1. The ratio of the surface area of a sphere and the curved surface area of the cylinder circumscribing the sphere is :
a) 1 : 1
b) 1 : 2
c) 2 : 1
d) 2 : 3
Explanation: Let the radius of the sphere be r
Then, radius of the cylinder = r
Height of the cylinder = 2r
Surface area of sphere = $$4\pi {{\text{r}}^2}$$
Surface area of the cylinder = $$2\pi {\text{r}}(2r) = 4\pi {{\text{r}}^2}$$
Required ratio :
= $$4\pi {{\text{r}}^2}$$ : $$4\pi {{\text{r}}^2}$$
= 1 : 1
2. A hemispherical bowl of internal radius 12 cm contains liquid. This liquid is to be filled into cylindrical container of diameter 4 cm and height 3 cm. The number of containers that is necessary to empty the bowl is :
a) 80
b) 96
c) 100
d) 112
Explanation: Volume of hemispherical bowl :
$$ = \left( {\frac{2}{3} \times \pi \times 12 \times 12 \times 12} \right)c{m^3}$$
Volume of 1 cylindrical container :
$$ = \left( {\pi \times 2 \times 2 \times 3} \right)c{m^3}$$
Number of containers required :
$$\eqalign{ & = \frac{2}{3} \times \frac{{12 \times 12 \times 12}}{{2 \times 2 \times 3}} \cr & = 96 \cr} $$
3. Length of each edge of a regular tetrahedron is 1 cm. It volume is :
a) $$\frac{{\sqrt 3 }}{{12}}{\text{ }}c{m^3}$$
b) $$\frac{1}{4}\sqrt 3 {\text{ }}c{m^3}$$
c) $$\frac{{\sqrt 2 }}{6}{\text{ }}c{m^3}$$
d) $$\frac{1}{{12}}\sqrt 2 {\text{ }}c{m^3}$$
Explanation: Length of each edge of a regular tetrahedron = 1 cm
Volume of regular tetrahedron :
$$\eqalign{ & = \frac{{{a^3}}}{{6\sqrt 2 }}{\text{ c}}{{\text{m}}^3} \cr & = \frac{1}{{6\sqrt 2 }} \cr & = \frac{{\sqrt 2 }}{{6\sqrt 2 \times \sqrt 2 }}{\text{ c}}{{\text{m}}^3} \cr & = \frac{{\sqrt 2 }}{{12}}{\text{ Or }}\frac{1}{{12}}\sqrt 2 {\text{ c}}{{\text{m}}^3} \cr} $$
4. The base of a right prism is a trapezium whose lengths of two parallels sides are 10 cm and 6 cm and distance between them is 5 cm. If the heights of the prism is 8 cm, its volume is :
a) 320 cm3
b) 300 cm3
c) 310 cm3
d) 300.5 cm3
Explanation: Length of parallel sides of prism = 10 cm and 6 cm
Height of prism = 8 cm
Volume of prism :
$$\eqalign{ & = \frac{1}{2}\left( {10 + 6} \right) \times 5 \times 8 \cr & = \frac{1}{2} \times 16 \times 5 \times 8 \cr & = 320{\text{ c}}{{\text{m}}^3} \cr} $$
5. A rectangular water reservoir contains 42000 litres of water. If the length of reservoir is 6 m and breadth of the reservoir is 3.5 m, then the depth of the reservoir will be :
a) 2 m
b) 5 m
c) 6 m
d) 8 m
Explanation: Volume of the reservoir = 42000 litres = 42 m3
Let the depth of the reservoir be h metres
$$\eqalign{ & 6 \times 3.5 \times h = 42 \cr & h = \frac{{42}}{{6 \times 3.5}} = 2\,m \cr} $$
6. When a ball bounces, it rises to $$\frac{2}{3}$$ of the height from which it fell. If the ball is dropped from a height of 36 m, how high will it rise at the third bounce ?
