

  • By observing an angle we can only get the type of angle but to compare it properly we need to measure it.
  • An angle is measured in the “degree”. One complete revolution is divided into 360 equal parts so each part is one degree. We write it as 360° and read as “three hundred sixty degrees”.
  • We can measure the angle using a ready to use device called Protractor. 
  • It has a curved edge, which is divided into 180 equal parts. It starts from 0° to 180° from right to left and vice versa.

∗To measure an angle 72° using protractor-ts vi class measure of angle 72 degrees

  • Place the protractor on the angle in such a way that the midpoint of protractor comes on the vertex B of the angle.
  • Adjust it so that line BC comes on the straight line of the protractor. 
  • Read the scale, which starts from 0° coinciding with the line BC. 
  • The point where the line AB comes on the protractor is the degree measure of the angle.

                Hence, ∠ABC = 72°.

Types of angles:

Type of angle

Measure

Zero angle

Right angle

90°

Straight angle

180°

Complete angle

360°

Acute angle

Between 0° to 90°

Obtuse angle

Between 90° to 180°

Reflex angle

Between 180° to 360°

 

Perpendicular Linesvi math perpendicular lines

If two lines intersect with each other and form an angle of 90° then they must be perpendicular to

 


6.INTEGERS

There several situations in our daily life, where we use these numbers to represent loss or profit; past or future; low or high temperature. The numbers on the left side of zero are called negative numbers.

     TS VI INTEGERS BASICS ON NUMBER LINE

Integers: The numbers which are positive, zero and negative numbers together are called as integers and they re denoted by I or Z.

                      Z = {…, -3, -2, -1, 0, 1, 2, 3…}.

Representation of integers on a number line: –

TS VI MATH REPRESENTATION OF INTEGERS ON THE NUMBER LINE

  • The numbers which are on the right side of zero are positive numbers and which are on the left side of zero are negative numbers.
  • 0 is neither positive nor negative.
  • On a number line, the number increases as we move to right and decrease as we move to the left.

     ∴ -3 < -2 <   -1 <   0   < 1 <   2   < 3 <   4   <  5  so on.

  • Note: – 1. Any positive integer is always greater than any negative integer
  1. Zero is less than every positive integer.
  2. Zero is greater than every negative integer.
  3.   Zero doesn’t come in any of the negative and positive integers.

Addition and subtraction of integers:

1.If two integers have same sign, then add the integers and put that sign before the result.

Ex: – 3 + 2 =5, −3 – 2 = −5.

2.If two integers have different sign, then subtract smaller one from bigger and put the bigger one sign before the result.

Ex: – 3 − 2 =1, −3 + 2 = −1, −10 + 5 = −5.

Addition of integers on a number line:

Add 3 and 4

TS VI MATH ADDITION OF INTEGERS ON A NUMBER LINE

  • On the number line, we first move three steps to the right of 0 to reach 3, then we move 4 steps to the right of 3 and to reach 7

               ∴ 3 + 4 = 7                                                                                                                                                                  
Add −3 and −4

TS VI MATH ADDITION OF INTEGERS ON A NUMBERLINE 1

  • On the number line, we first move three steps to the left of 0 to reach −3, then we move 4 steps to the left of −3 and to reach −7.

∴ − 3 − 4 = −7

∗ Any two distinct numbers that give zero when added to each other are additive inverse each other.

Subtraction of integers on a number line:

Subtract 3 from 6

TS VI MATH SUBTRACTION OF INTEGERS ON A NUMBER LINE

  • On the number line, we first move 6 steps to the right of 0 to reach 6, then we move 3 steps to the left of 6 and to reach 3.

∴ 6 − 3 = 3.

Subtract −3 from 6

 

   On the number line, we first move 6 steps to the right of 0 to reach 6. For – 3 we have to move left but for – ( −3) we move in the opposite direction. Thus, we move 3 steps to the left of 6 and to reach 9.

∴ 6 – (−3) = 9.

• Subtraction of integers is the same as the addition of their additive inverse.


7. FRACTIONS AND DECIMALS

A fraction means a part of a group of a whole.

The ‘whole’ here could be an object or the group of objects. But all the parts of the whole must be equal. The ‘whole’ here could be an object or the group of objects. However, all the parts of the whole must be equal.

TS vi Math fractions and decimals 1

• Fig(i) is the whole. The complete circle.

• In Fig (ii), we divide the circle into two equal parts, then the shaded portion is the half ie., TS vi Math fractions and decimals 2of the circle.

