RD SHARMA Solutions for Class 10 Maths Chapter 13 - Areas Related to Circles

Chapter 13 - Areas Related to Circles Exercise Ex. 13.1

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5

Solution 6

Solution 7

Solution 8

Solution 9

Solution 10

Solution 11

 

Area of a circle = πr2 = (22/7) × 28 × 28 = 2464 cm2

Solution 12

Solution 13



Solution 14

Solution 15

Solution 16

Solution 17

Solution 18

Solution 19

Solution 20

 

 

Solution 21

Solution 22

Solution 23

Solution 24

 

 

 

Solution 25

 

Solution 26

Solution 27

 

 

 

Solution 28

table attributes columnalign left end attributes row cell text Area   of   the   surrounding   path  =  end text pi left parenthesis R squared minus r squared right parenthesis end cell row cell R equals left parenthesis 17.5 divided by 2 right parenthesis plus 2 equals 8.75 plus 2 equals 10.75 text   m   is   the   outer   radius. end text end cell row cell r equals 17.5 divided by 2 equals 8.75 text   m   is   the   inner   radius. end text end cell row cell text Area   of   the   path  =  end text pi left parenthesis R squared minus r squared right parenthesis equals pi left parenthesis 10.75 squared minus 8.75 squared right parenthesis end cell row cell equals 3.14 cross times 39 equals 122.46 text   m end text to the power of text 2 end text end exponent end cell row cell text Total   cost  =  122.46 end text cross times text 25  =   Rs. 3061.50   end text end cell end table

Solution 29



Solution 30

Solution 31



Chapter 13 - Areas Related to Circles Exercise Ex. 13.2

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5

Solution 6

Solution 7

Solution 8

Solution 9

Solution 10

Solution 11

Solution 12



Solution 13


Solution 14

Solution 15

Solution 16

Solution 17

Solution 18

Solution 19

Solution 20

 

table attributes columnalign left end attributes row cell text A   minute   hand   covers   360 end text to the power of text o end text end exponent text   in   60   minutes. end text end cell row cell text Hence ,  the   minute   hand   covers   end text fraction numerator text 360 end text over denominator text 60 end text end fraction equals 6 degree text   in   one   minute. end text end cell row cell text The   minute   hand   covers   6 end text cross times text 5  =  30 end text to the power of text o end text end exponent text   in   5   minutes. end text end cell row cell text The   length   of   minute   hand   is   14   cm .  end text end cell row cell text Area   swept = end text text 30 end text to the power of text o end text end exponent over text 360 end text to the power of text o end text end exponent cross times pi R squared equals 1 over 6 cross times 22 over 7 cross times 14 squared equals 102.67 text   cm end text to the power of text 2 end text end exponent end cell end table


 

*Answer does not match with textbook answer.

Solution 21

 

table attributes columnalign left end attributes row cell left parenthesis text i end text right parenthesis text   Length   of   an   arc end text equals fraction numerator theta over denominator 360 degree end fraction cross times 2 pi R end cell row cell equals fraction numerator 60 degree over denominator 360 degree end fraction cross times 2 cross times 22 over 7 cross times 21 end cell row cell equals 1 over 6 cross times 2 cross times 22 over 7 cross times 21 equals 22 text   cm end text end cell end table



table attributes columnalign left end attributes row cell left parenthesis i i right parenthesis text   Area   of   the   sector = end text fraction numerator theta over denominator 360 degree end fraction cross times pi R squared end cell row cell equals fraction numerator 60 degree over denominator 360 degree end fraction cross times pi cross times 21 squared equals 1 over 6 cross times 22 over 7 cross times 21 squared equals 231 text   cm end text to the power of text 2 end text end exponent end cell end table


Solution 22

 

 

*Note: Answer given in the book is incorrect.

Solution 23

 

 

 

Solution 24

Solution 25

Solution 26

Solution 27




Chapter 13 - Areas Related to Circles Exercise Ex. 13.3

Solution 1

Solution 2

Solution 3

Solution 4

 

Solution 5

Solution 6

Solution 7

Solution 8

Solution 9

Solution 10

Chapter 13 - Areas Related to Circles Exercise Ex. 13.4

Solution 1

Solution 2



Solution 3



Solution 4



Solution 5





Solution 6

Solution 7

 

 

 

Solution 8

 

 

Solution 9

Solution 10




Solution 11

Solution 12

Solution 13


table attributes columnalign left end attributes row cell text Area   of   the   rectangle   ABCD  =  AB end text cross times text AD end text end cell row cell text AD   is   the   hypotenuse   of   the   right   angled   end text capital delta text AED. end text end cell row cell A D equals square root of A E squared plus E D squared end root equals square root of 9 squared plus 12 squared end root equals square root of 225 equals 15 text   cm end text end cell row cell text Hence ,  area   of   the   rectangle  =  end text 20 cross times 15 equals 300 text   cm end text to the power of text 2 end text end exponent end cell row cell text Area   of   the   right   end text capital delta text AED  =  end text 1 half cross times A E cross times D E equals 1 half cross times 9 cross times 12 end cell row cell equals 54 text   cm end text to the power of text 2 end text end exponent end cell row cell text Area   of   the   semicircle  =  end text fraction numerator text 1 end text over denominator text 2 end text end fraction cross times pi cross times open parentheses 15 over 2 close parentheses squared end cell row cell equals 1 half cross times 3.14 cross times 7.5 squared equals 88.31 text   cm end text to the power of text 2 end text end exponent end cell row cell text Area   of   the   shaded   region end text end cell row cell text = Area   of   the   rectangle  +  Area   of   the   semicircle end text end cell row cell negative text Area   of   the   right   triangle end text end cell row cell text = end text 300 plus 88.31 minus 54 equals 334.31 text   cm end text to the power of text 2 end text end exponent end cell end table

Solution 14

Solution 15

Solution 16

Solution 17

Solution 18

 

