Sunday, 18 June 2017

Factoring of quadratic expression - 2

factoring of quadratic exprssions - 1 me maine aapko x²+bx+c type k question ka factor karna shikhaya tha.

Matalab ki humne iss part me quadratic expression ax²+bx+c me a= 1 type k question ka factor karna shikha tha.
 Aaj mai aapko ax²+bx+c type k question ka factor karna shikhaunga. Matalab ki aaj mai aapko unn question ka factor karna shikhaunga , jink liye quadratic expression ax²+bx+c me a= 2,3,4,5..... ho .

Factoring of quadratic expression - 2

Also read -:

• Trigonometric ratio se related Questions with solutions
• Basic of trigonometric ratio

Chaliye aaj ka lesson start karte hai.

Factoring of quadratic expression - 2

Quadratic expression k 2nd type k question ka factor karne k liye maine niche working rule diya hai. Aap step-by-step follow kariye.

Working rule -:

Suppose karte hai ki, ax²+bx+c ek quadratic expression hai to

Step-1. Sabse pahle a aur c ka multiply ( a×c ) kariye.

Also read -:



Step-2. Agar a×c ki value positive ho. Means a×c = +ac to iss multiplication result k aise gununkhand ( factor ) karenge ki unka sum ( add) "b" k equal ho.

For example -: Maan lete hai ki 4x²+12x+5 ek quadratic expression hai. to a = 4 , b = 12 aur c = 5 aur 
a×c = 4×5 = 20 
Ab hume 20 k aise factor karna hai ki , unn factor ko add karne par "b" equal ho. Means 12 k equal ho.

Factor of a×c = 20 = 10×2
 aur            b = 12 = 10+2

Step-3. Agar a×c ki value  negative  ho.Means a×c = -ac to iss multiplication result k aise gununkhand ( factor ) karenge ki unka subtraction (minus) "b" k equal ho.

For example -:
 Maan lete hai ki 2x²+9x-5 ek quadratic expression hai. to  a = 2 , b = 9 aur c = -5 aur 
a×c = 2×(-5) = -10
Ab hume 10 k aise factor karna hai ki , unn factor ko subtration karne par "b" equal ho. Means 10 k equal ho.

Factor of a×c = -10 = 10×(-1)
 aur            b = 9 = 10-1      

Also read -:


Step-4.  Ab aap "b" ki value quadratic expression ax²+bx+c me rakh kark 2 parts me divide kark factor ko find kar sakte hai.

Aayiye ab kuch question ka factor karte hai .

Some questions of quadratic expression with factors


Q-1.  49x²+14x+1 ka gununkhand kijiye.

Sol.-: Diya hai , 49x²+14x+1 quadratic expression hai.

a = 49 , b = 14 aur  c = 1

Ab "a×c" = 49×1 = 49

Factor of  "a×c" = 49 = 7×7
aur              " b " = 14 = 7+7

49x²+14x+1 = 49x²+(7+7)x+1
                     = 49x²+7x+7x+1
                     = (49x²+7x)+(7x+1)
                     = 7x(7x+1)+1(7x+1)
                     = (7x+1)(7x+1)     Ans.

Q-2.   9x²-12x+4 ka factors ( gununkhand ) kijiye.

Sol.-:      a = 9 , b = -12 aur  c = 4

Ab "a×c" = 9×4 = 36
Factor of  "a×c" = 36 = 6×6
aur              " b " = 12 = 6+6

9x²-12x+4 = 9x²-(6+6)x+4
                     = 9x²-6x-6x+4
                     = (9x-6x)-(6x-4)
                     = 3x(3x-2)-2(3x-2)
                     = (3x-2)(3x-2)     Ans.

Q-3.  6x²-31x+35 gununkhand kariye. 

Sol.-:      a = 6 , b = -31 aur  c = 35

Ab "a×c" = 6×35 = 210
Factor of  "a×c" = 315 = 21×10
aur              " b " = 31 = 21+10

6x²-31x+35 = 6x²-(21+10)x+35
                     = 6x²-21x-10x+35
                     = (6x-21x)-(10x-35)
                     = 3x(2x-7)-5(2x-7)
                     = (2x-7)(3x-5)     Ans.

Also read -:

• Trigonometry ka basic concept in hindi part-1
• Fast cube tricks 1se 99 tak in Hindi

Q-4.  14x²-29x+15 gununkhand gyat kijiye.

Sol.-:
a = 14 , b = -29 aur  c = 15

Ab "a×c" = 14×15 = 210
Factor of  "a×c" = 210 = 14×15
aur              " b " = 29 = 14+15

14x²-29x+15 = 14x²-(14+15)x+15
                     = 14x²-14x-15x+15
                     = (14x²-14x)-(15x-15)
                     = 14x(x-1)-15(x-1)
                     = (x-1)(14x+15)     Ans.