a) $$10\frac{1}{3}$$ m
b) $$10\frac{2}{3}$$ m
c) $$12\frac{1}{3}$$ m
d) $$12\frac{2}{3}$$ m
Explanation: Ball is dropped from the height of 36 m when the ball will rise at the third bounce
Required height :
$$\eqalign{ & = \frac{2}{3} \times \frac{2}{3} \times \frac{2}{3} \times 36 \cr & = \frac{{32}}{3} \cr & = 10\frac{2}{3}\,m \cr} $$
7. The sum of perimeters of the six faces of a cuboid is 72 cm and the total surface area of the cuboid is 16 cm2. Find the longest possible length that can be kept inside the cuboid :
a) 5.2 cm
b) 7.8 cm
c) 8.05 cm
d) 8.36 cm
Explanation: Sum of perimeters of the six faces :
$$\eqalign{ & = 2\left[ {2\left( {l + b} \right) + 2\left( {b + h} \right) + 2\left( {l + h} \right)} \right] \cr & = 4\left( {2l + 2b + 2h} \right) \cr & = 8\left( {l + b + h} \right) \cr} $$
Total surface area $$ = 2\left( {lb + bh + lh} \right)$$
$$\eqalign{ & 8\left( {l + b + h} \right) = 72 \cr & \Rightarrow l + b + h = 9 \cr & 2\left( {lb + bh + lh} \right) = 16 \cr & \Rightarrow lb + bh + lh = 8 \cr} $$
Now,
$${\left( {l + b + h} \right)^2} = {l^2} + {b^2} + {h^2} + 2$$ $$\left( {lb + bh + lh} \right)$$
$$\eqalign{ & \Rightarrow {\left( 9 \right)^2} = {l^2} + {b^2} + {h^2} + 16 \cr & {l^2} + {b^2} + {h^2} = 81 - 16 \cr & {l^2} + {b^2} + {h^2} = 65 \cr} $$
Required length :
$$\eqalign{ & = \sqrt {{l^2} + {b^2} + {h^2}} \cr & = \sqrt {65} \cr & = 8.05\,cm \cr} $$
8. The surface area of a cube is 150 cm2. Its volume is :
a) 64 $${\text{c}}{{\text{m}}^3}$$
b) 125 $${\text{c}}{{\text{m}}^3}$$
c) 150 $${\text{c}}{{\text{m}}^3}$$
d) 216 $${\text{c}}{{\text{m}}^3}$$
Explanation:
$$\eqalign{ & 6{a^2} = 150 \cr & {a^2} = 25 \cr & a = 5 \cr} $$
Volume :
$${a^3} = {5^3} = 125\,{\text{ c}}{{\text{m}}^3}$$
9. If three equal cubes are placed adjacently in a row, then the ratio of the total surface area of the new cuboid to the sum of the surface areas of the three cubes will be ?
a) 1 : 3
b) 2 : 3
c) 5 : 9
d) 7 : 9
Explanation: Let the length of each edge of each cube be a
Then, the cuboid formed by placing 3 cubes adjacently has the dimensions 3a , a and a
Surface area of the cuboid :
$$\eqalign{ & = 2\left[ {3a \times a + a \times a + 3a \times a} \right] \cr & = 2\left[ {3{a^2} + {a^2} + 3{a^2}} \right] \cr & = 14{a^2} \cr} $$
Sum of surface area of 3 cubes :
$$\eqalign{ & = \left( {3 \times 6{a^2}} \right) \cr & = 18{a^2} \cr} $$
Required ratio :
$$\eqalign{ & = 14{a^2}:18{a^2} \cr & = 7:9 \cr} $$
10. The curved surface area of a cylindrical pillar is 264 m2 and its volume is 924 m3. Find the ratio of its diameter to its height.
a) 3 : 7
b) 7 : 3
c) 6 : 7
d) 7 : 6
Explanation:
$$\eqalign{ & \frac{{\pi {r^2}h}}{{2\pi rh}} = \frac{{924}}{{264}} \cr & \Rightarrow r = \left( {\frac{{924}}{{264}} \times 2} \right) \cr & \Rightarrow r = 7\,m \cr} $$
$$\eqalign{ & \therefore 2\pi rh = 264 \cr & h = \left( {264 \times \frac{7}{{22}} \times \frac{1}{2} \times \frac{1}{7}} \right) \cr & h = 6\,m \cr} $$
Required ratio :
$$ = \frac{{2r}}{h} = \frac{{14}}{6} = 7:3$$