Solution 19

Solution 20

 

table attributes columnalign left end attributes row cell text Area   of   the   square  =  end text O A squared equals 21 squared equals 441 text   cm end text to the power of text 2 end text end exponent end cell row cell text Diagonal   of   the   square   OB  =  end text square root of text 2 end text end root O A equals 21 square root of 2 text   cm end text end cell row cell text Diagonal   of   the   square   is   equal   to   the   radius end text end cell row cell text of   the   circle. end text end cell row cell text Hence ,  area   of   the   quadrant  =  end text fraction numerator text 1 end text over denominator text 4 end text end fraction cross times pi r squared equals fraction numerator text 1 end text over denominator text 4 end text end fraction cross times 22 over 7 cross times left parenthesis 21 square root of 2 right parenthesis squared end cell row cell equals fraction numerator text 1 end text over denominator text 4 end text end fraction cross times 22 over 7 cross times 441 cross times 2 equals 693 end cell row cell text Hence ,  area   of   the   shaded   region end text end cell row cell text =  Area   of   the   quadrant end text minus text Area   of   the   square  =  693 end text minus 441 end cell row cell equals 252 text   cm end text to the power of text 2 end text end exponent end cell end table

Solution 21

Solution 22
Solution 23

Solution 24

Solution 25

*Answer is not matching with textbook.

Solution 26

Solution 27

Solution 28

Consider the following figure:

 

  

 

Solution 29

(i)

According to the figure in the question, there are 6 triangles.

Area of one triangle is 9 cm2.

Area of hexagon = 6 × 9 = 54 cm2

 

(ii)

Area of the equilateral triangle = 9 cm2

Area of the circle in which the hexagon is inscribed

=

=

=

= 65.26 cm2

 

NOTE: Answer not matching with back answer.

Solution 30

Solution 31



Solution 32

Solution 33

Solution 34


table attributes columnalign left end attributes row cell text Area   of   the   rectangular   card  =  14 end text cross times text 7  =  98   cm end text to the power of text 2 end text end exponent end cell row cell text Maximum   diameter   of   a   circle   inscribed   in   end text end cell row cell text the   given   rectangle  =  7   cm. end text end cell row cell text Area   of   the   circles  =  end text pi cross times left parenthesis 7 divided by 2 right parenthesis squared end cell row cell text Area   of   two   such   circles  =  2 end text cross times pi cross times left parenthesis 7 divided by 2 right parenthesis squared equals 77 text   cm end text to the power of text 2 end text end exponent end cell row cell text Area   of   the   remaining   card  =  98   cm end text to the power of text 2 end text end exponent minus 77 text   cm end text to the power of text 2 end text end exponent equals 21 text   cm end text to the power of text 2 end text end exponent end cell end table




Solution 35

Solution 36

 

table attributes columnalign left end attributes row cell text Perimeter   of   the   shaded   region = end text end cell row cell text Length   of   end text stack P S R with overparenthesis on top plus text   Length   of end text stack P A Q with overparenthesis on top text   end text plus text   length   of   end text stack Q T R with overparenthesis on top end cell row cell equals pi cross times 5 plus pi cross times 3.5 plus pi cross times 1.5 end cell row cell equals pi cross times 10 end cell row cell equals 31.4 text   cm end text end cell end table




Solution 37

Solution 38

Solution 39

Solution 40

Solution 41

Solution 42

Solution 43



Solution 44

Solution 45

Solution 46

Solution 47

Solution 48

Since the data given in the question seems incomplete and inconsistent with the figure, we make the following assumptions to solve it:

1. ABCD a symmetric trapezium with AD = BC

2. AD = BC = 14 cm (the distance between AB and CD is not 14 cm)

Draw perpendiculars to CD from A and B to divide the trapezium into one rectangle and two congruent right angled triangles.

The base of the right angled triangle=(CD - AB) ÷ 2

=(32 - 18) ÷ 2=7 cm

cosD = base ÷ hypotenuse = 7 ÷ 14 =1/2

mD = 60° 

Hence, mA = 120° 


table attributes columnalign left end attributes row cell s i n angle D equals fraction numerator p e r p e n d i c u l a r over denominator h y p o t e n u s e end fraction equals p over 14 equals fraction numerator square root of 3 over denominator 2 end fraction end cell row cell rightwards double arrow p equals 7 square root of 3 text   cm end text end cell row cell text Area   of   the   trapezium  =  end text fraction numerator text 1 end text over denominator text 2 end text end fraction p left parenthesis A B plus C D right parenthesis equals 1 half cross times 7 square root of 3 left parenthesis 18 plus 32 right parenthesis end cell row cell equals 175 square root of 3 text   cm end text to the power of text 2 end text end exponent end cell row cell text The   trapezium   has   four   parts   of   circles , end text end cell row cell text two   with   angles   60 end text degree text   at   centre   and   two   with   angles   120 end text degree text   at   centre. end text end cell row cell text So   in   total   it   has   2 end text cross times fraction numerator text 1 end text over denominator text 6 end text end fraction t h plus 2 cross times 1 third r d equals text one   full   circle. end text end cell row cell text Area   of   the   circle = end text pi R squared equals 22 over 7 cross times 7 squared equals 154 text   cm end text to the power of text 2 end text end exponent end cell row cell text Area   of   the   shaded   part end text end cell row cell text = Area   of   the   trapezium end text minus text Area   of   the   circle end text end cell row cell text = end text 175 square root of 3 text   cm end text to the power of text 2 end text end exponent minus 154 text   cm end text to the power of text 2 end text end exponent end cell row cell equals 149.1 text   cm end text to the power of text 2 end text end exponent end cell end table

*Answer is not matching with textbook answer.

Solution 49

Solution 50

Solution 51

 

Solution 52

According to the question,

Side of a square is 28 cm.

Radius of a circle is 14 cm.

Required area = Area of the square + Area of the two circles - Area of two quadrants …(i)

Area of the square = 282 = 784 cm2

Area of the two circles = 2πr2

= 

= 1232 cm2

Area of two quadrants =

=

= 308 cm2

Required area = 784 + 1232 - 308 = 1708 cm2

 

NOTE: Answer not matching with back answer.