• In Fig (iii), we divide the circle into three equal parts, then the shaded portion is the one third of the circle i.e.,TS vi Math fractions and decimals 3 of the circle.

• In Fig (iv), we divide the circle into four equal parts, then the shaded portion is the one fourth of the circle i.e.,TS vi Math fractions and decimals 4 of the circle.

The numerator and the denominator:

TS vi Math fractions and decimals 5

The upper part of the fraction is called ‘numerator’. It tells the no. of parts we have.

The lower part of the fraction is called ‘denominator’. It tells the total parts in whole.

Representing fractions pictorially:

TS vi Math fractions and decimals 9

Representing fractions on a number line:

Mark TS vi Math fractions and decimals 7 on a number line

TS vi Math fractions and decimals 8

Proper fractions: In a fraction if the numerator is less than denominator then, then it is called proper fraction. If we represent a proper fraction on a number line then it is always lies between 0 and 1.

Ex: –TS vi Math fractions and decimals 10

Improper fractions: In a fraction if the numerator is greater than denominator then, then it is called improper fraction.

Ex: –TS vi Math fractions and decimals 11

Mixed fractions: – The fraction made by the combination of whole number and a part is called mixed fraction.

Ex: –TS vi Math fractions and decimals 12

Note: Only improper fractions can be represented as mixed fractions.

A mixed fraction is in the form of TS vi Math fractions and decimals 13

We can convert it into improper fraction by  TS vi Math fractions and decimals 14

Ex: –   TS vi Math fractions and decimals 15

Equivalent fractions: – Equivalent fractions those fractions which represent the same part of whole.

TS vi Math fractions and decimals 16

  • Equivalent fractions are arising when we multiply both the numerator and denominator by the same number.
  • Equivalent fraction of TS vi Math fractions and decimals 2 are TS vi Math fractions and decimals 17   and so on

Standard form of a fraction (simplest or lowest form):- A fraction is said to be in standard form if both the numerator and denominator of that fraction have no common factor except 1.

Ex: –TS vi Math fractions and decimals 18

Like and Unlike fractions: The fractional numbers that have the same denominators are called fractional numbers and have not the same denominator are called unlike fractions.

Ex: –  TS vi Math fractions and decimals 19 are like fractions and TS vi Math fractions and decimals 20 are un like fractions.

Comparing fractions:

Like fractions: – We have to compare the like fractions with the numerator only, because the like fractions have same denominator. The fraction with greater numerator is greater and the fraction with smaller numerator is smaller.

Ex: – TS vi Math fractions and decimals 21 and so on.

Unlike fractions: –

With same numerator: For comparing unlike fractions, we have to compare denominators when the numerator is same. The fraction with greater denominator is smaller and the fraction with smaller denominator is smaller.

Ex: – TS vi Math fractions and decimals 22 and so on.

Note: – To find the equivalent fractions of both the fractions with the same denominator, we have to take the LCM of their denominators.

Ascending order and Descending order: –

When we write numbers in a form that they increase from the left to right then they are in the Ascending order. When we write numbers in a form that they decrease from the left to right then they are in the Descending order.

Ex: – For fractions: TS vi Math fractions and decimals 23are in ascending order and TS vi Math fractions and decimals 24are in descending order.       

Addition of fractions:

 Like fractions: –

TS vi Math fractions and decimals 25

Ex:  TS vi Math fractions and decimals 27TS vi Math fractions and decimals 26

Un like fractions: – For adding unlike fractions, first we have to find the equivalent fraction of given fractions and then add them as like fractions.

Ex: – TS vi Math fractions and decimals 28TS vi Math fractions and decimals 29

Subtraction of fractions 

Like fractions: –                            

TS vi Math fractions and decimals 30

Ex: –  TS vi Math fractions and decimals 31

Un like fractions: – First we have to find the equivalent fraction of given fractions and then subtract them as like fractions.

Ex: –TS vi Math fractions and decimals 32

Decimal fractions:

A fraction where the denominator is a power of ten is called decimal fraction. We can write decimal fraction with a decimal point (.). it makes easier to do addition, subtraction and multiplication on fractions.

Ex: – TS vi Math fractions and decimals 33


8. DATA HANDLING

Data: collection of information in the form of numbers or words is called data.

Recording data: Recording of data depends on the requirement of the data. We can record data in different ways.

Organization of data: –

  • Data is difficult to read.
  • We have to organize it.
  • Data can be organized in a tabular form.
  • Data is represented in tabular form using frequency distribution and the tally marks.
  • Frequency tells the no. of times the observations is happened.
  • Tally marks show the frequency of the data.

