1. A closed aquarium of dimensions 30 m × 25 cm × 20 cm is made up entirely of glass plates held together with tapes. The total length of tape required to hold the plates together (ignore the overlapping tapes) is :
a) 75 cm
b) 120 cm
c) 150 cm
d) 300 cm
Explanation: Total length of tape required :
= Sum of lengths of edges
= (30 × 4 + 25 × 4 + 20 × 4) cm
= 300 cm
2. A swimming pool 9 m wide and 12 m long is 1 m deep on the shallow side and 4 m deep on the deeper side. It volume is :
a) 208 m3
b) 270 m3
c) 360 m3
d) 408 m3
Explanation: Volume :
$$\eqalign{ & = \left[ {12 \times 9 \times \left( {\frac{{1 + 4}}{2}} \right)} \right]{m^3} \cr & = \left( {12 \times 9 \times 2.5} \right){m^3} \cr & = 270\,{m^3} \cr} $$
3. An aluminium sheet 27 cm long, 8 cm broad and 1 cm thick is melted into a cube. The difference in the surface areas of the two solids would be :
a) Nil
b) 284 cm2
c) 286 cm2
d) 296 cm2
Explanation: Volume of cube = Volume of sheet = (27 × 8 × 1) cm3 = 216 cm3
Edge of cube :
$$\root 3 \of {216} \,cm = 6\,cm$$
Surface area of sheet :
$$\eqalign{ & = 2\left( lb + bh + lh \right) \cr & = 2\left( {27 \times 8 + 8 \times 1 + 27 \times 1} \right){\text{ c}}{{\text{m}}^2} \cr & = \left( {216 + 8 + 27} \right){\text{ c}}{{\text{m}}^2} \cr & = 502{\text{ c}}{{\text{m}}^2} \cr} $$
Surface area of cube :
$$\eqalign{ & = 6{a^2} \cr & = \left( {6 \times {6^2}} \right){\text{ c}}{{\text{m}}^2} \cr & = 216{\text{ c}}{{\text{m}}^2} \cr} $$
Required difference :
$$\eqalign{ & = \left( {502 - 216} \right){\text{ c}}{{\text{m}}^2} \cr & = 286{\text{ c}}{{\text{m}}^2} \cr} $$
4. The volumes of two cubes are in the ratio 8 : 27. The ratio of their surface areas is :
a) 2 : 3
b) 4 : 9
c) 12 : 9
d) None of these
Explanation: Let their edges be a and b
$$\eqalign{ & \frac{{{a^3}}}{{{b^3}}} = \frac{8}{{27}} \cr & {\left( {\frac{a}{b}} \right)^3} = {\left( {\frac{2}{3}} \right)^3} \cr & \frac{a}{b} = \frac{2}{3} \cr & \frac{{{a^2}}}{{{b^2}}} = \frac{4}{9} \cr & \frac{{6{a^2}}}{{6{b^2}}} = \frac{4}{9}\,Or\,4:9 \cr} $$
5. The height of a right circular cylinder is 6 m. If three times the sum of the areas of its two circular faces is twice the area of the curved surface, then the radius of its base is :
a) 1 m
b) 2 m
c) 3 m
d) 4 m
Explanation:
$$\eqalign{ & 3 \times 2\pi {r^2} = 2 \times 2\pi rh \cr & \Rightarrow 6r = 4h \cr & r = \frac{2}{3}h \cr & r = \left( {\frac{2}{3} \times 6} \right)m \cr & r = 4\,m \cr} $$
6.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:
$$\eqalign{ & \frac{{\pi \,{r^2}h}}{{2\pi \,rh}} = \frac{{924}}{{264}}\, \cr & \Rightarrow r = {\frac{{924}}{{264}} \times 2} = 7m \cr & {\text{And}},\,2\pi rh = 264 \cr & \Rightarrow h = {264 \times \frac{7}{{22}} \times \frac{1}{2} \times \frac{1}{7}} = 6m \cr & {\text{Required}}\,{\text{ratio}} \cr & = \frac{{2r}}{h} = \frac{{14}}{6} = 7:3 \cr} $$
7. A cistern of capacity 8000 litres measures externally 3.3 m by 2.6 m by 1.1 m and its walls are 5 cm thick. The thickness of the bottom is:
a) 90 cm
b) 1 dm
c) 1 m
d) 1.1 cm
Explanation: Let the thickness of the bottom be x cm
$${\mkern 1mu} {\left( {330 - 10} \right) \times \left( {260 - 10} \right) \times \left( {110 - x} \right)} = $$ $$8000 \times $$ $$1000$$
$$ \Rightarrow 320 \times 250 \times \left( {110 - x} \right) = 8000 \times 1000$$
$$\eqalign{ & \left( {110 - x} \right) = \frac{{8000 \times 1000}}{{320 \times 250}} = 100 \cr & x = 10\,{\text{cm}} = 1\,{\text{dm}} \cr} $$