Solution 53

According to the question,

For a cylindrical tank

d = 2 m, r = 1 m, h = 5 m

Volume of the tank = πr2h

  =

 =

After recycling, this water is used irrigate a park of a hospital with length 25 m and breadth 20 m.

If the tank is filled completely, then

Volume of cuboidal park = Volume of tank

h = 0.0314 m = 3.14 cm = p cm

Solution 54

Join OB.

Here,   is a right triangle.

By Pythagoras theorem,

Therefore, radius of the circle (r)

Area of the square

Area of the quadrant of a circle

Area of the shaded region = Area of quadrant - Area of square

= 128.25 cm2

Solution 55

Join AC.

Here,   is a right triangle.

By Pythagoras theorem,

Therefore, diameter of the circle = 4 cm

So, the radius of the circle (r) = 2 cm

Area of the square

Area of the circle

Area of the shaded region = Area of the circle - Area of square

= 4.56 cm2

Chapter 13 - Areas Related to Circles Exercise 13.69

Solution 1

Correct Option :- (D)

begin mathsize 12px style Let space radivs space of space circle space be space straight r.
circumferance space of space circle space equals space 2 πr
area space of space circle space equals space πr squared
Given comma area space equals space circumference
space space space space space space space space space space space space space space space space space space space space space rightwards double arrow πr squared equals 2 πr
space space space space space space space space space space space space space space space space space space space space space rightwards double arrow space box enclose straight r equals 2 end enclose space
diameter space equals space 2 straight r
space space space space space space space space space space space space space space space space space equals space 4 space space space space space space space space space space space space space space end style

Solution 2

According to the question,

Circumference of a circle =

 =

 = 44 cm 

Solution 3

Correct option (c)

begin mathsize 12px style L e n g t h space o f space w i r e space equals space C i r c u m t e r n c e space o f space C i r c l e
left parenthesis l right parenthesis space equals space 2 πr
straight l equals 2 straight pi cross times left parenthesis 56 right parenthesis
equals space 2 cross times 22 over 7 cross times 56
equals 352 space cm
If space bent space in space form space of space square space then space lenth space of space wire space equals space Perimeter space of space squre space if space side space of space squre space is space straight a space
then space straight l space equals space 49
rightwards double arrow straight a space equals space 352 over 4
rightwards double arrow straight a space equals space 88 space cm
area space of space square space equals space straight a squared
equals space 7744 cm squared


end style

Solution 4

correct option - (c)

begin mathsize 12px style a r e a space o f space s q u a r e space equals space 81 space c m squared
s i d e space o f space s q u a r e space equals space square root of 81
equals space 9 space c m
L e n g t h space o f space w i r e space equals space p e r i m e t e r space o f space s q u a r e
equals space 4 cross times 9
equals 36 space c m
w i r e space i s space b e n t space i n t o space s e m i minus c i r c l e space t h e n space
L e n g h t space o f space w i r e space equals space h a l f space o f space c i r c u m f e r n c e space plus 2 r
rightwards double arrow 36 equals πr plus 2 straight r
rightwards double arrow box enclose straight r equals fraction numerator 36 over denominator straight pi plus 2 end fraction end enclose
area space of space semi minus circle space equals space πr squared over 2
equals space straight pi over 2 cross times open parentheses fraction numerator 36 over denominator straight pi plus 2 end fraction close parentheses squared
equals straight pi over 2 open parentheses fraction numerator 36 over denominator straight pi plus 2 end fraction close parentheses squared
equals straight pi over 2 open parentheses fraction numerator 36 cross times 7 over denominator 36 end fraction close parentheses squared
equals fraction numerator 22 over denominator 7 cross times 2 end fraction cross times 7 cross times 7
equals 77 cm squared end style

Solution 5

correct option - (b)

begin mathsize 12px style L e t space o u t e r space r a d i u s space i s space b space a n d space i n n e r space r a d i u s space i s space a
G i v e n space comma space 2 pi b minus 2 pi a equals 132
rightwards double arrow 2 pi open parentheses b minus a close parentheses equals 132
rightwards double arrow open parentheses b minus a close parentheses space equals space fraction numerator 132 over denominator 2 cross times 22 end fraction cross times 7
w i d t h space o f space p a t h space rightwards double arrow space open parentheses b minus a close parentheses equals 21 end style

Solution 6

Correct Option: d

begin mathsize 12px style Let space wheel space take space straight n space revolutions
space After space 1. space revolution space distance space covered space by space wheel space equals space 2 πr
rightwards double arrow After space straight n space revolution space distance space covered space by space wheel space equals space 2 πrn
rightwards double arrow space Given comma space 2 πrn space equals space 11000 space straight m
rightwards double arrow 2 straight pi space cross times space open parentheses 0.25 close parentheses straight n space equals space 11000
rightwards double arrow straight n space equals space fraction numerator 11000 over denominator 2 cross times begin display style 22 over 7 end style cross times begin display style 1 fourth end style end fraction
straight n equals space 7000 end style

Solution 7

Correct Option: (c)

Perimeter space of space circular space path space equals space 2 space πr
Let space outer space radius space be space straight b space and space inner space radius space be space straight a space
rightwards double arrow space given space fraction numerator 2 πb over denominator 2 πa end fraction equals 23 over 22 rightwards double arrow straight b over straight a equals 23 over 22 rightwards double arrow box enclose straight b equals 23 over 22 straight a end enclose
also space width space comma space straight b minus straight a space equals space 5
from space circle enclose 1 comma circle enclose 2
23 over 22 straight a minus straight a space equals space 5
rightwards double arrow straight a over 22 equals 5
rightwards double arrow straight a space equals space 110 space straight m
diameter space equals space 2 straight a
equals space 220 space straight m

Solution 8

Correct option - (c)

begin mathsize 12px style 2 πr space equals space 100
rightwards double arrow straight r space equals space 50 over straight pi cm
Image space Pending space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space
OA squared plus OB squared equals space AB squared
OA space equals space OB space equals space straight r
rightwards double arrow 2 straight r squared equals AB squared
rightwards double arrow AB space equals space square root of 2 straight r end root
equals space fraction numerator 50 square root of 2 over denominator straight pi end fraction cm
end style