∗ Example for representing tally marks:

TS vi Math Data handling 1

Pictograph:

If the data is represented by the picture of objects instead if numbers, then it is called pictograph. Pictures make it is easy to understand the data and answer the questions to related it by observing the pictures.

Example for representing data by pictograph

TS vi Math Data handling 2

  • Drawing a pictograph is difficult to draw some difficult pictures.
  • For understanding every one, e must use proper symbols.
Bar graph:

• Bar graphs are used to represent the independent observations with frequencies.

• In a bar graph, bars of uniform width are drawn horizontally or vertically with equal spacing between them.

Construction of bar graph: – 

TS vi Math Data handling 3  TS vi Math Data handling 4

Steps to construction: –

1.Draw two perpendicular lines one horizontal (x – axis) and one vertical (y – axis).

2.Along the x- axis mark ‘items’ and along the x – axis mark ‘cost of items’.

3.Select a suitable scale 1cm = 10(rupees).

4.Calculate the heights of the bars by dividing the frequencies with the scale

70 ÷ 10 = 7, 40 ÷ 10 = 4 and so on.

5.Draw rectangular vertical bars of same width on the x- axis with heights calculated above.


9. INTRODUCTION TO ALGEBRA

Algebra is the use of letters or symbols to represent number. It helps us to study about un known quantities.

Patterns:

To make a triangle, 3 matchsticks are used   TS vi Math Introduction to Algebra 1

For making 2 triangles we have six matchsticks  TS vi Math Introduction to Algebra 2

For making 3 triangles we have nine matchsticksTS vi Math Introduction to Algebra 3

  • of matchsticks required for making 1 triangle = 3 = 3 × 1
  • of matchsticks required for making 2 triangles = 6 = 3 × 2
  • of matchsticks required for making 3 triangles = 9 = 3 × 3

Thus the no. of matchsticks for making ‘n’ triangles = 3 × n = 3n. 

Variable: Variable is a unknown quantity that may change. It is a dependent term.

In the above pattern, the rule is 3n, here ‘n’ is the variable.

  • We can use lower case alphabets are used as variable.
  • Numbers cannot use as variables, since they have fixed value.
  • Variables help us to solve other problems also.
  • Variables can take different values; they have no fixed value.
  • Mathematical operations addition, subtraction, multiplication and division can be done on the variables.

Use of variables:

perimeter of a polygon is the sum of the lengths of all its sides.

 Perimeter square = 4s, s is the variable

 Perimeter of rectangle = 2 (l + b); l, b are variables.

To find the nth term from the given pattern: 3, 6, 9…

         Number

3

6

9

12

15

       Pattern

3×1

3×2

3×3

3×4

3×5

From the table we observe that, the first number is 3×1, the second number is 3×2, the third number is 3×3 and so on.

∴ the nth term of pattern 3, 6, 9, 12, = 3n, here n is variable.

Simple equation: simple equation is a condition to be satisfied by the variables. Equation has equality sign between its two sides.

Ex: 5m = 10, 2x + 1 = 0 etc.

L.H.S and R.H.S of an equation:

The expression which is at the left of equal sign of an equation is called Left Hand Side (L.H.S)

The expression which is at the right side of equal sign of an equation is called Right Hand Side (R.H.S)

Ex: 4y = 20

L.H.S = 4y and R.H.S = 20

Solution of an equation (Root of the equation):

Solution or Root of an equation is the values of variable for which L.H.S and R.H.S are equal.

Ex: 3x = 15  

        If x = 5; LHS = 3×5 = 15

       RHS = 15

       ∴ solution of above equation is 5

Trial and error method:

By using this method, we get the solution of given equation.

Ex: solve 2n = 10

Substituting value of n

Value of L. H. S

Value of R. H. S

Whether LHS and RHS are equal

1

2×1 = 2

10

Not equal

2

2×2 = 4

10

Not equal

3

2×3 = 6

10

Not equal

4

2×4 = 8

10

Not equal

5

2×5 = 10

10

 Equal

When n = 5, LHS = RHS ∴ solution of equation is 5.


10. PERIMETER & AREA

Perimeter: Perimeter is the distance covered along the boundary forming a closed figure when you go around the figure once.
Perimeter of a Rectangle:

TS vi Math Perimeter and Area 1

Length of the rectangle = l, breadth = b

Perimeter of rectangle = sum of the lengths of its sides.

                                             = l + l +b + b

P = 2 (l + b) units.

Perimeter of a Square:   

TS vi Math Perimeter and Area 2                                

Length of the side  of a square   = a

Perimeter of rectangle = sum of the lengths of its sides.

