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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Which of the following is equal to 1 kilolitre?
Answer: D
Work through the metric chain of volume units.
\[ 1 \text{ litre} = 1 \text{ dm}^{3} \]
\[ 1 \text{ kilolitre} = 1000 \text{ litres} = 1000 \text{ dm}^{3} \]
That is also 1 cubic metre, since \( 1 \text{ m}^{3} = 1000 \text{ dm}^{3} \). The 1000 millilitres on offer is only one litre - a thousandth of the answer.
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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 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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Two identical cubes each of total surface area of 6 cm² are joined end to end. Which of the following is the total surface area of the cuboid so formed?
Answer: C
Each cube has total surface area 6 cm², so a single face is 1 cm² and the edge is 1 cm.
Joined end to end they form a cuboid of \( 2 \times 1 \times 1 \):
\[ \text{TSA} = 2(lb + bh + hl) = 2(2 + 1 + 2) = 10 \text{ cm}^{2} \]
The two faces that meet are sealed inside and no longer count. Adding the cubes' areas would give 12 cm², which is exactly 2 cm² too many - the pair of hidden faces.
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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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If the sides of a triangle are 26 cm, 24 cm and 10 cm, what is its area?
Answer: A
Check the sides first: \( 10^{2} + 24^{2} = 100 + 576 = 676 = 26^{2} \), so this is a right-angled triangle with legs 10 cm and 24 cm.
\[ A = \frac{1}{2} \times 10 \times 24 = 120 \text{ cm}^{2} \]
Heron's formula agrees: with \( s = 30 \), \( \sqrt{30 \times 4 \times 6 \times 20} = \sqrt{14400} = 120 \). Spotting the 5–12–13 triple doubled saves that work.
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The perimeter of a triangle is 28 cm and the inradius of the triangle is 2.5 cm. What is the area of the triangle?
Answer: D
The inradius relates the area to the semi-perimeter:
\[ A = r \times s, \qquad s = \frac{28}{2} = 14 \]
\[ A = 2.5 \times 14 = 35 \text{ cm}^{2} \]
35 cm² is not among the choices, so the answer is none of these. The 70 cm² on offer comes from using the full perimeter instead of the semi-perimeter.
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Find the area of trapezium whose parallel sides are 20 cm and 18 cm long, and the distance between them is 15 cm.
Answer: C
\[ A = \frac{1}{2}(a + b)h = \frac{1}{2}(20 + 18) \times 15 \]
\[ = \frac{1}{2} \times 38 \times 15 = 285 \text{ cm}^{2} \]
The two parallel sides are averaged, then multiplied by the perpendicular distance between them.
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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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