Q-5.  18x²+53x-35 factors kare .

Sol.-:

a = 18 , b = 53 aur  c = -35

Ab "a×c" = 18×(-35) = -630

Factor of  "a×c" = -630 = 10×63
aur              " b " = 53 = 63-10

18x²+53x-35 = 18x²+(63-10)x-35
                     = 18x²+63x-10x-35
                     = (18x²+63x)-(10x+35)
                     = 9x(2x+7)-5(2x+7)
                     = (2x+7)(9x-5)     Ans.



Q-6.  4-x-5x² find factors ?

Sol.-: sabse pahle iss question ko standard form me change karenge.

-5x²-x+4 usk baad

a = -5 , b = -1 aur  c = 4

Ab "a×c" = -5×4 = -20

Factor of  "a×c" = -20 = 5×(-4)
aur              " b " = 1 = 5-4

-5x²-x+4 = -5x²-(5-4)x+4
                     = -5x²-5x+4x+4
                     = (-5x²-5x)+(4x+4)
                     = -5x(x+1)+4(x+1)
                     = (x+1)(4-5x)     Ans.

Also read -:

• Fast multiply by 11 in Hindi
• Fast multiply by 99,999,9999 of any numbers in hindi

Q-7.  20-9x-20x² factors kya hoga ?

Sol.-: Sabse pahle iss question ko standard form me change karenge.

-20x²-9x+20  Ab

a = -20 , b = -9 aur  c = 20

Ab "a×c" = -20×20 = -400

Factor of  "a×c" = -400 = 16×(-25)
aur              " b " = 9 = 25 - 16

-20x²-9x+20 = -20x²-(25-16)x+20
                     = -20x²-25x+16x+20
                     = (-20x²-25x)+(16x+20)
                     = -5x(4x+5)+4(4x+5)
                     = (4x+5)(4-5x)     Ans.

Also read -:

• Fast calculate square of numbers ending with 5
• square root in few seconds

Q-8.  10x²+3x-4 factors kya hoga ?

Sol.-:
a = 10 , b = 3 aur  c = -4

Ab "a×c" = 10×(-4) = -40

Factor of  "a×c" = -40 = 8×(-5)
aur              " b " = 3 = 8-5

10x²+3x-4 = 10x²+(8-5)x-4
                     = 10x²+8x-5x-4
                     = (10x²+8x)-(5x+4)
                     = 2x(5x+4)-1(5x+4)
                     = (5x+1)(2x-1)     Ans.

Q-9.   7x²-48x-7 ka gununkhand kariye.

Sol.-:
a = 7 , b = -48 aur  c = -7

Ab "a×c" = 7×(-7) = -49

Factor of  "a×c" = -49 = 1×(-49)
aur              " b " = -48 = 1-49

7x²-48x-7 = 7x²-(1-49)x-7
                     = 7x²-x+49x-7
                     = (7x²-x)+(49x-7)
                     = x(7x-1)+7(7x-1)
                     = (7x-1)(x+7)     Ans.

Also read -:

•  Iss fast trick se kariye simple equation instantly solve
• Mathematics operation

Q-10.  4x²+12x+5 ka factors maalum kijiye.

Sol.-:  
a = 4 , b = 12 aur  c = 5

Ab "a×c" = 4×5 = 20

Factor of  "a×c" = 20 = 10×2
aur              " b " = 12 = 10+2

4x²+12x+5 = 4x²+(10+2)x+5
                     = 4x²+10x+2x+5
                     = (4x²+10x)+(2x+5)
                     = 2x(2x+5)+1(2x+5)
                     = (2x+5)(2x+1)     Ans.



Aaj k liye bas itana hi friends . Mujhe ummid hai ki , ab aap quadratic expression ka factor karna shikh gaye honge.

Also read -:

• Kya aapko koi help mathematics me chahiye?
• Top 10 math tricks post october 2016

Agar aapko koi problem ho to aap comment kark puchh sakte hai. Mujhe aapki help kark bahot khushi hogi.

Agar aap meri help karna chhahate hai to bas iss topic ko share kare.

Sunday, 11 June 2017

Factoring of quadratic exprssions - 1

Aaj mai factoring of quadratic expression k baare me baat karunga. quadratic exprssion ka factoring , means factor kaise karte hai ? Ya quadratic expression ka gununkhand kaise karte hai?  Ye kya hota hai ? inn sabhi k baare me mai full details me aapko bataunga.


Isse hindi me hum "द्विघातीय व्यंजको का गुणनखंड " ( dvighaateey vyanjako ka gunankhand ) khate hai.
General quadratic expression ko hum ax²+bx+c  se denote karte hai. Yaha ye yaad rakhe ki a, b  aur c numbers (constant) hota hai aur a is not equal to zero  hota hai.
Matlab ki "a" ki value "0" ( zero ) nhi ho sakti.