                                           = a + a + a + a

                                         = 4a units.

Perimeter of an Equilateral Triangle:              

TS vi Math Perimeter and Area 3                 

Length of each side   = a

Perimeter of rectangle = sum of the lengths of its sides.

                                           = a + a + a

                                           = 3a units.

Polygon: A polygon is a simple closed figure bounded by line segments.

Regular polygon: A polygon which has equal side and equal angles, is called Regular polygon.

The perimeter of regular polygon of ‘n’ sides whose length ‘a’ = na.

Area: The amount of surface enclosed by a closed figure is called its area.

 Area of a Rectangle:

TS vi Math Perimeter and Area 1

Length of the rectangle = l, breadth = b

Area of the Rectangle = l × b square units.

Area of a Square:

TS vi Math Perimeter and Area 2

Length of the side of a square   = a

Area of a Square = a × a = a2 square units.

Note:  The area of the square is more than the area of any other rectangle having the same perimeter.


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11. RATIO AND PROPORTION

Ratio:  Ratio is the comparison of two quantities of same kind.

 The ratio of two quantities a and b is written as a: b and read as ‘a is to b’.

 ‘a’ is called first term or antecedent and ‘b’ is called second term or consequent.

Simplest form of ratio:

If a ratio is written in terms of whole numbers with no common factors other than 1, then the ratio is said to be in the ‘simplest form’ or in the ‘lowest terms’.

Ex: the simplest form of 5 : 15 is 1 : 3.

Division of a given quantity in a given ratio:

Let us suppose that, if a quantity ‘c’ divided into two parts in the ratio a: b, then

Total parts = a + b

First part =TS vi math Ratio and proportion 1and second part =TS vi math Ratio and proportion 2

Ex: Divide 1200 in the ratio 2 : 3

Ans: Total parts = 2 + 3 = 5

 First part = TS vi math Ratio and proportion 3= 2 × 240 = 480

Second part =TS vi math Ratio and proportion 4  = 3× 240 = 720.

Proportion:

Equalities of ratios is called proportion.

If a : b = c : d, then a ,b ,c and dare in proportion. This is represented as a : b ∷ c : d.

If a, b, c and d are in proportion, then ad = bc.

Unitary method:

 In this method, first we find the value of one unit and then the value of the required number of units.

Ex: If the cost of 5 pens is ₹ 20, then find the cost of 12 pens.

Sol:  Given that cost of 5 pens = ₹20

                               Cost of one pen = 20 ÷ 5= 4

                                Cost of 12 pens = 4 × 12 = 48

     ∴ cost of 12 pens = ₹ 48.  


12. SYMMETRY

Symmetry:

The word symmetry comes from Greek word. It means ‘to measure together’.

Symmetry is the mirror image of an object.

Symmetry means that one object becomes exactly like another when we move it in some way: turn, flip or slide.
Ex:

TS vi math Symmetry 1

Line of symmetry:

A line along which you can fold a figure so that two parts of it coincide exactly is called a ‘line of symmetry’.

Line of symmetry can be horizontal, vertical or diagonal.

Ex:

TS vi math Symmetry 2

TS vi math Symmetry 3The English alphabet which have

  • Vertical line of symmetry: A, H, I, M, O, U, V, W and X
  • Horizontal line of symmetry: B, C, D, E, H, I, K, O and X
  • No line of symmetry: D, G, J, L, N, P, Q, R, S, Y AND Z.

13. PRACTICAL GEOMETRY

The following instruments from a geometry box are used to construct figures:

1.A Ruler (Scale)

2.The compasses

3.The divider

4.Protractor

∗ The ruler is used to measure lines.

∗ The compasses is used for constructing.

∗ The protractor is used for measuring angles.

∗ Divider is used to make equal line segments or mark point on a line.

Construction of a line segment of a given length:

We can construct a line segment in two ways: 1) By using Ruler 2) By using the Compasses

1.By using Ruler: –

Let us suppose we want to draw a line segment AB of length 3.5 cm

TS vi math Practical Geometry 1

Steps of construction: –

Step-1: Place the ruler on a paper and hold it firmly.

Step-2: Mark a point with sharp edged pencil against ‘0cm’ mark of the ruler.

Step-3: Name the point as A. Mark another point against 5 small divisions just after the 3cm mark. Name this point as B

Step-4: Join A and B along the edge of the ruler. AB is the required line segment of length 3.5cm.

1.By using the Compasses: –

  Let us suppose we want to draw a line segment AB of length 3.5 cm


Steps of construction: –

Step-1: draw a line l. Mark a point on the line l.

