A Quiz Of Mathematics For You

A Quiz Of Mathematics For You 






There are Total 10 questions in all .Each question is assigned with 2 marks. Attempt all Question. All the questions are from matrices and Determinants . Score will be Displayed at the end of Quiz.




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HOW TO FIND AREA BOUNDED BY THREE LINES AND CIRCLES , AREA UNDER CURVES BY INTEGRATION METHOD



How to find common area of three lines and one circles which are  intersecting at different  points with the help of an example.

 Given Lines and Curves

Consider one circle and three lines whose equations are  given below
x - 1 )2 y2 12            ......................(1)

      y = x                            .....................(2)

y = -√3 (x-2)                      .....................(3)

y = 0                                   .....................(4)



Let us draw these lines and circle in coordinate planes, We can compare the equation of circle  with standard form of circle to find  the coordinate of  centre of the circle is  (1,0)  and radius of both the circles is 1.


How To Draw Figure



HOW TO FIND AREA  BOUNDED BY  THREE LINES AND  CIRCLES1st of all check whether these lines  intersect with circle or not . And if these lines intersect with each other or with circle then what are their coordinates of points of intersections.


Solve (1) and (2)


Putting the value of  'y' from (2) in equation (1), we get
x - 1 )2 x2 12   

    x2  + 12 -2×1×x +x2 12   

     -2x +x2 = 0
    ⇒   2x(-1+x) = 0
Either  2x= 0 or (-1+x) = 0
              x = 0 or x = 1
Now putting the values of x in (2) we get 
x = 0 when x = 0  and y = 1 when x = 1
Therefore points of intersection of (1) and (2) are
O( 0 , 0 ) and A( 1, 1 )


Solve (1) and (3)


Putting the value of  'y' from (3) in equation (1), we get
x - 1 )2 [-√3 (x-2)]2 12 
⇒  x 2 1 2  - 2x +  3 (x-2)2 = 1


  x 2 1- 2x +  3 [ x 2 +4 - 4x] =1

   x 2 1- 2x +  3x 2 + 12 - 12x -1 = 0 
    4x 2 - 14x +12 = 0 

     2x 2 - 7x + 6 = 0 ,    By Factorisation Method

     2x 2 - 4x - 3x  + 6 = 0 



   2x(x-2) -3( x - 2) = 0


     (x-2)( 2x - 3) = 0
Either (x-2) = 0   or ( 2x - 3) = 0
   x = 2 and x = 3/2
To find the values of y , put both the values of  'x' in (3) .i.e. in  y = -√3 (x-2) 

when x = 2 ,    then  y = 0
and    x = 3/2 , then  y = √3 /2

Therefore points of intersection of (1) and (3) are
C( 2 , 0 ) and B( 3/2√3 /2 )

Solve (2) and (3)


is the coordinate of Point of intersection of Line (2) and (3)


Same problem with the help of this Video ⇊




How to Find Required Area


Required Area = Shaded Area = Area of ΔOAL + Area of Curve ABMLA+ Area of Δ BCM
HOW TO FIND AREA  BOUNDED BY  THREE LINES AND  CIRCLES
After simplification , we get  
HOW TO FIND AREA  BOUNDED BY  THREE LINES AND  CIRCLES



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Final words 



Thanks for visiting this website and spending your valuable time to read this post regarding how to find area bounded by three lines and circle  .If you liked this post , don't forget to   share it with your friends to benefit them also ,we shall meet in next post , till then bye and take care......


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HOW TO FIND COMMON AREA OF TWO PARABOLAS ,AREA UNDER TWO PARABOLAS , AREA UNDER CURVES

How to find common area of two parabolas  , Area under two parabolas , Area of region bounded by two parabolas .


Let us consider two parabolas whose equations are given by
y2 =  4ax  --------------  (1)
x2 =  4ay ----------------  (2)


To check whether these parabolas intersect with each others or not And if they intersect then what is/are their point/s of intersection.

