What is the total surface area of a cube whose edge measures 6 cm ?
Answer: D
Correct answer: 216 sq. cm
A cube has six identical square faces.
\[\text{Total surface area} = 6a^2 = 6 \times 6^2 = 6 \times 36 = 216 \text{ sq. cm}\]
The volume happens to be \(6^3 = 216\) cu. cm as well, which is a coincidence of the edge being 6.
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If base area of a room 12 m² and height is 3 m then its volume is:
Answer: B
\[ V = \text{base area} \times \text{height} = 12 \times 3 = 36 \text{ m}^{3} \]
The individual length and breadth are not needed; only their product, which the base area already gives.
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The length of a rectangular floor is more than its breadth by 200%. If Rs. 324 is required to paint the floor at the rate of Rs. 3 per sq m, then what would be the length of the floor?
Answer: C
Let the length and the breadth of the floor be l m and b m respectively.
l = b + 200% of b = l + 2b = 3b
Area of the floor = 324/3 = 108 sq m
l b = 108 i.e., l * l/3 = 108
l2 = 324 => l = 18.
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If the dimensions of a room are \( l \), \( b \) and \( h \) (length, breadth and height respectively), then which of the following is the area of its four walls?
Answer: A
The four walls form a band running round the floor at height \( h \). Two of them measure \( l \times h \) and two \( b \times h \):
\[ A = 2lh + 2bh = 2h(l + b) \]
Equivalently it is the floor's perimeter \( 2(l+b) \) multiplied by the height. The floor and ceiling are excluded, so \( lb \) never appears.
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If the area of a circle, inscribed in an equilateral triangle is 4π cm², then what is the area of the triangle?
Answer: A
Area of a circle = 4π cm²
πr² = 4π
r² = 4
r = 2 cm.
WKT, Radius of incircle = a/2√3
2 = a/2√3
a = 4√3.
WKT, Area of equilateral triangle = (√3/4)a²
Area of equilateral triangle = (√3/4)(4√3)²
= 12√3 cm².
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A hemisphere is made of a sheet of a metal 1 cm thick. If the outer radius is 5 cm, what is the weight of the hemisphere (1 cm³ of the metal weigh 9 g)?
Answer: B
The sheet is 1 cm thick, so the outer radius is 5 cm and the inner 4 cm. The metal is the shell between them.
\[ V = \frac{2}{3}\pi\left(R^{3} - r^{3}\right) = \frac{2}{3}\pi(125 - 64) \]
\[ = \frac{2}{3}\pi(61) = \frac{122\pi}{3} \text{ cm}^{3} \]
\[ \text{weight} = \frac{122\pi}{3} \times 9 = 366\pi \text{ g} \]
Treating the hemisphere as solid would give \( \tfrac{2}{3}\pi(125) \times 9 = 750\pi \) g - the cavity must be subtracted.
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Find the area of a parallelogram with base 24 cm and height 16 cm.
Answer: B
\[ A = \text{base} \times \text{height} = 24 \times 16 = 384 \text{ cm}^{2} \]
There is no factor of one half - that belongs to the triangle, and using it would give the 192 cm² on offer.
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Given that one cubic cm of marble weighs 25 gms, the weight of a marble block 28 cm in width and 5 cm thick is 112 kg. The length of the block is:
Answer: B
Turn the weight into a volume, then divide out the two known dimensions.
\[ 112 \text{ kg} = 112{,}000 \text{ g} \quad\Longrightarrow\quad V = \frac{112000}{25} = 4480 \text{ cm}^{3} \]
\[ \text{length} = \frac{4480}{28 \times 5} = \frac{4480}{140} = 32 \text{ cm} \]
The density is per gram, so the kilograms must be converted first.
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If the angle of a sector is \( z^{\circ} \), then the formula used to calculate the length of its arc is
Answer: D
An arc is the fraction of the circumference that its central angle is of a full turn.
\[ \text{arc length} = \frac{z^{\circ}}{360^{\circ}} \times \text{circumference} = \frac{z}{360} \times 2\pi r \]
The same fraction applied to the area of the circle gives the sector's area instead:
\[ \text{sector area} = \frac{z}{360} \times \pi r^{2} \]
Length comes from circumference and area from area - matching the two is what the options are testing.
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Find the length of the longest pole that can be placed in a room 12 m long, 8m broad and 9 m high.
Answer: B
The longest pole runs along the space diagonal of the room.
\[ d = \sqrt{l^{2} + b^{2} + h^{2}} \]
\[ = \sqrt{12^{2} + 8^{2} + 9^{2}} = \sqrt{144 + 64 + 81} = \sqrt{289} \]
\[ = 17 \text{ m} \]
The floor diagonal alone would be \( \sqrt{144 + 64} = 14.4 \) m; the height adds the third term.
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