TS vi math Practical Geometry 3
Step-2:
place the metal pointer of the compasses on the zero mark of the ruler. open the compasses so that the pencil point touches the 3.5cm mark on the ruler.

TS vi math Practical Geometry 2

Step-4: on the line l, we got the line segment AB of length 3.5cm.

TS vi math Practical Geometry 5
Step-3:
place the pointer on A on the line l and draw arc to cut the line. Mark the point where the arc cuts the line as B.

Construction of a circle:

  Let us suppose we want to draw a circle of radius 3 cm

TS vi math Practical Geometry 6

Steps of construction: –

Step-1: Open the compasses for radius 3 cm

Step-2: Mark a point with sharp edged pencil. This is the centre.

Step-3: Place the pointer of the compasses firmly at the centre.

Step-4: Without moving its metal point, slowly rotate the pencil and till it come back to the straight point.

Construction of perpendicular bisector a line segment:

Steps of construction: –

Step-1: Draw a line segment AB.TS vi math Practical Geometry 7
Step-2:
Set the compasses as radius more than half of the length of line segment AB.
TS vi math Practical Geometry 8
Step-3: With A as centre, draw arcs below and above the line segment

TS vi math Practical Geometry 9

 Step-4: With same radius and B as the centre draw two arcs above and below the line segment to cut the previous arcs. Name the intersecting points of arcs as M and N.

TS vi math Practical Geometry 10

Step-5: Join the points M and N. then, the line MN is the required perpendicular bisector of the line segment AB.

Construction of perpendicular to a line, through a point which is not on it:

Steps of construction: –

Step-1: Draw a line l and a point A not on it TS vi math Practical Geometry 11

Step-2: With A as centre draw an arc which intersects the given line at two points M and N.

TS vi math Practical Geometry 12

 Step-3: Using the same radius and with M and N as centres construct two arcs that intersect at a point B on the other side of the line.

TS vi math Practical Geometry 13Step-4: Join A and B. AB is the perpendicular of the given line l.     

TS vi math Practical Geometry 14Construction of Angles using Protractor:

Let us suppose we want to construct ∠ABC = 500

Steps of construction: –
Step-1:
Draw a ray BC of any length.TS vi math Practical Geometry 15

Step-2: Place the centre point of the protractor at B and the line aligned with theTS vi math Practical Geometry 16

Step-3: Mark a point A at 500

TS vi math Practical Geometry 17Step-4: join AB. ∠ABC is the required angle.

Constructing a copy an of Angle of un known measure:

Let ∠A is given, measure is not known

Steps of construction: –
Step-1:
Draw a line TS vi math Practical Geometry 4 and choose a point A on it.TS vi math Practical Geometry 18

Step-2: Now place the compasses at A and draw an arc to cut the rats AC and AB.

TS vi math Practical Geometry 20
Step-3:  Use the same compasses setting to draw an arc with P as centre, cutting l at Q.

TS vi math Practical Geometry 21

Step-4: Set your compasses with BC as the radius.

TS vi math Practical Geometry 22

Step-5: Place the compasses pointer at Q and draw an arc to cut the existing arc at R.

                      TS vi math Practical Geometry 23                                                                                       

Step-6: Join PR. It has the same measure as ∠BAC.TS vi math Practical Geometry 24

Construction to bisect a given angle:

Let an angle say ∠AOB be given

Steps of construction: – TS vi math Practical Geometry 25

Step-1: With O as the centre and ray convenient radius, draw an arc PQ cutting OA and OB at P and Q respectively.

TS vi math Practical Geometry 27

Step-2: With P as the centre and any radius slightly more than half of the length of PQ, draw an arc in the interior of the given angle.

TS vi math Practical Geometry 26

Step-3: With Q as the centre and without alternating radius draw another arc in the interior of ∠AOB.

                Let two arcs intersects at S

TS vi math Practical Geometry 28

Step-4: Draw ray , thenTS vi math Practical Geometry 29 is the bisector of ∠AOB

                Observe ∠AOS = ∠SOB

CONSTRUCTION ANGLES OF SPERCIAL MEASURES:

Construction of 600 angle: –

Steps of construction: –

Step-1: Draw a line l and mark a point O on it. TS vi math Practical Geometry 30

Step-2: Place the pointer at O and draw an arc of convenient radius which cuts the line at P (say).

TS vi math Practical Geometry 31

Step-3:  With the centre P and the same radius as in the step-2. Now draw an arc that passes through O.

Step-4:  Let the two arcs intersects at Q. Join OQ. We get ∠POQ = 600.