How to find Points of Intersection

To find coordinate of points of intersection ,we have to solve equation (1) and (2)

Consider   eq (2) 
 x2 =  4ay 
⇒ y  x2 /4a   ---------------(3)

Putting the value of "y" in equation  (1) ,we get


HOW TO FIND COMMON AREA OF TWO PARABOLAS
Area under Curves
(x2 /4a)2 =  4ax


     x4/16a2 =  4ax

⇒    x4=  64xa3
    x464xa3= 0

Taking 'x' common 

x(x364a3) = 0
Either  x = 0    or x364a3 = 0
⇒ x = 0    or    (x)3(4a)3 = 0 
⇒ x = 0    or    (x)3(4a)3 = 0 

⇒ x = 0    or    (x-4a)[ (x)2(4a)2 + (x)(4a)]   = 0 

⇒ x = 0    or    (x-4a) =  0  or  [ (x)2(4a)2 + (x)(4a)]   = 0 
⇒ x = 0    or    x =  4a  or   (x)2(4a)2 + (x)(4a)   = 0 

Since   x2+ 4a.x + 16a2   = 0   have no real  roots ,because its discriminant is negative, therefore this quadratic equation have complex roots. And these roots are rejected .

To find values of y

Now putting both  values of  "x"  in eq (3) i. e .  x2 /4a   ,we get
1st  put x = 0 
y = 0 / 4a = 0         when x = 0 then y = 0
and put x = 4 
y =  (4a)2 /(4a)
y =  4a                    ⇒ when x = 4a then y = 4a

Hence two points of intersection of (1) and (2)   O(0,0) and A (4a , 4a) .
Now draw two parabolas using their points of intersections as drawn in given picture.

How to Find Required Area

Now to find the area enclosed between two Parabolas.
Required Area = shaded Area =Area OLAMO - Area ONAMO

HOW TO FIND COMMON AREA OF TWO PARABOLAS

Watch this video to remove your doubts if any




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Final words 



Thanks for visiting this website and spending your valuable time to read this post regarding how to find area bounded by two parabolas  .If you liked this post , do share it with your friends to benefit them also ,we shall meet in next post , till then bye and take care......

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HOW TO FIND SAMPLE SPACE FOR TOSSING OF FIVE COINS

Let us discuss the coin toss probability formula, sample space of tossing 1 coin , sample space of tossing 2 coins , sample space of tossing 5 coins , How to make a tree diagram.

To find the sample space for tossing of one  coin

HOW TO FIND SAMPLE SPACE FOR TOSSING OF one COIN


we know that there are two outcomes in tossing of coin . Out of two outcome one is H and second is T.
So  Put H and T in sample space 'S'
S = {H , T}

To find the sample space for tossing of two coins

HOW TO FIND SAMPLE SPACE FOR TOSSING OF two COINS1st put  22   = 4  elements in a set of sample space,

1st character of 1st two elements must be H and 1st character of last two elements must be T .
2nd character of all the  elements must be H and T alternatively


 S={ HH , HT , TH , TT}

To find the sample space for tossing of three coins


1    Put  23  = 8  elements in a set of sample space,
2    1st character of 1st four elements must be H and 1st character of last four elements must be T .
HOW TO FIND SAMPLE SPACE FOR TOSSING OF FOUR COINS
3   2nd character of 1st ,2nd ,5th and 6th  elements must be H and  2nd character of 3rd, 4th, 7th and 8th  elements must be T alternatively
4    Last character of all the  elements must be H and T alternatively.
Therefore sample  space  'S'  is given by

S = { HHH , HHT , HTH , HTT ,THH, THT , TTH , TTT  }



To find the sample space for tossing of Four coins

1    Put  24  = 16  elements in a set ,
2    1st character of 1st eight elements must be H and 1st character of last eight elements must be T.

3   2nd character of 1st four and 9th to 12th  elements must be H and  2nd character of 5th, 6th , 7th and 8th and last four elements must be T alternatively.
4  3rd character for 1st , 2nd , 5th , 6th , 9th , 10th , 13th and 14th elements must be H and for remaining characters ( 3rd , 4th ,7th ,8th , 11th ,12th ,15th and 16th)  it must be T.
5    Last character of all the  elements must be H and T alternatively.

Hence sample space ' S' is given by 

S = { HHHH , HHHT , HHTH , HHTT , HTHH , HTHT , HTTH , HTTT , THHH , THHT , THTH , THTT , TTHH , TTHT , TTTH , TTTT }


To find the sample space for tossing of Five Coins

1    Put  25  = 32  elements in a set ,
2    1st character of 1st sixteen elements must be H and 1st character of last sixteen elements must be T.

