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1.	If the circumference of a circle and the perimeter of a square are equal, thenA. Area of the circle = Area of the squareB. Area of the circle `gt` Area of the squareC. Area of the cirlce `lt` Area of the squareD. Nothing definite can be said about the between the areas of the circle and square
Answer» Correct Answer - B Accordign to the given condition, Circumterence of a circle = Perimeter of square `2pi ` r = 4a [where, r and a are radius of circle and side of square respectively] `rArr (22)/(7)r = 2a rArr 11r = 7a` `rArr a =(11)/(7)r rArr r=(7a)/(11)` ....(i) Now, area of circle, `A_(1) = pir^(2)` = `pi((7a)/(11))^(2)=(22)/(7)xx(49a^(2))/(121)` [from Eq. (i)] = `(14a^(2))/(11)` ...(ii) and area of square, `A_(2)= (a)^(2)` ...(iii) From Eqs. (ii) and (iii), `A_(1)= (14)/(11)A_(2)` `:. A_(1) gt A_(2)` Hence, Area of the circle `gt` Area of the square.

2.	The numerical value of the area of a circla is greater than the numerical value of its circumference. Is this statement true ? Why ?
Answer» Correct Answer - False If `0lt r lt 2`, then numerical value of circumference is greater than numerical value of area of circle and if `r lt 2` , area is greater than circumference.

3.	Find the radius of a circle whose circumference is equal to the sum of the circumference of two circles of radii 15 cm and 18 cm .
Answer» Let the radius of a circle be r. `:.` Circumference of a circle = `2pir` Let the radii of two circles are `r_(1) and r_(2)` whose values are 15 cm and 18 cm respectively. i.e., `r_(1) ` = 15 cm and `r_(2)` = 18 cm Now, by given condition , Circumference of circle= Circumference of first circle + Circumfernce of second circle `rArr 2pir = 2pir_(1) + r_(2)` `rArr r = r_(1) + r_(2)` `rArr r= 15 + 18` `:.` r = 33cm Hence, required radius of a circle is 33 cm .

4.	If the sum of the areas of two circles with radii `R_(1)` and `R_(2)` is equal to the area of a circle of radius R, thenA. `R_(1) + R_(2) = R`B. `R_(1)^(2)+R_(2)^(2) = R^(2)`C. `R_(1) + R_(2) lt R`D. `R_(1)^(2) + R_(2)^(2) lt R^(2)`
Answer» Correct Answer - B According to the given condition, Area of circle = Area of first circle + Area of second circle `:. piR^(2)= piR_(1)^(2) + piR_(2)^(2)` `rArr R^(2)= R_(1)^(2) + R_(2)^(2)`

5.	The radius of a circle whose circumference is equal to the sum of the circumferences of the two circles of diameters 36 cm and 20 cm isA. 56 cmB. 42 cmC. 28 cmD. 16 cm
Answer» Correct Answer - C `because` Circumference of first circle = `2pir = pid_(1) = 36 pi ` cm [given, `d_(1)` = 36 cm] and circumference of second circle = `pid_(2) = 20 pi cm` [given, `d_(2)` = 20cm] According to the given condition, Circumference of circle= Circumference of first circle + Circumference of second circle `rArr piD = 36pi + 20pi` [where, D is diameter of a circle ] `rArr` D = 56cm So, diameter of a circle is 56 cm. `:.` Required radius of circle = `(56)/(2)=28cm`

6.	If the sum of the circumferences of two circles with radii `R_(1)` and `R_(2)` is equal to the circumference of a circle of radius R, thenA. `R_(1) + R_(2) = R`B. `R_(1) + R_(2) gt R`C. `R_(1) +R_(2) lt R`D. Nothing definite can be said about the relation among `R_(1), R_(2)` and R.
Answer» Correct Answer - A According to the given condition, Circumference of circle = Circumference of first circle + Circumference of second circle `:. 2piR= 2piR_(1)+ piR_(2)` `rArr R = R_(1) + R_(2)`

7.	If is proposed to build a single circular park equal in area to the sum of areas of two circular parks of diameters 16 m and 12 m in a locality. The radius of the new park would beA. 10 mB. 15 mC. 20 mD. 24 m
Answer» Correct Answer - A Area of first circular park, whose diameter is 16m = `pir^(2) = pi((16)/(2))^(2)= 64 pim^(2)` [`because r=(d)/(2)=(16)/(2)=8m`] Area of second circular park, whose diameter is 12m `pi((16)/(2))^(2)= 64 pim^(2)` [`because r=(d)/(2)=(16)/(2)=8m`] According to the given condition, Area of single circular park = Area of first circular park + Area of second circular park `piR^(2)=64 pi +36pi` [`because` R be the radius of single circular park] `rArr piR^(2)= 100pi rArr R^(2)= 100` `:.` R = 10m

8.	Is it true that the distance travelled by a cirular wheel of diameter d cm in one revolution is `2pid` cm ? Why ?
Answer» Correct Answer - False Because the distance travelled by the wheel in one revolution is equal to its circumference i.e., `pid`. i.e., `pi(2r) = 2pir` = Circumference of wheel [`because` d= 2r]

9.	Find the number of revolutions made by a circular wheel of area `1.54m^(2)` in rolling a distance of 176 m.
Answer» Let the number of revolutions made by a circular wheel be n and the radius of circular wheel be r. Given that, area of circular wheel = `1.54m^(2)` `rArrpir^(2)=1.54` [`because` area of circular = `pir^(2)`] `rArr r^(2)= (1.54)/(22)xx7rArrr^(2) = 0.49` `:.r = 0.7m` So, the radius of the wheel is `0.7m`. Distance travelled by a circlular wheel in one revolution = Circumference of circular wheel = `2pir` = `2xx(22)/(7)xx0.7=(22)/(5)=4.4m` [`because` circumference of a circle = `2pir`] Since, distance travelled by a circular wheel = 176m `:.` Number of revolutions = `("Total distance")/("Distance in one revolution")=(176)/(4.4)=40` Hence, the required number of revolutions made by a circular wheel is 40.

10.	A piece of wire 20 cm long is bent into the from of an arc of a circle, subtending an angle of `60^(@)` at its centre. Find the radius of the circle.
Answer» Length of arc of circle = 20 cm Here, central angle `theta = 60^(@)` `:.` Length of arc = `(theta)/(360^(@))xx2pir` `rArr 20=(60^(@))/(360^(@))xx2pirrArr(20xx6)/(2pi) =r` `r = (60)/(pi)cm` Hence, the radius of circle is `(60)/(pi) cm`.