Construction of 1200 angle: –
Steps of construction: –

Step-1: Draw a ray OATS vi math Practical Geometry 34

Step-2: Place the pointer of the compasses at O. With O as the centre and any convenient radius draw an arc cutting OA at P.

TS vi math Practical Geometry 35

Step-3: With P as the centre and the same radius as in the step-2 draw an arc which cuts the first arc at Q.

TS vi math Practical Geometry 36Step-4: With Q as the centre and the same radius as in the step-2 draw an arc which cuts the first arc at R.

TS vi math Practical Geometry 37

Step-5: Join OR. Then ∠POR = 1200.


14. UNDERSDTANDING 3D AND 2D SHAPES


Cuboid:
3D- shapes or Solids:

The object which have a length, breadth and height (or depth) are called ‘three dimensional’ or ‘3D- shapes’ or ‘Solids’.

Cuboid:TS vi math 3D and 2D shapes 1

Objects like match boxes, erasers are the examples for cuboid

A cuboid has 6- Faces, 8- Vertices and 12 – Edges

Cube:TS vi math 3D and 2D shapes 2

A dice is an Example for cube.

A cube has 6- Faces, 8- Vertices and 12 – Edges

Cylinder:TS vi math 3D and 2D shapes 3

Objects like wooden log, a piece of pipe are the examples for cylinder

the top and base of the cylinder are circular in shape.

Cone:TS vi math 3D and 2D shapes 4

joker cap is the example for cone

base of the cone is a circle.

Sphere:TS vi math 3D and 2D shapes 5

Balls, laddoos are the examples for globe.

Triangular prism:TS vi math 3D and 2D shapes 6

If the base of a prism is triangle, then it is called triangular prism.

 

Pyreamid:

A pyramid is a solid shape with a base and point vertex.TS vi math 3D and 2D shapes 7

If a pyramid has triangular base, then it is called triangular pyramidTS vi math 3D and 2D shapes 8

If a pyramid has  square base, then it is called square pyramid.

Polygon: A polygon is a closed figure made with linesegments.

RegularPolygon: A polygon with all equal sides and all equal angles is called a regular polygon.


 

 


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TS 10th class maths concept

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Studying mathematics successfully meaning that, children take responsibility for their own learning and learn to apply the concepts to solve problems.

This note is designed by the ‘Basic In Maths’ team. These notes to do help students fall in love with mathematics and overcome fear.


1. REAL NUMBERS

• Rational number: The number, which is written in the form of p/q where p,q are integers q not equal to zero is called a rational number. It is denoted by Q.

• Irrational numbers:- the number, which is not rational is called an irrational number. It is denoted by Q’ or S.

• Euclid division lemma:- For any positive integers a and b, then q, r are integers exists uniquely satisfying the rules a = bq + r, 0 ≤ r < b.

• Prime number:- The number which has only two factors 1 and itself is called a prime number. (2, 3, 5, 7 …. Etc.)

• Composite number:- the number which has more than two factors is called a composite number. (4, 6, 8, 9, 10,… etc.)

• Co-prime numbers:- Two numbers  said to be co-prime numbers if they have no common factor except 1. [Ex: (1, 2), (3, 4), (4, 7)…etc.]

• To find HCF, LCM by using prime factorization method:  H. C.F= product of the smallest power of each common prime factors of given numbers. L.C.M = product of the greatest power of each prime factor of given numbers.

  • In p/q, if prime factorisation of q is in form 2m 5n, then p/q is terminating decimal. Otherwise non-terminating repeating decimal.
  • Decimal numbers with the finite no. of digits is called terminating Decimal numbers with the infinite no. of digits is called non-terminating decimal. In a decimal, a digit or a sequence of digits in the decimal part keeps repeating itself infinitely. Such decimals are called non-terminating repeating decimals.

• ‘p’ is a prime number and ‘a’ is a positive integer, if p divides a2, then p divides a.

• If ax = N then x = {log_{a}}^{N}

(i) log (xy) = log(x) + log(y)  (ii) log (x/y) = log( x) – log( y) (iii) log (xm ) = m log (x)

 

 

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TS 10th class maths concept

 

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ICSE 8th Maths Concept

ICSE 8th Maths Concept| Basics In Maths

3.SQUARES AND SQUARE ROOTS, CUBES AND CUBE ROOTS

Square:  Square number is the number raised to the power 2. The number obtained by the number multiplied by itself.

Ex: – 1) square of 5 = 52 = 5 × 5 = 25, 2) square of 3 = 32 = 3× 3 = 9

∗If a natural number p can be expressed as q2, where q is also natural, then p is called a square number.

