If the edge of a cube is 1 cm then which of the following is its total surface area?
Answer: C
A cube has six identical square faces.
\[ \text{TSA} = 6a^{2} = 6 \times 1^{2} = 6 \text{ cm}^{2} \]
The 1 cm² option is the area of a single face and the 4 cm² the four side faces, leaving out the top and bottom.
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Water flows into a tank 200 m × 150 m through a rectangular pipe 1.5 m × 1.25 m @ 20 kmph. In what time (in minutes) will the water rise by 2 meters?
Answer: D
Work out the water needed and the rate at which it arrives.
\[ V = 200 \times 150 \times 2 = 60{,}000 \text{ m}^{3} \]
\[ \text{pipe cross-section} = 1.5 \times 1.25 = 1.875 \text{ m}^{2} \]
\[ \text{flow} = 1.875 \times 20{,}000 = 37{,}500 \text{ m}^{3}\text{/h} \]
\[ t = \frac{60{,}000}{37{,}500} = 1.6 \text{ h} = 96 \text{ minutes} \]
The speed of 20 kmph becomes 20,000 metres per hour, which is the length of water column delivered each hour.
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Considering radian, value of tan 1.1 is
Answer: A
The angle is in radians, so convert it to degrees first.
\[ 1.1 \text{ rad} = 1.1 \times \frac{180}{\pi} = 63.03^{\circ} \]
\[ \tan 63.03^{\circ} = 1.964 \]
Since \( 63^{\circ} \) is above \( 45^{\circ} \), the tangent must exceed 1 - which eliminates every other option at once.
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Two metallic right circular cones having their heights 4.1 cm and 4.3 cm and the radii of their bases 2.1 cm each, have been melted together and recast into a sphere. Find the diameter of the sphere.
Answer: C
Melting conserves volume, and both cones share the same base radius, so their volumes add through the heights.
\[ V = \frac{1}{3}\pi (2.1)^{2}(4.1 + 4.3) = \frac{1}{3}\pi (4.41)(8.4) \]
\[ = \frac{1}{3}\pi (37.044) = 12.348\pi \text{ cm}^{3} \]
Setting this equal to the sphere's volume:
\[ \frac{4}{3}\pi R^{3} = 12.348\pi \quad\Longrightarrow\quad R^{3} = 9.261 \]
\[ R = 2.1 \text{ cm} \quad\Longrightarrow\quad d = 4.2 \text{ cm} \]
The question asks for the diameter, so the radius must be doubled.
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If the radius of a sphere is increased by 50%, find the increase per cent in its surface area.
Answer: D
Surface area varies as the square of the radius.
\[ \frac{S'}{S} = \left(\frac{1.5r}{r}\right)^{2} = 2.25 \]
\[ \text{increase} = 2.25 - 1 = 1.25 = 125\% \]
The volume, which varies as the cube, would rise by \( 1.5^{3} - 1 = 2.375 \), or 237.5% - a much larger jump for the same change in radius.
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Half cubic meter of the gold sheet is extended by hammering so as to cover an area of one hectare. The thickness of the sheet is:
Answer: B
Hammering preserves the volume, which spreads over the given area.
\[ V = 0.5 \text{ m}^{3} = 500{,}000 \text{ cm}^{3} \]
\[ A = 1 \text{ hectare} = 10{,}000 \text{ m}^{2} = 100{,}000{,}000 \text{ cm}^{2} \]
\[ \text{thickness} = \frac{500{,}000}{100{,}000{,}000} = 0.005 \text{ cm} \]
Both conversions matter: a cubic metre is \( 10^{6} \) cm³ and a square metre \( 10^{4} \) cm².
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What is the are of an equilateral triangle of side 16 cm?
Answer: D
\[ A = \frac{\sqrt{3}}{4}a^{2} = \frac{\sqrt{3}}{4} \times 16^{2} \]
\[ = \frac{\sqrt{3}}{4} \times 256 = 64\sqrt{3} \text{ cm}^{2} \]
The side is squared, so doubling it would quadruple the area.
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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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