Factoring of quadratic equations

Aaj mai unn quadratic expression ka factor karna bataunga jink liye quadratic expression  ax²+bx+c me
a = 1 ho.
a = 2,3,4.... k liye mai next post me bataunga.
Matalab ki aaj mai aapko x²+bx+c type k quadratic expression ka factor karna shikhaunga.

Also read -:

• Top 10 math tricks post november 2016
• Subtraction trick without using borrowing for 2-digit number

Aayiye aaj ka topic start karte hai.

Factoring of quadratic expressions


Aapko koi problem naa iss liye maine isska working rule niche diya hai . Aap step-by-step rule ko follow kareiye.

Working rules for factoring of quadratic expression

Suppose karte hai ki , hume x²+bx+c ka factor karna hai to

Step-1. Number "c"  ka factor kariye.

Step-2. "c" k 2 aise factor choose kare . jinko Add ya subtract  karene par result "b" k equal ho.

For ex.-: suppose karte "c" k 2 factor s and t hai . Tab c = s×t aur

           b = s+t   ya    b = s - t   

Yaha ye yaad rakhe ki, agar quadratic expression me  "c"  ki value positive (+c) di ho to b= s+t ka use aur negative ( - c) ho to , b= s-t ka use karenge.

Step-3.  Ab diye gaye quadratic expression me "b" ki jagah par b = s+t   ya    b = s - t rakhe.

For ex.-:      x²+(s+t)x+c

Step-4. Ab 2-2 ki group banaye . Yahi bana group diye huye quadratic expression ka factors hai.

Also read -:

• Triangle and its types and defination in hindi
• 10 seconds me area of the triangle question solve

Aayiye Ab hum kuch question ko solve karte hai .

Quadratic expression se related question with solution


Q-1.     x²+5x+6 k gununkhand ( factor ) gyat kijiye.

Sol.-:  quadratic expression x²+5x+6 me , b = 5 aur c = 6

Factor of  "c" = 6 = 2×3
 aur          " b" = 5 = 2+3

x²+5x+6 = x²+(2+3)x+6
              = x²+2x+3x+6
              = (x²+2x)+(3x+6)
              = x(x+2)+3(x+2)
              = (x+2)(x+3)      Answer.

Q-2. Find the factor of quadratic expression x²+15x+56

Sol.-:  b = 15 aur c = 56
Factors of  "c" = 56 = 7×8
aur            "b" = 15 = 7+8

x²+15x+56 = x²+(7+8)x+56
              = x²+7x+8x+56
              = (x²+7x)+(8x+56)
              = x(x+7)+8(x+7)
              = (x+7)(x+8)      Answer.

Q-3.  x²+3x-10 k gununkhand gyat kijiye.

Sol.-:  b = 3 aur c = -10
Factors of  "c" = 10 = 5×2
aur             "b" = 3 = 5 - 2

x²+3x-10 = x²+(5-2)x-10
              = x²+5x-2x-10
              = (x²+5x)-(2x+10)
              = x(x+5)-2(x+5)
              = (x-2)(x+5)      Answer.


Q-4.  x²-25x-84 ka factor kariye.

Sol.-: b = -25 aur c = -84
Factors of  "c" = 84 = 21×4
aur            " b " = 25 = 21+4

x²-25x-84 = x²-(21+4)x-84
              = x²-21x-4x-84
              = (x²-21x)-(4x+84)
              = x(x-21)-4(x-21)
              = (x-21)(x-4)      Answer.


Q-5.  x²+4x-45 ka gununkhand find kijiye.

Sol.-: b = 4 aur c = -45
Factors of  "c" = 45 = 9×5
aur            " b " = 4 = 9-5

x²+4x-45 = x²+(9-5)x-45
              = x²+9x-5x-45
              = (x²+9x)-(5x+45)
              = x(x+9)-5(x+9)
              = (x+9)(x-5)      Answer.



Q-6.  20+9x+x² ka factors gyat kijiye.

Sol.-: Sabse pahle isse standard form me likhe -:
                          x²+9x+20

b = 9 aur c = 20
Factors of  "c" = 20 = 5×4
aur            " b " = 9 = 5+4

x²+9x+20 = x²+(5+4)x+20
              = x²+5x+4x+20
              = (x²+5x)+(4x+20)
              = x(x+5)+4(x+5)
              = (x+5)(x+4)      Answer.



Q-7.  12-7x+x² ka factors gyat kijiye.

Sol.-: sabse pahle isse Standerd form me likhe -:
                             x²-7x+12

b = -7 aur c = 12
Factors of  "c" = 12 = 3×4
aur            " b " = 7 = 3+4

x²-7x+12 = x²-(3+4)x+12
              = x²-3x-4x+12
              = (x²-3x)-(4x-12)
              = x(x-3)-4(x-3)
              = (x-3)(x-4)      Answer.