Solution 9

Correct option (c)

begin mathsize 12px style Imege space pending space
AB space equals space 42 space cm
straight E space is space mid space point space of space AB
Hence space EB space equals space 21 space cm
Also space less than ABC space equals space 60 to the power of 0
But space less than space ABO space equals space 1 half less than ABC
equals space 30 to the power of 0
In space increment OEB
tan space 30 to the power of 0 equals space OE over EB
rightwards double arrow OE space equals space EB space tan space 30 to the power of 0
equals space fraction numerator 21 over denominator square root of 3 end fraction
radius space space space straight r space equals space OE
straight r space equals space fraction numerator 21 over denominator square root of 3 end fraction
area space of space circle space equals space πr squared
equals space straight pi open parentheses fraction numerator 21 over denominator square root of 3 end fraction close parentheses squared
equals space straight pi cross times fraction numerator 21 cross times 21 over denominator 3 end fraction
equals space 22 over 7 cross times fraction numerator 21 cross times 21 over denominator 3 end fraction
equals space 462 space cm squared end style

Solution 10

Correct Option ( d )

begin mathsize 12px style πr squared equals 154
straight r squared space equals space fraction numerator 154 over denominator straight pi space end fraction space space equals space 154 over 22 cross times 7
straight r squared space equals space 49
straight r space equals space 7 space cm
OE space equals space straight r
Image space pending space
AB equals BC equals space AC
OE perpendicular AB space and space AE space equals EB
and space angle space EBC space equals space 60 to the power of 0
But space angle space EBO space equals space 1 half space angle space EBC space equals space 30 to the power of 0
In space increment space OEB space comma space tan space 30 to the power of 0 equals space OE over EB
rightwards double arrow EB space equals space fraction numerator OE over denominator tan space 30 to the power of 0 end fraction
equals 7 square root of 3
rightwards double arrow AB space equals space 2 EB
equals space 14 space square root of 3
perimeter space equals space AB space plus BC space plus CA
equals space 3 space AB
equals space 42 square root of 3
equals space 72.7 space cm
end style

Solution 11

Correct option (a)

begin mathsize 12px style Image space pending space space
For space the space largest space triangle. space with space largest space area. space we space need space that space base space of space triangle space must space be space large.
hence space AB space is space Base space of space triangle
Let space straight C space be space 3 rd space vertex space of space triangle.
area space equals 1 half cross times AB cross times space straight h
equals space 1 half cross times 2 straight r space cross times straight h
equals space straight r space straight h
largest space possible space value space of space straight h space is space straight r space
Hence space area space equals space straight r space cross times straight r
equals space straight r squared end style

Solution 12

Let r be the radius of the circle.

2pr = 88

Perimeter of a triangle = 30 cm

Semi-perimeter = 15 cm

Hence,

Area of a triangle = r × s …(r = incircle radius, s =semi perimeter)

= 14 × 15

= 210 cm2 

Solution 13

Correct option - (c)

begin mathsize 12px style πr squared equals space 220
rightwards double arrow straight r squared space equals fraction numerator space 220 over denominator 22 end fraction cross times 7
straight r squared equals space 70
straight r space equals space square root of 70 cm
Image space Pending space
OA space equals space OB equals straight r
ABCD space is space straight a space squre.
OA squared plus OB squared equals space AB squared
rightwards double arrow AB to the power of 2 space equals space 2 straight r squared end exponent
rightwards double arrow AB space equals space square root of 2 straight r end root
equals square root of 140
area space equals space left parenthesis AB right parenthesis squared
equals space 140 space cm squared

end style

Chapter 13 - Areas Related to Circles Exercise 13.70

Solution 14

begin mathsize 12px style Correct space option colon space open parentheses straight d close parentheses
Initially comma space 2 πr subscript 1 space equals space 4 straight pi
straight r subscript 1 space equals space 2
finally comma space 2 πr subscript 2 space equals space 8 straight pi
straight r subscript 2 space equals space 4
Initially space area space equals space πr subscript 1 superscript 2 space equals space 4 straight pi
final space area space equals space πr subscript 2 superscript 2
space space space space space space space space space space space space space space space space space space space space equals space 16 space straight pi
fraction numerator final space area over denominator Initial space area end fraction space equals space fraction numerator 16 straight pi over denominator 4 straight pi end fraction space equals space 4
area space is space quadrupled. end style

Solution 15

begin mathsize 12px style Coerrect space option colon space open parentheses straight b close parentheses
Initial space radius space equals space straight r
final space radius space equals space diminshed space by space 10 percent sign
space space space space space space space space space space space space space space space space space space space space space space space space equals space 90 percent sign space of space initial
space space space space space space space space space space space space space space space space space space space space space space space space equals space 0.9 space straight r
Initial space area space equals space πr squared
final space area space equals space 0.81 space straight r squared straight pi
area space is space diminished space by space equals space fraction numerator Initial space area space space minus space final space area space over denominator Initial space area end fraction space cross times space 100
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space.19 space cross times space 100
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space 19 percent sign
end style

Solution 16

begin mathsize 12px style Correct space option colon space open parentheses straight b close parentheses
area space of space square space equals space straight a squared
area space of space circle space equals space πr squared
given space straight a squared space equals space πr squared
straight a space equals space square root of straight pi straight r space space space space space space space space..... open parentheses 1 close parentheses
fraction numerator perimeter space of space square over denominator perimeter space of space circle space end fraction space equals space fraction numerator 4 straight a over denominator 2 πr end fraction
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space fraction numerator 4 square root of straight pi straight r over denominator 2 πr end fraction space space space space space space space space from space open parentheses 1 close parentheses
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space fraction numerator 2 over denominator square root of straight pi end fraction end style

