Question
The area of an equilateral triangle is $\dfrac {\sqrt {243}}{4} cm^2$. The length of each side of the triangle will be.
$3$ cm
$3\sqrt3$cm
$9$ cm
$\sqrt6$ cm

Solution
Correct option is $3$ cm

$\Rightarrow$ Side = $\sqrt{\dfrac{4}{\sqrt3}\times \dfrac{\sqrt{243}}{4}}$ $= \sqrt{\sqrt{\dfrac{243}{3 }}} = \sqrt9$ $= 3$cm

Question
The diameters of two cylinders, whose volumes are equal, are in the ratio 3 : 2. Their height will be in the ratio?
$4 : 9$
$5 : 6$
$5 : 8$
$8 : 9$

Solution
Correct option is $4 : 9$

$\Rightarrow \dfrac{\pi {R}^2_1H}{\pi {R}^2_2h} =1$ $\Rightarrow \dfrac{3^2 \times H}{2^2 \times h} =1$ $\Rightarrow \dfrac{H}{h} = \dfrac49$

Question
Find the length of the largest rod which can be placed in a room 16 m long, 12 m wide and $10\dfrac23$ m high.
$123$ m
$66$ m
$22 \dfrac23$ m
$22 \dfrac13$ m

Solution
Correct option is $22 \dfrac23$ m

$\Rightarrow$ Since the room is cuboid shape Length of largest rod = Diagonal of cuboid $= \sqrt{L^2 + b^2 + h^2}$ $= \sqrt{16^2 + 12^2 + \dfrac{32^2}{3^2}}$ $= \sqrt{256 + 144 + \dfrac{1024}{9}}$ $= \sqrt{\dfrac{2304 + 1296 + 1024}{9}}$ $= \sqrt{\dfrac{4624}{9}} = \dfrac{68}{3} = 22 \dfrac23$ m

Question
The diameter of a cylinder is 7 cm and its height is 16 cm. Using the value of $\pi = \dfrac {22} {7}$, the lateral surface area of the cylinder is
352 cm $^ 2$
350 cm $^ 2$
355 cm $^ 2$
348 cm $^ 2$

Solution
Correct option is 352 cm $^ 2$

$\Rightarrow$ Lateral surface area of the cylinder = $2\pi rh$ $= 2 \times \dfrac{22}{7} \times \dfrac72 \times 16 = 352$ sq.cm

Question
From a right circular cylinder of radius 10 cm and height 21 cm, a right circular cone of same base radius is removed. If the volume of the remaining part is 4400 cm $^3$, then the height of the removed cone is$\left( \space \pi = \dfrac{22}{7}\right)$?
15 cm
18 cm
21 cm
24 cm

Solution
Correct option is 21 cm

$\Rightarrow$ Volume of the cylinder = $\pi r^2 h$ $= \dfrac{22}{7} \times 10 \times 10 \times 21 = 6600$ cu. cm Volume of cone = $6600 - 4400 = 2200$cu.cm $\therefore 2200 = \dfrac13 \pi r \times 10^2 \times h$ $\Rightarrow 2200 = \dfrac{2200}{21} \times h \Rightarrow h = 21$cm .

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