Also read -:

• square root in few seconds
• Fast calculate square of numbers ending with 5

Q-8.  x²+2x-63 ka gununkhand gyat kijiye.

Sol. -: b = 2 aur c = -63
Factors of  "c" = 63 = 9×7
aur            " b " = 2 = 9-7

x²+2x-63 = x²+(9-7)x-63
              = x²+9x-7x-63
              = (x²+9x)-(7x+63)
              = x(x+9)-7(x+9)
              = (x+9)(x-7)      Answer.


Q-9.  Find the factors of x²-14x-48 

Sol.-: b = -14 aur c = -48
Factors of "c" = 48 = 6×8
aur            "b" = 14 = 6+8

x²-14x-48 = x²-(6+8)x-48
              = x²-6x-8x-48
              = (x²-6x)-(8x+48)
              = x(x-6)-8(x-6)
              = (x-6)(x-8)      Answer.


Q-10.  x²-14x+49 ka gununkhand kijiye.  

Sol.-:   b = -14 aur c = 49
Factors of "c" = 49 = 7×7
aur           "b" = 14 = 7+7

x²-14x+49 = x²-(7+7)x+49
              = x²-7x-7x+49
              = (x²-7x) - (7x-49)
              = x(x-7)-7(x-7)
              = (x-7)(x-7)      Answer.
Or 
              = (x-7)²              Answer.

Also read -:

• Fast multiply by 11 in Hindi
• Fast multiply by 99,999,9999 of any numbers in hindi

Aaj k liye bus itana hi friends,  mujhe puri umeed hai ki aap quadratic expression k factors karna shikh gaye honge.
Agar aapko koi problem ho to aap mujhe comment kark puchh sakte hai. Aapki help kark mujhe bahut khushi hogi.
Aagar aap humari help karna chhahte hai to bas iss post ko share kare. 

Monday, 5 June 2017

Trigonometric ratio se related Questions with solutions


                Trigonometry - 4     


Hello friends ! Maine last post me aapko Basic of trigonometric ratio k baare me full details me bataya tha. Ussi topic ko aage badate huye aaj  mai aapko trigonometric ratio se related question with solution bataunga.

Also read -:

• square root
• what are prime numbers and composite numbers ? 

Shirf bataunga hi nhi balki aapko solve karna bhi shikhaunga. Aur Mujhe puri ummid hai ki, agar aapne carefully samjh liya to aap bhi trigonometric ratio se related question ko easily kar paayenge. To  kya aap ready hai ? aayiye shikhate hai -:

Trigonometric ratio se related questions with solution 


Q-1.  ∆ABC me /_B = 90° hai. AC = 10 aur BC = 6 ,  to "sinθ" ki value batayiye.
Trigonometric ratios se related question-1
Figure-1


Solution-:  Figure-1 , me aap dekh sakte hai ki , lamb BC = 6 aur karn AC = 10 . Hum jaante hai ki ,

                 sinθ = lamb/karn
                          = BC/AC
                          = 6/10
                          = 3/5

Isliye , sinθ = 3/5      Answer.

Also read -:

• Fast multiply by 99,999,9999 of any numbers in hindi
• Fast multiply by 11 in Hindi

Q-2. ∆ABC me /_B = 90° hai.  AB = 3 aur BC = 7 , to "tanθ" = ??
Trigonometric ratios se related question-2
Figure-2


Solution -: Figure-2 se, lamb BC = 7 aur Aadhar AB = 3 
Formula use karte hai -:

              tanθ = lamb/Aadhar
                        = BC/AB
                        = 7/3

Isliye , tanθ = 7/3      Answer.

Q-3. ∆ABC me /_B = 90° hai. AC = 5 and BC = 3 ho to "cosθ"  ki value batayiye.
Trigonometric ratios se related question-3
Figure-3

Solution-: Figure-3 me "θ" , /_C par hai. Isliye "θ" k front side AB ko lamb kahenge aur side BC aadhar hoga.
Ab formula use karte hai -:

                  cosθ = Aadhar/karn
                            = BC/AC
                            = 3/5

Isliye , cosθ = 3/5     Answer.

Also read -:

• square root in few seconds
• Fast calculate square of numbers ending with 5

Q-4.  Figure-4 ki help se "cosec A" ki value bataye.
trigonometric ratio questions-4
Figure-4


Solution-:  Hume cosec A ki value batani hai aur /_A ki front side lamb BC aur Aadhar AB hoga.

Fig. me diya hai -:

Karn AC = 13
Lamb BC = 5

Ab hum jaante hai ki ,

             cosec A = karn/lamb
                            = 13/5

Isliye , cosec A = 13/5     Answer. 

Q-5. Fig.-5 ki help se "sec C" ki value find kariye.
trigonometric ratio question-5
Figure-5


Solution-: /_ C k liye karn AC = 12 aur Aadhar BC = 6 hoga.