Solution 17

begin mathsize 12px style Correct space option colon space open parentheses straight b close parentheses
For space the space largest space triangle. space with space largest space area. space we space need space that space base space of space triangle space must space be space large.
hence space AB space is space Base space of space triangle
Let space straight C space be space 3 rd space vertex space of space triangle.
area space equals 1 half cross times AB cross times space straight h
equals space 1 half cross times 2 straight r space cross times straight h
equals space straight r space straight h
largest space possible space value space of space straight h space is space straight r space
Hence space area space equals space straight r space cross times straight r
equals space straight r squared end style

Solution 18

begin mathsize 12px style Correct space option colon space open parentheses straight b close parentheses
Let space radius space of space circle space equals space straight r
Let space side space of space triangle space equals space straight a
given comma space 2 straight r space equals space straight a
fraction numerator area space of space circle over denominator area space of space equilateral space triangle end fraction space equals space fraction numerator πr squared over denominator begin display style fraction numerator square root of 3 over denominator 4 end fraction end style straight a squared end fraction
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space fraction numerator πr squared over denominator begin display style fraction numerator square root of 3 over denominator 4 end fraction end style space cross times space 4 straight r squared end fraction
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space bevelled fraction numerator straight pi over denominator square root of 3 end fraction
end style

Solution 19

begin mathsize 12px style Correct space option colon space open parentheses straight b close parentheses
area space of space circle space 1 space equals space πr subscript 1 superscript 2
area space of space circle space 2 space equals space πr subscript 2 superscript 2
area space of space circle space 3 space equals space πr squared
Given comma space space πr subscript 1 superscript 2 space plus space space πr subscript 2 superscript 2 space equals space πr squared
straight r subscript 1 superscript 2 space plus space straight r subscript 2 superscript 2 space equals space straight r squared end style

Solution 20

begin mathsize 12px style Correct space option colon space open parentheses straight c close parentheses
perimeter space of space semicircle space equals space πr space plus space 2 straight r
2 straight r space plus space πr space equals space 36
straight r space equals space fraction numerator 36 over denominator straight pi space plus space 2 end fraction
diameter space equals space 2 straight r
space space space space space space space space space space space space space space space space space space space equals space fraction numerator 72 over denominator straight pi space plus space 2 end fraction
space space space space space space space space space space space space space space space space space space space equals space 14 space cm end style

Solution 21

begin mathsize 12px style Correct space option colon space open parentheses straight a close parentheses
OA space equals space 7 space cm
circle space with space angle space 2 straight pi space has space circumference
equals space 2 πr
straight pi over 3 space radian space equals space 60 degree
for space angle space 1 space radian space equals space straight r
for space straight pi over 3 space radian space equals space straight pi over 3 straight r
straight pi over 3 space cross times space 7
22 over 7 space cm
perimeter space equals space OA space plus space OB space plus space circumference
space space space space space space space space space space space space space space space space space space space space space equals space 7 space plus space 7 space plus space 22 over 3
space space space space space space space space space space space space space space space space space space space space space equals space 14 space plus space 22 over 3
space space space space space space space space space space space space space space space space space space space space space equals space 64 over 3 space cm
end style

Solution 22

begin mathsize 12px style Correct space option space colon space open parentheses straight b close parentheses
any space sector space of space circle space with space radius space straight r.
circumference space open parentheses AB close parentheses space equals space open parentheses fraction numerator straight theta over denominator 2 straight pi end fraction close parentheses space 2 πr
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space θr space space space space space space space space space space...... open parentheses 1 close parentheses
area space equals space open parentheses fraction numerator straight theta over denominator 2 straight pi end fraction close parentheses πr squared
space space space space space space space space space space space equals space straight theta over 2 space straight r squared space space space space space...... open parentheses 2 close parentheses
perimeter space of space OAB space equals space OA space plus space AB space plus space OB
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space straight r space plus space straight r space plus space AB
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space 2 straight r space plus space AB
2 straight a space equals space 13 space space plus space AB
AB space equals space 16
From space open parentheses 1 close parentheses space θr space equals space 16
straight theta space equals space fraction numerator 16 over denominator 6.5 end fraction space radian
from space open parentheses 2 close parentheses
area space equals space straight theta over 2 space straight r squared
space space space space space space space space space space space equals space fraction numerator 16 over denominator 6.5 space cross times space 2 end fraction space cross times space 6.5 space cross times 6.5
space space space space space space space space space space space equals space 8 space cross times space 6.5
space space space space space space space space space space space equals space 52 space cm squared

end style

Solution 23

begin mathsize 12px style Correct space option colon space open parentheses straight c close parentheses
arc space equals space 5 straight pi comma space area space equals space 20 straight pi
We space know comma space length space of space arc space equals space θr
and space area space of space sector space equals space straight theta over 2 space straight r squared
Hence space θr space equals space 5 space straight pi space and space straight theta over 2 straight r squared space equals space 20 straight pi
fraction numerator begin display style straight theta over 2 end style straight r squared over denominator θr end fraction space equals space fraction numerator 20 straight pi over denominator 5 straight pi end fraction
straight r over 2 space equals space 4
straight r space equals space 8 space cm end style

Solution 24

Correct option: (d)

Diameter of circle = side of square

2r = 10

r = 5 cm

Area of circle = πr2 = 25 π cm2

Solution 25

begin mathsize 12px style Correct space option colon space open parentheses straight a close parentheses
2 πr space minus space straight r space equals space 37
straight r space equals space fraction numerator 37 over denominator 2 straight pi space minus space 1 end fraction equals space fraction numerator 37 over denominator 2 space cross times space begin display style 22 over 7 end style space minus space 1 end fraction space equals space fraction numerator 37 over denominator begin display style 37 over 7 end style end fraction
straight r space equals space 7 space cm
area space equals space πr squared
space space space space space space space space space space space equals space 49 space straight pi space cm squared
space space space space space space space space space space space equals space 154 space cm squared end style

Chapter 13 - Areas Related to Circles Exercise 13.71

Solution 26

Correct option: (b)

Inner radius = r

outer radius = r + h

area of shaded region = area of outer circle - area of inner circle

= π (r + h)2 - πr2

= π {(r + h)2 - r}

= π (r + h - r) (r + h + r)