8. What is the total surface area of a right circular cone of height 14 cm and base radius 7 cm?
a) 344.35 cm2
b) 462 cm2
c) 498.35 cm2
d) None of these
Explanation:
$$\eqalign{ & h = 14\,cm,\,r = 7\,cm \cr & {\text{So}},\,l = \sqrt {{{\left( 7 \right)}^2} + {{\left( {14} \right)}^2}} = \sqrt {245} = 7\sqrt 5 \,cm \cr & {\text{Total}}\,{\text{surface}}\,{\text{area}} \cr & = \pi \,rl + \pi \,{r^2} \cr & = \left( {\frac{{22}}{7} \times 7 \times 7\sqrt 5 + \frac{{22}}{7} \times 7 \times 7} \right)c{m^2} \cr & = \left[ {154\left( {\sqrt 5 + 1} \right)} \right]c{m^2} \cr & = \left( {154 \times 3.236} \right)c{m^2} \cr & = 498.35\,c{m^2} \cr} $$
9. A large cube is formed from the material obtained by melting three smaller cubes of 3, 4 and 5 cm side. What is the ratio of the total surface areas of the smaller cubes and the large cube?
a) 2 : 1
b) 3 : 2
c) 25 : 18
d) 27 : 20
Explanation:
$$\eqalign{ & {\text{Volume}}\,{\text{of}}\,{\text{the}}\,{\text{large}}\,{\text{cube}} \cr & = \left( {{3^3} + {4^3} + {5^3}} \right) = 216\,c{m^3} \cr & {\text{Let}}\,{\text{the}}\,{\text{edge}}\,{\text{of}}\,{\text{the}}\,{\text{large}}\,{\text{cube}}\,{\text{be}}\,a \cr & So,\,{a^3} = 216\,\,\,\,\, \Rightarrow \,\,\,\,\,a = 6\,cm \cr & {\text{Required}}\,{\text{ratio}} = {\frac{{6 \times \left( {{3^2} + {4^2} + {5^2}} \right)}}{{6 \times {6^2}}}} \cr & = \frac{{50}}{{36}} \cr & = 25:18 \cr} $$
10. How many bricks, each measuring 25 cm x 11.25 cm x 6 cm, will be needed to build a wall of 8 m x 6 m x 22.5 cm?
a) 5600
b) 6000
c) 6400
d) 7200
Explanation:
$$\eqalign{ & {\text{Number}}\,{\text{of}}\,{\text{bricks}} = \frac{{{\text{Volume}}\,{\text{of}}\,{\text{the}}\,{\text{wall}}}}{{{\text{Volume}}\,{\text{of}}\,{\text{1}}\,{\text{brick}}}} \cr & = {\frac{{800 \times 600 \times 22.5}}{{25 \times 11.25 \times 6}}} \cr & = 6400 \cr} $$