Formula -:  
            sec C = Karn/Aadhar
                       = 12/6
                       = 2

Isliye , sec C = 2    Answer.

Q-6. Fig.-6 se "cot C" ka maan gyat kijiye.
Trigonometric ratios se related question-6
Figure-6

Solution -: /_ C k liye Aadhar BC = 6 aur lamb AB = 15 hoga.
Formula -:
              cot C = Aadhar/lamb
                         = 6/15
                         = 2/5

Isliye , cot C = 2/5     Answer.

Also read -:

• Trigonometry ka basic concept in hindi part-1
• Fast cube tricks 1se 99 tak in Hindi

Q-7. fig.-7 ki help se "sin A" , "cos C aur "tan A" ki value bataye.
Figure-7

Solution-:   from fig.-7
/_A k liye , lamb BC = 8 aur Aadhar AB = 6 .
Sin A = l/k
           = BC/AC
           = 8/10
           = 4/5

Isliye , sin A = 4/5     Answer.

tan A = l/A
           = BC/AB
           = 8/6
           = 4/3

Isliye , tan A = 4/3       Answer.

/_C k liye, Lamb AB = 6 aur Aadhar BC = 8 .

cos C = A/K
           = BC/AC
           = 8/10
           = 4/5

Isliye , cos C = 4/5     Answer.

Q-8.  fig.-7 se , "sin C × cosec A "  ki value bataye.

Solution-: from fig.-7
/_C k liye , Lamb AB = 6 aur Aadhar BC = 8 .
Sin C = L/K
           = AB/AC
           = 6/10
           = 3/5

cosec A = K/L
               = AC/BC
               = 10/8
               = 5/4
Ab sin C × cosec A = (3/5)×(5/4)
                                   = 3/4

Isliye , sin C × cosec A = 3/4    Ans.

Also read -:

• Triangle and its types and defination in hindi
• 10 seconds me area of the triangle question solve


Q-9. Fig.-7 ki help se 
"tan A + sec C = ??

Solution-:   tan A = L/A
                                = BC/AB
                                = 8/6
                                = 4/3
sec C = K/A
          = AC/BC
          = 10/8
          = 5/4

tan A + sec C = (4/3)+(5/4)
                         = (16+15)/12
                         = 31/12
Isliye ,  tan A + sec C = 31/12    Ans.


Q-10. Fig.-7 ki help se ,
 cos C - cot A = ??


Solution -: from Q-7 ,  cos C = 4/5
cot A = A/L
          = AB/BC
          = 6/8
          = 3/4

Ab  cos C - cot A =  (4/5)-(3/4)
                               = (16-15)/20
                               = 1/20

Isliye, cos C - cot A = 1/20  Ans.


Also read -:

• Top 10 math tricks post november 2016
• Subtraction trick without using borrowing for 2-digit number

Aaj k liye bas itana hi friends . Aapko agar koi problem ho to aap comment kark puchh sakte hai.
Mai answer jarur dunga.
Agar aapko ye topic pasand aaya to please Facebook , Twitter , whatsapp par share kare.

Thanks for reading  !!!

Sunday, 28 May 2017

Basic of Trigonometric ratios in hindi


               Trigonometry - 3

Aaj mai trigonometry ka next part  lekar aaya hu. Ab tak maine iss topic par 2 part ( part-1 aur part-2) likha hai.
Agar aap part-1 aur part-2 nhi read kiya hai to maine niche diya hai . Aap read kar sakte hai.

Trigonometry part - 1
Trigonometry part - 2

Aaj mai aapko trigonometric ratio k baare me bataunga. Chaliye start karte hai.

Trigonometric ratio Sabse pahle aap niche diye right angled triangle ABC ko dekhiye.
Trig ratio
Figure-1

Uppar diye ∆ABC me aap dekh sakte hai , issk 3 sides hai -:
1. Hypotenuse
2.Opposite
3.Adjacent


1. Hypotenuse (AC) -:
Right angle triangle ABC me side AC  ko "Hypotenuse" kahate hai. Ye side , other 2 sides (AB aur BC) se badi ( big ) hoti hai.
Hypotenuse (AC) ko hindi me karn (कर्ण) kahate hai.
Apni suvidha k liye hum , karn ko "K" se denote karenge.

Also read -:

• Fast multiply by 11 in Hindi
• Fast cube tricks 1se 99 tak in Hindi
• 10 seconds me area of the triangle question solve

2. Opposite (BC) -:
Side BC ko opposite side kahate hai. Humesha yaad rakhe ki , θ k saamne ( front ) wale side ko hi "opposite side" kahenge.
Jaise niche wale right angle triangle ABC k liye opposite side AB  hai. Kyuki isme θ , C par hai.
Trig ratio
Figure-2

Iss side ko hindi me lamb (लम्ब ) kahate hai.
Lamb ko hum "L" se denote karenge.