= π (2r + h) h

Solution 27

begin mathsize 12px style Correct space option space colon space open parentheses straight d close parentheses
OO apostrophe space is space perpendicular space bisector space of space AB.
Hence space AO apostrophe space equals space 1 half AB
space space space space space space space space space space space space space space space space space space space space space space equals space fraction numerator 5 square root of 3 over denominator 2 end fraction
OA space equals space 5 space cm
In space triangle space OAO apostrophe comma space sin space straight theta space equals space fraction numerator AO apostrophe over denominator AO end fraction space equals space fraction numerator 5 square root of 3 over denominator 5 space cross times space 2 end fraction space equals space fraction numerator square root of 3 over denominator 2 end fraction
straight theta space equals space 60 degree
angle AOB space equals space 2 space angle AOO apostrophe
angle AOB space equals space 120 degree
space space space space space space space space space space space space space space equals space fraction numerator 2 straight pi over denominator 3 end fraction space radian
area space of space sector space with space angle space straight theta space equals space straight theta over 2 straight r squared
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space open parentheses fraction numerator 2 straight pi over denominator 3 end fraction close parentheses space cross times space 1 half space cross times space open parentheses 5 close parentheses squared
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space straight pi over 3 cross times space 25
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space fraction numerator 25 straight pi over denominator 3 end fraction end style

Solution 28

Correct option: (b)

area = circumference

πr2 = 2πr

r = 2 units

area = πr2

       = 4π sq. units

Solution 29

begin mathsize 12px style Correct space option colon space open parentheses straight a close parentheses
Initial space diameter space equals space straight d
Final space diameter space equals space straight d space plus space 40 percent sign space of space straight d
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space straight d space plus space 40 over 100 straight d space equals space 1.4 straight d
Initial space area space equals space straight pi open parentheses straight d over 2 close parentheses squared space equals space πd squared over 4
Final space area space equals space straight pi open parentheses fraction numerator 1.4 space straight d over denominator 2 end fraction close parentheses squared space equals space 1.96 πd squared over 4
percent sign space increase space in space area space equals space fraction numerator Final space minus space initial over denominator initial end fraction space cross times space 100
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space 96 percent sign space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space end style

Solution 30

** img pending

begin mathsize 12px style Correct space option colon space open parentheses straight b close parentheses
area space of space OAB space sector space equals space straight theta over 2 space straight r squared
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space straight pi over 4 straight r squared
area space of space triangle OAB space equals space 1 half space cross times space OA space cross times space OB
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space straight r squared over 2
area space of space shaded space region space equals space area space of space sector space minus space area space of space OAB
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space straight pi over 4 straight r squared space space minus space straight r squared over 2
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space open parentheses fraction numerator straight pi space minus space 2 over denominator 4 end fraction close parentheses straight r squared
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space space open parentheses fraction numerator straight pi space minus space 2 over denominator 4 end fraction close parentheses space open parentheses 10 close parentheses squared
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space 25 open parentheses straight pi space minus space 2 close parentheses space cm squared space space space space space space space space space space space space space space space space space space space space end style

Solution 31

begin mathsize 12px style Correct space option space colon space open parentheses straight b close parentheses
same space as space straight Q space 30 comma
Here space radius space is space straight a space
Hence space straight a squared over 4 open parentheses straight pi space minus space 2 close parentheses end style

Solution 32

begin mathsize 12px style Correct space option colon space open parentheses straight d close parentheses
straight theta space equals space 120 degree
space space space space equals space fraction numerator 2 straight pi over denominator 3 end fraction space radian
area space of space sector space OACBO space equals space straight theta over 2 space straight r squared
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space straight pi over 3 space straight r squared
space area space of space triangle OAB
In space triangle space OO apostrophe straight A
sin space 30 degree space equals space fraction numerator OO apostrophe over denominator OA end fraction
OO apostrophe space equals space straight r over 2
and space tan space 30 degree space equals space fraction numerator OO apostrophe over denominator AO apostrophe end fraction
AO apostrophe space equals space fraction numerator OO apostrophe over denominator tan space 30 degree end fraction space equals space fraction numerator square root of 3 straight r over denominator 2 end fraction
AB space equals space 2 space AO apostrophe
space space space space space space equals space square root of 3 straight r space
area space of space space triangle OAB space equals space 1 half space cross times space AB space cross times space OO apostrophe
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space 1 half space cross times space square root of 3 straight r space cross times space straight r over 2 space equals space fraction numerator square root of 3 straight r squared over denominator 4 end fraction
area space of space segment space ACB space equals space straight pi over 3 straight r squared space minus space fraction numerator square root of 3 over denominator 4 end fraction straight r squared
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space open parentheses straight pi over 3 space minus space fraction numerator square root of 3 over denominator 4 end fraction close parentheses straight r squared end style

Chapter 13 - Areas Related to Circles Exercise 13.72

Solution 33

begin mathsize 12px style Correct space option colon space open parentheses straight c close parentheses
length space of space arc space equals space θr
area space of space sector space equals space straight theta over 2 straight r squared
θr space equals space 5 space and space straight theta over 2 straight r squared space equals space 20 straight pi
fraction numerator θr over denominator begin display style straight theta over 2 end style straight r squared end fraction equals space fraction numerator 5 straight pi over denominator 20 straight pi end fraction
2 over straight r equals space 1 fourth

straight r space equals space 8 space cm end style

Solution 34

begin mathsize 12px style Correct space option colon space open parentheses straight d close parentheses
straight theta space of space sector space straight S subscript 1 space equals space 120 degree
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space fraction numerator 2 straight pi over denominator 3 end fraction space radian
area space equals space open parentheses straight theta over 2 close parentheses straight r squared
space space space space space space space space space space space equals space straight pi over 3 straight r squared
straight theta space of space sector space straight S subscript 2 space equals space 150 degree
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space fraction numerator 5 straight pi over denominator 6 end fraction space radian
area space equals space open parentheses fraction numerator 5 straight pi over denominator 12 end fraction close parentheses straight r squared
fraction numerator area space straight S subscript 1 over denominator area space straight S subscript 2 end fraction space equals space fraction numerator open parentheses begin display style straight pi over 3 end style close parentheses straight r squared over denominator open parentheses begin display style fraction numerator 5 straight pi over denominator 12 end fraction end style close parentheses straight r squared end fraction space equals space 12 over 15 space equals space 4 over 5 end style