Ex: – 1,4,9, …etc.

Test for a number to be a perfect square:

If a number is expressed as the product of pairs of equal factors, then it is called a perfect square.

Ex: – 36    test for perfect square

Prime factors of 36 = 2× 2× 3× 3

36 can be expressed as the product of pairs of equal factors.

∴ 36 is a perfect square.

Square Root: the square root of a number x is that number when multiplied by itself gives x as the product. The square root of x is denoted by sqrt x

example for sqrt

Methods of Finding Square root of given Number

Prime factorization method: –

Steps:

  1. Resolve the given number into prime factors.
  2. Make pairs of similar factors.
  3. The product of prime factors, choosing one out of every pair gives the square root of the given number.

Ex: – To find the square root of 16

Prim factors of 16 = 2 ×2× 2× 2

= 2 × 2 = 4

∴ square root of 16 = 4

Division method: –

Steps:

  1. Mark off the digits in pairs starting with the unit place. Each pair and remaining one digit are called a period.
  2. Think of the largest number whose square is equal to or just less than the first period. Take this number as the divisor as well as quotient.
  3. Subtract the product of divisor and quotient from the first period and bring down the next period to the right of the remainder. this becomes the new dividend.
  4. Now, the new divisor is obtained by taking twice the quotient and annexing with it a suitable digit which is also taken as the next digit of the quotient, chosen in such a way that the product of the new divisor and this digit is equal to or just less than the new dividend.

Repeat steps 2, 3, and 4 till all the periods have been taken up. Thus, the obtained quotient is the required square root.

Ex: – To find the square root of 225

icse class 8, square root of a number

Properties of a perfect square:

1. The square of an even number is always an even number.

Ex: – 22 = 4 (4 is even), 62 = 36 (36 is even), here 2, 6 are an even number.

2. The square of an odd number is always an odd number.

Ex: – 32 = 9 (9 is even), 152 = 225 (225 is even), here 3, 15 are an odd number.

3. The square of a proper fraction is a proper fraction less than the given fraction.

Ex: –square of a proper fractin

4. The square of decimal fraction less than 1 is smaller than the given decimal.

Ex: – (0.3)2 = 0.09 < 0.03.

5. A number ending with 2, 3, 7, or 8 is never a perfect square.

Ex: – 72, 58, 23 are not perfect squares.

6. A number ending with an odd no. of zeros is never a perfect square

Ex: – 20, 120,1000 and so on.

The square root of a number in decimal form

Make the no. of decimal places even, by affixing a zero, if necessary. Now periods and find out the square root by the long division method.

Put the decimal point in the square root as soon as the integral part is exhausted.

Ex: – To find the square root of 79.21

icse viii maths square root of decimal number

The square root of a decimal number which is not perfect square:

if the square root is required to correct up to two places of decimal, we shall find it up 3 places of decimal and then round it off up to two decimal places.

if the square root is required to correct up to three places of decimal, we shall find it up 4 places of decimal and then round it off up to three decimal places.

Ex: – To find the square root of 0.8 up to two decimal places

icse viii math square root of decimal number which is not perfect square

Cube of a number:

The cube of a number is that number raised to the power 3.

Ex: – cube of 0.3 = 0.33 = 0.027

Cube of 2 = 23 = 8

Perfect cube:

If a number is a perfect cube, then it can be written as the cube of some natural numbers.

Ex: – 1, 8, 27, and so on.

Cube root:

The cube root of a number x is that number which when multiplied by itself three times gives x as the product.

Cube root of x is denoted by Cube root x

example for cube root

Methods of finding the cube root of the given Number

Prime factorization method: –

Steps:

  1. Resolve the given number into prime factors.
  2. Make triplets of similar factors.
  3. The product of prime factors, choosing one out of every triplet gives the cube root of the given number.

Ex: – 27

Prim factors of 27 = 3×3×3

= 3

∴ cube root of 27 = 3

Test for a number to be perfect cube:

A given number is a perfect cube if it can be expressed as the product of triplets of equal factors.

Ex: – 2744  test-for-a-number-to-be-perfect-cub

Prime factors of 2744 = 2×2×2 × 7×7×7

∴ 2744 is a perfect cube.


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ICSE IX CLASS MATHS CONCEPT FEATURE IMAGE

ICSE IX Class Maths Concept

ICSE IX Class Maths Concept

 

ICSE IX Class Maths Concept: This note is designed by the ‘Basics in Maths’ team. These notes to do help the ICSE 9th class Maths students fall in love with mathematics and overcome their fear.