3. Adjacent (AB) -:
Figure-1 me, θ se attached wale side AB  ko "Adjacent side" kahate hai.
Figure-2 me, side BC ko "Adjacent side " kahenge.
Adjacent side ko hindi me aadhar ( आधार  ) kahate hai.
Aadhar ko hum "A" se denote karenge.

Means,         L = lamb (लम्ब)
                      A = aadhar ( आधार )
                      K = karn ( कर्ण )

Also read -:

Subtraction trick without using borrowing for 2-digit number
• Top 10 math tricks post november 2016

Ab mai batata hu ki , trigonometric ratio kya hota hai .

What is trigonometric ratio ?

Trigonometric ratio ko hindi me " trikonmitiya anupat ( त्रिकोणमितीय  अनुपात ) kahate hai. Ab mai isska definition batata hu.

Definition of trigonometric ratio

Kisi right angled triangle k koi bhi 2 sides k ratio ko " trigonometric ratio kahate hai.

Also read -:

• Iss fast trick se kariye simple equation instantly solve
• Fast calculate square of numbers ending with 5
• square root in few seconds
• Fast multiply by 99,999,9999 of any numbers in hindi

Types of trigonometric ratio

Figure-1 se ,  Suppose karte hai ki ABC ek right angled triangle hai .
Jisme /_ABC = 90° aur  /_CAB = θ ek acute angle hai.
To isliye trigonometric ratio 6 types k hote hai -:
1. sine θ
2. cosine θ
3. tangent θ
4. cotangent θ
5. secant θ
6. cosecant θ



Aayiye ab ek- ek kark inn sabhi k baare me jaante hai.

1. sine θ -:

Opposite side AB aur Hypotenuse side BC k ratio ko sineθ kahate hai. Isse sinθ se denote karte hai.

   sinθ = opposite / hypotenuse
            = BC/AC

Hindi me ,    
                  sinθ = Lamb/karn
Or
                  sinθ = L/K   

2. cosineθ  -:

Adjacent side AB aur Hypotenuse side AC k ratio ko cosineθ kahate hai. Isse cosθ se denote karte hai.

      cosθ = Adjacent/Hypotenuse 
                = AB/AC

Hindi me ,
                   cosθ = Aadhar/karn
Or

                  cosθ = A/K

3. tangentθ -:

Opposite side BC aur Adjacent side AB k ratio ko tangentθ kahate hai. Isse tanθ se denote karte hai.

         tanθ = Opposite/Adjacent
                   = BC/AB

Hindi me ,
                   tanθ = Lamb/Aadhar
Or
                   tanθ = L/A    

4.cotangent -:

Adjacent side AB aur opposite side BC k ratio ko cotangent kahate hai. Isse cotθ se denote karte hai.

        cotθ = Adjacent/Opposite
                 = AB/BC

Hindi me ,
                  cotθ = Aadhar/Lamb
Or
                    cotθ = A/L  

5.  secantθ -:

Hypotenuse AC aur Adjacent AB k ratio ko secantθ kahate hai. Isse secθ  se denote karte hai.

       secθ = Hypotenuse/Adjacent
                 = AC/AB

Hindi me ,
                  secθ = Karn/Aadhar
Or
                   secθ = K/A       

6. cosecantθ -:

Hypotenuse AC aur Opposite AB k ratio ko cosecantθ kahte hai . Isse cosecθ se denote karte hai.

   cosecθ = Hypotenuse/Opposite
                 = AC/BC

Hindi me ,
                  cosecθ = karn/lamb
Or
                  cosecθ = K/L   

Also read -:

• Top-10 math tricks post october 2016
• Kya aapko koi help mathematics me chahiye?
• mathematics operation

Trigonometric ratio ko easily kaise learn kare ?

Mai aapko trigonometric ratio learn ka easy method bataunga. Jisse aapko ye sabhi ratio aashani se learn ho jaayega.

Agar aap hindi me yaad rakhna chahate hai , to aap bas niche diye formula ko yaad rakhe.

                         L A L    
                         K K A

Aap isse लाल / कक्का  k name se learn kar lijiye. Ab apko sabhi ratio yaad ho jaayenge.
Aap kahenge kaise ?
Aayiye dikhata hu ,

sinθ , cosθ aur tanθ ko formula k upper likhe jaisa maine likha.
Ab up to down ⬇aap read kariye. Aap 1st 3 ratio ka formula milega.

               sinθ       cosθ       tanθ
                 L            A             L       
                 K            K             A

cosecθ , secθ aur cotθ k liye down to up ⬆aap read kare .

                 L            A             L       
                 K            K             A     
          cosecθ       secθ         cotθ


Friends , aaj k liye itana hi. apne next part-4 me iss topic se related question ko solve karna shikhenge.