Solution 35

begin mathsize 12px style Correct space option colon space open parentheses straight c close parentheses
area space of space sector space equals space straight theta over 2 straight r squared
area space of space circle space equals space πr squared
Given space fraction numerator begin display style straight theta over 2 end style straight r squared over denominator πr squared end fraction equals space 5 over 18
fraction numerator straight theta over denominator 2 straight pi end fraction equals space 5 over 18
straight theta space equals space fraction numerator 5 straight pi over denominator 9 end fraction
space space space space space equals space 100 degree end style

Solution 36

begin mathsize 12px style Correct space option space colon space open parentheses straight d close parentheses
fraction numerator begin display style straight theta over 2 end style straight r squared over denominator πr squared end fraction equals 7 over 20
fraction numerator straight theta over denominator 2 straight pi end fraction equals 7 over 20 rightwards double arrow straight theta space equals space fraction numerator 7 straight pi over denominator 10 end fraction
straight theta space equals space 126 degree end style

Solution 37

begin mathsize 12px style Correct space option colon space open parentheses straight a close parentheses
If space angle BAC space equals space straight theta
then space angle BOC space equals space 2 straight theta
But space angle BAC space equals space 60 degree
as space ABC thin space is space equilateral space
Hence space straight theta space equals space 60 degree
In space triangle OBC
sin space straight theta space equals space BE over OB
BE space equals space OB space sin space 60 degree
space space space space space space equals space fraction numerator square root of 3 straight r over denominator 2 end fraction
and space BC space equals space 2 BE
space space space space space space space space space space space space space space space space equals space square root of 3 straight r
Also space cos space straight theta space equals space OE over OB
OE space equals space OB space cos space 60 degree
space space space space space space space equals space straight r over 2
area space of space triangle OBC space equals space 1 half space cross times space BC space cross times space OE
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space 1 half space cross times space square root of 3 space straight r space cross times space straight r over 2 space equals space fraction numerator square root of 3 space straight r squared over denominator 4 end fraction
area space of space sector space OBC space equals space straight theta over 2 space straight r squared
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space θr squared
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space straight pi over 3 space straight r squared
area space of space shaded space region space equals space straight pi over 3 space straight r squared space minus space fraction numerator square root of 3 over denominator 4 end fraction space straight r squared
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space open parentheses straight pi over 3 space minus space fraction numerator square root of 3 over denominator 4 end fraction close parentheses space straight r squared space space space space space end style

Chapter 13 - Areas Related to Circles Exercise 13.73

Solution 38

begin mathsize 12px style Correct space option colon space left parenthesis straight a right parenthesis
PA space equals space 3 space cm
angle straight A space plus space angle straight B space plus space angle straight C space plus space angle straight D space equals space 360 degree
angle straight A space plus space 90 degree space plus space 90 degree space plus space 60 degree space equals space 360 degree
angle straight A space equals space 120 degree
space space space space space space space equals space fraction numerator 2 straight pi over denominator 3 end fraction space space radian
area space of space sector space with space angle space fraction numerator 2 straight pi over denominator 3 end fraction space equals space straight theta over 2 straight r squared
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space open parentheses fraction numerator 2 straight pi over denominator 3 end fraction close parentheses space 1 half space straight r squared
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space straight pi over 3 open parentheses PA close parentheses squared
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space straight pi over 3 open parentheses 3 close parentheses squared
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space space space 3 straight pi space cm squared space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space end style

Solution 39

begin mathsize 12px style Correct space option space colon space open parentheses straight b close parentheses
perimeter space of space circle space equals space perimeter space of space square
2 πr space equals space 4 straight a
straight a space equals space πr over 2
fraction numerator area space of space circle over denominator area space of space square end fraction space equals space πr squared over straight a squared space equals space fraction numerator πr squared over denominator begin display style fraction numerator straight pi squared straight r squared over denominator 4 end fraction end style end fraction equals space 4 over straight pi
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space fraction numerator 4 over denominator begin display style 22 over 7 end style end fraction space equals space 14 space colon space 11
end style

Solution 40

begin mathsize 12px style Correct space option colon space open parentheses straight b close parentheses
OA space equals space 20 space cm
area space of space 4 space parts space are space equal
Area space of space straight C 4 space equals space πr squared
space space space space space space space space space space space space space space space space space space space space space space space equals space straight pi open parentheses OA close parentheses squared
space space space space space space space space space space space space space space space space space space space space space space space equals space 400 space straight pi
area space of space each space part space equals space 1 fourth space cross times space 400 space straight pi
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space 100 space straight pi
straight C 1 space circle space area space equals space straight pi space open parentheses OD close parentheses squared
100 space straight pi space equals space straight pi space open parentheses OD close parentheses squared
open parentheses OD close parentheses squared space equals space 100
OD space equals space 10 space cm
area space of space straight C 2 space equals space 2 space cross times space area space of space circle space straight C 1
straight pi open parentheses OC close parentheses squared space equals space 2 space cross times space 100 straight pi
open parentheses OC close parentheses squared space equals space 2 space cross times space 100
OC space equals space 10 square root of 2
area space of space straight C 3 space equals space 3 space cross times space area space of space circle space straight C 1
straight pi open parentheses OB close parentheses squared space equals space 3 space cross times space 100 space straight pi
OB space equals space 10 square root of 3 space cm
end style

Solution 41

begin mathsize 12px style Correct space option colon space open parentheses straight C close parentheses
area space of space sector space equals space straight theta over 2 space straight r squared
perimeter space equals space θr space plus space 2 straight r
θr space plus space 2 straight r space equals space 4 straight r
θr space equals space 2 straight r
straight theta space equals space 2
area space equals space 2 over 2 space straight r squared
space space space space space space space space space space equals space straight r squared end style