These notes cover all the topics covered in the ICSE 9th class Maths syllabus and include plenty of formulae and concept to help you solve all the types of ICSE 9th

Math problems are asked in the CBSE board and entrance examinations.


1. RATIONAL AND IRRATIONAL NUMBERS

Natural numbers: counting numbers 1, 2, 3… called Natural numbers. The set of natural numbers is denoted by N.

N = {1, 2, 3…}

Whole numbers: Natural numbers including 0 are called whole numbers. The set of whole numbers denoted by W.

W = {0, 1, 2, 3…}

Integers: All positive numbers and negative numbers including 0 are called integers. The set of integers is denoted by I or Z.

Z = {…-3, -2, -1, 0, 1, 2, 3…}

Rational number: The number, which is written in the form of, where p, q are integers and q ≠ o is called a rational number. It is denoted by Q.

∗ In a rational number, the numerator and the denominator both can be positive or negative, but our convenience can take a positive denominator.

Ex: – \inline \fn_jvn -\frac{2}{3} can be written as \inline \fn_jvn \frac{-2}{3}=\frac{2}{-3}  but our convenience we can take \inline \fn_jvn \frac{-2}{3}

Equal rational numbers:

For any 4 integers a, b, c, and d (b, d ≠ 0), we have \inline \fn_jvn \frac{a}{b}=\frac{c}{d} ⇒ ad = bc

The order of Rational numbers:

If  are two rational numbers such that b> 0 and d > 0 then \inline \fn_jvn \frac{a}{b}> \frac{c}{d} ⇒ ad > bc

The absolute value of rational numbers:

The absolute value of a rational number is always positive. The absolute value of  \inline \fn_jvn \frac{a}{b} is denoted by \inline \fn_jvn \left | \frac{a}{b} \right | .

Ex: – absolute value of \inline \fn_jvn -\frac{2}{3}=\frac{2}{3}

To find rational number between given numbers:

  • Mean method: – A rational number between two numbers a and b is \inline \dpi{120} \fn_jvn \frac{a + b}{2}

  Ex: – insert two rational number between 1 and 2

1 <  \inline \dpi{120} \fn_jvn \frac{1 + 2}{2} < 2   ⟹     1 <  \inline \dpi{120} \fn_jvn \frac{3}{2}  < 2

1 <  \inline \dpi{120} \fn_jvn \frac{3}{2} \inline \dpi{120} \fn_jvn < \frac{\frac{3}2{+2}}{2}< 2   ⟹   1 < \inline \dpi{120} \fn_jvn \frac{3}{2}< \frac{7}{4} \dpi{120} \fn_jvn <  2

To rational numbers in a single step: –

Ex:- insert two rational numbers between 1 and 2

To find two rational numbers, we 1 and 2 as rational numbers with the same denominator 3

(∵ 1 + 2 = 3)

1 =   \fn_jvn \frac{1\times 3}{3}  and 2 = \inline \dpi{120} \fn_jvn \frac{2\times 3}{3}

\inline \dpi{120} \fn_jvn \frac{3}{3}\left ( 1 \right )< \frac{4}{3}< \frac{5}{3}< \frac{6}{3}\left ( 2 \right )

 

Note: – there are infinitely many rational numbers between two numbers.

The decimal form of rational numbers

icse ix terminating desimals

icse ix class nomterminating repeating decimal form

∗ Every rational number can be expressed as a terminating decimal or a non-terminating repeating decimal.

Converting decimal form into \dpi{120} \fn_jvn \frac{p}{q}  the form:

1. Terminating decimals: –

1.2 =\inline \dpi{120} \fn_jvn \frac{12}{10}=\frac{6}{5}

1.35 =\inline \dpi{120} \fn_jvn \frac{135}{100}=\frac{27}{20}

2. Non-Terminating repeating decimals: –

CONVERT INTO PB Y Q FORM 1

CONVERT INTO P BY Q FORM 2

Irrational numbers:

  • The numbers which are not written in the form of  \dpi{120} \fn_cm \frac{p}{q}, where p, q are integers, and q ≠ 0 are called rational numbers. Rational numbers are denoted by QI or S.
  • Every irrational number can be expressed as a non-terminating decimal or non-repeating decimal.

Ex:- \dpi{120} \fn_cm \sqrt{2},\, \sqrt{5},\pi and so on.

  • Calculation of square roots:
  • There is a reference of irrationals in the calculation of square roots in Sulba Sutra.
  • Procedure to finding \dpi{120} \fn_cm \sqrt{2} value:

finding sqrt of 2


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