Also read -:

• square root
• what are prime numbers and composite numbers ?
• Cube root just in seconds in Hindi
• Top-10 math tricks post January 2017

Aapko trigonometric ratio ka ye article kaisa laga ? comment kark jarur bataye .
Agar aapko iss topic se koi bhi problem ho rahi ho to aap abhi comment kark puchh sakte. Mujhe aapk comment ka intajaar rahega.

Agar ye article se kisi ki help ho sakti hai ? to please share share kare.


Tuesday, 16 May 2017

Fast multiplication shortcut trick for 3-digit numbers

Fast multiplication shortcut trick for 3-digit numbers
Hello friends ! Aaj mai aapko 3-digit wale numbers ka fast multiply karne ka ek best shortcut trick bataunga.
Iss trick ki help se aap kisi bhi 3-digit wale numbers ka multiply jaldi se jaldi kar sakte hai.

Fast 3-digit multiplication shortcut trick


Also read -:

• square root
• what are prime numbers and composite numbers ?
•  Cube root just in seconds in Hindi

Friends aap to jaante hi honge ki 3-digit k number ka multiply karna kitana jyada tough hota hai. Issme time aur mehanat dono jyada lagta hai.

But meri iss trick ka use kark aapki mehanat aur time dono bachegi. Aap kuch hi seconds me hi question ko solve kar lenge.
Kewal aapko thodi si practice karni padegi.
Iss trick ko start karne se pahle mai aapko kahana chaahunga ki aap bhi ek pen and paper lekar  mere saath-saath karte jaaye. Jisse aapko koi problem nhi hogi.

Also read -:

• Top-10 math tricks post January 2017
• Top-10 math tricks post october 2016
• Kya aapko koi help mathematics me chahiye?

To aayiye aaj ka trick start karte hai.

Fast multiplication shortcut trick for 3-digit numbers


Friends iss trick ka ek simple sa working rule hai. Jisse mai ek example k saath aapko batata hu.


Working rules for 3-digit numbers multiplication shortcut trick

1.   Sabse pahle jis numbers ka multiply karna uss number ko jaise maine niche likha hai waise likh lijiye.

Ex-1.      102 × 104 = ???
Trick -:     
                     1 0 2
                  × 1 0 4       
                                     
2. Upar wale number 102 k 2 ka multiply 104 k 4 se kariye.
(See the image )
3. Ab 102 k 0 ka multiply 104 k 4 se aur 102 k 2 ka multiply 104 k 0 se kark dono k results ko add kar dijiye. (See the image)
Isse answer me likh dijiye.


4. Ab niche diye image ki tarah se multiply kare aur sabhi multiplication results ko add kar dijiye. Addition results ko answer me likhe.

5.   Ab 1×0+0×1 = 0 likhiye. (See the image)
6. Iss step me hume 102 k 1 ka multiply 104 k 1 se karna hai.
Isliye , 102×104 = 10608 answer.

Aayiye kuch aur example dekhte hai -:

Ex-2.        123×112 = ??
Trick-:
1.                   123          [ 3×2=6]
                    ×112      
                        6

2.                   123          [ 2×2=4]
                    ×112          [ 3×1=3]
                         76          [ 4+3=7]

3.                   123             [ 1×2=2]
                    ×112             [ 3×1=3]
                      776             [ 2×1=2]
                                       [2+3+2=7]

4.                   123              [1×1=1]
                    ×112              [2×1=2]
                      3776            [1+2=3]

5.                   123          [ 1×1=1]
                    ×112      
                    13776

 123×1121 = 13,776 answer

 friends ab iss trick ko aur jyada short me karte hai.

Ex.-3.       153×402= ???

Step-1.     3×2 = 6

                 153
             ×  402     
                     6

Step-2.     5×2+3×0 = 10
                ( 0 likhe aur 1 carry next step k liye.)

                 153
             ×  402     
                    06

Step-3       1×2+3×4+5×0 = 14
                   14+1 = 15
                 ( 5 likhe aur 1 carry next step k liye )
         
                 153
             ×  402     
                  506

Step-4.   1×0+5×4 = 20
                20+1 = 21 ( 1 likhe aur 2 carry next step k liye)

                 153
             ×  402     
                1506

Step-5.     1×4 = 4
                 4+2=6

                 153
             ×  402     
              61506

153×402= 61,506 answer


Ex.-4.       216×318 = ???

Step-1.        6×8 = 48  ( 8 likhenge aur 4 carry )
                    216
                ×  318     
                         8

Step-2.      1×8+6×1=14
                  14+4=18 ( 8 likhenge aur 1 carry)
                    216
                ×  318     
                       88

Step-3.       2×8+6×3+1×1 = 35
                  35+1=36 ( 6 likhenge aur 3 carry )
                    216
                ×  318     
                     688

Step-4.      2×1+1×3=5
                  5+3=8

                    216
                ×  318     
                   8688

Step-5.       2×3 = 6

                    216
                ×  318     
                   68688

 216×318 = 68,688 answer


Ex.-5.       378×423 = ???