Solution 42

begin mathsize 12px style Correct space option colon space open parentheses straight C close parentheses
radius space equals space 28 space cm
area space of space sector space straight O space ABD space equals space straight theta over 2 space straight r squared
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space straight pi over 4 space straight r squared
area space of space circle space equals space πr squared
area space of space major space segment space equals space πr squared space minus space straight pi over 4 straight r squared
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space fraction numerator 3 straight pi over denominator 4 end fraction space straight r squared
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space 3 over 4 cross times 22 over 7 cross times space 28 space cross times space 28
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space 1848 space cm squared space space
end style

Solution 43

begin mathsize 12px style Correct space opton colon space open parentheses straight c close parentheses
ABC space is space an space equilateral space triangle
area space of space circle space equals space πr squared
πr squared space equals space 48 space straight pi
straight r squared space equals space 48
straight r space equals space 4 square root of 3
OE space equals space straight r space equals space 4 square root of 3
angle BCA space equals space 60 degree
and space angle ECO space equals space 1 half angle BCA
angle ECO space equals space 30 degree
In space triangle OEC
tan space 30 degree space equals space OE over EC
EC space equals space square root of 3 space OE space equals space 12
BC space equals space 2 space EC
space space space space space space equals space 24
perimeter space equals space AB space plus space BC space plus space CA
space space space space space space space space space space space space space space space space space space space space space equals space 3 space BC
space space space space space space space space space space space space space space space space space space space space space equals space 72 space end style

Chapter 13 - Areas Related to Circles Exercise 13.74

Solution 44

begin mathsize 12px style Correct space option colon space open parentheses straight b close parentheses
hour space hand space is space 6 space cm space long.
when space minute space hand space complete space 60 space minute space or space 2 straight pi space radian comma space hour space hand space lower space straight pi over 6 space radian
minute space hand space 2 straight pi space radian space rightwards arrow hour space hand space straight pi over 6 rad
minute space hand space 1 space radian space rightwards arrow space hour space hand space 1 over 12 rad
minute space hand space goes space 11 space colon space 20 space to space 11 space colon space 55 space equals space 35 space min
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space fraction numerator 2 straight pi over denominator 60 end fraction space cross times space 35
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space fraction numerator 35 straight pi over denominator 30 end fraction
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space fraction numerator 7 straight pi over denominator 6 end fraction space rad
minute space hand space fraction numerator 7 straight pi over denominator 6 end fraction space radian space rightwards arrow space hour space hand space 1 over 12 space cross times space fraction numerator 7 straight pi over denominator 6 end fraction space rad
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space rightwards arrow space fraction numerator 7 straight pi over denominator 72 end fraction space rad
area space equals straight Q over 2 straight r squared
space space space space space space space space space space space space space equals space open parentheses fraction numerator 7 straight pi over denominator 72 end fraction close parentheses space cross times space 1 half space cross times space open parentheses 6 close parentheses squared
space space space space space space space space space space space space space equals space fraction numerator 7 straight pi over denominator 144 end fraction space cross times space 36
space space space space space space space space space space space space space equals space fraction numerator 7 straight pi over denominator 4 end fraction space equals space 7 over 4 space cross times space 22 over 7 space equals space 5.5 space cm squared

space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space end style

Solution 45

begin mathsize 12px style Correct space option colon space open parentheses straight b close parentheses
AB space equals space 4 space cm
triangle CED space is space an space equilateral space triangle.
Hence space CD space equals space ED space equals space EC space equals space 4 space cm
In space triangle OED
sin space 30 degree space equals space OE over ED
OE space equals space ED over 2 space equals space 2 space cm
In space triangle EO apostrophe straight C
sin space 60 degree space equals space fraction numerator straight O apostrophe straight E over denominator EC end fraction
straight O apostrophe straight E space equals space EC space cross times space fraction numerator square root of 3 over denominator 2 end fraction
space space space space space space space space equals space 2 square root of 3
area space of space triangle EDC space equals space 1 half cross times DC cross times EO apostrophe
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space 1 half cross times 4 cross times 2 square root of 3
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space 4 square root of 3
area space of space triangle AED space equals space 1 half cross times space AD space cross times space OE
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space 1 half cross times space 4 space cross times space 2
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space 4
area space of space triangle ADC space equals space 1 half cross times space AD space cross times space DC
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space 1 half cross times space 4 space cross times space 4
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space 8
area space of space triangle AEC space equals space area space open parentheses triangle AED close parentheses space plus space area space open parentheses triangle DEC close parentheses space minus space area space open parentheses triangle ADC close parentheses
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space space 4 space plus space 4 square root of 3 space minus space 8
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space 4 square root of 3 space minus space 4
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space 4 open parentheses square root of 3 space minus space 1 close parentheses
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space end style

Solution 46

Correct option: (b)

radius of Circle = 5 cm

area = π (5)2

       = 25 π

rdius of circle 2 = 12 cm

area = π (12)2

       = 144 π

area of larger circle = 144 π + 25π

                            = 169 π

πr2 = 169 π

r2 = 169

r = 13

diameter = 2r

              = 26

Solution 47

begin mathsize 12px style Correct space option colon space open parentheses straight b close parentheses
radius space of space wheel space equals space diameter over 2 equals space 35 over 2 cm
distance space covered space by space wheel space in space one space revolution space equals space 2 πr
equals space 2 space cross times space 22 over 7 cross times 35 over 2
equals space 110 space cm
equals space 1.1 space straight m end style

Solution 48

begin mathsize 12px style Correct space option colon space open parentheses straight a close parentheses
Other space co minus ordinate space of space rectangle space is space open parentheses 0 comma space 0 close parentheses
Diagonals space of space rectangle space are space equal.
Hence space OC space equals space OB
OC space equals space square root of open parentheses 4 space minus space 0 close parentheses squared space plus space open parentheses 3 space minus space 0 close parentheses squared end root
space space space space space space space equals space square root of 16 space plus space 9 end root
space space space space space space space equals space square root of 25
space space space space space space space equals space 5 space space end style

 

Solution 49

 

 

Solution 50

Solution 51

Solution 52

Solution 53