Area - Study Mode

[#71] If the circumference of a circle is 100 units, then what will be the length of the arc described by an angle of 20 degrees ?
Correct Answer

(A) 5.55 units

Explanation

Solution: $$2pi r = 100$$ So, length of the arc : $$eqalign{
& = frac{{2pi r heta }}{{360}} cr
& = left( {frac{{100 imes 20}}{{360}}}
ight){ ext{units}} cr
& = left( {frac{{50}}{9}}
ight){ ext{ units}} cr
& = 5.55{ ext{ units}} cr} $$

[#72] If in a triangle, the area is numerically equal to the perimeter, then the radius of the inscribed circle of the triangle is :
Correct Answer

(C) 2

Explanation

Solution: $$eqalign{
& { ext{Radius}} = frac{{{ ext{Area}}}}{{{ ext{Semi - perimeter}}}} cr
& { ext{Radius}} = left( {{ ext{Area}} imes frac{2}{{{ ext{Area}}}}}
ight) cr
& { ext{Radius}} = 2 cr} $$

[#73] Two equal circle are drawn in square in such a way that a side of the square forms diameter of each circle. If the remaining area of the square is 42 cm 2 , how much will the diameter of the circle measure ?
Correct Answer

(C) 14 cm

Explanation

Solution: Let length of each side of square = 2π According to the question, $$eqalign{
& frac{{pi {r^2}}}{2} + frac{{pi {r^2}}}{2} + 42 = { ext{Area of square}} cr
& Rightarrow pi {r^2} + 42 = 4{r^2} cr
& Rightarrow 4{r^2} - pi {r^2} = 42 cr
& Rightarrow {r^2}left( {4 - frac{{22}}{7}}
ight) = 42 cr
& Rightarrow {r^2}left( {frac{{28 - 22}}{7}}
ight) = 42 cr
& Rightarrow frac{{6{r^2}}}{7} = 42 cr
& Rightarrow {r^2} = frac{{42 imes 7}}{6} cr
& Rightarrow {r^2} = 7 imes 7 cr
& Rightarrow r = 7 cr
& herefore 2r = 14,cm cr} $$

[#74] The base of triangle is 15 cm and height is 12 cm the height of another triangle of double the area having base 20 cm is :
Correct Answer

(C) 18 cm

Explanation

Solution: Given base of triangle and its height is 15 cm and 12 cm respectively Area of first triangle : $$eqalign{
& = frac{1}{2} imes { ext{Base}} imes { ext{Height}} cr
& = frac{1}{2} imes 15 imes 12 cr
& = 90{ ext{ sq}}{ ext{. cm}} cr} $$ According to the question, Let height of triangle be h cm Area of new triangle = 180 sq. cm Base = 20 sq. cm $$eqalign{
& Rightarrow 180 = frac{1}{2} imes 20 imes { ext{Height}} cr
& Rightarrow { ext{h}} = frac{{2 imes 180}}{{20}} cr
& Rightarrow { ext{h}} = 18{ ext{ cm}} cr} $$

[#75] The difference between the length and breadth of a rectangle is 23 m. If its perimeter is 206 m, then its area is :
Correct Answer

(D) 2520 m 2

Explanation

Solution: We have : (l - b) = 23 and 2(l + b) = 206 or (l + b) = 103 Solving the two equations, we get : l = 63 and b = 40 ∴ Area = (l × b) = (63 × 40)m 2 = 2520 m 2