Step-1.         8×3=24
                   ( 4 likhe aur 2 carry for next step)

                       378
                    × 423    
                            4

Step-2.       7×3+8×2 = 37
                   37+2=39   (9 likhe aur 3 carry)
                       378
                    × 423    
                          94

Step-3.       3×3+8×4+7×2 = 55
                   55+3 = 58 (8 likhe aur 5 carry)
                       378
                    × 423    
                       894

Step-4.       3×2+7×4 = 34
                   34+5=39 ( 9 likhe aur 3 carry)
                       378
                    × 423    
                       9894

Step-5.       3×4=12
                   12+3=15

                       378
                    × 423    
                    159894

378×423 = 1,59,894 answer


Ex-6.     502×768 = ???

Step-1.       2×8=16 ( 6 likhenge aur 1 carry)

                 502
              × 768     
                      6

Step-2.      0×8+2×6=12
                   12+1=13   ( 3 likhenge aur 1 carry)

                 502
              × 768     
                    36

Step-3.     5×8+2×7+0×6=54
                 54+1=55  ( 5 likhenge aur 5 carry)

                 502
              × 768     
                  536

Step-4.     5×6+0×7=30
                 30+5=35 ( 5 likhenge aur 3 carry)

                 502
              × 768     
                  5536

Step-5.    5×7=35
                35+3=38

                 502
              × 768     
             385536

502×768 = 3,85,536 Ans.

Ex.-7.     668×439=???

Step-1.     8×9=72  ( 2 likhenge aur 7 carry)
                      668
                    ×439     
                           2

Step-2.       6×9+8×3= 78
                   78+7=85   ( 5 likhenge aur 8 carry )

                      668
                    ×439     
                          52

Step-3.     6×9+8×4+6×3 = 104
                 104+8 = 112  ( 2 likhenge aur 11 carry)

                      668
                    ×439     
                        252

Step-4.       6×4+6×3 = 42
                   42+11=53   ( 3 likhenge aur 5 carry )

                      668
                    ×439     
                       3252

Step-5.     6×4 = 24
                 24+5 = 29

                      668
                    ×439     
                   293252


668×439= 2,93,252 Ans.


Ex.-8.     755×559=???

Step-1.       5×9 = 45  ( 5 likhenge aur 4 carry)
                     755
                   ×559     
                          5

Step-2.    5×9+5×5 = 70
                70+4=74  ( 4 likhenge aur 7 carry )
                     755
                   ×559     
                        45

Step-3.       7×9+5×5+5×5 = 113
                   113+7 = 120 ( 0 likhenge aur 12 carry)
                     755
                   ×559     
                      045

Step-4.     7×5+5×5 = 60
                 60+12=72    ( 2 likhenge aur 7 carry )

                     755
                   ×559     
                      2045

Step-5.     7×5 = 35
                 35+7 = 42

                     755
                   ×559     
                 422045

  755×559= 4,22,045 Ans.


Ex.-9.     884×744=???

Step-1.      4×4=16 ( 6 likhe aur 1 carry)
                      884
                  ×  744      
                           6

Step-2.      8×4+4×4=48
                  48+1=49 ( 9 likhenge aur 4 carry )
                      884
                  ×  744      
                         96

Step-3.     4×7+8×4+8×4= 92
                  92+4= 96 ( 6 likhenge aur 9 carry )

                      884
                  ×  744      
                       696

Step-4.      8×4+8×7=88
                  88+9 = 97  ( 7 likhenge aur 9 carry )
                      884
                  ×  744      
                      7696

Step-5.       8×7=56
                   56+9=65

                      884
                  ×  744      
                   657696

884×744= 6,57,696 Ans.


Ex.-10.   956×941=???

Step-1.     6×1=6

                       956
                   ×  941     
                            6

Step-2.      5×1+6×4=29 (9 likhenge aur 2 carry)
                       956
                   ×  941     
                          96

Step-3.       9×1+6×9+5×4 =83
                   83+2=85   ( 5 likhe aur 8 carry)
                       956
                   ×  941     
                         596

Step-4.      9×4+5×9=81
                  81+8=89 ( 9 likhenge aur 8 carry)
                       956
                   ×  941     
                        9596

Step-5.      9×9=81
                  81+8=89

                       956
                   ×  941     
                    899596

956×941= 8,99,596 Ans.


Agar aapko abhi bhi koi problem ho to ye video jarur dekhe.




To friends , aapne dekha iss shortcut trick ki help se humne kitana jadli-jaldi multiply karna shikh liya.

Also read -:

• square root in few seconds
• Fast multiply by 99,999,9999 of any numbers in hindi
• Fast multiply by 11 in Hindi

Aapko ye trick kaisa laga ? comment kark jarur bataye.
Aur haa apne dosto k saath jarur share kare.

Thanks for reading !