3   2nd character of 1st eight and  17th to 24th  elements must be H and  2nd character of 9th to 16th and 25th to 32nd  elements must be T alternatively.

4  3rd character for 1st four , 9th to 12th , 17th to 20th , 25th to 28th elements must be H , And  3rd character for 5th to 8th , 13th to 16th , 21st to 24th and 29th to 32nd elements  must be T.

5  4th character for 1st two ,5th and 6th ,9th and 10th , 13th and 14th , 17th and 18th, 21st and 22nd , 25th and 26th, 29th and 30th and 32nd must be H  And 3rd and 4th , 7th and 8th , 11th and 12th and 15th and 16th , 19th and 20th , 23rd and 24th , 27th and 28th , 31st and 32nd elements must be T.

6    Last character of all the  elements must be H and T alternatively.

Hence sample space ' S' is given by 


S = { HHHHH , HHHHT , HHHTH , HHHTT , HHTHH , HHTHT , HHTTH , HHTTT , HTHHH, HTHHT, HTHTH , HTHTT, HTTHH, HTTHT, HTTTH, HTTTT, THHHH , THHHT , THHTH , THHTT , THTHH , THTHT , THTTH , THTTT ,TTHHH , TTHHT, TTHTH , TTHTT, TTTHH, TTTHT , TTTTH , TTTTT }

Watch this video for finding sample space of coins

Final Words

Thanks for giving your precious time to read this post regarding  probability formula, sample space of tossing 1 coin ,  sample space of tossing 5 coins , How to make a tree diagram.. If you liked it then share it with your near and dear ones to benefit them. we shall meet in next post with another beneficial article till then bye ,take care.......
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USING METHOD OF INTEGRATION ,HOW TO FIND AREA OF TRIANGLE BOUNDED BY THREE LINES

Using method of integration find the area of triangle,using the method of integration find the area of the region bounded by the lines,area of triangle by integration method 

Using Method of Integration , How to find the area of triangle bounded by three lines 

2x + y = 0 , 3x - 2y = 6  and x - 3y  + 5 = 0

Solution


Given lines are
2x + y  = 4   --------  (1)
3x - 2y = 6  ---------  (2) and 
x - 3y  = -5  ---------  (3)


If these lines are intersecting then we have to find their coordinates of points of intersection .

To Find Coordinate of Point A

Multiply (1) by 2 and adding to (2) , we get
4x + 2y + 3x - 2y = 8 + 6
7x = 14 ⇒ x =2 
Putting x = 2 in (1) , we get 
USING METHOD OF INTEGRATION ,HOW TO FIND AREA OF TRIANGLE BOUNDED  BY THREE LINES
Area under Curve
2(2) + y = 4
4 + y = 4 ⇒ y = 0
∴ (1) and (2) meets at point A(2,0).


To Find Coordinate of Point B

To find point of intersection (2) and (3);
Multiply (3) by -3 and adding to (2) , we get
3x - 2y -3x +9y  = 6 +15 
7y = 21  ⇒ y = 3 

Putting y = 3 in (3) , we get 
x-3(3)  = -5  
⇒ x = -5 +9  ⇒ x = 4 
∴ (2) and (3) meets at point B(4,3).


To Find Coordinate of Point C

To find point of intersection (1) and (3);
Multiply (1) by 3 and adding to (3) , we get
6x + 3y + x - 3y = 12 - 5 
7x = 7  ⇒ x =1 

Putting x = 1 in (1) , we get 
2(1) + y  = 4  
⇒ y =4 - 2  ⇒ y = 2 
∴ (1) and (3) meets at point C(1,2).

we get points of intersection of (1) and (2)  A(2,0),  points of intersection of (2) and (3) B(4,3) and points of intersection of (1) and (3) C(1,2).


Required Area = Shaded Area   =  Area DCBED - Area DCAD -Area ABEA

How to find the area of triangle bounded by three lines


Also read my previous post  How to find area bounded by two circles 

For better understanding watch this video 



Conclusion 


 Thanks  for visiting this website and spending your precious time to read how to find area of triangle Using method of integration find the area of triangle,using the method of integration find the area of the region bounded by the lines,area of triangle by integration method.

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